Recursive Bisection Research Articles

Subdomain generation using emergent ant colony optimization

Finite elements mesh decomposition is a well known optimization problem and is used to split a computationally expensive finite elements mesh into smaller subdomains for parallel finite elements analysis. The ant colony optimization is a type of algorithm that seeks to model the emergent behaviour observed in ant colonies and utilize this behaviour to solve combinatorial problems. This technique has been applied to several problems, most of which are graph related because the ant colony metaphor can be most easily applied to such types of problems. This paper examines the application of ant colony optimization algorithm to the partitioning of unstructured adaptive meshes for parallel explicit time-stepping finite elements analysis. The concept of ant colony optimization technique in addition to the notion of swarm intelligence for finding approximate solutions to combinatorial optimization problems is described. This algorithm combines the features of the classical ant colony optimization technique with swarm intelligence to form a model which is an artificial system designed to perform a certain task. The application of the ant colony optimization for partitioning finite elements meshes based on triangular elements using the swarm intelligence concept is described. A recursive greedy algorithm optimization method is also presented as a local optimization technique to improve the quality of the solutions given by the ant colony optimization algorithm. The partitioning is based on the recursive bisection approach. The mesh partitioning is carried out using normal and predictive modes for which the predictive mode uses a trained multi-layered feedforward neural network that estimates the number of triangular elements that will be generated after finite elements mesh generation is carried out. The performance of the proposed hybrid approach for the recursive bisection of finite elements meshes is examined by decomposing two mesh examples and comparing them with a well known finite elements domain decomposer.

A design methodology for efficient application-specific on-chip interconnects

As the level of chip-integration continues to advance at a fast pace, the desire for efficient interconnects - whether on-chip or off-chip - is rapidly increasing. Traditional interconnects like buses, point-to-point wires, and regular topologies may suffer from poor resource sharing in the time and space domains, leading to high contention or low resource utilization. In this paper, we propose a design methodology for constructing networks for special-purpose computer systems with well-behaved (known) communication characteristics. A temporal and spatial model is proposed to define the sufficient condition for contention-free communication. Based upon this model, a design methodology using a recursive bisection technique is applied to systematically partition a parallel system such that the required number of links and switches is minimized while achieving low contention. Results show that the design methodology can generate more optimized on-chip networks with up to 60 percent fewer resources than meshes or tori while providing blocking performance closer to that of a fully connected crossbar.

PARALLEL SIMULATION ANALYSIS ON MESHLESS METHOD FOR SOLVING 3D BULK FORMING PROCESS

A computational implementation of modern explicit meshless methods is presented for explicit for 3D bulk forming process. The implemented parallel meshless method is based on the reproducing kernel particle method (RKPM) theory. Partitioning scheme modified recursive spectrum bisection (MRSB) procedure is applied for geometry model in preprocess. Explicit MPI message passing statements are used for coarse grain parallel program in analysis process. According to characterizes of contact search problem, the parallel contact search algorithm is also presented. The parallel program is coded and implemented into 3D bulk metal forming problems. The severe distortion mesh grid problem is settled and the simulation results demonstrated the efficiency of the suggested parallel RKPM algorithm.

A parallel 3D particle-in-cell code with dynamic load balancing

A parallel 3D electrostatic Particle-In-Cell (PIC) code including an algorithm for modelling Space Charge Limited (SCL) emission [E. Gjonaj, T. Weiland, 3D-modeling of space-charge-limited electron emission. A charge conserving algorithm, Proceedings of the 11th Biennial IEEE Conference on Electromagnetic Field Computation, 2004] is presented. A domain decomposition technique based on orthogonal recursive bisection is used to parallelize the computation on a distributed memory environment of clustered workstations. For problems with a highly nonuniform and time dependent distribution of particles, e.g., bunch dynamics, a dynamic load balancing between the processes is needed to preserve the parallel performance. The algorithm for the detection of a load imbalance and the redistribution of the tasks among the processes is based on a weight function criterion, where the weight of a cell measures the computational load associated with it. The algorithm is studied with two examples. In the first example, multiple electron bunches as occurring in the S-DALINAC [A. Richter, Operational experience at the S-DALINAC, Proceedings of the Fifth European Particle Accelerator Conference, 1996] accelerator are simulated in the absence of space charge fields. In the second example, the SCL emission and electron trajectories in an electron gun are simulated.

