Domain Decomposition Framework Research Articles

Local and Parallel Finite Element Algorithms Based on the Partition of Unity for the Stokes Problem

By combining the techniques of the two-grid method and the partition of unity, we derive two local and parallel finite element algorithms for the Stokes problem. The most interesting features of these algorithms are (1) the partition of unity technique introduces a framework for domain decomposition; (2) only a series of local residual problems need to be solved on these subdomains in parallel, meanwhile requiring very little communication; (3) a globally continuous finite element solution is constructed by combining all the local solutions via the partition of unity functions. The optimal error estimates in $L^2$ and energy norms are proved under some assumptions. Also, several numerical simulations are presented to demonstrate the effectiveness and flexibility of the new algorithms.

Bridging from Pore to Continuum: A Hybrid Mortar Domain Decomposition Framework for Subsurface Flow and Transport

Flow and transport in the subsurface occurs over a wide range of spatial scales (nanometer to kilometer). Modeling at the pore scale becomes imperative where scales are not separable. Since pore-scale models are computationally limited to small domain sizes, accurate field-scale modeling requires simulating parts of the reservoir at the pore scale and other parts at the continuum. The need for modeling large pore-scale domains for ascertaining macroscopic parameters is prevalent in the literature. We develop a hybrid mortar domain decomposition framework for parallel modeling (linear and nonlinear) flow and transport across scales and in large pore-scale domains. Novel and efficient mortar methods for coupling flow and transport are adapted and developed and comparisons are presented. Mortars are developed for pore-to-pore and pore-to-continuum interfaces and shown to be more suitable than the commonly used Lagrangian mortars for these interfaces. The methods are shown to produce accurate results and demo...

A hybridizable discontinuous Galerkin method combined to a Schwarz algorithm for the solution of 3d time-harmonic Maxwellʼs equation

A Schwarz-type domain decomposition method is presented for the solution of the system of 3d time-harmonic Maxwellʼs equations. We introduce a hybridizable discontinuous Galerkin (HDG) scheme for the discretization of the problem based on a tetrahedrization of the computational domain. The discrete system of the HDG method on each subdomain is solved by an optimized sparse direct (LU factorization) solver. The solution of the interface system in the domain decomposition framework is accelerated by a Krylov subspace method. The formulation and the implementation of the resulting DD-HDG (Domain Decomposed-Hybridizable Discontinuous Galerkin) method are detailed. Numerical results show that the resulting DD-HDG solution strategy has an optimal convergence rate and can save both CPU time and memory cost compared to a classical upwind flux-based DD-DG (Domain Decomposed-Discontinuous Galerkin) approach.

Domain decomposition methods to evaluate effective elastic properties of random fibre composites in the framework of the double-scale homogenization

The paper deals with the evaluation of the effective elastic properties in the framework of domain decomposition. The field of random fibre composites, for which the network of heterogeneities is complex and leads to several numerical shortcomings, is considered. 2D representative volume elements (RVEs) of the composite are generated and some elastic properties are estimated with the help of the double-scale homogenization. Such methodology is reliable but turns out to be potentially inefficient due to the required size of RVEs. Two adaptations of domain decomposition methods in the framework of double-scale homogenization are proposed to drastically reduce the calculation costs: a Schur complement method, and a mixed Schur complement and FETI-1 method. Several numerical tests are performed which highlight reliability and efficiency of the first one in the present context.

Auxiliary variables for 3D multiscale simulations in heterogeneous porous media

The multiscale control-volume methods for solving problems involving flow in porous media have gained much interest during the last decade. Recasting these methods in an algebraic framework allows one to consider them as preconditioners for iterative solvers. Despite intense research on the 2D formulation, few results have been shown for 3D, where indeed the performance of multiscale methods deteriorates. The interpretation of multiscale methods as vertex based domain decomposition methods, which are non-scalable for 3D domain decomposition problems, allows us to understand this loss of performance.We propose a generalized framework based on auxiliary variables on the coarse scale. These are enrichments of the coarse scale, which can be selected to improve the interpolation onto the fine scale. Where the existing coarse scale basis functions are designed to capture local sub-scale heterogeneities, the auxiliary variables are aimed at better capturing non-local effects resulting from non-linear behavior of the pressure field. The auxiliary coarse nodes fits into the framework of mass-conservative domain-decomposition (MCDD) preconditioners, allowing us to construct, as special cases, both the traditional (vertex based) multiscale methods as well as their wire basket generalization.

A domain decomposition method for concurrent coupling of multiscale models

SUMMARYMotivated by atomistic‐to‐continuum coupling, we consider a fine‐scale problem defined on a small region embedded in a much larger coarse‐scale domain and propose an efficient solution technique on the basis of the domain decomposition framework. Specifically, we develop a nonoverlapping Schwarz method with two important features: (i) the use of an efficient approximation of the Dirichlet‐to‐Neumann map for the interface conditions; and (ii) the utilization of the inherent scale separation in the solution. The paper includes a detailed formulation of the proposed interface condition, along with the illustration of its effectiveness by using simple but representative numerical experiments. Copyright © 2012 John Wiley & Sons, Ltd.

