Convex Optimization for Simplicial Mesh Improvement

Transcription

1 Convex Optimization for Simplicial Mesh Improvement Bryan Feldman December 9, 25 Abstract In this project I explore several strategies utilizing convex optimization for the purpose of improving the quality of a simplicial mesh. The intended application for the meshes is for performing finite element/volume simulations and therefore mesh quality and in particular the minimum element quality is of fundamental importance. In the first part I look at methods for creating valid meshes from ones that contain inverted elements. In the second part I look at improving the minimum element quality of the mesh. 2 Introduction Creating good quality meshes is important because poorly shaped elements can severely hinder the performance of finite element or finite volume simulations that use these meshes to discretize the simulation domain. A large number of researchers have investigated this problem using various strategies and within the literature several definitions for quality have arisen. The work of Schewcuk, [4] has shed some light on the quality of these various quality metrics and in what situations (if any) the metrics are appropriate. Quality can be improved in two ways, by optimizing the topology of elements or by optimizing the positions of the nodes. Delaunay triangulation (2D) or tetrahedralization (3D) is known to minimize several mesh quality metrics, and is widely regarded as the defacto standard method for generation of the underlying mesh connectivity. In this work I focus on the problem of optimizing the location of nodes in a mesh, referred to as mesh smoothing. Within the field of mesh smoothing there are generally two approaches, heuristic-based and optimization-based. The heuristic based methods are often relatively quick and simple. The most common method, Laplacian Smoothing, simply adjusts a node position to the average of its neighbors. However, heuristic methods offer no guarantees to their performance. In fact, these methods can decrease mesh quality even to the point of creating invalid meshes. Alternatively, optimization-based approaches, while slower, generally create nicer meshes as they are guided by the objective one wishes to maximize. Since the number of nodes in a

2 mesh is typically quite large, almost exclusively these algorithms operate by optimizing the mesh quality by moving a single node at a time, and repeating for all (or some subset of) nodes, perhaps several times. 3 Mesh Untangling In order to have any hope of a simulation producing reasonable results the mesh that the simulation is performed on need be valid, i.e. not contain any inverted or tangled components. One can envision obtaining an invalid mesh in the situation where mesh nodes are moved over a time step of the simulation without changing mesh topology such that some elements of the mesh become inverted. A quick indication for an inverted element is if the area (2D) or volume (3D) of the element is negative. In both 2D and 3D the area/volume can be computed by the determinant of a matrix formed by 2/3 edges of the element and so it is linear with respect to a single node position. As show in [2] a mesh can be untangled by optimizing the position nodes in the mesh belonging to a tangled elements. Since the optimization is performed for a single node at a time, several sweeps through the mesh may be required to untangle all nodes but this cost is substantially less than forming and optimizing the problem as a bundle. The area of the triangles surrounding a particular node, i at position, x i are the components of a vector, a which is found by a = A i x i + c Here obviously A i contains the linear multiples in the area calculations and c has all the required constant components. In [2], x i is optimized for an objective function that maximizes the minimum area max min j N a j This can be achieved by forming a matrix A c that is the concatenation of A i with a row of all negative ones, and solving for a vector, π which is the concatenation of x i and the minimum area (volume) local element. We can define slack variables s, as the difference between each local elements area and the minimum area (the last component of π) which from definitions yields: A c π + c = s With this we can maximize the minimum area element by the following LP maximize b T π subject to A c π + c = s s where b is a vector of all zeros except for the last term which is one. Since the last term of π is the minimum element area, maximizing this quantity achieves the desired optimization function. The constraints ensure that the difference between all element areas and the minimum is. 2

3 Additionally, I tested a method for untangling meshes that minimizes the two norm between the vector of local areas and some desired areas, â. Desired areas can be found by setting them to the average of existing local triangles, resulting in the following optimization, minimize A i x i + c â 2 where â = N ( T a ). This minimization can easily be solved as it is a standard Least-Squared problem. 3. Results The above described algorithms were implemented in MATLAB using the Convex Optimization package cvx As shown in Figure the methods used to untangle the meshes all successfully created valid meshes. They differ in the run time and ending quality of the meshes. There is apparently an inverse relationship between complexity of the algorithm and quality/speed of obtaining the results for these 2-D examples. Laplacian Smoothing, the dumbest and by far the fastest was able eventually untangle all test meshes I gave it. While it sometimes inverted a triangle subsequent applications of smoothing to either adjacent nodes or the same node in the next pass, seemed to undo the problem and resulted in a rather nice looking mesh. The algorithm which found the node position that made triangles have near equal area performed slightly worse (and slower) while the hard-core, maximize the minimum area algorithm created a rather poor quality mesh, although to be fair it did do the required job of untangling the mesh. I should point out that all tests are performed on 2-D meshes which the literature reports are substantially easier to handle than their notoriously fickle 3-D counterparts. Therefore, it stands to reason that in 3-D the more sophisticated algorithms may be required. This endeavor remains as an area for future work. 4 Node Position Optimization While the work in the previous section was able to successfully untangle initial meshes that included many severely inverted triangles, the quality of the triangles are often far from ideal. This shouldn t be surprising as area is not a good indicator of mesh quality, so optimizations involving only the area have no preferred tendency to be high quality (just not inverted). Thus, in this next part of the project I investigate a method for optimizing a sensible mesh quality metric. As reported in [4] there are a plethora of mesh quality metrics that have been used. One particularly nice metric suggested in the above mentioned work is q i = 4 3 A i l 2 rms boyd/cvx/ (2D) q i = V i A rms l rms (3D) 3

