Neighboring Subdomains Research Articles

Decreasing the memory of the discontinuous Galerkin volume integral equation method for scattering from inhomogeneous dielectric objects.

A memory-efficient implementation scheme for the discontinuous Galerkin volume integral equation method (DGVIE) using Schaubert-Wilton-Glisson (SWG) basis functions is proposed to analyze electromagnetic scattering from inhomogeneous dielectric objects. For this proposed scheme, almost no half-SWG basis functions are needed for the elements separating nonconformal meshes, while these half-SWG basis functions are indispensable for the conventional DGVIE-SWG method. This is realized by applying the divergence-free condition of the electric displacement vector explicitly for nonconformal meshes separating neighboring subdomains of an inhomogeneous dielectric body. Therefore, the number of unknowns of the conventional DGVIE method can be further reduced. As a result, the memory of the proposed DGVIE method is only about half of the conventional one for inhomogeneous dielectric problems. Meanwhile, the total solution time has been reduced by the use of the proposed scheme. Particularly, the proposed DGVIE-SWG method is efficient in memory usage not only for inhomogeneous dielectric cases with high contrast ratio but also for cases with relatively low contrast ratio.

A hybrid implicit-explicit discontinuous Galerkin spectral element time domain (DG-SETD) method for computational elastodynamics

SUMMARY A hybrid implicit-explicit (IMEX) discontinuous Galerkin spectral element time domain (DG-SETD) algorithm is proposed to simulate 3D elastic wave propagation in inhomogeneous media. In this method, the original problem can be divided into a number of well designed subdomains, and the mesh generation of different subdomains is completely independent, thus allowing flexible spatial discretization of complex geometry. The neighboring subdomains are connected by a Riemann transmission condition (RTC), and spectral elements with different interpolation orders can be used in different subdomains to maximize the computational efficiency of multiscale problems to facilitate parallel computing for different subdomains. In particular, the explicit or implicit time iteration scheme can be appropriately selected for a subdomain according to the size of its mesh elements to increase the global time step increment, thus giving higher computational efficiency: For subdomains with coarse meshes, the explicit time integration scheme is used and the time step increment is limited by the Courant−Friedrichs−Lewy (CFL) stability condition; for subdomain with fine structures and thus fine meshes, an implicit time integration scheme is used so that a large time step increment can be used without affecting the stability. In addition, a jump condition of displacement and velocity is introduced to accurately simulating fractures and faults, including lossless and viscous fractures with plane, curve or cross structures. This avoids the volume modeling of the extremely thin fracture structures, and effectively reduces the number of degrees of freedoms (DoFs) in the discretized system without the loss of accuracy. The accuracy, robustness and efficiency of the DG-SETD algorithm are demonstrated by quantitative comparisons of the waveforms with the commercial finite element software COMSOL.

Trace transfer-based diagonal sweeping domain decomposition method for the Helmholtz equation: Algorithms and convergence analysis

By utilizing the perfectly matched layer (PML) and source transfer techniques, the diagonal sweeping domain decomposition method (DDM) was recently developed for solving the high-frequency Helmholtz equation in Rn, which uses 2n sweeps along respective diagonal directions with checkerboard domain decomposition. Although this diagonal sweeping DDM is essentially multiplicative, it is highly suitable for parallel computing of the Helmholtz problem with multiple right-hand sides when combined with the pipeline processing since the number of sequential steps in each sweep is much smaller than the number of subdomains. In this paper, we propose and analyze a trace transfer-based diagonal sweeping DDM. A major advantage of changing from source transfer to trace transfer for information passing between neighbor subdomains is that the resulting diagonal sweeps become easier to analyze and implement and more efficient, since the transferred traces have only 2n cardinal directions between neighbor subdomains while the transferred sources come from a total of 3n−1 cardinal and corner directions. We rigorously prove that the proposed diagonal sweeping DDM not only gives the exact solution of the global PML problem in the constant medium case but also does it with at most one extra round of diagonal sweeps in the two-layered media case, which lays down the theoretical foundation of the method. Performance and parallel scalability of the proposed DDM as direct solver or preconditioner are also numerically demonstrated through extensive experiments in two and three dimensions.

Domain decomposition based on the spectral element method for frequency-domain computational elastodynamics

We propose a domain decomposition method based on the spectral element method (DDM-SEM) for elastic wave computation in frequency domain. It combines the high accuracy of the spectral element method and the high degree of parallelism of a domain decomposition technique, which makes this method suitable for accurate and efficient simulations of large scale problems in elastodynamics. In the DDM-SEM, the original large-scale problem is divided into a number of well designed subdomains. We use the spectral element method independently for each subdomain, and the neighboring subdomains are connected by a frequency-domain version of Riemann transmission condition (RTC) for elastic waves. For the proposed method, we can employ the non-conforming meshes and different interpolation orders in different subdomains to maximize the efficiency. By separating the internal and boundary unknowns of each subdomain, an efficient and naturally parallelizable block LDU direct solver is developed to solve the final system matrix. Numerical experiments verify its accuracy and efficiency, and show that the proposed DDM-SEM can be a promising numerical tool for accurately and effectively solving large and multi-scale problems of elastic waves. It is potentially valuable for the frequency domain seismic inversion where multiple source illuminations are required.

