[presentation] stokes flow finite element model

Introduction Mesh Generation Model Results Meshing Results Computation Time Results Conclusions

Unstructured adaptive mesh generation andsparse matrix storage applied to Stokes flow

around cylinders

Cameron Bracken1

E521May 1, 2008

1Humboldt State University

Introduction Mesh Generation Model Results Meshing Results Computation Time Results Conclusions

Project Goals

1. Solve the 2D Steady Stokes equations via finite elements

2. Investigate the placement of cylindrical obstructions in theflow field

3. Adaptively generate the finite element mesh

4. Utilize sparse matrix storage

Introduction Mesh Generation Model Results Meshing Results Computation Time Results Conclusions

Stokes equations - Simplification

Start with the NS equations:

ρVt − µ∇2V + ρ(V · ∇)V +∇p = 0

Assume steady,

µ∇2V + ρ(V · ∇)V +∇p = 0

Assume irrotational flow,

ν∇2V +∇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/orslow moving fluid (RE � 1,creeping flow).

Introduction Mesh Generation Model Results Meshing Results Computation Time Results Conclusions

Stokes equations - SimplificationStart with the NS equations:

ρVt − µ∇2V + ρ(V · ∇)V +∇p = 0

Assume steady,

µ∇2V + ρ(V · ∇)V +∇p = 0

Assume irrotational flow,

ν∇2V +∇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/orslow moving fluid (RE � 1,creeping flow).

Introduction Mesh Generation Model Results Meshing Results Computation Time Results Conclusions

Stokes equations - SimplificationStart with the NS equations:

ρVt − µ∇2V + ρ(V · ∇)V +∇p = 0

Assume steady,

µ∇2V + ρ(V · ∇)V +∇p = 0

Assume irrotational flow,

ν∇2V +∇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/orslow moving fluid (RE � 1,creeping flow).

Introduction Mesh Generation Model Results Meshing Results Computation Time Results Conclusions

Stokes equations - SimplificationStart with the NS equations:

ρVt − µ∇2V + ρ(V · ∇)V +∇p = 0

Assume steady,

µ∇2V + ρ(V · ∇)V +∇p = 0

Assume irrotational flow,

ν∇2V +∇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/orslow moving fluid (RE � 1,creeping flow).

Introduction Mesh Generation Model Results Meshing Results Computation Time Results Conclusions

Stokes equations - SimplificationStart with the NS equations:

ρVt − µ∇2V + ρ(V · ∇)V +∇p = 0

Assume steady,

µ∇2V + ρ(V · ∇)V +∇p = 0

Assume irrotational flow,

ν∇2V +∇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/orslow moving fluid (RE � 1,creeping flow).

Introduction Mesh Generation Model Results Meshing Results Computation Time Results Conclusions

Stokes equations - SimplificationStart with the NS equations:

ρVt − µ∇2V + ρ(V · ∇)V +∇p = 0

Assume steady,

µ∇2V + ρ(V · ∇)V +∇p = 0

Assume irrotational flow,

ν∇2V +∇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/orslow moving fluid (RE � 1,creeping flow).

Introduction Mesh Generation Model Results Meshing Results Computation Time Results Conclusions

Formal Problem Statement

Scalar Equations:

−ν ∂2u

∂x2− ν ∂

2u

∂y2+∂p

∂x= 0

−ν ∂2v

∂x2− ν ∂

2v

∂y2+∂p

∂y= 0

∂u

∂x+∂v

∂y= 0

∂u∂n = ∂v

∂n = 0

v = k x (1− x)u = 0

u=

v=

0 u=

v=

0

u = v = 0

p = 0

Introduction Mesh Generation Model Results Meshing Results Computation Time Results Conclusions

Formal Problem Statement

Scalar Equations:

−ν ∂2u

∂x2− ν ∂

2u

∂y2+∂p

∂x= 0

−ν ∂2v

∂x2− ν ∂

2v

∂y2+∂p

∂y= 0

∂u

∂x+∂v

∂y= 0

∂u∂n = ∂v

∂n = 0

v = k x (1− x)u = 0

u=

v=

0 u=

v=

0

u = v = 0

p = 0

Introduction Mesh Generation Model Results Meshing Results Computation Time Results Conclusions

Basis Functions

(xi , yi )

(xj , yj)

(xk , yk)

(xi , yi )

(xj , yj)

(xk , yk)

(xik , yik)

(xjk , yjk)

(xij , yij)

Linear Pressure Element. Quadratic Velocity Element.

