fluid flow topology optimization using … flow topology optimization using polyginal elements:...

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)

ICCES'15 Reno, NV, USA, July 20-24, 2015

ICCES'15

PolyTop Geometry & BC’s

PolyMesher‡ & PolyTop†

Polygonal element mesher and topology optimization implementation

in MATLAB PolyTop

† 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, 45:329–357, 2012. doi:10.1007/s00158-011-0696-x

Polygonal Mesh

INTRODUCTION

‡ 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, 45:309–328, 2012. doi:10.1007/s00158-011-0706-z

ICCES'15

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)

ICCES'15

POLYGONAL FINITE ELEMENTS

Q4 Elements Polygonal 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)

ICCES'15

POLYGONAL FINITE ELEMENTS

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):671-698, 2010.

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)

ICCES'15

Comparison with the 88-line code*

* 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, 2011. doi:10.1007/s00158-010-0594-7

Mesh Size 90x30 150x50 300x100 600x200

11.9 31.5 135.5 764.1

10.9 33.0 252.2 3092.9

PolyTop

88-line

(time in sec for 200 optimization iterations)

CODE EFFICIENCY

PolyTop: Efficiency

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CODE MODULARITY

PolyTop: Code Modularity and Flexibility

ICCES'15

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 (?)

ICCES'15

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 (?)

ICCES'15

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):

ICCES'15

• Material interpolation functions (e.g. SIMP, RAMP)

• Different optimizers (e.g. OC, MMA, SLP)

• Objective functions (e.g. Compliance, Compliant Mechanism)

• Different physics (?)

CODE MODULARITY

PolyTop: Code Modularity and Flexibility

ICCES'15

Governing equations for Stokes flow

STABILITY OF POLYGONAL FEs

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”

ICCES'15

• Numerical instabilities such the “checkerboard” problem could appear in

mixed variational formulation (pressure-velocity) of the Stokes flow

problems.

velocity

checkerboard on

pressure Lid-driven cavity problem

Q4 elements

STABILITY OF POLYGONAL FEs

ICCES'15

STABILITY OF POLYGONAL FEs

velocity

pressure

Lid-driven cavity problem

Polygonal elements

• Numerical instabilities such the “checkerboard” problem could appear in

mixed variational formulation (pressure-velocity) of the Stokes flow

problems.

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STABILITY OF POLYGONAL FEs

INF-SUP Test: compute the stability parameter

~ bh

where: is the space of pressure modes

for a sequence of progressively finer meshes.

ICCES'15

STABILITY OF POLYGONAL FEs

INF-SUP Test: compute the stability parameter

where: is the space of pressure modes

for a sequence of progressively finer meshes.

~ bh

Families of meshes:

Quadrilateral Hexagonal Random Voronoi Centroidal Voronoi (CVT)

ICCES'15

STABILITY OF POLYGONAL FEs

Computed values † of the stability parameter

remains bounded away from

zero under mesh refinement for

polygonal meshes*

~ bh

~ bh

† 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):134-151, 2014.

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PERFORMANCE AND ACCURACY

1 – Stokes flow on a unit square with known analytical solution

(smooth problem)

Quadrilateral Triangular (MINI)

Hexagonal

Random Voronoi Centroidal Voronoi (CVT)

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PERFORMANCE AND ACCURACY

H1- error in Velocity L2- error in Pressure

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“Given a level of error in pressure, the MINI elements require

almost two order of magnitude more DOFs than the CVT”

PERFORMANCE AND ACCURACY

H1- error in Velocity L2- error in Pressure

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PERFORMANCE AND ACCURACY

Uniform triangular Centroidal Voronoi (CVT) generated by PolyMesher

Representative example of the family of meshes (a) uniform triangular (b) uniform quadrilateral and (c) centroidal Voronoi (CVT)

Representative example of the family of meshes (a) uniform triangular (b) uniform quadrilateral and (c) centroidal Voronoi (CVT)

Representative example of the family of meshes (a) uniform triangular (b) uniform quadrilateral and (c) centroidal Voronoi (CVT)

