Robust Topology Optimization with Loading Magnitude and Direction Uncertainty Xiang Peng, Jianxiang Wang, Jiquan Li, Shaofei Jiang, Wanghui Bu, Bing Yi and Sihang Zhou
Abstract The robust topology optimization approach with uncertainties of loading magnitude and direction is investigated for continuum structures. The input loadings with uncertain magnitude and direction are decomposed using the second order Taylor series expansions, and then the response of statistical response of compliance is calculated using perturbation analysis method. The robust topology problem with minimize the compliance is solved using the modified SIMP (Solid Isotropic Material with Penalization) algorithm. The efficiency of the proposed methodology is verified using the examples of cantilever beam. Keywords Robust topology optimization Perturbation method
Loading uncertainty
X. Peng J. Wang J. Li S. Jiang (&) Key Laboratory of E&M, Zhejiang University of Technology, 310014 Hangzhou, China e-mail:
[email protected] X. Peng e-mail:
[email protected] W. Bu School of Mechanical Engineering, Tongji University, 201804 Shanghai, China B. Yi School of Traffic and Transportation Engineering, Central South University, Changsha 410027, China S. Zhou CRRC Hangzhou CO., LTD, Hangzhou 310014, China © Springer Nature Singapore Pte Ltd. 2018 J. Tan et al. (eds.), Advances in Mechanical Design, Mechanisms and Machine Science 55, https://doi.org/10.1007/978-981-10-6553-8_31
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1 Introduction Topology optimization determines the structural connectivity and material distribution in a given design domain, and has been applied to truss design [1], multi-material structural design [2], thermal design [3], micro electro-mechanical systems (MEMS) design [4], dynamics optimization design [5], multiscale composite structure design [6], additive manufacturing [7] etc. There are various sources of uncertainties in the real-world engineering application, such as loading uncertainty, material uncertainty, geometric and boundary uncertainty. Hence, robust topology optimization (RTO) techniques are used to decrease the influence of loading magnitude uncertainty, loading direction uncertainty, loading location uncertainty in continuum and truss structure. Kogiso [8] proposed a sensitivity-based robust topology optimization approach considering uncertainty of input loading direction. The robust objective function was defined as a combination of maximizing the output deformation under mean input load and minimizing variation in the output deformation. The proposed approach was validated with two compliant mechanism problems. Guest [9] transformed the truss topology optimization problem under uncertain nodal locations as optimization problem with equivalent uncertain loads, and then solved the material distribution problem using first-order elastic behavior and weighted average of multiple load patterns. Dunning [10, 11] solved a robust topology optimization problem under uncertainty of loading magnitude and applied direction. The expected compliance under these uncertainties was transformed to a total compliance under multiple load cases, and then the topological structural was optimized using the level-set method. Csébfalvi [12] introduced a framework for the worst-load-direction oriented structure topology optimization with uncertain loading directions, and used the proposed method for volume minimization topology design of 2D continuum structures and truss structures. Though many approaches of RTO have been proposed and applied to many engineering examples, a large amount of samples are used for the representation of random uncertainties, such as 107 samples are used in Reference [13], leading to complex and time-consuming quantification of the robustness of solutions. Therefore, a robust topology optimization approach considering uncertainty of loading magnitude and direction is presented using perturbation method, to increase the efficiency and decrease the calculation complexity. The objective of this study is to minimize the compliance considering the loading magnitude and direction uncertainties. The layout of this article is as follows. The perturbation-based uncertainty quantification of loadings is presented in
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Sect. 2, and the statistical response of compliance is calculated in Sect. 3. Section 4 develops an efficient robust topology optimization methodology. The examples of cantilever beam are presented to demonstrate the proposed approach in Sect. 5, followed by concluding remarks in Sect. 6.