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

Introduction Mesh Generation Model Results Meshing Results Computation Time Results Conclusions

Constrained Adaptive Mesh Generation

Mesh Generation

Unstructured grid generation 203

Fig. 8.16 Delaunay triangulation. Voronoi-segment algorithm.

Fig. 8.17 Delaunay triangulation for an airfoil. Voronoi-segment algorithm.

Fig. 8.18 Delaunay triangulation in an annulus. Voronoi-segment algorithm.

8.3 Advancing front technique (AFT)

8.3.1 Introduction

In certain problems, for example the computation of viscous flow solutions, the useof Delaunay triangulation for generation of unstructured grids may not be satisfactory.This could be due to the need to create triangular elements with high aspect ratio inboundary-layer regions, which would be difficult with Delaunay Triangulation alone.Another problem with Delaunay triangulation, as mentioned above, is that, even thoughboundary nodes will be vertices in the final triangulation, there is no guarantee that

Unstructured

Unstructured Adaptive

178 Basic Structured Grid Generation

1. Area.trid.fThis program solves the Euler-Lagrange equations of the form (6.51) for the areafunctional with weight function equal to a constant, so that the matrices in (6.48)are given by eqn (6.53) with S = 0. Figure 6.3 shows an example of the resultinggrid in a trapezium.

Fig. 6.3 Area functional.

Fig. 6.4 Area functional.

Fig. 6.5 Area-orthogonality.

Fig. 6.6 Area-orthogonality.

Structured

Structured Adaptive

[Basic Structured Grid Generation, Farrashkhalvat and Miles 2003]

Introduction Mesh Generation Model Results Meshing Results Computation Time Results Conclusions

Constrained Adaptive Mesh Generation

Mesh Generation

Unstructured grid generation 203

Fig. 8.16 Delaunay triangulation. Voronoi-segment algorithm.

Fig. 8.17 Delaunay triangulation for an airfoil. Voronoi-segment algorithm.

Fig. 8.18 Delaunay triangulation in an annulus. Voronoi-segment algorithm.

8.3 Advancing front technique (AFT)

8.3.1 Introduction

In certain problems, for example the computation of viscous flow solutions, the useof Delaunay triangulation for generation of unstructured grids may not be satisfactory.This could be due to the need to create triangular elements with high aspect ratio inboundary-layer regions, which would be difficult with Delaunay Triangulation alone.Another problem with Delaunay triangulation, as mentioned above, is that, even thoughboundary nodes will be vertices in the final triangulation, there is no guarantee that

Unstructured

Unstructured Adaptive

178 Basic Structured Grid Generation

1. Area.trid.fThis program solves the Euler-Lagrange equations of the form (6.51) for the areafunctional with weight function equal to a constant, so that the matrices in (6.48)are given by eqn (6.53) with S = 0. Figure 6.3 shows an example of the resultinggrid in a trapezium.

Fig. 6.3 Area functional.

Fig. 6.4 Area functional.

Fig. 6.5 Area-orthogonality.

Fig. 6.6 Area-orthogonality.

Structured

Structured Adaptive

[Basic Structured Grid Generation, Farrashkhalvat and Miles 2003]

Introduction Mesh Generation Model Results Meshing Results Computation Time Results Conclusions

Constrained Adaptive Mesh Generation

Mesh Generation

Unstructured grid generation 203

Fig. 8.16 Delaunay triangulation. Voronoi-segment algorithm.

