Non-overlapping Domain Decomposition Techniques Research Articles

A non-overlapping optimization-based domain decomposition approach to component-based model reduction of incompressible flows

We present a component-based model order reduction procedure to efficiently and accurately solve parameterized incompressible flows governed by the Navier-Stokes equations. Our approach leverages a non-overlapping optimization-based domain decomposition technique to determine the control variable that minimizes jumps across the interfaces between sub-domains. To solve the resulting constrained optimization problem, we propose both Gauss-Newton and sequential quadratic programming methods, which effectively transform the constrained problem into an unconstrained formulation. Furthermore, we integrate model order reduction techniques into the optimization framework, to speed up computations. In particular, we incorporate localized training and adaptive enrichment to reduce the burden associated with the training of the local reduced-order models. Numerical results are presented to demonstrate the validity and effectiveness of the overall methodology.

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.

Towards HPC simulations of billion-cell reservoirs by multiscale mixed methods

A three dimensional parallel implementation of Multiscale Mixed Methods based on non-overlapping domain decomposition techniques is proposed for multi-core computers and its computational performance is assessed by means of numerical experiments. As a prototypical method, from which many others can be derived, the Multiscale Robin Coupled Method is chosen and its implementation explained in detail. Numerical results for problems ranging from millions up to more than 2 billion computational cells in highly heterogeneous anisotropic rock formations based on the SPE10 benchmark are shown. The proposed implementation relies on direct solvers for both local problems and the interface coupling system. We find good weak and strong scalalability as compared against a state-of-the-art global fine grid solver based on Algebraic Multigrid preconditioning in single and two-phase flow problems.

Parallel and space-time adaptivity for the numerical simulation of cardiac action potentials

In this work, the study of parallel and space-time adaptivity for the numerical simulation of action potential propagation in cardiac electrophysiology is presented. The monodomain and bidomain models are employed for the numerical realization. This work is devoted to investigate the spatial adaptivity which is realized within multi finite element methods by using a gradient type a posteriori error estimator and the temporal adaptivity determined by a linearly implicit time integration techniques. In addition, the parallelization of such space-time adaptivity for the simulation of spiral wave dynamics based on non-overlapping domain decomposition techniques and dynamic load balancing is discussed. The numerical results demonstrate that the parallel simulations have been accelerated reasonably good using the adaptive space and time algorithm, with no significant loss in accuracy, for the single action potential wave propagation as well as reentry situations.

Domain decomposition and Uzawa-type iterative method for elliptic variational inequality

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.

Crack propagation analysis in composite materials by using moving mesh and multiscale techniques

A novel concurrent multiscale method for the crack propagation analysis in heterogeneous materials is proposed, based on a non-overlapping domain decomposition technique coupled with an adaptive zoom-in strategy. Both fiber/matrix interfacial debonding and matrix cracking are accounted for; the latter one is modeled by using an innovative shape optimization method coupling a moving mesh technique and a gradient-free optimization solver. Numerical applications are carried out with reference to the failure analysis of a single notched fiber-reinforced composite beam subjected to both mode-I and mixed-mode crack propagation conditions. The validity of the proposed method is assessed through original comparison models.

Partitioned solution strategies for coupled BEM–FEM acoustic fluid–structure interaction problems

This paper investigates two FEM–BEM coupling formulations for acoustic fluid–structure interaction (FSI) problems, using the Finite Element Method (FEM) to model the structure and the Boundary Element Method (BEM) to represent a linear acoustic fluid. The coupling methods described interconnect fluid and structure using classical or localized Lagrange multipliers, allowing the connection of non-matching interfaces. First coupling technique is the well known mortar method, that uses classical multipliers and is compared with a new formulation of the method of localized Lagrange multipliers (LLM) for FSI applications with non-matching interfaces. The proposed non-overlapping domain decomposition technique uses a classical non-symmetrical acoustic BEM formulation for the fluid, although a symmetric Galerkin BEM formulation could be used as well. A comparison between the localized methodology and the mortar method in highly non conforming interface meshes is presented. Furthermore, the methodology proposes an iterative preconditioned and projected bi-conjugate gradient solver which presents very good scalability properties in the solution of this kind of problems.

