Anisotropic quality measures and adaptation for polygonal meshes

Transcription

1 Anisotropic quality measures and adaptation for polygonal meshes Yanqiu Wang, Oklahoma State University joint work with Weizhang Huang Oct 2015, POEMs, Georgia Tech. 1 / 23

2 Consider the following function / 23

3 Goal: Find a good polygonal mesh for this function Solution: 3 / 23

4 Goal: Find a good polygonal mesh for this function Solution: 3 / 23

5 Method: Moving mesh algorithm Idea: Define an objective function I (T ) such that its minimal values occur on 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, / 23

6 Method: Moving mesh algorithm Idea: Define an objective function I (T ) such that its minimal values occur on 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, / 23

7 Method: Moving mesh algorithm Idea: Define an objective function I (T ) such that its minimal values occur on 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, / 23

8 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 L 2 interpolation error. (Can also be optimized for H 1 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 5 / 23

9 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 L 2 interpolation error. (Can also be optimized for H 1 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 5 / 23

10 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 L 2 interpolation error. (Can also be optimized for H 1 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 5 / 23

11 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 L 2 interpolation error. (Can also be optimized for H 1 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 5 / 23

12 Anisotropic metric: M is a SPD matrix Reference element T C and physical element T F ξ T C T x We say T under metric M T C under identity metric if σ e C i, e C j = e i, e j M which implies σ e C i, e C j = Je C i, Je C j M J t MJ = σi 6 / 23

13 Anisotropic metric: M is a SPD matrix Reference element T C and physical element T F ξ T C T x We say T under metric M T C under identity metric if σ e C i, e C j = e i, e j M which implies σ e C i, e C j = Je C i, Je C j M J t MJ = σi 6 / 23

14 Mesh quality measure Note that J t MJ = σi is equivalent to 1 2 trace(jt MJ) = det(j t MJ) 1/2 (Alignment) det(j) det(m) = σ (Equidistribution) Define q ali (x) = trace(jt MJ) 2 det(j t MJ) 1/2, q eq(x) = det(j) det(m) σ and Q ali = max x Ω q ali(x), Q eq = max x Ω q eq(x) 7 / 23

15 Mesh quality measure Note that J t MJ = σi is equivalent to 1 2 trace(jt MJ) = det(j t MJ) 1/2 (Alignment) det(j) det(m) = σ (Equidistribution) Define q ali (x) = trace(jt MJ) 2 det(j t MJ) 1/2, q eq(x) = det(j) det(m) σ and Q ali = max x Ω q ali(x), Q eq = max x Ω q eq(x) 7 / 23

16 Mesh quality measure To ensure T T = Ω, we set det(m) dx σ = Ω Ω det(j 1 ) dx = 1 det(m) dx T T C Ω It is not hard to see that Both Q ali and Q eq lie in [1, ) An ideally good mesh, with J t MJ = σi, has Q ali = 1 Q eq = 1 We can then set I (T ) = I (Q ali, Q eq ). 8 / 23

17 Mesh quality measure To ensure T T = Ω, we set det(m) dx σ = Ω Ω det(j 1 ) dx = 1 det(m) dx T T C Ω It is not hard to see that Both Q ali and Q eq lie in [1, ) An ideally good mesh, with J t MJ = σi, has Q ali = 1 Q eq = 1 We can then set I (T ) = I (Q ali, Q eq ). 8 / 23

18 How to compute? Recall that q ali (x) = trace(jt MJ) 2 det(j t MJ) 1/2, q eq(x) = det(j) det(m) σ and Q ali = max x Ω q ali(x), Q eq = max x Ω q eq(x) Two issues in the implementation: Find a proper set of reference polygons T C ; How to define the mapping F from T C to T? Or more precisely, how to compute J. 9 / 23

19 Polygonal mesh quality measures Method 1: least-squares fitting Use regular polygons as reference elements; By the Riemann mapping theorem, there exists F : T C T. Let ξ i and x i be vertices of T C and T, then x i = F (ξ i ) Compute a linear least squares fitting x = Aξ + c to F using the values on vertices. Matrix A gives a rough approximation to J. 10 / 23

20 Polygonal mesh quality measures Method 1: least-squares fitting Use regular polygons as reference elements; By the Riemann mapping theorem, there exists F : T C T. Let ξ i and x i be vertices of T C and T, then x i = F (ξ i ) Compute a linear least squares fitting x = Aξ + c to F using the values on vertices. Matrix A gives a rough approximation to J. 10 / 23

21 Polygonal mesh quality measures Method 1: least-squares fitting Use regular polygons as reference elements; By the Riemann mapping theorem, there exists F : T C T. Let ξ i and x i be vertices of T C and T, then x i = F (ξ i ) Compute a linear least squares fitting x = Aξ + c to F using the values on vertices. Matrix A gives a rough approximation to J. 10 / 23

22 Polygonal mesh quality measures Method 1: least-squares fitting Use regular polygons as reference elements; By the Riemann mapping theorem, there exists F : T C T. Let ξ i and x i be vertices of T C and T, then x i = F (ξ i ) Compute a linear least squares fitting x = Aξ + c to F using the values on vertices. Matrix A gives a rough approximation to J. 10 / 23

23 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, / 23

24 Polygonal mesh quality measures Method 3: affine mapping with special T C 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 T C such that the the in-radius of T C is greater than or equal to 1 n(n 1), and the outer-radius of T C is less than or equal to n 1 n. [ ] σ1 0 The affine mapping has J = U U 0 σ t / 23

25 Polygonal mesh quality measures Method 3: examples of reference polygons n=3 n=4 n=5 13 / 23

26 Comparing the quality measures on Lloyd iteration 14 / 23

27 Comparing the quality measures on Lloyd iteration Quality measures Lloyds iteration for 32x32 mesh Q ali,1 Q eq,1 Quality measures Lloyds iteration for 32x32 mesh Q ali,2 Q eq,2 Quality measures Lloyds iteration for 32x32 mesh Q ali,3 Q eq, Number of iterations Number of iterations Number of iterations 15 / 23

28 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). 16 / 23

29 Numerical results: Example 1 Example 1 u = f with exact solution u = tanh(40y 80x 2 ) tanh(40x 80y 2 ) / 23

30 Numerical results: Example 1 Using the MMPDE method gives 18 / 23

31 Numerical results: Example 1 History of Q ali and Q eq 3 History of alignment measure 3.5 History of equidistribution measure Q eq,1 Q ali,1 2.5 Q ali,2 Q ali,3 3 Q eq,2 Q eq,3 Q ali 2 Q eq MMPDE outer iter MMPDE outer iter. History of (u u h ) and u u h 8 H1 semi-norm of the error 0.06 L2 norm of the error error 4 error MMPDE outer iter MMPDE outer iter. 19 / 23

32 Numerical results: Example 2 Example 2 u = f with exact solution u = 0.5(r x) 0.25r 2 Corner singularity at (0, 0), and u H 3 2 ε (Ω) Convergence on quasi-uniform mesh L 2 norm H 1 semi-norm N error order error order e e e e e e e e e e / 23

33 Numerical results: Example 2 Using the MMPDE method gives 21 / 23

34 Numerical results: Example 2 MMPDE T (5), optimized for H 1 seminorm L 2 norm H 1 semi-norm N error order error order e e e e e e e e e e / 23

35 Thank you! 23 / 23