A Fully Nonlinear HOBEM with the Domain Decomposition Method for Simulation of Wave Propagation and Diffraction

A higher-order boundary element method(HOBEM) for simulating the fully nonlinear regular wave propagation and diffraction around a fixed vertical circular cylinder is investigated. The domain decomposition method with continuity conditions enforced on the interfaces between the adjacent sub-domains is implemented for reducing the computational cost. By adjusting the algorithm of iterative procedure on the interfaces, four types of coupling strategies are established, that is, Dirchlet/Dirchlet-Neumman/Neumman(D/D-N/N), Dirchlet-Neumman(D-N),Neumman-Dirchlet(N-D) and Mixed Dirchlet-Neumman/Neumman-Dirchlet(Mixed D-N/N-D). Numerical simulations indicate that the domain decomposition methods can provide accurate results compared with that of the single domain method. According to the comparisons of computational efficiency, the D/D-N/N coupling strategy is recommended for the wave propagation problem. As for the wave-body interaction problem, the Mixed D-N/N-D coupling strategy can obtain the highest computational efficiency.

Modified domain decomposition method for Hamilton-Jacobi-Bellman equations

This paper presents a modified domain decomposition method for the numerical solution of discrete Hamilton-Jacobi-Bellman equations arising from a class of optimal control problems using diffusion models. A convergence theorem is established. Numerical results indicate the effectiveness and accuracy of the method.

DOMAIN DECOMPOSITION METHODS FOR SOLVING PDE's ON MULTI-PROCESSORS

In this paper, we discuss the parallel domain decomposition method(DDM)for solving PDE's on parallel computers. Three types of DDM: DDM with overlapping, DDM without overlapping and DDM with fictitious component are discussed in a uniform framework. The eonvergence of the asynchronous parallel algorithms based on DDM are discussed.

Deep Domain Decomposition Methods:Helmholtz Equation

This paper proposes a deep-learning-based Robin-Robin domain decomposition method(DeepDDM)for Helmholtz equations.We first present the plane wave activation-based neural network(PWNN),which is more efficient for solving Helmholtz equations with constant coefficients and wavenumber k than finite difference methods(FDM).On this basis,we use PWNN to discretize the subproblems divided by domain decomposition methods(DDM),which is the main idea of DeepDDM.This paper will investigate the number of iterations of using DeepDDM for continuous and discontinuous Helmholtz equations.The results demonstrate that:DeepDDM exhibits behaviors consistent with conventional robust FDM-based domain decomposition method(FDM-DDM)under the same Robin parameters,i.e.,the number of iterations by DeepDDM is almost the same as that of FDM-DDM.By choosing suitable Robin parameters on different subdomains,the convergence rate is almost constant with the rise of wavenumber in both continuous and discontinuous cases.The performance of DeepDDM on Helmholtz equations may provide new insights for improving the PDE solver by deep learning.

A Robin-Robin Domain Decomposition Method for a Stokes-Darcy Structure Interaction with a Locally Modified Mesh

A new numerical method based on locally modified Cartesian meshes is proposed for solving a coupled system of a fluid flow and a porous media flow.The fluid flow is modeled by the Stokes equations while the porous media flow is modeled by Darcy’s law.The method is based on a Robin-Robin domain decomposition method with a Cartesian mesh with local modifications near the interface.Some computational examples are presented and discussed.

AN ITERATIVE PROCEDURE FOR DOMAIN DECOMPOSITION METHOD OF SECOND ORDER ELLIPTIC PROBLEM WITH MIXED BOUNDARY CONDITIONS

This paper is devoted to study of an iterative procedure for domain decomposition method of second order elliptic problem with mixed boundary conditions (i.e., Dirichlet condition on a part of boundary and Neumann condition on the another part of boundary). For the pure Dirichlet problem, Marini and Quarteroni [3], [4] considered a similar approach, which is extended to more complex problem in this paper.

Domain Decomposition Method for a System of Quasivariational Inequalities

We propose a domain decomposition method for a system of quasivariational inequalities related to the HJB equation. The monotone convergence of the algorithm is also established.

