We investigate a method of dividing an irregular mesh into equal-sized pieces with few interconnecting edges. The method’s novel feature is that it exploits the geometric coordinates of the mesh vertices. It is based on theoretical work of Miller, Teng, Thurston, and Vavasis, who showed that certain classes of “well-shaped” finite element meshes have good separators. The geometric method is quite simple to implement: we describe a Matlab code for it in some detail. The method is also quite efficient and effective: we compare it with some other methods, including spectral bisection.

Multilevel algorithms are a successful class of optimisation techniques which address the mesh partitioning problem. They usually combine a graph contraction algorithm together with a local optimisation method which refines the partition at each graph level. In this paper we present an enhancement of the technique which uses imbalance to achieve higher quality partitions. We also present a formulation of the Kernighan-Lin partition optimisation algorithm which incorporates load-balancing. The resulting algorithm is tested against a different but related state-ofthe -art partitioner and shown to provide improved results. Keywords: graph-partitioning, mesh partitioning, load-balancing, multilevel algorithms. 1 Introduction The need for mesh partitioning arises naturally in many finite element (FE) and finite volume (FV) applications. Meshes composed of elements such as triangles or tetrahedra are often better suited than regularly structured grids for representing completely general ge...

. We present a massively parallel algorithm for static and dynamic partitioning of unstructured FEM-meshes. The method consists of two parts. First a fast but inaccurate sequential clustering is determined which is used, together with a simple mapping heuristic, to map the mesh initially onto the processors of a massively parallel system. The second part of the method uses a massively parallel algorithm to remap and optimize the mesh decomposition taking several cost functions into account. It first calculates the amount of nodes that have to be migrated between pairs of clusters in order to obtain an optimal load balancing. In a second step, nodes to be migrated are chosen according to cost functions optimizing the amount and necessary communication and other measures which are important for the numerical solution method (like for example the aspect ratio of the resulting domains). The parallel parts of the method are implemented in C under Parix to run on the Parsytec GCel systems. R...

Identifying the parallelism in a problem by partitioning its data and tasks among the processors of a parallel computer is a fundamental issue in parallel computing. This problem can be modeled as a graph partitioning problem in which the vertices of a graph are divided into a specified number of subsets such that few edges join two vertices in different subsets. Several new graph partitioning algorithms have been developed in the past few years, and we survey some of this activity. We describe the terminology associated with graph partitioning, the complexity of computing good separators, and graphs that have good separators. We then discuss early algorithms for graph partitioning, followed by three new algorithms based on geometric, algebraic, and multilevel ideas. The algebraic algorithm relies on an eigenvector of a Laplacian matrix associated with the graph to compute the partition. The algebraic algorithm is justified by formulating graph partitioning as a quadratic assignment p...

New heuristic algorithms are proposed for the Graph Partitioning problem. A greedy construction scheme with an appropriate tie--breaking rule (MIN-MAX-GREEDY) produces initial assignments in a very fast time. For some classes of graphs, independent repetitions of MIN-MAX-GREEDY are sufficient to reproduce solutions found by more complex techniques. When the method is not competitive, the initial assignments are used as starting points for a prohibition-based scheme, where the prohibition is chosen in a randomized and reactive way, with a bias towards more successful choices in the previous part of the run. The relationship between prohibition-based diversification (Tabu Search) and the variable-depth Kernighan--Lin algorithm is discussed. Detailed experimental results are presented on benchmark suites used in the previous literature, consisting of graphs derived from parametric models (random graphs, geometric graphs, etc.) and of "realworld " graphs of large size. On the first series ...

Multilevel strategies have proven to be very powerful approaches in order to partition graphs efficiently. Their efficiency is dominated by two parts; the coarsening and the local improvement strategies. Several methods have been developed to solve these problems, but their efficiency has only been proven on an experimental basis. In this paper we present new and efficient methods for both problems, while satisfying certain quality measurements. For the coarsening part we develop a new approximation algorithm for maximum weighted matching in general edge-weighted graphs. It calculates a matching with an edge weight of at least 1 2 of the edge weight of a maximum weighted matching. Its time complexity is O(jEj), with jEj being the number of edges in the graph. Furthermore, we use the Helpful-Set strategy for the local improvement of partitions. For partitioning graphs with a regular degree of 2k into 2 parts, it guarantees an upper bound of k\Gamma1 2 jV j + 1 on the cut size of th...

The problem of partitioning a graph into a number of pieces is one of the fundamental tasks in computer science and has a number of applications e.g. in computational mechanics or VLSI design. Finding optimal partitions according to different measures is in most cases NP-complete. Nevertheless, a large number of efficient partitioning heuristics have been developed during recent years. The performance of these methods in terms of computation time as well as quality of approximation is heavily influenced by choices of parameters and certain implementation details. Fortunately, the partitioning problem itself is clearly defined and its description leads to a small interface. Thus, efficient implementations of approximation heuristics can be re-used for different applications. The PARTY partitioning library serves a variety of different partitioning methods in a very simple and easy way. Instead of implementing the methods directly, the user may take advantage of the ready implemented me...

this paper we discuss the mesh partitioning problem in the light of the coming generation of massively parallel machines and the resulting implications for such algorithms

We give an overview of some strategies for mapping unstructured meshes onto processor grids. Sample results show that the mapping can make a considerable difference to the communication overhead in the parallel solution time, particularly as the number of processors increase.

: Parallel implicit solution methods are increasingly important in aerodynamics, since reliable low-residual solutions at elevated CFL number are prerequisite to such large-scale applications of aerodynamic analysis codes as aeroelasticity and optimization. In this chapter, a class of nonlinear implicit methods and a class of linear implicit methods are defined and illustrated. Their composition forms a class of methods with strong potential for parallel implicit solution of aerodynamics problems. Newton-Krylov methods are suited for nonlinear problems in which it is unreasonable to compute or store a true Jacobian, given a strong enough preconditioner for the inner linear system that needs to be solved for each Newton correction. In turn, Krylov-Schwarz iterative methods are suited for the parallel implicit solution of multidimensional systems of linearized boundary value problems. Schwarz-type domain decomposition preconditioning provides good data locality for parallel implementatio...