*Microwave shielding effectiveness (SEA) for 10- and 20-dB bandwidth (BW) in near and far field in BaCoxTixFe(12−2x) O19 (x = 0.1, 0.3, 0.5, 0.7).*

#### *Investigation of Shielding Effectiveness of M-Type Ba-Co-Ti Hexagonal Ferrite and Composite… DOI: http://dx.doi.org/10.5772/intechopen.91204*

*Composite Materials*

in σac is seen with frequency in composite x = 0.1, 0.3, and 0.5; however, it remains nearly independent of frequency in x = 0.7. This increase in σac is ascribed to Koops-Wagner model, which explains ferrite comprising of heterogeneous structure [80]: ferrites owe layers of good conducting grains, effective at high frequencies, are separated by poor conducting grain boundaries that are effective at low frequencies.

693.6 MΩ cm, 2.8 kΩ cm, 0.5 kΩ cm, and 33.8 MΩ cm, respectively. The composite x = 0.1 has the highest resistivity but still a large σac attributed to the presence of more strength of Fe3+: electron hopping between Fe3+–Fe2+ ions is responsible for conduction in ferrites [81]. Among all composites, composite x = 0.5 (i) owe maximum σac besides with diminution in the number of Fe3+ ions and (ii) has the lowest DC resistivity. The competition between these factors altogether increases σac in this

*Change in skin depth (δ) with frequency for BaCoxTixFe(12−2x)O19 ferrite (x = 0.1, 0.3, 0.5, 0.7) with frequency* 

The composites x = 0.1, 0.3, 0.5, and 0.7 have DC resistivity (ρdc) of

**134**

**Figure 6.**

**Figure 5.**

*in X-band.*

*Plots of SEA versus (σac)0.5(S/m)0.5 for BaCoxTixFe(12−2x)O19 ferrite (x = 0.1, 0.3, 0.5, 0.7).*

composite. Similarly, steep fall of σac in x = 0.7 is associated with the least number of Fe3+ ions available for electron hopping and large DC resistivity.

The dependence of skin depth (δ) on frequency for a different level of substitution is shown in **Figure 5**. The decrement trend in δ is observed with frequency, and x = 0.7 and 0.5 exhibit large and small δ respectively among the composites in the frequency regime. The large conduction loss, as shown in σac (**Figure 4**), causes minimum δ, which attenuates the propagating microwave signal in the composite and vice versa; thus further penetration of signal is not possible inside the thickness of composite: the signal is attenuated more in x = 0.5 due to highest σac depicted in **Figure 4**, thereby causing lowest δ.

The dependence of shielding effectiveness (SEA) on AC conductivity (σac0.5) for different levels of doping is shown in **Figure 6**: it increases with doping from x = 0.1 to x = 0.5 and steep decrement is seen thereafter in x = 0.7. All composites display a monotonic trend of increase in SEA with σac0.5 and x = 0.5 owe maximum value while x = 0.7 stay at lowest one.

**Table 1** shows bandwidth (10 dB and 20 dB) of SEA for both near and far field versus doping: 10 and 20 dB means 90% and 99% absorption respectively. For near field, x = 0.1, 0.3, and 0.7 exhibit 10-dB bandwidth of 2.23, 2.34, and 2.12 GHz respectively whereas 20-dB bandwidth of 1.54, 0.89, and 3.60 GHz is observed in x = 0.1, 0.3 and 0.5 respectively. For far field, x = 0.1, 0.3, and 0.5 show 10 dB-bandwidth of 3.20, 3.70, and 0.50 GHz respectively, and 20-dB bandwidth of 4.70 GHz is seen in x = 0.5 only.

#### **4. Conclusions**

For near and far field, microwave shielding effectiveness in BaCoxTixFe(12−2x)O19 ferrite is governed by absorption and doping of Co2+ and Ti4+ ion increases SEA from x = 0.1, 0.3, and 0.5. Composite x = 0.5 owes the highest SEA of 38.9 dB at 10.26 GHz and 3.4 mm thickness; σac0.5, ρdc and δ are the contributing factors and same composite carries with highest SEA of 44.6 dB at σac0.5 of 4.5 (Ohm.cm)−0.5 for far field; s-parameter is the deciding factor. Furthermore, SEA increases monotonically with frequency and it can be tuned by varying intrinsic and extrinsic parameters. Composite x = 0.5 has far field and near field wideband of 4.70 and 3.60 GHz respectively for 20 dB SEA. The studied composites have the potential for practical absorber applications. The applications of these composite materials or other composite materials are very an important subject and more research is needed to find the optimum properties and optimum materials for X-band microwave applications.

#### **Acknowledgements**

The author *IA Abdel-Latif*, is thankful to the Deanship of Scientific Research in Najran University for their financial support NU/ESCI/16/063 in the frame of the local scientific research program support.

**137**

**Author details**

Charanjeet Singh1

Amritsar, Punjab, India

Kingdom of Saudi Arabia

Saudi Arabia

Cairo, Egypt

University, Phagwara, Punjab, India

\*, S. Bindra Narang2

\*Address all correspondence to: rcharanjeet@gmail.com;

provided the original work is properly cited.

charanjeet2003@rediffmail.com and ihab\_abdellatif@yahoo.co.uk

and Ihab A. Abdel-Latif3,4,5\*

1 Department of Electronics and Communication Engineering, Lovely Professional

2 Department of Electronics Technology, Guru Nanak Dev University,

3 Physics Department, College of Science and Arts, Najran University, Najran,

4 Advanced Materials and Nano-Research Centre, Najran University, Najran,

5 Reactor Physics Department, NRC, Atomic Energy Authority, Abou Zabaal,

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

*Investigation of Shielding Effectiveness of M-Type Ba-Co-Ti Hexagonal Ferrite and Composite…*

*DOI: http://dx.doi.org/10.5772/intechopen.91204*

*Investigation of Shielding Effectiveness of M-Type Ba-Co-Ti Hexagonal Ferrite and Composite… DOI: http://dx.doi.org/10.5772/intechopen.91204*

#### **Author details**

*Composite Materials*

**Figure 4**, thereby causing lowest δ.

while x = 0.7 stay at lowest one.

is seen in x = 0.5 only.

**4. Conclusions**

applications.

**Acknowledgements**

local scientific research program support.

composite. Similarly, steep fall of σac in x = 0.7 is associated with the least number

The dependence of skin depth (δ) on frequency for a different level of substitution is shown in **Figure 5**. The decrement trend in δ is observed with frequency, and x = 0.7 and 0.5 exhibit large and small δ respectively among the composites in the frequency regime. The large conduction loss, as shown in σac (**Figure 4**), causes minimum δ, which attenuates the propagating microwave signal in the composite and vice versa; thus further penetration of signal is not possible inside the thickness of composite: the signal is attenuated more in x = 0.5 due to highest σac depicted in

The dependence of shielding effectiveness (SEA) on AC conductivity (σac0.5) for different levels of doping is shown in **Figure 6**: it increases with doping from x = 0.1 to x = 0.5 and steep decrement is seen thereafter in x = 0.7. All composites display a monotonic trend of increase in SEA with σac0.5 and x = 0.5 owe maximum value

**Table 1** shows bandwidth (10 dB and 20 dB) of SEA for both near and far field versus doping: 10 and 20 dB means 90% and 99% absorption respectively. For near field, x = 0.1, 0.3, and 0.7 exhibit 10-dB bandwidth of 2.23, 2.34, and 2.12 GHz respectively whereas 20-dB bandwidth of 1.54, 0.89, and 3.60 GHz is observed in x = 0.1, 0.3 and 0.5 respectively. For far field, x = 0.1, 0.3, and 0.5 show 10 dB-bandwidth of 3.20, 3.70, and 0.50 GHz respectively, and 20-dB bandwidth of 4.70 GHz

For near and far field, microwave shielding effectiveness in BaCoxTixFe(12−2x)O19

The author *IA Abdel-Latif*, is thankful to the Deanship of Scientific Research in Najran University for their financial support NU/ESCI/16/063 in the frame of the

ferrite is governed by absorption and doping of Co2+ and Ti4+ ion increases SEA from x = 0.1, 0.3, and 0.5. Composite x = 0.5 owes the highest SEA of 38.9 dB at 10.26 GHz and 3.4 mm thickness; σac0.5, ρdc and δ are the contributing factors and same composite carries with highest SEA of 44.6 dB at σac0.5 of 4.5 (Ohm.cm)−0.5 for far field; s-parameter is the deciding factor. Furthermore, SEA increases monotonically with frequency and it can be tuned by varying intrinsic and extrinsic parameters. Composite x = 0.5 has far field and near field wideband of 4.70 and 3.60 GHz respectively for 20 dB SEA. The studied composites have the potential for practical absorber applications. The applications of these composite materials or other composite materials are very an important subject and more research is needed to find the optimum properties and optimum materials for X-band microwave

of Fe3+ ions available for electron hopping and large DC resistivity.

**136**

Charanjeet Singh1 \*, S. Bindra Narang2 and Ihab A. Abdel-Latif3,4,5\*

1 Department of Electronics and Communication Engineering, Lovely Professional University, Phagwara, Punjab, India

2 Department of Electronics Technology, Guru Nanak Dev University, Amritsar, Punjab, India

3 Physics Department, College of Science and Arts, Najran University, Najran, Kingdom of Saudi Arabia

4 Advanced Materials and Nano-Research Centre, Najran University, Najran, Saudi Arabia

5 Reactor Physics Department, NRC, Atomic Energy Authority, Abou Zabaal, Cairo, Egypt

\*Address all correspondence to: rcharanjeet@gmail.com; charanjeet2003@rediffmail.com and ihab\_abdellatif@yahoo.co.uk

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### **References**

[1] Gismelseed AM, Khalaf KAM, Elzain ME, Widatallah HM, Al-Rawas AD, Yousif AA. The structural and magnetic behavior of the MgFe2<sup>−</sup> xCrxO4 spinel ferrite. Hyperfine Interactions. 2012. DOI: 10.1007/ s10751-011-0529-8

[2] Roumaih K, Manapov RA, Sadykov EK, Pyataev AV. Mossbauer studies of Cu1−xNixFeMnO4 spinel ferrites. Journal of Magnetism and Magnetic Materials. 2005;**288**:267

[3] Abdel Latif IA. Fabrication of nano-size nickel ferrites for gas sensors applications. Journal of Physics. 2012;**1**(2):50-53

[4] Khalaf KA, Al-Rawas A, Gismelseed A, Al-Ruqeishi M, Al-Ani S, Al-Jubouri A, et al. Effects of Zn substitution on structure factors, Debye-Waller factors and related structural properties of the Mg1-xZnxFeNiO4 spinels. Advances in Materials. 2019;**8**(2):70-93. DOI: 10.11648/j. am.20190802.15

[5] Maensiri S, Sangmanee M, Wiengmoon A. Magnesium ferrite (MgFe2O4) nanostructures fabricated by electrospinning. Nanoscale Research Letters. 2009;**4**:221

[6] Fayek MK, Ata-Allah SS. 57Fe Mossbauer and electrical studies of the (NiO)–(Cr2O3)x–(Fe2O3)2−<sup>x</sup> system. Physica Status Solidi A: Applications and Material Science. 2003;**198**(457-464)

[7] Tatarchuk T, Bououdina M, Judith VJ, John Kennedy L. Spinel ferrite nanoparticles: Synthesis, crystal structure, properties, and perspective applications. In: Fesenko O, Yatsenko L, editors. Nanophysics, Nanomaterials, Interface Studies, and Applications. NANO 2016. Springer Proceedings in Physics, vol 195. Springer, Cham. 2017

[8] Zaki HM et al. Synthesis and characterization of nanocrystalline MgAlxFe2-xO4 ferrites. Journal of Materials Research. 2012;**27**(21):2798

[9] Al-Maashani M, Gismelseed AM, Khalaf KAM, Yousif AA, Al-Rawas AD, Widatallah HM, et al. Structural and Mossbauer study of nanoparticles CoFe2O4 prepared by sol-gel autocombustion and subsequent sintering. Hyperfine Interactions. 2018;**239**:15. DOI: 10.1007/s10751-018-1491-5

[10] Yousif AA et al. Study on Mossbauer and magnetic properties of strontium doped neodymium ferrimanganites perovskite-like structure. AIP Conference Proceedings. 2011;**1370**:103

[11] Abdel-Latif IA, Saleh SA. Effect of iron doping on the physical properties of europium Manganites. Journal of Alloys and Compounds. 2012;**530**:116

[12] Bashkirov S et al. Crystal structure, electric and magnetic properties of ferrimanganite NdFexMn1−xO3. Izv. RAS, Physical Series. 2003;**67**:1072

[13] Abdel-Latif IA et al. The influence of tilt angle on the CMR in Sm0.6Sr0.4MnO3. Journal of Alloys and Compounds. 2008;**452**:245

[14] Abdel-Latif IA et al. Magnetocaloric effect, electric, and dielectric properties of Nd0.6Sr0.4MnxCo1−xO3 composites. Journal of Magnetism and Magnetic Materials. 2018;**457**:126

[15] Bouziane KA et al. Electronic and magnetic properties of SmFe1<sup>−</sup> xMnxO3 orthoferrites (x = 0.1, 0.2 and 0.3). Journal of Applied Physics. 2005;**97**(10A):504

[16] Abdel-Latif IA. Study on the effect of particle size of strontium-ytterbium manganites on some physical properties. AIP Conference Proceedings. 2011;**1370**:108

**139**

*Investigation of Shielding Effectiveness of M-Type Ba-Co-Ti Hexagonal Ferrite and Composite…*

[28] Verma S, Pradhan SD, Pasricha R,

[29] Aen F, Ahmad M, Rana MU. Current

[30] Singh J et al. Elucidation of phase evolution, microstructural, Mossbauer and magnetic properties of CO2-Al3 doped M-type Ba-Sr hexaferrites synthesized by a ceramic method. Journal of Alloys and Compounds.

[31] Singh C, Narang SB, Hudiara IS, Bai Y, Marina K. Hysteresis analysis of Co–Ti substituted M-type Ba–Sr hexagonal ferrite. Materials Letters.

[32] Sharbati A, Choopani S, Azar A-M, Senna M. Structure and electromagnetic

SrMgxZrxFe12-2xO19 in the 8-12 GHz frequency range. Journal of Solid State Communications. 2010;**150**:2218-2222

[33] Reimann T, Schmidt T, Töpfer J. Phase stability and magnetic properties of SrFe18O27 W-type hexagonal ferrite. Journal of the American Ceramic Society. 2019. DOI: 10.1111/jace.16726

[34] Arjunwadkar PR, Salunkhe MY, Dudhe CM. Structural, electrical,

SrNi2+(Li1+Fe3+)0.5Fe16O27 ferrite. Journal of Solid State Physics. 2013. DOI:

[35] Mukhtar A, Grossinger R, Kriegisch M, Kubel F, Rana MU. Characterization of Sr-substituted W-type hexagonal ferrites synthesized by sol–gel autocombustion method. Journal of Magnetism and Magnetic Materials.

[36] Meshram MR, Agrawal NK,

Sinha B, Misra PS. Journal of Magnetism and Magnetic Materials. 2004;**271**:207

and magnetic study of

10.1155/2013/471472

2013;**332**:137-145

Sainkar SR, Joy PA. Journal of the American Ceramic Society.

Applied Physics. 2013;**13**:41-46

2005;**88**:2597-2259

2017;**695**:1112-1121

2009;**63**:1921-1924

behavior of nanocrystalline

*DOI: http://dx.doi.org/10.5772/intechopen.91204*

[18] Ahmed Farag IS et al. Preparation and structural characterization of Eu0.65Sr0.35Mn1-xFexO3. Egyptian Journal

[19] Abdel-Latif IA. Study on structure, electrical and dielectric properties of Eu 0.65Sr0.35Fe0.3Mn0.7O3. IOP Conference

[17] Parfenov VV, Bashkirov SS, Abdel-Latif IA, Marasinskaya AV. Russian Physics Journal. 2003;**46**:979-

of Solids. 2007;**30**(1):149

Series: Materials Science and Engineering. 2016;**146**:012003

[20] Ding J, Yang H, Miao WF, Mccormick PG, Street R. Journal of Alloys and Compounds. 1995;**221**:70-73

[21] Niaz Akhtar M et al. Y3Fe5O12 nano particulate garnet ferrites: Comprehensive study on the synthesis and characterization fabricated by various routes. Journal of

Magnetism and Magnetic Materials.

[22] Yu H, Zeng L, Lu C, Zhang W, Xu G. Materials Characterization.

[24] Sanchez RD, Rivas J, Vaqueiro P, Quintela MAL, Caeiro D. Journal of Magnetism and Magnetic Materials.

[25] Liu CP, Li MW, Cui Z, Huang JR, Tian YL, Lin T, et al. Journal of Materials

Khalid K, Maarof M. European Journal of Scientific Research. 2009;**36**:154-160

Joy PA, Bhattacharya AK. Journal of Magnetism and Magnetic Materials.

Science. 2007;**42**:6133-6138

[27] Rajendran M, Deka S,

2006;**301**:212-219

[26] Abbas Z, Al-habashi RM,

[23] Rastogi AC, Moorthy VN. Materials Science and Engineering.

2014;**368**:393-400

2011;**62**:378-381

2002;**B95**:131-136

2002;**247**:92-98

983

*Investigation of Shielding Effectiveness of M-Type Ba-Co-Ti Hexagonal Ferrite and Composite… DOI: http://dx.doi.org/10.5772/intechopen.91204*

[17] Parfenov VV, Bashkirov SS, Abdel-Latif IA, Marasinskaya AV. Russian Physics Journal. 2003;**46**:979- 983

[18] Ahmed Farag IS et al. Preparation and structural characterization of Eu0.65Sr0.35Mn1-xFexO3. Egyptian Journal of Solids. 2007;**30**(1):149

[19] Abdel-Latif IA. Study on structure, electrical and dielectric properties of Eu 0.65Sr0.35Fe0.3Mn0.7O3. IOP Conference Series: Materials Science and Engineering. 2016;**146**:012003

[20] Ding J, Yang H, Miao WF, Mccormick PG, Street R. Journal of Alloys and Compounds. 1995;**221**:70-73

[21] Niaz Akhtar M et al. Y3Fe5O12 nano particulate garnet ferrites: Comprehensive study on the synthesis and characterization fabricated by various routes. Journal of Magnetism and Magnetic Materials. 2014;**368**:393-400

[22] Yu H, Zeng L, Lu C, Zhang W, Xu G. Materials Characterization. 2011;**62**:378-381

[23] Rastogi AC, Moorthy VN. Materials Science and Engineering. 2002;**B95**:131-136

[24] Sanchez RD, Rivas J, Vaqueiro P, Quintela MAL, Caeiro D. Journal of Magnetism and Magnetic Materials. 2002;**247**:92-98

[25] Liu CP, Li MW, Cui Z, Huang JR, Tian YL, Lin T, et al. Journal of Materials Science. 2007;**42**:6133-6138

[26] Abbas Z, Al-habashi RM, Khalid K, Maarof M. European Journal of Scientific Research. 2009;**36**:154-160

[27] Rajendran M, Deka S, Joy PA, Bhattacharya AK. Journal of Magnetism and Magnetic Materials. 2006;**301**:212-219

[28] Verma S, Pradhan SD, Pasricha R, Sainkar SR, Joy PA. Journal of the American Ceramic Society. 2005;**88**:2597-2259

[29] Aen F, Ahmad M, Rana MU. Current Applied Physics. 2013;**13**:41-46

[30] Singh J et al. Elucidation of phase evolution, microstructural, Mossbauer and magnetic properties of CO2-Al3 doped M-type Ba-Sr hexaferrites synthesized by a ceramic method. Journal of Alloys and Compounds. 2017;**695**:1112-1121

[31] Singh C, Narang SB, Hudiara IS, Bai Y, Marina K. Hysteresis analysis of Co–Ti substituted M-type Ba–Sr hexagonal ferrite. Materials Letters. 2009;**63**:1921-1924

[32] Sharbati A, Choopani S, Azar A-M, Senna M. Structure and electromagnetic behavior of nanocrystalline SrMgxZrxFe12-2xO19 in the 8-12 GHz frequency range. Journal of Solid State Communications. 2010;**150**:2218-2222

[33] Reimann T, Schmidt T, Töpfer J. Phase stability and magnetic properties of SrFe18O27 W-type hexagonal ferrite. Journal of the American Ceramic Society. 2019. DOI: 10.1111/jace.16726

[34] Arjunwadkar PR, Salunkhe MY, Dudhe CM. Structural, electrical, and magnetic study of SrNi2+(Li1+Fe3+)0.5Fe16O27 ferrite. Journal of Solid State Physics. 2013. DOI: 10.1155/2013/471472

[35] Mukhtar A, Grossinger R, Kriegisch M, Kubel F, Rana MU. Characterization of Sr-substituted W-type hexagonal ferrites synthesized by sol–gel autocombustion method. Journal of Magnetism and Magnetic Materials. 2013;**332**:137-145

[36] Meshram MR, Agrawal NK, Sinha B, Misra PS. Journal of Magnetism and Magnetic Materials. 2004;**271**:207

**138**

*Composite Materials*

**References**

s10751-011-0529-8

2012;**1**(2):50-53

am.20190802.15

Letters. 2009;**4**:221

2003;**198**(457-464)

[5] Maensiri S, Sangmanee M, Wiengmoon A. Magnesium ferrite (MgFe2O4) nanostructures fabricated by electrospinning. Nanoscale Research

[6] Fayek MK, Ata-Allah SS. 57Fe Mossbauer and electrical studies of the (NiO)–(Cr2O3)x–(Fe2O3)2−<sup>x</sup> system. Physica Status Solidi A: Applications and Material Science.

[7] Tatarchuk T, Bououdina M, Judith VJ, John Kennedy L. Spinel ferrite nanoparticles: Synthesis, crystal structure, properties, and perspective applications. In: Fesenko O, Yatsenko L, editors. Nanophysics, Nanomaterials, Interface Studies, and Applications. NANO 2016. Springer Proceedings in Physics, vol 195. Springer, Cham. 2017

[1] Gismelseed AM, Khalaf KAM, Elzain ME, Widatallah HM,

[2] Roumaih K, Manapov RA, Sadykov EK, Pyataev AV. Mossbauer studies of Cu1−xNixFeMnO4 spinel ferrites. Journal of Magnetism and Magnetic Materials. 2005;**288**:267

[3] Abdel Latif IA. Fabrication of nano-size nickel ferrites for gas sensors applications. Journal of Physics.

[4] Khalaf KA, Al-Rawas A, Gismelseed A, Al-Ruqeishi M, Al-Ani S, Al-Jubouri A, et al. Effects of Zn substitution on structure factors, Debye-Waller factors and related structural properties of the Mg1-xZnxFeNiO4 spinels. Advances in Materials. 2019;**8**(2):70-93. DOI: 10.11648/j.

Al-Rawas AD, Yousif AA. The structural and magnetic behavior of the MgFe2<sup>−</sup> xCrxO4 spinel ferrite. Hyperfine Interactions. 2012. DOI: 10.1007/

[8] Zaki HM et al. Synthesis and characterization of nanocrystalline MgAlxFe2-xO4 ferrites. Journal of Materials Research. 2012;**27**(21):2798

[9] Al-Maashani M, Gismelseed AM, Khalaf KAM, Yousif AA, Al-Rawas AD, Widatallah HM, et al. Structural and Mossbauer study of nanoparticles CoFe2O4 prepared by sol-gel autocombustion and subsequent sintering. Hyperfine Interactions. 2018;**239**:15. DOI: 10.1007/s10751-018-1491-5

[10] Yousif AA et al. Study on Mossbauer and magnetic properties of strontium doped neodymium ferrimanganites perovskite-like structure. AIP

Conference Proceedings. 2011;**1370**:103

[11] Abdel-Latif IA, Saleh SA. Effect of iron doping on the physical properties of europium Manganites. Journal of Alloys and Compounds. 2012;**530**:116

[12] Bashkirov S et al. Crystal structure, electric and magnetic properties of ferrimanganite NdFexMn1−xO3. Izv. RAS,

Physical Series. 2003;**67**:1072

[13] Abdel-Latif IA et al. The

Compounds. 2008;**452**:245

Materials. 2018;**457**:126

2005;**97**(10A):504

2011;**1370**:108

[15] Bouziane KA et al. Electronic and magnetic properties of SmFe1<sup>−</sup> xMnxO3 orthoferrites (x = 0.1, 0.2 and 0.3). Journal of Applied Physics.

[16] Abdel-Latif IA. Study on the effect of particle size of strontium-ytterbium manganites on some physical properties.

AIP Conference Proceedings.

influence of tilt angle on the CMR in Sm0.6Sr0.4MnO3. Journal of Alloys and

[14] Abdel-Latif IA et al. Magnetocaloric effect, electric, and dielectric properties of Nd0.6Sr0.4MnxCo1−xO3 composites. Journal of Magnetism and Magnetic

[37] Goldman A. Modern Ferrite Technology. 2nd ed. Springer Publication; 2006. pp. 83

[38] Amer MA, Hemeda OM. Hyperfine Interactions. 1995;**96**:99

[39] Hosaka N et al. Crystal structure and magnetic properties of X-type hexagonal ferrite Ba2Ni2Fe28O46. Journal of the Japan Society of Powder and Powder Metallurgy. 2010;**57**:41-45

[40] Gu BX. Magnetic properties of *X*-type Ba2Me2Fe28O46 (Me = Fe, Co, and Mn) hexagonal ferrites. Journal of Applied Physics. 1992;**71**:5103. DOI: 10.1063/1.350613

[41] Gu BX. Magnetic properties of Ba2Me2Fe28O46 (Me2-*X*, Me = Ni, Cu, Mg, and Zn) hexaferrites. Journal of Applied Physics. 1991;**70**:372. DOI: 10.1063/1.350284

[42] Ben-Xi G, Huai-Xian LU, You-Wei DU. Magnetic properties and Mössbauer spectra of X type hexagonal ferrites. Journal of Magnetism and Magnetic Materials. 1983;**31-34**(Part 2):803-804

[43] Reddy MB, Reddy PV. Lowfrequency dielectric behaviour of mixed Li-Ti ferrites. Journal of Physics D: Applied Physics. 1991;**24**:975

[44] Bayrakdar H. Fabrication, magnetic and microwave absorbing properties of Ba2Co2Cr2Fe12O22 hexagonal ferrites. Journal of Alloys and Compounds. 2016;**675**:185-188

[45] Bai Y, Zhou J, Gui Z, Yue Z, Li L. Complex Y-type hexagonal ferrites: An ideal material for high-frequency chip magnetic components. Journal of Magnetism and Magnetic Materials. 2003;**264**(1):44-49

[46] Lee SG, Kwon SJ. Saturation magnetizations and Curie temperatures of Co-Zn Y-type ferrites. Journal of

Magnetism and Magnetic Materials. 1996;**153**:279-284

[47] Smit J, Wijn HPJ. Ferrites. Eindhoven, The Netherlands: Philips Technical Library; 1965. p. 177

[48] Sugimoto M. In: Wohlfarth EP, editor. Ferromagnetic Materials. Vol. 3. Amsterdam: North-Holland; 1982. p. 393

[49] Lubitz P, Rachford FJ. Z type Ba hexagonal ferrites with tailored microwave properties. Journal of Applied Physics. 2002;**91**:7613. DOI: 10.1063/1.1453932

[50] Nakamura T, Hankui E. Control of high-frequency permeability in polycrystalline (Ba,Co)-Ztype hexagonal ferrite. Journal of Magnetism and Magnetic Materials. 2003;**257**(2-3):158-164

[51] Narang SB, Kaur P, Bahel S, Singh C. Microwave characterization of Co–Ti substituted barium hexagonal ferrites in X-band. Journal of Magnetism and Magnetic Materials. 2016;**405**:17-21

[52] Zhang BS, Feng Y, Xiong J, Yang Y, Lu HX. Microwave-absorbing properties of de-aggregated flake-shaped carbonyl-iron particle composites at 2-18GHz. IEEE Transactions on Magnetics. 2006;**42**:1778-1781

[53] Singh P, Babbar VK, Razdan A, Srivastava SL, Puri RK. Complex permeability and permittivity, and microwave absorption studies of Ca(CoTi)xFe12−2xO19 hexaferrite composites in X-band microwave frequencies. Materials Science and Engineering. 1999;**B67**:132-138

[54] Sugimoto S, Haga K, Kagotani T, Inomata K. Microwave absorption properties of BaM-type ferrite prepared by a modified co precipitation method. Journal of Magnetism and Magnetic Materials. 2005;**290**:1188-1191

**141**

2012

*Investigation of Shielding Effectiveness of M-Type Ba-Co-Ti Hexagonal Ferrite and Composite…*

Conference (AP-RASC'10) Toyama.

[66] Al-Saleh MH, Sundararaj U. X-band EMI shielding mechanisms and shielding effectiveness of high structure carbon black/polypropylene composites. Journal of Physics D: Applied Physics.

[67] Cui R-B, Zhang C, Zhang J-Y, Xue W, Hou Z-L. Highly dispersive GO-based supramolecular absorber: Chemical-reduction optimization for impedance matching. Journal of Alloys

and Compounds. 2020;**155122**

materials: A review. Journal of Electronic Materials. 2019;**48**(5):2601-2634

2009;**4**(4):327-334

2018;**10**(6):582

s11051-020-4763-3

[73] Abdel-Latif IA. The particle size effect of Yb0.8R0.2MnO3 (R is

[68] Raveendran A, Sebastian MT, Raman S. Applications of microwave

[69] Pande S, Singh BP, Mathur RB, Dhami TL, Saini P, Dhawan SK. Improved electromagnetic interference shielding properties of MWCNT? PMMA composites using layered structures. Nanoscale Research Letters.

[70] Kashi S, Hadigheh SA, Varley R. Microwave attenuation of graphene modified thermoplastic poly

(butylene adipate-co-terephthalate)

[71] Abdel-Latif IA. Crystal structure and electrical transport of nanocrystalline strontium-doped

neodymium ortho-ferrites. Journal of Nanoparticle Research. 2020;**22**:60

[72] Saleh S, Abdel-Latif I, Hakeem AA, et al. Structural and frequency-dependent dielectric properties of (SnO2)1−*x* (Fe2O3)*x*. Journal of Nanoparticle Research. 2020;**22**:44. DOI: 10.1007/

nanocomposites. Polymers.

