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A domain decomposition method for large-scale elastic–plastic problems is proposed. The proposed method is based on a quasi-Newton method in conjunction with a balancing domain decomposition preconditioner. The use of a quasi-Newton method overcomes two problems associated with the conventional domain decomposition method based on the Newton–Raphson method: (1) avoidance of a double-loop iteration algorithm, which generally has large computational complexity, and (2) consideration of the local concentration of nonlinear deformation, which is observed in elastic–plastic problems with stress concentration. Moreover, the application of a balancing domain decomposition preconditioner ensures scalability. Using the conventional and proposed domain decomposition methods, several numerical tests, including weak scaling tests, were performed. The convergence performance of the proposed method is comparable to that of the conventional method. In particular, in elastic–plastic analysis, the proposed method exhibits better convergence performance than the conventional method., source:https://doi.org/10.1007/s00466-018-1579-4

This article is focused on the study of a micro-macro LaTIn based Domain Decomposition Method (LaTIn-DDM) for the prediction of the nonlinear behavior of slender composite structures subjected to bending, buckling and delamination. Previous studies have shown that an adequate selection of the iterative parameters (search directions and macroscopic space) allow to improve the convergence rate and ensure scalability (i.e. number of iterations is independent of the number of subdomains) of the iterative schema. To obtain precise solutions, only the size reduction of the subdomains' discretization has been addressed (h-refinement), disregarding the option of increasing the polynomial degree of the finite elements (p-refinement) and ignoring their underlying effects on the information's transmission through the interfaces between subdomains. In this work and using linear and quadratic finite elements, h and p refinements on the subdomains and local h-refinement only along the edges of the subdomains were investigated. It is conclude that the prefinement in the whole subdomain not only enables to reach more exact solutions than using global or local h-refinement, but also the convergence rate is improved. These enhancements allow more complex simulations but using less degrees of freedom and less calculation time, even up to 97% faster., Peer Reviewed

The need for increasing fidelity and accuracy in mission-critical electromagnetic applications have pushed the problem sizes toward extreme computational scales. Therefore, there is a large premium placed on the investigation of parallel and scalable integral equation simulators, a particularly useful class of computational electromagnetics, to meet this demand. This dissertation will focus on three interrelated areas: i) quasi-optimal, well-conditioned integral equation based domain decomposition methods, ii) geometry-adaptive, multi-scale discontinuous Galerkin boundary element methods, and iii) high-performance and scalable algorithms to reduce the computational complexity of extreme-scale simulations with the aid of parallel computing architectures. This dissertation will first develop the mathematical underpinnings of a geometry-adaptive, integral equation domain decomposition method. Then the parallelization technique will be discussed along with many numerical experiments to profile its performance. Finally, two real-world applications will be discussed that demonstrate the utility of a high performance-enabled solution. First, a solution to the challenging problem of the solution and prototyping of antennas on large platforms will be discussed. Finally, the code will be used to solve a channel modeling problem that has many important applications for the fifth generation (5G) of wireless telecommunication.

Domain decomposition and convex optimization play fundamental roles in current computation and analysis in many areas of science and engineering. These methods have been well developed and studied in the past thirty years, but they still require further study and improving not only in mathematics but in actual engineering computation with exponential increase of computational complexity and scale. The main goal of this thesis is to develop some efficient and powerful algorithms based on domain decomposition method and convex optimization. The topicsstudied in this thesis mainly include two classes of convex optimization problems: optimal control problems governed by time-dependent partial differential equations and general structured convex optimization problems. These problems have acquired a wide range of applications in engineering and also demand a very high computational complexity. The main contributions are as follows: In Chapter 2, the relevance of an adequate inner loop starting point (as opposed to a sufficient inner loop stopping rule) is discussed in the context of a numerical optimization algorithm consisting of nested primal-dual proximal-gradient iterations. To study the optimal control problem, we obtain second order domain decomposition methods by combining Crank-Nicolson scheme with implicit Galerkin method in the sub-domains and explicit flux approximation along inner boundaries in Chapter 3. Parallelism can be easily achieved for these explicit/implicit methods. Time step constraints are proved to be less severe than that of fully explicit Galerkin finite element method. Based on the domain decomposition method in Chapter 3, we propose an iterative algorithm to solve an optimal control problem associated with the corresponding partial differential equation with pointwise constraint for the control variable in Chapter 4. In Chapter 5, overlapping domain decomposition methods are designed for the wave equation on account of prediction-correction" s, Doctorat en Sciences, info:eu-repo/semantics/nonPublished

