Parallel pseudospectral domain decomposition techniques

The influence of interface boundary conditions on the ability to parallelize pseudospectral multidomain algorithms is investigated. Using the properties of spectral expansions, a novel parallel two-domain procedure is generalized to an arbitrary number of domains each of which can be solved on a separate processor. This interface boundary condition considerably simplifies influence matrix techniques.     

Parallel efficiency of domain decomposition methods

Data communication among processors is a key problem to increase performance of parallel algorithms for multiprocessors. In this paper, we investigate different types of domain decomposition methods on the distributed memory multiprocessor and give the estimates of their speedup and efficiency based on the computational complexity and the communication overheads.     

Particle-field decomposition and domain decomposition in parallel particle-in-cell beam dynamics simulation

Particle-in-cell (PIC) simulation is widely used in many branches of physics and engineering. In this paper, we give an analysis of the particle-field decomposition method and the domain decomposition method in parallel particle-in-cell beam dynamics simulation. The parallel performance of the two decomposition methods was studied on the Cray XT4 and the IBM Blue Gene/P Computers. The domain decomposition method shows better scalability but is slower than the particle-field decomposition in most cases (up to a few thousand processors) for macroparticle dominant applications. The particle-field decomposition method also shows less memory usage than the domain decomposition method due to its use of perfect static load balance. For applications with a smaller ratio of macroparticles to grid points, the domain decomposition method exhibits better scalability and faster speed. Application of the particle-field decomposition scheme to high-resolution macroparticle-dominant parallel beam dynamics simulation for a future light source linear accelerator is presented as an example.     

Automated domain decomposition for computational fluid dynamics

Automation of flow-field zoning in two-dimensions is an important step towards easing the three-dimensional grid generation bottleneck in computational fluid dynamics. A knowledge-based approach works well, but several aspects of flow-field zoning make the use of such an approach challenging. A proposed model and language to describe the process of zoning a flow field are presented, followed by a discussion of the implementation of EZGrid, a knowledge-based two-dimensional (2-D) flow-field zoner. Results are shown for representative 2-D aerodynamic configurations. Finally, an approach to the evaluation of flow-field zonings is described and used to compare the performance of EZGrid with that of a human expert.     

Non-overlapping domain decomposition methods in structural mechanics

Summary The modern design of industrial structures leads to very complex simulations characterized by nonlinearities, high heterogeneities, tortuous geometries... Whatever the modelization may be, such an analysis leads to the solution to a family of large ill-conditioned linear systems. In this paper we study strategies to efficiently solve to linear system based on non-overlapping domain decomposition methods. We present a review of most employed approaches and their strong connection. We outline their mechanical interpretations as well as the practical issues when willing to implement and use them. Numerical properties are illustrated by various assessments from academic to industrial problems. An hybrid approach, mainly designed for multifield problems, is also introduced as it provides a general framework of such approaches.     

The non-overlapping domain decomposition multiplicative schwarz method

Once a decomposition of the finite element space V into two or more subspaces is given, e.g., via domain decomposition, a specific Multiplicative Schwarz Method (MSM) and Additive Schwarz Method (ASM) is defined. In this paper, we analyse the MSM for the decomposition induced by the approximate discrete harmonic finite element basis which was introduced in a joint paper of the authors with A. Meyer (1990). The main theorem of the present paper states that a special symmetric version of the MSM with approximate orthoprojections is equivalent to some ASM with specially chosen basic transformation and block preconditioners. From this observation we can benefit twice. Indeed, the MSM-DD-preconditioner can be analysed in the MSM framework and implemented as specific ASM-DD-preconditioner in the parallel PCG method studied previously. Emphasis that we look at the ASM and MSM as techniques for defining and analysing parallel DD preconditioners used then in a parallelized version of the PCG-method which is well suited for computations on MIMD computers with local memory and message passing principle.     