Racoon: A parallel mesh-adaptive framework for hyperbolic conservation laws

We report on the development of a computational framework for the parallel, mesh-adaptive solution of systems of hyperbolic conservation laws like the time-dependent Euler equations in compressible gas dynamics or Magneto-Hydrodynamics (MHD) and similar models in plasma physics. Local mesh refinement is realized by the recursive bisection of grid blocks along each spatial dimension, implemented numerical schemes include standard finite-differences as well as shock-capturing central schemes, both in connection with Runge–Kutta type integrators. Parallel execution is achieved through a configurable hybrid of POSIX-multi-threading and MPI distribution with dynamic load balancing. One-, two- and three-dimensional test computations for the Euler equations have been carried out and show good parallel scaling behavior. The Racoon framework is currently used to study the formation of singularities in plasmas and fluids.

Mixed block placement via fractional cut recursive bisection

Recursive bisection is a popular approach for large scale circuit placement problems, combining a high degree of scalability with good results. In this paper, we present a bisection-based approach for both standard cell and mixed block placement; in contrast to prior work, our horizontal cut lines are not restricted to row boundaries. This technique, which we refer to as a fractional cut, simplifies mixed block placement and also avoids a narrow region problem encountered in standard cell placement. Our implementation of these techniques in the placement tool Feng Shui 2.6 retains the speed and simplicity for which bisection is known, while making it competitive with leading methods on standard cell designs. On mixed block placement problems, we obtain substantial improvements over recently published work. Half perimeter wire lengths are reduced by 29% on average, compared to a flow based on Capo and Parquet; compared to mPG-ms, wire lengths are reduced by 26% on average.

Unifying mesh- and tree-based programmable interconnect

We examine the traditional, symmetric, Manhattan mesh design for field-programmable gate-array (FPGA) routing along with tree-of-meshes (ToM) and mesh-of-trees (MoT) based designs. All three networks can provide general routing for limited bisection designs (Rent's rule with p<1) and allow locality exploitation. They differ in their detailed topology and use of hierarchy. We show that all three have the same asymptotic wiring requirements. We bound this tightly by providing constructive mappings between routes in one network and routes in another. For example, we show that a (c,p) MoT design can be mapped to a (2c,p) linear population ToM and introduce a corner turn scheme which will make it possible to perform the reverse mapping from any (c,p) linear population ToM to a (2c,p) MoT augmented with a particular set of corner turn switches. One consequence of this latter mapping is a multilayer layout strategy for N-node, linear population ToM designs that requires only /spl Theta/(N) two-dimensional area for any p when given sufficient wiring layers. We further show upper and lower bounds for global mesh routes based on recursive bisection width and show these are within a constant factor of each other and within a constant factor of MoT and ToM layout area. In the process we identify the parameters and characteristics which make the networks different, making it clear there is a unified design continuum in which these networks are simply particular regions.

A high-performance lattice Boltzmann implementation to model flow in porous media

We examine the problem of simulating single and multiphase flow in porous medium systems at the pore scale using the lattice Boltzmann (LB) method. The LB method is a powerful approach, but one which is also computationally demanding; the resolution needed to resolve fundamental phenomena at the pore scale leads to very large lattice sizes, and hence substantial computational and memory requirements that necessitate the use of massively parallel computing approaches. Common LB implementations for simulating flow in porous media store the full lattice, making parallelization straightforward but wasteful. We investigate a two-stage implementation consisting of a sparse domain decomposition stage and a simulation stage that avoids the need to store and operate on lattice points located within a solid phase. A set of five domain decomposition approaches are investigated for single and multiphase flow through both homogeneous and heterogeneous porous medium systems on different parallel computing platforms. An orthogonal recursive bisection method yields the best performance of the methods investigated, showing near linear scaling and substantially less storage and computational time than the traditional approach.