2 Perturbation Analysis of Loading Uncertainties The general 2D topology optimization problem is considered to determine the optimum distribution of the element density q that minimizes the compliance C which is subjected to the volume constraint V Vmax : The optimization function is formally written as follows: Min C ðqÞ ¼ FT dðqÞ s:t: KðqÞdðqÞ ¼ F; n P qe ve Vmax ;
ð1Þ
e¼1
0\qmin qe qmax ;
where F is the input loading vector, d is the displacement vector, K is the global stiffness matrix, n is the total number of elements, qe is the density of element e; ve is the volume of the element e; Vmax is the maximization of the material volume, qmax is the upper bound for the element density qe ; and qmin ¼ 103 is the lower bound for preventing the singularity of the equilibrium problem. In the 2D topology optimization problem, the input loading vector F¼½F1 ; F2 ; . . .; F2n includes the horizontal directional loading F2i1 and vertical directional loading F2i ; which are functions of the magnitude l2i1 ¼ l2i and direction h2i1 ¼ h2i at i-th element ði ¼ 1; . . .; nÞ: They are given in Eq. (2). F2i1 ¼ l2i1 cos h2i1 F2i ¼ l2i sin h2i
ð2Þ
The uncertain magnitude li and direction hi are composed of the deterministic parts l0i ; h0i and zero mean random fluctuation parts Dli ; Dhi ; respectively, Using the perturbation method, the uncertain input loading vector F with uncertain magnitude vector l and direction vector h is replaced by their second order Taylor series expansions. Fi ðli ; hi Þ f0;i þ f1;i Dli þ f2;i Dhi þ f3;i ðDli Þ2 þ f4;i Dli Dhi þ f5;i ðDhi Þ2 ;
ð3Þ
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where
f0;i ¼ Fi ðl0i ; h0i Þ;
f2;i
@Fi li ¼l0i ; ¼ @hi hi ¼h0i
f4;i ¼
@ 2 Fi li ¼l0i ; @li @hi hi ¼h0i
f1;i
@Fi ; ¼ @li li ¼ l0i h ¼h i 0i
f3;i
1 @ 2 Fi li ¼l0i ; ¼ 2 @l2i hi ¼h0i
f5;i ¼
1 @ 2 Fi li ¼l0i : 2 @h2i hi ¼h0i
When only uncertainties of input loading are considered, the material properties and element nodes are determinate, so there are no uncertainties in the total stiffness matrix K: The finite-element equilibrium equation is written as follows. Kdðl; hÞ ¼ Fðl; hÞ:
ð4Þ
The series expansions for the displacement vector dðl; hÞ to the second order would be written: di ðli ; hi Þ d0;i þ d1;i Dli þ d2;i Dhi þ d3;i ðDli Þ2 þ d4;i Dli Dhi þ d5;i ðDhi Þ2 ;
ð5Þ
Substituting Eqs. (3) and (5) into Eq. (4), through comparing coefficients of different orders of Dl and Dh; the Taylor series of displacement vector d ðl; hÞ can be solved as Eq. (6) di ¼ K1 f i ði ¼ 0; . . .; 5Þ:
ð6Þ
3 Response of Statistics Calculation of Compliance Having computed the second order statistics of the input loading vector Fðl; hÞ and displacement vector dðl; hÞ; the compliance C is a function of uncertain parameter Dl and Dh:
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C ¼ Fðl; hÞT dðl; hÞ;
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ð7Þ
The expected mean value of compliance E ðCÞ is a combination of compliance under auxiliary loading Ai and displacement di ði ¼ 0; . . .; 5Þ; which is shown in Eq. (8). EðC Þ ¼ AT0 d0 þ AT1 d1 þ AT2 d2 þ AT3 d3 þ AT4 d4 þ AT5 d5 ; h i h i A0 ¼ f 0 þ f 3 E ðDlÞ2 þ f 5 E ðDhÞ2 ; h i h i A1 ¼ f 1 E ðDlÞ2 þ f 3 E ðDlÞ3 ; h i h i A2 ¼ f 2 E ðDhÞ2 þ f 5 E ðDhÞ3 ; h i h i A3 ¼ f 0 E ðDlÞ2 þ f 1 E ðDlÞ3 h i h i h i þ f 3 E ðDlÞ4 þ f 5 E ðDlÞ2 E ðDhÞ2 ; h i h i A4 ¼ f 4 E ðDlÞ2 E ðDhÞ2 ; h i h i A5 ¼ f 0 E ðDhÞ2 þ f 2 E ðDhÞ3 h i h i h i þ f 3 E ðDlÞ2 E ðDhÞ2 þ f 5 E ðDhÞ4 :