Paper E4-3; 2010

2012;**46**(3):035304

*DOI: http://dx.doi.org/10.5772/intechopen.91204*

[55] Singh C, Narang SB, Hudiara IS, Sudheendran K, James Raju KC. Complex permittivity and complex permeability of Sr ions substituted Ba ferrite at X-band. Journal of Magnetism and Magnetic Materials.

[56] Lima UR, Nasar MC, Rezende MC, Araugo JH. Journal of Magnetism and Magnetic Materials. 2008;**320**:1666

[57] Aphesteguy JC, Pamiani A, Digiovanni D, Jacobo SE. Physica B.

[58] Huang X, Zhang J, Lai M, Sang T. Journal of Alloys and Compounds.

[59] Koledintseva MY, Mikhailovsky LK, Kitaytsev AA. IEEE Transactions on Electromagnetic Compatibility.

[60] Meng P, Xiong K, Ju K, Li S, Xu G. Journal of Magnetism and Magnetic

[62] Meng P, Xiong K, Wang L, Li S, Cheng Y, Xu G. Journal of Alloys and

2008;**3**(20):1657-1665

2009;**404**:2713

2015;**627**:367

2000;**2**:773

Materials. 2015;**385**:407

Letters. 2015;**158**:53

[61] Liu J, Zhang J, Zhang P, Wang S, Lu C, Li Y, et al. Materials

Compounds. 2015;**628**:75

[63] Singh C, Bindra Narang S, Vikramjit Singh, Kotnala RK. IEEE 15th International Symposium on Antenna Technology and Applied Electromagnetics (ANTEM), Toulouse;

[64] Singh C, Bindra Narang S, Hudiara IS. IEEE XXX General Assembly and Scientific Symposium of the International Union of Radio Science (URSI), Istanbul; 2011

[65] Singh C, Bindra Narang S, Hudiara IS, Koledintseva MY,

Kitaitsev AA. Asia-Pacific Radio Science

*Investigation of Shielding Effectiveness of M-Type Ba-Co-Ti Hexagonal Ferrite and Composite… DOI: http://dx.doi.org/10.5772/intechopen.91204*

[55] Singh C, Narang SB, Hudiara IS, Sudheendran K, James Raju KC. Complex permittivity and complex permeability of Sr ions substituted Ba ferrite at X-band. Journal of Magnetism and Magnetic Materials. 2008;**3**(20):1657-1665

*Composite Materials*

[37] Goldman A. Modern Ferrite Technology. 2nd ed. Springer Publication; 2006. pp. 83

Interactions. 1995;**96**:99

10.1063/1.350613

10.1063/1.350284

2):803-804

2016;**675**:185-188

2003;**264**(1):44-49

[38] Amer MA, Hemeda OM. Hyperfine

Magnetism and Magnetic Materials.

[48] Sugimoto M. In: Wohlfarth EP, editor. Ferromagnetic Materials. Vol. 3. Amsterdam: North-Holland; 1982. p.

[49] Lubitz P, Rachford FJ. Z type Ba hexagonal ferrites with tailored microwave properties. Journal of Applied Physics. 2002;**91**:7613. DOI:

[50] Nakamura T, Hankui E. Control of high-frequency permeability in polycrystalline (Ba,Co)-Ztype hexagonal ferrite. Journal of Magnetism and Magnetic Materials.

[51] Narang SB, Kaur P, Bahel S, Singh C. Microwave characterization of Co–Ti substituted barium hexagonal ferrites in X-band. Journal of Magnetism and Magnetic Materials. 2016;**405**:17-21

[52] Zhang BS, Feng Y, Xiong J, Yang Y, Lu HX. Microwave-absorbing properties

of de-aggregated flake-shaped carbonyl-iron particle composites at 2-18GHz. IEEE Transactions on Magnetics. 2006;**42**:1778-1781

[53] Singh P, Babbar VK, Razdan A, Srivastava SL, Puri RK. Complex permeability and permittivity, and microwave absorption studies of Ca(CoTi)xFe12−2xO19 hexaferrite composites in X-band microwave frequencies. Materials Science and Engineering. 1999;**B67**:132-138

[54] Sugimoto S, Haga K,

2005;**290**:1188-1191

Kagotani T, Inomata K. Microwave absorption properties of BaM-type ferrite prepared by a modified co precipitation method. Journal of Magnetism and Magnetic Materials.

[47] Smit J, Wijn HPJ. Ferrites. Eindhoven, The Netherlands: Philips Technical Library; 1965. p. 177

1996;**153**:279-284

10.1063/1.1453932

2003;**257**(2-3):158-164

393

[39] Hosaka N et al. Crystal structure and magnetic properties of X-type hexagonal ferrite Ba2Ni2Fe28O46. Journal of the Japan Society of Powder and Powder Metallurgy. 2010;**57**:41-45

[40] Gu BX. Magnetic properties of *X*-type Ba2Me2Fe28O46 (Me = Fe, Co, and Mn) hexagonal ferrites. Journal of Applied Physics. 1992;**71**:5103. DOI:

[41] Gu BX. Magnetic properties of Ba2Me2Fe28O46 (Me2-*X*, Me = Ni, Cu, Mg, and Zn) hexaferrites. Journal of Applied Physics. 1991;**70**:372. DOI:

[42] Ben-Xi G, Huai-Xian LU,

[43] Reddy MB, Reddy PV. Low-

You-Wei DU. Magnetic properties and Mössbauer spectra of X type hexagonal ferrites. Journal of Magnetism and Magnetic Materials. 1983;**31-34**(Part

frequency dielectric behaviour of mixed Li-Ti ferrites. Journal of Physics D: Applied Physics. 1991;**24**:975

[44] Bayrakdar H. Fabrication, magnetic and microwave absorbing properties of Ba2Co2Cr2Fe12O22 hexagonal ferrites. Journal of Alloys and Compounds.

[45] Bai Y, Zhou J, Gui Z, Yue Z, Li L. Complex Y-type hexagonal ferrites: An ideal material for high-frequency chip magnetic components. Journal of Magnetism and Magnetic Materials.

[46] Lee SG, Kwon SJ. Saturation

magnetizations and Curie temperatures of Co-Zn Y-type ferrites. Journal of

**140**

[56] Lima UR, Nasar MC, Rezende MC, Araugo JH. Journal of Magnetism and Magnetic Materials. 2008;**320**:1666

[57] Aphesteguy JC, Pamiani A, Digiovanni D, Jacobo SE. Physica B. 2009;**404**:2713

[58] Huang X, Zhang J, Lai M, Sang T. Journal of Alloys and Compounds. 2015;**627**:367

[59] Koledintseva MY, Mikhailovsky LK, Kitaytsev AA. IEEE Transactions on Electromagnetic Compatibility. 2000;**2**:773

[60] Meng P, Xiong K, Ju K, Li S, Xu G. Journal of Magnetism and Magnetic Materials. 2015;**385**:407

[61] Liu J, Zhang J, Zhang P, Wang S, Lu C, Li Y, et al. Materials Letters. 2015;**158**:53

[62] Meng P, Xiong K, Wang L, Li S, Cheng Y, Xu G. Journal of Alloys and Compounds. 2015;**628**:75

[63] Singh C, Bindra Narang S, Vikramjit Singh, Kotnala RK. IEEE 15th International Symposium on Antenna Technology and Applied Electromagnetics (ANTEM), Toulouse; 2012

[64] Singh C, Bindra Narang S, Hudiara IS. IEEE XXX General Assembly and Scientific Symposium of the International Union of Radio Science (URSI), Istanbul; 2011

[65] Singh C, Bindra Narang S, Hudiara IS, Koledintseva MY, Kitaitsev AA. Asia-Pacific Radio Science Conference (AP-RASC'10) Toyama. Paper E4-3; 2010

[66] Al-Saleh MH, Sundararaj U. X-band EMI shielding mechanisms and shielding effectiveness of high structure carbon black/polypropylene composites. Journal of Physics D: Applied Physics. 2012;**46**(3):035304

[67] Cui R-B, Zhang C, Zhang J-Y, Xue W, Hou Z-L. Highly dispersive GO-based supramolecular absorber: Chemical-reduction optimization for impedance matching. Journal of Alloys and Compounds. 2020;**155122**

[68] Raveendran A, Sebastian MT, Raman S. Applications of microwave materials: A review. Journal of Electronic Materials. 2019;**48**(5):2601-2634

[69] Pande S, Singh BP, Mathur RB, Dhami TL, Saini P, Dhawan SK. Improved electromagnetic interference shielding properties of MWCNT? PMMA composites using layered structures. Nanoscale Research Letters. 2009;**4**(4):327-334

[70] Kashi S, Hadigheh SA, Varley R. Microwave attenuation of graphene modified thermoplastic poly (butylene adipate-co-terephthalate) nanocomposites. Polymers. 2018;**10**(6):582

[71] Abdel-Latif IA. Crystal structure and electrical transport of nanocrystalline strontium-doped neodymium ortho-ferrites. Journal of Nanoparticle Research. 2020;**22**:60

[72] Saleh S, Abdel-Latif I, Hakeem AA, et al. Structural and frequency-dependent dielectric properties of (SnO2)1−*x* (Fe2O3)*x*. Journal of Nanoparticle Research. 2020;**22**:44. DOI: 10.1007/ s11051-020-4763-3

[73] Abdel-Latif IA. The particle size effect of Yb0.8R0.2MnO3 (R is Sm, Nd, and Eu) on some physical properties. Journal of Nanoparticle Research. 2020;**22**:45. DOI: 10.1007/ s11051-020-4759-z

**Chapter 9**

**Abstract**

inclusions.

**143**

A New Boundary Element

Formulation for Modeling

and Optimization of

Three-Temperature

Microstructures

*Mohamed Abdelsabour Fahmy*

Nonlinear Generalized

Magneto-Thermoelastic

Problems of FGA Composite

The main purpose of this chapter is to propose a new boundary element formulation for the modeling and optimization of three-temperature nonlinear generalized magneto-thermoelastic functionally graded anisotropic (FGA) composite microstructures' problems, which is the gap of this study. Numerical results show that anisotropy and the functionally graded material have great influences on the nonlinear displacement sensitivities and nonlinear thermal stress sensitivities of composite microstructure optimization problem. Since, there are no available data for comparison, except for the problems with one-temperature heat conduction model, we considered the special case of our general study based on replacing threetemperature radiative heat conductions with one-temperature heat conduction. In the considered special case, numerical results demonstrate the validity and accuracy of the proposed technique. In order to solve the optimization problem, the method of moving asymptotes (MMA) based on the bi-evolutionary structural optimization method (BESO) has been implemented. A new class of composite microstructures

problems with holes or inclusions was studied. The two-phase magneto-

**Keywords:** boundary element method, modeling and optimization,

graded anisotropic, composite microstructures

thermoelastic composite microstructure which is studied in this chapter consists of two different FGA materials. Through this chapter, we investigated that the optimal material distribution of the composite microstructures depends strongly on the heat conduction model, functionally graded parameter, and shapes of holes or

three-temperature, nonlinear generalized magneto-thermoelasticity, functionally

[74] Khasim S. Polyaniline-graphene nanoplatelet composite films with improved conductivity for high performance X-band microwave shielding applications. Results in Physics. 2019;**12**:1073-1081

[75] Bal S, Saha S. Scheming of microwave shielding effectiveness for X band considering functionalized MWNTs/epoxy composites. IOP Conference Series: Materials Science and Engineering. 2016;**115**(1):012027

[76] Arora M, Wahab MA, Saini P. Permittivity and electromagnetic interference shielding investigations of activated charcoal loaded acrylic coating compositions. Journal of Polymers. 2014

[77] Das NC, Chaki TK, Khastgir D, Chakraborty A. Electromagnetic interference shielding effectiveness of conductive carbon black and carbon fiber‐filled composites based on rubber and rubber blends. Advances in Polymer Technology: Journal of the Polymer Processing Institute. 2001;**20**(3):226-236

[78] Das NC, Chaki TK, Khastgir D, Chakraborty A. Electromagnetic interference shielding effectiveness of ethylene vinyl acetate based conductive composites containing carbon fillers. Journal of Applied Polymer Science. 2001;**80**(10):1601-1608

[79] Colaneri NF, Shacklette LW. IEEE Transactions on Instrumentation and Measurement. 1992;**41**:291

[80] Maxwell JC. New York: Oxford University Press; 2004. p. 828

[81] Van Uitert LG. The Journal of Chemical Physics. 1955;**23**:1883

#### **Chapter 9**

*Composite Materials*

s11051-020-4759-z

Sm, Nd, and Eu) on some physical properties. Journal of Nanoparticle Research. 2020;**22**:45. DOI: 10.1007/

[74] Khasim S. Polyaniline-graphene nanoplatelet composite films with improved conductivity for high performance X-band microwave shielding applications. Results in Physics. 2019;**12**:1073-1081

[75] Bal S, Saha S. Scheming of microwave shielding effectiveness for X band considering functionalized MWNTs/epoxy composites. IOP Conference Series: Materials Science and Engineering. 2016;**115**(1):012027

[76] Arora M, Wahab MA, Saini P. Permittivity and electromagnetic interference shielding investigations of activated charcoal loaded acrylic coating compositions. Journal of

[77] Das NC, Chaki TK, Khastgir D, Chakraborty A. Electromagnetic interference shielding effectiveness of conductive carbon black and carbon fiber‐filled composites based on rubber and rubber blends. Advances in Polymer Technology: Journal of the Polymer Processing Institute.

[78] Das NC, Chaki TK, Khastgir D, Chakraborty A. Electromagnetic interference shielding effectiveness of ethylene vinyl acetate based conductive composites containing carbon fillers. Journal of Applied Polymer Science.

[79] Colaneri NF, Shacklette LW. IEEE Transactions on Instrumentation and

[80] Maxwell JC. New York: Oxford University Press; 2004. p. 828

[81] Van Uitert LG. The Journal of Chemical Physics. 1955;**23**:1883

Polymers. 2014

2001;**20**(3):226-236

2001;**80**(10):1601-1608

Measurement. 1992;**41**:291

**142**

A New Boundary Element Formulation for Modeling and Optimization of Three-Temperature Nonlinear Generalized Magneto-Thermoelastic Problems of FGA Composite Microstructures

*Mohamed Abdelsabour Fahmy*

#### **Abstract**

The main purpose of this chapter is to propose a new boundary element formulation for the modeling and optimization of three-temperature nonlinear generalized magneto-thermoelastic functionally graded anisotropic (FGA) composite microstructures' problems, which is the gap of this study. Numerical results show that anisotropy and the functionally graded material have great influences on the nonlinear displacement sensitivities and nonlinear thermal stress sensitivities of composite microstructure optimization problem. Since, there are no available data for comparison, except for the problems with one-temperature heat conduction model, we considered the special case of our general study based on replacing threetemperature radiative heat conductions with one-temperature heat conduction. In the considered special case, numerical results demonstrate the validity and accuracy of the proposed technique. In order to solve the optimization problem, the method of moving asymptotes (MMA) based on the bi-evolutionary structural optimization method (BESO) has been implemented. A new class of composite microstructures problems with holes or inclusions was studied. The two-phase magnetothermoelastic composite microstructure which is studied in this chapter consists of two different FGA materials. Through this chapter, we investigated that the optimal material distribution of the composite microstructures depends strongly on the heat conduction model, functionally graded parameter, and shapes of holes or inclusions.

**Keywords:** boundary element method, modeling and optimization, three-temperature, nonlinear generalized magneto-thermoelasticity, functionally graded anisotropic, composite microstructures

#### **1. Introduction**

In the last few years, there is significant interest in using advanced composite structures, and among the oldest examples of them, reinforced concrete, mixing concrete and steel, and plastics laminated with wood. The main benefit of the composite structures which consist of two or more different materials is that the properties of each material can be combined to form a single unit that performs better than the separate component parts. The most common form of a composite structure in construction is a steel and concrete composite, where concrete works well in pressure but has less resistance to tension. However, steel is extremely strong in tension, and when tied together, it results in a highly efficient and lightweight unit usually used for structures such as buildings and multistory bridges. Although fiberglass and carbon/epoxy composites are not yet as important as the oldest advanced composite structures in terms of tonnage or total revenue, they are very important in engineering, aerospace, transportation, bioengineering, optics, electronics, commodities, chemical plant, and energy industries, especially for the new airplanes that will concentrate on achieving major improvements in the fuel use, emissions, noise, transportation energy consumption, and other important issues to conserve the environment [1–21].

solving three-temperature nonlinear generalized magneto-thermoelastic problems of FGA composite microstructures, the problems become too complicated with no general analytical solution. Therefore, we propose a new boundary element modeling technique which has recently been successfully developed and implemented to obtain the approximate solutions for such problems. Now, the boundary element method (BEM), which is also called boundary integral equation method, has been widely adopted in a large variety of engineering and industrial applications. In the BEM, only the boundary of the solution domain needs to be discretized, so, it has a

*A New Boundary Element Formulation for Modeling and Optimization of Three-Temperature…*

discretization, such as the finite difference method (FDM) [50–52], discontinuous Galerkin method (DGM) [53], and finite element method (FEM) [54–57]. This advantage of BEM over domain methods has significant importance for modeling of nonlinear generalized thermoelastic problems which can be implemented using BEM with little cost and less input data [58–71]. Recently, scientists were convinced that only the FEM method could solve complex engineering problems. But now after the huge achievements of the BEM and its ability to solve complex engineering problems with high efficiency, it gets them to change their conviction. Also, they tried to combine FEM and BEM in the solution of their complex problems.

The main aim of this chapter is to propose a novel boundary element formulation for modeling and optimization of three-temperature nonlinear generalized thermoelastic problems of functionally graded anisotropic (FGA) composite microstructures. The proposed boundary element technique has been *implemented* successfully for solving several engineering, scientific and industrial applications due to its simplicity, efficiency, ease of use, and applicability [72–85]. The numerical results are presented graphically to show the influence of anisotropy and functionally graded materials on the sensitivities of displacements and thermal stresses. Also, numerical results show the effect of heat conduction model, functionally graded parameter, holes shape, and inclusions shape. Numerical results demonstrate the validity and accuracy of our proposed BEM formulation and technique. A brief summary of the chapter is as follows: Section 1 introduces an overview of the historical background for a better understanding of the nonlinear generalized magneto-thermoelastic problems and composite materials applications. Section 2 describes the physical modeling of the three-temperature nonlinear generalized thermoelastic problems of FGA composite microstructures. Section 3 outlines the BEM implementation for solving the governing equations of the considered problem to obtain the three temperatures and displacement fields. Section 4 outlines the topology optimization technique used to obtain the optimal composite microstructure with and without holes or inclusions of various shapes. Section 5 presents the new numerical results that describe the effects of anisotropy and functionally graded parameters on the problem's fields'sensitivities during the optimization

major advantage over other methods which require the whole domain

*DOI: http://dx.doi.org/10.5772/intechopen.93515*

process. Section 6 outlines the significant findings of this chapter.

Consider a Cartesian coordinates system *Ox*1*x*2*x*<sup>3</sup> as shown in **Figure 1**. We shall

consider a functionally graded anisotropic composite microstructure of a finite thickness *β* placed in a primary magnetic field *H*<sup>0</sup> acting in the direction of the

n o with functionally graded

*x*3-axis. The considered composite microstructure occupies the region

**2. Formulation of the problem**

**145**

*R* ¼ ð Þ *x*1, *x*2, *x*<sup>3</sup> : 0<*x*<sup>1</sup> <*α*, 0< *x*<sup>2</sup> <*β*, 0 <*x*<sup>3</sup> <*γ*

material properties in the thickness direction.

Microstructure has been known to play a major role in determining the behavior of material. Therefore, material engineers strive to control the microstructure by improving their properties with the aim of producing a uniform microstructure throughout the material. They also produced FGMs whose microstructures depend on the position by treating the microstructure as a position-dependent variable; the properties of different materials can be combined into one component to achieve an optimum performance in a specific application [22, 23].

In recent years, great attention has been directed toward the study of nonlinear generalized magneto-thermoelastic interactions in functionally graded anisotropic (FGA) structures due to its many applications in physics, geophysics, earthquake engineering, astronautics, aeronautics, mining engineering, military technologies, plasma, robotics, high-energy particle accelerators, nuclear reactors, automobile industries, nuclear plants, soil dynamics, and other engineering and industrial applications. Duhamel [24] and Neuman [25] proposed the classical thermoelasticity (CTE) theory which has the following two paradoxes: first, the infinite propagation speeds of thermal signals are predicted, and second, there is no any elastic term included in heat equation. Biot [26] invented the classical coupled thermoelasticity (CCTE) theory to beat the first paradox in CTE, but CTE and CCTE share the second paradox. Then, numerous generalized thermoelasticity theories have been introduced to overcome the two paradoxes inherent in CTE, such as the extended thermoelasticity (ETE) theory of Lord and Shulman [27]; temperature-rate-dependent thermoelasticity (TRDTE) theory of Green and Lindsay [28]; three linear generalized thermoelasticity theories of Green and Naghdi (GN) [29, 30]; namely I, II, and III, respectively [where, GN theory I is based on Fourier's law of heat conduction and is identical to CTE theory, GN theory II characterizes the thermoelasticity without energy dissipation (TEWOED), and GN theory III characterizes the thermoelasticity with energy dissipation (TEWED)]; dual phase-lag thermoelasticity (DPLTE) [31, 32]; and three-phase-lag thermoelasticity (TPLTE) [33].

A large amount of research has been done on the generalized problems of thermoelasticity [34–44]. Our interest in studying the three-temperature thermoelasticity [45–49] has increased due to its important low-temperature and high-temperature applications. Due to the computational difficulties, inherent in

#### *A New Boundary Element Formulation for Modeling and Optimization of Three-Temperature… DOI: http://dx.doi.org/10.5772/intechopen.93515*

solving three-temperature nonlinear generalized magneto-thermoelastic problems of FGA composite microstructures, the problems become too complicated with no general analytical solution. Therefore, we propose a new boundary element modeling technique which has recently been successfully developed and implemented to obtain the approximate solutions for such problems. Now, the boundary element method (BEM), which is also called boundary integral equation method, has been widely adopted in a large variety of engineering and industrial applications. In the BEM, only the boundary of the solution domain needs to be discretized, so, it has a major advantage over other methods which require the whole domain discretization, such as the finite difference method (FDM) [50–52], discontinuous Galerkin method (DGM) [53], and finite element method (FEM) [54–57]. This advantage of BEM over domain methods has significant importance for modeling of nonlinear generalized thermoelastic problems which can be implemented using BEM with little cost and less input data [58–71]. Recently, scientists were convinced that only the FEM method could solve complex engineering problems. But now after the huge achievements of the BEM and its ability to solve complex engineering problems with high efficiency, it gets them to change their conviction. Also, they tried to combine FEM and BEM in the solution of their complex problems.

The main aim of this chapter is to propose a novel boundary element formulation for modeling and optimization of three-temperature nonlinear generalized thermoelastic problems of functionally graded anisotropic (FGA) composite microstructures. The proposed boundary element technique has been *implemented* successfully for solving several engineering, scientific and industrial applications due to its simplicity, efficiency, ease of use, and applicability [72–85]. The numerical results are presented graphically to show the influence of anisotropy and functionally graded materials on the sensitivities of displacements and thermal stresses. Also, numerical results show the effect of heat conduction model, functionally graded parameter, holes shape, and inclusions shape. Numerical results demonstrate the validity and accuracy of our proposed BEM formulation and technique.

A brief summary of the chapter is as follows: Section 1 introduces an overview of the historical background for a better understanding of the nonlinear generalized magneto-thermoelastic problems and composite materials applications. Section 2 describes the physical modeling of the three-temperature nonlinear generalized thermoelastic problems of FGA composite microstructures. Section 3 outlines the BEM implementation for solving the governing equations of the considered problem to obtain the three temperatures and displacement fields. Section 4 outlines the topology optimization technique used to obtain the optimal composite microstructure with and without holes or inclusions of various shapes. Section 5 presents the new numerical results that describe the effects of anisotropy and functionally graded parameters on the problem's fields'sensitivities during the optimization process. Section 6 outlines the significant findings of this chapter.

#### **2. Formulation of the problem**

Consider a Cartesian coordinates system *Ox*1*x*2*x*<sup>3</sup> as shown in **Figure 1**. We shall consider a functionally graded anisotropic composite microstructure of a finite thickness *β* placed in a primary magnetic field *H*<sup>0</sup> acting in the direction of the *x*3-axis. The considered composite microstructure occupies the region *R* ¼ ð Þ *x*1, *x*2, *x*<sup>3</sup> : 0<*x*<sup>1</sup> <*α*, 0< *x*<sup>2</sup> <*β*, 0 <*x*<sup>3</sup> <*γ* n o with functionally graded material properties in the thickness direction.

**1. Introduction**

*Composite Materials*

issues to conserve the environment [1–21].

optimum performance in a specific application [22, 23].

applications. Duhamel [24] and Neuman [25] proposed the classical

the extended thermoelasticity (ETE) theory of Lord and Shulman [27]; temperature-rate-dependent thermoelasticity (TRDTE) theory of Green and Lindsay [28]; three linear generalized thermoelasticity theories of Green and Naghdi (GN) [29, 30]; namely I, II, and III, respectively [where, GN theory I is based on Fourier's law of heat conduction and is identical to CTE theory, GN theory II characterizes the thermoelasticity without energy dissipation (TEWOED), and

GN theory III characterizes the thermoelasticity with energy dissipation

thermoelasticity (TPLTE) [33].

**144**

(TEWED)]; dual phase-lag thermoelasticity (DPLTE) [31, 32]; and three-phase-lag

A large amount of research has been done on the generalized problems of thermoelasticity [34–44]. Our interest in studying the three-temperature

thermoelasticity [45–49] has increased due to its important low-temperature and high-temperature applications. Due to the computational difficulties, inherent in

thermoelasticity (CTE) theory which has the following two paradoxes: first, the infinite propagation speeds of thermal signals are predicted, and second, there is no any elastic term included in heat equation. Biot [26] invented the classical coupled thermoelasticity (CCTE) theory to beat the first paradox in CTE, but CTE and CCTE share the second paradox. Then, numerous generalized thermoelasticity theories have been introduced to overcome the two paradoxes inherent in CTE, such as

In the last few years, there is significant interest in using advanced composite structures, and among the oldest examples of them, reinforced concrete, mixing concrete and steel, and plastics laminated with wood. The main benefit of the composite structures which consist of two or more different materials is that the properties of each material can be combined to form a single unit that performs better than the separate component parts. The most common form of a composite structure in construction is a steel and concrete composite, where concrete works well in pressure but has less resistance to tension. However, steel is extremely strong in tension, and when tied together, it results in a highly efficient and lightweight unit usually used for structures such as buildings and multistory bridges. Although fiberglass and carbon/epoxy composites are not yet as important as the oldest advanced composite structures in terms of tonnage or total revenue, they are very important in engineering, aerospace, transportation, bioengineering, optics, electronics, commodities, chemical plant, and energy industries, especially for the new airplanes that will concentrate on achieving major improvements in the fuel use, emissions, noise, transportation energy consumption, and other important

Microstructure has been known to play a major role in determining the behavior of material. Therefore, material engineers strive to control the microstructure by improving their properties with the aim of producing a uniform microstructure throughout the material. They also produced FGMs whose microstructures depend on the position by treating the microstructure as a position-dependent variable; the properties of different materials can be combined into one component to achieve an

In recent years, great attention has been directed toward the study of nonlinear generalized magneto-thermoelastic interactions in functionally graded anisotropic (FGA) structures due to its many applications in physics, geophysics, earthquake engineering, astronautics, aeronautics, mining engineering, military technologies, plasma, robotics, high-energy particle accelerators, nuclear reactors, automobile industries, nuclear plants, soil dynamics, and other engineering and industrial

**Figure 1.** *Computational domain of considered structure.*

The unified governing equations of three-temperature nonlinear generalized magneto-thermoelasticity for FGA composite microstructures can be expressed as follows [45–49]:

$$
\sigma\_{ab,b} + \tau\_{ab,b} = \rho(\varkappa + 1)^m \ddot{u}\_a \tag{1}
$$

where *σab*, *τab*, *uk*, *Tα*, and *T<sup>α</sup>*<sup>0</sup> are the mechanical stress tensor, Maxwell's electromagnetic stress tensor, displacement vector, temperature, and reference temperature, respectively; *Cabfg Cabfg* <sup>¼</sup> *Cfgab* <sup>¼</sup> *Cbafg* and *<sup>β</sup>ab <sup>β</sup>ab* <sup>¼</sup> *<sup>β</sup>ba* ð Þ are, respectively, the constant elastic moduli and stress-temperature coefficients of the

*A New Boundary Element Formulation for Modeling and Optimization of Three-Temperature…*

field; *α*ð Þ *α* ¼ *e*, *i*, *p* are the thermal conductivity coefficients; *Cvα*ð Þ *α* ¼ *e*, *i*, *p* are specific heat coefficients; *e*, *i*, and *p* denote electron, ion, and phonon, respectively;

ei is the electron-ion energy coefficient; *ep* is the electron-phonon energy

*<sup>P</sup>* <sup>¼</sup> *Pe* <sup>þ</sup> *Pi* <sup>þ</sup> *Pp*, *Pe* <sup>¼</sup> *ceTe*, *Pi* <sup>¼</sup> *ciTi*, *Pp* <sup>¼</sup> <sup>1</sup>

¼ 0, *α* ¼ *e*, *i*, *Tr*

*ta*ð Þ¼ *x*, *y*, *τ δ <sup>f</sup>*ð Þ *x*, *y*, *τ* for ð Þ *x*, *y* ∈*C*4, *τ* >0,*C* ¼ *C*<sup>3</sup> ∪*C*4,*C*<sup>3</sup> ∩*C*<sup>4</sup> ¼ ∅ (14)

The above governing Eqs. (1)–(4) can be reduced to the different theories of three-temperature nonlinear generalized magneto-thermoelasticity for FGA com-

By using the fundamental solution that satisfies the following equation:

*<sup>∂</sup><sup>n</sup>* ¼ �*<sup>δ</sup> <sup>r</sup>* � *pi*

 *C*1

*u <sup>f</sup>*ð Þ¼ *x*, *y*, 0 *u*\_ *<sup>f</sup>*ð Þ¼ *x*, *y*, 0 0 for ð Þ *x*, *y* ∈*R* ∪*C* (12)

*u <sup>f</sup>*ð Þ¼ *x*, *y*, *τ* Ψ *<sup>f</sup>*ð Þ *x*, *y*, *τ* for ð Þ *x*, *y* ∈*C*<sup>3</sup> (13)

