Discrete Multi-material Topology Optimization under Total Mass Constraint

Transcription

1 Discrete Multi-material Topology Optimization under Total Mass Constraint Xingtong Yang Ming Li State Key Laboratory of CAD&CG, Zhejiang University Solid and Physical Modeling Bilbao, June 12, 2018

2 Problem statement Overview Problem statement Multi-material optimization under total mass constraint Discrete design domain: Ω = {Ω e, e = 1... N}; Candidate materials: (E j, ρ j ), j = 1... m. E 1... E m and ρ 1... ρ m of Young s modulus E i and density ρ i ; Material distribution: x = {x e}, x e = {x ej }, e = 1,..., N, j = 1,..., m, m j=1 x ej = 1. Classic compliance minimization problem min x s.t. c(x) = 1 2 ut K(x)u K(x)u = f, u U M(x) M. Xingtong Yang, Ming Li (ZJU) Discrete Multi-material TO SPM 2018, Bilbao 1 / 25

3 Problem statement Overview Problem statement Multi-material optimization under total mass constraint Discrete design domain: Ω = {Ω e, e = 1... N}; Candidate materials: (E j, ρ j ), j = 1... m. E 1... E m and ρ 1... ρ m of Young s modulus E i and density ρ i ; Material distribution: x = {x e}, x e = {x ej }, e = 1,..., N, j = 1,..., m, m j=1 x ej = 1. Classic compliance minimization problem min x s.t. c(x) = 1 2 ut K(x)u K(x)u = f, u U M(x) M. Xingtong Yang, Ming Li (ZJU) Discrete Multi-material TO SPM 2018, Bilbao 1 / 25

4 Challenges Overview Problem statement Challenges: Multi-materials, total mass constraint, discrete variable Multiple materials VS solid-void material Larger design space, difficult convergence control and material representation Total mass constraint VS single material mass constraint The incompressibility constraint: an additional usage on the summation of the total materials Discrete variable VS continuous variable Essentially of combinatorial complexity Xingtong Yang, Ming Li (ZJU) Discrete Multi-material TO SPM 2018, Bilbao 2 / 25

5 Challenges Overview Problem statement Challenges: Multi-materials, total mass constraint, discrete variable Multiple materials VS solid-void material Larger design space, difficult convergence control and material representation Total mass constraint VS single material mass constraint The incompressibility constraint: an additional usage on the summation of the total materials Discrete variable VS continuous variable Essentially of combinatorial complexity Xingtong Yang, Ming Li (ZJU) Discrete Multi-material TO SPM 2018, Bilbao 2 / 25

6 Challenges Overview Problem statement Challenges: Multi-materials, total mass constraint, discrete variable Multiple materials VS solid-void material Larger design space, difficult convergence control and material representation Total mass constraint VS single material mass constraint The incompressibility constraint: an additional usage on the summation of the total materials Discrete variable VS continuous variable Essentially of combinatorial complexity Xingtong Yang, Ming Li (ZJU) Discrete Multi-material TO SPM 2018, Bilbao 2 / 25

7 Overview Problem statement Overall approach Key observation Material density update only depends on structural compliance, once the quantity of each material is specified. Approach overview 1: Evolutionary mass reduction at total mass M i 2: Exploring density x k at total mass constraint M i // smooth transition 3: Update x k to x k+1 based on elemental compliance ranking Xingtong Yang, Ming Li (ZJU) Discrete Multi-material TO SPM 2018, Bilbao 3 / 25

8 Overview Problem statement Overall approach Key observation Material density update only depends on structural compliance, once the quantity of each material is specified. Approach overview 1: Evolutionary mass reduction at total mass M i 2: Exploring density x k at total mass constraint M i // smooth transition 3: Update x k to x k+1 based on elemental compliance ranking Xingtong Yang, Ming Li (ZJU) Discrete Multi-material TO SPM 2018, Bilbao 3 / 25

9 Theory Update Theory: update only depends on compliance ranking Lemma 1 Let x M 1 be an optimal solution to the original problem under mass bound M 1, and M 2 be another mass bound satisfying M 1 M 2, M 1 M 2 δ, δ > 0. We have x M 2 = arg max M(x) M 2c(x M 1 ) T x. The solution is determined by choosing the elements of top compliances c(x M 1 ) e whose sum of masses is less than M 2. Xingtong Yang, Ming Li (ZJU) Discrete Multi-material TO SPM 2018, Bilbao 4 / 25