伝搬方向への構造変化が緩やかで反射波を無視できる光導波路の解析にはビーム伝搬法(BPM) が有効であり，様々なビーム伝搬法が開発されてきた．その中で，横方向の界の離散化に有限要素法(FEM) を用いる有限要素ビーム伝搬法(FE-BPM) は任意形状導波路への適用性に優れているが，伝搬方向に有限要素メッシュを更新する必要があることに加えて一般に連立一次方程式を解く必要があるため，伝搬方向に構造が連続的に変化する導波路に対して必ずしも計算が効率的とは言えない．本研究では，横方向の離散化に用いるFEM 解析を領域分割法を用いて並列化することでFE-BPM 解析を高速化するための検討を行なっている．領域分割法では，境界成分を厳密に求める方法と，領域同士を重ねることにより境界成分の計算を省略する方法の2 通りについて検討している．, Beam Propagation Method (BPM) is effective for the analysis of the waveguide which is longitudinally varying and could ignore the reflection wave. Therefore various BPM had developed now. Finite Element Beam Propagation Method (FE-BPM) which use Finite Element Method (FEM) has applicability for arbitrary shape of waveguides. However, FE-BPM is not enough efficient for the waveguides which is longitudinally varying, because Finite Element mesh need to be updated and generally, this method need to solve the primary simultaneous equa-tions. In this study, FEM analysis which is used for discretization waveguides to transverse direction is parallelized by domain decomposition method to speed up FE-BPM analysis. In domain decomposition method, two approaches are concerned, one way is strict calculating the boundary fields, another is omitting the strict calculation boundary fields by iterating between domains.

In this paper the first excursion problem of linear structures subjected to stochastic wind loads is considered. The geometry of the corresponding failure domain is further explored, with the finding that rotational relationship exists for the quadratic elementary failure domains comprising the overall failure domain. Besides, procedures using the gradient projection method are developed for obtaining the design point of the quadratic elementary failure domain, which can be used to modify the previously developed simulation method. An illustrative example shows the accuracy and the enhanced efficiency of the modified simulation method, compared with its original version. © 2016 Elsevier Ltd. All rights reserved.

In this paper the first excursion problem of linear structures subjected to stochastic wind loads is considered. The geometry of the corresponding failure domain is further explored, with the finding that rotational relationship exists for the quadratic elementary failure domains comprising the overall failure domain. Besides, procedures using the gradient projection method are developed for obtaining the design point of the quadratic elementary failure domain, which can be used to modify the previously developed simulation method. An illustrative example shows the accuracy and the enhanced efficiency of the modified simulation method, compared with its original version.

The Time Domain Boundary Element Method (TDBEM) has its advantages in the analysis of transient electromagnetic fields (wake fields) induced by a charged particle beam with curved trajectory in a particle accelerator. On the other hand, the TDBEM has disadvantages of huge required memory and computation time compared with those of the Finite Difference Time Domain (FDTD) method or the Finite Integration Technique (FIT). This paper presents 4-D domain decomposition method for the TDBEM based on an initial value problem formulation and application to a full model simulation of the bunch compressor section of the high-energy particle accelerators, in which the charged particle beam takes curved trajectory and the trajectory is sensitively affected by the wake fields.

This paper presents an all-direct domain decomposition approach for large-scale structural analysis. The proposed approach achieves computational robustness and efficiency by enforcing the compatibility of the displacement field across the subdomain boundaries via local Lagrange multipliers and augmented Lagrangian formulation (ALF). The proposed domain decomposition approach was compared to the existing FETI approach in terms of the computational time and memory usage. The parallel implementation of the proposed algorithm was described in detail. Finally, a preliminary validation was attempted for the proposed approach, and the numerical results of two- and three-dimensional problems were compared to those obtained through a dual-primal FETI approach. The results indicate an improvement in the performance as a result of the implementing the proposed approach.

