ccopyright by Xiangyang Li 1999

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1 DYNAMIC LOAD BALANCING FOR PARALLEL ADAPTIVE MESH REFINEMENT BY XIANGYANG LI B.Eng., Tsinghua University, BeiJing, 1995 B.Eco., Tsinghua University, BeiJing, 1995 THESIS Submitted in partial fulllment of the requirements for the degree of Master of Science in Computer Science in the Graduate College of the University of Illinois at Urbana-Champaign, 1999 Urbana, Illinois

2 ccopyright by Xiangyang Li 1999

3 Dynamic Load Balancing for Parallel Adaptive Mesh Renement Xiangyang Li, M.S. Department of Computer Science University of Illinois at Urbana-Champaign, 1999 Shang-Hua Teng, Advisor A key step in the nite element method is to generate a high quality mesh that is as small as possible for an input domain. Several meshing methods and heuristics have been developed and implemented. Methods based on advancing front, Delaunay triangulations, and quadtrees/octrees are among the most popular ones. Advancing front uses simple data structures and is ecient. Unfortunately, in general, it does not provide any guarantee on the size and quality of the mesh it produces. On the other hand, the sphere-packing based Delaunay methods generate a well-shaped mesh whose size is within a constant factor of the optimal. Adaptive mesh renement is a key problem in large-scale numerical calculations. The need of adaptive mesh renement could introduce load imbalance among processors, where the load measures the amount of work required by renement itself as well as by numerical calculations thereafter. We present a dynamic load balancing algorithm to ensure that the work for renement and computation thereafter are balanced while the communication overhead (including the overhead caused by moving submeshes around) is minimized. The main ingredient of our method is a technique for the estimation of the size and the element distribution of the rened mesh before we actually generate the rened mesh. Base on this estimation, we can reduce the dynamic load balancing problem to a collection of static partitioning problems, one for each processor. In parallel each processor could then locally apply a static partitioning algorithm to generate the basic units of submeshes for load rebalancing. We then model the communication cost of moving submeshes by a condensed and much smaller subdomain graph, and apply a static partitioning algorithm to generate the nal partition. iii

4 TO MY FAMILY iv

5 Acknowledgements First and foremost, I would like to thank Professor Shang-Hua Teng for being a wonderful advisor. His insights and invaluable help and guidance presented in this thesis enlightened me in various aspects of the work. I would like to acknowledge the constant assistance and encouragements from my best friends. The special thanks goes to Alper Ungor, who was also conducting research in the area of the mesh generation algorithm, for his educating comments and perfect-pursuing habits. The discussion with him not only enhances my research abilities, but also improves my English a lot. I also thank Marcia, my ocemate and friend for making the oce so nice place to stay. I would also like to thank the Department of Computer Science and the CSAR project at the University of Illinois for the computing resources. Last but not least, my parents, my wife, and my daughter deserves the particular recognition for being the driving force in my life. I thank my wife Chen Min for her love and support, my parents and sister for their unconditional encouragement and belief in my ongoing studies and my sweet daughter Sophia for ensuring that there was no a dull moment when writing this thesis. v

6 Table of Contents Chapter 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Mesh Generation Graph Partition Parallel Mesh Renement and Load Balancing Thesis Outline Mesh Generation : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Introduction Structured and Unstructured Mesh Quality of Mesh Control Space Geometric Features Numerical Spacing Control Spacing Function Conformality and Size of Mesh Delaunay Triangulation Mesh Generation Method Advancing Front Methods Sphere Packing Methods Graph Partition : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Introduction Background and Good Separators Graph Partition Algorithm Level-structure Partitioning The Spectral Partitioning Algorithm Geometric Approach A Multilevel Algorithm Load Balancing : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Introduction Dynamic Balanced Quadtrees Modeling Adaptive Renement with Dynamic Quadtree Reduce Dynamic Load Balancing to Static Partitioning Subdomain Size Estimation vi

7 4.5.1 Aected Boxes of Rening a Box Size Estimation of Balanced Quadtree Sampling Boxes from T Subdomain Partitioning Subdomain Redistribution Remeshing Unstructured Meshes Conclusions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 56 Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 58 Vita : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 63 vii

8 List of Figures 2.1 A typical example of a unstructured mesh The Delaunay mesh from a random point set The 16-way partition of its' vertices The 16-way partition of its' triangle elements A balanced quadtree T in two dimensions, and a 4-way partition quadtree T 0 after rening T, and T after balancing T 0. (a): assume the splitting depth of b 1, b 2, b 3 and b 4 are 1, 1, 2 and 2 respectively. (b): assume after balancing T 0, we maintain the current partition of T The two templates for splitting boxes in region(b), and how the renement of b inuence the splitting of leaf-boxes contained in region(b) The three scenarios for two edge neighbor leaf-boxes. (a): (b; b 1 ) = 1; (b): (b; b 1 ) = 0; (c): (b; b 1 ) =? The ve scenarios fro two corner neighbor leaf box b 1 and b. (a): (b; b 1 ) = 2; (b): (b; b 1 ) = 1; (c): (b; b 1 ) = 0; (d): (b; b 1 ) =?1; (e): (b; b 1 ) =? an example of the pressures of leaf-box b, and the examples of the edge-boxes, corner-boxes, center-boxes A typical example of subdomain partition for T Constructing subdomain graph from subdomain partition An example of subdomain redistribution and renement of subdomain viii

