Effect of Modeling Parameters in SIMP Based Stress Constrained Structural Topology Optimization

Transcription

1 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:17 No:06 32 Effect of Modeling arameters in SIM Based Stress Constrained Structural Topology Optimization Hailu Shimels Gebremedhen 1,a,Dereje Engida Woldemichael 1,b* and Fakhruldin Mohd Hashim 1,c 1 Universiti Teknologi ETRONAS, Seri Iskandar, erak Darul Ridzuan, Malaysia a hshimels278@gmail.com, b* dereje.woldemichael@utp.edu.my and c fakhruldin_mhashim@utp.edu.my Abstract This paper presents the effect of modeling parameters, namely minimum filtering radius () and penalization factor (enal), on the computational efficiency (iteration number), maximum stress induced, and optimal layouts of SIM based stress constrained topology optimization. Matlab was used to generate optimal topologies and output parameters for the feasible region of modeling parameters. Response surface methodology using MINITAB 14.1 statistical software was used to analyze combined effect of these parameters. The simulation results show variations in penalization factor and minimum filtering radius has a significant effect on the number of iteration to converge and optimal plot. The effect of these modeling parameters on maximum stress induced and weight percentage reduction is insignificant compared to their effect on iteration number and optimal material distribution. The results also showed that the combination of these parameters in their upper range (1.7< <3 and enal >3) is the best option while considering iteration number, maximum stress induced and optimal material plots as an output of the optimization problem. Based on the numerical result and statistical analysis, the computational time which is associated with iteration number can be reduced with careful selection of modeling parameters Index Term Topology optimization; stress constraints; penalization factor; filtering radius; SIM method. I. INTRODUCTION Topology optimization is a mathematical approach which seeks optimal material distribution within a given design domain to sustain applied load under specific boundary conditions. It includes determination of connectivity, geometries of cavities and location of voids in the design domain. Topology optimization has a great implication in the conceptual design stage where various modifications are made and 80% of the cost of a given product/design is determined [1]. The changes in the design at the conceptual design stage affect the performance and manufacturability of the final structure. Unlike size and shape optimization, in topology optimization the number and location of void shapes and solid elements are not known prior to the optimization process. This gives the designer more freedom to distribute the material optimally within the design domain. The definition of any topology optimization includes selection of design variables and formulation of objective and constraint functions. Different formulation approaches have been suggested for formulating and solving topology optimization problem. Among problem formulation approaches the SIM (solid Isotropic Material with enalization) approach is common due to its conceptual simplicity and high computational efficiency [2]. Most of the researches related to structural topology optimization are focused on formulating and solving compliance minimization problems [2-4]. Though this approach becomes more popular, there are few challenges including variation of results with the amount of material distributed, unable to consider stress and displacement in the optimization process which may lead the results to be infeasible in the real world applications [2]. For engineering problems where ductile materials are considered stress is a major failure criterion. Structural topology optimization problems have been formulated and solved to include this parameter [5-8]. Though formulating this type of problems is more realistic compared to compliance based formulations, it has some challenges associated with the stress constraints, namely local nature of stress constraints, singularity phenomenon associated with void materials and high nonlinear dependence of stress constraint. Distinct methods have been proposed to address these challenges including; global constraint, block aggregation[9] and enhanced aggregation[6] to alleviate local nature of stress constraints, and different relaxation techniques to address singularity issue [10]. One of the challenges associated with the local nature of stress constraints is higher computational time due to stress evaluation for elements in the design domain. To address this limitation, more emphasis was given to grouping stress constraints and determination of parameters in aggregation function. In modeling stress based topology optimization problems using SIM method, there are two major parameters namely penalization factor and minimum filtering radius (R min). The values of these parameters were adopted from compliance based formulations. enalization factor, commonly taken as 3, is introduced to penalize design domain variables into solid/void element variables. Minimum filtering radius, commonly taken as 1.2 or 1.5, is usually considered to remove checkerboard effect and mesh dependency effects on optimal topologies [3, 11-13]. There is no study so far which investigates the effect of these modeling parameters in stress based topology optimization. This paper aims to study the effect of these modeling parameters on generated optimal topologies, induced maximum stress, weight percentage

