We study distributed load balancing problem on arbitrary graphs. First Order (FO) and Second Order (SO) schemes are popular local diffusive schedules for this problem. To use these schemes, several parameters have to be chosen carefully. Determining the ''optimal'' parameters analytically is difficult, and on a practical level, despite the widespread use of these schemes, little is known on how relevant parameters must be set.We employ systematic experiments to engineer the choice of relevant parameters in first and second order schemes. Some of our contributions here are as follows. We present a centralized polynomial time algorithm for choosing the ''optimal'' FO scheme based on semidefinite programming. Based on the empirical evidence from our implementation of this algorithm, we pose conjectures on the closed-form solution of optimal FO schemes for various graphs. We also present a heuristic algorithm to locally estimate relevant parameters in the FO and SO schemes; our estimates are fairly accurate compared to those based on expensive global communication. Finally, we show that the FO and SO schemes that use approximate values rather than the optimal parameters, can be improved using a new iterative scheme that we introduce here; this scheme is of independent interest. The software we have developed for our implementations can serve as a platform for experimental research in this area and it will be made available freely. Our methods are being included in PadFEM, the Paderborn Finite Element Library [1].

The domain decomposition method embodies large potentials for a parallelization of FEM methods. In this data-parallel approach, the domain of interest is partitioned into smaller subdomains, either before the mesh generation or afterwards (mesh partitioning). The subdomains are assigned to processors that calculate the corresponding part of the approximation. The mesh decomposition and assignments of subdomains to processors can be modeled as graph embedding problem where a large graph (the mesh) has to be mapped onto a smaller one (the processor network). Unfortunately, this mapping problem is NP-complete and there exist almost no eecient sequential or parallel heuristics that solve this problem suuciently 1]. With growing performance of interconnection networks and especially with the establishment of independent routing networks, it is appropriate to reduce the mapping problem to the task of partitioning the graph (the FEM-mesh) into as many equal sized (or weighted) clusters as there are numbers of processors and to minimize the number of edges crossing the partition boundaries. For the special application of partitioning FEM-meshes heuristics have to be exible with respect to the measures they optimize. Graph partitioning in general is a combinatorial optimization problem. Normal partitioning heuristics minimize the total cut size, i.e. the number of edges crossing partition boundaries. This makes sense in order to reduce the amount of necessary communication of the parallel FEM-simulation. But there are several other measures that are often much more important than cut size and that depend very much on the numerical solution method used for the simulation 4]. Examples are the aspect ratio of subdomains or their convexity. A large number of eecient graph partitioning heuristics have been designed in the past, most of them for recursive bisection but also some for direct k-partitioning. See 2, 3, 6] for overviews. Many of the most eecient methods have been collected into the Chaco library by Hendrickson and Leland 7]. Farhat describes in 4] a simple and eecient sequential algorithm for the partitioning of FEM-meshes. This front-technique is a breath-rst-search based method which is widely used by engineers and innuenced the development of a number of other partitioning tools (e.g. 9]). Recursive Orthogonal Bisection (ROB) and Unbalanced Recursive Bisection (URB) use node-coordinates to partition a mesh and neglect the graph structure 8]. Both methods are fast and easy to implement whereas URB ooers larger exibility. Walshaw and Berzins apply Recursive Spectral Bisection (RSB) which was introduced by Simon et. …

This paper deals with the measure of Aspect Ratio for mesh partitioning and gives hints why, for certain solvers, the Aspect Ratio of partitions plays an important role. We define and rate different kinds of Aspect Ratio, present a new center-based partitioning method which optimizes this measure implicitly and rate several existing partitioning methods and tools under the criterion of Aspect Ratio.

We present a dynamic distributed load balancing algorithm for parallel, adaptive finite element simulations using preconditioned conjugate gradient solvers based on domain-decomposition. The load balancer is designed to maintain good partition aspect ratios. It can calculate a balancing flow using different versions of diffusion and a variant of breadth first search. Elements to be migrated are chosen according to a cost function aiming at the optimization of subdomain shapes. We show how to use information from the second step to guide the first. Experimental results using Bramble's preconditioner and comparisons to existing state-ot-the-art load balancers show the benefits of the construction.