9 Chapter 1 Introduction Many problems in computational science and engineering, geographic information system, and computer graphics are based on structured or/and unstructured meshes in two or three dimensions. The meshes can be quite large, often containing millions of elements. The size of meshes is usually determined by the size of the machine available to solve the problem, and the accuracy requirement of the problem. An essential step in numerical simulation is to nd a proper discretization of a continuous domain. This is the problem of mesh generation [4, 40], which is a key component in computer simulation of physical and engineering problems. The following six basic steps are usually used to conduct a numerical simulation by nite element, nite dierence, and nite volume methods. 1. Mathematical modeling: dene the continuous domain and partial dierential equations (PDE) over the domain that accurately model the physical and engineering problem; 2. Geometric modeling: approximate the continuous domain with a discrete description. 3. Mesh generation: decompose the interior of the domain into a mesh M of simple and \well-shaped" elements such as boxes and simplices. 4. Numerical approximation: construct a system of linear or non-linear equations over M for the governing PDEs. 5. Numerical solution: solve the system of equations and estimate the error of the solution; 1

10 6. Adaptive renement: based on the error estimation, if necessary, rene the mesh and repeat the above steps 5 and 6 over the rened meshes. 1.1 Mesh Generation Meshing [4, 40, 26, 27, 28, 43, 44] can be dened as the process of breaking up a physical domain into smaller sub-domains (elements) in order to facilitate the numerical solution of a partial dierential equation. While meshing can be used for a wide variety of applications, such as solid modeling, computer aided design, graphical rendering, and scientic computation. The principal application of interest is the nite element method. For problems with complex geometry boundaries and with solutions that change rapidly, we need to use an unstructured mesh with a varying local topology and spacing in order to reduce the problem size. A good unstructured meshing algorithm uses elements of properly chosen size and shape that adapt to the complex geometry and solution accuracy. In doing so, it generates meshes that are numerically sound and that are also as small as possible. Several meshing methods and heuristics have been developed, implemented, and applied to various applications such as steady state and transient compressible inviscid ow simulations. Over the years, several meshing methods such as those based on advancing front, Delaunay triangulations, and quadtrees/octrees have become popular due to their eectiveness in practical applications. However, these methods do not come with equal strengths. For example, advancing front [6, 30, 31] uses simple data structures and is ecient and relatively easy to implement. It oers a high quality of point placement strategy and the integrity of the boundary. Unfortunately, it does not provide any general guarantee on the size and quality of the mesh it produces. On the other hand, more sophisticate methods such as quadtree/octree renement [4, 40, 52] and Delaunay methods [8, 9, 10, 35, 43, 44] generate a well-shaped mesh whose size is within a constant factor of the optimal. Li et. al. [28] recently developed a new mesh generation algorithm called biting method. It combines the strengths of advancing front and these provably good meshing methods. It not only guarantees the quality of the generated mesh, but also solve the problems caused by the advancing front method. The particular type of Delaunay method that they use in conjunction with advancing front is the sphere packing method. It rst constructs a well-spaced point set by computing a sphere 2

11 packing of the domain and then uses the Delaunay triangulation of this point set as the nal mesh. Two methods have been developed to generate the well-spaced point set. The rst one applies particle simulation [45, 46, 47] to nd a stable conguration of a set of energetic spheres. The second one uses quadtree/octree renement to obtain an oversample of the input domain, and then applies a properly dened maximal independent set to create the sphere packing [3, 5, 27, 37, 50]. Both in theory and in practice, the second approach is faster. 1.2 Graph Partition Parallel computing has become a critical component of the computing technology of this decade. To reduce the time spent waiting at synchronization events, the work load of each processor should be balanced. The process of balancing the workload and reducing the synchronization wait time consists of four parts [12]. 1. Identifying enough concurrency in decomposition and overcoming Amdahl's Law, 2. Deciding how to manage the concurrency { statically or dynamically, 3. Determining the granularity at which to exploit the concurrency, 4. Reducing the serialization and synchronization cost. The computation and data dependency of a lot of scientic computing problem can be modeled by a graph ( directed graph or undirected graph): the vertices of graph is the basic data unit or commutating tasks; two vertices are connected if there are a data dependency between them or there are computation dependency between tasks. Then we can use the technique of the graph partition to identify the concurrency in a given parallel computing problem. A partition of a graph into subgraphs leads to a decomposition of the data and/or tasks associated with a computational problem and the subgraphs can be mapped to the processors of a multiprocessor. Graph partition also plays an important role in serial algorithm design especially for the problem with strong divide and conquer property. For parallel computing, the ideal assignment of the data and/or tasks to processors is that the work load of every processor are approximately equal and the communication during the computation is minimized. Hence, the following two objectives are usually stated in the 3