This paper presents an all-direct domain decomposition approach for large-scale structural analysis. The proposed approach achieves computational robustness and efficiency by enforcing the compatibility of the displacement field across the subdomain boundaries via local Lagrange multipliers and augmented Lagrangian formulation (ALF). The proposed domain decomposition approach was compared to the existing FETI approach in terms of the computational time and memory usage. The parallel implementation of the proposed algorithm was described in detail. Finally, a preliminary validation was attempted for the proposed approach, and the numerical results of two- and three-dimensional problems were compared to those obtained through a dual-primal FETI approach. The results indicate an improvement in the performance as a result of the implementing the proposed approach.

This paper presents an all-direct domain decomposition approach for large-scale structural analysis. The proposed approach achieves computational robustness and efficiency by enforcing the compatibility of the displacement field across the subdomain boundaries via local Lagrange multipliers and augmented Lagrangian formulation (ALF). The proposed domain decomposition approach was compared to the existing FETI approach in terms of the computational time and memory usage. The parallel implementation of the proposed algorithm was described in detail. Finally, a preliminary validation was attempted for the proposed approach, and the numerical results of two- and three-dimensional problems were compared to those obtained through a dual-primal FETI approach. The results indicate an improvement in the performance as a result of the implementing the proposed approach.

Domain composition methods (DCM) consist of obtaining a solution to a problem on a domain, from the formulations of the same problem expressed on various subdomains. These methods have therefore the oppostive objective to domain decomposition methods (DDM). However, they are sometimes based on the same methodology as after decomposing, DDM have to recompose. As a consequence, in the literature, the term DDM has many times been substituted for DCM. In contrast to DCM, DDM are usually applied to matching meshes as their purpose consists mainly of distributing the work in parallel environments. DCM are powerful techniques that can be used for different purposes: to simplify the meshing of a complex geometry by dividing it into different meshable pieces; to perform local refinement to adapt to local mesh requirements; to treat subdomains in relative motion (Chimera, sliding mesh), etc. The term DCM is generic and does not give any clue about how the fragmented solutions on the different subdomains are composed into a global one. In the literature, many methodologies have been proposed: they are mesh-based, formulation-based, or algebraic-based. In mesh-based formulations, the coupling is achieved at the mesh level, before the governing equations are assembled into an algebraic system (mesh conforming, shear-slip mesh update, HERMESH). The formulation-based counterpart recomposes the solution from the strong or weak formulation itself, and are implemented during the assembly of the algebraic system on the subdomain meshes. The different coupling techniques can be formulated at the strong and continous level, at the weak and continous or discrete levels (iteration-by-subdomains, mortar element, mesh free interpolation). Although the different methods usually lead to the same solutions at the continuous level, which usually coincides with the solution of the problem on the original domain, they have very different behaviour at the discrete level and accept very different i, Peer Reviewed, Postprint (author's final draft)

In this paper the problem of reliability-based optimal design of linear structures subjected to stochastic excitations is considered. A global optimization method based on Transitional Markov chain Monte Carlo (TMCMC) is used to address the problem, where the optimization problem is converted into the task of generating sample points (designs) according to a probability density function (PDF) suitably constructed on the feasible space of designs satisfying all the constraints. TMCMC is used for generating sample points, in order to get higher convergence rate of the stationary distribution of the Markov chain states to the constructed PDF. The generation of sample points uniformly distributed in the feasible space, which is required at the initial stage of TMCMC, is achieved by using Subset Simulation. To apply Subset Simulation and TMCMC in the concerned reliability-based optimization problem, Domain Decomposition Method (DDM) is used to examine the reliability constraint, that is, whether the failure probability at a given design exceeds a specified threshold. Based on the statistical properties of the failure probability estimator given by DDM, a 'minimum' computational effort, in terms of providing a reliable judgment on the reliability constraint, is defined so that a further reduction in the computational cost can be achieved in the proposed reliability-based optimization (RBO) algorithm. Illustrative examples are presented to show the application and the advantages of the proposed global RBO algorithm. (C) 2014 Elsevier Ltd. All rights reserved.