2 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:17 No:06 33 reduction and iteration number to converge for a given design domain. II. ROBLEM FORMULATION A stress constrained topology optimization to minimize weight of a structure using SIM method based on Von Mises failure theory can be formulated as shown in (1). N minv ( ) ( ) v e e e e 1 subjected to, vm Y 1 KU F 0 1 Where V is volume of a structure to be minimized, N is total number of elements within the design domain, design variable, v e is elemental volume, and are induced stress and yield stress, respectively. K, U and F are global stiffness matrix, global displacement and force vector, respectively. σvm In von Mises stress failure theory, it is assumed that a material will fail when the von Mises stress induced in the material exceeds the yield strength of a material as shown in (2). (1) e σy is 1 v 0 E D 0 v 1 0 (5) 2 1 v 1 v Where, v is the oisson s ratio of an isotropic material and E is a young s modulus of solid material which can be related to the young s modulus of base material using SIM method as shown in (6). E E 0 x Where E 0 is the young s modulus of base material, penalization factor ( x) B e u e (7) Substituting (4) and (7) into (3) the stress at any material point with the given design domain can be expressed as shown in (8). p q (x) x Do B e u e (8) From the above derivations and relations, stress based topology optimization problem defined in (1), can be expressed as shown in (9). N minv ( ) ( ) v e e e e 1 (6) (9) [ ( ) 3 12 ] vm (2) 2 In order to have a full controll on the stress measure in each element in the design domain, all the stress constraints are defined at element level using an interpolation proposed by Duysinx and Bendsoe [14] as shown in (3). D e ( x) ( x) (x) q (3) x Where, (x) is local stress at a material point, D e(x) is macroscopic elastic tensor which can be related with the constitutive elasticity tensor D o by a power law approach as shown in (4), is the average strain of a material point which can be expressed in terms of strain displacement matrix B e (x) and elemental displacement vector as shown in (7). The exponent q 1 is a constant to preserve physical consistency in the modeling of a porous SIM material model. p De ( x) x Do (4) Where D 0 is the constitutive matrix with a unit Young s modulus. The unit constitutive matrix is given by: u e subjected to pq x D 0 B e u e Y KU F 0 1 III. NUMERICAL RESULT 1 To analyse the effect of formualtion parameters for the problem defined in Eq.9, different benchmark problems are considered. All the deisgn domains considered are disctritized by a square rectangular finite element. The material cosiderd for all cases considered in this paper has a Young s modulus of E 0 = 1Ma, a oison s ratio of v = 0.3 and a von Mises stress of 1.2Mpa subjected to a unit load [15]. Default values for penalization factor and minimum filtering raius are 3 and 1.2, respectively. Individual effect of these paramters on optimal plots and iteration number for convergence was studied through developing a Matlab code using optimality criteria method as a slover. A range of penalization factor and minimum filtering radius listed in Table 1 were considered for analyzing effect of values of these modeling parameters. Matlab is used to simulate the model developed. Once the feasible range for the paramters was determined a surface

3 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:17 No:06 34 methodology was used to analayze combined effect of the paramaters on different output paramaters. Table I RANGE OF MODELING ARAMETERS enal R min TABLE II EFFECT OF VARIATION OF ENAL-VALUES ON OTIMAL LOTS enal= 5.0 ena l=4.0 The response surface methodology and Minitab statistical software is used to develop a regression model to predict number of iteration, maximum stress induced and weight percentage reduction within given benchmark design domains. Analysis of variance was used to evaluate and test the significance of the developed models. Using the developed models, a contour and 3D surface plots were plotted to investigate the interactions of modelling parameters on the response and find out the minimum number of iteration with a reasonable percentage weight reduction and induced maximum stress. Coefficient of determination R 2, an indicator on how close the data are to the fitted regression line, is usually taken 0.8 and above. A 95% confidence level () which gives α value 0.05, was used to obtain the -value to identify significant and insignificant modelling parameters for the model. Usually the parameter with -value less than α value is considered as significant in the model [16]. enal= 4.5 enal= 2.5 ena l=3.5 ena l=3.0 The effect of values for penalization factor on iteration number is plotted in Fig.3, which shows the default value (enal=3) and enal=4.5 have less iteration number. From the perspective of iteration number, optimal plot and maximum stress, enal=4.5 is the best alternative for the defined design domain for this specific case study. A. cantilever Beam with predefined shape In this section, effect of penalization factor and minimum filtering radius will be discussed. A classical cantilever beam with predefined shape with loading and boundary conditions discretized into 120 by 60 rectangular elements as shown in Fig.1 was used to analyze the effect of these parameters. Fig. 2. Effect of enal values on maximum induced stress Fig. 1. Boundary and Loading condition Fig.2 shows variation of induced maximum stress for the range of penalization power considered. Considering the value of the penalization factor greater than the default value (3), the maximum stress induced is less than that of the default value and the optimal material distribution plot is less complex as shown in Fig.2 and Table 2, respectively. From the simulation result it was difficult to generate optimal topologies using the lower range of penalization factor. Generated optimal plots are less complex when penalization factor greater than the default values are used. Fig. 3. Effect of enal - Values on Number of Iteration The number of iteration to converge decreases for R min values in the upper range (R min >1.2) as shown in Fig.4. Even if the iteration number decreases in the upper range of R min value the optimal material distribution is full of transition elements, which does not represent any material as shown in Table.3. This leads the manufacturing of optimal layouts difficult. The R min value between can be taken as the best range for this case study. Fig.5 shows variation of induced maximum