12 partitioning algorithm. Partition a given graph into a specied number of subgraphs such that the subgraphs have roughly equal number of vertices and few edges join dierent subgraphs to each other ( this edges are the cut of the partition). Here we assume that the vertex model the basic workload unit. In the context of the parallel computation, the size of a subgraph determines the computational work load that a processor has to perform, and the number of cut edges is the measure of the communication volume in the algorithm. In a serial algorithm, equal-sized subgraphs lead to the lowest worst-case running time, and the number of the cut edges measures the cost of combining the partial solutions to compute the global solution. More general objective functions need to be considered for many complicated problems. For example, the work load is not always proportional to the number of vertices in the subgraph. Hence the work associated with a subgraph may be modeled more accurately by attaching a weight to each vertex. Then load balancing requirement can be modeled as to partition the graph to subgraph such that subgraphs has approximately equal summation weights of vertices assigned to it. Note that for data communication, we can use the message packing technique, i.e., to pack all the messages which will be sent to same processor. The communication costs in the algorithm might be modeled by the number of subgraphs a given subgraph is connected to, or the number of boundary vertices of the subgraph, or similar variants. If the graph is a geometry graph, sometimes the shape of the subgraph is also important to get a good solution to the problem. In other words, the shape of the subgraphs (e.g., the aspect ratios) may be an important parameter in some algorithms (e.g., the convergence of a domain decomposition method). In some applications, it may be essential to partition into connected subgraphs. 1.3 Parallel Mesh Renement and Load Balancing Once we have a discretization (mesh) of the domain, dierential equations for ow, waves, and heat distribution are then approximated by nite dierence, nite element, or nite volume formulations. To properly approximate a continuous function, in addition to the conditions that a mesh must conform to the boundary of the region and be ne enough, each individual element of the mesh must be well shaped. A common shape criteria for elements is the condition that the angles of each element are not too small or the aspect ratio of each element is bounded 4

13 [49]. The aspect ratio of a simplex is dened as its maximum side-length divided by its minimum altitude. We consider issues and algorithms for adaptive mesh renement, (cf, Step 6 in the paradigm 1). The general scenario is the following. We start with an initial well-shaped mesh M 0 for an input domain and dierential equations. Then we form the numerical system from the mesh M 0 and the original dierential equations. We partition mesh M 0 and map the submeshes and their fraction of the numerical system onto a parallel machine. By solving the numerical system in parallel, we obtain an initial numerical solution S 0. If the solution S 0 is within the error tolerance, we then return the solution. Otherwise, an error-estimation of S 0 generates a renement spacing-function h 1 over the domain, which denes the expected size of mesh elements at a particular region in the domain. In other words, if in some region the error estimation of S 0 is too large compared to the tolerant error or average error, we have to rene the mesh element in that region, such that the next computation will generate more accurate solution in this region. Therefore, we need to properly rene the initial mesh M 0 according to h 1 to generate another well-shaped mesh M 1. The requirement for renement introduces load imbalance among processors in the parallel machine. Some submeshes, after renement, might be much larger than others. The work-load of a processor in the next stage computation is determined by the summation of the time the processor needs to spend in rening its submesh and the time it needs to solve its fraction of the numerical system over the rened mesh M 1. In this thesis, we present a dynamic load balancing algorithm to ensure that the computation at each stage of the renement is balanced and optimized. The basic idea of our algorithm is as following. Our algorithm estimates the size and distribution of M 1 before it is actually generated. Based on this estimation, we can compute a quality partition of M 1 before we generate it. The partition of M 1 can be projected back to M 0, which divides the submesh on each processor into one or more subsubmeshes. Our algorithm rst moves these subsubmeshes to proper processors before performing the renement. This is more ecient than moving M 1 because M 0 is usually smaller than M 1. Note that this approach considers the mesh renement cost as one of the the work load to be balanced. In partitioning M 1, we take into account of the communication cost of moving these subsubmeshes as well as the communication cost in solving the numerical system over M 1. 5

14 1.4 Thesis Outline The thesis is organized as following. Chapter 2 introduces the basic concepts of the mesh generation and the quality measure of mesh. It also introduces some basic mesh generation algorithm. Graph partition concepts and some widely used algorithm are introduced in Chapter 3 Chapter 4 introduces an abstract problem to model parallel adaptive mesh renement. An algorithm to estimate the size and distribution of the rened mesh before its generation is also presented in this chapter. It also presents a technique to reduce dynamic load balancing for mesh renement to a collection of static partitioning problems. This reduction makes use of the estimation information generated by the size estimation algorithm. It rst applies static partitioning to divide the submesh (subdomain) on each processor into a set of subsubmeshes (subsubdomains) according to the projection of the nal partition onto the current mesh. We introduce a notion of subdomain graph to incorporate the communicational cost in moving these subsubmeshes with the communicational cost in solving the subsequent numerical system. We then apply a static partitioning algorithm to complete the denition of the partition of the rened mesh. We also extends ours algorithm from the abstract problem to unstructured meshes. Chapter 5 concludes the thesis with a discussion of some future research directions. 6

