**2. Basics of approach**

Latin Hypercube type experimental designs first proposed by Vilnis Eglajs in his work (Audze & Eglajs, 1977), then by McKay (McKay et al., 1979) and used by many other investigators, as well as its improvements (Auzins, 2004) are a very essential aspect utilized in the proposed method. The significance of approximations for the solution of optimization problems proposed by Lucien Schmit in his early works (Schmit & Farcshi, 1971) and nowadays generally recognized as the Response Surface Method is also the foundation of current approach. The use of planned computer experiments and the metamodeling (surrogate model) approach (Forrester et al., 2008; Sacks et al., 1989) ensures great economy of computing time, especially for finite element (FEM) calculations. First of all let us demonstrate our approach on the simple test problems.

Shape Optimization of Mechanical Components for Measurement Systems 245

Fig. 2. Shapes of cutout in compliance with design of experiment.

squares method:

kernel (weight function):

characterizes significance.

crossvalidation:

In the case of locally weighted polynomial approximation, coefficients

where *u* is Euclidian distance from *x*0 to current point and

depend on point *x*0 where prediction is calculated and are obtained using the weighted least

<sup>0</sup> argmin ( ) ( ( ))

where the significance of neighboring points in the set *NX* is taken into account by Gaussian

<sup>2</sup> *wu u* exp

Quality of approximation is estimated by relative crossvalidation error using leave-one-out

1

*i err <sup>n</sup>*

*n*

where root mean squared prediction error stands in numerator and mean square deviation of response from its average value stands in denominator, *n* is the total number of

*n*

*n*

100%

1

*i*

<sup>1</sup> ( ) <sup>1</sup>

*i*

*y y*

<sup>1</sup> ( () )

 

*i i i*

*yx y*

*β wx x y yx*

*X*

*j N*

*L*

(3)

is a coefficient that

(4)

2

*jj j*

(2)

2

2

#### **2.1 Test problem of plate bending**

A clamped square plate is considered under a concentrated load of 500 N applied at centre in a direction orthogonal to its main surface. The isotropic material properties are: the Young's modulus *E* = 200 GPa, the Poisson's ratio = 0.3 and dimensions are 400x400x4.2118 mm. The shape optimization of the plate with constant thickness is carried out to minimize its volume in the case of a single displacement constraint *δ* = 0.5 mm. The cutout shape of the plate is defined by subsequent techniques shown in Figure 1: 1) with the points that are connected with straight lines; 2) with the NURBS knot points; 3) with the control points of NURBS polygon. Due to symmetry only ⅛ of the plate is considered for cutout definition and ¼ of the plate for problem solution by FEM.

Fig. 1. Techniques for definition of cutout.

Three parameters are stated to define location of points. Parameters are varied in the following ranges: 100≤ *X*1≤170; 100≤*X*2≤210; 100≤*X*3≤230 mm for the first two variants and 100≤*X*1≤180; 100≤*X*2≤235; 100≤*X*3≤230 mm for third variant of definition. In the last case at both end points two continuity vectors are defined additionally with direction normal to the side and to symmetry axis of the corresponding plate and with fixed length of 19 and 3 mm. The design of experiment for 3 factors and 40 trial points is calculated with mean-square error (MSE) criterion (Auzins & Janushevskis, 2002; Auzins, et al., 2006) value 0.4262 using EDAOpt (Auzins & Janushevskis, 2007) - software for design of experiments, approximation and optimization developed in the Riga Technical University. The geometrical models are developed using SolidWorks (SW) for all variants. The shapes for the third variant of definition are shown in Figure 2. In the next step responses of these models are calculated by SW Simulation (Lombard, 2009), using elements with a global size 4 mm and total number of DoF ~100000. Then these responses are used for approximation by EDAOpt. For example, for approximation of response *y* by quadratic polynomial the following expression (see, for example, Auzins & Janushevskis, 2007) is used:

$$\stackrel{\frown}{y} = \beta\_0 + \sum\_{i=1}^{d} \beta\_i \mathbf{x}\_i + \sum\_{i=1}^{d-1} \sum\_{j=i+1}^{d} \beta\_{ij} \mathbf{x}\_i \mathbf{x}\_j + \sum\_{i=1}^{d} \beta\_{ij} \mathbf{x}\_i^{\ 2} + \varepsilon \tag{1}$$

where there are *d* variables *x*1,…, *xd*, *L* = (*d*+1)(*d*+2)/2 unknown coefficients and the errors are assumed independent with zero mean and constant variance 2.