In this paper the problem of reliability-based optimal design of linear structures subjected to stochastic excitations is considered. A global optimization method based on Transitional Markov chain Monte Carlo (TMCMC) is used to address the problem, where the optimization problem is converted into the task of generating sample points (designs) according to a probability density function (PDF) suitably constructed on the feasible space of designs satisfying all the constraints. TMCMC is used for generating sample points, in order to get higher convergence rate of the stationary distribution of the Markov chain states to the constructed PDF. The generation of sample points uniformly distributed in the feasible space, which is required at the initial stage of TMCMC, is achieved by using Subset Simulation. To apply Subset Simulation and TMCMC in the concerned reliability-based optimization problem, Domain Decomposition Method (DDM) is used to examine the reliability constraint, that is, whether the failure probability at a given design exceeds a specified threshold. Based on the statistical properties of the failure probability estimator given by DDM, a 'minimum' computational effort, in terms of providing a reliable judgment on the reliability constraint, is defined so that a further reduction in the computational cost can be achieved in the proposed reliability-based optimization (RBO) algorithm. Illustrative examples are presented to show the application and the advantages of the proposed global RBO algorithm. (C) 2014 Elsevier Ltd. All rights reserved.

In this paper the problem of reliability-based optimal design of linear structures subjected to stochastic excitations is considered. A global optimization method based on Transitional Markov chain Monte Carlo (TMCMC) is used to address the problem, where the optimization problem is converted into the task of generating sample points (designs) according to a probability density function (PDF) suitably constructed on the feasible space of designs satisfying all the constraints. TMCMC is used for generating sample points, in order to get higher convergence rate of the stationary distribution of the Markov chain states to the constructed PDF. The generation of sample points uniformly distributed in the feasible space, which is required at the initial stage of TMCMC, is achieved by using Subset Simulation. To apply Subset Simulation and TMCMC in the concerned reliability-based optimization problem, Domain Decomposition Method (DDM) is used to examine the reliability constraint, that is, whether the failure probability at a given design exceeds a specified threshold. Based on the statistical properties of the failure probability estimator given by DDM, a 'minimum' computational effort, in terms of providing a reliable judgment on the reliability constraint, is defined so that a further reduction in the computational cost can be achieved in the proposed reliability-based optimization (RBO) algorithm. Illustrative examples are presented to show the application and the advantages of the proposed global RBO algorithm. (C) 2014 Elsevier Ltd. All rights reserved.

Isogeometric collocation methods are very recent and promising numerical schemes that preserve the advantages of isogeometric analysis but often exhibit better performances than their Galerkin counterparts. In the present paper, an additive overlapping Schwarz method for isogeometric collocation discretizations is introduced and studied. The resulting preconditioner, accelerated by GMRES, is shown to be scalable with respect to the number of subdomains and very robust with respect to the isogeometric discretization parameters such as the mesh size and polynomial degree, as well as with respect to the presence of discontinuous elliptic coefficients and domain deformations

This work presents an analysis methodology based on the use of the Finite Element Method (FEM) nowadays considered one of the main numerical tools for solving Boundary Value Problems (BVPs). The proposed methodology, so-called cg-FEM (Cartesian grid FEM), has been implemented for fast and accurate numerical analysis of 2D linear elasticity problems. The traditional FEM uses geometry-conforming meshes; however, in cg-FEM the analysis mesh is not conformal to the geometry. This allows for defining very efficient mesh generation techniques and using a robust integration procedure, to accurately integrate the domain's geometry. The hierarchical data structure used in cg-FEM together with the Cartesian meshes allow for trivial data sharing between similar entities. The cg-FEM methodology uses advanced recovery techniques to obtain an improved solution of the displacement and stress fields (for which a discretization error estimator in energy norm is available) that will be the output of the analysis. All this results in a substantial increase in accuracy and computational efficiency with respect to the standard FEM. cg-FEM has been applied in structural shape optimization showing robustness and computational efficiency in comparison with FEM solutions obtained with a commercial code, despite the fact that cg-FEM has been fully implemented in MATLAB.