On Topological Derivatives for Contact Problems in Elasticity

In this article, a general method for shape-topology sensitivity analysis of contact problems is proposed. The method uses domain decomposition combined with specific properties of minimizers for the energy functional. The method is applied to the static problem of an elastic body in frictionless contact with a rigid foundation. The contact model allows a small interpenetration of the bodies in the contact region. This interpenetration is modeled by means of a scalar function that depends on the normal component of the displacement field on the potential contact zone. We present the asymptotic behavior of the energy shape functional when a spheroidal void is introduced at an arbitrary point of the elastic body. For the asymptotic analysis, we use a nonoverlapping domain decomposition technique and the associated Steklov---Poincare pseudodifferential operator. The differentiability of the energy with respect to the nonsmooth perturbation is established, and the topological derivative is presented in the closed form.

Parallel algebraic domain decomposition solver for the solution of augmented systems

We consider the parallel iterative solution of indefinite linear systems given as augmented systems. Our numerical technique is based on an algebraic non-overlapping domain decomposition technique that only exploits the graph of the sparse matrix. This approach to high-performance, scalable solution of large sparse linear systems in parallel scientific computing, is to combine direct and iterative methods. We report numerical and parallel performance of the scheme on large matrices arising from the finite element discretization of linear elasticity in structural mechanics problems.

An efficient S-DDM iterative approach for compressible contamination fluid flows in porous media

In this paper, we develop an efficient splitting domain decomposition method (S-DDM) for compressible contamination fluid flows in porous media over multiple block-divided sub-domains by combining the non-overlapping domain decomposition, splitting, linearization and extrapolation techniques. The proposed S-DDM iterative approach divides the large domain into multiple block sub-domains. In each time interval, the S-DDM scheme is applied to solve the water head equation, in which an efficient local multilevel scheme is used for computing the values of water head on the interfaces of sub-domains, and the splitting implicit scheme is used for computing the interior values of water head in sub-domains; and the S-DDM scheme is then proposed to solve the concentration equation by combining the upstream volume technique. Numerical experiments are performed and analyzed to illustrate the efficiency of the S-DDM iterative approach for simulating compressible contamination fluid flows in porous media. The developed method takes the excellent attractive advantages of both the non-overlapping domain decomposition and the splitting technique, and reduces computational complexities, large memory requirements and long computational durations.

A posteriori error analysis for two non-overlapping domain decomposition techniques

This paper is devoted to the construction of fast solvers for penalty domain decomposition techniques, based upon a posteriori error analysis. We introduce a penalty non-overlapping domain decomposition method (ddm) motivated by the a posteriori error analysis of the method proposed by Chacón and Chacón in [T. Chacón Rebollo, E. Chacón Vera, A non-overlapping domain decomposition method for the Stokes equations via a penalty term on the interface, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1–16]. In the new method a H 00 1 / 2 ( Γ ) penalty term replaces the L 2 ( Γ ) one in the original method. The number of iterations needed by the new ddm to yield a solution with an error of the same order as the discretization error is remarkably reduced. We develop an a posteriori error analysis that we use to determine an optimal value of the penalty parameter for a given grid, and also to jointly determine an optimal grid and a penalty parameter to reduce the error below a targeted value. Several numerical tests for model problems exhibit the good performances of our approach and provide to a numerical comparison of the two penalty methods.

An extension theorem for polynomials on triangles

We present an extension theorem for polynomial functions that proves a quasi-optimal bound for a lifting from L 2 on edges onto a fractional-order Sobolev space on triangles. The extension is such that it can be further extended continuously by zero within the trace space of H 1. Such an extension result is critical for the analysis of non-overlapping domain decomposition techniques applied to the p-and hp-versions of the finite and boundary element methods for elliptic problems of second order in three dimensions.

