Anisotropic quality measures and adaptation for polygonal meshes - - PowerPoint PPT Presentation

Anisotropic quality measures and adaptation for polygonal meshes

Yanqiu Wang, Oklahoma State University joint work with Weizhang Huang Oct 2015, POEMs, Georgia Tech.

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Consider the following function

1 0.8 0.6 0.4 0.2 0.5 3 5 4.5 4 3.5 1 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

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Goal: Find a good polygonal mesh for this function

Solution:

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Goal: Find a good polygonal mesh for this function

Solution:

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Method: Moving mesh algorithm

Idea:

Define an objective function I(T ) such that its minimal values occur

• n ideally “good” meshes;

Start from a random initial mesh, and use an iterative method to approximate the minimization problem by moving vertices of the mesh.

References:

Adaptive Moving Mesh Methods, W. Huang and R.D. Russell, Springer 2011 Adaptivity with moving grids, C.J. Budd, W. Huang, and R.D., Russel. Acta Numerica, 2009.

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Method: Moving mesh algorithm

Idea:

Define an objective function I(T ) such that its minimal values occur

• n ideally “good” meshes;

Start from a random initial mesh, and use an iterative method to approximate the minimization problem by moving vertices of the mesh.

References:

Adaptive Moving Mesh Methods, W. Huang and R.D. Russell, Springer 2011 Adaptivity with moving grids, C.J. Budd, W. Huang, and R.D., Russel. Acta Numerica, 2009.

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Method: Moving mesh algorithm

Idea:

Define an objective function I(T ) such that its minimal values occur

• n ideally “good” meshes;

Start from a random initial mesh, and use an iterative method to approximate the minimization problem by moving vertices of the mesh.

References:

Adaptive Moving Mesh Methods, W. Huang and R.D. Russell, Springer 2011 Adaptivity with moving grids, C.J. Budd, W. Huang, and R.D., Russel. Acta Numerica, 2009.

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Question: What is a “good” mesh?

The answer depends on the applications:

To minimize the interpolation error? To minimize the condition number of stiffness matrix? Or others?

We focus on minimizing the L2 interpolation error. (Can also be optimized for H1 seminorm) In isotropic case, a “good” mesh should have elements

regular in shape uniform in size (equidistribution, de Boor 1973)

In anisotropic case, a “good” mesh should have shape-regularity and uniformity measured in an anisotropic metric

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Question: What is a “good” mesh?

The answer depends on the applications:

To minimize the interpolation error? To minimize the condition number of stiffness matrix? Or others?

We focus on minimizing the L2 interpolation error. (Can also be optimized for H1 seminorm) In isotropic case, a “good” mesh should have elements

regular in shape uniform in size (equidistribution, de Boor 1973)

In anisotropic case, a “good” mesh should have shape-regularity and uniformity measured in an anisotropic metric

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Question: What is a “good” mesh?

The answer depends on the applications:

To minimize the interpolation error? To minimize the condition number of stiffness matrix? Or others?

We focus on minimizing the L2 interpolation error. (Can also be optimized for H1 seminorm) In isotropic case, a “good” mesh should have elements

regular in shape uniform in size (equidistribution, de Boor 1973)

In anisotropic case, a “good” mesh should have shape-regularity and uniformity measured in an anisotropic metric

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Question: What is a “good” mesh?

The answer depends on the applications:

To minimize the interpolation error? To minimize the condition number of stiffness matrix? Or others?