ð8Þ
ð9Þ
Using a similar method, the variance of compliance VarðC Þ is also the function of uncertain parameters Dl and Dh: After the variance calculation of high order function of Dl and Dh; VarðCÞ is calculated as follows. VarðC Þ ¼ dT0 B0 d0 þ dT1 B1 d1 þ dT2 B2 d2 þ dT3 B3 d3 þ dT4 B4 d4 þ dT5 B5 d5 ;
ð10Þ
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B0 ¼ f 1
Var½Dlf T1
Var½Dhf T2
þ f2 h i þ f 3 Var ðDlÞ2 f T3 þ f 4 Var½DlDhf T4 h i þ f 5 Var ðDhÞ2 f T5 h i B1 ¼ f 0 Var½Dlf T0 þ f 1 Var ðDlÞ2 f T1 h i þ f 2 Var½Dl Dhf T2 þ f 3 Var ðDlÞ3 f T3 h i h i þ f 4 Var ðDlÞ2 Dh f T4 þ f 5 Var Dl ðDhÞ2 f T5 ; B2 ¼ f 0 Var½Dhf T0 þ f 1 Var½Dl Dhf T1 h i h i þ f 2 Var ðDhÞ2 f T2 þ f 3 Var ðDlÞ2 Dh f T3 h i h i þ f 4 Var Dl ðDhÞ2 f T4 þ f 5 Var ðDhÞ3 f T5 ; h i h i B3 ¼ f 0 Var ðDlÞ2 f T0 þ f 1 Var ðDlÞ3 f T1 h i h i þ f 2 Var ðDlÞ2 Dh f T2 þ f 3 Var ðDlÞ4 f T3 h i h i þ f 4 Var ðDlÞ3 Dh f T4 þ f 5 Var ðDlÞ2 ðDhÞ2 f T5 ; h i B4 ¼ f 0 Var½Dl Dhf T0 þ f 1 Var ðDlÞ2 Dh f T1 h i h i 2 T 3 T þ f 2 f 2 þ f 2 2 Var Dl ðDhÞ 3 Var ðDlÞ Dh f 3 h i h i þ f 4 Var ðDl DhÞ2 f T4 þ f 5 Var Dl ðDhÞ3 f T5 ; h i h i B5 ¼ f 0 Var ðDhÞ2 f T0 þ f 1 Var Dl ðDhÞ2 f T1 h i h i 2 2 T þ f 22 Var ðDhÞ3 f T2 þ f 2 Var ð Dl Þ ð Dh Þ f3 3 h i h i þ f 4 Var Dl ðDhÞ3 f T4 þ f 5 Var ðDhÞ4 f T5 :
ð11Þ
4 Robust Topology Optimization of Compliance Using the perturbation analysis method, the mean E ðC Þ and variance VarðCÞ of compliance considering loading uncertainties are calculated using Eq. (8) and (10), respectively. After solving the Taylor expansion of input loading F and displacement d in Eqs. (3)–(6), the constraint Kd ¼ F is decomposed into Kdi ¼ Fi ði ¼ 0; . . .; 5Þ: The robust topology optimization function considering loading uncertainties, which can be described in Eq. (12).
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^ ð qÞ ¼ E ð C Þ þ a min C q
s:t: n P
KðqÞdi ðqÞ ¼ f i q v Vmax e e
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pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi VarðC Þ ði ¼ 0; ; 5Þ
ð12Þ
e¼1
0\qmin qe qmax ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where a is the weighting coefficient to balance E ðCÞ and VarðC Þ: It sets to be a ¼ 3 in the following examples according to Ref [13]. The influence of a on the optimum solution will be researched in the future. After the robust optimization function and sensitivity analysis are performed, the sensitivity analysis between the material density and robust design objective is implemented using the Lagrange multiplier method. The material densities are characterized using SIMP method [14] and updated using the Method of Moving Asymptotes [15]. The steps are summarized as follows: Step 1 Uncertain perturbation decomposition of input loading. The uncertain input loading vector F is decomposed into the second Taylor expansion of uncertain magnitude l and direction h using Eq. (3). Then the uncertain components f 0 ; f 1 ; f 2 ; f 3 ; f 4 and f 5 are solved. Step 2 Statistic analysis of uncertain magnitudes and directions. The expected mean E½ðDlÞm ; E½ðDhÞm and variance Var½ðDlÞm ; Var½ðDhÞm are computed based on the specified probability distribution function of magnitude l and direction h; respectively. h i hLet Dl and iDh be mutually
independent, then E ðDlÞm ðDhÞl and Var ðDlÞm ðDhÞl are calculated.