We study a distributed load balancing problem on arbitrary graphs. First Order (FO) and Second Order (SO) schemes are popular local diffusive schedules for this problem. To use them, several parameters have to be chosen carefully. Determining the “optimal” parameters analytically is difficult, and on a practical level, despite the widespread use of these schemes, little is known on how relevant parameters must be set. We employ systematic experiments to engineer the choice of relevant parameters in first and second order schemes. We present a centralized polynomial time algorithm for choosing the “optimal” FO scheme based on semidefinite programming. Based on the empirical evidence from our implementation of this algorithm, we pose conjectures on the closed-form solution of optimal FO schemes for various graphs. We also present a heuristic algorithm to locally estimate relevant parameters in the FO and SO schemes; ourestimates are fairly accurate compared to those based on expensive global communication. Finally, we show that the FO and SO schemes that use approximate values rather than the optimal parameters, can be improved using a new iterative scheme that we introduce here; this scheme is of independent interest. The software we have developed for our implementations is available freely, and can serve as a platform for experimental research in this area. Our methods are being included in PadFEM, the Paderborn Finite Element Library [1].

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. Results on up to 64 processors are presented and compared to those of other existing methods.

We present a modular toolbox for parallel finite element simulations on distributed memory systems. The library named PadFEM includes a graphical editor for specifying domains with boundary conditions, automatic mesh generation, automatic mesh partitioning and mapping onto the processors of a MIMD-system. The parallel FEM-simulation uses the preconditioned conjugate gradient method. The parallelization is based on PVM which guarantees portable parallel code. The current implementation is able to solve the two-dimensional Poisson equation in arbitrary domains with mixed Dirichlet- and Neumann-type boundary conditions using triangular meshes. The modular structure based on the idea of programming frames allows an easy extension to other problems.

Modern flow control techniques used for massively parallel computers have made network capacity a more important parameter for the application performance than network latency. Network latency is usually rather low as long as the injection rate is below a specific value. Nowadays the maximal injection rate is usually approximated by the bisection bandwith of the network. We will describe the state of the art in determining the bisection bandwith of interconnection systems. Unfortunately the bisection bandwith leads only to very vague approximations of the communication capacity of a network. We will describe some methods aiming at modeling the maximal network capacity by using probabilistic models. Especially we will present results for the multistage interconnection network which is often used in parallel computing and more general communication applications. The presented results show a rather close relation to results gained by simulations and therefore have the potential to replace them. We argue that theoretical investigations, leading to close term expressions or polynomial algorithms computing the exact network throughput can be of great help for engineers usually determing the throughput of a given network by time consuming simulations.

We design a general mathematical framework to analyze the properties of nearest neighbor balancing algorithms of the diffusion type. Within this framework we develop a new optimal polynomial scheme (OPS) which we show to terminate within a finite number m of steps, where m only depends on the graph and not on the initial load distribution. We show that all existing diffusion load balancing algorithms, including OPS determine a flow of load on the edges of the graph which is uniquely defined, independent of the method and minimal in the l 2-norm. This result can also be extended to edge weighted graphs. The l 2-minimality is achieved only if a diffusion algorithm is used as preprocessing and the real movement of load is performed in a second step. Thus, it is advisable to split the balancing process into the two steps of first determining a balancing flow and afterwards moving the load. We introduce the problem of scheduling a flow and present some first results on the approximation quality of local greedy heuristics.