4 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:17 No:06 35 stress for the range of R min values considered. The maximum stress induced in the design domain has minimum value for higher range including the default value of R min value. =6.0 =10.0 Fig. 4. Effect of values on number of iteration =0.5 =0.9 TABLE III EFFECT OF RMINVALUE ON OTIMAL LOTS =1.0 =1.2 Fig. 5. Effect of values on induced stress B. Simply supported beam The second case study considered was simply supported beam discretized by 240x60 rectangular finite elements under loading and boundary conditions defined as shown in Fig.6. The same procedure and methodology used to solve the previous case study was followed to solve this case study. =1.5 =1.7 =1.9 =2.3 =2.1 =2.5 Fig. 6. Boundary and Loading condition Fig.7 shows effect of variation of penalization factor on induced maximum stress. From the figure, the maximum stress induced in the design domain is less when the penalization factor is in the upper range. The optimization is divergent when the penalization factor in the lower range (<3) is considered. The optimal plots are less complex when the penalization factor in the upper range (>3) are considered. =3.0 =4.0 Fig. 7. Effect of penal values on Max stress induced

5 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:17 No:06 36 Fig.8. shows effect of variation of R min on maximum induced stress. Fig. 9 shows effect of variation of R min value on the iteration number. From the figure, the iteration number is less when the R min value is taken in the lower and upper range. Even if the lower range of R min yields less number of iteration the optimal plots from this range are full of transition elements as shown in Table.4. Therefore, the feasible region for this design domain lays in the upper range ( ). =0.9 =1.0 =1.9 =2.1 =1.2 Fig. 8. Effect of on Max stress induced Once the feasible range of penalization factor and minimum filtering radius is found out combined effect of these parameters is analysed using Response surface methodology Table 5, Table 6 and Table 7 present the coefficient of iteration number, weight reduction percentage and maximum stress induced along with the significant of linear, quadratic and the interaction of terms of modelling parameters, respectively using numerical data from case study one. Table V Model coefficient and Analysis of variance for number of iteration taken to converge Estimated Regression Coefficients For Iter Term Estimated regression coefficient Constant enal Fig. 9. Effect of R min values on Max stress induced TABLE IV EFFECT OF RMIN VALUES ON OTIMAL LOTS =0.5 =1.5 * enal*enal *enal R-Sq 94.6% R-Sq(adj) 84.8% Analysis of Variance Source =0.7 =1.7 Regression Linear 0.25 Square 0.05 Interaction Lack-of-Fit 0.06

6 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:17 No:06 37 Table VI Model coefficient and Analysis of variance for weight reduction percentage Estimated Regression Coefficients For Weight reduction % Term Estimated regression coefficient Constant enal * enal*enal *enal 0 1 R-Sq 84.8% R-Sq(adj) 76.7% Source Analysis of Variance Regression Linear 0.07 Square Interaction 1 Lack-of-Fit 0.08 Fig.10 (a), (b) and (c) shows the normal probability plot of residuals for maximum stress induced, weight reduction percentage and iteration number taken to converge, respectively. Since the residuals for the three responses were approximately distributed along a straight line, the normality assumption is satisfied. Thus, the responses can be taken as normally distributed. Therefore, the proposed models can be used to predict maximum stress induced, weight reduction percentage and iteration number taken to converge. (a) Table VII Model coefficient and Analysis of variance for maximum stress induced Estimated Regression Coefficients For max stress Term Estimated regression coefficient Constant enal * enal*enal *enal R-Sq 96.5% R-Sq(adj) 94.5% Source Analysis of Variance Regression 0 Linear 0 Square 0 Interaction Lack-of-Fit 0.45 (c) Fig. 10. (a) Normal probability plot of the residuals (a) maximum stress induced, (b) weight reduction percentage and (c) iteration number taken to converge (b)