*δ τ*ð Þ � *<sup>r</sup>* , *<sup>D</sup>* <sup>¼</sup> *<sup>α</sup>*

CTE : *j* ¼ 1, Å ¼ 0 and *τ*<sup>0</sup> ¼ *τ*<sup>1</sup> ¼ *τ*<sup>2</sup> ¼ 0 (16) CCTE : *j* ¼ 1, Å ¼ 1 and *τ*<sup>0</sup> ¼ *τ*<sup>1</sup> ¼ *τ*<sup>2</sup> ¼ 0 (17)

ETE : *j* ¼ 1, Å ¼ 1 and *τ*<sup>1</sup> ¼ *τ*<sup>2</sup> ¼ 0 (18) TRDTE : *j* ¼ 1, Å ¼ 1 and *τ*<sup>0</sup> ¼ 0 (19)

*<sup>α</sup>* is the second order tensor associated with the TEWED and TEWOED theories;

coefficient; *cα*ð Þ *α* ¼ *e*, *i*, *p* are constants; *ρ*, *τ*, and Å are the density, time, and unified parameter which introduced to consolidate all theories into a unified equations system, respectively; *τ*0, *τ*1, and *τ*<sup>2</sup> are the relaxation times; and *m* is a functionally graded parameter. Also, *g*1, *g*2, Ψ *<sup>f</sup>* , and *δ <sup>f</sup>* are suitably prescribed functions; *ta* are the tractions defined by *ta* ¼ *σabnb*; and *δ*1*<sup>j</sup>* and *δ*2*<sup>j</sup>* are the Kronecker delta functions. A superposed dot denotes the differentiation with respect to the time, and a comma followed by a subscript denotes partial differentiation with respect to the

*h* is the perturbed magnetic

4 *cpT*<sup>4</sup>

*<sup>α</sup>*ð Þ¼ *x*, *y g*1ð Þ *x*, *τ* (9)

¼ 0, *α* ¼ *e*, *i*, *p* (11)

*ρc*

¼ *g*2ð Þ *x*, *τ* (10)

*<sup>p</sup>* (8)

(15)

anisotropic medium; *μ* is the magnetic permeability; ~

*DOI: http://dx.doi.org/10.5772/intechopen.93515*

The unit mass total energy can be written as

*<sup>α</sup> ∂T<sup>α</sup> ∂n C*1

*D*∇<sup>2</sup>

where *pi* are singular points.

posite microstructures as follows [77]:

**147**

*T<sup>α</sup>* þ

*∂T*<sup>∗</sup> *α*

By using the following initial and boundary conditions:

*<sup>α</sup> ∂T<sup>α</sup> ∂n C*2

*<sup>T</sup>α*ð Þ¼ *<sup>x</sup>*, *<sup>y</sup>*, 0 *<sup>T</sup>*<sup>0</sup>

<sup>∗</sup>

corresponding coordinates.

$$
\sigma\_{ab} = \left(\mathbf{x} + \mathbf{1}\right)^{m} \left[\mathbf{C}\_{ab\text{fg}}\boldsymbol{\mu}\_{f\text{g}} - \beta\_{ab}\left(T - T\_0 + \tau\_1 \dot{T}\right)\right] \tag{2}
$$

$$
\pi\_{ab} = \mu(\varkappa + 1)^m \left( \tilde{h}\_a H\_b + \tilde{h}\_b H\_a - \delta\_{ba} \left( \tilde{h}\_f H\_f \right) \right) \tag{3}
$$

The *2D-3T* radiative heat conduction Eqs. (7)–(9) can be expressed as follows:

$$\nabla \left[ \left( \delta\_{\mathbf{l}\dot{\jmath}} \mathbb{K}\_a + \delta\_{\mathbf{l}\dot{\jmath}} \mathbb{K}\_a^\* \right) \nabla T\_a(r, \tau) \right] - \overline{\mathcal{W}}(r, \tau) = \mathbb{C}\_{va} \rho(\mathbf{x} + \mathbf{1})^m \delta\_{\mathbf{l}\dot{\jmath}} \frac{\partial T\_a(r, \tau)}{\partial \tau} \tag{4}$$

where

$$\overline{\boldsymbol{W}}(\boldsymbol{r},\boldsymbol{\varepsilon})=\begin{cases} \rho\overline{\boldsymbol{W}}\_{\boldsymbol{\varepsilon}}(\boldsymbol{T}\_{\boldsymbol{\varepsilon}}-\boldsymbol{T}\_{\boldsymbol{i}})+\rho\overline{\boldsymbol{W}}\_{\boldsymbol{\varepsilon}\boldsymbol{\varepsilon}}(\boldsymbol{T}\_{\boldsymbol{\varepsilon}}-\boldsymbol{T}\_{\boldsymbol{p}})+\overline{\overline{\boldsymbol{W}}},\boldsymbol{a}=\boldsymbol{e},\delta\_{1}=1\\\\ -\rho\,\overline{\boldsymbol{W}}\_{\boldsymbol{\varepsilon}\boldsymbol{\varepsilon}}(\boldsymbol{T}\_{\boldsymbol{\varepsilon}}-\boldsymbol{T}\_{\boldsymbol{i}})+\overline{\overline{\boldsymbol{W}}},&\boldsymbol{a}=\boldsymbol{i},\delta\_{1}=1\\\\ -\rho\,\overline{\boldsymbol{W}}\_{\boldsymbol{\varepsilon}\boldsymbol{\varepsilon}}\left(\boldsymbol{T}\_{\boldsymbol{\varepsilon}}-\boldsymbol{T}\_{\boldsymbol{p}}\right)+\overline{\overline{\boldsymbol{W}}},&\boldsymbol{a}=\boldsymbol{p},\delta\_{1}=\boldsymbol{T}\_{\boldsymbol{p}}^{3}\end{cases},\boldsymbol{\varepsilon}=\begin{cases}\boldsymbol{c}\_{\boldsymbol{\varepsilon}}&\boldsymbol{a}=\boldsymbol{c}\\\\ \boldsymbol{c}\_{\boldsymbol{i}}&\boldsymbol{a}=\boldsymbol{c}\_{\boldsymbol{i}}\\ \boldsymbol{c}\_{\boldsymbol{p}}\boldsymbol{T}\_{\boldsymbol{p}}^{3}&\boldsymbol{a}=\boldsymbol{p}\end{cases}\tag{5}$$

in which

$$\begin{aligned} \overline{\overline{\boldsymbol{W}}}(\boldsymbol{r},\boldsymbol{\tau}) &= -\delta\_{\overline{\mathbf{j}}} \mathbb{K}\_{a} \dot{\boldsymbol{T}}\_{a,ab} + \beta\_{ab} \boldsymbol{T}\_{a0} (\boldsymbol{x} + \mathbf{1})^{m} \left[ \mathbf{\dot{A}} \delta\_{\mathbf{i}l} \dot{\boldsymbol{u}}\_{a,b} + (\boldsymbol{\tau}\_{0} + \boldsymbol{\delta}\_{\mathbf{i}i}) \ddot{\boldsymbol{u}}\_{a,b} \right] \\ &+ \rho \boldsymbol{c}\_{a} (\boldsymbol{x} + \mathbf{1})^{m} \left[ \left( \boldsymbol{\tau}\_{0} + \boldsymbol{\delta}\_{\mathbf{i}\overline{\boldsymbol{r}}} \boldsymbol{\tau}\_{2} + \boldsymbol{\delta}\_{\mathbf{i}\overline{\boldsymbol{r}}} \right) \ddot{\boldsymbol{T}}\_{a} \right] \end{aligned} \tag{6}$$

and

$$\mathbb{W}\_{\text{ci}} = \rho \mathbb{A}\_{\text{ci}} T\_{\text{e}}^{-2/3}, \mathbb{W}\_{\text{cr}} = \rho \mathbb{A}\_{\text{cr}} T\_{\text{e}}^{-1/2}, \mathbb{K}\_{a} = \mathbb{A}\_{a} T\_{a}^{5/2}, a = e, i, \mathbb{K}\_{p} = \mathbb{A}\_{p} T\_{p}^{3+\mathbb{B}} \tag{7}$$

*A New Boundary Element Formulation for Modeling and Optimization of Three-Temperature… DOI: http://dx.doi.org/10.5772/intechopen.93515*

where *σab*, *τab*, *uk*, *Tα*, and *T<sup>α</sup>*<sup>0</sup> are the mechanical stress tensor, Maxwell's electromagnetic stress tensor, displacement vector, temperature, and reference temperature, respectively; *Cabfg Cabfg* <sup>¼</sup> *Cfgab* <sup>¼</sup> *Cbafg* and *<sup>β</sup>ab <sup>β</sup>ab* <sup>¼</sup> *<sup>β</sup>ba* ð Þ are, respectively, the constant elastic moduli and stress-temperature coefficients of the anisotropic medium; *μ* is the magnetic permeability; ~ *h* is the perturbed magnetic field; *α*ð Þ *α* ¼ *e*, *i*, *p* are the thermal conductivity coefficients; *Cvα*ð Þ *α* ¼ *e*, *i*, *p* are specific heat coefficients; *e*, *i*, and *p* denote electron, ion, and phonon, respectively; <sup>∗</sup> *<sup>α</sup>* is the second order tensor associated with the TEWED and TEWOED theories; ei is the electron-ion energy coefficient; *ep* is the electron-phonon energy coefficient; *cα*ð Þ *α* ¼ *e*, *i*, *p* are constants; *ρ*, *τ*, and Å are the density, time, and unified parameter which introduced to consolidate all theories into a unified equations system, respectively; *τ*0, *τ*1, and *τ*<sup>2</sup> are the relaxation times; and *m* is a functionally graded parameter. Also, *g*1, *g*2, Ψ *<sup>f</sup>* , and *δ <sup>f</sup>* are suitably prescribed functions; *ta* are the tractions defined by *ta* ¼ *σabnb*; and *δ*1*<sup>j</sup>* and *δ*2*<sup>j</sup>* are the Kronecker delta functions.

A superposed dot denotes the differentiation with respect to the time, and a comma followed by a subscript denotes partial differentiation with respect to the corresponding coordinates.

The unit mass total energy can be written as

$$P = P\_\epsilon + P\_i + P\_p,\\ P\_\epsilon = c\_\epsilon T\_\epsilon,\\ P\_i = c\_i T\_i,\\ P\_p = \frac{1}{4} c\_p T\_p^4 \tag{8}$$

By using the following initial and boundary conditions:

$$T\_a(\mathbf{x}, \mathbf{y}, \mathbf{0}) = T\_a^0(\mathbf{x}, \mathbf{y}) = \mathbf{g}\_1(\mathbf{x}, \mathbf{r})\tag{9}$$

$$\left. \mathbb{K}\_a \frac{\partial T\_a}{\partial n} \right|\_{C\_1} = 0, a = e, i, T\_r \Bigg|\_{C\_1} = \mathbf{g}\_2(\mathbf{x}, \mathbf{r}) \tag{10}$$

$$\left. \mathbb{K}\_a \frac{\partial T\_a}{\partial n} \right|\_{C\_2} = 0, a = e, i, p \tag{11}$$

$$
\mu\_f(\mathbf{x}, y, \mathbf{0}) = \dot{u}\_f(\mathbf{x}, y, \mathbf{0}) = \mathbf{0} \text{ for } (\mathbf{x}, y) \in \mathbb{R} \cup \mathcal{C} \tag{12}
$$

$$\mu\_f(\mathbf{x}, y, \tau) = \Psi\_f(\mathbf{x}, y, \tau) \text{ for } (\mathbf{x}, y) \in \mathcal{C}\_3 \tag{13}$$

$$\overline{\mathfrak{t}}\_{a}(\mathbf{x}, \mathbf{y}, \boldsymbol{\pi}) = \delta\_{f}(\mathbf{x}, \mathbf{y}, \boldsymbol{\pi}) \text{ for } (\mathbf{x}, \mathbf{y}) \in \mathbf{C}\_{4}, \boldsymbol{\pi} > \mathbf{0}, \mathbf{C} = \mathbf{C}\_{3} \cup \mathbf{C}\_{4}, \mathbf{C}\_{3} \cap \mathbf{C}\_{4} = \boxtimes \tag{14}$$

By using the fundamental solution that satisfies the following equation:

$$D\nabla^2 T\_a + \frac{\partial T\_a^\*}{\partial \mathbf{n}} = -\delta(r - p\_i)\delta(\mathbf{r} - r),\\D = \frac{\mathbb{K}\_a}{\rho \mathbf{c}}\tag{15}$$

where *pi* are singular points.

The above governing Eqs. (1)–(4) can be reduced to the different theories of three-temperature nonlinear generalized magneto-thermoelasticity for FGA composite microstructures as follows [77]:

$$\text{CTE}: j = \mathbf{1}, \dot{\mathbf{A}} = \mathbf{0} \text{ and } \tau\_0 = \tau\_1 = \tau\_2 = \mathbf{0} \tag{16}$$

$$\text{CCTE}: j = \mathbf{1}, \mathbf{\dot{A}} = \mathbf{1} \text{ and } \tau\_0 = \tau\_1 = \tau\_2 = \mathbf{0} \tag{17}$$

$$\text{ATE}: j = \mathbf{1}, \dot{\mathbf{A}} = \mathbf{1} \text{ and } \tau\_1 = \tau\_2 = \mathbf{0} \tag{18}$$

$$\text{TRDTE}: j = \mathbf{1}, \mathbf{\dot{A}} = \mathbf{1} \text{ and } \mathbf{\tau}\_0 = \mathbf{0} \tag{19}$$

The unified governing equations of three-temperature nonlinear generalized magneto-thermoelasticity for FGA composite microstructures can be expressed as

*haHb* <sup>þ</sup> <sup>~</sup>

� �∇*Tα*ð Þ *<sup>r</sup>*, *<sup>τ</sup>* � � � ð Þ¼ *<sup>r</sup>*, *<sup>τ</sup> Cv<sup>α</sup> <sup>ρ</sup>*ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *<sup>m</sup>δ*1*δ*1*<sup>j</sup>*

� � <sup>þ</sup> , *<sup>α</sup>* <sup>¼</sup> *<sup>e</sup>*, *<sup>δ</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup>

�*ρei*ð Þþ *Te* � *Ti* , *α* ¼ *i*, *δ*<sup>1</sup> ¼ 1 ,*Cv<sup>α</sup>* ¼

� � <sup>þ</sup> , *<sup>α</sup>* <sup>¼</sup> *<sup>p</sup>*, *<sup>δ</sup>*<sup>1</sup> <sup>¼</sup> *<sup>T</sup>*<sup>3</sup>

ð Þ¼� *<sup>r</sup>*, *<sup>τ</sup> <sup>δ</sup>*2*<sup>j</sup>αT*\_ *<sup>α</sup>*,*ab* <sup>þ</sup> *<sup>β</sup>abT<sup>α</sup>*0ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *<sup>m</sup>* <sup>Å</sup>*δ*1*iu*\_ *<sup>a</sup>*,*<sup>b</sup>* <sup>þ</sup> ð Þ *<sup>τ</sup>*<sup>0</sup> <sup>þ</sup> *<sup>δ</sup>*2*<sup>i</sup> <sup>u</sup>*€*<sup>a</sup>*,*<sup>b</sup>*

� �*T*€ *<sup>α</sup>*

<sup>þ</sup>*ρcα*ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *<sup>m</sup> <sup>τ</sup>*<sup>0</sup> <sup>þ</sup> *<sup>δ</sup>*1*<sup>j</sup>τ*<sup>2</sup> <sup>þ</sup> *<sup>δ</sup>*2*<sup>j</sup>*

*ei* <sup>¼</sup> *<sup>ρ</sup>eiT*�2*=*<sup>3</sup> *<sup>e</sup>* ,*er* <sup>¼</sup> *<sup>ρ</sup>erT*�1*=*<sup>2</sup> *<sup>e</sup>* , *<sup>α</sup>* <sup>¼</sup> *αT*<sup>5</sup>*=*<sup>2</sup>

The *2D-3T* radiative heat conduction Eqs. (7)–(9) can be expressed as follows:

*<sup>τ</sup>ab* <sup>¼</sup> *<sup>μ</sup>*ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *<sup>m</sup>* <sup>~</sup>

*<sup>σ</sup>ab*,*<sup>b</sup>* <sup>þ</sup> *<sup>τ</sup>ab*,*<sup>b</sup>* <sup>¼</sup> *<sup>ρ</sup>*ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *mu*€*<sup>a</sup>* (1)

*h fH <sup>f</sup>*

*<sup>∂</sup>Tα*ð Þ *<sup>r</sup>*, *<sup>τ</sup>*

*p*

� �

*<sup>α</sup>* , *<sup>α</sup>* <sup>¼</sup> *<sup>e</sup>*, *<sup>i</sup>*, *<sup>p</sup>* <sup>¼</sup> *pT*<sup>3</sup>þ

� � (6)

*<sup>∂</sup><sup>τ</sup>* (4)

8 >>< >>: *ce α* ¼ *e ci α* ¼ *i cpT*<sup>3</sup> *<sup>p</sup> α* ¼ *p*

(5)

*<sup>p</sup>* (7)

(3)

*<sup>σ</sup>ab* <sup>¼</sup> ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *<sup>m</sup> Cabfgu <sup>f</sup>*,*<sup>g</sup>* � *<sup>β</sup>ab <sup>T</sup>* � *<sup>T</sup>*<sup>0</sup> <sup>þ</sup> *<sup>τ</sup>*1*T*\_ � � � � (2)

� � � �

*hbHa* � *<sup>δ</sup>ba* <sup>~</sup>

follows [45–49]:

*Composite Materials*

**Figure 1.**

<sup>∇</sup> *<sup>δ</sup>*1*<sup>j</sup><sup>α</sup>* <sup>þ</sup> *<sup>δ</sup>*2*<sup>j</sup>*<sup>∗</sup>

�*ρer Te* � *Tp*

where

8 >>>>>>>><

>>>>>>>>:

in which

and

**146**

ð Þ¼ *r*, *τ*

*α*

*Computational domain of considered structure.*

*ρei*ð Þþ *Te* � *Ti ρer Te* � *Tp*

$$\text{TEWED}: j = 2, \dot{\mathbf{A}} = \mathbf{0} \text{ and } \tau\_0 = \mathbf{0} \tag{20}$$

By applying integration by parts to Eq. (28) and using the sifting property with

*A New Boundary Element Formulation for Modeling and Optimization of Three-Temperature…*

*daβab*ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *mTnb*

By implementing the WRM and integration by parts, we can write Eq. (25) in

*<sup>q</sup>* <sup>∗</sup> ¼ �*α<sup>T</sup>* <sup>∗</sup>

*<sup>q</sup>* <sup>∗</sup> *<sup>T</sup>* � *qT*<sup>∗</sup> ð Þ*dC* �

The field Eqs. (32) and (37) can be written in one equation of the form:

*daβab*ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *mnb*

3 5*dR*

The generalized thermoelastic vectors and tensors can be written in contracted

*UA* <sup>¼</sup> *ua <sup>a</sup>* <sup>¼</sup> *<sup>A</sup>* <sup>¼</sup> 1, 2, 3 *T A* ¼ 4

*TA* <sup>¼</sup> *ta <sup>a</sup>* <sup>¼</sup> *<sup>A</sup>* <sup>¼</sup> 1, 2, 3 *q A* ¼ 4

0 *d* ¼ *D* ¼ 1, 2, 3; *A* ¼ 4 0 *D* ¼ 4; *a* ¼ *A* ¼ 1, 2, 3

�*T*<sup>∗</sup> *<sup>D</sup>* <sup>¼</sup> 4; *<sup>A</sup>* <sup>¼</sup> <sup>4</sup>

*da d* ¼ *D* ¼ 1, 2, 3; *a* ¼ *A* ¼ 1, 2, 3

" # *ua*

�*f ab*

�

�

*u*∗

8 >>>><

>>>>:

Based on the sifting property, we can express Eq. (34) as follows:

*T*ð Þ¼ *ξ*

*t* ∗ *da* �*u*<sup>∗</sup>

*u*∗ *da* 0

<sup>0</sup> �*T*<sup>∗</sup>

" # *f gb*

ð

*C*

<sup>0</sup> �*<sup>q</sup>* <sup>∗</sup>

2 4 ð

*C*

ð

*R*

*T*

" #

( ) " #

þ

*u*∗ *da* 0

<sup>0</sup> �*T*<sup>∗</sup>

" # *ta*

*q*

*dC*

(38)

(39)

(40)

(41)

ð

*f gbu*<sup>∗</sup>

*<sup>q</sup>* <sup>∗</sup> *<sup>T</sup>* � *qT*<sup>∗</sup> ð Þ*dC* (34)

*:bna* (36)

*f abT*<sup>∗</sup> *dR* (37)

*dadR* (32)

*R*

*LabT*<sup>∗</sup> ¼ �*δ*ð Þ *<sup>x</sup>*, *<sup>ξ</sup>* (33)

*q* ¼ �*αt:bna* (35)

� �*dC* �

Eqs. (29) and (31), we obtain

ð

*u*∗ *data* � *<sup>t</sup>* <sup>∗</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.93515*

ð

*R*

*daua* <sup>þ</sup> *<sup>u</sup>*<sup>∗</sup>

*LabTT*<sup>∗</sup> � *LabT* <sup>∗</sup> ð Þ *<sup>T</sup> dR* <sup>¼</sup>

The fundamental solution *T*<sup>∗</sup> can be expressed as

*C*

*ud*ð Þ¼ *ξ*

the following form:

where

*ud*ð Þ*ξ*

¼ ð

*C* �

� ð

*R*

*U* <sup>∗</sup> *DA* ¼

" #

*T*ð Þ*ξ*

notation as follows:

**149**

$$\text{TEWOED}: j = 2, \dot{\text{A}} = \mathbf{0}, \tau\_0 = \mathbf{0} \text{ and } \mathbb{K}\_a \to \mathbf{0} \tag{21}$$

#### **3. BEM implementation**

By using Eqs. (2) and (3), we can write Eq. (1) as follows:

$$L\_{\rm gb}u\_f = \rho \ddot{u}\_a - D\_a T = f\_{\rm gb} \tag{22}$$

where

$$L\_{gb} = D\_{abf} \frac{\partial}{\partial \mathbf{x}\_b} + D\_{af} + \Lambda D\_{df}, \\ D\_{abf} = C\_{abfg} e, \\ e = \frac{\partial}{\partial \mathbf{x}\_g},$$

$$D\_{af} = \mu H\_0^2 \left(\frac{\partial}{\partial \mathbf{x}\_a} + \delta\_{a1} \Lambda \right) \frac{\partial}{\partial \mathbf{x}\_f}, \\ D\_{a} = -\beta\_{ab} \left(\frac{\partial}{\partial \mathbf{x}\_b} + \delta\_{b1} \Lambda + \tau\_1 \left(\frac{\partial}{\partial \mathbf{x}\_b} + \Lambda \right) \frac{\partial}{\partial \tau} \right),$$

$$\Lambda = \frac{m}{\varkappa + 1}, \\ f\_{gb} = \rho \ddot{u}\_a - D\_a T. \tag{23}$$

The field equations can be written in the following operator form:

$$L\_{\mathfrak{g}^b} u\_f = f\_{\mathfrak{g}^b} \tag{24}$$

$$L\_{ab}T = f\_{\;\;ab} \tag{25}$$

where the operators *Lgb* and *f gb* are defined above in Eq. (23), and the operators *Lab* and *f ab* are defined as follows:

$$L\_{ab} = \nabla \left( \delta\_{\sharp \circ} \mathbb{K}\_a^\* \right) \nabla \tag{26}$$

$$f\_{\
abla} = -\nabla \left(\delta\_{\mathbf{l}\dot{\mathbf{j}}} \mathbb{K}\_a \right) \nabla + c\_a \rho \delta\_{\mathbf{l}} \delta\_{\mathbf{l}\dot{\mathbf{j}}} (\varkappa + \mathbf{1})^m \frac{\partial T\_a(r, \tau)}{\partial \tau} + \overline{\mathbb{W}}(r, \tau) \tag{27}$$

By applying the weighted residual method (WRM) to the differential Eq. (24), we obtain

$$\int\_{R} (L\_{\mathfrak{g}b} u\_f - f\_{\mathfrak{g}b}) u\_{da}^\* dR = \mathbf{0} \tag{28}$$

Now, we can choose the fundamental solution *u*<sup>∗</sup> *df* as weighting function as

$$L\_{\rm gb} u\_{\rm df}^{\*} = -\delta\_{\rm ad} \delta(\propto, \xi) \tag{29}$$

The corresponding traction field can be expressed as

$$t\_{da}^{\*} = \mathbf{C}\_{ab\text{fg}}(\mathfrak{x} + \mathbf{1})^{m} u\_{df\text{g}}^{\*} n\_{b} \tag{30}$$

The traction vector can be expressed as

$$t\_d = \frac{\overline{t}\_d}{(\varkappa + 1)^m} = (\varkappa + 1)^m \left( \mathcal{C}\_{ab\text{f\"}\text{g\"}} \boldsymbol{\mu}\_{f\text{\"}\text{g}} - \beta\_{ab} \left( T + \tau\_1 \dot{T} \right) \right) \boldsymbol{n}\_b \tag{31}$$

*A New Boundary Element Formulation for Modeling and Optimization of Three-Temperature… DOI: http://dx.doi.org/10.5772/intechopen.93515*

By applying integration by parts to Eq. (28) and using the sifting property with Eqs. (29) and (31), we obtain

$$u\_d(\xi) = \int\_C \left( u\_{da}^\* t\_a - t\_{da}^\* u\_a + u\_{da}^\* \beta\_{ab} (\varkappa + \mathbf{1})^m T n\_b \right) d\mathcal{C} - \int\_R f\_{gb} u\_{da}^\* d\mathcal{R} \tag{32}$$

The fundamental solution *T*<sup>∗</sup> can be expressed as

$$L\_{ab}T^\* = -\delta(\mathfrak{x}, \mathfrak{f})\tag{33}$$

By implementing the WRM and integration by parts, we can write Eq. (25) in the following form:

$$\int\_{R} (L\_{ab}TT^\* - L\_{ab}T^\*T)dR = \int\_{C} (q^\*T - qT^\*)dC \tag{34}$$

where

TEWED : *j* ¼ 2, Å ¼ 0 and *τ*<sup>0</sup> ¼ 0 (20)

*Lgbu <sup>f</sup>* ¼ *ρu*€*<sup>a</sup>* � *DaT* ¼ *f gb* (22)

þ *δ*b1Λ þ *τ*<sup>1</sup>

*∂xg* ,

*∂ ∂xb* þ Λ � � *∂*

� �

, *f gb* ¼ *ρu*€*<sup>a</sup>* � *DaT:* (23)

*Lgbu <sup>f</sup>* ¼ *f gb* (24) *LabT* ¼ *f ab* (25)

� �∇ (26)

*dadR* ¼ 0 (28)

*df* as weighting function as

*df*,*gnb* (30)

*df* ¼ �*δadδ*ð Þ *x*, *ξ* (29)

ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *<sup>m</sup>* <sup>¼</sup> ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *<sup>m</sup> Cabfgu <sup>f</sup>*,*<sup>g</sup>* � *<sup>β</sup>ab <sup>T</sup>* <sup>þ</sup> *<sup>τ</sup>*1*T*\_ � � � � *nb* (31)

*<sup>∂</sup><sup>τ</sup>* <sup>þ</sup> ð Þ *<sup>r</sup>*, *<sup>τ</sup>* (27)

*∂τ*

,

TEWOED : *j* ¼ 2, Å ¼ 0, *τ*<sup>0</sup> ¼ 0 and *<sup>α</sup>* ! 0 (21)

<sup>þ</sup> *Daf* <sup>þ</sup> <sup>Λ</sup>*Da*1*<sup>f</sup>* , *Dabf* <sup>¼</sup> *Cabfgε*, *<sup>ε</sup>* <sup>¼</sup> *<sup>∂</sup>*

*∂ ∂xb*

**3. BEM implementation**

*Composite Materials*

*Lgb* ¼ *Dabf*

*∂ ∂xa*

where

we obtain

**148**

*Daf* <sup>¼</sup> *<sup>μ</sup>H*<sup>2</sup>

0

*Lab* and *f ab* are defined as follows:

*f ab* ¼ �∇ *δ*1*<sup>j</sup><sup>α</sup>*

ð

*R*

The corresponding traction field can be expressed as

*t* ∗

The traction vector can be expressed as

*ta* <sup>¼</sup> *ta*

Now, we can choose the fundamental solution *u*<sup>∗</sup>

By using Eqs. (2) and (3), we can write Eq. (1) as follows:

*∂ ∂xb*

*∂x <sup>f</sup>*

<sup>Λ</sup> <sup>¼</sup> *<sup>m</sup> x* þ 1

, *Da* ¼ �*βab*

where the operators *Lgb* and *f gb* are defined above in Eq. (23), and the operators

*α*

*Lab* <sup>¼</sup> <sup>∇</sup> *<sup>δ</sup>*2*<sup>j</sup>*<sup>∗</sup>

*Lgbu <sup>f</sup>* � *f gb* � �

*Lgbu*<sup>∗</sup>

*da* <sup>¼</sup> *Cabfg* ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *mu*<sup>∗</sup>

� �<sup>∇</sup> <sup>þ</sup> *<sup>c</sup>αρδ*1*δ*1*<sup>j</sup>*ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *<sup>m</sup> <sup>∂</sup>Tα*ð Þ *<sup>r</sup>*, *<sup>τ</sup>*

By applying the weighted residual method (WRM) to the differential Eq. (24),

*u*∗

The field equations can be written in the following operator form:

þ *δ<sup>a</sup>*1Λ � � *∂*

$$q = -\mathbb{K}\_a t\_. b \mathfrak{n}\_a \tag{35}$$

$$q^\* = -\mathbb{K}\_a T\_{\;b}^\* n\_a \tag{36}$$

Based on the sifting property, we can express Eq. (34) as follows:

$$T(\xi) = \int\_{C} (q^\*T - qT^\*)d\mathbf{C} - \int\_{R} f\_{ab}T^\* \, d\mathbf{R} \tag{37}$$

The field Eqs. (32) and (37) can be written in one equation of the form:

$$
\begin{bmatrix} u\_d(\xi) \\ T(\xi) \end{bmatrix} = \int\_{\tilde{C}} \left\{ -\begin{bmatrix} t\_{da}^\* & -u\_{da}^\* \beta\_{ab} (\mathbf{x} + \mathbf{1})^m u\_b \\ \mathbf{0} & -q^\* \end{bmatrix} \begin{bmatrix} u\_a \\ T \end{bmatrix} + \begin{bmatrix} u\_{da}^\* & \mathbf{0} \\ \mathbf{0} & -T^\* \end{bmatrix} \begin{bmatrix} t\_a \\ q \end{bmatrix} \right\} d\mathbf{C} \tag{38}
$$

$$
$$

The generalized thermoelastic vectors and tensors can be written in contracted notation as follows:

$$U\_A = \begin{cases} u\_a & a = A = 1,2,3 \\ T & A = 4 \end{cases} \tag{39}$$

$$T\_A = \begin{cases} t\_a & a = A = 1,2,3 \\ q & A = 4 \end{cases} \tag{40}$$

$$U\_{DA}^{\*} = \begin{cases} u\_{da}^{\*} & d=D=1,2,3; a=A=1,2,3\\ 0 & d=D=1,2,3; A=4\\ 0 & D=4; a=A=1,2,3\\ -T^{\*} & D=4; A=4 \end{cases} \tag{41}$$

$$
\bar{T}\_{DA}^\* = \begin{cases}
t\_{da}^\* & d=D=1,2,3; a=A=1,2,3\\ 
0 & D=4; a=A=1,2,3\\ 
$$

$$
\bar{u}\_d^\* = u\_{da}^\* \beta\_{af} n\_f \tag{43}
$$

þ *δ*2*<sup>j</sup><sup>α</sup>*

þ

2 4

*∂ ∂xa*

*<sup>ρ</sup>*ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *mu*€*<sup>a</sup>*

unknown coefficients *α<sup>q</sup>*

mental solutions *u*<sup>∗</sup>

follows:

equation as

**151**

*∂ ∂xb*

*DOI: http://dx.doi.org/10.5772/intechopen.93515*

�*ρcα*ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *<sup>m</sup> <sup>τ</sup>*<sup>0</sup> <sup>þ</sup> *<sup>δ</sup>*1*jτ*<sup>2</sup> <sup>þ</sup> *<sup>δ</sup>*2*<sup>j</sup>*

�*T<sup>α</sup>*0*βfg* ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *<sup>m</sup> <sup>τ</sup>*<sup>0</sup> <sup>þ</sup> *<sup>δ</sup>*2*<sup>j</sup>*

*UD*ð Þ¼ *ξ*

*E*:

ð

*U* <sup>∗</sup>

*DATA* � *<sup>T</sup>*<sup>~</sup> <sup>∗</sup>

*C*

*da* and *T*<sup>∗</sup> .