We focus on minimizing the L2 interpolation error. (Can also be optimized for H1 seminorm) In isotropic case, a “good” mesh should have elements

regular in shape uniform in size (equidistribution, de Boor 1973)

In anisotropic case, a “good” mesh should have shape-regularity and uniformity measured in an anisotropic metric

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Anisotropic metric: M is a SPD matrix

Reference element TC and physical element T

F ξ x T T

C

We say “ T under metric M ⇐ ⇒ TC under identity metric” if σeC

i , eC j = ei, ejM

which implies σeC

i , eC j = JeC i , JeC j M

⇒ JtMJ = σI

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Anisotropic metric: M is a SPD matrix

Reference element TC and physical element T

F ξ x T T

C

We say “ T under metric M ⇐ ⇒ TC under identity metric” if σeC

i , eC j = ei, ejM

which implies σeC

i , eC j = JeC i , JeC j M

⇒ JtMJ = σI

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Mesh quality measure

Note that JtMJ = σI is equivalent to   

1 2trace(JtMJ) = det(JtMJ)1/2

(Alignment) det(J)

• det(M) = σ

(Equidistribution) Define qali(x) = trace(JtMJ) 2 det(JtMJ)1/2 , qeq(x) = det(J)

• det(M)

σ and Qali = max

x∈Ω qali(x),

Qeq = max

x∈Ω qeq(x)

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Mesh quality measure

Note that JtMJ = σI is equivalent to   

1 2trace(JtMJ) = det(JtMJ)1/2

(Alignment) det(J)

• det(M) = σ

(Equidistribution) Define qali(x) = trace(JtMJ) 2 det(JtMJ)1/2 , qeq(x) = det(J)

• det(M)

σ and Qali = max

x∈Ω qali(x),

Qeq = max

x∈Ω qeq(x)

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Mesh quality measure

To ensure

T |T| = |Ω|, we set

σ =

• Ω
• det(M) dx
• Ω det(J−1) dx =

1

• T |TC|
• Ω
• det(M) dx

It is not hard to see that Both Qali and Qeq lie in [1, ∞) An ideally “good” mesh, with JtMJ = σI, has Qali = 1 Qeq = 1 We can then set I(T ) = I(Qali, Qeq).

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Mesh quality measure

To ensure

T |T| = |Ω|, we set

σ =

• Ω
• det(M) dx
• Ω det(J−1) dx =

1

• T |TC|
• Ω
• det(M) dx

It is not hard to see that Both Qali and Qeq lie in [1, ∞) An ideally “good” mesh, with JtMJ = σI, has Qali = 1 Qeq = 1 We can then set I(T ) = I(Qali, Qeq).

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How to compute?

Recall that qali(x) = trace(JtMJ) 2 det(JtMJ)1/2 , qeq(x) = det(J)

• det(M)

σ and Qali = max

x∈Ω qali(x),

Qeq = max

x∈Ω qeq(x)

Two issues in the implementation: Find a proper set of reference polygons TC; How to define the mapping F from TC to T? Or more precisely, how to compute J.

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Polygonal mesh quality measures

Method 1: least-squares fitting Use regular polygons as reference elements; By the Riemann mapping theorem, there exists F : TC → T. Let ξi and xi be vertices of TC and T, then xi = F(ξi) Compute a linear least squares fitting x = Aξ + c to F using the values on vertices. Matrix A gives a rough approximation to J.

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Polygonal mesh quality measures

Method 1: least-squares fitting Use regular polygons as reference elements; By the Riemann mapping theorem, there exists F : TC → T. Let ξi and xi be vertices of TC and T, then xi = F(ξi) Compute a linear least squares fitting x = Aξ + c to F using the values on vertices. Matrix A gives a rough approximation to J.

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Polygonal mesh quality measures

Method 1: least-squares fitting Use regular polygons as reference elements; By the Riemann mapping theorem, there exists F : TC → T. Let ξi and xi be vertices of TC and T, then xi = F(ξi) Compute a linear least squares fitting x = Aξ + c to F using the values on vertices. Matrix A gives a rough approximation to J.

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Polygonal mesh quality measures

Method 1: least-squares fitting Use regular polygons as reference elements; By the Riemann mapping theorem, there exists F : TC → T. Let ξi and xi be vertices of TC and T, then xi = F(ξi) Compute a linear least squares fitting x = Aξ + c to F using the values on vertices. Matrix A gives a rough approximation to J.