This work presents an analysis methodology based on the use of the Finite Element Method (FEM) nowadays considered one of the main numerical tools for solving Boundary Value Problems (BVPs). The proposed methodology, so-called cg-FEM (Cartesian grid FEM), has been implemented for fast and accurate numerical analysis of 2D linear elasticity problems. The traditional FEM uses geometry-conforming meshes; however, in cg-FEM the analysis mesh is not conformal to the geometry. This allows for defining very efficient mesh generation techniques and using a robust integration procedure, to accurately integrate the domain's geometry. The hierarchical data structure used in cg-FEM together with the Cartesian meshes allow for trivial data sharing between similar entities. The cg-FEM methodology uses advanced recovery techniques to obtain an improved solution of the displacement and stress fields (for which a discretization error estimator in energy norm is available) that will be the output of the analysis. All this results in a substantial increase in accuracy and computational efficiency with respect to the standard FEM. cg-FEM has been applied in structural shape optimization showing robustness and computational efficiency in comparison with FEM solutions obtained with a commercial code, despite the fact that cg-FEM has been fully implemented in MATLAB.

We propose a reduced order modelling technique based on a partitioning of the domain of study in the context of para- metric nonlinear problems. A formulation of the reduction of the displacement and of the interface tractions linking subdomains to each others will be performed in a FETI context.

We propose a reduced order modelling technique based on a partitioning of the domain of study in the context of para- metric nonlinear problems. A formulation of the reduction of the displacement and of the interface tractions linking subdomains to each others will be performed in a FETI context.

We propose a reduced order modelling technique based on a partitioning of the domain of study in the context of para- metric nonlinear problems. A formulation of the reduction of the displacement and of the interface tractions linking subdomains to each others will be performed in a FETI context.

Domain Decomposition Methods (DDMs) are techniques that divide the solution of a PDE on a domain into smaller solutions on smaller subdomains coupling them using a certain strategy. They are used for essentially two purposes: designing parallel solvers and/or coupling subdomains with different meshes, different numerical approximations, etc. In this paper we are interested in this last category. One example of application is the Chimera method. In that sense, the Chimera method can be viewed as a preprocess technique plus a DDM on overlapping and non-conforming subdomains. The coupling technique of DDM is usually achieved via transmission conditions to impose the continuities of the unknown and its flux across the subdomain boundaries. We propose in this work an alternative coupling strategy, intervening as a preprocess method. It consists in connecting the nodes of one subdomain with the nodes of the adjacent subdomains via newly created elements. In this way, the multi-domain character of a DDM disappears, making it a parallel, implicit and versatile method. We discuss in this paper the relation between the proposed method and the existing coupling strategies. We also present some convergence results as well as some applications to the Navier-Stokes equations and other PDE's. © 2012 Elsevier Ltd.

Práce se zabývá paralelní implementací TFETI metody rozložení oblasti, která je vyvíjena na Katedře aplikované matematiky VŠB - TU Ostrava pro řešení rozsáhlých inženýrských problémů, a dále její aplikací na řešení eliptických okrajových úloh. Po diskretizaci oblasti pomocí metody konečných prvků se oblast rozdělí na nepřekrývající se podoblasti tvořené elementy sítě. Toto vede na soustavu rovnic s blokovou maticí tuhosti, což usnadňuje paralelizaci jejího řešení. Pomocí Lagrangeových multiplikátorů jsou pak vynucovány nejen lepící podmínky, ale i Dirichletovy okrajové podmínky. Metoda je tak ve srovnání s metodou FETI efektivnější a snažší na implementaci. Je zde také provedena efektivní regularizace matic tuhosti podoblastí a jsou zavedeny ortogonální projektory, které zlepšují podmíněnost duální úlohy s Lagrangeovými multiplikátory. Na numerických experimentech je ověřena numerická i paralelní škálovatelnost metody., The bachelor thesis deals with parallel implementation of the TFETI domain decomposition method, which is developed at the Department of Applied Mathematics VŠB - TU Ostrava to solve large engineering problems. It deals also with application of TFETI on the solution of elliptic boundary value problems. After finite element space discretization we decompose the domain into nonoverlapping subdomains defined by mesh elements. This leads to a system of linear equations with block-diagonal system matrix which enables its effective parallel solution. The gluing conditions and even the Dirichlet boundary conditions are enforced by the Lagrange multipliers, so the method is more effective and simpler for implementation than FETI. The effective regularization of the stiffness matrices of the subdomains is performed and the condition number of the dual problem with Lagrange multipliers is improved by using orthogonal projectors. Both numerical and parallel scalability of TFETI are tested in the numerical experiments., Ve zpracování, Import 03/08/2012