A clamped square plate is considered under a concentrated load of 500 N applied at centre in a direction orthogonal to its main surface. The isotropic material properties are: the

400x400x4.2118 mm. The shape optimization of the plate with constant thickness is carried out to minimize its volume in the case of a single displacement constraint *δ* = 0.5 mm. The cutout shape of the plate is defined by subsequent techniques shown in Figure 1: 1) with the points that are connected with straight lines; 2) with the NURBS knot points; 3) with the control points of NURBS polygon. Due to symmetry only ⅛ of the plate is considered for

Three parameters are stated to define location of points. Parameters are varied in the following ranges: 100≤ *X*1≤170; 100≤*X*2≤210; 100≤*X*3≤230 mm for the first two variants and 100≤*X*1≤180; 100≤*X*2≤235; 100≤*X*3≤230 mm for third variant of definition. In the last case at both end points two continuity vectors are defined additionally with direction normal to the side and to symmetry axis of the corresponding plate and with fixed length of 19 and 3 mm. The design of experiment for 3 factors and 40 trial points is calculated with mean-square error (MSE) criterion (Auzins & Janushevskis, 2002; Auzins, et al., 2006) value 0.4262 using EDAOpt (Auzins & Janushevskis, 2007) - software for design of experiments, approximation and optimization developed in the Riga Technical University. The geometrical models are developed using SolidWorks (SW) for all variants. The shapes for the third variant of definition are shown in Figure 2. In the next step responses of these models are calculated by SW Simulation (Lombard, 2009), using elements with a global size 4 mm and total number of DoF ~100000. Then these responses are used for approximation by EDAOpt. For example, for approximation of response *y* by quadratic polynomial the following expression (see, for

1

1 11 1

*i i ji i*

*i i ij i j ij i*

 

*d dd d*

 *xx x* 

2

2. and the errors

 

(1)

= 0.3 and dimensions are

**2.1 Test problem of plate bending** 

Fig. 1. Techniques for definition of cutout.

example, Auzins & Janushevskis, 2007) is used:

0

*y x* 

are assumed independent with zero mean and constant variance

where there are *d* variables *x*1,…, *xd*, *L* = (*d*+1)(*d*+2)/2 unknown coefficients

Young's modulus *E* = 200 GPa, the Poisson's ratio

cutout definition and ¼ of the plate for problem solution by FEM.

Fig. 2. Shapes of cutout in compliance with design of experiment.

In the case of locally weighted polynomial approximation, coefficients *L* depend on point *x*0 where prediction is calculated and are obtained using the weighted least squares method:

$$\boldsymbol{\beta} = \arg\min\_{\boldsymbol{\beta}} \sum\_{j \in \mathcal{N}\_{\boldsymbol{\Sigma}}} w(\mathbf{x}\_0 - \boldsymbol{\alpha}\_j) \times (\boldsymbol{y}\_j - \boldsymbol{\hat{y}}(\mathbf{x}\_j))^2 \tag{2}$$

where the significance of neighboring points in the set *NX* is taken into account by Gaussian kernel (weight function):

$$w(\mu) = \exp\left(-\alpha u^2\right) \tag{3}$$

where *u* is Euclidian distance from *x*0 to current point and is a coefficient that characterizes significance.

Quality of approximation is estimated by relative crossvalidation error using leave-one-out crossvalidation:

$$\sigma\_{crr} = 100\% \frac{\sqrt{\frac{1}{n} \sum\_{i=1}^{n} (\hat{y}\_{-i}(\mathbf{x}\_{i}) - y\_{i})^2}}{\sqrt{\frac{1}{n-1} \sum\_{i=1}^{n} (y\_{i} - \overline{y})^2}}\tag{4}$$

where root mean squared prediction error stands in numerator and mean square deviation of response from its average value stands in denominator, *n* is the total number of

Shape Optimization of Mechanical Components for Measurement Systems 247

It should be mentioned that the predicted volume for variant e is in good agreement with the actual value. At the same time the total number of FEM problem calculations (i.e. number of trials of computer experiment) is less on 10 than by using the homogenization method. The distribution of resultant displacement for plate of optimal shape (variant e) is

A square plate with dimensions 1300x1300x0.0001 mm is considered under two axial stretching loads of p = 0.65 N/mm2 and p/2 accordingly (see Figure 6). The isotropic material

optimization of the plate with constant thickness is carried out to minimize its volume in case of a single constraint on maximal level of equivalent stress *σmax* = 4.38 MPa. Due to symmetry only ¼ of the plate is considered for cutout definition and for problem solution by FEM. The cutout shape of the plate is defined by five coordinates of points 2, 3, 4, 5 and 6 situated on straight lines that make angles of 0; 22.5; 45 and 90 degrees with horizontal axis as shown in Figure 5. The initial cutout shape is a circle with radius 250 mm and the volume of plate *v* is

= 0.3. The shape

properties are: the Young's modulus *E* = 210 GPa, the Poisson's ratio

shown in fig. 4.