ð

*u*∗ *dat q an* � *<sup>t</sup>* <sup>∗</sup> *dauq an* � �*dC* �

*<sup>q</sup>* <sup>∗</sup> *<sup>T</sup><sup>q</sup>* � *<sup>q</sup><sup>q</sup> <sup>T</sup>* <sup>∗</sup> ð Þ*dC* �

*AN* � *<sup>T</sup>* <sup>∗</sup>

The elastic and thermal representation formulae can be combined in one single

*DAU<sup>q</sup> AN* � �*dC* �

By substituting from Eq. (58) into Eq. (53), the coupled thermoelastic represen-

*C*

*C*

ð

*U* <sup>∗</sup> *DAT<sup>q</sup>*

*C*

*uq dn*ð Þ¼ *ξ*

> *Tq* ð Þ¼ *ξ* ð

*Uq DN*ð Þ¼ *ξ*

tation formula can be expressed as follows:

� *<sup>c</sup>αρδ*1*δ*1*j*ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *<sup>m</sup>*

� �*u*€ *<sup>f</sup>*,*<sup>g</sup>*

*T*\_ *α*

3 5

<sup>5</sup> � *<sup>β</sup>ab*ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *mTα*0Å*δ*1*<sup>j</sup>*

*N*

ð

*R U* <sup>∗</sup> *DA f q ANdRα<sup>q</sup>*

*q*¼1

ð

*R u*∗ *da f q*

ð

*R f q*

ð

*R U* <sup>∗</sup> *DA f q* 0

2 4

*<sup>N</sup>* (52)

*an* (54)

*pj* (55)

*an dR* (56)

*T* <sup>∗</sup> *dR* (57)

*ANdR* (58)

*u*\_ *<sup>f</sup>*,*<sup>g</sup>*

3 5

> *q AE* and

*<sup>N</sup>* (53)

(51)

2 4

*A New Boundary Element Formulation for Modeling and Optimization of Three-Temperature…*

0

3

2 4

*T*€ *α*

3 5

In order to transform the domain integral in Eq. (44) to the boundary, we

approximate the source vector *SA* by a series of given known functions *f*

*SA* <sup>≈</sup> <sup>X</sup> *N*

*q*¼1 *f q ANα<sup>q</sup>*

Thus, the thermoelastic representation formula (44) can be expressed as

*DAUA* � �*dC* �<sup>X</sup>

By applying the WRM to the following elastic and thermal equations:

*Lgbu<sup>q</sup> fn* ¼ *f q*

*LabT<sup>q</sup>* <sup>¼</sup> *<sup>f</sup>*

Now, the weighting functions were chosen as the elastic and thermal funda-

Then, the representation formulae of elastic and thermal fields are given as

*q*

� � 0

� �

The thermoelastic representation formula (38) can be written in contracted notation as

$$U\_D(\xi) = \int\_C (U\_{DA}^\* \mathcal{T}\_A - \tilde{T}\_{DA} U\_A) d\mathcal{C} - \int\_R U\_{DA}^\* \mathcal{S}\_A d\mathcal{R} \tag{44}$$

The vector *SA* can be splitted as

$$\mathbf{S}\_{A} = \mathbf{S}\_{A}^{T} + \mathbf{S}\_{A}^{T} + \mathbf{S}\_{A}^{T} + \mathbf{S}\_{A}^{\dot{u}} + \mathbf{S}\_{A}^{\ddot{u}} \tag{45}$$

where *S<sup>T</sup> <sup>A</sup>* ¼ ω*AFUF* with

$$\rho\_{AF} = \begin{cases} -D\_a & A = 1, 2, 3; F = 4 \\\\ \nabla \left( \delta\_3 \mathbb{K}\_a^\* \right) \nabla T\_a + \begin{cases} \rho \mathbb{W}\_{\varepsilon t} \left( T\_\varepsilon - T\_i \right) + \rho \mathbb{W}\_{\sigma \tau} \left( T\_\varepsilon - T\_p \right), & a = \varepsilon, \delta\_1 = 1 \\ -\rho \mathbb{W}\_{\varepsilon t} \left( T\_\varepsilon - T\_i \right), & a = i, \delta\_1 = 1 \\ -\rho \mathbb{W}\_{\sigma \tau} \left( T\_\varepsilon - T\_p \right), & a = p, \delta\_1 = \frac{4}{\rho} T\_p^3 \end{cases} & \text{otherwise} \end{cases} \tag{46}$$

$$S\_A^\uparrow = \left(-\delta\_{\mathbf{j}\mathbf{j}}\mathbb{K}\_a \frac{\partial}{\partial \mathbf{x}\_a} \frac{\partial}{\partial \mathbf{x}\_b} + c\_a \rho \delta\_1 \delta\_{\mathbf{j}\mathbf{j}} (\mathbf{x} + \mathbf{1})^m \right) \delta\_{AF} \dot{U}\_F$$
 
$$\text{with } \delta\_{AF} = \begin{cases} 1 & A = 4; F = 4\\ 0 & \text{otherwise} \end{cases} \tag{47}$$

$$\mathbf{S}\_A^T = -\rho \mathbf{c}\_a (\mathbf{x} + \mathbf{1})^m \left(\mathbf{r}\_0 + \delta\_{\mathbf{i}\mathbf{j}} \mathbf{r}\_2 + \delta\_{\mathbf{2}\mathbf{j}}\right) \delta\_{AF} \ddot{U}\_F \tag{48}$$

$$\mathbf{S}\_A^{\dot{a}} = -\beta\_{ab} (\varkappa + \mathbf{1})^m T\_{a0} \mathbf{\mathring{A}} \delta\_{\mathbf{1}\dot{\jmath}} \dot{U}\_F \tag{49}$$

$$\begin{aligned} S\_A^i &= \therefore \ddot{U}\_F \text{ with } A\\ &= \begin{cases} \rho (\mathbf{x} + \mathbf{1})^m & A = \mathbf{1}, 2, 3; \mathbf{F} = \mathbf{1}, 2, 3, \\\ -T\_{a0} \rho\_{\text{f\!\!g}} (\mathbf{x} + \mathbf{1})^m (\mathbf{r}\_0 + \delta\_{\text{\!\!2}}) A = \mathbf{4}; f = F = 4 \end{cases} \end{aligned} \tag{50}$$

The thermoelastic representation formula (38) can also be expressed as follows:

$$\begin{aligned} [\mathbf{S}\_{\mathbf{A}}] &= \begin{bmatrix} -\mathbf{D}\_{a}T\_{a} \\\\ \nabla \left(\delta\_{\mathbf{\hat{j}}} \mathbb{K}\_{a}^{\*}\right) \nabla T\_{a} + \begin{cases} \rho \mathbb{W}\_{\text{er}} \left(T\_{\epsilon} - T\_{i}\right) + \rho \mathbb{W}\_{\text{er}} \left(T\_{\epsilon} - T\_{p}\right), & a = e, \delta\_{\mathbf{i}} = 1 \\\\ -\rho \mathbb{W}\_{\text{er}} \left(T\_{\epsilon} - T\_{i}\right), & a = i, \delta\_{\mathbf{i}} = 1 \\\\ -\rho \mathbb{W}\_{\text{er}} \left(T\_{\epsilon} - T\_{p}\right), & a = p, \delta\_{\mathbf{i}} = \frac{4}{\rho} T\_{p}^{3} \end{bmatrix} \end{aligned}$$

*A New Boundary Element Formulation for Modeling and Optimization of Three-Temperature… DOI: http://dx.doi.org/10.5772/intechopen.93515*

$$\begin{aligned} &+ \left(\delta\_{\mathcal{Y}}\mathbb{K}\_{a}\frac{\partial}{\partial \mathbf{x}\_{a}}\frac{\partial}{\partial \mathbf{x}\_{b}} - c\_{a}\rho\delta\_{1}\delta\_{\mathcal{Y}}(\mathbf{x}+\mathbf{1})^{m}\right) \begin{bmatrix} \mathbf{0} \\\\ \dot{T}\_{a} \end{bmatrix} \\ &- \rho c\_{a}(\mathbf{x}+\mathbf{1})^{m} \left(\mathbf{r}\_{0} + \delta\_{\mathcal{Y}}\mathbf{r}\_{2} + \delta\_{\mathcal{Y}}\right) \begin{bmatrix} \mathbf{0} \\\\ \ddot{T}\_{a} \end{bmatrix} - \beta\_{ab}(\mathbf{x}+\mathbf{1})^{m}T\_{a0}\dot{\mathbf{A}}\delta\_{\mathcal{Y}} \begin{bmatrix} \mathbf{0} \\\\ \dot{\boldsymbol{u}}\_{f\mathcal{S}} \end{bmatrix} \\ &+ \begin{bmatrix} \rho(\mathbf{x}+\mathbf{1})^{m}\ddot{\boldsymbol{u}}\_{d} \\\\ -T\_{a0}\rho\delta\_{\mathcal{Y}}(\mathbf{x}+\mathbf{1})^{m}(\mathbf{r}\_{0}+\boldsymbol{\delta}\_{\mathcal{Y}})\ddot{\boldsymbol{u}}\_{f\mathcal{S}} \end{bmatrix} \end{aligned} \tag{51}$$

In order to transform the domain integral in Eq. (44) to the boundary, we approximate the source vector *SA* by a series of given known functions *f q AE* and unknown coefficients *α<sup>q</sup> E*:

$$S\_A \approx \sum\_{q=1}^{N} f\_{AN}^q a\_N^q \tag{52}$$

Thus, the thermoelastic representation formula (44) can be expressed as

$$U\_D(\xi) = \int\_C \left( U\_{DA}^\* T\_A - \bar{T}\_{DA}^\* U\_A \right) d\mathcal{C} - \sum\_{q=1}^N \int\_R U\_{DA}^\* f\_{AN}^q d\mathcal{R} a\_N^q \tag{53}$$

By applying the WRM to the following elastic and thermal equations:

$$L\_{\rm gb}u^q\_{fn} = f^q\_{an} \tag{54}$$

$$L\_{ab}T^q = f^q\_{\;pj} \tag{55}$$

Now, the weighting functions were chosen as the elastic and thermal fundamental solutions *u*<sup>∗</sup> *da* and *T*<sup>∗</sup> .

Then, the representation formulae of elastic and thermal fields are given as follows:

$$u\_{dn}^q(\xi) = \int\_C (u\_{da}^\* t\_{an}^q - t\_{da}^\* u\_{an}^q) \, d\mathcal{C} - \int\_R u\_{da}^\* f\_{an}^q \, d\mathcal{R} \tag{56}$$

$$T^q(\xi) = \int\_C (q^\* \, T^q - q^q T^\*) \, dC - \int\_R f^q T^\* \, dR \tag{57}$$

The elastic and thermal representation formulae can be combined in one single equation as

$$\mathcal{U}\_{DN}^{q}(\xi) = \int\_{C} \left( U\_{DA}^{\*} T\_{AN}^{q} - T\_{DA}^{\*} U\_{AN}^{q} \right) d\mathcal{C} - \int\_{R} U\_{DA}^{\*} f\_{AN}^{q} d\mathcal{R} \tag{58}$$

By substituting from Eq. (58) into Eq. (53), the coupled thermoelastic representation formula can be expressed as follows:

*T*~ ∗ *DA* ¼

notation as

*Composite Materials*

where *S<sup>T</sup>*

∇ *δ*2*<sup>j</sup>*<sup>∗</sup> *α* � �∇*T<sup>α</sup>* <sup>þ</sup>

with

8 >>>>>><

>>>>>>:

*Su*€

½ �¼ *SA*

**150**

*<sup>A</sup>* <sup>¼</sup> <sup>Ⅎ</sup>*U*€ *<sup>F</sup>* with <sup>Ⅎ</sup>

�*DaT<sup>α</sup>*

∇ *δ*2*<sup>j</sup>*<sup>∗</sup> *α* � �∇*T<sup>α</sup>* <sup>þ</sup>

(

ω*AF* ¼

*t* ∗

8 >>>>><

>>>>>:

ð

*U* <sup>∗</sup>

*SA* <sup>¼</sup> *ST*

�*ρer Te* � *Tp*

*∂ ∂xa*

*∂ ∂xb*

�

*<sup>A</sup>* ¼ �*ρcα*ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *<sup>m</sup> <sup>τ</sup>*<sup>0</sup> <sup>þ</sup> *<sup>δ</sup>*1*<sup>j</sup>τ*<sup>2</sup> <sup>þ</sup> *<sup>δ</sup>*2*<sup>j</sup>*

� �

with *<sup>δ</sup>AF* <sup>¼</sup> <sup>1</sup> *<sup>A</sup>* <sup>¼</sup> 4; *<sup>F</sup>* <sup>¼</sup> <sup>4</sup>

<sup>¼</sup> *<sup>ρ</sup>*ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *<sup>m</sup> <sup>A</sup>* <sup>¼</sup> 1, 2, 3; F <sup>¼</sup> 1, 2, 3,

The thermoelastic representation formula (38) can also be expressed as follows:

*ρei*ð Þþ *Te* � *Ti ρer Te* � *Tp*

�*ρer Te* � *Tp*

� �*<sup>A</sup>* <sup>¼</sup> 4; *<sup>f</sup>* <sup>¼</sup> *<sup>F</sup>* <sup>¼</sup> <sup>4</sup>

*C*

*UD*ð Þ¼ *ξ*

8 >>>><

>>>>:

*ST*€

�*T<sup>α</sup>*0*βfg* ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *<sup>m</sup> <sup>τ</sup>*<sup>0</sup> <sup>þ</sup> *<sup>δ</sup>*2*<sup>j</sup>*

*<sup>A</sup>* ¼ �*δ*2*<sup>j</sup><sup>α</sup>*

*Su*\_

8 >>>>><

>>>>>:

*ST*\_

The vector *SA* can be splitted as

*<sup>A</sup>* ¼ ω*AFUF*

�*u*<sup>∗</sup>

*da d* ¼ *D* ¼ 1, 2, 3; *a* ¼ *A* ¼ 1, 2, 3

*daβaf n <sup>f</sup>* (43)

*DASAdR* (44)

*<sup>A</sup>* (45)

otherwise

(46)

(47)

(50)

*ρ T*3 *p*

*ρ T*3 *p*

*δAFU*\_ *<sup>F</sup>*

� �*δAFU*€ *<sup>F</sup>* (48)

� �, *<sup>α</sup>* <sup>¼</sup> *<sup>e</sup>*, *<sup>δ</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup>

�*ρei*ð Þ *Te* � *Ti* , *α* ¼ *i*, *δ*<sup>1</sup> ¼ 1

� �, *<sup>α</sup>* <sup>¼</sup> *<sup>p</sup>*, *<sup>δ</sup>*<sup>1</sup> <sup>¼</sup> <sup>4</sup>

*<sup>A</sup>* ¼ �*βab*ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *mTα*0Å*δ*1*jU*\_ *<sup>F</sup>* (49)

ð

*R U* <sup>∗</sup>

*<sup>A</sup>* <sup>þ</sup> *Su*€

� �, *<sup>α</sup>* <sup>¼</sup> *<sup>e</sup>*, *<sup>δ</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup>

(42)

*<sup>d</sup> d* ¼ *D* ¼ 1, 2, 3; *A* ¼ 4

0 *D* ¼ 4; *a* ¼ *A* ¼ 1, 2, 3

The thermoelastic representation formula (38) can be written in contracted

*DA*T*<sup>A</sup>* � *<sup>T</sup>*~*DAUA* � �*dC* �

*<sup>A</sup>* <sup>þ</sup> *<sup>S</sup><sup>T</sup>*\_

*ρei*ð Þþ *Te* � *Ti ρer Te* � *Tp*

*<sup>A</sup>* <sup>þ</sup> *ST*€

*<sup>A</sup>* <sup>þ</sup> *<sup>S</sup><sup>u</sup>*\_

�*Da A* ¼ 1, 2, 3; *F* ¼ 4

� �, *<sup>α</sup>* <sup>¼</sup> *<sup>p</sup>*, *<sup>δ</sup>*<sup>1</sup> <sup>¼</sup> <sup>4</sup>

<sup>þ</sup> *<sup>c</sup>αρδ*1*δ*1*<sup>j</sup>*ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *<sup>m</sup>*

0 otherwise

�*ρei*ð Þ *Te* � *Ti* , *α* ¼ *i*, *δ*<sup>1</sup> ¼ 1

�*<sup>q</sup>* <sup>∗</sup> *<sup>D</sup>* <sup>¼</sup> 4; *<sup>A</sup>* <sup>¼</sup> <sup>4</sup>

*u*~ ∗ *<sup>d</sup>* <sup>¼</sup> *<sup>u</sup>*<sup>∗</sup>

$$U\_D(\xi) = \int\_C \left( U\_{DA}^\* T\_A - \ddot{T}\_{DA}^\* U\_A \right) d\mathcal{C} + \sum\_{q=1}^N \left( U\_{DN}^q(\xi) + \int\_{\mathcal{C}} \left( T\_{DA}^\* U\_{AN}^q - U\_{DA}^\* T\_{AN}^q \right) d\mathcal{C} \right) a\_N^q \tag{59}$$

By differentiation of Eq. (59) with respect to *ξl*, we obtain

$$\begin{aligned} \frac{\partial U\_D(\xi)}{\partial \xi\_l} &= -\int\_C \left( U\_{DA,l}^\* T\_A - \breve{T}\_{DA,l}^\* U\_A \right) d\mathcal{C} \\\\ &+ \sum\_{q=1}^N \left( \frac{\partial U\_{DN}^q(\xi)}{\partial \xi\_l} - \int\_C \left( T\_{DA,l}^\* U\_{AN}^q - U\_{DA,l}^\* T\_{AN}^q \right) d\mathcal{C} \right) d\mathcal{C}\_N^q \end{aligned} \tag{60}$$

According to the procedure described in Fahmy [78], the boundary integral Eq. (59) can be expressed as

$$
\tilde{\zeta}U - \eta T = (\zeta \check{U} - \eta \check{\wp})a \tag{61}
$$

where

*S T*,*q AD* ¼ *SAF f*

*S u*\_,*q AD* ¼ *SFA f*

*S* ¼ *Jα*, *U* ¼ *J*

Now, we can write the coefficients *α* in terms of nodal values of the

and (63), we obtain the following equation system:

*DOI: http://dx.doi.org/10.5772/intechopen.93515*

Solving the system (70) for *α*, *γ*, and ~*γ* yields

*α* ¼ *J*

�<sup>1</sup> �

�

þ1Þ

þ *δ*2*<sup>j</sup><sup>α</sup>*

�

*α* ¼ *J* �1� *Sγ* ¼ *J* 0�<sup>1</sup>

*<sup>S</sup>*<sup>0</sup> <sup>þ</sup> *BTJ*

*mTα*0Å*δ*1*jJ*

*M* z}|{

*<sup>T</sup>*€ <sup>þ</sup> <sup>A</sup> z}|{

, ¼ *ηT* þ *VS*

X z}|{

> �1 , *M*

*∂ ∂xb*

z}|{ <sup>¼</sup> *<sup>T</sup>*0*βab*ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *<sup>m</sup>*Å*δ*1*<sup>j</sup>*,

*α ∂ ∂xa*

z}|{, *K* z}|{ , A z}|{

0�<sup>1</sup>

*∂ ∂xa*

z}|{ <sup>¼</sup> *<sup>δ</sup>*2*<sup>j</sup>*<sup>∗</sup>

z}|{, <sup>Γ</sup>

where

*K*

Γ

*<sup>V</sup>* <sup>¼</sup> *<sup>η</sup>*℘� � *<sup>ζ</sup>U*� � �*<sup>J</sup>*

z}|{ <sup>¼</sup> <sup>~</sup>*<sup>ζ</sup>* <sup>þ</sup> *VBTJ*

z}|{ <sup>¼</sup> *<sup>V</sup> <sup>α</sup>*

where *V*, *M*

and z}|{

**153**

A

displacements, *U*, velocities, *U*\_ , and accelerations, *U*€ as follows:

*∂ ∂xa*

0�1 *U* �

> *∂ ∂xb*

0�1 � *U*\_

þ �*cαρ*ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *<sup>m</sup> <sup>τ</sup>*<sup>0</sup> <sup>þ</sup> *<sup>δ</sup>*1*<sup>j</sup>τ*<sup>2</sup> <sup>þ</sup> *<sup>δ</sup>*2*<sup>j</sup>* � �*δAF* � �*U*€

> *<sup>U</sup>*€ <sup>þ</sup> <sup>Γ</sup> z}|{

z}|{ <sup>¼</sup> *VA*~, X

� *<sup>c</sup>αρ*ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *<sup>m</sup>δ*1*<sup>j</sup>*

*∂ ∂xb*

*<sup>T</sup>*\_ <sup>þ</sup> <sup>B</sup> z}|{

> *^*0 , B

, and B z}|{

damping, stiffness, capacity, and conductivity matrices, respectively; *U*€ , *U*\_ , *U*, *T*,

represent the acceleration, velocity, displacement, temperature, and

� �*δAF* � *<sup>T</sup>*0Å*δ*1*<sup>j</sup>βfg* ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *mJ*

� �,

� *<sup>ρ</sup>cα*ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *<sup>m</sup>δ*1*<sup>j</sup>*

*q*

*q*

By applying the point collocation procedure of Gaul et al. [10] to Eqs. (52), (62),

*A New Boundary Element Formulation for Modeling and Optimization of Three-Temperature…*

0 *<sup>γ</sup>*, *<sup>U</sup>*\_ <sup>¼</sup> *<sup>J</sup>* 0

*U* ~*γ* ¼ *J*

� *<sup>c</sup>αρδ*1*δ*1*<sup>j</sup>*ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *<sup>m</sup>* � �*δAF* � *<sup>β</sup>ab*ð*<sup>x</sup>*

By substituting from Eq. (72) into Eq. (61) and implementing implicit-implicit staggered algorithm of Farhat et al. [86], the governing equations can be rewritten as

> *<sup>U</sup>*\_ <sup>þ</sup> *<sup>K</sup>* z}|{

> > *T* ¼ z}|{

z}|{ <sup>¼</sup> *<sup>δ</sup>*1*<sup>j</sup><sup>α</sup>* <sup>þ</sup> *<sup>δ</sup>*2*<sup>j</sup>*<sup>∗</sup>

� �*δAF:* (75)

*U* ¼ z}|{

> *<sup>U</sup>*€ <sup>þ</sup> z}|{

z}|{ ¼ �*ρcα*ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *<sup>m</sup> <sup>τ</sup>*<sup>0</sup> <sup>þ</sup> *<sup>δ</sup>*1*<sup>j</sup>τ*<sup>2</sup> <sup>þ</sup> *<sup>δ</sup>*2*<sup>j</sup>*

z}|{ <sup>¼</sup> *<sup>T</sup><sup>α</sup>*0*βab*ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *<sup>m</sup> <sup>τ</sup>*<sup>0</sup> <sup>þ</sup> *<sup>δ</sup>*2*<sup>j</sup>*

*α* ,

are represent the volume, mass,

0�<sup>1</sup>

�

*FD*,*<sup>q</sup>* (68)

*FD*,*<sup>g</sup>* (69)

~*γ* (70)

*U*\_ (71)

(72)

(73)

*U*\_ (74)

� �,

� �,

0�<sup>1</sup>

According to the technique of Partridge et al. [68], the displacements *UF* and velocities *U*\_ *<sup>F</sup>* can be approximated as

$$U\_F \approx \sum\_{q=1}^{N} f\_{FD}^q(\mathbf{x}) \chi\_D^q \tag{62}$$

$$\dot{U}\_F \approx \sum\_{q=1}^{N} f\_{FD}^q(\mathbf{x}) \tilde{\boldsymbol{\eta}}\_D^q \tag{63}$$

where *f q FD* are known functions, and *γ q <sup>D</sup>* and ~*γ q <sup>D</sup>* are unknown coefficients. The gradients of the displacement and velocity can be approximated as

$$U\_{\mathbf{F}, \mathfrak{g}} \approx \sum\_{q=1}^{N} f\_{FD, \mathfrak{g}}^{q}(\mathbf{x}) \eta\_{K}^{q} \tag{64}$$

$$\dot{U}\_{F,\mathfrak{g}} \approx \sum\_{q=1}^{N} f\_{FD,\mathfrak{g}}^{q}(\mathfrak{x}) \widetilde{\eta}\_{D}^{q} \tag{65}$$

By substituting from Eqs. (62) and (63) into Eqs. (46) and (49), the corresponding source terms can be expressed as

$$\mathbf{S}\_A^T = \sum\_{q=1}^N \mathbf{S}\_{AD}^{T,q} \mathbf{y}\_D^q \tag{66}$$

$$\mathbf{S}\_A^{\dot{\mathbf{u}}} = -\beta\_{ab}(\mathbf{x} + \mathbf{1})^m T\_{a0} \mathbf{\check{A}} \delta\_{\mathbf{l}\dot{\mathbf{j}}} \sum\_{q=1}^N \mathbf{S}\_{AD}^{\dot{a}q} \boldsymbol{\tilde{j}}\_D^q \tag{67}$$

*A New Boundary Element Formulation for Modeling and Optimization of Three-Temperature… DOI: http://dx.doi.org/10.5772/intechopen.93515*

where

*UD*ð Þ¼ *ξ*

ð

*Composite Materials*

*U* <sup>∗</sup>

*DATA* � *<sup>T</sup>*� <sup>∗</sup>

� �

*DAUA*

*dC* <sup>þ</sup><sup>X</sup> *N*

> *DA*,*l UA*

� ð

*C*

*UF* <sup>≈</sup> <sup>X</sup> *N*

*<sup>U</sup>*\_ *<sup>F</sup>* <sup>≈</sup> <sup>X</sup> *N*

*q*¼1 *f q FD*ð Þ *x γ q*

*q*¼1 *f q FD*ð Þ *x* ~*γ q*

The gradients of the displacement and velocity can be approximated as

*q*¼1 *f q FD*,*<sup>g</sup>* ð Þ *x γ q*

*q*¼1 *f q FD*,*<sup>g</sup>* ð Þ *x* ~*γ q*

> *q*¼1 *S T*,*q ADγ q*

> > X *N*

*q*¼1 *S u*\_,*q AD*~*γ q*

*UF*,*<sup>g</sup>* <sup>≈</sup> <sup>X</sup> *N*

*<sup>U</sup>*\_ *<sup>F</sup>*,*<sup>g</sup>* <sup>≈</sup> <sup>X</sup> *N*

By substituting from Eqs. (62) and (63) into Eqs. (46) and (49), the

*ST <sup>A</sup>* <sup>¼</sup> <sup>X</sup> *N*

*<sup>A</sup>* ¼ �*βab*ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *mTα*0Å*δ*1*<sup>j</sup>*

*q <sup>D</sup>* and ~*γ q*

*T* ∗ *DA*,*l Uq*

According to the procedure described in Fahmy [78], the boundary integral

According to the technique of Partridge et al. [68], the displacements *UF* and

By differentiation of Eq. (59) with respect to *ξl*, we obtain

*TA* � *<sup>T</sup>*� <sup>∗</sup>

� �

*q*¼1

*Uq DN*ð Þþ *ξ*

*dC*

*AN* � *<sup>U</sup>* <sup>∗</sup>

� �*dC*

*DA*,*l Tq AN*

<sup>~</sup>*ζ<sup>U</sup>* � *<sup>η</sup><sup>T</sup>* <sup>¼</sup> *<sup>ζ</sup>U*� � *<sup>η</sup>*℘� � �*<sup>α</sup>* (61)

0

B@

ð

*T*∗ *DAU<sup>q</sup>*

*AN* � *<sup>U</sup>* <sup>∗</sup>

1 CA*αq N*

*<sup>D</sup>* (62)

*<sup>D</sup>* (63)

*<sup>K</sup>* (64)

*<sup>D</sup>* (65)

*<sup>D</sup>* (66)

*<sup>D</sup>* (67)

*<sup>D</sup>* are unknown coefficients.