Fig. 8.17 Delaunay triangulation for an airfoil. Voronoi-segment algorithm.

Fig. 8.18 Delaunay triangulation in an annulus. Voronoi-segment algorithm.

8.3 Advancing front technique (AFT)

8.3.1 Introduction

In certain problems, for example the computation of viscous flow solutions, the useof Delaunay triangulation for generation of unstructured grids may not be satisfactory.This could be due to the need to create triangular elements with high aspect ratio inboundary-layer regions, which would be difficult with Delaunay Triangulation alone.Another problem with Delaunay triangulation, as mentioned above, is that, even thoughboundary nodes will be vertices in the final triangulation, there is no guarantee that

Unstructured

Unstructured Adaptive

178 Basic Structured Grid Generation

1. Area.trid.fThis program solves the Euler-Lagrange equations of the form (6.51) for the areafunctional with weight function equal to a constant, so that the matrices in (6.48)are given by eqn (6.53) with S = 0. Figure 6.3 shows an example of the resultinggrid in a trapezium.

Fig. 6.3 Area functional.

Fig. 6.4 Area functional.

Fig. 6.5 Area-orthogonality.

Fig. 6.6 Area-orthogonality.

Structured

Structured Adaptive

[Basic Structured Grid Generation, Farrashkhalvat and Miles 2003]

Introduction Mesh Generation Model Results Meshing Results Computation Time Results Conclusions

Constrained Adaptive Mesh Generation

Mesh Generation

Unstructured grid generation 203

Fig. 8.16 Delaunay triangulation. Voronoi-segment algorithm.

Fig. 8.17 Delaunay triangulation for an airfoil. Voronoi-segment algorithm.

Fig. 8.18 Delaunay triangulation in an annulus. Voronoi-segment algorithm.

8.3 Advancing front technique (AFT)

8.3.1 Introduction

In certain problems, for example the computation of viscous flow solutions, the useof Delaunay triangulation for generation of unstructured grids may not be satisfactory.This could be due to the need to create triangular elements with high aspect ratio inboundary-layer regions, which would be difficult with Delaunay Triangulation alone.Another problem with Delaunay triangulation, as mentioned above, is that, even thoughboundary nodes will be vertices in the final triangulation, there is no guarantee that

Unstructured

Unstructured Adaptive

178 Basic Structured Grid Generation

1. Area.trid.fThis program solves the Euler-Lagrange equations of the form (6.51) for the areafunctional with weight function equal to a constant, so that the matrices in (6.48)are given by eqn (6.53) with S = 0. Figure 6.3 shows an example of the resultinggrid in a trapezium.

Fig. 6.3 Area functional.

Fig. 6.4 Area functional.

Fig. 6.5 Area-orthogonality.

Fig. 6.6 Area-orthogonality.

Structured

Structured Adaptive

[Basic Structured Grid Generation, Farrashkhalvat and Miles 2003]

Introduction Mesh Generation Model Results Meshing Results Computation Time Results Conclusions

Constrained Adaptive Mesh Generation

Mesh Generation

Unstructured grid generation 203

Fig. 8.16 Delaunay triangulation. Voronoi-segment algorithm.

Fig. 8.17 Delaunay triangulation for an airfoil. Voronoi-segment algorithm.

Fig. 8.18 Delaunay triangulation in an annulus. Voronoi-segment algorithm.

8.3 Advancing front technique (AFT)

8.3.1 Introduction

In certain problems, for example the computation of viscous flow solutions, the useof Delaunay triangulation for generation of unstructured grids may not be satisfactory.This could be due to the need to create triangular elements with high aspect ratio inboundary-layer regions, which would be difficult with Delaunay Triangulation alone.Another problem with Delaunay triangulation, as mentioned above, is that, even thoughboundary nodes will be vertices in the final triangulation, there is no guarantee that

Unstructured

Unstructured Adaptive

178 Basic Structured Grid Generation

1. Area.trid.fThis program solves the Euler-Lagrange equations of the form (6.51) for the areafunctional with weight function equal to a constant, so that the matrices in (6.48)are given by eqn (6.53) with S = 0. Figure 6.3 shows an example of the resultinggrid in a trapezium.