373.4 mm3.

Fig. 4. Displacement of plate for variant e.

Fig. 5. Scheme of ¼ of plate with initial cutout.

**2.2 Stretched plate test problem** 

experimental trials and ( )*<sup>i</sup> <sup>i</sup> <sup>y</sup> <sup>x</sup>* denotes approximated value for response in *i*-th point, calculated without taking into account the *i*-th experimental point.

Using the obtained locally weighted polynomial approximations by global search procedure (Janushevskis et. al, 2004), implemented in EDAOpt, the optimal cutout shape is obtained (see Figure 3) for the different aforementioned techniques. In table 1 the results are summarized and compared with volume obtained in work (Liang et al., 2001) using the homogenization method. Variants correspond to shapes shown in figure 3. The value of Gaussian kernel parameter of the local quadratic polynomial approximation is chosen to minimize relative leave-one-out crossvalidation error *σerr* of approximations of appropriate responses, i.e. deflection *δ* and volume *v* of plate. *vp* is the predicted volume calculated using approximations and *va* is the actual volume calculated using a geometrical model. *vp* and *va* in % show comparison of appropriate volume in respect to the volume obtained in (Liang et al., 2001). Best results are achieved with the technique using the control points of NURBS polygon. This allows reducing the volume of the plate by 1.38 % (Fig. 3 e) in comparison with the homogenization method.

Fig. 3. Shapes of plate obtained by (a) homogenization method (Liang et al., 2001); and by the current approach using different techniques: (b) the points that are connected with straight lines; (c) with the NURBS knot points; (d) with the control points of NURBS polygon; (e) same as "d" but with additionally optimized tangent weighting at the spline endpoints; (f) same as "e" but with circle added.


Table 1. Quantitative indices of the shape optimization of cutout for the plate bending problem.

It should be mentioned that the predicted volume for variant e is in good agreement with the actual value. At the same time the total number of FEM problem calculations (i.e. number of trials of computer experiment) is less on 10 than by using the homogenization method. The distribution of resultant displacement for plate of optimal shape (variant e) is shown in fig. 4.

Fig. 4. Displacement of plate for variant e.

#### **2.2 Stretched plate test problem**

246 Advanced Topics in Measurements

Using the obtained locally weighted polynomial approximations by global search procedure (Janushevskis et. al, 2004), implemented in EDAOpt, the optimal cutout shape is obtained (see Figure 3) for the different aforementioned techniques. In table 1 the results are summarized and compared with volume obtained in work (Liang et al., 2001) using the homogenization method. Variants correspond to shapes shown in figure 3. The value of

minimize relative leave-one-out crossvalidation error *σerr* of approximations of appropriate responses, i.e. deflection *δ* and volume *v* of plate. *vp* is the predicted volume calculated using approximations and *va* is the actual volume calculated using a geometrical model. *vp* and *va* in % show comparison of appropriate volume in respect to the volume obtained in (Liang et al., 2001). Best results are achieved with the technique using the control points of NURBS polygon. This allows reducing the volume of the plate by 1.38 % (Fig. 3 e) in comparison

Fig. 3. Shapes of plate obtained by (a) homogenization method (Liang et al., 2001); and by the current approach using different techniques: (b) the points that are connected with straight lines; (c) with the NURBS knot points; (d) with the control points of NURBS polygon; (e) same as "d" but with additionally optimized tangent weighting at the spline

> *vp mm3*

a - - - - 68750.00 - b 17 9.81 0.03 69414.78 69331.58 1 0.84 c 17 9.81 0.03 68988.67 68815.88 0.4 0.096 d 15.6 9.81 0.03 68862.32 68721.98 0.16 -0.04 e 3.2 0.79 0.16 67797.524 67800.975 -1.385 -1.38 Table 1. Quantitative indices of the shape optimization of cutout for the plate bending problem.

