FLUID FLOW TOPOLOGY OPTIMIZATION USING POLYGINAL ELEMENTS: STABILITY AND COMPUTATIONAL IMPLEMENTATION IN PolyTop

Transcription

1 FLUID FLOW TOPOLOGY OPTIMIZATION USING POLYGINAL ELEMENTS: STABILITY AND COMPUTATIONAL IMPLEMENTATION IN PolyTop Anderson Pereira (Tecgraf/PUC-Rio) Cameron Talischi (UIUC) - Ivan Menezes (PUC-Rio) - Glaucio Paulino (GATech) Reno, NV, USA, July 20-24, 2015

2 INTRODUCTION PolyMesher & PolyTop Polygonal element mesher and topology optimization implementation in MATLAB PolyTop Geometry & BC s Polygonal Mesh PolyTop Talischi, C., Paulino, G.H., Pereira, A., and Menezes, I.F.M., PolyMesher: a general-purpose mesh generator for polygonal elements written in Matlab, JSMO, 45: , doi: /s z Talischi, C., Paulino, G.H., Pereira, A., and Menezes, I.F.M., PolyTop: a Matlab implementation of a general topology optimization framework using unstructured polygonal finite element meshes, JSMO, 45: , doi: /s x

3 POLYGONAL FINITE ELEMENTS Provide great flexibility in discretizing complex domains Naturally exclude checkerboard layouts and one-node connections Not biased by the standard FEM simplex geometry (triangles and quads)

4 POLYGONAL FINITE ELEMENTS Provide great flexibility in discretizing complex domains Naturally exclude checkerboard layouts and one-node connections Not biased by the standard FEM simplex geometry (triangles and quads) Q4 Elements Polygonal Elements

5 POLYGONAL FINITE ELEMENTS Provide great flexibility in discretizing complex domains Naturally exclude checkerboard layouts and one-node connections Not biased by the standard FEM simplex geometry (triangles and quads) POLYGONAL FINITE ELEMENTS T6 Elements Polygonal Elements Talischi, C., Paulino, G.H., Pereira, A. and Menezes, I.F.M., Polygonal Finite Elements for Topology Optimization: A Unifying Paradigm, IJNME, 82(6): , 2010.

6 CODE EFFICIENCY PolyTop: Efficiency Comparison with the 88-line code* Mesh Size 90x30 150x50 300x x200 PolyTop 88-line (time in sec for 200 optimization iterations) * Andreassen E., Clausen A., Schevenels M., Lazarov B., Sigmund O., Efficient topology optimization in MATLAB using 88 lines of code, JSMO, 43(1):1 16, doi: /s

7 CODE MODULARITY PolyTop: Code Modularity and Flexibility

8 CODE MODULARITY PolyTop: Code Modularity and Flexibility Material interpolation functions (e.g. SIMP, RAMP) Different optimizers (e.g. OC, MMA, SLP) Objective functions (e.g. Compliance, Compliant Mechanism) Different physics (?)

9 CODE MODULARITY PolyTop: Code Modularity and Flexibility Material interpolation functions (e.g. SIMP, RAMP) Different optimizers (e.g. OC, MMA, SLP) Objective functions (e.g. Compliance, Compliant Mechanism) Different physics (?)

10 CODE MODULARITY PolyTop: Code Modularity and Flexibility Material interpolation functions (e.g. SIMP, RAMP) Different optimizers (e.g. OC, MMA, SLP) Objective functions (e.g. Compliance, Compliant Mechanism) Different physics (?) Example (Compliant Mechanism):

11 CODE MODULARITY PolyTop: Code Modularity and Flexibility Material interpolation functions (e.g. SIMP, RAMP) Different optimizers (e.g. OC, MMA, SLP) Objective functions (e.g. Compliance, Compliant Mechanism) Different physics (?)

12 STABILITY OF POLYGONAL FEs Governing equations for Stokes flow Stability is a critical issue concerning mixed FE formulations and it is well-known that it is dictated by the INF-SUP condition It delineates the appropriate balance between the velocity and pressure approximations

13 STABILITY OF POLYGONAL FEs Numerical instabilities such the checkerboard problem could appear in mixed variational formulation (pressure-velocity) of the Stokes flow problems. velocity Lid-driven cavity problem checkerboard on pressure Q4 elements

14 STABILITY OF POLYGONAL FEs Numerical instabilities such the checkerboard problem could appear in mixed variational formulation (pressure-velocity) of the Stokes flow problems. velocity Lid-driven cavity problem pressure Polygonal elements

15 STABILITY OF POLYGONAL FEs ~ INF-SUP Test: compute the stability parameter b h where: is the space of pressure modes for a sequence of progressively finer meshes.

16 STABILITY OF POLYGONAL FEs ~ INF-SUP Test: compute the stability parameter b h where: is the space of pressure modes for a sequence of progressively finer meshes. Families of meshes: Quadrilateral Hexagonal Random Voronoi Centroidal Voronoi (CVT)

17 STABILITY OF POLYGONAL FEs Computed values of the stability parameter ~ b h ~ b h remains bounded away from zero under mesh refinement for polygonal meshes* Talischi, C., Pereira, A., Paulino, G.H., Menezes, I.F.M., and Carvalho, M.S., Polygonal Finite Elements for Incompressible Fluid Flow, IJNMF, 74(2): , 2014.