Fig. 6.3 Area functional.

Fig. 6.4 Area functional.

Fig. 6.5 Area-orthogonality.

Fig. 6.6 Area-orthogonality.

Structured

Structured Adaptive

[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 ofInitial Nodes

Mesh Generation

Solve Stokes EquationsIdentify

Elements toRefine

Elements torefine <m

Refine Mesh

no

Stopyes

Introduction Mesh Generation Model Results Meshing Results Computation Time Results Conclusions

Adaptive mesh framework (a.k.a. Error controlled FE)Generate

Sparse Set ofInitial Nodes

Mesh Generation

Solve Stokes EquationsIdentify

Elements toRefine

Elements torefine <m

Refine Mesh

no

Stopyes

Introduction Mesh Generation Model Results Meshing Results Computation Time Results Conclusions

Adaptive mesh framework (a.k.a. Error controlled FE)Generate

Sparse Set ofInitial Nodes

Mesh Generation

Solve Stokes Equations

IdentifyElements to

Refine

Elements torefine <m

Refine Mesh

no

Stopyes

Introduction Mesh Generation Model Results Meshing Results Computation Time Results Conclusions

Adaptive mesh framework (a.k.a. Error controlled FE)Generate

Sparse Set ofInitial Nodes

Mesh Generation

Solve Stokes EquationsIdentify

Elements toRefine

Elements torefine <m

Refine Mesh

no

Stopyes

Introduction Mesh Generation Model Results Meshing Results Computation Time Results Conclusions

Adaptive mesh framework (a.k.a. Error controlled FE)Generate

Sparse Set ofInitial Nodes

Mesh Generation

Solve Stokes EquationsIdentify

Elements toRefine

Elements torefine <m

Refine Mesh

no

Stopyes

Introduction Mesh Generation Model Results Meshing Results Computation Time Results Conclusions

Adaptive mesh framework (a.k.a. Error controlled FE)Generate

Sparse Set ofInitial Nodes

Mesh Generation

Solve Stokes EquationsIdentify

Elements toRefine

Elements torefine <m

Refine Mesh

no

Stopyes

Introduction Mesh Generation Model Results Meshing Results Computation Time Results Conclusions

Adaptive mesh framework (a.k.a. Error controlled FE)Generate

Sparse Set ofInitial Nodes

Mesh Generation

Solve Stokes EquationsIdentify

Elements toRefine

Elements torefine <m

Refine Mesh

no

Stopyes

Introduction Mesh Generation Model Results Meshing Results Computation Time Results Conclusions

Adaptive mesh framework (a.k.a. Error controlled FE)Generate

Sparse Set ofInitial Nodes

Mesh Generation

Solve Stokes EquationsIdentify

Elements toRefine

Elements torefine <m

Refine Mesh

no

Stopyes

How is this efficient?

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 anindicator of elemental error.

I These measures are called “a posteriori” (after the fact)estimates because we must solve the problem before we getan 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

Ee =max

(θei , θ

ej , θ

ek

)−min

(θei , θ

ej , θ

ek

)1

nele

∑nelen=1

[max

(θni , θ

nj , θ

nk

)−min

(θni , θ

nj , θ

nk

)]

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 anindicator of elemental error.

I These measures are called “a posteriori” (after the fact)estimates because we must solve the problem before we getan 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

Ee =max

(θei , θ

ej , θ

ek

)−min

(θei , θ

ej , θ

ek

)1

nele

∑nelen=1

[max

(θni , θ

nj , θ

nk

)−min

(θni , θ

nj , θ

nk

)]

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 anindicator of elemental error.