*va mm3*

*vp %* 

*va %* 

calculated without taking into account the *i*-th experimental point.

denotes approximated value for response in *i*-th point,

of the local quadratic polynomial approximation is chosen to

experimental trials and ( )*<sup>i</sup> <sup>i</sup> <sup>y</sup> <sup>x</sup>*

Gaussian kernel parameter

with the homogenization method.

endpoints; (f) same as "e" but with circle added.

*σerr <sup>δ</sup> %* 

*σerr <sup>v</sup> %* 

*Variant* 

A square plate with dimensions 1300x1300x0.0001 mm is considered under two axial stretching loads of p = 0.65 N/mm2 and p/2 accordingly (see Figure 6). The isotropic material properties are: the Young's modulus *E* = 210 GPa, the Poisson's ratio = 0.3. The shape optimization of the plate with constant thickness is carried out to minimize its volume in case of a single constraint on maximal level of equivalent stress *σmax* = 4.38 MPa. Due to symmetry only ¼ of the plate is considered for cutout definition and for problem solution by FEM. The cutout shape of the plate is defined by five coordinates of points 2, 3, 4, 5 and 6 situated on straight lines that make angles of 0; 22.5; 45 and 90 degrees with horizontal axis as shown in Figure 5. The initial cutout shape is a circle with radius 250 mm and the volume of plate *v* is 373.4 mm3.

Fig. 5. Scheme of ¼ of plate with initial cutout.

Shape Optimization of Mechanical Components for Measurement Systems 249

10.5 % (Fig. 6 d) in comparison with the volume obtained in (Papadrakakis et al., 1998).

Distribution of equivalent von Mises stress for case d is shown in Figure 7.

Fig. 7. Equivalent von Mises stress levels in the plate for variant d.

At the present time special tensometric wheel pairs are used for the wheel - rail system monitoring. For each type of rolling stock these wheel pairs must fit the vehicle's wheel wearing condition, diameter and bearing box connection type. Using and delivering the tensometric wheel pairs is expensive and takes a lot of time for preparing strength – dynamics tests. In this work removable equipment for monitoring is proposed for mounting on the ordinary wheel pairs. The monitoring wireless system (Grigorov, 2004; Hart, 1986) for 80 tons wagon (freight car) is taken for prototype. The movable part of the equipment (Fig. 8) consists of a removable disk, two transmitters and a transmitting antenna as well as strain gauges bonded to the wheel at defined places. The removable disk is fixedly attached to the wheel pair's axis. A circular transmitting antenna and two transmitters are mounted on the

Removable equipment must be lightweight to minimize distortions of measurements and at the same time it must be with appropriate durability. During testing dynamical loads caused by rail joints, railroad switches and other irregularities as well as due to defects of wheel geometry are transmitted from wheel pairs to the removable disk which is rigidly mounted on the wheelset axis. Therefore shape optimization of the mounting disk that is the main heavy-weight part of the equipment is very important to reduce its total weight.

**3. Tensometric wheel pairs** 

outside of the disk.

The aforementioned five parameters are varied in the range from 250 to 640 mm. The design of experiment for 5 factors and 112 trial points is calculated with MSE criterion value 0.5394. In the next step responses of these models are calculated by SW Simulation using shell elements with global size 7.5 mm and total number of DoF ~108000. Then these responses are used for approximation by EDAOpt. Using obtained locally weighted polynomial approximations by global search procedure, the optimal cutout shape is obtained (see Figure 6) for the different aforementioned techniques. In table 2 the results are summarized and compared with the volume obtained in work (Papadrakakis et al., 1998). Variants correspond to shapes shown in figure 6.

Fig. 6. The obtained plate shapes. Plate shapes obtained (a) in (Papadrakakis et al., 1998); and by current approach using different techniques: (b) the points that are connected with straight lines; (c) with the NURBS knot points; (d) with the control points of NURBS polygon.


Table 2. Quantitative indices of the shape optimization of cutout for the plate stretching problem.

The value of Gaussian kernel parameter of the local quadratic polynomial approximation is chosen to minimize relative leave-one-out crossvalidation error *σerr* of approximations of appropriate responses, i.e. maximal equivalent stress *σmax* and volume *v* of plate. *vp* is predicted volume calculated using approximations and *va* is actual volume calculated using full FEM and geometrical models. Again the best results are achieved with the technique using the control points of NURBS polygon. This allows reducing volume of the plate by

10.5 % (Fig. 6 d) in comparison with the volume obtained in (Papadrakakis et al., 1998). Distribution of equivalent von Mises stress for case d is shown in Figure 7.

Fig. 7. Equivalent von Mises stress levels in the plate for variant d.