4 .5 tangled nodepos Freitag nodepos Laplacian smoothing nodepos avg Area nodepos Figure : Comparison of methods to untangle mesh. Top Left) original tangled mesh. Top Right)Untangling by maximizing minimum area. Lower Left) Laplacian Smoothing to untangle. Lower Right) Minimizing distance from average area. where q i is the quality of element i, l is length of an edge, A is area, V volume and the subscript rms indicates the root mean squared of all edges, or areas belonging to that element. Also note that these metrics include normalization factors such that the ideal (equilateral) elements has a quality of one. This metric is convex, smooth, and differentiable and actually has a nice intuitive meaning as being the condition number of the matric which transforms the element to the ideal element. While overall (or average) mesh quality is important, the accuracy of a simulation can be ruined by a single bad element as it has the capability to ill-condition matrices. Therefore, it is important to create meshes that have elements of at least some minimal quality. Using ideas presented in Optimization Strategies in Unstructured Mesh Generation [5] I implemented a system to maximize an objective function, Ψ(x), the minimum quality element in the mesh, Ψ(x) = min i=...n (q i) 4

5 with my implementation using a different quality metric than the one in [5]. In this form the problem is particularly nasty for a number of reasons. First, the problem size can be quite large, meshes can contain hundreds of thousands or even millions of nodes, rendering any optimization intractable. The second nastiness of this problem is the fact that the objective is non-differentiable. In order to cope with these problems [5] suggest several relaxations to the problem. In order to reduce the dimension of the problem, first the current worst quality element is found and only the positions of the nodes in some local neighborhood are used in the optimization, x. In order to deal with the non-differentiability of the objective they suggest a modified Steepest Decent Algorithm that is taken from [3]. The approach relaxes an indicator variable, I, to be a real valued vector, µ. The steps of the optimization are:. Compute the step direction δ = m µ i q i (x ) where µ is computed as the solution of Quadratic Programing sub-problem i µ = argmin [ m i µ k [Ψ(x ) q i (x )]] 2 + L 2 m i µ k q i (x ) 2 s.t. µ T µ = with L set to an average local edge length size. 2. Compute the step size, λ using Armijo s rule λ = max λ = {β k (k =,,...) Ψ(x + λδ) Ψ(x ) λα δ 2 } 3. Take step (adjust positions of local neighborhood of nodes) x = x + λδ Because solving the QP sub-problem is expensive another relaxation is used to reduce the dimension of the problem. It is assumed that µ is zero for all local elements with q(x ) Ψ(x ) ɛ with a typical value for ɛ =.. This reduces the size of the QP substantially, potentially even removing the need for it if no other elements have low quality. This results in a significant speedup of the optimization. 4. Results The algorithm successfully improved many of the poor quality elements. As shown in Figure 2 the resulting mesh looks substantially better than the initial one, although in fact some of the worst elements in the initial mesh are difficult to see since they are so thin. Also in Figure 3 we see that the minimum element quality is increased from.78 to.443. Simulation on 5

6 the initial mesh would probably create problems while the improved mesh should do fine. Comparing the optimized mesh with the one obtained using Laplacian Smoothing, again we see that that Laplacian Smoothing results in a superior mesh. This brings to mind a quip from John Strain of our math department who said, You should resist the urge to be clever. In the case of 2-D meshes it seems his message rings true. In the defense of cleverness I must again point out that the examples take place in 2-D and that complexity in 3-D is so much more that there is probably a place for some cleverness when dealing with 3-D meshes. In particular, a good system for mesh improvement would probably include both the simple and robust method. One can envision using Laplacian Smoothing to quickly improve large portions of the mesh, and then relying on the more robust approaches of optimization-based methods to deal with problems remaining/created by the heuristic-based methods. before after Figure 2: Mesh before and after optimization by method described in Section 4 References [] L. Freitag and P. Knupp. Tetrahedral mesh improvement of the element condition number. Inter. Journal for Numerical Methods in Engineering, 53:377 39, 22. [2] L. Freitag and P. Plassmann. Local optimization-based simplicial mesh untangling and improvement. 2nd Symposium on Trends in Unstructured Mesh Generation,

7 min mean Figure 3: Improvement in minimum and mean quality [3] Elijah Polak. On the mathematical foundations of nondifferentiable optimization in engineering design. Technical Report UCB/ERL M85/7, EECS Department, University of California, Berkeley, 985. [4] J.R. Shewchuk. What is a good linear element? interpolation, conditioning and quality measures. th International Meshing Roundtable, 22. [5] P.D. Zavattieri, E.A. Dari, and G.C. Buscaglia. Optimization strategies in unstructured mesh generation. Int. Journal of Numerical Methods, 39:255 27,