10 Theory Update Proof to Lemma The density update from total M 1 to M 2 for M 2 < M 1 and close enough. c(x M 2 ) = c(x M 1 (x M 1 x M 2 )) = c(x M 1 ) c (x M 1 )(x M 1 x M 2 ) + O(x M 1 x M 2 ) c(x M 1 ) c (x M 1 )(x M 1 x M 2 ). In this situation, the minimizing problem in the last Lemma becomes arg min c(x M 2 ) = arg min c(x M 2 ) c(x M 1 ) arg min c (x M 1 ) T (x M 1 x M 2 ). Further taking into account of c(x) x ej = c(x ej ) and noticing that x M 1 is constant, there is arg min c(x M 2 ) arg min c(x M 1 ) T (x M 1 x M 2 ) = arg min c(x M 1 ) T x M 2 = arg max c(x M 1 ) T x M 2. Xingtong Yang, Ming Li (ZJU) Discrete Multi-material TO SPM 2018, Bilbao 5 / 25

11 Theory Searching strategy Theory: three-stage density exploration To prevent an abrupt density variation during the iteration. Decomposed into three sub-iterations for full exploration: preprocessing iteration, upwards searching iteration and downwards searching iteration. Figure 1: Left: overall exploration process; Right: inner three-state updating. Xingtong Yang, Ming Li (ZJU) Discrete Multi-material TO SPM 2018, Bilbao 6 / 25

12 Numerical approach Two-material Numerical approach: Two-material Ideas: constraints determine quantity of each material Constraints: N elements & the total mass M : { { n 1 + n 2 = N n 1 ρ 1 + n 2 ρ 2 M where n i is the number of elements filled with material i. n 1 M Nρ 2 ρ 1 ρ 2 n 2 = N n 1 n 1 = M Nρ 2 ρ 1 ρ 2, n 2 = N n 1. Xingtong Yang, Ming Li (ZJU) Discrete Multi-material TO SPM 2018, Bilbao 7 / 25

13 Numerical approach Two-material Numerical approach: Two-material algorithm Two-material procedure Step 1: Input material properties, total mass constraint M, evolutionary ratio ER, domain size nelx, nely, and initial density consisting of only material 1. Step 2: Set the next target mass as follows M i+1 = max(m i (1 ER), M ). Step 3: Calculate elemental compliances {c M i e } via FE analysis. Step 4: Determine the quantity of material 1 and 2 in the design domain n 1 = M i+1 Nρ 2 ρ 1 ρ 2, n 2 = N n 1. Step 5: Update density x i+1 via filling the top n 1 elements of large compliances {c M i e } by material 1, and others by material 2. Step 6: Repeat steps 2-5 until the objective mass is achieved and the convergent criterion is satisfied error = xi+1 x i x i+1 τ Xingtong Yang, Ming Li (ZJU) Discrete Multi-material TO SPM 2018, Bilbao 8 / 25

14 Numerical approach Three-material Numerical approach: Three-material Two challenges: Two constraints (total material number/mass) determine two equality: the left freedom is to explore all choices. Exploration strategy: Smooth material distribution transition. Xingtong Yang, Ming Li (ZJU) Discrete Multi-material TO SPM 2018, Bilbao 9 / 25

15 Numerical approach Three-material Numerical approach: material quantity equality The possible maximum number n 1 max of material 1 and the minimum n 3 min of material 3 can be calculated by { { n 1 + n 3 = N n 1 ρ 1 + n 3 ρ 3 M i+1 n 1 max = M i+1 Nρ 3 ρ 1 ρ 3 n 3 min = N n 1 max. { n2 + n 3 = N n 1 n 2 ρ 2 + n 3 ρ 3 M i+1 n 1 ρ 1 { n 2 = M i+1 n 1 ρ 1 (N n 1 )ρ 3 ρ 2 ρ 3. n 3 = N n 1 n 2 where E 1 E 2 E 3, ρ 1 ρ 2 ρ 3, the mass constraint is M i+1 in current iteration. In each mass deduction iteration, the proposed method will construct and search a set of feasible material combinations Ω = Ω 1 Ω 2 Ω 3, mainly confined by the feasible range Ω 1 = [n 1 min, n 1 max] of material 1. Xingtong Yang, Ming Li (ZJU) Discrete Multi-material TO SPM 2018, Bilbao 10 / 25