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Polygonal mesh quality measures

Method 2: generalized barycentric mapping Use regular polygons as reference elements; Let F be a generalized barycentric mapping

Pick a generalized barycentric coordinate λ(ξ) F is the composite mapping: ξ − → λ − → x Examples: piecewise linear barycentric mapping; Wachspress barycentric mapping, ...

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Polygonal mesh quality measures

Method 3: affine mapping with special TC In order to make F an affine mapping, We need to redefine “reference polygons”

Lemma

Using SVD of vertex matrices, each convex n-gon T is affine similar to a reference n-gon TC such that the the in-radius of TC is greater than or equal to

• 1

n(n−1), and the outer-radius of TC is less than or equal to

• n−1

n .

The affine mapping has J = U σ1 σ2

• Ut.

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Polygonal mesh quality measures

Method 3: examples of reference polygons

n=3 n=4 n=5

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Comparing the quality measures on Lloyd iteration

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Comparing the quality measures on Lloyd iteration

Number of iterations 10 20 30 40 50 Quality measures 1 1.5 2 2.5 3 3.5 Lloyds iteration for 32x32 mesh

Qali,1 Qeq,1

Number of iterations 10 20 30 40 50 Quality measures 2 3 4 5 6 7 8 Lloyds iteration for 32x32 mesh

Qali,2 Qeq,2 Number of iterations 10 20 30 40 50 Quality measures 1 1.5 2 2.5 3 3.5 4 Lloyds iteration for 32x32 mesh Qali,3 Qeq,3

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Moving Mesh PDE

To solve a second order elliptic equation:

1 Initialization: Given an initial physical mesh T (0) in Ω; 2 Outer iteration (k = 0, 1, ...): 1

Update the metric tensor M(k) based on the information available at the current iteration. The information includes the current mesh T (k) and the physical solution u(k) that is obtained by solving the underlying PDE on the current mesh T (k);

2

Use the moving mesh method to get a new mesh T (k+1) has a better quality under the metric M(k).

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Numerical results: Example 1

Example 1 −∆u = f with exact solution u = tanh(40y − 80x2) − tanh(40x − 80y2)

1 0.8 0.6 0.4 0.2 0.2 0.4 0.6 2

• 2
• 1

1 0.8 1

0.5 1 0.2 0.4 0.6 0.8 1

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Numerical results: Example 1

Using the MMPDE method gives

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Numerical results: Example 1

History of Qali and Qeq

MMPDE outer iter. 5 10 Qali 1 1.5 2 2.5 3 History of alignment measure

Qali,1 Qali,2 Qali,3

MMPDE outer iter. 5 10 Qeq 1.5 2 2.5 3 3.5 History of equidistribution measure

Qeq,1 Qeq,2 Qeq,3

History of ∇(u − uh) and u − uh

MMPDE outer iter. 5 10 error 2 4 6 8 H1 semi-norm of the error MMPDE outer iter. 5 10 error 0.01 0.02 0.03 0.04 0.05 0.06 L2 norm of the error

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Numerical results: Example 2

Example 2 −∆u = f with exact solution u =

• 0.5(r − x) − 0.25r2

Corner singularity at (0, 0), and u ∈ H

3 2 −ε(Ω)

Convergence on quasi-uniform mesh L2 norm H1 semi-norm N error

• rder

error

• rder

8 2.90e-3 8.99e-2 16 8.43e-4 1.8 6.09e-2 0.6 32 3.00e-4 1.5 4.43e-2 0.5 64 1.19e-4 1.3 3.15e-2 0.5 128 4.55e-5 1.4 2.35e-2 0.4

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Numerical results: Example 2

Using the MMPDE method gives

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Numerical results: Example 2

MMPDE T (5), optimized for H1 seminorm L2 norm H1 semi-norm N error

• rder

error

• rder

8 2.17e-03 6.83e-02 16 5.18e-04 2.1 3.55e-02 0.9 32 1.43e-04 1.9 1.95e-02 0.9 64 3.55e-05 2.0 1.08e-02 0.9 128 9.28e-06 1.9 6.36e-03 0.8

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Thank you!

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