15 Chapter 2 Mesh Generation 2.1 Introduction Decomposition of a geometric input into simpler objects is fundamental in many areas, such as solid modeling, computer aided design, graphical rendering, and scientic computation. For example, an essential step in numerical simulation of physical and engineering problems is to nd a proper discretization of a continuous domain. This is the problem of mesh generation [4, 40, 26, 27, 43, 44]. Finding the optimal triangulation is a particular mesh generation problem in computational geometry. The most often used optimization criteria [4] include maximizing the minimum angle among all elements of the partition (solved by the wellknown Delaunay triangulation [48]), minimizing the maximum angle [14], minimizing a maximum mincontainment ellipse [13], and minimizing total length (an outstanding open problem in the eld [17]). Variants of these problems allow one to add extra vertices, called Steiner points, in order to further improve the quality of the solution. Mesh generation is a great example of inter-disciplinary research. Its development is built upon advances in computational and combinatorial geometry, data structures, numerical analysis, and scientic applications. Its success is justied not only by mathematical proofs about the quality and the numerical relevancy of geometry-based meshing algorithms, but also by the performance of meshing software in real applications. It embraces both provably good algorithms and practical heuristics. 7

16 2.2 Structured and Unstructured Mesh The simplest form of a mesh is a structured mesh. Structured meshes are widely used by the early stage scientic computing because the easiness to construct the structured meshes and to form the linear systems for the mesh sometimes. There are two types of structured meshes: geometrically structured and topologically structured meshes. Examples of geometrically structured meshes are regular Cartesian grids and uniform hexagon grids. In these meshes, all elements are geometrically alike: the size of the mesh element are similar and the shape of the element are also similar. The domain that a geometrically structured mesh can be applied can not be too complicated because of the grid property of the elements it used. A mesh is topologically structured if its topological structure is isomorphic to that of a geometrically structured mesh. For example, we can apply a conformal transformation to a structured grid to generate a topologically structured mesh. The topological structured mesh enhance the ability to approximate the complicate input domain. Structured grids are easy to generate and manipulate, which facilitate the use of simple data structures to reduce the complexity of programming. In addition, the numerical theory about these types of discretization is well understood. However, it is not easy to apply the structured mesh to approximate the complicated domain or the numerical systems that changes rapidly. The use of structured regular grids limits the applicability of numerical methods to problems whose domains are simple and whose solution functions are smooth. The other type of the mesh is unstructured mesh. It has varying local topology and spacing in order to reduce the problem size. For problems with complex geometry boundaries and with solutions that change rapidly, we need to use an unstructured mesh. For the example of modeling the combustion of the material in the rocket, we need a much denser and accurate discretization the boundary of the combustion while it is desirable not to waste mesh points in regions with low activities. The adaptability of unstructured meshes comes with new challenges, especially for 3D problems; the numerical theory becomes more dicult and the algorithmic design becomes harder. The most general and versatile mesh is an unstructured triangular mesh in which each element is a simplex, i.e. a triangle in 2D or a tetrahedron in 3D. In general, a k-simplex is a convex polytope of k + 1 points of dimension k. A triangular mesh is a triangulation of 8

17 the input domain (e.g., a polygon), along with some extra points, called Steiner points. A triangulation or a simplicial complex is a decomposition of a domain into a collection of interior disjointed simplices so that two simplices can only intersect at a lower dimensional simplex, i.e., neighboring elements are conformal at their boundaries. Combinatorially, a triangulation T can be expressed as a PLS of a set S T of simplices: if s is a simplex in S T then all of its lower dimensional faces, which themselves are simplices as well, also belong to S T. Following Figure 2.1 gives a typical example of the unstructured mesh. It is from the paper of Borouchaki. Figure 2.1: A typical example of a unstructured mesh. A triangulation T conforms to the boundary of a domain if each boundary polytope of is a union of some simplices in S T. A triangulation T is the constrained triangulation of domain if all mesh vertices are the domain vertices, and each domain boundary polytope of is a union of some simplices in S T. A triangulation T covers a domain if each domain boundary 9