16 Numerical approach Three-material Numerical approach: material quantity equality The possible maximum number n 1 max of material 1 and the minimum n 3 min of material 3 can be calculated by { { n 1 + n 3 = N n 1 ρ 1 + n 3 ρ 3 M i+1 n 1 max = M i+1 Nρ 3 ρ 1 ρ 3 n 3 min = N n 1 max. { n2 + n 3 = N n 1 n 2 ρ 2 + n 3 ρ 3 M i+1 n 1 ρ 1 { n 2 = M i+1 n 1 ρ 1 (N n 1 )ρ 3 ρ 2 ρ 3. n 3 = N n 1 n 2 where E 1 E 2 E 3, ρ 1 ρ 2 ρ 3, the mass constraint is M i+1 in current iteration. In each mass deduction iteration, the proposed method will construct and search a set of feasible material combinations Ω = Ω 1 Ω 2 Ω 3, mainly confined by the feasible range Ω 1 = [n 1 min, n 1 max] of material 1. Xingtong Yang, Ming Li (ZJU) Discrete Multi-material TO SPM 2018, Bilbao 10 / 25

17 Numerical approach Three-material Three-stage strategy Preprocessing stage The preprocessing step gradually find x i Ω 1 via iterating different numbers of domain material compositions, specifically, 1 Moving from (n i 1, ni 2, ni 3 ) at mass constraint M i, 2 via to an intermediate composition (n i 1, 0, N ni 1 ) at mass M i+1, 3 till reaching (n 1 max, 0, N n 1 max) at mass M i+1. Upwards and Downwards stage 1 Once the n 1 of an initial structure lies in Ω 1. 2 Upwards searching procedure from n 1 to n 1 max. 3 Downwards searching procedure from n 1 to n 1 min. 4 The best of them gives the next update density. Xingtong Yang, Ming Li (ZJU) Discrete Multi-material TO SPM 2018, Bilbao 11 / 25

18 Numerical approach Three-material Three-stage strategy Preprocessing stage The preprocessing step gradually find x i Ω 1 via iterating different numbers of domain material compositions, specifically, 1 Moving from (n i 1, ni 2, ni 3 ) at mass constraint M i, 2 via to an intermediate composition (n i 1, 0, N ni 1 ) at mass M i+1, 3 till reaching (n 1 max, 0, N n 1 max) at mass M i+1. Upwards and Downwards stage 1 Once the n 1 of an initial structure lies in Ω 1. 2 Upwards searching procedure from n 1 to n 1 max. 3 Downwards searching procedure from n 1 to n 1 min. 4 The best of them gives the next update density. Xingtong Yang, Ming Li (ZJU) Discrete Multi-material TO SPM 2018, Bilbao 11 / 25

19 Numerical approach Three-material Three-stage strategy Preprocessing stage The preprocessing step gradually find x i Ω 1 via iterating different numbers of domain material compositions, specifically, 1 Moving from (n i 1, ni 2, ni 3 ) at mass constraint M i, 2 via to an intermediate composition (n i 1, 0, N ni 1 ) at mass M i+1, 3 till reaching (n 1 max, 0, N n 1 max) at mass M i+1. Upwards and Downwards stage 1 Once the n 1 of an initial structure lies in Ω 1. 2 Upwards searching procedure from n 1 to n 1 max. 3 Downwards searching procedure from n 1 to n 1 min. 4 The best of them gives the next update density. Xingtong Yang, Ming Li (ZJU) Discrete Multi-material TO SPM 2018, Bilbao 11 / 25

20 Numerical results Numerical results Initial value Fully-filled with the material of the largest Youngs modulus and density. Structural compliance c str Actual value of the optimized structure during each iteration step. Estimated compliance c est The one computed as its approximation, c est = c(x i ) T x i+1. Iteration gap The step size in the iteration process as involved in the three sub-processing stages. step = 0.3%N. Default settings Evolutionary ratio ER = Filter: r min = 2.5. Mesh: Environment Matlab2017b, Windows10, Intel Core i GHz CPU, 16GB RAM. Xingtong Yang, Ming Li (ZJU) Discrete Multi-material TO SPM 2018, Bilbao 12 / 25

21 Numerical results Two-material Two-material Problem definition Both gradually decrease the mass from the initial maximal one till the target. Same mass fraction curve. Similar value of the objective function. Solid-void problem iteration process Xingtong Yang, Ming Li (ZJU) Discrete Multi-material TO SPM 2018, Bilbao 13 / 25

22 Numerical results Two-material Two-material Problem definition Structure converges to a stable topology after 30 iterations. c est is larger and shows less smooth convergence, but leads the iterative procedure to an expected structure. Two non-zero problem process Xingtong Yang, Ming Li (ZJU) Discrete Multi-material TO SPM 2018, Bilbao 14 / 25

23 Numerical results Three-material Three-material: iteration process Involve an iterative searching at the same mass constraint: preprocessing, upwards, downwards searching stages. Main-loop produces an optimal structure at certain mass constraint. Sub-loop at each main-loop step, search for the best structure for all the possible candidate numbers of materials at a given mass constraint. Inner iteration: controlled by the quantity of material A; lack regularity. Xingtong Yang, Ming Li (ZJU) Discrete Multi-material TO SPM 2018, Bilbao 15 / 25

