Balancing Domain Decomposition Method Research Articles

Balancing domain decomposition method for large-scale analysis of an assembly structure having millions of multipoint constraints

A method by which to incorporate multipoint constraints (MPCs) into the balancing domain decomposition (BDD) method proposed in a previous study (BDD-MPC method) is investigated, improved, and applied to realistic and complex problems. First, the effects of the penalty term to take into account the MPCs in the coarse-grid problem of the BDD method are investigated. Then, improved implementation techniques for the BDD-MPC method are proposed. The computation speed of the projection procedure in the conjugate projected gradient method is improved using the block diagonal structure of the constraint matrix. The parallel solution procedure for the coarse-grid problem considering the MPCs that are projected to the coarse space is also improved. The improved MPC-BDD method is implemented in ADVENTURE_Solid, which is a parallel finite element structural analysis code. The code is parallelized using an MPI-OpenMP hybrid parallel implementation framework. In the previous study, the BDD-MPC method was applied to only simple numerical examples. In the present study the performance of the improved method is demonstrated by solving super-high-rise building models having very complex geometry and several million MPCs.

Acceleration of a parallel BDDC solver by using graphics processing units on subdomains

An approach to accelerating a parallel domain decomposition (DD) solver by graphics processing units (GPUs) is investigated. The solver is based on the Balancing Domain Decomposition Method by Constraints (BDDC), which is a nonoverlapping DD technique. Two kinds of local matrices are required by BDDC. First, dense matrices corresponding to local Schur complements of interior unknowns are constructed by the sparse direct solver. These are further used as part of the local saddle-point problems within BDDC. In the next step, the local matrices are copied to GPUs. Repeated multiplications of local vectors with the dense matrix of the Schur complement are performed for each subdomain. In addition, factorizations and backsubstitutions with the dense saddle-point subdomain matrices are also performed on GPUs. Detailed times of main components of the algorithm are measured on a benchmark Poisson problem. The method is also applied to an unsteady problem of incompressible flow, where the Krylov subspace iterations are performed repeatedly in each time step. The results demonstrate the potential of the approach to speed up realistic simulations up to 5 times with a preference towards large subdomains.

Implementation of balancing domain decomposition method for parallel finite element analysis involving inactive elements

AbstractThe present study developed a numerical method to implement domain decomposition (DD) solvers with the diagonal‐scaling preconditioner, the balancing domain decomposition (BDD) preconditioner and the BDD with diagonal scaling (BDD‐DIAG) preconditioner, for inactive elements. The inactive element, which is a finite element having zero stiffness, is used in several fields such as multi‐pass welding analysis, additive manufacturing analysis, damage analysis, and topology optimization. For this sort of analysis, we adopted the one‐time decomposition approach, in which the DD process is performed once at the beginning of the analysis. Based on this approach, we formulated the matrix–vector multiplication, the preconditioning and the vector operations in the algorithm of the conjugate gradient method, along with the inactive elements and floating degrees of freedom caused by the inactive elements. Consideration of the inactive elements is enabled by the slight modifications of matrices and vectors in the algorithm. Numerical examples confirmed the scalability of the BDD and BDD‐DIAG preconditioners with the present implementation method. Moreover, the capability of the present method for damage analysis, topology computation, and thermal elastic–plastic analysis of metal additive manufacturing problems was demonstrated.

離散deRham系列を満たす高次多角形要素を用いた静磁場問題に対する領域分割法

A Balancing Domain Decomposition (BDD) method is considered as the preconditioner of an iterative Domain Decomposition Method (DDM) for perturbed magnetostatic problems, where the magnetic vector potential is regarded as an unknown function approximated by the Ned´ elec curl conforming finite element. To reduce the number of the Degrees´ Of Freedom (DOF) of coarse spaces in BDD methods, Polynomial Element Methods (PEMs) are introduced. Owing to the introduction of PEMs and the result of BDD method originally proposed by Mandel, the condition number of coefficient matrices derived from the iterative DDM is evaluated. Therefore, the number of iterations of the iterative DDM can be kept even when the number of the subdomains becomes larger. Moreover, the approximate coarse space by PEMs can admit that each subdomain becomes more general polyhedron. The results in case of higher order PEMs are also shown.

領域分割法と簡略化熱源モデルを用いた金属積層造形問題の熱弾塑性解析とバリデーション

The metal additive manufacturing problems can be modeled by the one way coupling problem from a heat conduction to a thermal elastic plastic analysis. It costs a lot of time and temperature steps to analyze the problems. To reduce the numbers of these steps, our previous studies used the balancing domain decomposition method as a parallel strategy, and a simplified heat source model. The model allows to sequentially heat an entire layer one by one, and this process continues until all layers are simulated. However, it was only applied to a simulation of depositing a plate. Thus, this study focuses on applying the model to a shape other than a plate with the balancing domain decomposition method. Especially, a half pipe model was considered. The paper presents how to find parameters of the simplified heat source model on the half pipe model. One of numerical results showed a speedup of more than one hundred times. They were also verified and validated.

