[presentation] Stokes Flow Finite Element Model

Introduction

Mesh Generation

Model Results

Meshing Results

Computation Time Results

Unstructured adaptive mesh generation and sparse matrix storage applied to Stokes flow around cylinders Cameron Bracken1

E521 May 1, 2008

1

Humboldt State University

Conclusions

Introduction

Mesh Generation

Model Results

Meshing Results

Computation Time Results

Project Goals

1. Solve the 2D Steady Stokes equations via finite elements 2. Investigate the placement of cylindrical obstructions in the flow field 3. Adaptively generate the finite element mesh 4. Utilize sparse matrix storage

Conclusions

Introduction

Mesh Generation

Model Results

Meshing Results

Stokes equations - Simplification

Computation Time Results

Conclusions

Introduction

Mesh Generation

Model Results

Meshing Results

Computation Time Results

Stokes equations - Simplification Start with the NS equations: ρVt − µ∇2 V + ρ(V · ∇)V + ∇p = 0

Conclusions

Introduction

Mesh Generation

Model Results

Meshing Results

Computation Time Results

Stokes equations - Simplification Start with the NS equations: ρVt − µ∇2 V + ρ(V · ∇)V + ∇p = 0 Assume steady, µ∇2 V + ρ(V · ∇)V + ∇p = 0

Conclusions

Introduction

Mesh Generation

Model Results

Meshing Results

Computation Time Results

Conclusions

Stokes equations - Simplification Start with the NS equations: ρVt − µ∇2 V + ρ(V · ∇)V + ∇p = 0 Assume steady, µ∇2 V + ρ(V · ∇)V + ∇p = 0 Assume irrotational flow, ν∇2 V + ∇p = 0 And we arrive at the Stokes equations.

(1)

Introduction

Mesh Generation

Model Results

Meshing Results

Computation Time Results

Conclusions

Stokes equations - Simplification Start with the NS equations: ρVt − µ∇2 V + ρ(V · ∇)V + ∇p = 0 Assume steady, µ∇2 V + ρ(V · ∇)V + ∇p = 0 Assume irrotational flow, ν∇2 V + ∇p = 0 And we arrive at the Stokes equations. And for the continuity equation, assume incompressible,

(1)

Introduction

Mesh Generation

Model Results

Meshing Results

Computation Time Results

Conclusions

Stokes equations - Simplification Start with the NS equations: ρVt − µ∇2 V + ρ(V · ∇)V + ∇p = 0 Assume steady, µ∇2 V + ρ(V · ∇)V + ∇p = 0 Assume irrotational flow, ν∇2 V + ∇p = 0

(1)

And we arrive at the Stokes equations. And for the continuity equation, assume incompressible, ∇·V =0

(2)

We have enough information to describe any very viscous and/or slow moving fluid (RE  1,creeping flow).

Introduction

Mesh Generation

Model Results

Formal Problem Statement

Scalar Equations: −ν

∂2u ∂ 2 u ∂p − ν + =0 ∂x 2 ∂y 2 ∂x

−ν

∂2v ∂p ∂2v − ν + =0 2 2 ∂x ∂y ∂y ∂u ∂v + =0 ∂x ∂y

Meshing Results

Computation Time Results

Conclusions

Introduction

Mesh Generation

Model Results

Meshing Results

Computation Time Results

Formal Problem Statement ∂u ∂n

=

∂v ∂n

=0

∂2u ∂ 2 u ∂p − ν + =0 ∂x 2 ∂y 2 ∂x

−ν

∂2v ∂p ∂2v − ν + =0 2 2 ∂x ∂y ∂y ∂u ∂v + =0 ∂x ∂y

u=v =0

p=0

v = k x (1 − x) u=0

u=v =0

−ν

u=v =0

Scalar Equations:

Conclusions

Introduction

Mesh Generation

Model Results

Meshing Results

Computation Time Results

Conclusions

Basis Functions (xk , yk )

(xk , yk ) (xik , yik ) (xjk , yjk )

(xi , yi )

(xi , yi ) (xij , yij ) (xj , yj )

Linear Pressure Element.

(xj , yj ) Quadratic Velocity Element.

How, you ask, do we generate a mesh which accommodates holes and adaptively refines itself?

Introduction

Mesh Generation

Model Results

Meshing Results

Computation Time Results

Conclusions

Constrained Adaptive Mesh Generation Mesh Generation

[Basic Structured Grid Generation, Farrashkhalvat and Miles 2003]

Introduction

Mesh Generation

Model Results

Unstructured grid generation Meshing Results

203

Computation Time Results

Conclusions

Constrained Adaptive Mesh Generation Mesh Generation

nay triangulation. Voronoi-segment algorithm.