We construct and analyze an overlapping Schwarz preconditioner for elliptic problems discretized with isogeometric analysis. The preconditioner is based on partitioning the domain of the problem into overlapping subdomains, solving local isogeometric problems on these subdomains, and solving an additional coarse isogeometric problem associated with the subdomain mesh. We develop an h-analysis of the preconditioner, showing in particular that the resulting algorithm is scalable and its convergence rate depends linearly on the ratio between subdomain and "overlap sizes" for fixed polynomial degree p and regularity k of the basis functions. Numerical results in two- and three-dimensional tests show the good convergence properties of the preconditioner with respect to the isogeometric discretization parameters h, p, k, number of subdomains N, overlap size, and also jumps in the coefficients of the elliptic operator. © 2012 Society for Industrial and Applied Mathematics.

This thesis addresses the construction of some algorithms for numerically solving optimal feedback control problems. Optimal control deals with the problem of finding a control law for a given system such that a certain optimality criterion is achieved. More precisely, optimal control problems involve a dynamic system with input quantities, called controls, and some quantity, called cost, to be minimized. An optimal control is a set of differential equations describing the paths of the control variables that optimise the cost. Finding solutions to problems of this nature involves a significantly high degree of difficulty in terms of cost and power compared with the related task of solving optimal open-loop control problems. Moreover, stability is a major problem in the feedback control problem, which may tend to overcorrect errors that can cause oscillations of constant or changing amplitude. A feedback control problem essentially depends on both state and time variables, and so its determination by numerical schemes has one serious drawback, it is the so called curse of dimensionality. Therefore, efficient numerical methods are needed for the accurate determination of optimal feedback controls. There are essentially two equivalent ways in widespread use today to solve optimal feedback control problems. In the first approach, often referred to as the direct approach, the optimal feedback control problem is approximated by considering the optimisation of an objective functional with respect to the control function. This optimisation is subject to the system dynamics and numerous constraints on the state and control variables. In the second approach, the optimal feedback control problem is transformed into a first order terminal value problem by formulating the problem as a nonlinear hyperbolic partial differential equation, known as the Hamilton-Jacobi-Bellman (HJB) equation. In this thesis we consider some numerical algorithms for solving the HJB equation, based on Radial Basis Functions (RBFs). We present a new adaptive least-squares collocation RBFs method for solving a HJB equation. The method involves the use of the least squares method using a set of RBFs in space variables, combined with the implicit backward Euler finite difference method in time, to create an unconditionally stable solution scheme. We also present some of the more theoretical aspects related to the solution of the HJB equation using the adaptive least-squares collocation RBFs method, especially, the relevant existence, uniqueness and stability results. We demonstrate the accuracy and effectiveness of this method by performing numerical experiments on test problems with up to three states and two control variables. Furthermore, we construct another numerical method based on a domain decomposition method using a matrix inversion technique for solving HJB equation. In this method, we propose a new formula for inverting nonsymmetric and full dense coefficient matrix faster than the classical matrix inversion techniques. We also investigate the accuracy of the numerical solution, condition numbers of the system matrix, and the computational time when increasing the number of subdomains. We perform some numerical experiments to illustrate the usefulness and accuracy of the method., Thesis (Ph.D.)--University of Western Australia, 2010

