[presentation] stokes flow finite element model
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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
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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.
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