� �*dC*

*DAT<sup>q</sup> AN* 1 CA*αq N*

(59)

(60)

*C*

*C*

*<sup>∂</sup>UD*ð Þ*<sup>ξ</sup> ∂ξl*

¼� ð

*C*

þ X *N*

velocities *U*\_ *<sup>F</sup>* can be approximated as

Eq. (59) can be expressed as

where *f*

**152**

*q*

*q*¼1

*U* <sup>∗</sup> *DA*,*l*

0

B@

*∂U<sup>q</sup> DN*ð Þ*ξ ∂ξl*

*FD* are known functions, and *γ*

corresponding source terms can be expressed as

*Su*\_

$$\mathbf{S}\_{AD}^{T,q} = \mathbf{S}\_{AF} f\_{FD,q}^{q} \tag{68}$$

$$\mathbf{S}\_{\rm AD}^{\dot{\alpha},q} = \mathbf{S}\_{\rm FA} f\_{\rm FD,g}^q \tag{69}$$

By applying the point collocation procedure of Gaul et al. [10] to Eqs. (52), (62), and (63), we obtain the following equation system:

$$
\check{\mathbf{S}} = Ja, \mathbf{U} = \mathbf{f}'\boldsymbol{\chi}, \dot{\mathbf{U}} = \mathbf{f}'\boldsymbol{\tilde{\chi}}\tag{70}
$$

Solving the system (70) for *α*, *γ*, and ~*γ* yields

$$a = \mathbf{J}^{-1}\ddot{\mathbf{S}}\mathbf{y} = \mathbf{J}'^{-1}\mathbf{U}\ \ddot{\mathbf{y}} = \mathbf{J}'^{-1}\dot{\mathbf{U}}\tag{71}$$

Now, we can write the coefficients *α* in terms of nodal values of the displacements, *U*, velocities, *U*\_ , and accelerations, *U*€ as follows:

$$\begin{aligned} \alpha &= f^{-1} \left( \ddot{S}^0 + B^T f'^{-1} U \\ &+ \left[ \left( \delta\_{\mathfrak{Z}\dot{\mathfrak{Z}}a} \frac{\partial}{\partial \mathfrak{x}\_a} \frac{\partial}{\partial \mathfrak{x}\_b} - c\_a \rho \delta\_1 \delta\_{\mathfrak{I}\dot{\mathfrak{I}}} (\mathfrak{x} + \mathfrak{1})^m \right) \delta\_{AF} - \beta\_{ab} (\mathfrak{x} \\ &+ \mathfrak{1})^m T\_{a0} \dot{\Lambda} \delta\_{\mathfrak{I}\dot{\mathfrak{J}}} f'^{-1} \right] \dot{U} \\ &+ \left[ -c\_a \rho (\mathfrak{x} + \mathfrak{1})^m \left( \tau\_0 + \delta\_{\mathfrak{I}\dot{\mathfrak{I}}} \tau\_2 + \delta\_{\mathfrak{Z}\dot{\mathfrak{I}}} \right) \delta\_{AF} \right] \ddot{U} \end{aligned} \tag{72}$$

By substituting from Eq. (72) into Eq. (61) and implementing implicit-implicit staggered algorithm of Farhat et al. [86], the governing equations can be rewritten as

$$
\widehat{\phantom{M}}\ddot{\bar{U}} + \widehat{\phantom{\Gamma}}\widehat{\bar{U}} + \widehat{\phantom{K}}\widehat{\phantom{U}U} = \widehat{\phantom{\mathbb{Q}}}\widehat{\phantom{\Gamma}}\tag{73}
$$

$$
\widehat{\mathbf{X}}^T \ddot{T} + \widehat{\mathbf{A}}^T \dot{T} + \widehat{\mathbf{B}}^T \mathbf{T} = \widehat{\mathbb{Z}}^T \ddot{U} + \widehat{\mathbb{R}}^T \dot{U} \tag{74}
$$

where

*<sup>V</sup>* <sup>¼</sup> *<sup>η</sup>*℘� � *<sup>ζ</sup>U*� � �*<sup>J</sup>* �1 , *M* z}|{ <sup>¼</sup> *VA*~, X z}|{ ¼ �*ρcα*ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *<sup>m</sup> <sup>τ</sup>*<sup>0</sup> <sup>þ</sup> *<sup>δ</sup>*1*<sup>j</sup>τ*<sup>2</sup> <sup>þ</sup> *<sup>δ</sup>*2*<sup>j</sup>* � �, *K* z}|{ <sup>¼</sup> <sup>~</sup>*<sup>ζ</sup>* <sup>þ</sup> *VBTJ* 0�<sup>1</sup> , ¼ *ηT* þ *VS ^*0 , B z}|{ <sup>¼</sup> *<sup>δ</sup>*1*<sup>j</sup><sup>α</sup>* <sup>þ</sup> *<sup>δ</sup>*2*<sup>j</sup>*<sup>∗</sup> *α* , Γ z}|{ <sup>¼</sup> *<sup>V</sup> <sup>α</sup> ∂ ∂xa ∂ ∂xb* � *<sup>c</sup>αρ*ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *<sup>m</sup>δ*1*<sup>j</sup>* � �*δAF* � *<sup>T</sup>*0Å*δ*1*<sup>j</sup>βfg* ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *mJ* 0�<sup>1</sup> � �, z}|{ <sup>¼</sup> *<sup>T</sup>*0*βab*ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *<sup>m</sup>*Å*δ*1*<sup>j</sup>*, z}|{ <sup>¼</sup> *<sup>T</sup><sup>α</sup>*0*βab*ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *<sup>m</sup> <sup>τ</sup>*<sup>0</sup> <sup>þ</sup> *<sup>δ</sup>*2*<sup>j</sup>* � �, A z}|{ <sup>¼</sup> *<sup>δ</sup>*2*<sup>j</sup>*<sup>∗</sup> *α ∂ ∂xa ∂ ∂xb* � *<sup>ρ</sup>cα*ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> *<sup>m</sup>δ*1*<sup>j</sup>* � �*δAF:* (75)

where *V*, *M* z}|{, <sup>Γ</sup> z}|{, *K* z}|{ , A z}|{ , and B z}|{ are represent the volume, mass, damping, stiffness, capacity, and conductivity matrices, respectively; *U*€ , *U*\_ , *U*, *T*, and z}|{ represent the acceleration, velocity, displacement, temperature, and

external force vectors, respectively, Xz}|{ is a Green and Lindsay material constants vector, and z}|{ and z}|{ are coupling matrices.

Hence, the governing equations lead to the following coupled system of differential-algebraic equations (DAEs) as in Farhat et al. [86]:

$$
\widehat{\phantom{M}M}\ddot{\boldsymbol{U}}\_{n+1} + \widehat{\phantom{\Gamma}\boldsymbol{U}}\dot{\boldsymbol{U}}\_{n+1} + \widehat{\phantom{K}K}\boldsymbol{U}\_{n+1} = \widehat{\phantom{\mathbb{Q}}\boldsymbol{U}}\_{n+1}^{p} \tag{76}
$$

*Tn*þ<sup>1</sup> ¼ *Tn* þ

Δ*τ* 2

<sup>¼</sup> *Tn* <sup>þ</sup> <sup>Δ</sup>*τT*\_ *<sup>n</sup>* <sup>þ</sup>

*<sup>T</sup>*\_ *<sup>n</sup>*þ<sup>1</sup> <sup>¼</sup> <sup>γ</sup>�<sup>1</sup> *<sup>T</sup>*\_ *<sup>n</sup>* <sup>þ</sup>

where <sup>γ</sup> <sup>¼</sup> *<sup>I</sup>* <sup>þ</sup> <sup>1</sup>

*Tn*þ<sup>1</sup> <sup>¼</sup> *Tn* <sup>þ</sup> <sup>Δ</sup>*τT*\_ *<sup>n</sup>* <sup>þ</sup>

� A

� A

and (87), respectively.

stresses sensitivities.

*<sup>T</sup>*€ *<sup>n</sup>*þ<sup>1</sup> <sup>¼</sup> <sup>X</sup> z}|{�<sup>1</sup>

ture field.

ature.

**155**

z}|{ <sup>γ</sup>�<sup>1</sup> *<sup>T</sup>*\_ *<sup>n</sup>* <sup>þ</sup>

 z}|{*U*€ *<sup>n</sup>*þ<sup>1</sup> <sup>þ</sup>

z}|{ <sup>γ</sup>�<sup>1</sup> *<sup>T</sup>*\_ *<sup>n</sup>* <sup>þ</sup>

�

From Eq. (83), we have

*<sup>T</sup>*\_ *<sup>n</sup>*þ<sup>1</sup> <sup>þ</sup> *<sup>T</sup>*\_ *<sup>n</sup>* � �

*DOI: http://dx.doi.org/10.5772/intechopen.93515*

Δ*τ*<sup>2</sup> 4

Δ*τ* 2

<sup>2</sup> A z}|{

> Δ*τ* 2 X z}|{�<sup>1</sup>

Δ*τ* 2 X z}|{�<sup>1</sup>

**4. Design sensitivity and optimization**

Δ*τ*<sup>2</sup> 4

X z}|{�<sup>1</sup>

Δ*τ* X � � z}|{�<sup>1</sup>

On substitution of Eq. (85) in Eq. (84), we obtain

*<sup>T</sup>*€*<sup>n</sup>* <sup>þ</sup> <sup>X</sup> z}|{�<sup>1</sup>

z}|{*U*\_ *<sup>n</sup>*þ<sup>1</sup>

First step. Predict the displacement field: *U<sup>p</sup>*

� �

 z}|{

.

 z}|{*U*€ *<sup>n</sup>*þ<sup>1</sup> <sup>þ</sup>

 z}|{*U*€ *<sup>n</sup>*þ<sup>1</sup> <sup>þ</sup>

On substitution of *<sup>T</sup>*\_ *<sup>n</sup>*þ<sup>1</sup> from Eq. (85) in Eq. (77), we get

 z}|{*U*€ *<sup>n</sup>*þ<sup>1</sup> <sup>þ</sup>

*<sup>T</sup>*€*<sup>n</sup>* <sup>þ</sup> <sup>X</sup> z}|{�<sup>1</sup>

 z}|{

*A New Boundary Element Formulation for Modeling and Optimization of Three-Temperature…*

*<sup>U</sup>*€ *<sup>n</sup>*þ<sup>1</sup> <sup>þ</sup> z}|{

*<sup>U</sup>*€ *<sup>n</sup>*þ<sup>1</sup> <sup>þ</sup> z}|{

� �

� � � � (85)

z}|{*U*\_ *<sup>n</sup>*þ<sup>1</sup>

z}|{*U*\_ *<sup>n</sup>*þ<sup>1</sup> � <sup>B</sup>

z}|{*U*\_ *<sup>n</sup>*þ<sup>1</sup> � <sup>B</sup>

� �

Now, our algorithm for the solution of Eqs. (81) and (86) is obtained as follows:

Second step. Substituting for *<sup>U</sup>*\_ *<sup>n</sup>*þ<sup>1</sup> from Eq. (78) and substituting for *<sup>U</sup>*€ *<sup>n</sup>*þ<sup>1</sup> from Eq. (76). Then, by using the resulted equations in Eq. (86) to obtain the tempera-

Third step. Correct the displacement field (81) by using the computed temper-

Fourth step. Compute *<sup>U</sup>*\_ *<sup>n</sup>*þ1, *<sup>U</sup>*€ *<sup>n</sup>*þ1, *<sup>T</sup>*\_ *<sup>n</sup>*þ1, and *<sup>T</sup>*€ *<sup>n</sup>*þ<sup>1</sup> from Eqs. (80), (82), (85),

According to Fahmy [77, 78], the design sensitivities of the nonlinear temperature field and nonlinear displacement field can be performed by the implicit differentiation of Eqs. (76) and (77), respectively, which describe the structural response with respect to the design variables, then we can compute the nonlinear thermal

In order to solve our topology optimization problem, the method of moving asymptotes (MMA) [87] has been implemented as an optimizer in our topology optimization program. The benefit of MMA algorithm is that it replaces the original nonlinear, non-convex optimization problem by a sequence of approximating convex subproblems which are much easier to solve. The implemented MMA is based on the bi-directional evolutionary structural optimization (BESO), which is the evolutionary topology optimization approach that allows modification of the

� �

� � � � � �

� � � � � �

*<sup>U</sup>*\_ *<sup>n</sup>*þ<sup>1</sup> � <sup>A</sup> z}|{

� � � �

*<sup>U</sup>*\_ *<sup>n</sup>*þ<sup>1</sup> � <sup>B</sup> z}|{ *Tn*þ<sup>1</sup>

z}|{*Tn*þ<sup>1</sup>

z}|{*Tn*þ<sup>1</sup>

*<sup>n</sup>*þ<sup>1</sup> ¼ *Un*.

<sup>þ</sup> *<sup>T</sup>*€*<sup>n</sup>*

<sup>þ</sup> *<sup>T</sup>*€ *<sup>n</sup>*

*<sup>T</sup>*\_ *<sup>n</sup>*þ<sup>1</sup> � <sup>B</sup> z}|{ *Tn*þ<sup>1</sup>

<sup>þ</sup> *<sup>T</sup>*€ *<sup>n</sup>*

� B z}|{*Tn*þ<sup>1</sup>

� B z}|{*Tn*þ<sup>1</sup> (84)

�� (86)

�� (87)

$$\widehat{\mathbf{X}^\*}\ddot{T}\_{n+1} + \widehat{\mathbf{A}^\*}\dot{T}\_{n+1} + \widehat{\mathbf{B}^\*}T\_{n+1} = \widehat{\mathbf{Z}^\*}\ddot{U}\_{n+1}\widehat{\mathbf{R}^\*}\dot{U}\_{n+1} \tag{77}$$

where z}|{*<sup>p</sup> <sup>n</sup>*þ<sup>1</sup> <sup>¼</sup> *<sup>η</sup>T<sup>p</sup> <sup>n</sup>*þ<sup>1</sup> <sup>þ</sup> *<sup>V</sup>*� *S* <sup>0</sup> and *T<sup>p</sup> <sup>n</sup>*þ1. By integrating Eq. (73) and using Eq. (76), we get

$$\begin{split} \dot{U}\_{n+1} &= \dot{U}\_{n} + \frac{\Delta \tau}{2} (\ddot{U}\_{n+1} + \ddot{U}\_{n}) \\ &= \dot{U}\_{n} + \frac{\Delta \tau}{2} \Big[ \ddot{U}\_{n} + \widetilde{\phantom{\phantom{\Delta \tau}}}^{-1} \left( \widetilde{\phantom{\bigtriangleup}}\_{n+1}^{p} - \widetilde{\phantom{\frown} \phantom{\hboxarrow{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\pi}}}}}}}}}}}}}}}}}}}}} \right{\hat{\mathring{\phantom{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\pi}}}}}}}}}}}}}{\dot{I}}}}{\ddot{U}\_{n+1}}} \end{\dot{U}\_{n+1}} \cdots \dot{U}\_{n}} \dot{U}\_{n+1} - \widetilde{\phantom{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\pi}}}}}}}}}}}}}}}}} \right{\hat{U}\_{n+1}}} \dot{U}\_{n+1}} \cdots \dot{U}\_{n}} \end{split}} \end{split}$$

$$\begin{split} U\_{n+1} &= U\_n + \frac{\Delta t}{2} \left( \dot{U}\_{n+1} + \dot{U}\_n \right) \\ &= U\_n + \Delta \tau \dot{U}\_n + \frac{\Delta \tau^2}{4} \left[ \ddot{U}\_n + \widehat{\phantom{\phantom{\alpha} \Delta \tau}}^{-1} \left( \widehat{\phantom{\frown} \nabla}\_{n+1} - \widehat{\phantom{\frown} \nabla} \dot{U}\_{n+1} - \widehat{\phantom{\frown} \nabla} U\_{n+1} \right) \right] \end{split} \tag{79}$$

From Eq. (78) we obtain

$$\dot{U}\_{n+1} = \overline{\gamma}^{-1} \left[ \dot{U}\_n + \frac{\Delta x}{2} \left[ \ddot{U}\_n + \widehat{\widetilde{M}}^{-1} \left( \widehat{\widetilde{\mathbb{Q}}}\_{n+1} - \widehat{\widetilde{K}}^n U\_{n+1} \right) \right] \right] \tag{80}$$

where <sup>γ</sup> <sup>¼</sup> *<sup>I</sup>* <sup>Δ</sup>*<sup>τ</sup>* <sup>2</sup> *M* z}|{�<sup>1</sup> Γ z}|{ � �. Substitution of Eq. (80) in Eq. (79), we obtain

$$\begin{aligned} U\_{n+1} &= U\_n + \Delta t \dot{U}\_n \\ &+ \frac{\Delta t^2}{4} \left[ \ddot{U}\_n + \widetilde{\mathcal{M}}^{-1} \left( \widetilde{\mathbb{Q}}\_{n+1} - \widetilde{\Gamma} \, \widetilde{\mathcal{V}}^{-1} \left[ \dot{U}\_n + \frac{\Delta t}{2} \left[ \ddot{U}\_n + \widetilde{\mathcal{M}}^{-1} \left( \widetilde{\mathbb{Q}}\_{n+1} - \widetilde{\mathcal{K}}^n U\_{n+1} \right) \right] \right] - \widetilde{\mathcal{K}}^n U\_{n+1} \right) \right] \end{aligned} \tag{81}$$

Substituting *<sup>U</sup>*\_ *<sup>n</sup>*þ<sup>1</sup> from Eq. (80) into Eq. (76), we obtain

$$\ddot{\mathcal{U}}\_{\boldsymbol{n}+1} = \widehat{\boldsymbol{\mathcal{M}}}^{-1} \left[ \widehat{\boldsymbol{\heron}}\_{\boldsymbol{n}+1}^{\boldsymbol{p}} - \widehat{\boldsymbol{\Gamma}}^{\boldsymbol{n}} \left[ \overline{\boldsymbol{\eta}}^{-1} \Big[ \dot{\boldsymbol{U}}\_{\boldsymbol{n}} + \frac{\Delta \boldsymbol{\pi}}{2} \Big[ \ddot{\boldsymbol{U}}\_{\boldsymbol{n}} + \widehat{\boldsymbol{\mathcal{M}}}^{\boldsymbol{n}-1} \Big( \widehat{\boldsymbol{\heron}}\_{\boldsymbol{n}+1}^{\boldsymbol{p}} - \widehat{\boldsymbol{\mathcal{K}}}^{\boldsymbol{n}} \boldsymbol{U}\_{\boldsymbol{n}+1} \Big] \Big] \right] - \widehat{\boldsymbol{\mathcal{K}}}^{\boldsymbol{n}} \boldsymbol{U}\_{\boldsymbol{n}+1} \Big] \right] \tag{82}$$

Integrating the heat Eq. (74) using the trapezoidal rule and Eq. (77), we get

$$\begin{split} \dot{T}\_{n+1} &= \dot{T}\_n + \frac{\Delta \tau}{2} \left( \ddot{T}\_{n+1} + \ddot{T}\_n \right) \\ &= \dot{T}\_n + \frac{\Delta \tau}{2} \left( \widehat{\mathbf{X}}^{-1} \left[ \widehat{\mathbf{Z}}^{-1} \ddot{\mathbf{U}}\_{n+1} + \widehat{\mathbf{R}}^{-1} \dot{\mathbf{U}}\_{n+1} - \widehat{\mathbf{A}}^{-1} \ddot{T}\_{n+1} - \widehat{\mathbf{B}}^{-1} \boldsymbol{T}\_{n+1} \right] + \ddot{T}\_n \right) \end{split} \tag{83}$$

*A New Boundary Element Formulation for Modeling and Optimization of Three-Temperature… DOI: http://dx.doi.org/10.5772/intechopen.93515*

$$\begin{split} T\_{n+1} &= T\_n + \frac{\Delta \tau}{2} \left( \dot{T}\_{n+1} + \dot{T}\_n \right) \\ &= T\_n + \Delta \tau \dot{T}\_n + \frac{\Delta \tau^2}{4} \left( \ddot{T}\_n + \widehat{\phantom{\rm S}}^{-1} \left[ \widehat{\phantom{\rm S}} \dddot{U}\_{n+1} + \widehat{\phantom{\rm S}}^{-1} \dot{U}\_{n+1} - \widehat{\phantom{\rm S}} \dddot{T}\_{n+1} - \widehat{\phantom{\rm S}} \dddot{T}\_{n+1} \right] \right) \end{split} \tag{84}$$

From Eq. (83), we have

external force vectors, respectively, Xz}|{

and z}|{

> *M* z}|{

*<sup>T</sup>*€ *<sup>n</sup>*þ<sup>1</sup> <sup>þ</sup> <sup>A</sup> z}|{

> *<sup>n</sup>*þ<sup>1</sup> <sup>þ</sup> *<sup>V</sup>*� *S*

<sup>2</sup> *<sup>U</sup>*€ *<sup>n</sup>*þ<sup>1</sup> <sup>þ</sup> *<sup>U</sup>*€ *<sup>n</sup>* � �

z}|{�<sup>1</sup>

*<sup>U</sup>*€ *<sup>n</sup>* <sup>þ</sup> *<sup>M</sup>* z}|{�<sup>1</sup>

<sup>2</sup> *<sup>U</sup>*€ *<sup>n</sup>* <sup>þ</sup> *<sup>M</sup>*

z}|{γ�<sup>1</sup> *<sup>U</sup>*\_ *<sup>n</sup>* <sup>þ</sup>

Δ*τ*

*<sup>U</sup>*€ *<sup>n</sup>*þ<sup>1</sup> <sup>þ</sup> z}|{

.

<sup>2</sup> *<sup>U</sup>*€ *<sup>n</sup>* <sup>þ</sup> *<sup>M</sup>*

By integrating Eq. (73) and using Eq. (76), we get

*X* z}|{

*<sup>n</sup>*þ<sup>1</sup> <sup>¼</sup> *<sup>η</sup>T<sup>p</sup>*

Δ*τ*

Δ*τ*

<sup>2</sup> *<sup>U</sup>*\_ *<sup>n</sup>*þ<sup>1</sup> <sup>þ</sup> *<sup>U</sup>*\_ *<sup>n</sup>* � �

> Δ*τ*<sup>2</sup> 4

> > Δ*τ*

Γ

*<sup>n</sup>*þ<sup>1</sup> � Γ

Substituting *<sup>U</sup>*\_ *<sup>n</sup>*þ<sup>1</sup> from Eq. (80) into Eq. (76), we obtain

<sup>γ</sup>�<sup>1</sup> *<sup>U</sup>*\_ *<sup>n</sup>* <sup>þ</sup>

 z}|{

Substitution of Eq. (80) in Eq. (79), we obtain

 z}|{*<sup>p</sup>*

are coupling matrices. Hence, the governing equations lead to the following coupled system of

> *<sup>T</sup>*\_ *<sup>n</sup>*þ<sup>1</sup> <sup>þ</sup> <sup>B</sup> z}|{

<sup>0</sup> and *T<sup>p</sup>*

*<sup>U</sup>*\_ *<sup>n</sup>*þ<sup>1</sup> <sup>þ</sup> *<sup>K</sup>* z}|{

*<sup>n</sup>*þ1.

 z}|{*<sup>p</sup>*

z}|{�<sup>1</sup>

*<sup>n</sup>*þ<sup>1</sup> � Γ z}|{

 z}|{*<sup>p</sup>*

� � � � � �

Δ*τ* <sup>2</sup> *<sup>U</sup>*€ *<sup>n</sup>* <sup>þ</sup> *<sup>M</sup>*

<sup>2</sup> *<sup>U</sup>*€ *<sup>n</sup>* <sup>þ</sup> *<sup>M</sup>*

Integrating the heat Eq. (74) using the trapezoidal rule and Eq. (77), we get

differential-algebraic equations (DAEs) as in Farhat et al. [86]:

*<sup>U</sup>*€ *<sup>n</sup>*þ<sup>1</sup> <sup>þ</sup> <sup>Γ</sup> z}|{

vector, and

*Composite Materials*

where

*Un*þ<sup>1</sup> ¼ *Un* þ

z}|{*<sup>p</sup>*

*<sup>U</sup>*\_ *<sup>n</sup>*þ<sup>1</sup> <sup>¼</sup> *<sup>U</sup>*\_ *<sup>n</sup>* <sup>þ</sup>

<sup>¼</sup> *<sup>U</sup>*\_ *<sup>n</sup>* <sup>þ</sup>

Δ*τ*

<sup>¼</sup> *Un* <sup>þ</sup> <sup>Δ</sup>*τU*\_ *<sup>n</sup>* <sup>þ</sup>

From Eq. (78) we obtain

*<sup>U</sup>*€ *<sup>n</sup>* <sup>þ</sup> *<sup>M</sup>* z}|{�<sup>1</sup>

> z}|{*<sup>p</sup>*

> > Δ*τ* 2

> > Δ*τ* 2

*<sup>n</sup>*þ<sup>1</sup> � Γ z}|{

> *<sup>T</sup>*€*<sup>n</sup>*þ<sup>1</sup> <sup>þ</sup> *<sup>T</sup>*€*<sup>n</sup>* � �

> > X z}|{�<sup>1</sup>

where <sup>γ</sup> <sup>¼</sup> *<sup>I</sup>* <sup>Δ</sup>*<sup>τ</sup>*

*Un*þ<sup>1</sup> <sup>¼</sup> *Un* <sup>þ</sup> <sup>Δ</sup>*τU*\_ *<sup>n</sup>*

þ Δ*τ*<sup>2</sup> 4

*<sup>U</sup>*€ *<sup>n</sup>*þ<sup>1</sup> <sup>¼</sup> *<sup>M</sup>*

**154**

z}|{�<sup>1</sup>

*<sup>T</sup>*\_ *<sup>n</sup>*þ<sup>1</sup> <sup>¼</sup> *<sup>T</sup>*\_ *<sup>n</sup>* <sup>þ</sup>

<sup>¼</sup> *<sup>T</sup>*\_ *<sup>n</sup>* <sup>þ</sup>

*<sup>U</sup>*\_ *<sup>n</sup>*þ<sup>1</sup> <sup>¼</sup> <sup>γ</sup>�<sup>1</sup> *<sup>U</sup>*\_ *<sup>n</sup>* <sup>þ</sup>

<sup>2</sup> *M* z}|{�<sup>1</sup>

z}|{ � �

z}|{

is a Green and Lindsay material constants

*<sup>n</sup>*þ<sup>1</sup> (76)

*<sup>U</sup>*\_ *<sup>n</sup>*þ<sup>1</sup> (77)

(80)

� *K* z}|{*Un*þ<sup>1</sup>

� *K* z}|{ *Un*þ<sup>1</sup>

<sup>þ</sup> *<sup>T</sup>*€*<sup>n</sup>*

(83)

(81)

(82)

*Un*þ<sup>1</sup> ¼

*Tn*þ<sup>1</sup> ¼ z}|{ z}|{*<sup>p</sup>*

*<sup>U</sup>*€ *<sup>n</sup>*þ<sup>1</sup> z}|{

*<sup>U</sup>*\_ *<sup>n</sup>*þ<sup>1</sup> � *<sup>K</sup>* z}|{ *Un*þ<sup>1</sup>

" # � � (79)

*<sup>n</sup>*þ<sup>1</sup> � *K* z}|{ *Un*þ<sup>1</sup>

z}|{�<sup>1</sup>

� � ��

z}|{�<sup>1</sup>

� � � � � � � �

*<sup>U</sup>*\_ *<sup>n</sup>*þ<sup>1</sup> � <sup>A</sup> z}|{

� �

� �

� �

 z}|{*<sup>p</sup>*

� � � � � �

 z}|{*<sup>p</sup>*

*<sup>n</sup>*þ<sup>1</sup> � *K* z}|{ *Un*þ<sup>1</sup>

*<sup>T</sup>*\_ *<sup>n</sup>*þ<sup>1</sup> � <sup>B</sup> z}|{ *Tn*þ<sup>1</sup>

*<sup>n</sup>*þ<sup>1</sup> � *K* z}|{*Un*þ<sup>1</sup>

*<sup>U</sup>*\_ *<sup>n</sup>*þ<sup>1</sup> � *<sup>K</sup>* z}|{ *Un*þ<sup>1</sup>

" # � � (78)

*<sup>n</sup>*þ<sup>1</sup> � Γ z}|{

 z}|{*<sup>p</sup>*

$$\dot{T}\_{n+1} = \gamma^{-1} \left[ \dot{T}\_n + \frac{\Delta \tau}{2} \left( \widehat{\mathbf{X}}^{-1} \left[ \widehat{\mathbf{Z}}^{\cdot} \ddot{\mathbf{U}}\_{n+1} + \widehat{\mathbf{R}}^{\cdot} \dot{\mathbf{U}}\_{n+1} - \widehat{\mathbf{B}}^{\cdot} \boldsymbol{T}\_{n+1} \right] + \ddot{T}\_n \right) \right] \tag{85}$$

where <sup>γ</sup> <sup>¼</sup> *<sup>I</sup>* <sup>þ</sup> <sup>1</sup> <sup>2</sup> A z}|{ Δ*τ* X � � z}|{�<sup>1</sup> .

On substitution of Eq. (85) in Eq. (84), we obtain

$$\begin{split} T\_{n+1} &= T\_n + \Delta \boldsymbol{\tau} \dot{\boldsymbol{T}}\_n + \frac{\Delta \mathbf{r}^2}{4} \Big[ \bar{\boldsymbol{T}}\_n + \widehat{\mathbf{X}}^{-1} \Big[ \widehat{\boldsymbol{\nabla}\boldsymbol{U}}\_{n+1} + \widehat{\mathbf{R}}^{\boldsymbol{\tau}} \dot{\boldsymbol{U}}\_{n+1} \\ &- \widehat{\mathbf{A}}^{-1} \Big( \boldsymbol{\eta}^{-1} \Big[ \dot{\boldsymbol{T}}\_n + \frac{\Delta \mathbf{r}}{2} \Big( \widehat{\mathbf{X}}^{-1} \left[ \widehat{\boldsymbol{\nabla}\boldsymbol{U}}\_{n+1} + \widehat{\mathbf{R}}^{\boldsymbol{\tau}} \dot{\boldsymbol{U}}\_{n+1} - \widehat{\mathbf{B}}^{\boldsymbol{\tau}} \boldsymbol{T}\_{n+1} \Big] + \dot{\boldsymbol{T}}\_n \Big) \Big] \Big) - \widehat{\mathbf{B}}^{-1} \boldsymbol{T}\_{n+1} \Big] \Big] \end{split} \tag{86}$$

On substitution of *<sup>T</sup>*\_ *<sup>n</sup>*þ<sup>1</sup> from Eq. (85) in Eq. (77), we get

$$\begin{split} \ddot{T}\_{\boldsymbol{n}+1} &= \widehat{\mathbf{X}}^{-1} \Big[ \widehat{\mathbf{Z}}^{-1} \ddot{\mathbf{U}}\_{\boldsymbol{n}+1} + \widehat{\mathbf{R}}^{\prime} \dot{\mathbf{U}}\_{\boldsymbol{n}+1} \\ &- \widehat{\mathbf{A}}^{-1} \Big( \boldsymbol{\eta}^{-1} \Big[ \dot{T}\_{\boldsymbol{n}} + \frac{\Delta \mathbf{r}}{2} \Big( \widehat{\mathbf{X}}^{-1} \Big[ \widehat{\mathbf{Z}}^{-1} \ddot{\mathbf{U}}\_{\boldsymbol{n}+1} + \widehat{\mathbf{R}}^{\prime} \dot{\mathbf{U}}\_{\boldsymbol{n}+1} - \widehat{\mathbf{B}}^{\prime} \boldsymbol{T}\_{\boldsymbol{n}+1} \Big] + \ddot{T}\_{\boldsymbol{n}} \Big) \Big] \Big) - \widehat{\mathbf{B}}^{-1} \boldsymbol{T}\_{\boldsymbol{n}+1} \Big] \end{split} \tag{87}$$

Now, our algorithm for the solution of Eqs. (81) and (86) is obtained as follows: First step. Predict the displacement field: *U<sup>p</sup> <sup>n</sup>*þ<sup>1</sup> ¼ *Un*.