Object Oriented Development of Parallel 3D Free Mesh Method

Free Mesh Method (FMM) is a kind of node-based finite element method, which does not require mesh information. Therefore it has a main advantage of meshless method cutted cost of calculation by omitting mesh creation. Then it seems suitable for an adaptive method, a crack propagation problem, and a large-scale problem. Moreover, it still has merits of finite element method, i.e., it can use many existent techniques developed for finite element method. Although FMM has been demonstrated to be applied to many fields of research, no general purpose program based on FMM has been developed yet. So we designed Free Mesh Method program for 3D problem. Some of main features of this software are : (1) Large-scale : It can be executed in parallel environment, such as PC cluster with domain decomposition method (DDM) (2) Expandable : It is designed by object-oriented approach and new functions can be added very easily. Here we describe the local mesh generation of FMM, generalized recursive bisection method for DDM, the object-oriented design of the software, and the object-oriented parallel matrix solver.

Effective free space management for cut-based placement via analytical constraint generation

IP blocks and large macro cells are becoming more prevalent in the physical layout of a design, actually causing an increase in the available free space. We observe that top-down placement based on recursive bisection with multilevel partitioning performs poorly on these porous designs since it lacks a global view of the ideal placement. However, the strength of analytic placement methods lies in their ability to ascertain this global view. Consequently, we propose an enhancement to cut-based placement called analytic constraint generation (ACG). ACG utilizes an analytic engine to distribute available free space appropriately by determining balance constraints for each partitioning step. For one-dimensional placements, our experiments illustrate the large gap between analytic engines, traditional cut-based placement, and ACG as a design becomes increasingly sparse. We also show that for real industry designs, ACG significantly improves the performance of cut-based placement, particularly timing perspective, as implemented within a state-of-the-art industrial placer.

Clustering analysis of microarray gene expression data by splitting algorithm

A clustering method based on recursive bisection is introduced for analyzing microarray gene expression data. Either or both dimensions for the genes and the samples of a given microarray dataset can be classified in an unsupervised fashion. Alternatively, if certain prior knowledge of the genes or samples is available, a supervised version of the clustering analysis can also be carried out. Either approach may be used to generate a partial or complete binary hierarchy, the dendrogram, showing the underlying structure of the dataset. Compared to other existing clustering methods used for microarray data analysis (such as hierarchical and K-means), the method presented here has the advantage of much improved computational efficiency while retaining effective separation of data clusters under a distance metric, a straightforward parallel implementation, and useful extraction and presentation of biological information. Clustering results of both synthesized and experimental microarray data are presented to demonstrate the performance of the algorithm.

Parallelization of a treecode

I describe here the performance of a parallel treecode with individual particle timesteps. The code is based on the Barnes–Hut algorithm and runs cosmological N-body simulations on parallel machines with a distributed memory architecture using the MPI message-passing library. For a configuration with a constant number of particles per processor the scalability of the code was tested up to P=128 processors on an IBM SP4 machine. In the large P limit the average CPU time per processor necessary for solving the gravitational interactions is ∼10% higher than that expected from the ideal scaling relation. The processor domains are determined every large timestep according to a recursive orthogonal bisection, using a weighting scheme which takes into account the total particle computational load within the timestep. The results of the numerical tests show that the load balancing efficiency L of the code is high (≳90%) up to P=32, and decreases to L∼80% when P=128. In the latter case it is found that some aspects of the code performance are affected by machine hardware, while the proposed weighting scheme can achieve a load balance as high as L∼90% even in the large P limit.