18 polytope of is a union of some simplices in S T and each boundary segment of is an edge of some simplices of S T. An area of great potential is the automatic adaptation of the mesh, without intervention by the analyst. It automatically continues solution until the required accuracy has been reached. Such mesh adaptation generally means that in certain areas of the input domain the size of the elements is decreased (or increased) and the order of the elements may be increased (or decreased). In concept, such adaptation is most appealing, but there are diculties when complex practical situations are considered. 2.3 Quality of Mesh Numerical approximation errors depend on the quality of the mesh, while the time and the space requirements of numerical algorithms are a function of the number of mesh elements. To properly approximate a continuous function, in addition to the conditions that a mesh must conform to the boundaries of the region and be ne enough, each individual element of the mesh must be well-shaped. A common shape criterion for elements is the condition that the angles of each element are not too small, or the aspect ratio of each element is bounded [1, 4, 49]. Certain requirements about the size and neighborhood relations of the mesh vertices should be imposed in order to get a high quality mesh with small number of elements. As Babuska and Aziz [1] justied one should avoid the large angles for a high quality mesh. Specically they showed that the nite element method convergences if the maximum angle of all mesh elements is bounded above from, i.e., there exist a constant 0 such that every angle is at most? 0. Strang and Fix [49] also showed that the nite element method convergences if the smallest angle of the mesh elements is bounded below from a constant. Note the above two conditions can be reduced to bound the smallest angle of all mesh element. The following denition of the aspect-ratio, popularized by Mitchell and Vavasis [40], is uniformly dened for any dimension. Denition (aspect-ratio) The aspect ratio of a simplex T is the ratio R T =r T, where R T and r T are the radii of the smallest ball containing T and the largest ball contained in T, respectively. The aspect ratio of a triangular mesh M is the largest aspect ratio among its elements. M is well shaped for a constant > 1 if its aspect ratio is at most. 10

19 In this thesis, we measure the quality of a triangular mesh by the radius-edge aspect ratio dened by Miller, Talmor, Teng, and Walkington [35, 36]. Denition (radius-edge ratio) The radius-edge aspect-ratio of a simplex is the ratio of the circum-radius to the length of the shortest edge to of the simplex. A mesh M is -wellshaped for a constant > 1 if the radius-edge aspect-ratio is bounded from above by. In two dimensions, these denitions are equivalent in the sense that if a triangle is bounded away from being an ill-shaped triangle under one aspect-ratio, it is bounded away under the others as well. In three dimensions, they are not equivalent. Silver of the three dimension has good radius edge ratio, but it does not have a good aspect ratio. If there are silver in the mesh, the nite element method may be not convergent to the solution. The control volume method guarantees the convergence for the mesh having silver. 2.4 Control Space The rst stage of an adaptive nite element scheme consists in creating an initial mesh of a given domain A size specication eld is deduced before the mesh generation, i.e., at the vicinity of each mesh vertex, the desired mesh element size is specied. The size specication can be derived from the previous numerical results, or from the local geometry feature, or from the geometric error which indicates the gap between the facetization and the real surface in case of surfaces meshes. As shown in [35, 43], the spacing function for a well-shaped mesh should be smooth in the sense that it changes slowly as a function of distance. Formally, a function f is Lipschitz with a constant if for any two points x; y in the domain, jf(x)? f(y)j jjx? yjj. Each domain and a dierential equation u denes a desired local spacing within a domain to specify, for example, the expected element size in a given neighborhood or point densities near a point. In this section, we discuss how to determine the local spacing from the geometry of and the numerical condition of u Geometric Features The geometry of the boundary of also contributes to the local spacing of a well-shaped mesh. In two dimensions, we assume that is given as a planar-straight-line graph (PSLG), which is 11

20 a collection of line segments and points in the plane, closed under intersection. Suppose is described by a PSLG S. Ruppert [43] introduced the following concept called local feature size. Denition Given a PSLG S, the local feature size at a point x, lfs S (x), or simply lf s(x), is the radius of the smallest disk centered at x that intersects two non-incident vertices or segments of S. 1 Note that adding new Steiner vertices does not change the value of lfs() function, since it is determined by the input. Ruppert has observed that lf s changes slowly within the domain. Formally, a function f() is Lipschitz with a coeciency if for any two points x, y in the domain, jf(x)? f(y)j jjx? yjj. Then the Lipschitz coeciency of lfs is bounded from above by 1 [43]. In addition, lfs is the maximum in the following sense. Lemma If f is a 1-Lipschitz function over a domain such that for each point x f(x) lfs (x), then for every x 2, f(x) lfs (x). There are several ways to describe the spacing function of a well-shaped mesh M over a domain : Denition (Edge-length function, el M ) For each point x 2, el M (x) is equal to the length of the longest edges of all mesh simplex elements that contain x (note that points on the lower dimensional faces of a simplex are contained in more than one element). Denition (Nearest-neighbor function, nn M ) Let x be a point in, there are two cases. (1) if x is a mesh point, then nn M (x) is equal to the distance of x to the nearest mesh point in M. (2) if x is not a mesh point, then nn M (x) is equal to the distance to the second closest mesh point in M. Lemma ([35]) If M is an -well-shaped, then there exists constants c 1 and c 2 depending only on such that for all point x 2, c 1 el M (x) nn M (x) c 2 el M (x): 1 Ruppert also gave a modied denition by using the geodesic distance to the 2 nearest non-incident portions of the input to handle the two arms situation [43]. The geodesic distance is measured along the shortest path that stays within the domain to be triangulated. 12