Disposal of highly saline industrial by-products in landfills is not permitted in member states of the European Union, such as Germany. Large amounts of such by-products thus have to be disposed of in alternative ways. In many countries bare potash mining residue mounds, consisting almost entirely of rock salt (NaCl), pose environmental problems. Covering such mounds with soil or soil-like material could help to reduce the yearly amount of briny runoff. A fine-granular saline aluminum recycling by-product (ALRP) has been proposed as a soil substitute to cover rock salt residue mounds. Use of this by-product as a combined soil substitute and surface barrier is not considered to be a landfill disposal, but as a beneficial by-product reuse. To judge the feasibility of ALRP for this purpose, its properties must be known. In this study physical characteristics of an industrially produced ALRP, mixed with the flue gas desulfurization by-product (FGDP) of a coal combustion power plant, were determined. It was found that the texture of both ALRP and ALRP-FGDP mix was silt loam. Bulk densities of ALRP and ALRP-FGDP were 0.93 and 0.88 Mg x m(-3) and the corresponding salt contents were 50.0 and 35.5%, respectively. The erodibility factor K of pure ALRP was estimated as 0.65 Mg h x ha (-1)N(-1). Because of the stabilizing effect of FGDP, this factor was reduced considerably in ALRP-FGDP. The water-holding capacity of unwashed ALRP was 44.5% and of washed ALRP-FGDP 61.8%. In view of its physical properties, ALRP-FGDP seems to be suitable as an evaporation enhancing, runoff reducing cover material for potash mine residue mounds, even on steep slopes. Use of ALRP, mixed with FGDP, as a soil substitute in a surface barrier, thus seems to be environmentally meaningful. However, the high salt content initially prevents plant growth. With time, after the salt has been leached, the material seems able to support plant growth, which would further reduce runoff. The physical and hydraulic parameters determined in this study may serve future users of similar by-products.

† Paderborn Center for Parallel Computing, Germany, http://www.upb.de/pc2/ Abstract. An efficient and accurate contact algorithm is essential for FE-simulation of several moving bodies or fragmentation processes of brittle material. The calculation time needed to check and correct unphysical penetrations in the case of large displacements is mainly determined by the search phase of the contact algorithm. In this paper, a brief description of a contact formulation used within an explicit, dy- namic 2D FE-code is given. The contact algorithm is a modified Lagrangian method, i. e. it ensures non-penetration after each step of the explicit time integration. It combines straight-line and C1-continuous (smoothed) boundary approximations. A flexible dynamic data structure allows dynamically growing surfaces which are needed to simulate fragmen- tation processes. A new global contact search algorithm based on position codes is investigated. It builds an artificial ordering of the nodes resting upon a space filling curve. An analysis shows the theoretical benefits of the new algorithm and experimental comparisons with two other global search algorithms show that these benefits carry over to the practice as well.

An efficient contact search for dynamic explicit finite element (FE) simulations with several moving bodies is essential to avoid unacceptable costs. The time spent for contact algorithms is mainly determined by the cost of the search phase. We present a variant of the position code algorithm for efficient global contact search. Instead of a row-wise ordering we propose a numbering that follows a space filling curve. An analysis proves that the new ordering is more efficient in case of long surface segments. The two variants of the position code algorithm and another widely used algorithm, which is based on a hierarchical ordering, are implemented in an industrial FE code and applied to problems in fastening and demolition technology. Experimental results show that the presented method is well suited for all types of meshes and meets the great demands on efficiency for the mentioned field while the other algorithms show disadvantages in some cases.

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 12 of the edge weight of a maximum weighted matching. Its time complexity is O(|E|), with |E| 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 two parts, it guarantees an upper bound of ((k−1)/2)|V|+1 on the cut size of the partition, with |V| being the number of vertices. These quality methods used for the two parts of the multilevel approach lead to an efficient graph-partitioning concept.

We present a dynamic distributed load balancing algorithm for parallel, adaptive Finite Element simulations in which we use preconditioned Conjugate Gradient solvers based on domain-decomposition. The load balancing is designed to maintain good partition aspect ratio and we show that cut size is not always the appropriate measure in load balancing. Furthermore, we attempt to answer the question why the aspect ratio of partitions plays an important role for certain solvers. We define and rate different kinds of aspect ratio and present a new center-based partitioning method of calculating the initial distribution which implicitly optimizes this measure. During the adaptive simulation, the load balancer calculates a balancing flow using different versions of the diffusion algorithm and a variant of breadth first search. Elements to be migrated are chosen according to a cost function aiming at the optimization of subdomain shapes. Experimental results for Bramble's preconditioner and comparisons to state-of-the-art load balancers show the benefits of the construction.