A second-order continuity domain decomposition technique based on integrated Chebyshev polynomials for two-dimensional elliptic problems

This paper presents a second-order continuity non-overlapping domain decomposition (DD) technique for numerically solving second-order elliptic problems in two-dimensional space. The proposed DD technique uses integrated Chebyshev polynomials to represent the solution in subdomains. The constants of integration are utilized to impose continuity of the second-order normal derivative of the solution at the interior points of subdomain interfaces. To also achieve a C 2 function at the intersection of interfaces, two additional unknowns are introduced at each intersection point. Numerical results show that the present DD method yields a higher level of accuracy than conventional DD techniques based on differentiated Chebyshev polynomials.

Mortar methods with curved interfaces

We analyze a nonoverlapping domain decomposition technique with curvilinear boundaries. The weak coupling at the curved interfaces is carried out in terms of Lagrange multiplier spaces. We use the abstract framework of mortar and blending elements to obtain a priori results for this nonconforming discretization scheme. Introducing a mesh dependent jump on the curved interfaces based on piecewise linear approximations of the interfaces, the consistency error for the piecewise linear approximation can be decomposed into a consistency error for blending elements and a variational crime. Numerical results illustrate the performance of the method.

Domain decomposition techniques for reaction–diffusion equations in two-dimensional regions with re-entrant corners

A system of two non-linear reaction–diffusion equations is solved numerically by means of linearized θ-methods and both overlapping and non-overlapping domain decomposition techniques in two-dimensional regions with re-entrant corners. Two numerical methods based on either approximate factorization (AF) or the bi-conjugate-gradient-stabilized (BiCGstab) technique are employed. A study of the effects of the number of overlapping grid lines on both the accuracy and numerical efficiency is presented. For non-overlapping domain decomposition techniques, the unknown values at the common interface between adjacent subdomains have been updated by means of Dirichlet, Neumann and Robin couplings, and combinations thereof. It is shown that non-overlapping domain techniques are less accurate than overlapping ones for domains with re-entrant corners because the interfaces between adjacent subdomains are evaluated by imposing continuity of the unknowns and their normal derivatives there, and, therefore, the partial differential equations are not solved at the interfaces between adjacent subdomains. Nevertheless, the accuracy of these techniques increases as the grid spacing is decreased, although they still exhibit large errors near the re-entrant corners.

The Block Stride Reduction (B.S.R.) AlgorithmWith Application To Domain DecompositionMethods

In this paper we consider the application of the Block Stride Reduction algorithm for the solution of separable self-adjoint elliptic equations on rectangular domains. Non-overlapping domain decomposition techniques are applied to reduce the differential operator to block diagonally bordered linear systems which can then be solved by the Preconditioned Conjugate Gradient method. Finally, a new preconditioner for such systems is outlined.

Parallel multidomain iterative algorithms for the Helmholtz wave equation

In this paper, we consider parallel iterative algorithms for solving the Helmholtz wave equation employing nonoverlapping domain decomposition techniques. A modified Robin interface condition incorporated with an iteration parameter is used to communicate the data near the interfaces. An automatic and non-expensive strategy for finding efficient iteration parameters is discussed in detail. Numerical results carried out on an nCUBE2 are given to demonstrate the effectiveness of the method.

Domain decomposition and parallel processing of a finite element model of the shallow water equations

We present nonoverlapping domain decomposition techniques applied to a two-stage Numerov-Galerkin finite element model of the shallow water equations over a limited-area domain. The Schur complement matrix formulation is employed and a modified interface matrix approach is proposed to handle the coupling between subdomains. The resulting nonsymmetric Schur complement matrices, modified interface matrices as well as the subdomain coefficient matrices are solved using Preconditioned Conjugate Gradient Squared (PCGS) non-symmetric iterative solvers. Various stages of the finite element solution are parallelized and the code is implemented ona four processor CRAY Y-MP supercomputer applying multitasking techniques in a dedicated environment.