21 For convenience, if the context is clear, we will use nn(x) other than nn M (x) to denote the nearest neighbor value of point x in mesh M Numerical Spacing The numerical condition is usually obtained from an a priori error analysis, or an a posteriori error analysis based on an initial numerical simulation. It denes a numerical spacing functions, denoted by nsf(x), for each point x in the domain. The value of nsf(x), from the interpolation viewpoint, is determined by the eigenvalues of the Hessian matrix of u [49]. Locally at point x, u can be approximated by a quadratic function u(x + dx) = 1 2 (xhxt ) + x 5 u(x) + u(x); where H is the Hessian matrix of u, the matrix of the second partial derivatives. The spacing of the mesh points, required by the accuracy of the discretization near x should depend on the reciprocal of the square root of the largest eigenvalues of H at x. For example, in adaptive numerical simulation, we estimate the eigenvalue of the Hessian matrix at a certain set of points in based on the solution of the previous iteration, and then expand the spacing requirement from these points to the entire domain. From the new spacing and the old spacing function deduced from the previous mesh, we can get the renement or coarsening factor for mesh points. We can then use the simultaneous renement and coarsening method of Li et al. [27] to generate the mesh that satises the new control space requirement Control Spacing Function The local feature size lf s and the numerical condition nsf together denes the global control spacing function. Notice however, that the Lipschitz coeciency of nsf may not be bounded by a constant. Using the technique of Miller, Talmor, and Teng [32], we can dene a new numerical spacing function nsf() as the following: for each point x, nsf(x) = min(nsf(x); min y2 (nsf(y) + jjx? yjj)): (2.1) 13

22 The Lipschitz coeciency of nsf() is at most 1. In addition, nsf() is the best possible in the sense that for any 1-Lipschitz function g over the domain, if g(x) nsf(x) point-wise in, then g(x) nsf(x) point-wise. The global control spacing function gns() can then be dened as gns(x) = min(lf s(x); nsf(x)): (2.2) Where gns stands for Geometric and Numerical Spacing[51]. The function gns() captures both the numerical and the geometric requirements for a well-shaped adaptive mesh. Lemma If f() and g() are 1 and 2 -Lipschitz respectively over, then f() + g() is Lipschitz, and min(f(); g()) and max(f(); g()) are max( 1 ; 2 )-Lipschitz. Therefore, gns is 1-Lipschitz. For mesh generation, we do not need to compute these spacing functions exactly. A common approach to approximate gns() is to store discrete values on the vertices of a background mesh such as a quadtree/octree decomposition of the domain. When we need to evaluate the function at an arbitrary point in the domain, we simply interpolate these discrete values. 2.5 Conformality and Size of Mesh Given the control function specication f, a generated mesh M must conform the control spacing f, in addition to be well shaped. From the generated mesh M, we use the nearest neighbor value nn(x) to denote the spacing of point x in the generated mesh M. Then we dene the conformality of point x as the following. Denition [Point Conformality C f;m ] Assume that f is the control spacing for generating mesh M. For any point x in the domain, let nn(x) be the nearest neighbor value of point x in M. Then the conformality of point x is min( nn(x) f(x) ; f(x) ): (2.3) nn(x) Then to dene the conformality of the mesh M, we can use the minimal conformality value of all mesh vertices; we also can use the average conformality value of all mesh vertices as the 14

23 conformality of the mesh M. If a mesh M conforms the control spacing well, the conformality of every point x in the mesh should be bounded below from a constant. If the constant is closer to 1, the mesh M conforms the control spacing better. We say that a mesh perfectly conform the control spacing, if for every mesh vertex p of M, the conformality of p is equal to 1. The following lemma of Miller et al. [34] estimates the size of the generated well shaped mesh. Lemma (Size of a Well-shaped Mesh [34]) If M is an -well-shaped mesh of n elements, then Z n = ( da ): (2.4) nn 2 M For any well-shaped mesh M, if it conforms the given spacing function f, then the following lemma bounds the number of mesh elements of M. Lemma (Size of Mesh Respect to the Space Control [51]) There exists a constant c such that if M is a well-shaped mesh of n elements over a domain that satises the control spacing function f(), then Z da n c f : (2.5) 2 Here, a mesh M satises the control spacing means that the conformality of the mesh is bounded below from a constant. 2.6 Delaunay Triangulation For mesh generation, there are variety of optimization objective. Especially, the nite element method requires that the minimal angle of the mesh must be bounded below from a constant. To generate a mesh, there are two approaches: the rst approach is to generate some points in the domain, then use some technique to connect these points to generate the nal mesh; the second approach is to generate the mesh elements when generate some mesh vertices. The typical example of the rst approach is the sphere packing method [3, 5, 27, 37, 50, 28]. The advancing front methods [6, 30, 31] are the most widely used second approach. 15