A Balancing Domain Decomposition Method by Constraints (BDDC) is constructed and analyzed for the Reissner-Mindlin plate bending problem discretized with Mixed Interpolation of Tensorial Components (MITC) finite elements. This BDDC algorithm is based on selecting the plate rotations and deflection degrees of freedom at the subdomain vertices as primal continuity constraints. After the implicit elimination of the interior degree s of freedom in each subdomain, the resulting plate Schur complement is solved by the preconditioned conjugate gradient method. The preconditioner is based on the solution of local Reissner-Mindlin plate problems on each subdomain with clamping conditions at the primal degrees of freedom and on the solution of a coarse Reissner- Mindlin plate problem for the primal degrees of freedom. The main results of the paper are the proof and numerical verification that the proposed BDDC plate algorithm is scalable, quasi-optimal, and, most important, robust with respect to the plate thickness. While this result is due to an underlying mixed formulation of the problem, both the interface plate problem and the preconditioner are positive definite. The numerical results also show that the proposed algorithm is robust with respect to discontinuities of the material properties. © 2010 Society for Industrial and Applied Mathematics.

This paper is concerned with the development of a parallel algorithm on cluster of workstation with Parallel Virtual Machine (PVM) for solving the finite difference Navier-Stokes and energy equations. The numerical procedure is based on SIMPLE (Semi Implicit Method for Pressure Link Equations) developed by Spalding. The governing equations are transformed into finite difference forms using the control volume approach. The hybrid scheme which is a combination of the central difference and up wind scheme was used in obtaining a profile assumption for parameter variations between the grids points. The Domain Decomposition Method (DDM) was used to decompose the domain into a small domain and each of the domains was solved by different processors. The accuracy of the parallelization method was done by comparing with a benchmark solution of a standardized problem related to the two dimensional buoyancy flow in a square enclosure. The results are shown in the forms of contour maps of non-dimensional temperature and velocities.

This paper is concerned with the development of a parallel algorithm on cluster of workstation with Parallel Virtual Machine (PVM) for solving the finite difference Navier-Stokes and energy equations. The numerical procedure is based on SIMPLE (Semi Implicit Method for Pressure Link Equations) developed by Spalding. The governing equations are transformed into finite difference forms using the control volume approach. The hybrid scheme which is a combination of the central difference and up wind scheme was used in obtaining a profile assumption for parameter variations between the grids points. The Domain Decomposition Method (DDM) was used to decompose the domain into a small domain and each of the domains was solved by different processors. The accuracy of the parallelization method was done by comparing with a benchmark solution of a standardized problem related to the two dimensional buoyancy flow in a square enclosure. The results are shown in the forms of contour maps of non-dimensional temperature and velocities.

This paper is concerned with the development of a parallel algorithm on cluster of workstation with Parallel Virtual Machine (PVM) for solving the finite difference Navier-Stokes and energy equations. The numerical procedure is based on SIMPLE (Semi Implicit Method for Pressure Link Equations) developed by Spalding. The governing equations are transformed into finite difference forms using the control volume approach. The hybrid scheme which is a combination of the central difference and up wind scheme was used in obtaining a profile assumption for parameter variations between the grids points. The Domain Decomposition Method (DDM) was used to decompose the domain into a small domain and each of the domains was solved by different processors. The accuracy of the parallelization method was done by comparing with a benchmark solution of a standardized problem related to the two dimensional buoyancy flow in a square enclosure. The results are shown in the forms of contour maps of non-dimensional temperature and velocities.