We design a general mathematical framework to analyze the properties of nearest neighbor balancing algorithms of the diffusion type. Within this framework we develop a new Optimal Polynomial Scheme (OPS) which we show to terminate within a finite number m of steps, where m only depends on the graph and not on the initial load distribution.We show that all existing diffusion load balancing algorithms, including OPS, determine a flow of load on the edges of the graph which is uniquely defined, independent of the method and minimal in the l2-norm. This result can also be extended to edge weighted graphs.The l2-minimality is achieved only if a diffusion algorithm is used as preprocessing and the real movement of load is performed in a second step. Thus, it is advisable to split the balancing process into the two steps of first determining a balancing flow and afterwards moving the load. We introduce the problem of scheduling a flow and present some first results on its complexity and the approximation quality of local greedy heuristics.

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. To date these algorithms have been used almost exclusively to minimise the cut-edge weight, however it has been shown that for certain classes of solution algorithm, the convergence of the solver is strongly influenced by the subdomain aspect ratio. In this paper therefore, we modify the multi level algorithms in order to optimise a cost function based on aspect ratio. Several variants of the algorithms are tested and shown to provide excellent results.

Multilevel algorithms are a successful class of optimization techniques that address the mesh partitioning problem for mapping meshes onto parallel computers. They usually combine a graph contraction algorithm together with a lo-cal optimization method that refines the partition at each graph level. To date, these algorithms have been used al-most exclusively to minimize the cut-edge weight in the graph with the aim of minimizing the parallel communication overhead. However, it has been shown that for certain classes of problems, the convergence of the underlying solution algorithm is strongly influenced by the shape or aspect ratio of the subdomains. Therefore, in this paper, the authors modify the multilevel algorithms to optimize a cost function based on the aspect ratio. Several variants of the algorithms are tested and shown to provide excellent results.

In parallel adaptive finite element simulations the work load on the individual processors may change frequently. To (re)distribute the load evenly over the processors a load balancing heuristic is needed. Common strategies try to minimise subdomain dependencies by optimising the cutsize of the partitioning. However for certain solvers cutsize only plays a minor role, and their convergence is highly dependent on the subdomain shapes. Degenerated subdomain shapes cause them to need significantly more iterations to converge. In this work a new parallel load balancing strategy is introduced which directly addresses the problem of generating and conserving reasonably good subdomain shapes in a dynamically changing Finite Element Simulation. Geometric data is used to formulate several cost functions to rate elements in terms of their suitability to be migrated. The well known diffusive method which calculates the necessary load flow is enhanced by weighting the subdomain edges with the help of these cost functions. The proposed methods have been tested and results are presented.

In this paper we present a decentralized remapping method for data parallel applications on distributed memory multiprocessors. The method uses a generalized dimension exchange (GDE) algorithm periodically during the execution of an application to balance (remap) the system's workload. We implemented this remapping method in parallel WaTor simulations and parallel image thinning applications, and found it to be effective in reducing the computation time. The average performance gain is about 20% in the WaTor simulation of a 256 × 256 ocean grid on 16 processors, and up to 8% in the thinning of a typical image of size 128 × 128 on eight processors. The performance gains due to remapping in the image thinning case are reasonably substantial given the fact that the application by its very nature does not necessarily favor remapping. We also implemented this remapping method, using up to 32 processors, for partitioning and re-partitioning of grids in computational fluid dynamics. It was found that the GDE-based parallel refinement policy, coupled with simple geometric strategies, produces partitions that are comparable in quality to those from the best serial algorithms. © 1997 John Wiley & Sons, Ltd.