24 If we have the set of points in the domain, Delaunay triangulation [48] of the point set minimizes the smallest angle for two dimension domain. Assume that P is a point set in IR d. The simplex dened by (d + 1) anely independent points from P is a Delaunay simplex if the circum-sphere of the simplex contains no point from P in its interior. The union of all Delaunay simplices forms the Delaunay diagram DT (P ). If P is not degenerate, then DT (P ) is a triangulation of the convex hull of P. Let DB(P ) denote the set of circum-spheres of the simplices of DT (P ). By denition, there is no point in P that lies in the interior of any sphere from DB(P ). The geometric dual of the Delaunay Diagram is the Voronoi Diagram, which consists of a set of polyhedra V 1 V n, one for each point in P. V i is called the Voronoi cell of p i and p i is called the center of V i. Geometrically, V i is the set of points in IR d, whose distance to p i is less than or equal to that of any other point in P. Delaunay triangulation has some desired properties for mesh generation. For example, among all triangulations of a point set in 2D, the Delaunay triangulation maximizes the minimum angle. In any dimension, it always contains the nearest neighbor graph of the point set, i.e. in the Delaunay triangulation, every point is directly connected with its nearest neighbors. The Delaunay triangulation also contains the minimum spanning tree connecting the point set. 2.7 Mesh Generation Method A mesh generator usually does two things: Point set generation: Places Steiner points within or on the boundary of the domain. Element generation: Forms elements of the mesh by triangulating the point set or by using some other element formation procedures. Some mesh generation algorithms construct the point set and then triangulate it, but most mesh generation algorithms merge these two functions, and generate the point set implicitly as a part of the mesh generation phase. We will discuss both of these methods in the next two sections. 16

25 2.7.1 Advancing Front Methods Advancing front methods construct a mesh of a domain by moving a front from its boundary towards its interior. It rst generates an initial front typically by constructing a surface mesh for the boundary of the domain. It then creates new elements one at a time or a layer at a time and updates the front with these created faces [16, 15, 6, 30, 31]: In the one element-at-a-time model, it chooses a face of the current front and introduces a new mesh element with it as the base face. It can use another vertex on the front or insert a new Steiner point in the interior as the additional vertex of the new element. The base face and potentially some other faces on the front (if the additional vertex is an existing one) are removed from the front, and some faces of the new element are added to the front. This process is repeated until the front is empty, i.e., all fronts have merged upon each other and the domain is fully meshed. Hence the method involves the simultaneous generation of eld points and their connectivity. The initial front is constructed by triangulating the boundary of the domain. Note that initial front does not have to be a single component. For example, for a domain with holes, the initial front can be built for the boundary of each hole as well. The selection of the base face and the placement of the new mesh vertex are the two key steps of any advancing front method. These two steps must ensure that the new mesh element is valid and well-shaped and keep the front in good condition to allow the creation of graceful future elements. The faces of the clefts and the small faces are given priority to be picked as the base faces to satisfy these requirements. Hence, once the base face is chosen, we need to decide where to place the new vertex. Recall that in a well-shaped triangular mesh, points must be well-spaced [50, 33], which implies that for each base face, we can only place the Steiner point in a particular region near the base face so that the new element is well-shaped. Call this region the feasible region. Some points in the feasible region will make the new element slightly larger (by a constant factor) than some other points do. This is where the control spacing function can be used. It helps us to decide whether we should go for a larger new element or a smaller one. Paving [6] is one of the popular advancing front methods. It uses a number of operations to tightly controlled the moving front to ensure the mesh validity and quality. These operations include row choice, closure check, row generation, smooth, seam, row adjustment, intersection, 17

26 and cleanup [6]. The size of the elements in the mesh is determined by the spacing of the nodes on the paving boundary as it propagates. The spacing on the paving boundary is initially dened by the xed node spacing on the corresponding exterior boundary. Advancing front methods can be combined with Delaunay or quadtree/octree renements. For instance, these renement techniques can be used to generate a pretty-good domain decomposition of the input domain and then advancing front can be applied to get a mesh for each subdomain. We can also use quadtree/octree renement to generate the set of points for the creation of the new elements Sphere Packing Methods At a high level, the sphere-packing method lls an input domain with a set of spheres whose centers provide a good vertex set for a quality Delaunay mesh. It can be used to generate meshes for various quality conditions. For example, Bern, Mitchell, and Ruppert [5] use sphere packing to triangulate a n-vertex polygonal region (potentially with holes) so that no element has angle larger than =2. They show that one can do so with O(n) triangles, improving a previous result that uses O(n 2 ) triangles [2]. The algorithm rst packs a set of spheres within the domain such that the gaps between them are surrounded by at most four tangent spheres 2. It then denes the mesh points as the union of centers of these spheres, the tangency points, and one point within each gap, and locally triangulates these points. Notice that for nonobtuse triangulation, one does not need to consider the control spacing function. Therefore, their mesh may have elements with very bad aspect ratio. A similar sphere-packing based method has been developed by Bern and Eppstein [3] for quadrilateral meshes. Shimada and Gossard [46] have developed a sphere-packing method called bubble mesh to generate triangular meshes for two and three dimensions. Their packing scheme is based on the simulation of the particles that interact each other with repulsive/attractive forces. They rst dene a proximity based force among the spheres, and then nd a stable conguration by 2 This thesis uses the word sphere in its general meaning; it denotes a circle for two-dimensional domains and a (d? 1)-dimensional sphere for d-dimensional domains. 18