A New Paradigm for Parallel Adaptive Meshing Algorithms

We present a new approach to the use of parallel computers with adaptive finite element methods. This approach addresses the load balancing problem in a new way, requiring far less communication than current approaches. It also allows existing sequential adaptive PDE codes such as PLTMG and MC to run in a parallel environment without a large investment in recoding. In this new approach, the load balancing problem is reduced to the numerical solution of a small elliptic problem on a single processor, using a sequential adaptive solver, without requiring any modifications to the sequential solver. The small elliptic problem is used to produce a posteriori error estimates to predict future element densities in the mesh, which are then used in a weighted recursive spectral bisection of the initial mesh. The bulk of the calculation then takes place independently on each processor, with no communication, using possibly the same sequential adaptive solver. Each processor adapts its region of the mesh independent...

Multiresolution Compression and Visualization of Global Topographic Data

We present a multiresolution model for terrain surfaces which is able to handle large-scale global topographic data. It is based on a hierarchical decomposition of the sphere by a recursive bisection triangulation in geographic coordinates. Error indicators allow the representation of the data at various levels of detail and enable data compression by local omission of data values. The resulting adaptive hierarchical triangulation is stored using a bit code of the underlying binary tree and additionally, relative pointers which allow a selective tree traversal. This way, it is possible to work directly on the compressed data. We show that significant compression rates can be obtained already for small threshold values. In a visualization application, adaptive triangulations which consist of hundreds of thousands of shaded triangles are extracted and drawn at interactive rates.

A parallel framework for the analysis of metal-flanged rectangular-aperture arrays

Computer-aided design of horn arrays, even by using state-of-the-art electromagnetic (EM) formulations, requires a considerable numerical effort and is, therefore, an ideal candidate for parallelization. We introduce a rigorous methodology, based on Petri nets and recursive bisection, in order to migrate a standard integral equation code toward parallel multiprocessor machines, following a data-flow design and a multiple-level parallelism. Significant (quasi-linear) speed-ups are demonstrated.

Parallel Computing in Water Network Analysis and Leakage Minimization

In this paper a parallel computing based software demonstrator for the simulation and leakage minimization of water networks is presented. This demonstrator, based on the EPANET package, tackles three different types of problems making use of parallel computing. First, the solution of the hydraulic problem is treated by means of the gradient method. The key point in the parallelization of the method is the solution of the underlying linear systems, which is carried out by means of a multifrontal Choleski method. Second, the water quality simulation problem is approached by using the discrete volume element method. The application of parallel computing is based on dividing the water network in several parts using the multilevel recursive bisection graph partitioning algorithm. Finally, the problem of leakage minimization using pressure reducing valves is approached. This results in the formulation of an optimization problem for each time step, which is solved by means of sequential quadratic programming. Because these subproblems are independent of each other, they can be solved in parallel.

A New Paradigm for Parallel Adaptive Meshing Algorithms

We present a new approach to the use of parallel computers with adaptive finite element methods. This approach addresses the load balancing problem in a new way, requiring far less communication than current approaches. It also allows existing sequential adaptive PDE codes such as PLTMG and MC to run in a parallel environment without a large investment in recoding. In this new approach, the load balancing problem is reduced to the numerical solution of a small elliptic problem on a single processor, using a sequential adaptive solver, without requiring any modifications to the sequential solver. The small elliptic problem is used to produce a posteriori error estimates to predict future element densities in the mesh, which are then used in a weighted recursive spectral bisection of the initial mesh. The bulk of the calculation then takes place independently on each processor, with no communication, using possibly the same sequential adaptive solver. Each processor adapts its region of the mesh independently, and a nearly load-balanced mesh distribution is usually obtained as a result of the initial weighted spectral bisection. Only the initial fan-out of the mesh decomposition to the processors requires communication. Two additional steps requiring boundary exchange communication may be employed after the individual processors reach an adapted solution, namely, the construction of a global conforming mesh from the independent subproblems, followed by a final smoothing phase using the subdomain solutions as an initial guess. We present a series of convincing numerical experiments which illustrate the effectiveness of this approach. The justification of the initial refinement prediction step, as well as the justification of skipping the two communication-intensive steps, is supported by some recent [J. Xu and A. Zhou, Math. Comp., to appear] and not so recent [J. A. Nitsche and A. H. Schatz, Math. Comp., 28 (1974), pp. 937--958; A. H. Schatz and L. B. Wahlbin, Math. Comp., 31 (1977), pp. 414--442; A. H. Schatz and L. B. Wahlbin, Math. Comp., 64 (1995), pp. 907--928] results on local a priori and a posteriori error estimation.