7 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:17 No:06 38 Fig. 11 (a) and (b) shows 3D surface and 2D contour plots of the combined effect of penalization factor and minimum filtering radius on iteration number taken to converge using numerical data from the first case study. From the figure, the number of iteration taken for convergence can be altered for different combination of penalization factor and minimum filtering radius. Combining the common value of the penalization factor to the lower and upper range of minimum filtering radius leads to convergence of the optimization problem faster. The common minimum filtering radius leads to better convergence when it is combine with those penalization factors in the upper range Fig. 12 (a) and (b) shows 3D surface and 2D contour plots of the interactive effect of penalization factor and minimum filtering radius on iteration number taken to converge (computational time). Combining the default value of the penalization factor to the lower and upper range of minimum filtering radius leads to convergence of the optimization problem faster. The common minimum filtering radius leads to better convergence when it is combined with those penalization factors in the lower range. (a) (a) (b) Fig. 11. (a) 3D surface plot and (b) 2D contour plot for iteration number The weight percentage reduction and maximum stress induced for the design domain considered falls between % and % and 0.69 and 0.73, respectively. This shows that the significance of two modelling parameters on the weight reduction percentage and maximum stress induced is less compared to the effect of parameters on convergence. Similar methodology and procedure has been followed for the second case study and the following results were obtained: The value of R 2 for iteration number, weight percentage reduction and maximum stress induced is 94.1%, 81.1% and 96.5%, respectively. This shows the proposed model fits well for all response parameters. (b) Fig. 12. (a) 3D surface plot and (b) 2D contour plot for weight reduction percentage The weight percentage reduction and maximum stress induced for the design domain considered falls between 38.25% and 40.25% and and 0.679, respectively. This show that the significance of two modelling parameters on the weight reduction percentage and maximum stress induced is less compared to the effect of parameters on convergence. But here the significance of modelling parameters on weight percentage reduction is higher than the first case study. The results from simulation and response surface methodology shows careful selection of modelling parameters has significance effect on the computational efficiency of stress based topology optimization. The results also show, variations of these parameters have higher significance on computational time compared to other output parameters. The feasible range for minimum filtering radius falls between 1.7 and 2.5, those values less than the default values yields optimal material distribution highly affected by checkerboard effect. Those

8 International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:17 No:06 39 values greater than the default value yields optimal plots full of transition elements which will be a challenge in the manufacturing of optimal plots since it does not represent any material. The feasible range for the penalization factor falls in the upper range (>3) from maximum stress induced, optimal materials distribution and iteration number point of view. Combination of these parameters in their upper range is the best option while considering iteration number, maximum stress induced and optimal material plots as an output of the optimization problem. IV. CONCLUSION In this paper, effect of modeling parameter (penalization factor and minimum filtering radius) while modeling stress based topology optimization using SIM method for benchmark problems is numerically investigated. The numerical experiment indicates without major variation of maximum stress induced and weight reduction percentage, the computational efficiency can be reduced and optimal material distribution can be enhanced. From the simulation results of the case studies defined and solved the following conclusions can be drawn: Even if the simulation converges faster lower range values of minimum filtering radius ( <1.2) are not feasible solutions due to high checkerboard effect. >3 leads to topologies full of transition elements which makes manufacturing of optimal topologies difficult. The lower range penalization factor (<3) leads the optimization be divergent. Numerical investigation shows combining the penalization factor and minimum filtering radius in the above range is preferable to reduce computational time and generate optimal plots. The recommended combination of minimum filtering radius and penalization factor based on the numerical investigation falls between and , respectively. In this paper, stress constraints are defined and evaluated at element level. A future work will be carried out with global definition of stress constraints, aggregation of stress constraints and other currently available proposed solutions to enhance computational efficiency of stress based topology optimization. ACKNOWLEDGMENT The authors acknowledge Universiti Teknologi ETRONAS for the financial support to produce this paper. The work is partially supported by Ministry of Higher Education (MOHE) Malaysia under FRGS grant FRGS /1/2014/TK01/UT/02/1. environment. Structural and Multidisciplinary Optimization, (3): p [5] Cai, S. and W. Zhang, Stress constrained topology optimization with free-form design domains. Computer Methods in Applied Mechanics and Engineering, : p [6] Luo, Y., M.Y. Wang, and Z. Kang, An enhanced aggregation method for topology optimization with local stress constraints. 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