27 moving or deleting spheres. However, their method does not provide a theoretical bound on the time of the algorithm nor the quality of the mesh that they generate. Miller et al. [35, 36] have designed a sphere-packing based meshing method which combines two well-known methods, quadtree and Delaunay renements. First, they apply a balanced quadtree renement to approximate the spacing function f(). Second, they oversample a set of points in the domain to dene a set of overlapping spheres. Then, they compute a maximal set of non-overlapping spheres from this set to obtain a sphere packing. Finally, they compute the Delaunay triangulation of the centers of these spheres. Suppose f() is the desired edge-length or nearest-neighbor function of a well-shaped mesh for a domain. We now introduce some denitions to capture the quality of sphere packing. Let B(x; r) denote the sphere centered at point x with radius r. Denition (-Packing) Let be a positive real constant. A set S of spheres is a - packing with centers P of with respect to a spacing function f() if For each point p of P, B(p; f(p)=2) 2 S; The interiors of any two spheres s 1 and s 2 in S do not overlap; and For each point q 2, there is a sphere in S that overlaps with B(q; f(q)=2). The following structure theorem of Miller, Talmor, and Teng [34] states an equivalence relationship between -sphere packing and well-shaped meshes. Theorem (Sphere Packing and Well-Shaped Meshes) 1. For any positive constant, there exists a constant depending only on such that if f() is a spacing function of Lipschitz constant 1 over a domain and S is a -packing with respect to f(), then the Delaunay triangulation M of the centers of S is an well-shaped mesh; in addition, for each point p in, nn M (p) = (f(p)), where the constant in depends only on. 2. For any positive constant, there exists a constant depending only on such that if M is an well-shaped mesh, then the set of spheres S = fb(p; nn M (p)=2) : for all mesh point p 2 Mg; is a -packing with respect to nn M =2. 19

28 Chapter 3 Graph Partition 3.1 Introduction Identifying enough concurrency in decomposition of data and/or tasks is one of the main objective in parallel algorithm design. The computation and data dependency of a lot of scientic computing problem can be modeled by a graph ( directed graph or undirected graph): the vertices of graph is the basic data unit or computing tasks; two vertices are connected if there are a data dependency between them or there are computation dependency between tasks. Graph partition plays an important role in parallel computing by identifying the concurrency in a given problem. A partition of a graph into subgraphs leads to a decomposition of the data and/or tasks associated with a computational problem and the subgraphs can be mapped to the processors of a multiprocessor. Graph partition also plays an important role in serial algorithm design by means the divide and conquer paradigm. For parallel computing, the ideal assignment of the data and/or tasks to processors is that the work load of every processor are approximately equal and the communication during the computation is minimized. Hence, the following two objectives are usually stated in the partitioning algorithm. First, the subgraphs partitioned have roughly equal number of vertices. Second, there are few edges join dierent subgraphs to each other, i.e., the partition has small cut. In the context of the parallel computation, we assume that every vertex of the original graph denotes the basic unit of the computation overload. Hence the size of a subgraph determines 20

29 the computational work load that a processor has to perform. The number of cut edges is the measure of the communication volume caused by this partition. In a serial algorithm, equalsized subgraphs lead to the lowest worst-case running time, and the number of the cut edges measures the cost of combining the partial solutions to compute the global solution. More general objective functions may need to be considered for many problems. The work associated with a subgraph may be modeled more accurately by attaching a weight to each vertex, and then equipartitioning the weights. The communication costs in the algorithm might be modeled more accurately by the number of subgraphs a given subgraph is connected to, or the number of boundary vertices, or similar variants. In addition, in some algorithms, the shape of the subgraphs may be an important parameter. For example, the convergence of a domain decomposition method dependents on the shape quality of each partition. In some applications, it may be essential to partition into connected subgraphs. In Figure 3.1, 3.2, 3.3, we show the partition of a mesh into sixteen subgraphs computed by the METIS partitioning algorithm. This mesh was generated using the mesh generator written by Xiangyang Li at University of Illinois at Urbana-Champaign. It generates the Delaunay triangulation of an input point set. It also support the constrained edges and holes in the input and also provide the functions to do Delaunay renement, Functional Coarsening and Function Renement idea from the paper [26] of Li et. al.. The Figure shows 3.2 that each subgraph in the partitioned mesh is connected, and it appears to the eye as a good partition of the mesh. Quantitative measures such as the number of edges cut by the partition show that the METIS partition algorithm does generate partitions of good quality for many nite element meshes. 3.2 Background and Good Separators There are two dierent type of partition for a graph G = (V; E): edge separator, vertex separator. A set of edges of E is an edges separator of G, if removal of these edges will partition G to at least two unconnected subgraphs. A set of vertices of V is a vertex separator of G, if after removing these vertices and the edges connected to them, G can be divided into two disconnected subgraphs. Given a graph G = (V; E), and a vertex set B V, let B denote the vertices from V not in B, i.e., V? B. A partition of a connected graph G = (V; E) is a division of its vertices into 21