Step 3 Construction of the response statistics function of compliance. The response mean function E ðC Þ and variance function VarðCÞ of compliance are constructed using Eqs. (8) and (10), respectively. The coefficients A0 A5 and B0 B5 are analyzed based on magnitude and directional uncertainty of input loadings, and then calculated using Eqs. (9) and (11), respectively. Step 4 Initial material distribution. Start with an initial guess of material density distribution q: The homogeneous distribution of material is assumed, and the density for all elements is qe ¼ V0 ðe ¼ 1; . . .; nÞ; where V0 is the maximum volume ratio. Step 5 Finite element analysis. The global stiffness matrix K is constructed based on the element density q and element stiffness matrix K0 : The finite element equations Kdi ¼ f i ði ¼ 0; . . .; 5Þ are solved, and the auxiliary displacements di ði ¼ 0; . . .; 5Þ are calculated. Step 6 Calculation of compliance. The statistical mean EðC Þ and variance VarðCÞ of compliance are calculated using Eqs. (8) and (10), respectively. Step 7 Sensitivity calculation. The sensitivity of the robust objective function is calculated using Lagrange multipliers and the ad-joint method.
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Step 8 Update material density variable q using Method of Moving Asymptotes method. Step 9 Converge analysis. If the difference between the original density distribution in Step 4 and the updated density distribution in Step 9 is higher than 0.01, Step 4–Step 9 are repeated using the updated density distribution.
5 Topology Optimization of a Cantilever Beam The examples of the cantilever beam are used to verify the influence of loading magnitude and direction uncertainty on the optimum topological structure. The codes are developed through extending the standard 88 line code [16] of determinate topology optimization using Matlab® R2015a. The Young’s moduli of 1 GPa and 109 GPa are used for solid and void materials, respectively, Poisson ratio of 0.3 is used for all materials. Firstly, a qualitative comparison of cantilever beam topology design is carried out with some equivalent results from Ref [17] to demonstrate the efficiency of the presented algorithm. The length and width of the beam shown in Fig. 1 is 68 mm and 42.5 mm, respectively. The mean magnitude of F is 1 kN with standard deviation of 0.1 kN, and there is no directional uncertainty. Considering the maximize volume ratio V0 ¼ 0:3 and V0 ¼ 0:4; respectively, the optimum topologies are shown in Fig. 2a, b which are almost the same as results in Ref [17]. The deterministic optimum structure is obtained using the standard 88 line code [16]. Through comparing the deterministic solution and the proposed solution, the compliance under V0 ¼ 0:3 decreases from 80.02 to 74.02 kN mm, and the compliance under V0 ¼ 0:4 decreases from 54.23 to 51.89 kN mm. These results display the correctness of the proposed method under input loading with uncertain magnitude. Fig. 1 Design domain and boundary condition of beam1
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Fig. 2 Topology optimum structure of beam1 under different volume ratio
(a) V0 = 0.3
(b) V0 = 0.4
Fig. 3 Design domain and boundary conditions of beam2
Then, the proposed method is used for beam topology design with uncertainties of loading magnitude and direction. The design domain and boundary are shown in Fig. 3. The length and width are 120 mm and 40 mm, respectively. The magnitude of F is normal distribution with mean ll ¼ 1 kN and standard deviation rl ¼ 0:1 kN, and the direction of F is normal distribution with mean lh ¼ p=2 and standard deviation rh ¼ p=12: The design domain is meshed with 120 40 elements, and the filter radius is set to rmin ¼ 3: The optimum topological structures under volume ratio 0.3, 0.4, 0.5, 0.6 are shown in Fig. 4. Results show the proposed methodology is effective for robust topology optimization under both magnitude and directional uncertainties simultaneously. As the increase of volume ratio, the main construction of the optimum structure is almost the same, the thickness of the steadying bar increases, and some tiny structures appear. The optimum compliance of the proposed method and deterministic method under volume ration V0 = 0.3– 0.6 are plotted in Fig. 5. Results show that the expected compliance is effectively decreased under uncertain concentrated loading.