Geometry-aware domain decomposition for T-spline-based manifold modeling

This paper presents a new and effective method to construct manifold T-splines of complicated topology/geometry. The fundamental idea of our novel approach is the geometry-aware object segmentation, by which an arbitrarily complicated surface model can be decomposed into a group of disjoint components that comprise branches, handles, and base patches. Such a domain decomposition simplifies objects of arbitrary topological type into a family of genus-zero/one open surfaces, each of which can be conformally parameterized into a set of rectangles. In contrast to the conventional decomposition approaches, our method can guarantee that the cutting locus are consistent on the parametric domain. As a result, the resultant T-splines of decomposed components are automatically glued and have high-order continuity everywhere except at the extraordinary points. We show that the number of extraordinary points of the domain manifold is bounded by the number of segmented components. Furthermore, the entire mesh-to-spline data conversion pipeline can be implemented with full automation, and thus, has potential in shape modeling and reverse engineering applications of complicated real-world objects.     

Applying domain decomposition to wind field calculation

Forest fire are natural hazards that every year cause significant looses. Predicting the evolution of a forest fire is a critical issue in mitigating its effects. Such predictions must accomplish strict real time constraints to be effective. Wind field calculation is a key issue in providing accurate forest fire propagation predictions. However, it implies solving large linear systems with 10⁵ to 10⁸ variables that takes too long using conventional methods. Therefore, the domain decomposition Schur method has been applied to accelerate wind field calculation. Using the Schur method, the linear system is significantly reduced and several phases can be parallelised exploiting cluster computing capabilities. Results show that the execution time for the wind field calculation of a map of 800 × 800 cells has been reduced from 400 s to 90 s using 10 nodes.     

Two-constraint domain decomposition with Space Filling Curves

In scientific computing, Space Filling Curves are a widely used tool for one-constraint domain decomposition. They provide a mechanism to sort multi-dimensional data in a locality preserving way, and, in this way, a (one dimensional) list of mesh elements is established which is subsequently split into 3 partitions under consideration of the constraint. This procedure has a runtime of O(NlogN) (N is the number of mesh elements) while nearly perfect load balancing can be established with reasonable partition surface sizes.In this work, we discuss the extensibility of this procedure to two-constraint settings which is desirable, since the methodology is extremely fast. Here, the splitting operation is subject to two constraints, and, unlike to the one-constraint case, obtaining near perfect balancing is often hard to establish, and is, even more as in the one-constraint case, in conflict with the induced surface sizes (or edge-cuts). We discuss multiple strategies to tackle the splitting, and we present a fast, O(NlogN) splitting heuristic algorithm which provides an integer σ that allows to trade off between balancing and surface sizes which results in a O(NlogN) two-constraint decomposition method. Results are compared to the multi-constraint domain decomposition abilities implemented in the Metis software package, and show that the method produces higher surface sizes, but is orders of magnitudes faster which makes the method superior for certain applications.     

Convergence analysis of a domain decomposition paradigm

We describe a domain decomposition algorithm for use in several variants of the parallel adaptive meshing paradigm of Bank and Holst. This algorithm has low communication, makes extensive use of existing sequential solvers, and exploits in several important ways data generated as part of the adaptive meshing paradigm. We show that for an idealized version of the algorithm, the rate of convergence is independent of both the global problem size N and the number of subdomains p used in the domain decomposition partition. Numerical examples illustrate the effectiveness of the procedure.     

A domain decomposition algorithm for inverse welding problems

A mathematical model and a numerical scheme for the inverse determination of heat sources generated by means of a welding process is presented in this paper. The accuracy of the heat source retrieval is discussed. Received: 30 January 2001 / Accepted: 30 May 2001     

Solution domain decomposition method for viscous incompressible flow

A solution domain decomposition method is developed for steady state solution of the biharmonic-based Navier-Stokes equations. It consists of a domain decomposition in conjunction with Chebyshev collocation for spatial discretization. The interactions between subdomains are effectively decoupled by means of a superposition of auxiliary solutions to yield a set of independent elementary problems which can be solved concurrently on multiprocessor computers. Assessments are carried out to a number of test problems including the two-dimensional steady flow in a driven square cavity. Illustrative examples indicate a good performance of the proposed methodology which does not affect the convergence and stability of the discretization scheme. Spectral accuracy is retained with absolute error decaying in an exponential fashion. The numerical solutions for the driven cavity compare favorably against previously published numerical results except for a slight overprediction in the vertical velocity component at Reynolds number of 400. TheC ³ continuity is speculated to be its cause.     