Image-Space Decomposition Algorithms for Sort-First Parallel Volume Rendering of Unstructured Grids

Twelve adaptive image-space decomposition algorithms are presented for sort-first parallel direct volume rendering (DVR) of unstructured grids on distributed-memory architectures. The algorithms are presented under a novel taxonomy based on the dimension of the screen decomposition, the dimension of the workload arrays used in the decomposition, and the scheme used for workload-array creation and querying the workload of a region. For the 2D decomposition schemes using 2D workload arrays, a novel scheme is proposed to query the exact number of screen-space bounding boxes of the primitives in a screen region in constant time. A probe-based chains-on-chains partitioning algorithm is exploited for load balancing in optimal 1D decomposition and iterative 2D rectilinear decomposition (RD). A new probe-based optimal 2D jagged decomposition (OJD) is proposed which is much faster than the dynamic-programming based OJD scheme proposed in the literature. The summed-area table is successfully exploited to query the workload of a rectangular region in constant time in both OJD and RD schemes for the subdivision of general 2D workload arrays. Two orthogonal recursive bisection (ORB) variants are adapted to relax the straight-line division restriction in conventional ORB through using the medians-of-medians approach on regular mesh and quadtree superimposed on the screen. Two approaches based on the Hilbert space-filling curve and graph-partitioning are also proposed. An efficient primitive classification scheme is proposed for redistribution in 1D, and 2D rectilinear and jagged decompositions. The performance comparison of the decomposition algorithms is modeled by establishing appropriate quality measures for load-balancing, amount of primitive replication and parallel execution time. The experimental results on a Parsytec CC system using a set of benchmark volumetric datasets verify the validity of the proposed performance models. The performance evaluation of the decomposition algorithms is also carried out through the sort-first parallelization of an efficient DVR algorithm.

Parallel volume meshing using face removals and hierarchical repartitioning

Parallel unstructured three-dimensional mesh generation is a challenging problem for many reasons, the most obvious coming from the complexity of ‘partitioning’ the problem such that meshing is load balanced at all times. In this paper, a parallel volume meshing procedure whose input is a surface mesh (distributed or not) is presented. In order to help drive the parallel execution, a distributed octree is built considering the surface mesh and meshing size attributes. The tree is partitioned in parallel using the Recursive Bisection methodology and ‘portions of space’ are handed out to processors for meshing. Volume meshing operates on two techniques: (i) octant template meshing for interior octants; and (ii) face removals (advancing front) to fill the space in between the surface mesh and the ‘templated’ octants. Due to the mesh being distributed, domains corresponding to the interfaces of the initial partitioning are left unmeshed. To complete volume meshing, it is necessary to repartition the interface domains. A hierarchical repartitioning procedure has been developed to effectively mesh face, edge, and vertex interfaces. It is the focus of this paper.

Normal forms and stability in nonsmooth potential elastostatics

The numerical solution of the steady two-dimensional Euler and Navier-Stokes equations using unstructured meshes with triangular elements is considered. A finite volume vertex-centered scheme is used. The Roe upwind scheme is implemented for the inviscid fluxes calculation, coupled with a monotonic extrapolation scheme (MUSCL) for higher-order accuracy. Viscous fluxes are calculated via a finite-element consistent scheme. The discretized equations are solved by means of a point-Jacobi procedure and the algorithm is ported on a distributed memory computer. The recursive spectral bisection algorithm provides patched or overlapping subdomains which are assigned to different processors. Overlapping or non-overlapping subdomain solutions are compared in terms of (a) computer cost for the partitioning, (b) efficiency of the parallel processing of the flow equations and (c) susceptibility of artifices for improving speed-up.