[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

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

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).

Stokes equations - SimplificationStart with the NS equations:

ρVt − µ∇2V + ρ(V · ∇)V +∇p = 0

Assume steady,

µ∇2V + ρ(V · ∇)V +∇p = 0

ν∇2V +∇p = 0 (1)

∇ · V = 0 (2)

ρVt − µ∇2V + ρ(V · ∇)V +∇p = 0

Assume steady,

µ∇2V + ρ(V · ∇)V +∇p = 0

ν∇2V +∇p = 0 (1)

∇ · V = 0 (2)

ρVt − µ∇2V + ρ(V · ∇)V +∇p = 0

Assume steady,

µ∇2V + ρ(V · ∇)V +∇p = 0

ν∇2V +∇p = 0 (1)

And we arrive at the Stokes equations.

And for the continuity equation, assume incompressible,

∇ · V = 0 (2)

ρVt − µ∇2V + ρ(V · ∇)V +∇p = 0

Assume steady,

µ∇2V + ρ(V · ∇)V +∇p = 0

ν∇2V +∇p = 0 (1)

∇ · V = 0 (2)

ρVt − µ∇2V + ρ(V · ∇)V +∇p = 0

Assume steady,

µ∇2V + ρ(V · ∇)V +∇p = 0

ν∇2V +∇p = 0 (1)

∇ · V = 0 (2)

Formal Problem Statement

Scalar Equations:

−ν ∂2u

∂x2− ν ∂

∂y2+∂p

∂x= 0

−ν ∂2v

∂x2− ν ∂

∂y2+∂p

∂y= 0

∂x+∂v

∂y= 0

∂u∂n = ∂v

∂n = 0

v = k x (1− x)u = 0

u = v = 0

Formal Problem Statement

Scalar Equations:

−ν ∂2u

∂x2− ν ∂

∂y2+∂p

∂x= 0

−ν ∂2v

∂x2− ν ∂

∂y2+∂p

∂y= 0

∂x+∂v

∂y= 0

∂u∂n = ∂v

∂n = 0

v = k x (1− x)u = 0

u = v = 0

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?

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. 6.5 Area-orthogonality.

Structured

Structured Adaptive

[Basic Structured Grid Generation, Farrashkhalvat and Miles 2003]

Mesh Generation

8.3.1 Introduction

Unstructured

Structured

Structured Adaptive

Mesh Generation

8.3.1 Introduction

Unstructured

Structured

Structured Adaptive

Mesh Generation

8.3.1 Introduction

Unstructured

Structured

Structured Adaptive

Mesh Generation

8.3.1 Introduction

Unstructured

Structured

Structured Adaptive

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

Stopyes

Mesh Generation

Elements toRefine

Refine Mesh

Stopyes

Mesh Generation

Solve Stokes Equations

IdentifyElements to

Refine

Refine Mesh

Stopyes

Mesh Generation

Elements toRefine

Refine Mesh

Stopyes

Mesh Generation

Elements toRefine

Refine Mesh

Stopyes

Mesh Generation

Elements toRefine

Refine Mesh

Stopyes

Mesh Generation

Elements toRefine

Refine Mesh

Stopyes

Mesh Generation

Elements toRefine

Refine Mesh

Stopyes

How is this efficient?

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

)−min

(θei , θ

ej , θ

∑nelen=1

(θni , θ

nj , θ

)−min

(θni , θ

nj , θ

Ee =max

(θei , θ

ej , θ

)−min

(θei , θ

ej , θ

∑nelen=1

(θni , θ

nj , θ

)−min

(θni , θ

nj , θ

Ee =max

(θei , θ

ej , θ

)−min

(θei , θ

ej , θ

∑nelen=1

(θni , θ

nj , θ

)−min

(θni , θ

nj , θ

Ee =max

(θei , θ

ej , θ

)−min

(θei , θ

ej , θ

∑nelen=1

(θni , θ

nj , θ

)−min

(θni , θ

nj , θ

Ee =max

(θei , θ

ej , θ

)−min

(θei , θ

ej , θ

∑nelen=1

(θni , θ

nj , θ

)−min

(θni , θ

nj , θ

Simulation Model Results - Verification

0 0.5 1

0 0.1 0.2 0.3 0.4

(a) Velocity Field

0 0.5 10

0.1 0.2 0.3 0.4

(b) Velocity Con-tours

0 0.5 10

−4 −2 0 2

(c) Streamlines andPressure Contours

Simulation Model Results - Verification

0 0.5 1

0 0.1 0.2 0.3 0.4

(d) Velocity Field

0 0.5 10

0.1 0.2 0.3 0.4

(e) Velocity Con-tours

0 0.5 10

−4 −2 0 2

(f) Streamlines andPressure Contours

1 Obstruction

0 0.5 1

0 0.2 0.4

(a) Velocity Field

0 0.5 10

0.1 0.2 0.3 0.4 0.5

0 0.5 10

0 5 10 15 20

5 Obstructions

0 0.5 1

0 0.2 0.4 0.6 0.8

(a) Velocity Field

0 0.5 10

0.2 0.4 0.6 0.8

0 0.5 10

0 20 40 60 80

13 obstructions

0 0.5 1

(a) Velocity Field

0 0.5 10

0.2 0.4 0.6 0.8

0 0.5 10

0 100 200

Additional Verification

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

b*- -.

'1, : , ' : : '

[TQ Education and Training Ltd.]

Meshing ResultsPressure Horizontal velocity Vertical velocity

Initial Mesh

Refinement 1

Refinement 2

Refinement 3

Refinement 4

Refinement 5

Done Done

Refinement 6

Done Done Done

Meshing Results 5 Hole

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

Vertical velocity as error indicator

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

Pressure as error indicator

Meshing Results 13 HolesHorizontal velocity as error indicator

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

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

●●

1 2 3 4 5 6 7

Mesh Refinement Step

● ●

●● ●

Horizontal velocityVertical velocityPressure

●●

1 2 3 4 5

●●

1 Obstruction 5 Obstructions

1 2 3 4 5

●●

Error Results

13 Obstructions

Computation Time Results - Sparse vs. Dense Solvers

1000 2000 3000 4000 5000 6000 7000

# of Unknowns

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

Sparseness Computation Time

Computation Time Results - Sparse vs. Dense Solvers

1000 2000 3000 4000 5000 6000 7000

# of Unknowns

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

Sparseness Computation Time

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.

[presentation] stokes flow finite element model

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