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Fig. 4 Topology optimum structure of beam2 under different volume ratio
(a) V0 = 0.3
(b) V0 = 0.4
(c) V0 = 0.5
(d) V0 = 0.6 Fig. 5 Compliance comparison under different volume ratio
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6 Conclusions The robust topology optimization design methodology with loading magnitude and direction uncertainty is proposed. The input loadings with uncertain magnitude and direction are decomposed using the second order Taylor series expansions, and then the uncertain compliance is calculated directly using an auxiliary certain loading and displacement with perturbation. The optimum topological structures are calculated using the modified SIMP method. Results of cantilever beam show that the proposed method can improve the performance under uncertain loading magnitude and direction. However, there are some limitations of the proposed method. (1) The proposed method can be used for known probability distribution of loading, but it cannot be used for RTO under bounded uncertainty of loading. (2) The proposed method can be used only under loading magnitude and direction uncertainty, but it cannot be used for uncertainty of material and loading location simultaneously. Acknowledgements This work was supported by the National Natural Science Foundation of China under grant [number No. 51505421, 51375451, U1610112], and Open Project Program of the State Key Lab of CAD&CG under grant [number: A1716].
References 1. Savsani VJ, Tejani GG, Patel VK. Truss topology optimization with static and dynamic constraints using modified subpopulation teaching–learning-based optimization. Eng Optim. 2016;48(11):1990–2006. 2. Park J, Sutrandhar A. A multi-resolution method for 3D multi-material topology optimization. Comput Methods Appl Mech Eng. 2015;285:571–86. 3. Vicente WM, Picelli R, Pavanello R, et al. Topology optimization of frequency responses of fluid–structure interaction systems. Finite Element Anal Design. 2015;98:1–13. 4. Ramirez-Gil FJ, Silva ECN, Montealegre-Rubio W. Topology optimization design of 3D electrothermomechanical actuators by using GPU as a co-processor. Comput Methods Appl Mech Eng. 2016;302:44–69. 5. Kim BJ, Yun DK, Lee SH, et al. Topology optimization of industrial robots for system-level stiffness maximization by using part-level metamodels. Struct Multidiscip Optim. 2016;54 (4):1061–71. 6. Coelho PG, Guedes JM, Rodrigues HC. Multiscale topology optimization of bi-material laminated composite structures. Compos Struct. 2015;132:495–505. 7. Smith CJ, Gilbert M, Todd I, et al. Application of layout optimization to the design of additively manufactured metallic components. Struct Multidiscip Optim. 2016;54(5):1297– 313. 8. Kogiso N. Robust topology optimization for compliant mechanisms considering uncertainty of applied loads. J Adv Mech Design Syst Manuf. 2008;2(1):96–107. 9. Guest JK, Igusa A. Structural optimization under uncertain loads and nodal locations. Comput Methods Appl Mech Eng. 2008;198(1):116–24. 10. Dunning PD, Kim HA, Mullineux G. Introducing loading uncertainty in topology optimization. AIAA J. 2011;49(4):760–8.
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11. Dunning PD, Kim HA. Robust topology optimization: minimization of expected and variance of compliance. AIAA J. 2013;51(11):2656–64. 12. Csebfalvi A. Structural optimization under uncertainty in loading directions: benchmark results. Adv Eng Softw. 2016;000:1–11. 13. Zhao J, Wang C. Robust topology optimization under loading uncertainty based on linear elastic theory and orthogonal diagonalization of symmetric matrices. Comput Methods Appl Mech Eng. 2014;273:204–18. 14. Bendsoe MP, Sigmund O. Topology optimization: theory, methods, and applications. Berlin: Springer Science & Business Media; 2013. 15. Svanberg K. The method of moving asymptotes—A new method for structural optimization. Int J Numer Meth Eng. 1987;24:359–73. 16. Andreassen E, Clausen A, Schevenels M, et al. Efficient topology optimization in MATLAB using 88 lines of code. Struct Multidiscip Optim. 2011;43(1):1–16. 17. Richatdson JN, Coelho RF, Adriaenssens S. A unified stochastic framework for robust topology optimization of continuum and truss-like structures. Eng Optim. 2016;48(2):334–50.
Author Biographies Xiang Peng born in 1989, is currently a lecturer at College of Mechanical Engineering, Zhejiang University of Technology, China. He received his Ph.D. degree from Zhejiang University, China, in 2014. His research interest is design methodology of complex product. Shaofei Jiang born in 1975, is currently a professor at College of Mechanical Engineering, Zhejiang University of Technology, China. He received his Ph.D. degree from Zhejiang University, China, in 2003. His research interests are design methodology, precision mold manufacturing.
