Approximate Reanalysis in Topology Optimization

1. Approximate Reanalysis in Topology Optimization Oded Amir, PhD student DTU - Technical University of Denmark Department of Mathematics July 1, 2008 WCCM8 – ECCOMAS 2008

2. Topology optimization: the nested approach The topology optimization problem in the nested approach (e.g. minimum compliance): The main computational bottleneck: repeated solution of the analysis equations Approximate Reanalysis in Topology Optimization

3. Approximate reanalysis in topology optimization A possible way to reduce the computational effort: approximate reanalysis • The approximation results from a reduced basis expression. • The y vector contains only a small number of unknowns. • Nested equations are: the reduced equation system and the equation systems for generating the basis vectors (no factorization required!) Approximate Reanalysis in Topology Optimization

4. Approximate reanalysis A known solution from a certain stage K0u0 = f The reanalysis form of the system for a later stage Ku = f → (K0 + ΔK) u = f This system will be solved approximately We refer to this procedure as approximate reanalysis. The aim of the approximation: perform fewer matrix factorizations. Approximate Reanalysis in Topology Optimization

5. Reanalysis by Combined Approximations Combined Approximations for structural reanalysis (Kirsch 1991): terms from a local approximation (series expansion) are used in a global approximation (reduced basis) solution. Recurrence relation K0 (k+1)u = f – ΔK (k)u The so-called binomial series u =(I – B + B2– B3+...) u1 B = K0-1 ΔK Focusing on the series terms u1 = K0-1 f u2 = - K0-1 ΔK u1 ui = - K0-1 ΔK ui-1 Using the factorized matrix U0TU0u1 = f U0TU0 ui = - ΔKui-1 Approximate Reanalysis in Topology Optimization

6. Reanalysis by Combined Approximations A small number of series terms will now serve as basis vectors in a reduced basis solution: Only a small system needs to be solved! • Insertion of the app. vector K RB y = f • Pre-multiplication RBTK RB y = RBT f • Reduced system (s unknowns) KRy = fR • Orthonormalization of the basis RB → VB Approximate Reanalysis in Topology Optimization

7. Topology optimization: the nested approach When solving the full system: When approximating: Each formulation requires a corresponding sensitivity analysis. Approximate Reanalysis in Topology Optimization

8. Important notes about the sensitivities Using the “standard” sensitivities requires the approximation to be very accurate ==> not efficient, not practical for large problems ==> deriving consistent sensitivities is favorable. Applying a sensitivity filter to the consistent sensitivities leads to significant inaccuracies. Deriving consistent sensitivities and applying a density filter lead to an accurate solution and an efficient procedure. Approximate Reanalysis in Topology Optimization

9. Sensitivity analysis for approximate reanalysis Sensitivity analysis is performed using the adjoint method. For this purpose we add several zero terms to the objective function: Then we define: Approximate Reanalysis in Topology Optimization

10. Sensitivity analysis for approximate reanalysis The sensitivity with respect to each design variable: The adjoint vectors for the basis vectors: • Some interesting results: • “Standard” term • “Correction” term • Force residual is taken into account • Adjoints are calculated in a reverse sequence • First basis vector does not need an adjoint since it is not sensitive to the design changes Approximate Reanalysis in Topology Optimization

11. Solution scheme Start • Possible control: • Fixed frequency of factorization updates. • Adaptive frequency according to the magnitude of the design changes. • Adaptive frequency according to the “convergence” of the approximate iterations. Control decision CA reanalysis Full FE analysis CA sensitivity analysis Sensitivity analysis Optimizer (MMA) Convergence? End Approximate Reanalysis in Topology Optimization

12. Example 1: 2D MBB-beam, fixed control MMA move limits: move = 0.3; xmin = max(0,x_0-move,xval-move/2); xmax = min(1,x_0+move,xval+move/2); 60 x 20 elements; 2562 DOF Volume fraction 0.5 Penalization power 3 Linear density filter size 2.1 MAXNBV = 4; MINFREQ = 10 Full solution: 92 iterations; c = 221.55 • Approximate solution: 190 iterations; 19 factorizations; c = 221.55 Approximate Reanalysis in Topology Optimization

13. Example 1: 2D MBB-beam, fixed control Large errors in the reanalysis can be tolerated Changes in the design can be used to control the procedure Approximate Reanalysis in Topology Optimization

14. Example 2: 2D MBB-beam, adaptive control MMA move limits: move = 0.3; xmin = max(0,x_0-move,xval-move/2); xmax = min(1,x_0+move,xval+move/2); 60 x 20 elements; 2562 DOF Volume fraction 0.5 Penalization power 3 Linear density filter size 2.1 MAXNBV = 4; MINFREQ = 20 Full solution: 92 iterations; c = 221.55 • Approximate solution: 200 iterations; 15 factorizations; c = 221.57 Approximate Reanalysis in Topology Optimization

15. Computational savings • Approximate solution: CA 4/10 (fixed frequency of factorization updates). • Full solution obtained using direct sparse solver routines available in the Sun Performance Library. • Programmed in Fortran, executed on a standard Sun UNIX terminal. Approximate Reanalysis in Topology Optimization

16. Final remarks So far, the procedure was applied successfully to minimum compliance problems (2D and 3D) and linear compliant mechanism design (2D). Nested analysis problems with 105 DOF can be reduced to only 4 unknowns! The ultimate goal is to use approximate reanalysis in nonlinear, transient topology optimization problems. Approximate Reanalysis in Topology Optimization

17. Acknowledgements I wish to thank: My supervisors Martin P. Bendsøe and Ole Sigmund My colleague Michael Bogomolni Prof. KristerSvanbergfor the MMA code Approximate Reanalysis in Topology Optimization

18. Thank you for listening! Questions? Approximate Reanalysis in Topology Optimization