I These measures are called “a posteriori” (after the fact)estimates because we must solve the problem before we getan 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

Ee =max

(θei , θ

ej , θ

ek

)−min

(θei , θ

ej , θ

ek

)1

nele

∑nelen=1

[max

(θni , θ

nj , θ

nk

)−min

(θni , θ

nj , θ

nk

)]

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 anindicator of elemental error.

I These measures are called “a posteriori” (after the fact)estimates because we must solve the problem before we getan 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

Ee =max

(θei , θ

ej , θ

ek

)−min

(θei , θ

ej , θ

ek

)1

nele

∑nelen=1

[max

(θni , θ

nj , θ

nk

)−min

(θni , θ

nj , θ

nk

)]

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 anindicator of elemental error.

I These measures are called “a posteriori” (after the fact)estimates because we must solve the problem before we getan 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

Ee =max

(θei , θ

ej , θ

ek

)−min

(θei , θ

ej , θ

ek

)1

nele

∑nelen=1

[max

(θni , θ

nj , θ

nk

)−min

(θni , θ

nj , θ

nk

)]

Introduction Mesh Generation Model Results Meshing Results Computation Time Results Conclusions

Simulation Model Results - Verification

0 0.5 1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.1 0.2 0.3 0.4

(a) Velocity Field

0 0.5 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0.1 0.2 0.3 0.4

(b) Velocity Con-tours

0 0.5 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

−4 −2 0 2

(c) Streamlines andPressure Contours

Introduction Mesh Generation Model Results Meshing Results Computation Time Results Conclusions

Simulation Model Results - Verification

0 0.5 1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.1 0.2 0.3 0.4

(d) Velocity Field

0 0.5 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0.1 0.2 0.3 0.4

(e) Velocity Con-tours

0 0.5 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

−4 −2 0 2

(f) Streamlines andPressure Contours

Introduction Mesh Generation Model Results Meshing Results Computation Time Results Conclusions

1 Obstruction

0 0.5 1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.2 0.4

(a) Velocity Field

0 0.5 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0.1 0.2 0.3 0.4 0.5

(b) Velocity Con-tours

0 0.5 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 5 10 15 20

(c) Streamlines andPressure Contours

Introduction Mesh Generation Model Results Meshing Results Computation Time Results Conclusions

5 Obstructions

0 0.5 1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.2 0.4 0.6 0.8

(a) Velocity Field

0 0.5 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0.2 0.4 0.6 0.8

(b) Velocity Con-tours

0 0.5 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 20 40 60 80

(c) Streamlines andPressure Contours

Introduction Mesh Generation Model Results Meshing Results Computation Time Results Conclusions

13 obstructions

0 0.5 1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.5 1

(a) Velocity Field

0 0.5 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0.2 0.4 0.6 0.8

(b) Velocity Con-tours

0 0.5 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 100 200

(c) Streamlines andPressure Contours

Introduction Mesh Generation Model Results Meshing Results Computation Time Results Conclusions

Additional Verification

'; ""':"""" 'i'l'

- t-

b*- -.

'1, : , ' : : '

[TQ Education and Training Ltd.]

Introduction Mesh Generation Model Results Meshing Results Computation Time Results Conclusions

Meshing ResultsPressure Horizontal velocity Vertical velocity

Initial Mesh

Introduction Mesh Generation Model Results Meshing Results Computation Time Results Conclusions

Meshing ResultsPressure Horizontal velocity Vertical velocity

Refinement 1

Introduction Mesh Generation Model Results Meshing Results Computation Time Results Conclusions

Meshing ResultsPressure Horizontal velocity Vertical velocity

Refinement 2

Introduction Mesh Generation Model Results Meshing Results Computation Time Results Conclusions

Meshing ResultsPressure Horizontal velocity Vertical velocity

Refinement 3

Done

Introduction Mesh Generation Model Results Meshing Results Computation Time Results Conclusions

Meshing ResultsPressure Horizontal velocity Vertical velocity

Refinement 4

Done

Introduction Mesh Generation Model Results Meshing Results Computation Time Results Conclusions