BDDC algorithms for advection-diffusion problems with HDG discretizations

In this paper, a preconditioned GMRES method is developed and analyzed for solving the linear system from advection-diffusion equations with the hybridizable discontinuous Galerkin (HDG) discretization. The preconditioner is the balancing domain decomposition methods (BDDC), one of the most popular nonoverlapping domain decomposition methods. For large viscosity, if the subdomain size is small enough, the number of iterations is independent of the number of subdomains and depends only slightly on the subdomain problem size. The convergence deteriorates when the viscosity decreases. These results are similar to those with the standard finite element discretizations. Numerical results of two examples in two dimensions are provided to confirm the theory.

Numerical Implementation of Inactive Elements in Balancing Domain Decomposition Method and Its Application to Metal Additive Manufacturing Simulation

Numerical Implementation of Inactive Elements in Balancing Domain Decomposition Method and Its Application to Metal Additive Manufacturing Simulation

局所座標系によりコースグリッドを定義したバランシング領域分割法

局所座標系によりコースグリッドを定義したバランシング領域分割法

局所座標系によりコースグリッドを定義したBDD法の性能評価

局所座標系によりコースグリッドを定義したBDD法の性能評価

An Adaptive MultiPreconditioned Conjugate Gradient Algorithm

This article introduces and analyzes a new adaptive algorithm for solving symmetric positive definite linear systems in cases where several preconditioners are available or the usual preconditioner is a sum of contributions. A new theoretical result allows us to select, at each iteration, whether a classical preconditioned conjugate gradient (CG) iteration is sufficient (i.e., the error decreases by a factor of at least some chosen ratio) or whether convergence needs to be accelerated by performing an iteration of multipreconditioned CG [4]. This is first presented in an abstract framework with the one strong assumption being that a bound for the smallest eigenvalue of the preconditioned operator is available. Then, the algorithm is applied to the balancing domain decomposition method and its behavior is illustrated numerically. In particular, it is observed to be optimal in terms of local solves, for both well-conditioned and ill-conditioned test cases, which makes it a good candidate to be a default par...

対角スケーリング前処理を伴ったバランシング領域分割法による複数材料モデルの有限要素解析

対角スケーリング前処理を伴ったバランシング領域分割法による複数材料モデルの有限要素解析

A robust coarse space for optimized Schwarz methods: SORAS-GenEO-2

Optimized Schwarz methods (OSM) are very popular methods that were introduced in [11] for elliptic problems and in [3] for propagative wave phenomena. We build here a coarse space for which the convergence rate of the two-level method is guaranteed regardless of the regularity of the coefficients. We do this by introducing a symmetrized variant of the ORAS (Optimized Restricted Additive Schwarz) algorithm [17] and by identifying the problematic modes using two different generalized eigenvalue problems instead of only one as in [16,15] for the ASM (Additive Schwarz method), BDD (Balancing Domain Decomposition [12]) or FETI (Finite-Element Tearing and Interconnection [6]) methods.

Numerical simulation of hydrogen dispersion behaviour in a partially open space by a stabilized balancing domain decomposition method

The dispersion behaviour of leaking hydrogen in a partially open space is simulated numerically by a balancing domain decomposition method in this work. An analogy of the thermal convection problems is applied. The linear systems of Navier–Stokes equations and the convection diffusion equation are symmetrized by a pressure stabilized domain decomposition method. The interface problem after domain decomposition is solved by a balancing preconditioned conjugate gradient method. Numerical results are validated by comparing with the experimental data and available numerical results. The transient behaviour of hydrogen dispersion and accumulation in the partially open space is discussed; the mechanism of air influence is investigated and safe region inside this partially open space is classified.

Large Scale Simulation of Hydrogen Dispersion by a Stabilized Balancing Domain Decomposition Method

The dispersion behaviour of leaking hydrogen in a partially open space is simulated by a balancing domain decomposition method in this work. An analogy of the Boussinesq approximation is employed to describe the connection between the flow field and the concentration field. The linear systems of Navier-Stokes equations and the convection diffusion equation are symmetrized by a pressure stabilized Lagrange-Galerkin method, and thus a balancing domain decomposition method is enabled to solve the interface problem of the domain decomposition system. Numerical results are validated by comparing with the experimental data and available numerical results. The dilution effect of ventilation is investigated, especially at the doors, where flow pattern is complicated and oscillations appear in the past research reported by other researchers. The transient behaviour of hydrogen and the process of accumulation in the partially open space are discussed, and more details are revealed by large scale computation.