Second step. Substituting for *<sup>U</sup>*\_ *<sup>n</sup>*þ<sup>1</sup> from Eq. (78) and substituting for *<sup>U</sup>*€ *<sup>n</sup>*þ<sup>1</sup> from Eq. (76). Then, by using the resulted equations in Eq. (86) to obtain the temperature field.

Third step. Correct the displacement field (81) by using the computed temperature.

Fourth step. Compute *<sup>U</sup>*\_ *<sup>n</sup>*þ1, *<sup>U</sup>*€ *<sup>n</sup>*þ1, *<sup>T</sup>*\_ *<sup>n</sup>*þ1, and *<sup>T</sup>*€ *<sup>n</sup>*þ<sup>1</sup> from Eqs. (80), (82), (85), and (87), respectively.

#### **4. Design sensitivity and optimization**

According to Fahmy [77, 78], the design sensitivities of the nonlinear temperature field and nonlinear displacement field can be performed by the implicit differentiation of Eqs. (76) and (77), respectively, which describe the structural response with respect to the design variables, then we can compute the nonlinear thermal stresses sensitivities.

In order to solve our topology optimization problem, the method of moving asymptotes (MMA) [87] has been implemented as an optimizer in our topology optimization program. The benefit of MMA algorithm is that it replaces the original nonlinear, non-convex optimization problem by a sequence of approximating convex subproblems which are much easier to solve. The implemented MMA is based on the bi-directional evolutionary structural optimization (BESO), which is the evolutionary topology optimization approach that allows modification of the

structure by either adding efficient material or removing inefficient material to or from the structure design [88–96]. This addition or removal depends upon the sensitivity analysis. Sensitivity analysis is the estimation of the response of the structure to the modification of the input design variables and is dependent upon the calculation of derivatives.

The homogenized vector of thermal expansion coefficients *α<sup>H</sup>* can be written in terms of the homogenized elastic matrix *D<sup>H</sup>* and homogenized stress-temperature coefficients vector *β<sup>H</sup>* as follows:

$$a^H = \left(D^H\right)^{-1} \beta^H \tag{88}$$

Mechanical temperature coefficient:

*DOI: http://dx.doi.org/10.5772/intechopen.93515*

Tensor of thermal conductivity:

Elasticity tensor:

*Cpjkl* ¼

**157**

*βpj* ¼

2 6 4

*kpj* ¼

17*:*77 3*:*78 3*:*76

3*:*78 19*:*45 4*:*13

3*:*76 4*:*13 21*:*79

0 00

0 00

0*:*03 1*:*13 0*:*38

2 6 4

*kpj* ¼

Oersted, *μ* ¼ 0*:*5 Gauss/Oersted, *h* ¼ 2, and Δ*τ* ¼ 0*:*0001.

at *<sup>x</sup>* <sup>¼</sup> <sup>0</sup> *<sup>∂</sup>u*<sup>1</sup>

at *<sup>x</sup>* <sup>¼</sup> *<sup>h</sup> <sup>∂</sup>u*<sup>1</sup>

at *<sup>y</sup>* <sup>¼</sup> <sup>0</sup> *<sup>∂</sup>u*<sup>1</sup>

2 6 4

Mechanical temperature coefficient:

*βpj* ¼

Tensor of thermal conductivity:

2 6 4

Mass density *<sup>ρ</sup>* <sup>¼</sup> 7820 kg*=*m3 and heat capacity *<sup>c</sup>* <sup>¼</sup> 461 J/(kg�K), *H*<sup>0</sup> ¼ 1000000 Oersted, *μ* ¼ 0*:*5 Gauss/Oersted, *h* ¼ 2, and Δ*τ* ¼ 0*:*0001.

> 0*:*001 0*:*02 0 0*:*02 0*:*006 0 0 00*:*05

> > 1 0*:*1 0*:*2 0*:*1 1*:*1 0*:*15 0*:*2 0*:*15 0*:*9

Mass density *<sup>ρ</sup>* <sup>¼</sup> 2216kg*=*m<sup>3</sup> and heat capacity *<sup>c</sup>* <sup>¼</sup> <sup>0</sup>*:*1 J/(kg�K), *<sup>H</sup>*<sup>0</sup> <sup>¼</sup> <sup>1000000</sup>

The initial and boundary conditions considered in the calculations are

*<sup>∂</sup><sup>x</sup>* <sup>¼</sup> *<sup>∂</sup>u*<sup>2</sup>

*<sup>∂</sup><sup>x</sup>* <sup>¼</sup> *<sup>∂</sup>u*<sup>2</sup>

*<sup>∂</sup><sup>y</sup>* <sup>¼</sup> *<sup>∂</sup>u*<sup>2</sup>

1*:*01 2*:*00 0 2*:*00 1*:*48 0 0 07*:*52

*A New Boundary Element Formulation for Modeling and Optimization of Three-Temperature…*

5*:*20 0 0 7*:*6 0 0 0 38*:*3

The physical data of the North Sea sandstone reservoir rock is given as follows:

3 7

> 3 7 5

0*:*24 �0*:*28 0*:*03 0 01*:*13

0 00*:*38

8*:*30 0*:*66 0

0*:*66 7*:*62 0

3 7

> 3 7 5

at *τ* ¼ 0 *u*<sup>1</sup> ¼ *u*<sup>2</sup> ¼ *u*\_ <sup>1</sup> ¼ *u*\_ <sup>2</sup> ¼ 0, *T* ¼ 0 (98)

*<sup>∂</sup><sup>x</sup>* <sup>¼</sup> 0, *<sup>∂</sup><sup>T</sup>*

*<sup>∂</sup><sup>x</sup>* <sup>¼</sup> 0, *<sup>∂</sup><sup>T</sup>*

*<sup>∂</sup><sup>y</sup>* <sup>¼</sup> 0, *<sup>∂</sup><sup>T</sup>*

0 07*:*77

<sup>5</sup> � 106N*=*Km<sup>2</sup> (93)

W*=*km (94)

<sup>5</sup> � <sup>10</sup><sup>6</sup> <sup>N</sup>*=*Km<sup>2</sup> (96)

W*=*km (97)

*<sup>∂</sup><sup>x</sup>* <sup>¼</sup> <sup>0</sup> (99)

*<sup>∂</sup><sup>x</sup>* <sup>¼</sup> <sup>0</sup> (100)

*<sup>∂</sup><sup>y</sup>* <sup>¼</sup> <sup>0</sup> (101)

GPa (95)

For the material design, the derivative of the homogenized thermal expansion coefficients vector can be expressed as

$$\frac{\partial \boldsymbol{a}^{H}}{\partial \mathbf{X}\_{kl}^{m}} = \left(\boldsymbol{D}^{H}\right)^{-1} \left(\frac{\partial \boldsymbol{\beta}^{H}}{\partial \mathbf{X}\_{kl}^{m}} - \frac{\partial \boldsymbol{D}^{H}}{\partial \mathbf{X}\_{kl}^{m}} \boldsymbol{a}^{H}\right) \tag{89}$$

where *<sup>∂</sup>DH ∂X<sup>m</sup> kl* and *<sup>∂</sup>β<sup>H</sup> ∂X<sup>m</sup> kl* for any *l*th material phase, can be calculated using the adjoint variable method [91] as

$$\frac{\partial D^H}{\partial X\_{kl}^m} = \frac{1}{|\Omega|} \int\_Y \left( I - B^m U^m \right)^T \frac{\partial D^m}{\partial X\_{kl}^m} (I - B^m U^m) \, dy \tag{90}$$

and

$$\begin{split} \frac{\partial \rho^{H}}{\partial X\_{kl}^{m}} &= \frac{1}{|\Omega|} \int\_{Y} (I - B^{m}U^{m})^{T} \frac{\partial D^{m}}{\partial X\_{kl}^{m}} (\alpha^{m} - B^{m}\rho^{m}) \, d\mathbf{y} \\ &+ \frac{1}{|\Omega|} \int\_{Y} (I - B^{m}U^{m})^{T} D^{m} \frac{\partial \alpha^{m}}{\partial X\_{kl}^{m}} \, d\mathbf{y} \end{split} \tag{91}$$

where, ∣Ω∣ is the volume of the base cell*.*

#### **5. Numerical examples, results, and discussion**

The proposed technique used in the current chapter should be applicable to any three-temperature nonlinear generalized magneto-thermoelastic problem. The application is for the purpose of illustration.

The two anisotropic materials considered in the calculation are monoclinic graphite-epoxy and North Sea sandstone reservoir rock, where the physical data of monoclinic graphite-epoxy material is given as follows:

Elasticity tensor:

$$\mathbf{C}\_{pjkl} = \begin{bmatrix} 430.1 & 130.4 & 18.2 & 0 & 0 & 201.3 \\ 130.4 & 116.7 & 21.0 & 0 & 0 & 70.1 \\ 18.2 & 21.0 & 73.6 & 0 & 0 & 2.4 \\ 0 & 0 & 0 & 19.8 & -8.0 & 0 \\ 0 & 0 & 0 & -8.0 & 29.1 & 0 \\ 201.3 & 70.1 & 2.4 & 0 & 0 & 147.3 \end{bmatrix} \text{GPa} \tag{92}$$

*A New Boundary Element Formulation for Modeling and Optimization of Three-Temperature… DOI: http://dx.doi.org/10.5772/intechopen.93515*

Mechanical temperature coefficient:

structure by either adding efficient material or removing inefficient material to or from the structure design [88–96]. This addition or removal depends upon the sensitivity analysis. Sensitivity analysis is the estimation of the response of the structure to the modification of the input design variables and is dependent upon

The homogenized vector of thermal expansion coefficients *α<sup>H</sup>* can be written in terms of the homogenized elastic matrix *D<sup>H</sup>* and homogenized stress-temperature

*β<sup>H</sup>* (88)

*<sup>I</sup>* � *BmUm* ð Þ*dy* (90)

(89)

(91)

*<sup>α</sup><sup>H</sup>* <sup>¼</sup> *DH* � ��<sup>1</sup>

<sup>¼</sup> *DH* � ��<sup>1</sup> *<sup>∂</sup>β<sup>H</sup>*

For the material design, the derivative of the homogenized thermal expansion

*∂X<sup>m</sup> kl*

*<sup>I</sup>* � *<sup>B</sup>mUm* ð Þ*<sup>T</sup> <sup>∂</sup>D<sup>m</sup>*

*<sup>I</sup>* � *BmUm* ð Þ*<sup>T</sup> <sup>∂</sup>D<sup>m</sup>*

� *<sup>∂</sup>D<sup>H</sup> ∂X<sup>m</sup> kl αH*

for any *l*th material phase, can be calculated using the adjoint

*∂X<sup>m</sup> kl dy*

*<sup>α</sup><sup>m</sup>* � *Bmφ<sup>m</sup>* ð Þ*dy*

GPa (92)

� �

*∂X<sup>m</sup> kl*

*∂X<sup>m</sup> kl*

*<sup>I</sup>* � *BmUm* ð Þ*TDm <sup>∂</sup>α<sup>m</sup>*

The proposed technique used in the current chapter should be applicable to any

three-temperature nonlinear generalized magneto-thermoelastic problem. The

The two anisotropic materials considered in the calculation are monoclinic graphite-epoxy and North Sea sandstone reservoir rock, where the physical data of

> *:*1 130*:*4 18*:*2 0 0 201*:*3 *:*4 116*:*7 21*:*0 0 0 70*:*1 *:*2 21*:*0 73*:*60 0 2*:*4 0 0 0 19*:*8 �8*:*0 0 0 00 �8*:*0 29*:*1 0 *:*3 70*:*1 2*:*4 0 0 147*:*3

the calculation of derivatives.

*Composite Materials*

coefficients vector *β<sup>H</sup>* as follows:

where *<sup>∂</sup>DH ∂X<sup>m</sup> kl*

and

variable method [91] as

coefficients vector can be expressed as

and *<sup>∂</sup>β<sup>H</sup> ∂X<sup>m</sup> kl*

> *∂DH ∂X<sup>m</sup> kl* ¼ 1 ∣Ω∣ ð *Y*

*∂β<sup>H</sup> ∂X<sup>m</sup> kl* <sup>¼</sup> <sup>1</sup> ∣Ω∣ ð *Y*

> þ 1 ∣Ω∣ ð *Y*

**5. Numerical examples, results, and discussion**

monoclinic graphite-epoxy material is given as follows:

where, ∣Ω∣ is the volume of the base cell*.*

application is for the purpose of illustration.

Elasticity tensor:

*Cpjkl* ¼

**156**

*∂α<sup>H</sup> ∂X<sup>m</sup> kl*

$$
\beta\_{pj} = \begin{bmatrix}
\mathbf{1.01} & \mathbf{2.00} & \mathbf{0} \\
\mathbf{2.00} & \mathbf{1.48} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{7.52}
\end{bmatrix} \cdot \mathbf{10}^6 \mathbf{N} / \text{K} \text{m}^2 \tag{93}
$$

Tensor of thermal conductivity:

$$k\_{pj} = \begin{bmatrix} 5.2 & 0 & 0 \\ 0 & 7.6 & 0 \\ 0 & 0 & 38.3 \end{bmatrix} \text{W/km} \tag{94}$$

Mass density *<sup>ρ</sup>* <sup>¼</sup> 7820 kg*=*m3 and heat capacity *<sup>c</sup>* <sup>¼</sup> 461 J/(kg�K),

*H*<sup>0</sup> ¼ 1000000 Oersted, *μ* ¼ 0*:*5 Gauss/Oersted, *h* ¼ 2, and Δ*τ* ¼ 0*:*0001. The physical data of the North Sea sandstone reservoir rock is given as follows: Elasticity tensor:

$$\mathbf{C}\_{pjkl} = \begin{bmatrix} 17.77 & 3.78 & 3.76 & 0.24 & -0.28 & 0.03 \\\\ 3.78 & 19.45 & 4.13 & 0 & 0 & 1.13 \\\\ 3.76 & 4.13 & 21.79 & 0 & 0 & 0.38 \\\\ 0 & 0 & 0 & 8.30 & 0.66 & 0 \\\\ 0 & 0 & 0 & 0.66 & 7.62 & 0 \\\\ 0.03 & 1.13 & 0.38 & 0 & 0 & 7.77 \end{bmatrix} \text{GPa} \tag{95}$$

Mechanical temperature coefficient:

$$
\boldsymbol{\beta}\_{pj} = \begin{bmatrix}
\mathbf{0.001} & \mathbf{0.02} & \mathbf{0} \\
\mathbf{0.02} & \mathbf{0.006} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{0.05}
\end{bmatrix} \cdot \mathbf{10^6} \,\mathrm{N/Km^2} \tag{96}
$$

Tensor of thermal conductivity:

$$k\_{pj} = \begin{bmatrix} \mathbf{1} & \mathbf{0.1} & \mathbf{0.2} \\ \mathbf{0.1} & \mathbf{1.1} & \mathbf{0.15} \\ \mathbf{0.2} & \mathbf{0.15} & \mathbf{0.9} \end{bmatrix} \text{W/km} \tag{97}$$

Mass density *<sup>ρ</sup>* <sup>¼</sup> 2216kg*=*m<sup>3</sup> and heat capacity *<sup>c</sup>* <sup>¼</sup> <sup>0</sup>*:*1 J/(kg�K), *<sup>H</sup>*<sup>0</sup> <sup>¼</sup> <sup>1000000</sup> Oersted, *μ* ¼ 0*:*5 Gauss/Oersted, *h* ¼ 2, and Δ*τ* ¼ 0*:*0001.

The initial and boundary conditions considered in the calculations are

$$\text{at } \tau = 0\\ \text{ } u\_1 = u\_2 = \dot{u}\_1 = \dot{u}\_2 = 0, T = 0 \text{ }\tag{98}$$

$$\text{at}\,\text{x} = \mathbf{0} \,\frac{\partial u\_1}{\partial \mathbf{x}} = \frac{\partial u\_2}{\partial \mathbf{x}} = \mathbf{0}, \frac{\partial T}{\partial \mathbf{x}} = \mathbf{0} \tag{99}$$

$$\text{at } \mathbf{x} = h \, \frac{\partial u\_1}{\partial \mathbf{x}} = \frac{\partial u\_2}{\partial \mathbf{x}} = \mathbf{0}, \frac{\partial T}{\partial \mathbf{x}} = \mathbf{0} \tag{100}$$

$$\text{at } y = 0 \,\, \frac{\partial u\_1}{\partial y} = \frac{\partial u\_2}{\partial y} = 0 \,, \frac{\partial T}{\partial y} = 0 \,\, \tag{101}$$

$$\text{at } y = b \frac{\partial u\_1}{\partial y} = \frac{\partial u\_2}{\partial y} = 0, \frac{\partial T}{\partial y} = 0 \tag{102}$$

**Figure 4.**

**Figure 5.**

**Figure 6.**

**159**

*Variation of the displacement u2 sensitivity with time τ.*

*DOI: http://dx.doi.org/10.5772/intechopen.93515*

*A New Boundary Element Formulation for Modeling and Optimization of Three-Temperature…*

*Variation of the thermal stress σ<sup>11</sup> sensitivity with time τ.*

*Variation of the thermal stress σ<sup>12</sup> sensitivity with time τ.*

In order to study the effects of anisotropy and functionally graded materials on composite microstructure, we consider the following four cases, namely, isotropic homogeneous (IH), isotropic functionally graded (IF), anisotropic homogeneous (AH), and anisotropic functionally graded (AF). Also, we considered total temperature *T T* ¼ *Te* þ *Ti* þ *Tp* as the considered temperature field in all calculations of this study.

**Figure 2** shows the variations of the nonlinear three-temperature Te, Ti, and Tp and total temperature T T ¼ Te þ Ti þ Tp , with the time τ through composite microstructure.

**Figures 3** and **4** show the variation of the nonlinear displacement sensitivities u1 and u2, with time *τ* for different cases IH, IF, AH, and AF. It was shown from these figures that the anisotropy and functionally graded material have great effects on the nonlinear displacement sensitivities through the FGA composite microstructure.

**Figures 5**–**7** show the variation of the nonlinear thermal stress sensitivities σ11, σ12, and σ22, respectively, with time τ for different cases IH, IF, AH, and AF. It was noted from these figures that the anisotropy and functionally graded material have

**Figure 2.** *Variation of the temperature sensitivity with time τ.*

**Figure 3.** *Variation of the displacement u1 sensitivity with time τ.*

*A New Boundary Element Formulation for Modeling and Optimization of Three-Temperature… DOI: http://dx.doi.org/10.5772/intechopen.93515*

**Figure 4.**

at *<sup>y</sup>* <sup>¼</sup> *<sup>b</sup> <sup>∂</sup>u*<sup>1</sup>

ature *T T* ¼ *Te* þ *Ti* þ *Tp*

and total temperature T T ¼ Te þ Ti þ Tp

*Variation of the temperature sensitivity with time τ.*

*Variation of the displacement u1 sensitivity with time τ.*

this study.

**Figure 2.**

**Figure 3.**

**158**

microstructure.

*Composite Materials*

*<sup>∂</sup><sup>y</sup>* <sup>¼</sup> *<sup>∂</sup>u*<sup>2</sup>

In order to study the effects of anisotropy and functionally graded materials on composite microstructure, we consider the following four cases, namely, isotropic homogeneous (IH), isotropic functionally graded (IF), anisotropic homogeneous (AH), and anisotropic functionally graded (AF). Also, we considered total temper-

as the considered temperature field in all calculations of

, with the time τ through composite

**Figure 2** shows the variations of the nonlinear three-temperature Te, Ti, and Tp

**Figures 3** and **4** show the variation of the nonlinear displacement sensitivities u1 and u2, with time *τ* for different cases IH, IF, AH, and AF. It was shown from these figures that the anisotropy and functionally graded material have great effects on the nonlinear displacement sensitivities through the FGA composite microstructure. **Figures 5**–**7** show the variation of the nonlinear thermal stress sensitivities σ11, σ12, and σ22, respectively, with time τ for different cases IH, IF, AH, and AF. It was noted from these figures that the anisotropy and functionally graded material have

*<sup>∂</sup><sup>y</sup>* <sup>¼</sup> 0, *<sup>∂</sup><sup>T</sup>*

*<sup>∂</sup><sup>y</sup>* <sup>¼</sup> <sup>0</sup> (102)

*Variation of the displacement u2 sensitivity with time τ.*

**Figure 5.** *Variation of the thermal stress σ<sup>11</sup> sensitivity with time τ.*

**Figure 6.** *Variation of the thermal stress σ<sup>12</sup> sensitivity with time τ.*

**Figure 7.** *Variation of the thermal stress σ<sup>22</sup> sensitivity with time τ.*

great influences on the nonlinear thermal stress sensitivities through the FGA composite microstructure.

The mean compliance has been minimized, to obtain the maximum stiffness for the composite microstructures made from two competitive materials and without holes or inclusions. Investigation of the effect of the functionally graded parameter on the optimal composite microstructure has been shown in **Table 1** for the 1*T* model and in **Table 2** for the 3*T* model. It is noticed from these tables that the heat conduction model and functionally graded parameter have a significant effect on

*A New Boundary Element Formulation for Modeling and Optimization of Three-Temperature…*

The mean compliance has been minimized to obtain the maximum stiffness for

*Investigation of the influence of functionally graded parameter,* m*, on the optimal composite microstructure for*

*Investigation of the influence of functionally graded parameter,* m*, on the optimal composite microstructure for*

the topology optimization process of the multi-material FGA composite

*Variation of the thermal stress σ<sup>11</sup> sensitivity along x-axis.*

*DOI: http://dx.doi.org/10.5772/intechopen.93515*

Example 2. Composite microstructures with circular or square holes.

the composite microstructures made from two competitive materials and with circular or square holes. Investigation of the effect of the functionally graded

microstructures.

**Table 1.**

**Table 2.**

**161**

*the 3T model.*

*the 1*T *model.*

**Figure 9.**

For comparison purposes with those of other studies, we only considered onedimensional numerical results of the considered three-temperature problem. In the considered special case, the nonlinear displacement *u*<sup>1</sup> and nonlinear thermal stress *σ*<sup>11</sup> results are plotted in **Figures 8** and **9**, respectively. It can be noticed from these that the BEM results, which are based on replacing one-temperature heat conduction with three-temperature heat conduction, are in excellent agreement when compared to results obtained from the finite difference method of Pazera and Jędrysiak [97] and the finite element method (FEM) of Xiong and Tian [98]. We thus demonstrate the validity and accuracy of our proposed BEM technique.

Three numerical examples of BESO topological optimization of composite microstructures are performed to illustrate the optimization results of this study [99]. In order to obtain the functionally graded parameter effects during the optimization process of the considered composite microstructure, we consider the following values *m* ¼ 0, 0*:*5, 0*:*75, and 1 in the one-temperature heat conduction model and the three-temperature radiative heat conduction model.

Example 1. Composite microstructures without holes or inclusions.

**Figure 8.** *Variation of the displacement u1 sensitivity along x-axis.*

*A New Boundary Element Formulation for Modeling and Optimization of Three-Temperature… DOI: http://dx.doi.org/10.5772/intechopen.93515*

**Figure 9.** *Variation of the thermal stress σ<sup>11</sup> sensitivity along x-axis.*

The mean compliance has been minimized, to obtain the maximum stiffness for the composite microstructures made from two competitive materials and without holes or inclusions. Investigation of the effect of the functionally graded parameter on the optimal composite microstructure has been shown in **Table 1** for the 1*T* model and in **Table 2** for the 3*T* model. It is noticed from these tables that the heat conduction model and functionally graded parameter have a significant effect on the topology optimization process of the multi-material FGA composite microstructures.

Example 2. Composite microstructures with circular or square holes.

The mean compliance has been minimized to obtain the maximum stiffness for the composite microstructures made from two competitive materials and with circular or square holes. Investigation of the effect of the functionally graded

**Table 1.**

great influences on the nonlinear thermal stress sensitivities through the FGA com-

For comparison purposes with those of other studies, we only considered onedimensional numerical results of the considered three-temperature problem. In the considered special case, the nonlinear displacement *u*<sup>1</sup> and nonlinear thermal stress *σ*<sup>11</sup> results are plotted in **Figures 8** and **9**, respectively. It can be noticed from these that the BEM results, which are based on replacing one-temperature heat conduction with three-temperature heat conduction, are in excellent agreement when compared to results obtained from the finite difference method of Pazera and Jędrysiak [97] and the finite element method (FEM) of Xiong and Tian [98]. We thus demonstrate the validity and accuracy of our proposed BEM technique. Three numerical examples of BESO topological optimization of composite microstructures are performed to illustrate the optimization results of this study [99]. In order to obtain the functionally graded parameter effects during the optimization process of the considered composite microstructure, we consider the following values *m* ¼ 0, 0*:*5, 0*:*75, and 1 in the one-temperature heat conduction

model and the three-temperature radiative heat conduction model. Example 1. Composite microstructures without holes or inclusions.

posite microstructure.

*Composite Materials*

*Variation of the thermal stress σ<sup>22</sup> sensitivity with time τ.*

**Figure 7.**

**Figure 8.**

**160**

*Variation of the displacement u1 sensitivity along x-axis.*

*Investigation of the influence of functionally graded parameter,* m*, on the optimal composite microstructure for the 1*T *model.*

#### **Table 2.**

*Investigation of the influence of functionally graded parameter,* m*, on the optimal composite microstructure for the 3T model.*

parameter on the optimal composite microstructure with circular holes has been shown in **Table 3** for the 1*T* model and in **Table 4** for the 3*T* model. Also, the investigation of the effect of the functionally graded parameter on the optimal composite microstructure with square holes has been shown in **Table 5** for the 1*T* model and in **Table 6** for the 3*T* model. It is noticed from these tables that the heat

conduction model, functionally graded parameter, and holes shape have a significant effect on the topology optimization process of the multi-material FGA com-

*A New Boundary Element Formulation for Modeling and Optimization of Three-Temperature…*

Example 3. Composite microstructures with circular or square inclusions. The mean compliance has been minimized to obtain the maximum stiffness for

the composite microstructures made from two competitive materials and with circular or square inclusions. Investigation of the effect of the functionally graded parameter on optimal composite microstructure with circular inclusions has been shown in **Table 7** for the 1*T* model and in **Table 8** for the 3*T* model. Also, the investigation of the effect of the functionally graded parameter on the optimal composite microstructure with square inclusions has been shown in **Table 9** for the 1*T* model and in **Table 10** for the 3*T* model. It is noticed from these tables that the heat conduction model, functionally graded parameter, and inclusions shape have a significant effect on the topology optimization process of the multi-material FGA

The BESO topology optimization problem implemented in the numerical examples to find the distribution of the two materials in the design domain that minimize

*Investigation of the influence of functionally graded parameter, m, on the optimal composite microstructure*

*Investigation of the influence of functionally graded parameter,* m*, on the optimal composite microstructure*

*Investigation of the influence of the functionally graded parameter,* m*, on the optimal composite microstructure*

posite microstructures.

*DOI: http://dx.doi.org/10.5772/intechopen.93515*

composite microstructures.

**Table 8.**

**Table 7.**

**Table 9.**

**163**

*with circular shape inclusions for the 3*T *model.*

*with circular shape inclusions for the 1*T *model.*

*with square shape inclusions for the 1T model.*

**Table 3.**

*Investigation of the influence of functionally graded parameter,* m*, on the optimal composite microstructure with circular shape holes for the 1*T *model.*

#### **Table 4.**

*Investigation of the influence of functionally graded parameter m on the optimal composite microstructure with circular shape holes for the 3*T *model.*

#### **Table 5.**

*Investigation of the influence of functionally graded parameter,* m*, on the optimal composite microstructure with square shape holes for the 1*T *model.*

#### **Table 6.**

*Investigation of the influence of functionally graded parameter,* m*, on the optimal composite microstructure with square shape holes for the 3*T *model.*

*A New Boundary Element Formulation for Modeling and Optimization of Three-Temperature… DOI: http://dx.doi.org/10.5772/intechopen.93515*

conduction model, functionally graded parameter, and holes shape have a significant effect on the topology optimization process of the multi-material FGA composite microstructures.

Example 3. Composite microstructures with circular or square inclusions.

The mean compliance has been minimized to obtain the maximum stiffness for the composite microstructures made from two competitive materials and with circular or square inclusions. Investigation of the effect of the functionally graded parameter on optimal composite microstructure with circular inclusions has been shown in **Table 7** for the 1*T* model and in **Table 8** for the 3*T* model. Also, the investigation of the effect of the functionally graded parameter on the optimal composite microstructure with square inclusions has been shown in **Table 9** for the 1*T* model and in **Table 10** for the 3*T* model. It is noticed from these tables that the heat conduction model, functionally graded parameter, and inclusions shape have a significant effect on the topology optimization process of the multi-material FGA composite microstructures.