Uniform Quadrilateral

Representative example of the family of meshes for the L-shaped problem

2 – Stokes flow on an L-shaped domain with known analytical solution

(non-smooth problem)

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PERFORMANCE AND ACCURACY

H1- error in Velocity L2- error in Pressure

Low order elements

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PERFORMANCE AND ACCURACY

H1- error in Velocity L2- error in Pressure

High order elements

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TOPOLOGY OPTIMIZATION FOR FLUIDS †

Governing BVP

Objective Function (“drag minimization problem”):

= inverse permeability function

(relates design to physics)

• since r is piecewise constant,

this is a discontinuous coefficient

• “porosity approach” *

* Borrvall, T., and Petersson, J., “Topology optimization of fluids in stokes flow”, IJNMF, 41, 1 (2003), 77–107

† 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: 10.1007/s00158-014-1182-z

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CHANGES IN POLYTOP CODE

25

20

14

116

14

(13.0%)

(10.5%)

(7.5%)

(61.5%)

(7.5%)

18

9 L

ine

s

Main Loop

Update Scheme (OC)

FE Analysis

Plotting Results

Objective Function

& Constraint

25

20

14

133

14

(12.0%)

(10.0%)

(6.5%)

(65.0%)

(6.5%)

206 L

ines

Elasticity Problems Fluid Flow Problems

22 Lines Changed

11 Lines Deleted

28 Lines Added

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NUMERICAL RESULTS

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NUMERICAL RESULTS

Diffuser - Problem description

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NUMERICAL RESULTS

Diffuser - Solution

Velocity Field Pressure Field Optimal solution

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NUMERICAL RESULTS

Bend - Problem description

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NUMERICAL RESULTS

Bend - Solution

Velocity Field Pressure Field Optimal solution

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NUMERICAL RESULTS

Double Pipe

Problem description Optimal solution

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NUMERICAL RESULTS

Double Pipe

Velocity Field Pressure Field

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Problem description Optimal solution

NUMERICAL RESULTS

Fluid Mechanism

(maximize the y-velocity at a specific location)

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Velocity field Pressure field

NUMERICAL RESULTS

Fluid Mechanism

(maximize the y-velocity at a specific location)

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CONCLUDING REMARKS

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• 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.

CONCLUDING REMARKS

ICCES'15

QUESTIONS ?

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Comparison with the 88-line code*

* 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, 2011. doi:10.1007/s00158-010-0594-7

CODE EFFICIENCY

† Design Volume

(OC Update Function)

Mesh Size 90x30 150x50 300x100 600x200

11.9 31.5 135.5 764.1 PolyTop

9.7 24.3 119.7 708.8 88-line†

(time in sec for 200 optimization iterations)

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NUMERICAL RESULTS

Diffuser - Results

Diffuser Problem 2,500 elements 10,000 elements

# iterations objective # iterations objective

Present work (curved domain) 18 31.31 19 30.64

Present work (square domain) 19 31.34 19 30.70

Borwall and Petersson (2003) 29 30.59 33 30.46

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NUMERICAL RESULTS

Bend - Results

Bend Problem 2,500 elements 10,000 elements

# iterations objective # iterations objective

Present work (curved domain) 36 10.11 31 9.77

Present work (square domain) 37 9.99 30 9.77

Borwall and Petersson (2003) 64 10.01 85 9.76

MM&FGM 2014

13TH INTERNATIONAL SYMPOSIUM ON MULTISCALE, MULTIFUNCTIONAL

AND FUNCTIONALLY GRADED MATERIALS 41

GENERATION OF POLYHEDRAL MESH

Seed and its

reflection have a

common edge

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, 45:309–328, 2012. doi:10.1007/s00158-011-0706-z

ICCES'15

STABILITY OF POLYGONAL FEs

Families of meshes:

Quadrilateral Hexagonal Random Voronoi Centroidal Voronoi (CVT)

† Beirão da Veiga, L. and Lipnikov, K., “A mimetic discretization of the Stokes

problem with selected edge bubles”, SIAM J Sci Comput, 32(2):875–893, 2010.

“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”