Multiscale mixed methods for two-phase flows in high-contrast porous media

The Multiscale Robin Coupled Method (MRCM) is a domain decomposition procedure that has been developed to efficiently approximate velocity and pressure fields for single-phase flows in highly heterogeneous porous media. It generalizes other well-established multiscale domain decomposition mixed methods and it adds great flexibility to the choice of interface spaces as well as in the boundary conditions for subdomain coupling. We investigate the approximation of two phase flows in porous media using the MRCM to compute velocity fields. We consider an operator splitting strategy, where a scalar conservation law for the saturation of one of the phases and the velocity field are updated sequentially. We find that the coupling of nearest neighbor subdomains through the imposition of a continuous pressure (respectively, normal fluxes) is the best strategy to approximate two-phase flows in the presence of high (resp., low) permeability channels (resp., regions). A new adaptivity strategy for setting an algorithmic parameter of the MRCM, that controls the relative importance of Dirichlet and Neumann boundary conditions in the coupling of subdomains, is proposed and tested in challenging, high-contrast permeability fields. Our numerical simulations of two-phase flows show that by switching between existing multiscale procedures we can observe unprecedented accuracy, in that we produce better solutions for problems with high-contrast permeability coefficients when compared to solutions obtained with some standard multiscale mixed methods.

Near-Field to Far-Field Transformation on Arbitrary Surfaces via Multi-Level Spherical Wave Expansion

This article presents a general technique for the far-field transformation of near fields measured over arbitrary surfaces and multimode probe correction in an efficient and accurate way. The use of not only one but several spherical wave expansions to model the antenna under test fields allows to incorporate full probe correction for arbitrary orientations with low computation cost. Combining this approach with a hierarchical subdomain decomposition strategy, the transformation problem is solved with a low computational complexity. In this method, the fields produced by neighbor subdomains are gradually aggregated and interpolated following a multilevel scheme, leading to a good scalability with frequency compared with other matrix-based transformation methods. The numerical and experimental results are provided to show the efficiency and capabilities of the proposed algorithm.

An Additive Overlapping Domain Decomposition Method for the Helmholtz Equation

In this paper, we propose and analyze an additive domain decomposition method (DDM) for solving the high-frequency Helmholtz equation with the Sommerfeld radiation condition. In the proposed method, the computational domain is partitioned into structured subdomains along all spatial directions, and each subdomain contains an overlapping region for source transferring. At each iteration all subdomain PML problems are solved completely in parallel, then all horizontal, vertical and corner directional residuals on each subdomain are passed to its corresponding neighbor subdomains as the source for the next iteration. This DDM method is highly scalable in nature and theoretically shown to produce the exact solution for the PML problem defined in ${\mathbb{R}}^2$ in the constant medium case. A slightly modified version of the method for bounded truncated domains is also developed for its use in practice and an error estimate is rigorously proved. Various numerical experiments in two and three dimensions are conducted on the supercomputer "Tianhe-2 Cluster" to verify the theoretical results and demonstrate excellent performance of the proposed method as an iterative solver or a preconditioner.

A Local Parallel Communication Algorithm for Polydisperse Rigid Body Dynamics

Abstract The simulation of large ensembles of particles is usually parallelized by partitioning the domain spatially and using message passing to communicate between the processes handling neighboring subdomains. The particles are represented as individual geometric objects and are assigned to the subdomains. Handling collisions and migrating particles between subdomains, as required for proper parallel execution, requires a complex communication protocol. Typically, the parallelization is restricted to handling only particles that are smaller than a subdomain. In many applications, however, particle sizes may vary drastically with some of them being larger than a subdomain. In this article we propose a new communication and synchronization algorithm that can handle the parallelization without size restrictions on the particles. Despite the additional complexity and extended functionality, the new algorithm introduces only minimal overhead. We demonstrate the scalability of the previous and the new communication algorithms up to almost two million parallel processes and for handling ten billion (1010) geometrically resolved particles on a state-of-the-art petascale supercomputer. Different scenarios are presented to analyze the performance of the new algorithm and to demonstrate its capability to simulate polydisperse scenarios, where large individual particles can extend across several subdomains.

Quasi-disjoint pentadiagonal matrix systems for the parallelization of compact finite-difference schemes and filters

This paper proposes a novel systematic approach for the parallelization of pentadiagonal compact finite-difference schemes and filters based on domain decomposition. The proposed approach allows a pentadiagonal banded matrix system to be split into quasi-disjoint subsystems by using a linear–algebraic transformation technique. As a result the inversion of pentadiagonal matrices can be implemented within each subdomain in an independent manner subject to a conventional halo-exchange process. The proposed matrix transformation leads to new subdomain boundary (SB) compact schemes and filters that require three halo terms to exchange with neighboring subdomains. The internode communication overhead in the present approach is equivalent to that of standard explicit schemes and filters based on seven-point discretization stencils. The new SB compact schemes and filters demand additional arithmetic operations compared to the original serial ones. However, it is shown that the additional cost becomes sufficiently low by choosing optimal sizes of their discretization stencils. Compared to earlier published results, the proposed SB compact schemes and filters successfully reduce parallelization artifacts arising from subdomain boundaries to a level sufficiently negligible for sophisticated aeroacoustic simulations without degrading parallel efficiency. The overall performance and parallel efficiency of the proposed approach are demonstrated by stringent benchmark tests.