Implementation and Scalability Analysis of Balancing Domain Decomposition Methods

In this paper we present a detailed description of a high-performance distributed-memory implementation of balancing domain decomposition preconditioning techniques. This coverage provides a pool of implementation hints and considerations that can be very useful for scientists that are willing to tackle large-scale distributed-memory machines using these methods. On the other hand, the paper includes a comprehensive performance and scalability study of the resulting codes when they are applied for the solution of the Poisson problem on a large-scale multicore-based distributed-memory machine with up to 4096 cores. Well-known theoretical results guarantee the optimality (algorithmic scalability) of these preconditioning techniques for weak scaling scenarios, as they are able to keep the condition number of the preconditioned operator bounded by a constant with fixed load per core and increasing number of cores. The experimental study presented in the paper complements this mathematical analysis and answers how far can these methods go in the number of cores and the scale of the problem to still be within reasonable ranges of efficiency on current distributed-memory machines. Besides, for those scenarios where poor scalability is expected, the study precisely identifies, quantifies and justifies which are the main sources of inefficiency.

Efficiency and scalability of a two‐level Schwarz algorithm for incompressible and compressible flows

SUMMARYThis paper studies the application of two‐level Schwarz algorithms to several models of computational fluid dynamics. The purpose is to build an algorithm suitable for elliptic and convective models. The subdomain approximated solution relies on the incomplete lower‐upper factorization. The algebraic coupling between the coarse grid and the Schwarz preconditioner is discussed. The deflation method and the balancing domain decomposition method are studied for introducing the coarse‐grid correction as a preconditioner. Standard coarse grids are built with the characteristic or indicator functions of the subdomains. The building of a set of smooth basis functions (analogous to smoothed‐aggregation methods) is considered. A first test problem is the Poisson problem with a discontinuous coefficient. The two options are compared for the standpoint of coarse‐grid consistency and for the gain in scalability of the global Schwarz iteration. The advection–diffusion model is then considered as a second test problem. Extensions to compressible flows (together with incompressible flows for comparison) are then proposed. Parallel applications are presented and their performance measured. Copyright © 2012 John Wiley & Sons, Ltd.

A Monolithic Approach Based on the Balancing Domain Decomposition Method for Acoustic Fluid-Structure Interaction

A scalable and efficient monolithic approach based on the Balancing Domain Decomposition (BDD) method for acoustic fluid-structure interaction problems is developed. The BDD method is a well-known domain decomposition method for non-overlapping sub-domains, which consists of Neumann-Neumann (NN) preconditioning and coarse grid correction. In this study, we derive four types of BDD method, considering two options for NN preconditioning (NN-I and NN-C) and two options for coarse grid correction (CGC-FULL and CGC-DIAG). From the results of numerical experiments, the combination of NN-I and CGC-FULL turns out to be the most efficient scheme, showing fast convergence property irrespective of the number of sub-domains, DOFs of fluid and solid domains, and the added-mass effect of fluid. The combination of NN-I and CGC-DIAG is also expected to be an efficient scheme in some situations in a parallel environment.

Robust BDDC Preconditioners for Reissner–Mindlin Plate Bending Problems and MITC Elements

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 degrees 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.

FETI and BDD preconditioners for Stokes–Mortar–Darcy Systems

We consider the coupling across an interface of a fluid flow and a porous media flow. The differential equations involve Stokes equations in the fluid region, Darcy equations in the porous region, plus a coupling through an interface with Beaver‐Joseph‐Saffman transmission conditions. The discretization consists of P2/P1 triangular Taylor‐Hood finite elements in the fluid region, the lowest order triangular Raviart‐Thomas finite elements in the porous region, and the mortar piecewise constant Lagrange multipliers on the interface. We allow for nonmatching meshes across the interface. Due to the small values of the permeability parameter of the porous medium, the resulting discrete symmetric saddle point system is very ill conditioned. We design and analyze preconditioners based on the finite element by tearing and interconnecting (FETI) and balancing domain decomposition (BDD) methods and derive a condition number estimate of order C1.1C.1=// for the preconditioned operator. In case the fluid discretization is finer than the porous side discretization, we derive a better estimate of order C2.. C 1/=. C.h p / 2 // for the FETI preconditioner. Here h p is the mesh size of the porous side triangulation. The constants C1 and C2 are independent of the permeability , the fluid viscosity , and the mesh ratio across the interface. Numerical experiments confirm the sharpness of the theoretical estimates.

Convergence analysis of a balancing domain decomposition method for solving a class of indefinite linear systems

AbstractA variant of balancing domain decomposition method by constraints (BDDC) is proposed for solving a class of indefinite systems of linear equations of the form (K−σ2M)u=f, which arise from solving eigenvalue problems when an inverse shifted method is used and also from the finite element discretization of Helmholtz equations. Here, both K and M are symmetric positive definite. The proposed BDDC method is closely related to the previous dual–primal finite element tearing and interconnecting method (FETI‐DP) for solving this type of problems (Appl. Numer. Math. 2005; 54:150–166), where a coarse level problem containing certain free‐space solutions of the inherent homogeneous partial differential equation is used in the algorithm to accelerate the convergence. Under the condition that the diameters of the subdomains are small enough, the convergence rate of the proposed algorithm is established, which depends polylogarithmically on the dimension of the individual subdomain problems and which improves with a decrease of the subdomain diameters. These results are supported by numerical experiments of solving a two‐dimensional problem. Copyright © 2009 John Wiley & Sons, Ltd.