Unstructured

nay triangulation for an airfoil. Voronoi-segment algorithm. [Basic Structured Grid Generation, Farrashkhalvat and Miles 2003]

Introduction

Mesh Generation

Model Results

Unstructured grid generation Meshing Results

203

Computation Time Results

Conclusions

Constrained Adaptive Mesh Generation Mesh Generation

nay triangulation. Voronoi-segment algorithm.

Unstructured

Unstructured Adaptive

nay triangulation for an airfoil. Voronoi-segment algorithm. [Basic Structured Grid Generation, Farrashkhalvat and Miles 2003]

Introduction

Mesh Generation

Unstructured grid generation 203 Meshing Results Computation Time Results Fig. 6.3 Area functional.

Model Results

Conclusions

Constrained Adaptive Mesh Generation Mesh Generation

nay triangulation. Voronoi-segment algorithm. Fig. 6.4 Area functional.

Unstructured

Structured

Unstructured Adaptive

Fig. 6.5 Area-orthogonality. nay triangulation for an airfoil. Voronoi-segment algorithm. [Basic Structured Grid Generation, Farrashkhalvat and Miles 2003]

Introduction

Mesh Generation

Unstructured grid generation 203 Meshing Results Computation Time Results Fig. 6.3 Area functional.

Model Results

Conclusions

Constrained Adaptive Mesh Generation Mesh Generation

nay triangulation. Voronoi-segment algorithm. Fig. 6.4 Area functional.

Unstructured

Structured

Unstructured Adaptive

Structured Adaptive

Fig. 6.5 Area-orthogonality. nay triangulation for an airfoil. Voronoi-segment algorithm. [Basic Structured Grid Generation, Farrashkhalvat and Miles 2003]

Introduction

Mesh Generation

Model Results

Meshing Results

Computation Time Results

Conclusions

Adaptive mesh framework (a.k.a. Error controlled FE) Generate Sparse Set of Initial Nodes

Introduction

Mesh Generation

Model Results

Meshing Results

Computation Time Results

Conclusions

Adaptive mesh framework (a.k.a. Error controlled FE) Generate Sparse Set of Initial Nodes Mesh Generation

Introduction

Mesh Generation

Model Results

Meshing Results

Computation Time Results

Conclusions

Adaptive mesh framework (a.k.a. Error controlled FE) Generate Sparse Set of Initial Nodes Mesh Generation Solve Stokes Equations

Introduction

Mesh Generation

Model Results

Meshing Results

Computation Time Results

Conclusions

Adaptive mesh framework (a.k.a. Error controlled FE) Generate Sparse Set of Initial Nodes Mesh Generation Solve Stokes Equations

Identify Elements to Refine

Introduction

Mesh Generation

Model Results

Meshing Results

Computation Time Results

Conclusions

Adaptive mesh framework (a.k.a. Error controlled FE) Generate Sparse Set of Initial Nodes Mesh Generation Solve Stokes Equations

Elements to refine <m

Identify Elements to Refine

Introduction

Mesh Generation

Model Results

Meshing Results

Computation Time Results

Conclusions

Adaptive mesh framework (a.k.a. Error controlled FE) Generate Sparse Set of Initial Nodes Mesh Generation Refine Mesh

Solve Stokes Equations

no

Elements to refine <m

Identify Elements to Refine

Introduction

Mesh Generation

Model Results

Meshing Results

Computation Time Results

Conclusions

Adaptive mesh framework (a.k.a. Error controlled FE) Generate Sparse Set of Initial Nodes Mesh Generation Refine Mesh

Solve Stokes Equations

no

Elements to refine <m

Identify Elements to Refine

Introduction

Mesh Generation

Model Results

Meshing Results

Computation Time Results

Conclusions

Adaptive mesh framework (a.k.a. Error controlled FE) Generate Sparse Set of Initial Nodes Mesh Generation Refine Mesh

Solve Stokes Equations

no

Elements to refine <m Stop

How is this efficient?

yes

Identify Elements to Refine

Introduction

Mesh Generation

Model Results

Meshing Results

Computation Time Results

Conclusions

The Error Indicator and Relative Error Measurement I

Typically we take some measure of the solution gradient as an indicator of elemental error.

Introduction

Mesh Generation

Model Results

Meshing Results

Computation Time Results

Conclusions

The Error Indicator and Relative Error Measurement I

Typically we take some measure of the solution gradient as an indicator of elemental error.

I

These measures are called “a posteriori” (after the fact) estimates because we must solve the problem before we get an estimate of error.

Introduction

Mesh Generation

Model Results

Meshing Results

Computation Time Results

Conclusions

The Error Indicator and Relative Error Measurement I

Typically we take some measure of the solution gradient as an indicator of elemental error.

I

These measures are called “a posteriori” (after the fact) estimates because we must solve the problem before we get an estimate of error.