Filters in topology optimization based on Helmholtz‐type differential equations

AbstractThe aim of this paper is to apply a Helmholtz‐type partial differential equation as an alternative to standard density filtering in topology optimization problems. Previously, this approach has been successfully applied as a sensitivity filter. The usual filtering techniques in topology optimization require information about the neighbor cells, which is difficult to obtain for fine meshes or complex domains and geometries. The complexity of the problem increases further in parallel computing, when the design domain is decomposed into multiple non‐overlapping partitions. Obtaining information from the neighbor subdomains is an expensive operation. The proposed filter technique requires only mesh information necessary for the finite element discretization of the problem. The main idea is to define the filtered variable implicitly as a solution of a Helmholtz‐type differential equation with homogeneous Neumann boundary conditions. The properties of the filter are demonstrated for various 2D and 3D topology optimization problems in linear elasticity, solved on serial and parallel computers. Copyright © 2010 John Wiley & Sons, Ltd.

영역/경계 분할법을 이용한 저복 문제의 효율적인 유한요소 해석

접촉 문제의 효율적인 유한요소 해석을 위하여 부영역, 공유면 및 접촉 공유면의 개념에 근거한 영역/경계 분할법을 제시하였다. 부영역과 공유면 또는 접촉 공유면을 결합하기 위한 등식 적합 조건을 벌칙 함수로 처리함으로써 모든 유효 강성 행렬이 양 정치화되므로, 역행렬과 같은 각종 행렬의 연산이 매우 간편해진다. 또한 전체 영역 형상이 복잡하더라도, 임의의 부영역으로 분할한 후 공유면에서의 절점 연속성을 고려하지 않고 각각의 부영역을 독립적으로 이산화할 수 있다. 간단한 수치 예제 해석을 통하여 본 기법의 기본적인 특성을 고찰하였다. new domain/boundary decomposition method is suggested to perform efficient finite element analyses of contact problems. A penalty method is used for connecting an interface or contact interfaces with neighboring subdomains that satisfy continuity conditions. As a result, the derived effective stiffness matrices are always positive definite, and computational efficiency can be improved to a considerable degree. Moreover, any complex-shaped domain can be divided into independently modeled subdomains without considering the conformity of meshes along the interface. Using a computer code based on the present method, these advantageous features are confirmed through a set of numerical examples.

A new parallel Gauss–Seidel method based on alternating group explicit method and domain decomposition method

A new parallel Gauss–Seidel method is presented for solution of system of linear equations related to finite difference discretization of partial differential equations. This method is based on domain decomposition method and local coupling between interfaces of neighbor sub-domains, same as alternating group explicit method. This method is convergent and number of iterations for achieving convergence criteria is near the original Gauss–Seidel method (sometimes better and sometimes worse but difference is very small). The convergence theory is discussed in details. Numerical results are given to justify the convergence and performance of the proposed iterative method.

An unconditionally stable parallel difference scheme for parabolic equations

An unconditionally stable difference scheme with intrinsic parallelism is constructed in this paper, taking the parabolic equation u t = u xx as an example. The main ideas are described as follows. We decompose the domain Ω into some overlapping subdomains, take values of the last time layer as values of the time layer on inner boundary points of subdomains, solve it with the fully implicit scheme inside each subdomain, and then take correspondent values of its neighbor subdomains as their values for inner boundary points of each subdomain and mean of its neighbor subdomain and itself at overlapping points. Its unconditional stability and convergence are proved. Though its truncation error is O(1), we will still prove that the estimate order is improved to O( τ+ h).

Parallel watershed transformation algorithms for image segmentation

The watershed transformation is a mid-level operation used in morphological image segmentation. Techniques applied on large images, which must often complete fast, are usually computationally expensive and complex entailing efficient parallel algorithms. Two distributed approaches of the watershed transformation are introduced in this paper. The algorithms survey in a Single Program Multiple Data (SPMD) model both local and global connectivity properties of the morphological gradient of a gray-scale image to label connected components. The sequentiality of the serial algorithm is broken in the parallel versions by exploiting the ordering relation between two neighboring pixels successively incorporated in the same region. Thus, a path is traced, for every unlabeled pixel, down to its region of inclusion (whose label is then propagated backwards); in the second algorithm, regions grow independently around their seeds. In both cases only pixels which satisfy the ordering relation are incorporated in any region. This way, not only different regions are explored in a parallel fashion, but also different parts of the same region, when the latter extends to neighboring subdomains, are treated likewise. Running time and relative speedup evaluated on a Cray T3D parallel computer are used to appreciate the performance of both algorithms.