18 PERFORMANCE AND ACCURACY 1 Stokes flow on a unit square with known analytical solution (smooth problem) Quadrilateral Hexagonal Triangular (MINI) Random Voronoi Centroidal Voronoi (CVT)

19 PERFORMANCE AND ACCURACY H 1 - error in Velocity L 2 - error in Pressure

20 PERFORMANCE AND ACCURACY H 1 - error in Velocity L 2 - error in Pressure Given a level of error in pressure, the MINI elements require almost two order of magnitude more DOFs than the CVT

21 PERFORMANCE AND ACCURACY 2 Stokes flow on an L-shaped domain with known analytical solution (non-smooth problem) Representative example of the family of meshes for the L-shaped problem Uniform triangular Uniform Quadrilateral Centroidal Voronoi (CVT) generated by PolyMesher Representative example example example of the of family the of the family of family meshes of meshes of meshes (a) uniform (a) uniform (a) uniform triangular triangular triangular (b) (b) (b) uniform uniform uniform quadrilateral and and (c) centroidal and (c) centroidal (c) centroidal Voronoi Voronoi Voronoi (CVT) (CVT) (CVT)

22 PERFORMANCE AND ACCURACY Low order elements H 1 - error in Velocity L 2 - error in Pressure

23 PERFORMANCE AND ACCURACY High order elements H 1 - error in Velocity L 2 - error in Pressure

24 TOPOLOGY OPTIMIZATION FOR FLUIDS Governing BVP = inverse permeability function (relates design to physics) since r is piecewise constant, this is a discontinuous coefficient porosity approach * Objective Function ( drag minimization problem ): Pereira, A., Talischi, C., Paulino, G.H., Menezes, I.F.M., and Carvalho, M.S., Fluid Flow Topology Optimization in PolyTop: Stability and Computational Implementation, JSMO, 2015, doi: /s z * Borrvall, T., and Petersson, J., Topology optimization of fluids in stokes flow, IJNMF, 41, 1 (2003),

25 189 Lines 206 Lines CHANGES IN POLYTOP CODE Elasticity Problems Fluid Flow Problems 25 (13.0%) Main Loop 25 (12.0%) 20 (10.5%) Objective Function & Constraint 20 (10.0%) 14 (7.5%) Update Scheme (OC) 14 (6.5%) FE Analysis 116 (61.5%) 22 Lines Changed 11 Lines Deleted 28 Lines Added 133 (65.0%) 14 (7.5%) Plotting Results 14 (6.5%)

26 NUMERICAL RESULTS

27 NUMERICAL RESULTS Diffuser - Problem description

28 NUMERICAL RESULTS Diffuser - Solution Optimal solution Velocity Field Pressure Field

29 NUMERICAL RESULTS Bend - Problem description

30 NUMERICAL RESULTS Bend - Solution Optimal solution Velocity Field Pressure Field

31 NUMERICAL RESULTS Double Pipe Problem description Optimal solution

32 NUMERICAL RESULTS Double Pipe Velocity Field Pressure Field

33 NUMERICAL RESULTS Fluid Mechanism (maximize the y-velocity at a specific location) Problem description Optimal solution

34 NUMERICAL RESULTS Fluid Mechanism (maximize the y-velocity at a specific location) Velocity field Pressure field

35 CONCLUDING REMARKS

36 CONCLUDING REMARKS The general framework of PolyTop emphasizes a modular code structure where the analysis routine, including sensitivity calculations with respect to analysis parameters, and the optimization algorithm are kept separated from quantities defining the design field. This separation in turn permits changing the topology optimization formulation, including the choice of material interpolation scheme and the complexity control mechanism (e.g. filters and other manufacturing constraints), without the need for modifying the analysis function. Because polygonal finite elements (from the original PolyTop code) are again employed for the fluid analysis, the basis function construction and element integration routines also remain intact. The PolyTop code, originally written for compliance minimization in elasticity, was easily extended to model the problem of minimizing dissipated power in Stokes flow: only a few lines of codes were involved.

37 QUESTIONS?

38 CODE EFFICIENCY Comparison with the 88-line code* Mesh Size 90x30 150x50 300x x200 PolyTop 88-line (time in sec for 200 optimization iterations) Design Volume (OC Update Function) * Andreassen E., Clausen A., Schevenels M., Lazarov B., Sigmund O., Efficient topology optimization in MATLAB using 88 lines of code, JSMO, 43(1):1 16, doi: /s

39 NUMERICAL RESULTS Diffuser - Results Diffuser Problem 2,500 elements 10,000 elements # iterations objective # iterations objective Present work (curved domain) Present work (square domain) Borwall and Petersson (2003)

40 NUMERICAL RESULTS Bend - Results Bend Problem 2,500 elements 10,000 elements # iterations objective # iterations objective Present work (curved domain) Present work (square domain) Borwall and Petersson (2003)

41 GENERATION OF POLYHEDRAL MESH A polygonal discretization can be obtained from the Voronoi diagram of a given set of seeds and their reflections Talischi, C., Paulino, G.H., Pereira, A., and Menezes, I.F.M., PolyMesher: a general-purpose mesh generator for polygonal elements written in Matlab, JSMO, 45: , doi: /s z Seed and its reflection have a common edge 41 MM&FGM TH INTERNATIONAL SYMPOSIUM ON MULTISCALE, MULTIFUNCTIONAL AND FUNCTIONALLY GRADED MATERIALS

42 STABILITY OF POLYGONAL FEs Families of meshes: Quadrilateral Hexagonal Random Voronoi Centroidal Voronoi (CVT) For meshes consisting of convex polygons, the results by Beirão da Veiga and Lipnikov guarantees the satisfaction of INF-SUP condition if every internal node in the mesh is connected to at most three edges Beirão da Veiga, L. and Lipnikov, K., A mimetic discretization of the Stokes problem with selected edge bubles, SIAM J Sci Comput, 32(2): , 2010.