The BESO topology optimization problem implemented in the numerical examples to find the distribution of the two materials in the design domain that minimize

#### **Table 7.**

parameter on the optimal composite microstructure with circular holes has been shown in **Table 3** for the 1*T* model and in **Table 4** for the 3*T* model. Also, the investigation of the effect of the functionally graded parameter on the optimal composite microstructure with square holes has been shown in **Table 5** for the 1*T* model and in **Table 6** for the 3*T* model. It is noticed from these tables that the heat

*Investigation of the influence of functionally graded parameter,* m*, on the optimal composite microstructure*

*Investigation of the influence of functionally graded parameter m on the optimal composite microstructure with*

*Investigation of the influence of functionally graded parameter,* m*, on the optimal composite microstructure*

*Investigation of the influence of functionally graded parameter,* m*, on the optimal composite microstructure*

**Table 3.**

*Composite Materials*

**Table 4.**

**Table 5.**

**Table 6.**

**162**

*with circular shape holes for the 1*T *model.*

*circular shape holes for the 3*T *model.*

*with square shape holes for the 1*T *model.*

*with square shape holes for the 3*T *model.*

*Investigation of the influence of functionally graded parameter,* m*, on the optimal composite microstructure with circular shape inclusions for the 1*T *model.*

#### **Table 8.**

*Investigation of the influence of functionally graded parameter, m, on the optimal composite microstructure with circular shape inclusions for the 3*T *model.*

#### **Table 9.**

*Investigation of the influence of the functionally graded parameter,* m*, on the optimal composite microstructure with square shape inclusions for the 1T model.*

#### **Table 10.**

*Investigation of the influence of functionally graded parameter,* m*, on the optimal composite microstructure with square shape inclusions for the 3*T *model.*

the compliance of the structure subject to a volume constraint in both phases can be stated as

Find *X<sup>M</sup>*

That minimize  $\mathbf{C}^{M} = \frac{1}{2} \left( P^{M} \right)^{T} u^{M} = \frac{1}{2} \left( f^{M, \text{ter}} + f^{M, \text{mec}} \right)^{T} u^{M}$ 

Subject to  $\mathbf{V}\_{j}^{M, \*} - \Sigma\_{i=1}^{N} \mathbf{V}\_{i}^{M} \mathbf{X}\_{ij}^{M} - \Sigma\_{i=1}^{j-1} \mathbf{V}\_{i}^{M, \*} = \mathbf{0}$ 

$$K^{M} u^{M} = P^{M}$$

$$X\_i^M = \mathfrak{x}\_{\min} \mathbf{V} \mathbf{1}; j = \mathbf{1}, \mathbf{2}$$

where *X<sup>M</sup>* is the design variable; *V<sup>M</sup>*, <sup>∗</sup> *<sup>j</sup>* is the volume of the *j*th material phase, where *i* and *j* denote the element *i*th which is made of *j*th material; *C<sup>M</sup>* is the mean compliance; *P* is the total load on the structure, which is the sum of mechanical and thermal loads; *uM* is the displacement vector; *V<sup>M</sup>*, <sup>∗</sup> is the volume of the solid material; *N* is the total number of elements; *K<sup>M</sup>* is the global stiffness matrix; *xmin* is a small value (e.g., 0.0001), which guarantees that none of the elements will be removed completely from design domain; *f <sup>M</sup>*,*mec* is the mechanical load vector; and *f <sup>M</sup>*,*ter* is the thermal load vector. Also, the BESO parameters considered in these examples can be seen in **Table 11**. The validity of our implemented BESO topology optimization technique has been demonstrated in our recent reference [100].

Example 4. Laminated composite microstructure with three different sets of boundary conditions are considered in this example to validate the BEM formulation of the current study. These boundary conditions are called: simply—simply supported (SS), clamped—clamped (CC), and clamped—simply supported (CS). One-temperature (1T) and three-temperature (3T) models of nonlinear thermal stresses sensitivities results have been compared with the finite element method (FEM) results of Rajanna et al. [101] as well as with the finite volume method (FVM) results of Fallah and Delzendeh [102], which are tabulated in **Table 12** for different types of boundary conditions and different methods. It can be observed that the BEM results for all the three types of boundary conditions are in excellent

*Models of 1T and 3T nonlinear thermal stresses' sensitivities for different types of boundary conditions and*

**Model Type Method** *σ***<sup>11</sup> sensitivity** *σ***<sup>12</sup> sensitivity** *σ***<sup>22</sup> sensitivity**

*A New Boundary Element Formulation for Modeling and Optimization of Three-Temperature…*

*DOI: http://dx.doi.org/10.5772/intechopen.93515*

IT SS FEM [101] 0.4084297 0.0509346 0.5332620

IT CC FEM [101] 0.3591487 0.0408259 0.3758618

IT CS FEM [101] 0.2518378 0.0307735 0.2613531

3T SS FEM [101] 0.3147696 0.0304364 0.4767923

3T CC FEM [101] 0.2432755 0.0204747 0.3052856

3T CS FEM [101] 0.1258947 0.0107824 0.2079734

BEM (present) 0.4084297 0.0509346 0.5332620

FVM [102] 0.4084297 0.0509346 0.5332620 BEM (present) 0.3591487 0.0408259 0.3758618

FVM [102] 0.3591487 0.0408259 0.3758618 BEM (present) 0.2518379 0.0307736 0.2613532

FVM [102] 0.2518379 0.0307736 0.2613532 BEM (present) 0.3147697 0.0304365 0.4767924

FVM [102] 0.3147697 0.0304365 0.4767924 BEM (present) 0.2432756 0.0204748 0.3052857

FVM [102] 0.2432756 0.0204748 0.3052857 BEM (present) 0.1258948 0.0107825 0.2079735

FVM [102] 0.1258948 0.0107825 0.2079737

The main aim of this chapter is to describe a new boundary element formulation for the modeling and optimization of the three-temperature nonlinear generalized magneto-thermoelastic functionally graded anisotropic (FGA) composite microstructures. The governing equations of the considered model are very difficult to solve analytically because of the nonlinearity and anisotropy. To overcome this, we propose a new boundary element formulation for solving such equations, where we used the three-temperature nonlinear radiative heat conduction equations combined with electron, ion, and phonon temperatures. Numerical results show the three-temperature distributions through composite microstructure. The effects of

agreement with FEM results of [101] and the FVM results of [102].

**6. Conclusion**

**165**

**Table 12.**

*different methods.*


**Table 11.** *Multi-material BESO parameters for minimization of a composite microstructure.*


*A New Boundary Element Formulation for Modeling and Optimization of Three-Temperature… DOI: http://dx.doi.org/10.5772/intechopen.93515*

#### **Table 12.**

the compliance of the structure subject to a volume constraint in both phases can be

*Investigation of the influence of functionally graded parameter,* m*, on the optimal composite microstructure*

<sup>2</sup> *f*

*<sup>i</sup>*¼<sup>1</sup> *<sup>V</sup><sup>M</sup>*, <sup>∗</sup>

*<sup>K</sup>MuM* <sup>¼</sup> *<sup>P</sup><sup>M</sup>*

*<sup>i</sup>* ¼ *xmin*V1; *j* ¼ 1, 2

where *i* and *j* denote the element *i*th which is made of *j*th material; *C<sup>M</sup>* is the mean compliance; *P* is the total load on the structure, which is the sum of mechanical and thermal loads; *uM* is the displacement vector; *V<sup>M</sup>*, <sup>∗</sup> is the volume of the solid material; *N* is the total number of elements; *K<sup>M</sup>* is the global stiffness matrix; *xmin* is a small value (e.g., 0.0001), which guarantees that none of the elements will be

*<sup>M</sup>*,*ter* is the thermal load vector. Also, the BESO parameters considered in these examples can be seen in **Table 11**. The validity of our implemented BESO topology optimization technique has been demonstrated in our recent reference [100].

**Variable name Variable description Variable value**

*<sup>f</sup>* <sup>1</sup> Final volume fraction of the material 1 for both interpolations 0.10

*<sup>f</sup>* <sup>2</sup> Final volume fraction of the material 2 for both interpolations 0.20 *ERM* Evolutionary ratio for interpolation 1 2% *ER<sup>M</sup>* Evolutionary ratio for interpolation 2 3%

*max* Volume addition ratio for interpolation 1 3%

*max* Volume addition ratio for interpolation 2 2%

*min* Filter ratio for interpolation 1 4 mm

*min* Filter ratio for interpolation 2 3 mm *τ* Convergence tolerance for both interpolations 0.01% *N* Convergence parameter for both interpolations 5

*Multi-material BESO parameters for minimization of a composite microstructure.*

*<sup>M</sup>*,*ter* <sup>þ</sup> *<sup>f</sup> <sup>M</sup>*,*mec <sup>T</sup>*

*<sup>i</sup>* ¼ 0; *j* ¼ 1, 2

*uM*

*<sup>j</sup>* is the volume of the *j*th material phase,

*<sup>M</sup>*,*mec* is the mechanical load vector; and

stated as Find *X<sup>M</sup>*

**Table 10.**

*Composite Materials*

*f*

*V<sup>M</sup>*

*V<sup>M</sup>*

*ARM*

*ARM*

*rM*

*rM*

**Table 11.**

**164**

That minimize *<sup>C</sup><sup>M</sup>* <sup>¼</sup> <sup>1</sup>

*<sup>j</sup>* � <sup>Σ</sup>*<sup>N</sup>*

*with square shape inclusions for the 3*T *model.*

where *X<sup>M</sup>* is the design variable; *V<sup>M</sup>*, <sup>∗</sup>

removed completely from design domain; *f*

Subject to *V<sup>M</sup>*, <sup>∗</sup>

<sup>2</sup> *PM <sup>T</sup>*

*<sup>i</sup>*¼<sup>1</sup>*V<sup>M</sup> <sup>i</sup> X<sup>M</sup>*

*uM* <sup>¼</sup> <sup>1</sup>

*ij* � <sup>Σ</sup> *<sup>j</sup>*�<sup>1</sup>

*X<sup>M</sup>*

*Models of 1T and 3T nonlinear thermal stresses' sensitivities for different types of boundary conditions and different methods.*

Example 4. Laminated composite microstructure with three different sets of boundary conditions are considered in this example to validate the BEM formulation of the current study. These boundary conditions are called: simply—simply supported (SS), clamped—clamped (CC), and clamped—simply supported (CS). One-temperature (1T) and three-temperature (3T) models of nonlinear thermal stresses sensitivities results have been compared with the finite element method (FEM) results of Rajanna et al. [101] as well as with the finite volume method (FVM) results of Fallah and Delzendeh [102], which are tabulated in **Table 12** for different types of boundary conditions and different methods. It can be observed that the BEM results for all the three types of boundary conditions are in excellent agreement with FEM results of [101] and the FVM results of [102].

#### **6. Conclusion**

The main aim of this chapter is to describe a new boundary element formulation for the modeling and optimization of the three-temperature nonlinear generalized magneto-thermoelastic functionally graded anisotropic (FGA) composite microstructures. The governing equations of the considered model are very difficult to solve analytically because of the nonlinearity and anisotropy. To overcome this, we propose a new boundary element formulation for solving such equations, where we used the three-temperature nonlinear radiative heat conduction equations combined with electron, ion, and phonon temperatures. Numerical results show the three-temperature distributions through composite microstructure. The effects of

anisotropy and functionally graded material on the three-temperature nonlinear displacement sensitivities and nonlinear thermal stress sensitivities through the composite microstructure are very significant and pronounced. Because there are no available results in the literature to confirm the validity and accuracy of our proposed technique except for one-temperature heat conduction, we replace the three-temperature radiative heat conduction with one-temperature heat conduction as a special case from our current general study. In the considered special case, the BEM results have been compared graphically with the FDM results and FEM results, and it can be noticed that the BEM results are in excellent agreement with the FDM and FEM results. These results thus demonstrate the validity and accuracy of our proposed technique.

Numerical examples are solved using the method of moving asymptotes (MMA) algorithm based on the bi-evolutionary structural optimization method (BESO), where we used the topological optimization to manufacture three-temperature magneto-thermoelastic composite microstructures to obtain the required specific engineering properties. A new class of FGA composite microstructures consisting of two competitive materials has been studied, taking into account existence of holes or inclusions. The effects of the heat conduction model, functionally graded parameter, and holes shape and inclusions shape on the optimal composite microstructure are investigated through the considered examples with great practical interest.

The ability to understand and manipulate composite microstructures has been fundamental to our technical development over time. Today, scientists and engineers recognize the importance of composite microstructures use for economic and environmental reasons. Based on the BEM implementation and its results, this study concluded that the boundary element technique is the most suitable technique for the manufacturing of FGA composite microstructures in the future works. This technique aimed to describe the behavior of FGA composite microstructures and achieves improvement in the composition optimization and mechanical properties of the resulting FGA composite microstructures.

Due to three-temperature and numerous low-temperature and hightemperature applications in laminated composites microstructures, as a future work and based on the findings obtained in the present study, we would suggest further research to develop numerical techniques for solving the three-temperature nonlinear thermoelastic wave propagation problems and for manufacturing of advanced laminated composites. The numerical results of our considered study can provide data references for mechanical engineers, computer engineers, geotechnical engineers, geothermal engineers, technologists, new materials designers, physicists, material science researchers, and those who are interested in novel technologies in the area of three-temperature magneto-thermoelastic FGA composite microstructures. Application of three-temperature theories in advanced manufacturing technologies, with the development of soft machines and robotics in biomedical engineering and advanced manufacturing, and nonlinear generalized magnetothermoelastic problems will be encountered more often where three-temperature radiative heat conduction will turn out to be the best choice for thermomechanical analysis in the design and analysis of advanced composite microstructures.

**Author details**

Saudi Arabia

Ismailia, Egypt

**167**

Mohamed Abdelsabour Fahmy1,2

provided the original work is properly cited.

1 Jamoum University College, Umm Al-Qura University, Jamoum, Makkah,

*A New Boundary Element Formulation for Modeling and Optimization of Three-Temperature…*

*DOI: http://dx.doi.org/10.5772/intechopen.93515*

2 Faculty of Computers and Informatics, Suez Canal University, New Campus,

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: mohamed\_fahmy@ci.suez.edu.eg

*A New Boundary Element Formulation for Modeling and Optimization of Three-Temperature… DOI: http://dx.doi.org/10.5772/intechopen.93515*

#### **Author details**

anisotropy and functionally graded material on the three-temperature nonlinear displacement sensitivities and nonlinear thermal stress sensitivities through the composite microstructure are very significant and pronounced. Because there are no available results in the literature to confirm the validity and accuracy of our proposed technique except for one-temperature heat conduction, we replace the three-temperature radiative heat conduction with one-temperature heat conduction as a special case from our current general study. In the considered special case, the BEM results have been compared graphically with the FDM results and FEM results, and it can be noticed that the BEM results are in excellent agreement with the FDM and FEM results. These results thus demonstrate the validity and accuracy

Numerical examples are solved using the method of moving asymptotes (MMA) algorithm based on the bi-evolutionary structural optimization method (BESO), where we used the topological optimization to manufacture three-temperature magneto-thermoelastic composite microstructures to obtain the required specific engineering properties. A new class of FGA composite microstructures consisting of two competitive materials has been studied, taking into account existence of holes or inclusions. The effects of the heat conduction model, functionally graded parameter, and holes shape and inclusions shape on the optimal composite microstructure are investigated through the considered examples with great practical

The ability to understand and manipulate composite microstructures has been fundamental to our technical development over time. Today, scientists and engineers recognize the importance of composite microstructures use for economic and environmental reasons. Based on the BEM implementation and its results, this study concluded that the boundary element technique is the most suitable technique for the manufacturing of FGA composite microstructures in the future works. This technique aimed to describe the behavior of FGA composite microstructures and achieves improvement in the composition optimization and mechanical properties

Due to three-temperature and numerous low-temperature and high-

analysis in the design and analysis of advanced composite microstructures.

temperature applications in laminated composites microstructures, as a future work and based on the findings obtained in the present study, we would suggest further research to develop numerical techniques for solving the three-temperature nonlinear thermoelastic wave propagation problems and for manufacturing of advanced laminated composites. The numerical results of our considered study can provide data references for mechanical engineers, computer engineers, geotechnical engineers, geothermal engineers, technologists, new materials designers, physicists, material science researchers, and those who are interested in novel technologies in the area of three-temperature magneto-thermoelastic FGA composite microstructures. Application of three-temperature theories in advanced manufacturing technologies, with the development of soft machines and robotics in biomedical engineering and advanced manufacturing, and nonlinear generalized magnetothermoelastic problems will be encountered more often where three-temperature radiative heat conduction will turn out to be the best choice for thermomechanical

of our proposed technique.

*Composite Materials*

of the resulting FGA composite microstructures.

interest.

**166**

Mohamed Abdelsabour Fahmy1,2

1 Jamoum University College, Umm Al-Qura University, Jamoum, Makkah, Saudi Arabia

2 Faculty of Computers and Informatics, Suez Canal University, New Campus, Ismailia, Egypt

\*Address all correspondence to: mohamed\_fahmy@ci.suez.edu.eg

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### **References**

[1] Pindera MJ, Arnold SM, Aboudi J, Hui D. Use of composites in functionally graded materials. Composites Engineering. 1994;**4**:1-145

[2] Pindera MJ, Aboudi J, Arnold SM, Jones WF. Use of composites in multiphased and functionally graded materials. Composites Engineering. 1995;**5**:743-974

[3] Yin HM, Paulino GH, Buttlar WG, Sun LZ. Effective thermal conductivity of two-phase functionally graded particulate composites. Journal of Applied Physics. 2005;**98**:063704

[4] Miyamoto Y, Kaysser WA, Rabin BH, Kawasaki A, Ford RG. Functionally Graded Materials: Design, Processing and Applications. New York: Springer US; 1999

[5] Noda N. Thermal stresses in functionally graded material. Journal of Thermal Stresses. 1999;**22**:477-512

[6] Kieback B, Neubrand A, Riedel H. Processing techniques for functionally graded materials. Materials Science and Engineering. 2003;**362**:81-106

[7] Kawasaki A, Watanabe R. Microstructural designing and fabrication of disk shaped functionally gradient materials by powder metallurgy. Journal of the Japan Society of Power and Powder Metallurgy. 1990; **37**:253-258

[8] Kiebact B, Neubrand A. Processing techniques for functionally graded materials. Materials Science and Engineering A. 2003;**362**:81-85

[9] Fahmy MA. A time-stepping DRBEM for 3D anisotropic functionally graded piezoelectric structures under the influence of gravitational waves. In: Rodrigues H, Elnashai A, Calvi G. editors. Facing the Challenges in

Structural Engineering. Sustainable Civil Infrastructures. 15-19 July 2017; Sharm El Sheikh, Egypt (GeoMEast 2017). Cham: Springer; 2018. pp. 350-365. DOI: 10.1007/978-3-319- 61914-9\_27

[15] Fahmy MA. A computerized DRBEM model for generalized magneto-thermo-visco-elastic stress waves in functionally graded anisotropic thin film/substrate structures. Latin American Journal of Solids and Structures. 2014;**11**:386-409

*DOI: http://dx.doi.org/10.5772/intechopen.93515*

of time fractional order dual phase lag

functionally graded tissues. Numerical Heat Transfer, Part A: Applications.

[22] Hyun S, Torquato S. Designing composite microstructures with

[23] Rodriguez R, Kelestemur MH. Processing and microstructural

targeted properties. Journal of Materials

characterization of functionally gradient Al A356/SiCp composite. Journal of Materials Science. 2002;**37**:1813-1821

[24] Duhamel J. Some memoire sur les phenomenes thermo-mechanique. Journal de l'École polytechnique. 1837;

[25] Neumann F. Vorlesungen Uber die theorie der elasticitat. Meyer: Brestau;

[27] Lord HW, Shulman Y. A generalized dynamical theory of thermoelasticity. Journal of the Mechanics and Physics of

Thermoelasticity. Journal of Elasticity.

undamped heat waves in an elastic solid. Journal of Thermal Stresses. 1992;**15**:

[31] Tzou DY. A unified field approach for heat conduction from macro to micro scales. ASME Journal of Heat

[26] Biot M. Thermoelasticity and irreversible thermo-dynamics. Journal of Applied Physics. 1956;**27**:249-253

Solids. 1967;**15**:299-309

1972;**2**:1-7

253-264

**31**:189-208

[28] Green AE, Lindsay KA.

[29] Green AE, Naghdi PM. On

[30] Green AE, Naghdi PM. Thermoelasticity without energy dissipation. Journal of Elasticity. 1993;

Transfer. 1995;**117**:8-16

bioheat transfer problems in

Research. 2001;**16**:280-285

2019;**75**:616-626

*A New Boundary Element Formulation for Modeling and Optimization of Three-Temperature…*

**15**:1-57

1885

[16] Fahmy MA, Salem AM,

[17] Fahmy MA, Salem AM,

2014;**4**:1010-1026

2017;**25**:1-20

5369-5382

2019;**3**:1-13

**169**

674-685

Metwally MS, Rashid MM. Computer implementation of the DRBEM for studying the classical coupled

thermoelastic responses of functionally graded anisotropic plates. Physical Science International Journal. 2014;**4**:

Metwally MS, Rashid MM. Computer implementation of the DRBEM for studying the generalized thermo elastic responses of functionally graded anisotropic rotating plates with two relaxation times. British Journal of Mathematics & Computer Science.

[18] Fahmy MA. DRBEM sensitivity analysis and shape optimization of rotating magneto-thermo-viscoelastic FGA structures using golden-section search algorithm based on uniform bicubic B-splines. Journal of Advances in Mathematics and Computer Science.

[19] Fahmy MA. A predictor-corrector time-stepping DRBEM for shape design

sensitivity and optimization of multilayer FGA structures.

BH, Sibih AM. A computerized boundary element algorithm for modeling and optimization of complex magneto-thermoelastic problems in MFGA structures. Journal of Engineering Research and Reports.

Transylvanian Review. 2017;**XXV**:

[20] Fahmy MA, Al-Harbi SM, Al-Harbi

[21] Fahmy MA. A new LRBFCM-GBEM modeling algorithm for general solution

[10] Fahmy MA. 3D DRBEM modeling for rotating initially stressed anisotropic functionally graded piezoelectric plates. In: Proceedings of the 7th European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2016); 5-10 June 2016; Crete Island, Greece. 2016. pp. 7640-7658

[11] Fahmy MA. Boundary element solution of 2D coupled problem in anisotropic piezoelectric FGM plates. In: Proceedings of the 6th International Conference on Computational Methods for Coupled Problems in Science and Engineering (Coupled Problems 2015); 18-20 May 2015; Venice, Italy. 2015. pp. 382-391

[12] Fahmy MA. The DRBEM solution of the generalized magneto-thermoviscoelastic problems in 3D anisotropic functionally graded solids. In: Proceedings of the 5th International Conference on Coupled Problems in Science and Engineering (Coupled Problems 2013); 17-19 June 2013; Ibiza, Spain. 2013. pp. 862-872

[13] Fahmy MA. Computerized Boundary Element Solutions for Thermoelastic Problems: Applications to Functionally Graded Anisotropic Structures. Saarbrücken: LAP Lambert Academic Publishing; 2017

[14] Fahmy MA. Boundary Element Computation of Shape Sensitivity and Optimization: Applications to Functionally Graded Anisotropic Structures. Saarbrücken: LAP Lambert Academic Publishing; 2017

*A New Boundary Element Formulation for Modeling and Optimization of Three-Temperature… DOI: http://dx.doi.org/10.5772/intechopen.93515*

[15] Fahmy MA. A computerized DRBEM model for generalized magneto-thermo-visco-elastic stress waves in functionally graded anisotropic thin film/substrate structures. Latin American Journal of Solids and Structures. 2014;**11**:386-409

**References**

*Composite Materials*

1995;**5**:743-974

[1] Pindera MJ, Arnold SM, Aboudi J, Hui D. Use of composites in functionally Structural Engineering. Sustainable Civil Infrastructures. 15-19 July 2017; Sharm El Sheikh, Egypt (GeoMEast 2017). Cham: Springer; 2018.

pp. 350-365. DOI: 10.1007/978-3-319-

[10] Fahmy MA. 3D DRBEM modeling

for rotating initially stressed anisotropic functionally graded piezoelectric plates. In: Proceedings of

the 7th European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2016); 5-10 June 2016; Crete Island,

Greece. 2016. pp. 7640-7658

[11] Fahmy MA. Boundary element solution of 2D coupled problem in anisotropic piezoelectric FGM plates. In: Proceedings of the 6th International Conference on Computational Methods for Coupled Problems in Science and Engineering (Coupled Problems 2015); 18-20 May 2015; Venice, Italy. 2015.

[12] Fahmy MA. The DRBEM solution of the generalized magneto-thermoviscoelastic problems in 3D anisotropic

functionally graded solids. In: Proceedings of the 5th International Conference on Coupled Problems in Science and Engineering (Coupled Problems 2013); 17-19 June 2013; Ibiza,

Spain. 2013. pp. 862-872

[13] Fahmy MA. Computerized Boundary Element Solutions for

Academic Publishing; 2017

Optimization: Applications to Functionally Graded Anisotropic Structures. Saarbrücken: LAP Lambert

Academic Publishing; 2017

Thermoelastic Problems: Applications to Functionally Graded Anisotropic Structures. Saarbrücken: LAP Lambert

[14] Fahmy MA. Boundary Element Computation of Shape Sensitivity and

61914-9\_27

pp. 382-391

[2] Pindera MJ, Aboudi J, Arnold SM, Jones WF. Use of composites in multiphased and functionally graded materials. Composites Engineering.

[3] Yin HM, Paulino GH, Buttlar WG, Sun LZ. Effective thermal conductivity of two-phase functionally graded particulate composites. Journal of Applied Physics. 2005;**98**:063704

[4] Miyamoto Y, Kaysser WA, Rabin BH, Kawasaki A, Ford RG. Functionally Graded Materials: Design, Processing and Applications. New York:

[5] Noda N. Thermal stresses in

Engineering. 2003;**362**:81-106

[7] Kawasaki A, Watanabe R. Microstructural designing and

gradient materials by powder

**37**:253-258

**168**

functionally graded material. Journal of Thermal Stresses. 1999;**22**:477-512

[6] Kieback B, Neubrand A, Riedel H. Processing techniques for functionally graded materials. Materials Science and

fabrication of disk shaped functionally

metallurgy. Journal of the Japan Society of Power and Powder Metallurgy. 1990;

[8] Kiebact B, Neubrand A. Processing techniques for functionally graded materials. Materials Science and Engineering A. 2003;**362**:81-85

[9] Fahmy MA. A time-stepping DRBEM for 3D anisotropic functionally graded piezoelectric structures under the influence of gravitational waves. In: Rodrigues H, Elnashai A, Calvi G. editors. Facing the Challenges in

Springer US; 1999

graded materials. Composites Engineering. 1994;**4**:1-145

> [16] Fahmy MA, Salem AM, Metwally MS, Rashid MM. Computer implementation of the DRBEM for studying the classical coupled thermoelastic responses of functionally graded anisotropic plates. Physical Science International Journal. 2014;**4**: 674-685

> [17] Fahmy MA, Salem AM, Metwally MS, Rashid MM. Computer implementation of the DRBEM for studying the generalized thermo elastic responses of functionally graded anisotropic rotating plates with two relaxation times. British Journal of Mathematics & Computer Science. 2014;**4**:1010-1026

> [18] Fahmy MA. DRBEM sensitivity analysis and shape optimization of rotating magneto-thermo-viscoelastic FGA structures using golden-section search algorithm based on uniform bicubic B-splines. Journal of Advances in Mathematics and Computer Science. 2017;**25**:1-20

[19] Fahmy MA. A predictor-corrector time-stepping DRBEM for shape design sensitivity and optimization of multilayer FGA structures. Transylvanian Review. 2017;**XXV**: 5369-5382

[20] Fahmy MA, Al-Harbi SM, Al-Harbi BH, Sibih AM. A computerized boundary element algorithm for modeling and optimization of complex magneto-thermoelastic problems in MFGA structures. Journal of Engineering Research and Reports. 2019;**3**:1-13

[21] Fahmy MA. A new LRBFCM-GBEM modeling algorithm for general solution

of time fractional order dual phase lag bioheat transfer problems in functionally graded tissues. Numerical Heat Transfer, Part A: Applications. 2019;**75**:616-626

[22] Hyun S, Torquato S. Designing composite microstructures with targeted properties. Journal of Materials Research. 2001;**16**:280-285

[23] Rodriguez R, Kelestemur MH. Processing and microstructural characterization of functionally gradient Al A356/SiCp composite. Journal of Materials Science. 2002;**37**:1813-1821

[24] Duhamel J. Some memoire sur les phenomenes thermo-mechanique. Journal de l'École polytechnique. 1837; **15**:1-57

[25] Neumann F. Vorlesungen Uber die theorie der elasticitat. Meyer: Brestau; 1885

[26] Biot M. Thermoelasticity and irreversible thermo-dynamics. Journal of Applied Physics. 1956;**27**:249-253

[27] Lord HW, Shulman Y. A generalized dynamical theory of thermoelasticity. Journal of the Mechanics and Physics of Solids. 1967;**15**:299-309

[28] Green AE, Lindsay KA. Thermoelasticity. Journal of Elasticity. 1972;**2**:1-7

[29] Green AE, Naghdi PM. On undamped heat waves in an elastic solid. Journal of Thermal Stresses. 1992;**15**: 253-264

[30] Green AE, Naghdi PM. Thermoelasticity without energy dissipation. Journal of Elasticity. 1993; **31**:189-208

[31] Tzou DY. A unified field approach for heat conduction from macro to micro scales. ASME Journal of Heat Transfer. 1995;**117**:8-16

[32] Chandrasekharaiah DS. Hyperbolic thermoelasticity: A review of recent literature. Applied Mechanics Reviews. 1998;**51**:705-729

[33] Roychoudhuri SK. On a thermoelastic three-phase-lag model. Journal of Thermal Stresses. 2007;**30**: 231-238

[34] Fahmy MA. A time-stepping DRBEM for magneto-thermoviscoelastic interactions in a rotating nonhomogeneous anisotropic solid. International Journal of Applied Mechanics. 2011;**3**:1-24

[35] Fahmy MA. A time-stepping DRBEM for the transient magnetothermo-visco-elastic stresses in a rotating non-homogeneous anisotropic solid. Engineering Analysis with Boundary Elements. 2012;**36**:335-345

[36] Fahmy MA. Numerical modeling of transient magneto-thermo-viscoelastic waves in a rotating nonhomogeneous anisotropic solid under initial stress. International Journal of Modeling. Simulation and Scientific Computing. 2012;**3**:1250002

[37] Fahmy MA. Transient magnetothermo-viscoelastic stresses in a rotating nonhomogeneous anisotropic solid with and without a moving heat source. Journal of Engineering Physics and Thermophysics. 2012;**85**:950-958

[38] Fahmy MA. Transient magnetothermo-elastic stresses in an anisotropic viscoelastic solid with and without moving heat source. Numerical Heat Transfer Part A: Applications. 2012;**61**: 547-564

[39] Fahmy MA. Transient magnetothermoviscoelastic plane waves in a non-homogeneous anisotropic thick strip subjected to a moving heat source. Applied Mathematical Modelling. 2012; **36**:4565-4578

[40] Fahmy MA. The effect of rotation and inhomogeneity on the transient magneto-thermoviscoelastic stresses in an anisotropic solid. ASME Journal of Applied Mechanics. 2012;**79**: 1015

structures. In: Awrejcewicz J, Grzelczyk D, editors. Dynamical Systems Theory. London, UK: IntechOpen; 2019. pp. 1-17

[47] Fahmy MA. Boundary element model for nonlinear fractional-order heat transfer in magneto-thermoelastic

*DOI: http://dx.doi.org/10.5772/intechopen.93515*

[53] Hu Q, Zhao L. Domain

1069-1100

*A New Boundary Element Formulation for Modeling and Optimization of Three-Temperature…*

1850043

decomposition preconditioners for the system generated by discontinuous Galerkin discretization of 2D-3T heat conduction equations. Communications in Computational Physics. 2017;**22**:

[54] Cho JR, Ha DY. Averaging and finite element discretization aproaches in the numerical analysis of functionally graded materials. Materials Science and Engineering A. 2001;**302**:187-196

Panda SK. Thermoacoustic behavior of laminated composite curved panels using higher-order finite-boundary element model. International Journal of Applied Mechanics. 2018;**10**:1850017

considering shear effects. International Journal of Applied Mechanics. 2019;**10**:

[57] Soliman AH, Fahmy MA. Range of applying the boundary condition at fluid/porous interface and evaluation of beavers and Joseph's slip coefficient using finite element method. Computation. 2020;**8**:14

[58] Fahmy MA. A new boundary element strategy for modeling and simulation of three temperatures nonlinear generalized micropolarmagneto-thermoelastic wave propagation problems in FGA

structures. Engineering Analysis with Boundary Elements. 2019;**108**:192-200

[59] Fahmy MA. A three-dimensional generalized magneto-thermoviscoelastic problem of a rotating functionally graded anisotropic solids with and without energy dissipation. Numerical Heat Transfer, Part A: Applications. 2013;**63**:713-733

[55] Sharma N, Mahapatra TR,

[56] Eskandari AH, Baghani M, Sohrabpour S. A time-dependent finite element formulation for thick shape memory polymer beams

[48] Fahmy MA. Boundary element mathematical modelling and boundary element numerical techniques for optimization of micropolar

thermoviscoelastic problems in solid deformable bodies. In: Sivasankaran S,

Bodies. London, UK: IntechOpen; 2020.