The time-harmonic acoustic scattering in multi-layered media is modeled with an inhomogeneous Helmholtz equation for the pressure field. For example, such problems result from acoustic geological surveys. The exterior problem is truncated into a rectangle and an absorbing boundary condition is posed on its boundaries. A low-order finite difference discretization is performed on a uniform grid. Subdomains which overlap only on the interfaces are defined by the domains where the material properties are constants. Each of these subdomains is embedded into a larger rectangular domains with absorbing boundary conditions. The subdomain preconditioner is a Schur complement matrix and problems with it can be solved using a fast direct solver. Based on these subdomain preconditioners a Schwarz-type multiplicative preconditioner is defined. Preconditioned systems are solved using the GMRES iterations which are reduced on a neighborhood of the interfaces. Numerical experiments demonstrate that problems with millions of unknowns can be solved in some tens of seconds on a PC. Furthermore, the convergence rate deteriorates only mildly when frequency is increased.

The time-harmonic acoustic scattering in multi-layered media is modeled with an inhomogeneous Helmholtz equation for the pressure field. For example, such problems result from acoustic geological surveys. The exterior problem is truncated into a rectangle and an absorbing boundary condition is posed on its boundaries. A low-order finite difference discretization is performed on a uniform grid. Subdomains which overlap only on the interfaces are defined by the domains where the material properties are constants. Each of these subdomains is embedded into a larger rectangular domains with absorbing boundary conditions. The subdomain preconditioner is a Schur complement matrix and problems with it can be solved using a fast direct solver. Based on these subdomain preconditioners a Schwarz-type multiplicative preconditioner is defined. Preconditioned systems are solved using the GMRES iterations which are reduced on a neighborhood of the interfaces. Numerical experiments demonstrate that problems with millions of unknowns can be solved in some tens of seconds on a PC. Furthermore, the convergence rate deteriorates only mildly when frequency is increased.

Рассматриваются аппроксимации краевой задачи для сингулярно возмущенного уравнения реакции-диффузии с условиями III рода на границе, допускающими условия Дирихле и Неймана; cтаршие производные уравнения содержат возмущающий параметр $\varepsilon^2$, где $\varepsilon$ принимает произвольные значения из (0,1]. На примере ОДУ строятся и исследуются континуальные и разностные (на кусочно-равномерных сетках, сгущающихся в пограничных слоях) схемы метода декомпозиции области. % в случае последовательных и параллельных вычислений. Приводятся условия, обеспечивающие $\varepsilon$-равномерную сходимость решений с ростом числа итераций. Проведен сравнительный анализ эффективности схем декомпозиции для последовательных и параллельных вычислений. Получены оценки снизу и сверху для погрешности и числа итераций. Показано (в отличие от рассмотренного ранее параболического уравнения конвекции-диффузии), что увеличение числа решателей в параллельных схемах приводит к ускорению параллельного метода по сравнению с последовательным без потери точности решения декомпозируемой схемы. Результаты обобщаются на случай сингулярно возмущенного эллиптического уравнения на прямоугольнике., We consider approximations to the boundary value problem for a singularly perturbed reaction-diffusion equation with the third-kind boundary conditions admitting both Dirichlet and Neumann conditions. The highest derivatives of the equation are multiplied by the perturbation parameter $\varepsilon^2\!$, where $\varepsilon$ takes any values in (0,1]. With an example of ODE, we construct and study continual and difference (on piecewise uniform grids condensing in the boundary layers) schemes based on an overlapping domain decomposition method. We give conditions that ensure the $\varepsilon$-uniform convergence of solutions with increasing the number of iterations. A comparative analysis of the efficiency of the decomposition schemes for sequential and parallel computations is made. Lower and upper bounds for the error and for the number of iterations are obtained. It is shown (in contrast to the parabolic convection-diffusion case considered earlier) that the growth in the number of solvers in the parallel schemes leads to the acceleration of the parallel method in comparison with the sequential one, without loss in the accuracy of the solution of the decomposed scheme. The results can be generalized to the case of a singularly perturbed elliptic equation on a rectangle.

© 2017, Pleiades Publishing, Ltd. Finite element approximation of an elliptic variational inequality with quaisilinear operator and constraints to the solution and its gradient is constructed. Corresponding discrete problem is splitted into subproblems by non-overlapping domain decomposition technique and constrained saddle point problem is constructed for the splitted problem. Block relaxation-Uzawa iterative solution method is applied to this resulting saddle point problem. Existence of a solution to saddle point problem and convergence of the iterative method are proved.