Domain Decomposition Method for the Forward-Backward Heat Equation

The forward-backward heat equation arises in a remarkable variety of physical applications. A non-overlaping domain decomposition method was constructed to obtain numerical solutions of the forward-backward heat equation. The primary advantage is that the method reduces the computation time tremendously. The convergence of the given method is established. The numerical performance shows that the domain decomposition method is effective.

NONOVERLAPPING DOMAIN DECOMPOSITION METHOD WITH MIXED ELEMENT FOR ELLIPTIC PROBLEMS

In this paper we consider the nonoverlapping domain decomposition method based on mixed element approximation for elliptic problems in two dimentional space. We give a kind of discrete domain decomposition iterative algorithm using mixed finite element, the subdomain problems of which can be implemented parallelly. We also give the existence, uniqueness and convergence of the approximate solution.

THE OPTIMAL PRECONDITIONING IN THE DOMAIN DECOMPOSITION METHOD FOR WILSON ELEMENT

This paper discusses the optimal preconditioning in the domain decomposition method for Wilson element. The process of the preconditioning is composed of the resolution of a small scale global problem based on a coarser grid and a number of independent local subproblems, which can be chosen arbitrarily. The condition number of the preconditioned system is estimated by some characteristic numbers related to global and local subproblems. With a proper selection, the optimal preconditioner can be obtained, while the condition number is independent of the scale of the problem and the number of subproblems.

A COMBINED TECHNIQUE FOR SOLUTION OF PDE＇s VIA THE GENERALIZED DOMAIN DECOMPOSITION METHOD

The Domain Decomposition Method(DDM) is a powerful approach to solving maily types of PDE's. DDM is especially suitable for massively Parallel computers. In the past, most research on DDM has focused on the domain splitting technique. In this paper. we focus our attention on use of a combination of techniques to solve each subproblem. The central question with DDM is that of how to doal with the pseodoboundary conditions. Here, we introduce a set of operators which act on the pseudo-boundaries in the solution process, referring to this new. procedure as the 'Generalized Domain Decomposition A.Jlethod(GDDM).' We have already obtained convergence factors for GDDM with certain classes of PDE's. These ctonvergence factors show that we can derive exact solutions of the whole problem for certain types of PDE's, and can get superior speed of convergence for other types.

THE ANALYSIS OF THE SEPARATED LAYERS ALGORITHM BY DOMAIN DECOMPOSITION METHOD

From the principle of of the Domain Decomposition Method (DDM), we analyse the 2nd-order linear elliptic partial differential problems and link the Separated-Layers Algorithm (SLA) with DDM. The mathematical properties of SLA and numerical example are presented to obtain satisfactory computation results. For general linear differential ones, also are the structure of SLA and its characteristics discussed.

TRACE AVERAGING DOMAIN DECOMPOSITION METHOD WITH NONCONFORMING FINITE ELEMENTS

We consider, in this paper, the trace averaging domain decomposition method for the second order self-adjoint elliptic problems discretized by a class of nonconforming finite elements, which is only continuous at the nodes of the quasi-uniform mesh. We show its geometric convergence and present the dependence of the convergence factor on the relaxation factor, the subdomain diameter H and the mesh parameter h. In essence;, this method is equivalent to the simple iterative method for the preconditioned capacitance equation. The preconditioner implied in this iteration is easily invertible and can be applied to preconditioning the capacitance matrix with the condition number no more than O((1 + In H/h)max(1 + H-2, 1 + In H/h)).

The Monotone Robin-Robin Domain Decomposition Methods for the Elliptic Problems with Stefan-Boltzmann Conditions

This paper is concerned with the elliptic problems with nonlinear StefanBoltzmann boundary condition.By combining with the monotone method,the RobinRobin domain decomposition methods are proposed to decouple the nonlinear interface and boundary condition.The monotone properties are verified for both the multiplicative and the additive domain decomposition methods.The numerical results confirm the theoretical analysis.

A DISTRIBUTED LATTICE BOLTZMANN METHOD

In this paper, an overlapping lattice Boltzmann model is introduced and its domain decomposition method, a distributed lattice Boltzmann method is presented. Parallel effectiveness of some programs based on the distributed lattice Boltzmann method are analyzed.