24 Numerical results Three-material Material iteration Fluctuate locally, converge ultimately The variations of the associated volume and mass fraction of each material during the main iteration process. The inner: materials update smoothly. The outer: material iteration curves fluctuate locally, but ultimately converge globally. Xingtong Yang, Ming Li (ZJU) Discrete Multi-material TO SPM 2018, Bilbao 16 / 25

25 Higher resolution Numerical results Three-material Mesh size: v.s Able to deal with high resolution problems. Xingtong Yang, Ming Li (ZJU) Discrete Multi-material TO SPM 2018, Bilbao 17 / 25

26 Numerical results Four-material Four-material: effect of material properties Usually: the more, the better Results of various types of material combinations, using four, three and two candidate materials under the same total mass constraint. The usage of more materials indeed produces structures of better compliance. Xingtong Yang, Ming Li (ZJU) Discrete Multi-material TO SPM 2018, Bilbao 18 / 25

27 Numerical results Four-material Four-material: effect of material properties Need further exploration Effect of the candidate material properties: the ranking of the Young s modulus or the modulus-density ratio. No clear evidence on the dependence of the final produced structure on property ranking of the materials. Xingtong Yang, Ming Li (ZJU) Discrete Multi-material TO SPM 2018, Bilbao 19 / 25

28 Numerical results Four-material Compared to SIMP Unpredictable optimized structures Problem definition Mesh size: Better compliance. Ability of avoiding being stuck in local minima. (To a certain extent) The single-material optimal structure: material C has the largest E/ρ? Xingtong Yang, Ming Li (ZJU) Discrete Multi-material TO SPM 2018, Bilbao 20 / 25

29 Numerical results Four-material Four-material: c est, the motivation Problem definition c est v.s. c str Estimated compliances v.s actual structural compliances. More than two materials: high similarity. Single material: strongly different. Estimated compliances works in all cases. Xingtong Yang, Ming Li (ZJU) Discrete Multi-material TO SPM 2018, Bilbao 21 / 25

30 3D Cantilever beam Numerical results 3D examples Two-material, Material E ρ A 1 1 B M c 0.4 c Three-material, Material E ρ A 1 1 B C M c 0.4 c Xingtong Yang, Ming Li (ZJU) Discrete Multi-material TO SPM 2018, Bilbao 22 / 25

31 3D Cantilever beam Numerical results 3D examples Four-material Material E ρ A 2 1 B C D M c 0.7 c Xingtong Yang, Ming Li (ZJU) Discrete Multi-material TO SPM 2018, Bilbao 23 / 25

32 Computation time Numerical results Computation time Problem Size M c m Our SIMP/ BESO Half MBB MBB Cantilever Multi-grid preconditioned conjugate gradients (MGCG) solver. Better in target value but worse in computation time, especially when m 3. Computation time: strongly effected by m and N. Xingtong Yang, Ming Li (ZJU) Discrete Multi-material TO SPM 2018, Bilbao 24 / 25

33 Conclusion Conclusion Contributions Discrete method on multi-material optimization under total mass constraint, seldom studied before. Theoretical finding to reduce multi-material to a series of two-material problems. Practical regulated approach to iterate optimization in two-material subproblems. Limitations & Future work 1 Efficiency improvement The traversal searching strategy is able to produce a better solution but inefficient. Advanced searching technique is to be developed. 2 Theoretical exploration Theoretical understanding on the number of material types in the final optimal solution. 3 Microstructure Extended cases multi-material microstructure optimization or to cases of large deformations. Xingtong Yang, Ming Li (ZJU) Discrete Multi-material TO SPM 2018, Bilbao 25 / 25

34 Conclusion Conclusion Contributions Discrete method on multi-material optimization under total mass constraint, seldom studied before. Theoretical finding to reduce multi-material to a series of two-material problems. Practical regulated approach to iterate optimization in two-material subproblems. Limitations & Future work 1 Efficiency improvement The traversal searching strategy is able to produce a better solution but inefficient. Advanced searching technique is to be developed. 2 Theoretical exploration Theoretical understanding on the number of material types in the final optimal solution. 3 Microstructure Extended cases multi-material microstructure optimization or to cases of large deformations. Xingtong Yang, Ming Li (ZJU) Discrete Multi-material TO SPM 2018, Bilbao 25 / 25

35 Thanks Thanks for listening! Xingtong Yang, Ming Li (ZJU) Discrete Multi-material TO SPM 2018, Bilbao 25 / 25