I

For the Stokes equations we have three indicators: u, v , p.

Introduction

Mesh Generation

Model Results

Meshing Results

Computation Time Results

Conclusions

The Error Indicator and Relative Error Measurement I

Typically we take some measure of the solution gradient as an indicator of elemental error.

I

These measures are called “a posteriori” (after the fact) estimates because we must solve the problem before we get an estimate of error.

I

For the Stokes equations we have three indicators: u, v , p.

Let θe be the indicator for an element e with vertex nodes i, j, k:

Introduction

Mesh Generation

Model Results

Meshing Results

Computation Time Results

Conclusions

The Error Indicator and Relative Error Measurement I

Typically we take some measure of the solution gradient as an indicator of elemental error.

I

These measures are called “a posteriori” (after the fact) estimates because we must solve the problem before we get an estimate of error.

I

For the Stokes equations we have three indicators: u, v , p.

Let θe be the indicator for an element e with vertex nodes i, j, k: Ee =

(Maximum nodal value − Minimum Nodal Value)e Average difference in max and min nodal values over all elements     max θie , θje , θke − min θie , θje , θke h    i Ee = 1 Pnele n , θ n , θ n − min θ n , θ n , θ n max θ n=1 i j i j k k nele

Introduction

Mesh Generation

Model Results

Meshing Results

Computation Time Results

Simulation Model Results - Verification

Conclusions

Introduction

Mesh Generation

Model Results

Meshing Results

Computation Time Results

Simulation Model Results - Verification 0

0.1

0.2

0.3

0.1

0.4

0.2

0.3

0.4

−4

0

2

2

0.2

1.8

1.8

0.4

1.6

1.6

0.6

1.4

1.4

0.8

1.2

1.2

1

1

1

1.2

0.8

0.8

1.4

0.6

0.6

1.6

0.4

0.4

1.8

0.2

0.2

2

0

0.5

1

0

0

0.5

(d) Velocity Field (e) Velocity tours

1

0

0

−2

0

0.5

2

1

Con- (f) Streamlines and Pressure Contours

Conclusions

Introduction

Mesh Generation

Model Results

Meshing Results

Computation Time Results

1 Obstruction 0

0.2

0.1

0.4

0.2

0.3

0.4

0

0.5

0

2

2

0.2

1.8

1.8

0.4

1.6

1.6

0.6

1.4

1.4

0.8

1.2

1.2

1

1

1

1.2

0.8

0.8

1.4

0.6

0.6

1.6

0.4

0.4

1.8

0.2

0.2

2

0

0.5

1

0

0

0.5

(a) Velocity Field (b) Velocity tours

1

0

0

5

10

0.5

15

20

1

Con- (c) Streamlines and Pressure Contours

Conclusions

Introduction

Mesh Generation

Model Results

Meshing Results

Computation Time Results

5 Obstructions 0

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0

0.8

2

0.2

1.8

0.4

1.6

1.6

0.6

1.4

1.4

0.8

1.2

1.2

1

1

1

0.8

0.8

1.4

0.6

0.6

1.6

0.4

0.4

1.8

0.2

0.2

2

0

1

60

80

1.8

1.2

0.5

40

2

0

0

20

0

0.5

(a) Velocity Field (b) Velocity tours

1

0

0

0.5

1

Con- (c) Streamlines and Pressure Contours

Conclusions

Introduction

Mesh Generation

Model Results

Meshing Results

Computation Time Results

13 obstructions 0

0.5

0.2

1

0.4

0.6

0.8

0

0

2

2

0.2

1.8

1.8

0.4

1.6

1.6

0.6

1.4

1.4

0.8

1.2

1.2

1

1

1

1.2

0.8

0.8

1.4

0.6

0.6

1.6

0.4

0.4

1.8

0.2

0.2

2

0

0

0.5

1

0

0.5

(a) Velocity Field (b) Velocity tours

1

0

0

100

0.5

200

1

Con- (c) Streamlines and Pressure Contours

Conclusions

Introduction

Mesh Generation

Model Results

Meshing Results

Additional Verification

';

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""':"""" t-

b* ' 1 , :, ' : : '

[TQ Education and Training Ltd.]

-.