Meshing ResultsPressure Horizontal velocity Vertical velocity

Refinement 5

Done Done

Introduction Mesh Generation Model Results Meshing Results Computation Time Results Conclusions

Meshing ResultsPressure Horizontal velocity Vertical velocity

Refinement 6

Done Done Done

Introduction Mesh Generation Model Results Meshing Results Computation Time Results Conclusions

Meshing Results 5 Hole

Introduction Mesh Generation Model Results Meshing Results Computation Time Results Conclusions

Pressure as error indicatorBRACKEN: ADAPTIVE MESHING AND SPARSE MATRIX STORAGE FOR STOKES FLOW 13

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

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

Meshing Results 5 HoleHorizontal velocity as error indicator

BRACKEN: ADAPTIVE MESHING AND SPARSE MATRIX STORAGE FOR STOKES FLOW 11

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

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

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

Vertical velocity as error indicator

BRACKEN: ADAPTIVE MESHING AND SPARSE MATRIX STORAGE FOR STOKES FLOW 11

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

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

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

Introduction Mesh Generation Model Results Meshing Results Computation Time Results Conclusions

Pressure as error indicator

BRACKEN: ADAPTIVE MESHING AND SPARSE MATRIX STORAGE FOR STOKES FLOW 15

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

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

Meshing Results 13 HolesHorizontal velocity as error indicator

14 BRACKEN: ADAPTIVE MESHING AND SPARSE MATRIX STORAGE FOR STOKES FLOW

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

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

Figure 19. Sample Refinement process with 5 obstructions using pressure as the error indicator.

Figure 20. Sample Refinement process with 13 obstructions using horizonal component of velocity as the error indicator.Vertical velocity as error indicatorBRACKEN: ADAPTIVE MESHING AND SPARSE MATRIX STORAGE FOR STOKES FLOW 15

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

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

Introduction Mesh Generation Model Results Meshing Results Computation Time Results Conclusions

●

●

●

●

●●

1 2 3 4 5 6 7

2025

3035

4045

50

Mesh Refinement Step

Ave

rage

Per

cent

Rea

lativ

e El

emen

tal E

rror

●

● ●

●

●● ●

●

●

●

●

Horizontal velocityVertical velocityPressure

●

●

●

●●

1 2 3 4 5

2530

3540

4550

Mesh Refinement Step

Ave

rage

Per

cent

Rea

lativ

e El

emen

tal E

rror

●

●

●

●

●

●

●

●●

Horizontal velocityVertical velocityPressure

1 Obstruction 5 Obstructions

●

●

●

●

1 2 3 4 5

2025

3035

4045

50

Mesh Refinement Step

Ave

rage

Per

cent

Rea

lativ

e El

emen

tal E

rror

●

●

●

●

●

●

●●

●

Horizontal velocityVertical velocityPressure

Error Results

13 Obstructions

Introduction Mesh Generation Model Results Meshing Results Computation Time Results Conclusions

Computation Time Results - Sparse vs. Dense Solvers

1000 2000 3000 4000 5000 6000 7000

05

1015

2025

30

# of Unknowns

Solu

tion

Tim

e (s

)

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

Sparseness Computation Time

Introduction Mesh Generation Model Results Meshing Results Computation Time Results Conclusions

Computation Time Results - Sparse vs. Dense Solvers

1000 2000 3000 4000 5000 6000 7000

05

1015

2025

30

# of Unknowns

Solu

tion

Tim

e (s

)

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

Sparseness Computation Time

Introduction Mesh Generation Model Results Meshing Results Computation Time Results Conclusions

Conclusions

I Reproduced well know flow patterns around cylinders at verylow Reynolds numbers.

I Implemented unstructured adaptive meshing algorithm.I Pressure performed well as an error indicatorI Different Error indicators produced very different meshes.

I Sparse matrix storage and equation solver drastically reducedsolution time.