**6. Conclusion**

**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

**Variable name Variable description Variable value**

*<sup>f</sup>* <sup>1</sup> Final volume fraction of the material 1 for both interpolations 0.25

*<sup>f</sup>* <sup>1</sup> Final volume fraction of the material 2 for both interpolations 0.25 *ERM* Evolutionary ratio for interpolation 1 3% *ERM* Evolutionary ratio for interpolation 2 3%

*max* Volume addition ratio for interpolation 1 1%

*max* Volume addition ratio for interpolation 2 1%

*min* Filter ratio for interpolation 1 4 mm

*min* Filter ratio for interpolation 2 4 mm *τ* Convergence tolerance for both interpolations 0.5 % *N* Convergence parameter for both interpolations 5

**Variable name Variable description Variable value**

*<sup>f</sup>* Final volume fraction 0.4 *ER<sup>M</sup>* Evolutionary ratio 1.2%

*max* Volume addition ratio 3%

*min* Filter ratio 0.19 mm *τ* Convergence tolerance 0.1% *N* Convergence parameter 5

*Multi-material BESO parameters for minimization of a roller-supported beam.*

*BESO parameters for minimization of a cantilever beam.*

*Multi-material BESO parameters for minimization of a MBB beam.*

*V<sup>M</sup>*

*Fractal Analysis - Selected Examples*

*V<sup>M</sup>*

*ARM*

*ARM*

*rM*

*rM*

**Table 2.**

*V<sup>M</sup>*

*V<sup>M</sup>*

*ARM*

*ARM*

*rM*

*rM*

**Table 3.**

*V<sup>M</sup>*

*AR<sup>M</sup>*

*rM*

**Table 4.**

**76**

The main purpose of this chapter is to describe a new boundary element formulation for modeling and optimization of 3T time fractional order nonlinear generalized thermoelastic multi-material ISMFGA structures subjected to moving heat source, where we used the three-temperature nonlinear radiative heat conduction equations combined with electron, ion, and phonon temperatures.

Numerical results show the influence of fractional order parameter on the sensitivities of the study's fields. The validity of the present method is examined and demonstrated by comparing the obtained outcomes with those known in the literature. Because there are no available data to confirm the validity and accuracy of our proposed technique, we replace the three-temperature radiative heat conduction with one-temperature heat conduction as a special case from our current general study of three-temperature nonlinear generalized thermoelasticity. In the considered special case of 3T time fractional order nonlinear generalized thermoelastic multi-material ISMFGA structures, the BEM results have been compared graphically with the FEM results; it can be noticed that the BEM results are in excellent agreement with the FEM results. These results thus demonstrate the validity and accuracy of our proposed technique. Numerical examples are solved using the multi-material topology optimization algorithm based on the bi-evolutionary structural optimization method (BESO). Numerical results of these examples show that the fractional order parameter affects the final result of optimization. The implemented optimization algorithm has proven to be an appropriate computational tool for material design.

Nowadays, the knowledge of 3T fractional order optimization of multi-material ISMFGA structures, can be utilized by mechanical engineers for designing heat exchangers, semiconductor nano materials, thermoelastic actuators, shape memory actuators, bimetallic valves and boiler tubes. As well as for chemists to observe the chemical processes such as bond breaking and bond forming.

### **Author details**

Mohamed Abdelsabour Fahmy Faculty of Computers and Informatics, Suez Canal University, 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.
