The use of second-order information in structural topology optimization Susana Rojas Labanda, PhD student Mathias Stolpe, Senior researcher
What is Topology Optimization? Optimize the design of a structure given certain constraints, loads and supports. The design domain is discretized. The variables denotes the presence of material at each element. The goal is to decide which elements should contain material and which ones not. It is a -1 discrete problem. Model as an optimization problem minimize x f (x) subject to g(x) apple h(x) =. Bendsøe, M. P and Sigmund, O. Topology optimization: Theory, methods and applications Springer 23. 2 DTU Wind Energy 11.3.215
Minimum compliance problem SAND formulation minimize t,u subject to f T u a T t apple V K(t)u f = apple t apple 1. NESTED formulation minimize t subject to u T (t)k(t)u(t) a T t apple V apple t apple 1. f 2 R d the force vector. a 2 R n the volume vector. V > is the upper volume fraction. 3 DTU Wind Energy 11.3.215
Minimum compliance problem SAND formulation minimize t,u subject to f T u a T t apple V K(t)u f = apple t apple 1. NESTED formulation minimize t subject to u T (t)k(t)u(t) a T t apple V apple t apple 1. f 2 R d the force vector. a 2 R n the volume vector. V > is the upper volume fraction. Use the SIMP material interpolation to penalize intermediate densities t i = t p i p > 1 Use Density filter to avoid checkerboards and mesh-dependency issues. Linear Elasticity 1 t e =  i2ne h ei  h ei t i i2n e h ei = max{, r min dist(e, i)} E(t i )=E v +(E 1 + E v ) t p i K(t) = E(t e )K e 3 DTU Wind Energy 11.3.215
Optimization methods Topology optimization problem OC: Optimality criteria method. MMA: Sequential convex approximations. GCMMA: Global convergence MMA. Andreassen, E and Clausen, A and Schevenels, M and Lazarov, B. S and Sigmund, O. Efficient topology optimization in MATLAB using 88 lines of code. Structural and Multidisciplinary Optimization, 43(1): 1 16, 211. Svanberg, K. The method of moving asymptotes a new method for structural optimization. International Journal for Numerical Methods in Engineering, 24(2): 359 373. 1987. Svanberg, K. A class of globally convergent optimization methods based on conservative convex separable approximations. SIAM Journal on Optimization, 12(2): 555-573, 22. 4 DTU Wind Energy 11.3.215
Optimization methods Topology optimization problem + non-linear problem OC: Optimality criteria method. MMA: Sequential convex approximations. GCMMA: Global convergence MMA. Interior point solvers: IPOPT, FMINCON,... Sequential quadratic programming:snopt,... Gill, P. E and Murray, W and Saunders, M. A. SNOPT: An SQP Algorithm for Large -Scale Constrained Optimization. SIAM Journal on Optimization, 47(4):99 131, 25. Wächter, A and Biegler, L. T. On the implementation of an interior point filter line-search algorithm for large-scale nonlinear programming. Mathematical Programming, 16(1):25 57, 26. 4 DTU Wind Energy 11.3.215
Sequential Quadratic Programming for topology optimization SQP for minimum compliance problems in the nested formulation Solve a sequence of approximate sub-problems Convex quadratic approximation of the Lagrangian function. Linearization of the constraints. Implementation of SQP+ = IQP + EQP Require: Define the starting point x, the initial Lagrangian multipliers l and the optimality tolerance w. repeat Define an approximation of the Hessian of the Lagrange function, B k such as B k r 2 L(x k, l k ). Solve IQP sub-problem. Determine the working set of the inequality constraints and the boundary conditions. Solve EQP sub-problem (using active constraints). Compute the contraction parameter b 2 (, 1] such as the linearized contraints of the sub-problem are feasible at the iterate point x k + d iq k + bdeq k. Acceptance/rejection step. Use of line search strategy in conjunction with a merit function. Update the primal and dual iterates. until convergence return Morales, J.L and Nocedal, J and Wu, Y. A sequential quadratic programming algorithm with an additional equality constrained phase. Journal of Numerical Analysis,32:553 579, 21. 5 DTU Wind Energy 11.3.215
Finding an approximate positive definite Hessian Sensitivity analysis for the minimum compliance problem r 2 f (t) =2F T (t)k 1 (t)f(t) Q(t) Ĥ k = 2F T (t k )K 1 (t k )F(t k ) 6 DTU Wind Energy 11.3.215