Iterative methods for model reduction by domain decomposition

It is proposed a method to reduce the computational effort to solve a partial differential equation on a given domain. The main idea is to split the domain of interest in two subdomains, and to use different approximation methods in each of the two subdomains. In particular, in one subdomain we discretize the governing equations by a canonical scheme, whereas in the other one we solve a reduced order model of the original problem. Different approaches to couple the low-order model to the usual discretization are presented. The effectiveness of these approaches is tested on numerical examples pertinent to non-linear model problems including Laplace equation with non-linear boundary conditions and compressible Euler equations.     

Hierarchical matrix techniques for a domain decomposition algorithm

Summary In this paper, we investigate the effectiveness of hierarchical matrix techniques when used as the linear solver in a certain domain decomposition algorithm. In particular, we provide a direct performance comparison between an algebraic multigrid solver and a hierarchical matrix solver which is based on nested dissection clustering within the software package PLTMG.      

On a mixed and multiscale domain decomposition method

This paper presents a reexamination of a multiscale computational strategy with homogenization in space and time for the resolution of highly heterogeneous structural problems, focusing on its suitability for parallel computing. Spatially, this strategy can be viewed as a mixed, multilevel domain decomposition method (or, more accurately, as a “structure decomposition” method). Regarding time, a “parallel” property is also described. We also draw bridges between this and other current approaches.     

Parallel Monte Carlo simulations by asynchronous domain decomposition

A simple general method for performing Metropolis Monte Carlo condensed matter simulations on parallel processors is examined. The method is based on the cyclic generation of temporary discrete domains within the system, which are separated by distances greater than the inter-particle interaction range. Particle configurations within each domain are then sampled independently by an assigned processor, whilst particles outside these domains are held fixed. Results for a simulated Lennard-Jones fluid confirm that the method rigorously satisfies the detailed balance condition, and that the efficiency of configurational sampling scales almost linearly with the number of processors. Furthermore, the number of iterations performed on a given processor can be essentially arbitrary, with very low levels of inter-process communication. Provided the CPU time per step is not state-dependent, the method can then be used to perform large calculations as unsupervised background tasks on heterogeneous networks.     

A domain decomposition method for almost incompressible flow

A domain decomposition method for solving the Navier-Stokes equations for almost incompressible flow is examined. At the price of a nonuniform decomposition of the domain, we have fast solvers in all subdomains. Hence, each iteration on the Schur complement system can be performed very efficiently. We have shown theoretically that the method requires much fewer memory positions and arithmetic operations than a direct method. Numerical experiments show that the iteration on the Schur complement system converges very fast. We also show that the spatial grid ratio might be crucial for the performance of the method. Moreover, we show that for a given discretization of the problem, the rate of efficiency is larger than 100% for the problem studied here, due to the very nice parallelization properties of the algorithm.     

Spectral domain decomposition techniques for viscous incompressible flows

Spectral domain decomposition methods are described for solving the equations governing the flow of viscous incompressible fluids in rectangularly decomposable domains. The domain of interest is divided into a number of rectangular subdomains on each of which a spectral approximation of the flow variables is sought. For Newtonian flows a stream function formulation is used whereas for nonNewtonian flows the components of the extra-stress tensor are also used. Efficient direct methods for the solution of the algebraic systems are discussed. Numerical results are presented for laminar flow through a channel contraction and for the stick-slip problem.     

An efficient large deformation method using domain decomposition

Efficiently simulating large deformations of flexible objects is a challenging problem in computer graphics. In this paper, we present a physically based approach to this problem, using the linear elasticity model and a finite elements method. To handle large deformations in the linear elasticity model, we exploit the domain decomposition method, based on the observation that each sub-domain undergoes a relatively small local deformation, involving a global rigid transformation. In order to efficiently solve the deformation at each simulation time step, we pre-compute the object responses in terms of displacement accelerations to the forces acting on each node, yielding a force–displacement matrix. However, the force–displacement matrix could be too large to handle for densely tessellated objects. To address this problem, we present two methods. The first method exploits spatial coherence to compress the force-displacement matrix using the clustered principal component analysis method; and the second method pre-computes only the force–displacement vectors for the boundary vertices of the sub-domains and resorts to the Cholesky factorization to solve the acceleration for the internal vertices of the sub-domains. Finally, we present some experimental results to show the large deformation effects and fast performance on complex large scale objects under interactive user manipulations.