A PARALLEL ITERATIVE DOMAIN DECOMPOSITION ALGORITHM FOR ELLIPTIC PROBLEMS

An iterative nonoverlapping domain decomposition procedure is proposed and analyzed for linear elliptic problems. At the interface of two subdomains, one subdomain problem requires that Dirichlet data be passed to it from the previous iteration level, while the other subdomain problem requires that Neumann data be passed to it. This procedure is suitable for parallel processing. A convergence analysis is established. Standard and mixed finite element methods are employed to give discrete versions of this domain decomposition algorithm. Numerical experiments arc conducted to show the effectiveness of the method.

Multi-domain Spectral Immersed Interface Method for Solving Elliptic Equation with a Global Description of Discontinuous Functions

This paper presents the extension of the global description approach of a discontinuous function, which is proposed in the previous paper, to a spectral domain decomposition method. This multi-domain spectral immersed interlace method（IIM） divides the whole computation domain into the smooth and discontinuous parts. Fewer points on the smooth domains are used via taking advantage of the high accuracy property of the spectral method, but more points on the discontinuous domains are employed to enhance the resolution of the calculation. Two that the domain decomposition technique can placed around the discontinuity. The present reached, in spite of the enlarged computational discontinuous problems are tested to verify the present method. The results show reduce the error of the spectral IIM, especially when more collocation points are method is t：avorable for the reason that the same level of the accuracy can be domain.

CONVERGENCE RESULTS FOR NON-OVERLAP SCHWARZ WAVEFORM RELAXATION ALGORITHM WITH CHANGING TRANSMISSION CONDITIONS

In this paper,we establish a new algorithm to the non-overlapping Schwarz domain decomposition methods with changing transmission conditions for solving one dimensional advection reaction diffusion problem.More precisely,we first describe the new algorithm and prove the convergence results under several natural assumptions on the sequences of parameters which determine the transmission conditions.Then we give a simple method to estimate the new value of parameters in each iteration.The interesting advantage of our method is that one may update the better parameters in each iteration to save the computational cost for optimizing the parameters after many steps.Finally some numerical experiments are performed to show the behavior of the convergence rate for the new method.

High-performance computing of 3D blasting wave propagation in underground rock cavern by using 4D-LSM on TianHe-3 prototype E class supercomputer

Parallel computing assigns the computing model to different processors on different devices and implements it simultaneously.Accordingly,it has broad applications in the numerical simulation of geotechnical engineering and underground engineering,of which models are always large-scale.With parallel computing,the computing time or the memory requirements will be reduced by splitting the original domain of the numerical model into many subdomains,which is thus named as the domain decomposition method.In this study,a cubic and equal volume domain decomposition strategy was utilized to realize the parallel computing on the distributed memory system of four-dimensional lattice spring model(4D-LSM)based on the message passing interface.With a more efficient communication strategy introduced,this study aimed at operating an one-billion-particle model on a supercomputer platform.The preprocessing procedure of the parallelized 4D-LSM was restructured and the particle generation strategy suitable for the supercomputer platform was employed to minimize the time consumption in preprocessing and calculation.On this basis,numerical calculations were performed on TianHe-3 prototype E class supercomputer at the National Supercomputer Center in Tianjin.Two fieldscale three-dimensional blasting wave propagation models were carried out,of which the numerical results verify the computing power and the advantage of the parallelized 4D-LSM in the simulation of large-scale three-dimension models.Subsequently,the time complexity and spatial complexity of 4D-LSM and other particle discrete element methods were analyzed.

A NUMERICAL STUDY OF THE VORTEX MOTION IN OSCILLATING FLOW AROUND A CIRCULAR CYLINDER AT LOW AND MIDDLE Kc NUMBERS

A new hybrid model, which is based on domain decomposition and proposed by the authors is used for calculating the flow around a circular cylinder at low and middle Keulegan-Carpenter numbers (Kc=2～18)respectively.The vortex motion patterns in asymmetric regime,single pair(or transverse)regime and double pair(or diagonal)regime are successfully simulated.The calculated drag and inertial force coefficients are in better agreement with experimental data than other recent computational results.