Reformulations of IQP and EQP sub-problems Approximate Hessian computationally expensive Use dual formulation for the IQP sub-problem minimize d subject to r f (x k ) T d + 1 2 dt (2Fk T K 1 k F k )d A k d apple b k minimize a,b 1 4 bt K k b + a T b k subject to A T k a FT k b = r f (x k) a. Expansion of the EQP system minimize (r f (x k )+H k d iq d k )T d + 1 2 dt (2Fk T K 1 k F k )d subject to A i d = i 2W. @ FT k A W F k 1/2K k A T W 1 A @ v k d eq k l eq k 1 A = @ r f (x k)+h k d iq 1 A 7 DTU Wind Energy 11.3.215
Reformulations of IQP and EQP sub-problems Approximate Hessian computationally expensive Use dual formulation for the IQP sub-problem minimize d subject to r f (x k ) T d + 1 2 dt (2Fk T K 1 k F k )d A k d apple b k minimize a,b 1 4 bt K k b + a T b k subject to A T k a FT k b = r f (x k) a. Expansion of the EQP system minimize (r f (x k )+H k d iq d k )T d + 1 2 dt (2Fk T K 1 k F k )d subject to A i d = i 2W. @ FT k A W F k 1/2K k A T W 1 A @ v k d eq k l eq k 1 A = @ r f (x k)+h k d iq 1 A 7 DTU Wind Energy 11.3.215
Benchmarking in topology optimization How? Using performance profiles. Evaluate the cumulative ratio for a performance metric. Represent for each solver, the percentage of instances that achieve a criterion for different ratio values. 1 Performance profile 9 r s (t) = 1 n size{p 2 P : r p,s apple t}, r p,s = iter p,s min{iter p,s : s 2 S}. %problems 8 7 6 5 4 3 2 1 Solver1 Solver2 1 1.2 1.4 1.6 1.8 1.1 1.12 1.14 1.16 1.18 1.2 τ (iterp,s = τ min{iterp}) Dolan, E. D and Moré, J. J. Benchmarking optimization software with performance profiles. Mathematical Programming,91:21 213, 22. 8 DTU Wind Energy 11.3.215
Benchmark set of topology optimization problems Total Problems: 225. From 4 to 4, number of elements. (up to 81, 2 dof). 3 different domains 9 DTU Wind Energy 11.3.215
Performance profiles Objective function value Number of iterations 1 1 9 9 8 8 7 7 %problems 6 5 4 %problems 6 5 4 3 2 1 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 τ SQP+ IPOPT N IPOPT S SNOPT GCMMA Performance profiles in a reduce test set of 194 instances. Penalization of problems with KKT error higher than w = 1e 4. 3 2 1.2.4.6.8 1 1.2 1.4 1.6 1.8 τ (log1) SQP+ IPOPT N IPOPT S SNOPT GCMMA 1 DTU Wind Energy 11.3.215
Performance profiles Objective function value Number of iterations 1 1 9 9 8 8 7 7 %problems 6 5 4 %problems 6 5 4 3 2 1 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 τ SQP+ IPOPT N IPOPT S SNOPT GCMMA Performance profiles in a reduce test set of 194 instances. Penalization of problems with KKT error higher than w = 1e 4. 3 2 1.2.4.6.8 1 1.2 1.4 1.6 1.8 τ (log1) SQP+ IPOPT N IPOPT S SNOPT GCMMA 11 DTU Wind Energy 11.3.215
Performance profiles Number of stiffness matrix assemblies Computational time 1 1 9 9 8 8 7 7 %problems 6 5 4 %problems 6 5 4 3 2 1.5 1 1.5 2 τ (log1) SQP+ IPOPT N IPOPT S SNOPT GCMMA Performance profiles in a reduce test set of 194 instances. Penalization of problems with KKT error higher than w = 1e 4. 3 2 1.5 1 1.5 2 2.5 3 τ (log1) SQP+ IPOPT N IPOPT S SNOPT GCMMA 12 DTU Wind Energy 11.3.215
Performance profiles Number of stiffness matrix assemblies Computational time 1 1 9 9 8 8 7 7 %problems 6 5 4 %problems 6 5 4 3 2 1.5 1 1.5 2 τ (log1) SQP+ IPOPT N IPOPT S SNOPT GCMMA Performance profiles in a reduce test set of 194 instances. Penalization of problems with KKT error higher than w = 1e 4. 3 2 1.5 1 1.5 2 2.5 3 τ (log1) SQP+ IPOPT N IPOPT S SNOPT GCMMA 13 DTU Wind Energy 11.3.215
What can we conclude from the performance profiles? GCMMA tends to obtain a design with large KKT error IPOPT-S produces the best designs followed by SQP+ IPOPT-S and SQP+ (exact Hessian) produce better design than IPOPT-N and SNOPT (BFGS approximations) SQP+ converge in the least number of iterations and stiffness assemblies (= function evaluations) SAND formulation requires a lot of computational time and memory Need to improve the computational time spent in SQP+ 14 DTU Wind Energy 11.3.215
THANK YOU!!! This research is funded by the Villum Foundation through the research project Topology Optimization the Next Generation (NextTop). 15 DTU Wind Energy 11.3.215