Computation Time Results

Conclusions

Introduction

Mesh Generation

Model Results

Meshing Results

Computation Time Results

Meshing Results Pressure

Initial Mesh

Horizontal velocity

Vertical velocity

Conclusions

Introduction

Mesh Generation

Model Results

Meshing Results

Computation Time Results

Meshing Results Pressure

Refinement 1

Horizontal velocity

Vertical velocity

Conclusions

Introduction

Mesh Generation

Model Results

Meshing Results

Computation Time Results

Meshing Results Pressure

Refinement 2

Horizontal velocity

Vertical velocity

Conclusions

Introduction

Mesh Generation

Model Results

Meshing Results

Computation Time Results

Meshing Results Pressure

Refinement 3

Done

Horizontal velocity

Vertical velocity

Conclusions

Introduction

Mesh Generation

Model Results

Meshing Results

Computation Time Results

Meshing Results Pressure

Refinement 4

Done

Horizontal velocity

Vertical velocity

Conclusions

Introduction

Mesh Generation

Model Results

Meshing Results

Computation Time Results

Meshing Results Pressure

Horizontal velocity

Refinement 5

Done

Done

Vertical velocity

Conclusions

Introduction

Mesh Generation

Model Results

Meshing Results

Computation Time Results

Meshing Results Pressure

Horizontal velocity

Vertical velocity

Refinement 6

Done

Done

Done

Conclusions

Introduction

Mesh Generation

Model Results

Meshing Results 5 Hole

Meshing Results

Computation Time Results

Conclusions

Introduction

Mesh Generation ModelAND Results Meshing Results Computation Time BRACKEN: ADAPTIVE MESHING SPARSE MATRIX STORAGE FOR STOKES FLOW 11 Results BRACKEN: ADAPTIVE MESHING AND SPARSE MATRIX STORAGE FOR STOKES FLOW

13

Pressure as error indicator

Figure 16. Sample Refinement process using pressure as the error indicator. Figure 16. Sample Refinement process using pressure as the error indicator.

Meshing Results 5 Hole

Horizontal velocity as error indicator Figure 17. Sample Refinement process with 5 obstructions using pressure as the error indicator.

Figure 17. Sample Refinement process with 5 obstructions using horizonal component of velocity as the error indicator.

Vertical as errorprocess indicator Figure 17. velocity Sample Refinement with 5 obstructions using horizonal component of velocity as the error indicator. Figure 18. Sample Refinement process with 13 obstructions using horizonal component of velocity as the error indicator.

Conclusions

Figure 17. 19. Sample Refinement process with 5Model obstructions using horizonal pressure ascomponent the error indicator. of velocity as Computation the error indicator. Introduction Mesh Generation Results Meshing Results Time Results

Conclusions

Figure 21. Sample Refinement process with 13 obstructions using vertical component of velocity as the error indicator.

Pressure as error indicator

Meshing Results 13 Holes

Figure 18. Sample Refinement process with 5 obstructions using vertical component of velocity as the error indicator.

Horizontal velocity as error indicator Figure 22. Sample Refinement process with 13 obstructions using pressure as the error indicator.

BRACKEN: ADAPTIVE MESHING AND SPARSE MATRIX STORAGE FOR STOKES FLOW

Vertical as errorprocess indicator Figure 20. velocity Sample Refinement with 13 obstructions using horizonal component of velocity as the error indicator.

15

●

● ●

25

● ● ●

●

●

●

50 45

Horizontal velocity Vertical velocity Pressure

40

●

●

35

●

35 30

●

Computation Time Results

●

● ● ● ● ●

30

40

●

Average Percent Realative Elemental Error

Horizontal velocity Vertical velocity Pressure

●

Meshing Results

●

● ●

25

● ●

Model Results

45

50

Mesh Generation

20

Average Percent Realative Elemental Error

Introduction

1

2

3

4

●

5

6

7

2

3

4

5

5 Obstructions

Horizontal velocity Vertical velocity Pressure

40

45

50

1 Obstruction ●

1

Mesh Refinement Step

Mesh Refinement Step

●

35

Error Results

30

● ● ●

●

●

25

●

●

●

●

20

Average Percent Realative Elemental Error

●

●

●

1

2

●

3 Mesh Refinement Step

13 Obstructions

●

4

5

Conclusions

Introduction

Mesh Generation

Model Results

Meshing Results

Computation Time Results

Conclusions

Computation Time Results - Sparse vs. Dense Solvers

Introduction

Mesh Generation

Model Results

Meshing Results

Computation Time Results

Conclusions

Direct Dense Solver (LAPACK's DGETRS) Iterative Sparse Solver (SLAP's DGMRES)

20 15 10 5 0

Solution Time (s)

25

30

Computation Time Results - Sparse vs. Dense Solvers

1000

2000

3000

4000

5000

6000

7000

# of Unknowns

Sparseness

Computation Time

Introduction

Mesh Generation

Model Results

Meshing Results

Computation Time Results

Conclusions

Conclusions

I

I

Reproduced well know flow patterns around cylinders at very low Reynolds numbers. Implemented unstructured adaptive meshing algorithm. I I

I

Pressure performed well as an error indicator Different Error indicators produced very different meshes.

Sparse matrix storage and equation solver drastically reduced solution time.