[49] Fahmy MA. Boundary element modeling and optimization based on fractional-order derivative for nonlinear generalized photo-

thermoelastic stress wave propagation in three-temperature anisotropic semiconductor structures. In: Sadollah A, Sinha TS, editors. Recent Trends in Computational Intelligence. London, UK: IntechOpen;

[50] El-Naggar AM, Abd-Alla AM, Fahmy MA, Ahmed SM. Thermal stresses in a rotating non-homogeneous orthotropic hollow cylinder. Heat and

Mass Transfer. 2002;**39**:41-46

[51] El-Naggar AM, Abd-Alla AM, Fahmy MA. The propagation of thermal stresses in an infinite elastic slab. Applied Mathematics and Computation.

[52] Abd-Alla AM, El-Naggar AM, Fahmy MA. Magneto-thermoelastic problem in non-homogeneous isotropic cylinder. Heat and Mass Transfer. 2003;

Nayak PK, Günay E, editors. Mechanics of Solid Deformable

pp. 1-21

2020. pp. 1-16

2003;**12**:220-226

**39**:625-629

**171**

FGA structures involving three temperatures. In: Ebrahimi F, editor. Mechanics of Functionally Graded Materials and Structures. London, UK:

IntechOpen; 2019. pp. 1-22

[41] Sharma N, Mahapatra TR, Panda SK. Thermoacoustic behavior of laminated composite curved panels using higher-order finite-boundary element model. International Journal of Applied Mechanics. 2018;**10**:1850017

[42] Othman MIA, Khan A, Jahangir R, Jahangir A. Analysis on plane waves through magneto-thermoelastic microstretch rotating medium with temperature dependent elastic properties. Applied Mathematical Modelling. 2019;**65**:535-548

[43] Ezzat MA, El-Karamany AS, El-Bary AA. On dual-phase-lag thermoelasticity theory with memorydependent derivative. Mechanics of Advanced Materials and Structures. 2017;**24**:908-916

[44] Ezzat MA, El-Karamany AS, El-Bary AA. Generalized thermoelasticity with memory-dependent derivatives involving two temperatures. Mechanics of Advanced Materials and Structures. 2016;**23**:545-553

[45] Fahmy MA. A computerized boundary element model for simulation and optimization of fractional-order three temperatures nonlinear generalized piezothermoelastic problems based on genetic algorithm. In: AIP Conference Proceedings 2138 of Innovation and Analytics Conference and Exihibiton (IACE 2019); 25-28 March 2019; Sintok, Malaysia. 2019. p. 030015

[46] Fahmy MA. A new computerized boundary element model for three-temperature nonlinear generalized thermoelastic stresses in anisotropic circular cylindrical plate

*A New Boundary Element Formulation for Modeling and Optimization of Three-Temperature… DOI: http://dx.doi.org/10.5772/intechopen.93515*

structures. In: Awrejcewicz J, Grzelczyk D, editors. Dynamical Systems Theory. London, UK: IntechOpen; 2019. pp. 1-17

[32] Chandrasekharaiah DS. Hyperbolic thermoelasticity: A review of recent literature. Applied Mechanics Reviews.

[40] Fahmy MA. The effect of rotation and inhomogeneity on the transient magneto-thermoviscoelastic stresses in an anisotropic solid. ASME Journal of Applied Mechanics. 2012;**79**:

[41] Sharma N, Mahapatra TR,

Panda SK. Thermoacoustic behavior of laminated composite curved panels using higher-order finite-boundary element model. International Journal of Applied Mechanics. 2018;**10**:1850017

[42] Othman MIA, Khan A, Jahangir R, Jahangir A. Analysis on plane waves through magneto-thermoelastic microstretch rotating medium with temperature dependent elastic properties. Applied Mathematical Modelling. 2019;**65**:535-548

[43] Ezzat MA, El-Karamany AS, El-

thermoelasticity theory with memorydependent derivative. Mechanics of Advanced Materials and Structures.

[44] Ezzat MA, El-Karamany AS, El-Bary AA. Generalized thermoelasticity with memory-dependent derivatives involving two temperatures. Mechanics of Advanced Materials and Structures.

[45] Fahmy MA. A computerized boundary element model for simulation and optimization of fractional-order three temperatures nonlinear generalized piezothermoelastic

problems based on genetic algorithm. In: AIP Conference Proceedings 2138 of Innovation and Analytics Conference and Exihibiton (IACE 2019); 25-28 March 2019; Sintok, Malaysia. 2019.

[46] Fahmy MA. A new computerized

generalized thermoelastic stresses in anisotropic circular cylindrical plate

boundary element model for three-temperature nonlinear

Bary AA. On dual-phase-lag

2017;**24**:908-916

2016;**23**:545-553

p. 030015

1015

thermoelastic three-phase-lag model. Journal of Thermal Stresses. 2007;**30**:

[34] Fahmy MA. A time-stepping DRBEM for magneto-thermoviscoelastic interactions in a rotating nonhomogeneous anisotropic solid. International Journal of Applied

[35] Fahmy MA. A time-stepping DRBEM for the transient magnetothermo-visco-elastic stresses in a rotating non-homogeneous anisotropic solid. Engineering Analysis with Boundary Elements. 2012;**36**:335-345

[36] Fahmy MA. Numerical modeling of transient magneto-thermo-viscoelastic waves in a rotating nonhomogeneous anisotropic solid under initial stress. International Journal of Modeling. Simulation and Scientific Computing.

[37] Fahmy MA. Transient magnetothermo-viscoelastic stresses in a rotating nonhomogeneous anisotropic solid with and without a moving heat source. Journal of Engineering Physics and Thermophysics. 2012;**85**:950-958

[38] Fahmy MA. Transient magnetothermo-elastic stresses in an anisotropic viscoelastic solid with and without moving heat source. Numerical Heat Transfer Part A: Applications. 2012;**61**:

[39] Fahmy MA. Transient magnetothermoviscoelastic plane waves in a non-homogeneous anisotropic thick strip subjected to a moving heat source. Applied Mathematical Modelling. 2012;

Mechanics. 2011;**3**:1-24

2012;**3**:1250002

547-564

**36**:4565-4578

**170**

1998;**51**:705-729

*Composite Materials*

231-238

[33] Roychoudhuri SK. On a

[47] Fahmy MA. Boundary element model for nonlinear fractional-order heat transfer in magneto-thermoelastic FGA structures involving three temperatures. In: Ebrahimi F, editor. Mechanics of Functionally Graded Materials and Structures. London, UK: IntechOpen; 2019. pp. 1-22

[48] Fahmy MA. Boundary element mathematical modelling and boundary element numerical techniques for optimization of micropolar thermoviscoelastic problems in solid deformable bodies. In: Sivasankaran S, Nayak PK, Günay E, editors. Mechanics of Solid Deformable Bodies. London, UK: IntechOpen; 2020. pp. 1-21

[49] Fahmy MA. Boundary element modeling and optimization based on fractional-order derivative for nonlinear generalized photothermoelastic stress wave propagation in three-temperature anisotropic semiconductor structures. In: Sadollah A, Sinha TS, editors. Recent Trends in Computational Intelligence. London, UK: IntechOpen; 2020. pp. 1-16

[50] El-Naggar AM, Abd-Alla AM, Fahmy MA, Ahmed SM. Thermal stresses in a rotating non-homogeneous orthotropic hollow cylinder. Heat and Mass Transfer. 2002;**39**:41-46

[51] El-Naggar AM, Abd-Alla AM, Fahmy MA. The propagation of thermal stresses in an infinite elastic slab. Applied Mathematics and Computation. 2003;**12**:220-226

[52] Abd-Alla AM, El-Naggar AM, Fahmy MA. Magneto-thermoelastic problem in non-homogeneous isotropic cylinder. Heat and Mass Transfer. 2003; **39**:625-629

[53] Hu Q, Zhao L. Domain decomposition preconditioners for the system generated by discontinuous Galerkin discretization of 2D-3T heat conduction equations. Communications in Computational Physics. 2017;**22**: 1069-1100

[54] Cho JR, Ha DY. Averaging and finite element discretization aproaches in the numerical analysis of functionally graded materials. Materials Science and Engineering A. 2001;**302**:187-196

[55] Sharma N, Mahapatra TR, Panda SK. Thermoacoustic behavior of laminated composite curved panels using higher-order finite-boundary element model. International Journal of Applied Mechanics. 2018;**10**:1850017

[56] Eskandari AH, Baghani M, Sohrabpour S. A time-dependent finite element formulation for thick shape memory polymer beams considering shear effects. International Journal of Applied Mechanics. 2019;**10**: 1850043

[57] Soliman AH, Fahmy MA. Range of applying the boundary condition at fluid/porous interface and evaluation of beavers and Joseph's slip coefficient using finite element method. Computation. 2020;**8**:14

[58] Fahmy MA. A new boundary element strategy for modeling and simulation of three temperatures nonlinear generalized micropolarmagneto-thermoelastic wave propagation problems in FGA structures. Engineering Analysis with Boundary Elements. 2019;**108**:192-200

[59] Fahmy MA. A three-dimensional generalized magneto-thermoviscoelastic problem of a rotating functionally graded anisotropic solids with and without energy dissipation. Numerical Heat Transfer, Part A: Applications. 2013;**63**:713-733

[60] Fahmy MA. A 2-D DRBEM for generalized magneto-thermoviscoelastic transient response of rotating functionally graded anisotropic thick strip. International Journal of Engineering and Technology Innovation. 2013;**3**:70-85

[61] Fahmy MA, Salem AM, Metwally MS, Rashid MM. Computer implementation of the DRBEM for studying the generalized thermoelastic responses of functionally graded anisotropic rotating plates with one relaxation time. International Journal of Applied Science and Technology. 2013; **3**:130-140

[62] Fahmy MA, Salem AM, Metwally MS, Rashid MM. Computer implementation of the DRBEM for studying the classical uncoupled theory of thermoelasticity of functionally graded anisotropic rotating plates. International Journal of Engineering Research and Applications. 2013;**3**: 1146-1154

[63] Fahmy MA. A Computerized Boundary Element Models for Coupled, Uncoupled and Generalized Thermoelasticity Theories of Functionally Graded Anisotropic Rotating Plates. UK: Book Publisher International; 2019

[64] Fahmy MA. A new computerized boundary element algorithm for cancer modeling of cardiac anisotropy on the ECG simulation. Asian Journal of Research in Computer Science. 2018;**2**: 1-10

[65] Brebbia CA, Telles JCF, Wrobel L. Boundary Element Techniques in Engineering. New York: Springer-Verlag; 1984

[66] Wrobel LC, Brebbia CA. The dual reciprocity boundary element formulation for nonlinear diffusion problems. Computer Methods in Applied Mechanics and Engineering. 1987;**65**:147-164

[67] Partridge PW, Brebbia CA. Computer implementation of the BEM dual reciprocity method for the solution of general field equations. Communications in Applied Numerical Methods. 1990;**6**:83-92

[74] Fahmy MA. A 2D time domain DRBEM computer model for magnetothermoelastic coupled wave propagation problems. International Journal of Engineering and Technology Innovation. 2014;**4**:138-151

*DOI: http://dx.doi.org/10.5772/intechopen.93515*

biothermomechanical behavior in anisotropic laser-induced tissue hyperthermia. Engineering Analysis with Boundary Elements. 2019;**101**:

[82] Fahmy MA. Design optimization for a simulation of rotating anisotropic viscoelastic porous structures using time-domain OQBEM. Mathematics and Computers in Simulation. 2019;**66**:

[83] Fahmy MA. A new convolution variational boundary element technique for design sensitivity analysis and topology optimization of anisotropic thermo-poroelastic structures. Arab Journal of Basic and Applied Sciences.

[84] Fahmy MA. Thermoelastic stresses

in a rotating non-homogeneous anisotropic body. Numerical Heat Transfer, Part A: Applications. 2008;**53**:

[85] Fahmy MA, El-Shahat TM. The

inhomogeneity on the thermoelastic stresses in a rotating anisotropic solid. Archive of Applied Mechanics. 2008;**78**:

[86] Farhat C, Park KC, Dubois-Pelerin Y. An unconditionally stable staggered algorithm for transient finite element analysis of coupled thermoelastic problems. Computer Methods in Applied Mechanics and Engineering.

[87] Svanberg K. The method of moving asymptotes a new method for structural optimization. International Journal of Numerical Methods in Engineering.

[88] Huang X, Xie Y. Convergent and mesh-independent solutions for the bi-directional evolutionary structural optimization method. Finite Elements in

effect of initial stress and

156-164

*A New Boundary Element Formulation for Modeling and Optimization of Three-Temperature…*

193-205

2020;**27**:1-12

1001-1011

431-442

1991;**85**:349-365

1987;**24**:359-373

[75] Fahmy MA, Al-Harbi SM, Al-Harbi BH. Implicit time-stepping DRBEM for design sensitivity analysis of magnetothermo-elastic FGA structure under initial stress. American Journal of Mathematical and Computational

[76] Fahmy MA. The effect of anisotropy on the structure optimization using golden-section search algorithm based on BEM. Journal of Advances in Mathematics and Computer Science.

[77] Fahmy MA. Shape design sensitivity

[78] Fahmy MA. Shape design sensitivity and optimization for two-temperature generalized magneto-thermoelastic problems using time-domain DRBEM. Journal of Thermal Stresses. 2018;**41**:

[79] Fahmy MA. Boundary element algorithm for modeling and simulation of dual-phase lag bioheat transfer and biomechanics of anisotropic soft tissues.

International Journal of Applied Mechanics. 2018;**10**:1850108

[80] Fahmy MA. Modeling and

2019;**44**:1671-1684

**173**

optimization of anisotropic viscoelastic porous structures using CQBEM and moving asymptotes algorithm. Arabian Journal for Science and Engineering.

[81] Fahmy MA. Boundary element modeling and simulation of

and optimization of anisotropic functionally graded smart structures using bicubic B-splines DRBEM. Engineering Analysis with Boundary

Elements. 2018;**87**:27-35

Sciences. 2017;**2**:55-62

2017;**25**:1-18

119-138

[68] Partridge PW, Brebbia CA, Wrobel LC. The Dual Reciprocity Boundary Element Method. Southampton: Computational Mechanics Publications; 1992

[69] Fahmy MA. Boundary element algorithm for nonlinear modeling and simulation of three temperature anisotropic generalized micropolar piezothermoelasticity with memorydependent derivative. International Journal of Applied Mechanics. 2020;**12**: 2050027

[70] Abd-Alla AM, Fahmy MA, El-Shahat TM. Magneto-thermo-elastic problem of a rotating non-homogeneous anisotropic solid cylinder. Archive of Applied Mechanics. 2008;**78**:135-148

[71] Fahmy MA. A New BEM for Modeling and Simulation of Laser Generated Ultrasound Waves in 3T Fractional Nonlinear Generalized Micropolar Poro-Thermoelastic FGA Structures. In: Valdman J, Marcinkowski L, editors. Modeling and Simulation in Engineering. London, UK: IntechOpen; 2020

[72] Fahmy MA. Implicit-explicit time integration DRBEM for generalized magneto-thermoelasticity problems of rotating anisotropic viscoelastic functionally graded solids. Engineering Analysis with Boundary Elements. 2013; **37**:107-115

[73] Fahmy MA. Generalized magnetothermo-viscoelastic problems of rotating functionally graded anisotropic plates by the dual reciprocity boundary element method. Journal of Thermal Stresses. 2013;**36**:1-20

*A New Boundary Element Formulation for Modeling and Optimization of Three-Temperature… DOI: http://dx.doi.org/10.5772/intechopen.93515*

[74] Fahmy MA. A 2D time domain DRBEM computer model for magnetothermoelastic coupled wave propagation problems. International Journal of Engineering and Technology Innovation. 2014;**4**:138-151

[60] Fahmy MA. A 2-D DRBEM for generalized magneto-thermoviscoelastic transient response of rotating functionally graded anisotropic thick strip. International Journal of Engineering and Technology Innovation. 2013;**3**:70-85

*Composite Materials*

[67] Partridge PW, Brebbia CA.

[68] Partridge PW, Brebbia CA, Wrobel LC. The Dual Reciprocity Boundary Element Method. Southampton: Computational Mechanics Publications; 1992

[69] Fahmy MA. Boundary element algorithm for nonlinear modeling and simulation of three temperature anisotropic generalized micropolar piezothermoelasticity with memorydependent derivative. International Journal of Applied Mechanics. 2020;**12**:

[70] Abd-Alla AM, Fahmy MA, El-Shahat TM. Magneto-thermo-elastic problem of a rotating non-homogeneous anisotropic solid cylinder. Archive of Applied Mechanics. 2008;**78**:135-148

[71] Fahmy MA. A New BEM for Modeling and Simulation of Laser Generated Ultrasound Waves in 3T Fractional Nonlinear Generalized Micropolar Poro-Thermoelastic FGA Structures. In: Valdman J, Marcinkowski L, editors. Modeling and Simulation in Engineering. London, UK: IntechOpen;

[72] Fahmy MA. Implicit-explicit time integration DRBEM for generalized magneto-thermoelasticity problems of rotating anisotropic viscoelastic

functionally graded solids. Engineering Analysis with Boundary Elements. 2013;

[73] Fahmy MA. Generalized magnetothermo-viscoelastic problems of

rotating functionally graded anisotropic plates by the dual reciprocity boundary element method. Journal of Thermal

of general field equations.

Methods. 1990;**6**:83-92

2050027

2020

**37**:107-115

Stresses. 2013;**36**:1-20

Computer implementation of the BEM dual reciprocity method for the solution

Communications in Applied Numerical

[61] Fahmy MA, Salem AM, Metwally MS, Rashid MM. Computer implementation of the DRBEM for studying the generalized thermoelastic responses of functionally graded anisotropic rotating plates with one relaxation time. International Journal of Applied Science and Technology. 2013;

**3**:130-140

1146-1154

1-10

Verlag; 1984

1987;**65**:147-164

**172**

[62] Fahmy MA, Salem AM,

Metwally MS, Rashid MM. Computer implementation of the DRBEM for studying the classical uncoupled theory of thermoelasticity of functionally graded anisotropic rotating plates. International Journal of Engineering Research and Applications. 2013;**3**:

[63] Fahmy MA. A Computerized Boundary Element Models for Coupled,

[64] Fahmy MA. A new computerized boundary element algorithm for cancer modeling of cardiac anisotropy on the ECG simulation. Asian Journal of Research in Computer Science. 2018;**2**:

[65] Brebbia CA, Telles JCF, Wrobel L. Boundary Element Techniques in Engineering. New York: Springer-

[66] Wrobel LC, Brebbia CA. The dual

reciprocity boundary element formulation for nonlinear diffusion problems. Computer Methods in Applied Mechanics and Engineering.

Uncoupled and Generalized Thermoelasticity Theories of Functionally Graded Anisotropic Rotating Plates. UK: Book Publisher

International; 2019

[75] Fahmy MA, Al-Harbi SM, Al-Harbi BH. Implicit time-stepping DRBEM for design sensitivity analysis of magnetothermo-elastic FGA structure under initial stress. American Journal of Mathematical and Computational Sciences. 2017;**2**:55-62

[76] Fahmy MA. The effect of anisotropy on the structure optimization using golden-section search algorithm based on BEM. Journal of Advances in Mathematics and Computer Science. 2017;**25**:1-18

[77] Fahmy MA. Shape design sensitivity and optimization of anisotropic functionally graded smart structures using bicubic B-splines DRBEM. Engineering Analysis with Boundary Elements. 2018;**87**:27-35

[78] Fahmy MA. Shape design sensitivity and optimization for two-temperature generalized magneto-thermoelastic problems using time-domain DRBEM. Journal of Thermal Stresses. 2018;**41**: 119-138

[79] Fahmy MA. Boundary element algorithm for modeling and simulation of dual-phase lag bioheat transfer and biomechanics of anisotropic soft tissues. International Journal of Applied Mechanics. 2018;**10**:1850108

[80] Fahmy MA. Modeling and optimization of anisotropic viscoelastic porous structures using CQBEM and moving asymptotes algorithm. Arabian Journal for Science and Engineering. 2019;**44**:1671-1684

[81] Fahmy MA. Boundary element modeling and simulation of

biothermomechanical behavior in anisotropic laser-induced tissue hyperthermia. Engineering Analysis with Boundary Elements. 2019;**101**: 156-164

[82] Fahmy MA. Design optimization for a simulation of rotating anisotropic viscoelastic porous structures using time-domain OQBEM. Mathematics and Computers in Simulation. 2019;**66**: 193-205

[83] Fahmy MA. A new convolution variational boundary element technique for design sensitivity analysis and topology optimization of anisotropic thermo-poroelastic structures. Arab Journal of Basic and Applied Sciences. 2020;**27**:1-12

[84] Fahmy MA. Thermoelastic stresses in a rotating non-homogeneous anisotropic body. Numerical Heat Transfer, Part A: Applications. 2008;**53**: 1001-1011

[85] Fahmy MA, El-Shahat TM. The effect of initial stress and inhomogeneity on the thermoelastic stresses in a rotating anisotropic solid. Archive of Applied Mechanics. 2008;**78**: 431-442

[86] Farhat C, Park KC, Dubois-Pelerin Y. An unconditionally stable staggered algorithm for transient finite element analysis of coupled thermoelastic problems. Computer Methods in Applied Mechanics and Engineering. 1991;**85**:349-365

[87] Svanberg K. The method of moving asymptotes a new method for structural optimization. International Journal of Numerical Methods in Engineering. 1987;**24**:359-373

[88] Huang X, Xie Y. Convergent and mesh-independent solutions for the bi-directional evolutionary structural optimization method. Finite Elements in Analysis and Design. 2007;**43**(14): 1039-1049

[89] Huang X, Xie Y. Evolutionary Topology Optimization of Continuum Structures. USA: John Wiley & Sons Ltd.; 2010

[90] Huang X, Xie YM. Bi-directional evolutionary topology optimization of continuum structures with one or multiple materials. Computational Mechanics. 2008;**43**(3):393

[91] Huang X, Zhou S, Xie Y, Li Q. Topology optimization of microstructures of cellular materials and composites for macrostructures. Computational Materials Science. 2013; **67**:397-407

[92] Sigmund O. Design of multiphysics actuators using topology optimization - Part I: One material structures. Computer Methods in Applied Mechanics and Engineering. 2001; **190**(49):6577-6604

[93] Sigmund O, Torquato S. Composites with extremal thermal expansion coefficients. Applied Physics Letters. 1996;**69**(21):3203-3205

[94] Sigmund O, Torquato S. Design of materials with extreme thermal expansion using a three-phase topology optimization method. Journal of the Mechanics and Physics of Solids. 1997; **45**(6):1037-1067

[95] Wang Y, Luo Z, Zhang N, Wu T. Topological design for mechanical metamaterials using a multiphase level set method. Structural and Multidisciplinary Optimization. 2016b; **54**:937-954

[96] Xu B, Huang X, Zhou S, Xie Y. Concurrent topological design of composite thermoelastic macrostructure and microstructure with multi-phase

material for maximum stiffness. Composite Structures. 2016;**150**:84-102

[97] Pazera E, Jędrysiak J. Effect of microstructure in thermoelasticity problems of functionally graded laminates. Composite Structures. 2018; **202**:296-303

[98] Xiong QL, Tian XG. Generalized magneto-thermo-microstretch response during thermal shock. Latin American Journal of Solids and Structures. 2015; **12**:2562-2580

[99] Krysko AV, Awrejcewicz J, Pavlov SP, Bodyagina KS, Krysko VA. Topological optimization of thermoelastic composites with maximized stiffness and heat transfer. Composites Part B Engineering. 2019; **158**:319-327

[100] Fahmy MA. A new BEM for modeling and optimization of 3T fractional nonlinear generalized magneto-thermoelastic multi-material ISMFGA structures subjected to moving heat source. In: Koprowski R, editor. Fractal Analysis. London, UK: IntechOpen; 2020

[101] Rajanna T, Banerjee S, Desai YM, Prabhakara DL. Effect of boundary conditions and non-uniform edge loads on buckling characteristics of laminated composite panels with and without cutout. International Journal for Computational Methods in Engineering Science and Mechanics. 2017;**18**:64-76

[102] Fallah N, Delzendeh M. Free vibration analysis of laminated composite plates using meshless finite volume method. Engineering Analysis with Boundary Elements. 2018;**88**: 132-144

Analysis and Design. 2007;**43**(14):

material for maximum stiffness. Composite Structures. 2016;**150**:84-102

[97] Pazera E, Jędrysiak J. Effect of microstructure in thermoelasticity problems of functionally graded laminates. Composite Structures. 2018;

[98] Xiong QL, Tian XG. Generalized magneto-thermo-microstretch response during thermal shock. Latin American Journal of Solids and Structures. 2015;

[99] Krysko AV, Awrejcewicz J, Pavlov SP, Bodyagina KS, Krysko VA.

maximized stiffness and heat transfer. Composites Part B Engineering. 2019;

[101] Rajanna T, Banerjee S, Desai YM, Prabhakara DL. Effect of boundary conditions and non-uniform edge loads on buckling characteristics of laminated composite panels with and without cutout. International Journal for

Computational Methods in Engineering Science and Mechanics. 2017;**18**:64-76

[102] Fallah N, Delzendeh M. Free vibration analysis of laminated composite plates using meshless finite volume method. Engineering Analysis with Boundary Elements. 2018;**88**:

[100] Fahmy MA. A new BEM for modeling and optimization of 3T fractional nonlinear generalized magneto-thermoelastic multi-material ISMFGA structures subjected to moving heat source. In: Koprowski R, editor. Fractal Analysis. London, UK:

Topological optimization of thermoelastic composites with

**202**:296-303

**12**:2562-2580

**158**:319-327

IntechOpen; 2020

132-144

[89] Huang X, Xie Y. Evolutionary Topology Optimization of Continuum Structures. USA: John Wiley & Sons

[90] Huang X, Xie YM. Bi-directional evolutionary topology optimization of continuum structures with one or multiple materials. Computational Mechanics. 2008;**43**(3):393

[91] Huang X, Zhou S, Xie Y, Li Q.

composites for macrostructures. Computational Materials Science. 2013;

Part I: One material structures. Computer Methods in Applied Mechanics and Engineering. 2001;

**190**(49):6577-6604

1996;**69**(21):3203-3205

**45**(6):1037-1067

**54**:937-954

**174**

microstructures of cellular materials and

[92] Sigmund O. Design of multiphysics actuators using topology optimization -

[93] Sigmund O, Torquato S. Composites with extremal thermal expansion coefficients. Applied Physics Letters.

[94] Sigmund O, Torquato S. Design of materials with extreme thermal

expansion using a three-phase topology optimization method. Journal of the Mechanics and Physics of Solids. 1997;

[95] Wang Y, Luo Z, Zhang N, Wu T. Topological design for mechanical metamaterials using a multiphase level set method. Structural and Multidisciplinary Optimization. 2016b;

[96] Xu B, Huang X, Zhou S, Xie Y. Concurrent topological design of

composite thermoelastic macrostructure and microstructure with multi-phase

Topology optimization of

1039-1049

*Composite Materials*

Ltd.; 2010

**67**:397-407

*Edited by Mohammad Asaduzzaman Chowdhury, José Luis Rivera Armenta, Mohammed Muzibur Rahman, Abdullah Asiri and Inamuddin*

This book presents information about composite materials, which have a variety of applications in engineering and aeronautics, transportation, construction, sports, and recreational activities, and so on. The first section evaluates the thermal and mechanical properties of thermoplastic and thermoset polymers reinforced with particles and fibers. The second section discusses new 2D composites such as thin films for their conductivity and shielding properties. In discussing the different materials, Composite Materials include information on the design of the materials, their structure, and their preparation methods.

Published in London, UK © 2021 IntechOpen © ktsimage / iStock

Composite Materials

Composite Materials

*Edited by Mohammad Asaduzzaman Chowdhury, José Luis Rivera Armenta, Mohammed Muzibur Rahman, Abdullah Asiri and Inamuddin*