Analysis of photonic crystals using the hybrid finite‐element/finite‐difference time domain technique based on the discontinuous Galerkin method

SUMMARYTwo‐dimensional photonic crystal structures are analyzed by a recently developed hybrid technique combining the finite‐element time‐domain (FETD) method and the finite‐difference time‐domain (FDTD) method. This hybrid FETD/FDTD method uses the discontinuous Galerkin method as framework for domain decomposition. To the best of our knowledge, this is the first hybrid FETD/FDTD method that allows non‐conformal meshes between different FETD and FDTD subdomains. It is also highly parallelizable. These properties are very suitable for the computation of periodic structures with curved surfaces. Numerical examples for the computation of the scattering parameters of two‐dimensional photonic bandgap structures are presented as applications of the hybrid FETD/FDTD method. Numerical results demonstrate the efficiency and accuracy of the proposed hybrid method. Copyright © 2012 John Wiley & Sons, Ltd.

A MATRIX SPLITTING DDM BASED ON SVE-BI FOR MULTIPLE CONDUCTING BODIES COATED BY THIN LAYER DIELECTRIC

A matrix splitting domain decomposition method based on hybrid shell vector element–boundary integral (MSDD-SVE-BI) for three dimensional electromagnetic scattering from multiple conducting bodies coated by thin layer dielectric is proposed. In the framework of domain decomposition, the whole computational domains are divided into a lot of sub-SVE-domains and boundary element domains. For conducting body coated with thin-layer dielectric, the shell vector element is used instead of traditional tetrahedral elements to reduce the number of unknowns. Further, a block Gauss-Seidel type preconditioner is applied to attain fast matrix splitting formulation for the matrix connecting surface electric field and surface magnetic field. By this method, only sub-matrix inversion is required in the SVE-BI method, the computational time for connecting matrix can be reduced greatly. Several numerical examples prove the accuracy and efficiency of the present method.

A Frozen Jacobian Multiscale Mortar Preconditioner for Nonlinear Interface Operators

We present an efficient approach for preconditioning systems arising in multiphase flow in a parallel domain decomposition framework known as the mortar mixed finite element method. Subdomains are coupled together with appropriate interface conditions using mortar finite elements. These conditions are enforced using an inexact Newton--Krylov method, which traditionally required the solution of nonlinear subdomain problems on each interface iteration. A new preconditioner is formed by constructing a multiscale basis on each subdomain for a fixed Jacobian and time step. This basis contains the solutions of nonlinear subdomain problems for each degree of freedom in the mortar space and is applied using an efficient linear combination. Numerical experiments demonstrate the relative computational savings of recomputing the multiscale preconditioner sparingly throughout the simulation versus the traditional approach.

A monolithic energy conserving method to couple heterogeneous time integrators with incompatible time steps in structural dynamics

A new hybrid multi-time method for multi-time scales structural dynamics simulations is described. A monolithic method in a Schur dual domain decomposition framework is proposed and allows to consider heterogeneous time integrators with their own time discretization and possible large ratio between the time steps for each subdomain. In the proposed method, zero numerical dissipation is ensured at the interface. This implies that the global stability of the coupling method is governed by the stability of each time integrator without influence of the interface. For that purpose, velocity continuity is ensured in a weak sense at the interfaces, and time integrators (Newmark, HHT, Simo, Krenk, Verlet) are introduced in a unified framework (incremental velocity formulation). Furthermore, dynamics governing equations are introduced from a weak formulation in time. In other words, equilibrium equation is no more ensured in a strong sense at a given time step, but rather on average on a time interval. Some numerical examples illustrate the efficiency and the robustness of the proposed method, for ratio of time scales close to 1000 without any numerical dissipation at the interfaces.

On the control of the load increments for a proper description of multiple delamination in a domain decomposition framework

AbstractIn quasi‐static non‐linear time‐dependent analysis, the choice of the time discretization is a complex issue. The most basic strategy consists in determining a value of the load increment that ensures the convergence of the solution with respect to time on the base of preliminary simulations. In more advanced applications, the load increments can be controlled for instance by prescribing the number of iterations of the non‐linear resolution procedure, or by using an arc‐length algorithm. These techniques usually introduce a parameter whose correct value is not easy to obtain. In this paper, an alternative procedure is proposed. It is based on the continuous control of the residual of the reference problem over time, whose measure is easy to interpret. This idea is applied in the framework of a multiscale domain decomposition strategy in order to perform 3D delamination analysis. Copyright © 2010 John Wiley & Sons, Ltd.

An efficient domain decomposition framework for accurate representation of geodata in distributed hydrologic models

Physically‐based, fully‐distributed hydrologic models simulate hydrologic state variables in space and time while using information regarding heterogeneity in climate, land use, topography and hydrogeology. Since fine spatio‐temporal resolution and increased process dimension will have large data requirements, there is a practical need to strike a balance between descriptive detail and computational load for a particular model application. In this paper, we present a flexible domain decomposition strategy for efficient and accurate integration of the physiographic, climatic and hydrographic watershed features. The approach takes advantage of different GIS feature types while generating high‐quality unstructured grids with user‐specified geometrical and physical constraints. The framework is able to anchor the efficient capture of spatially distributed and temporally varying hydrologic interactions and also ingest the physical prototypes effectively and accurately from a geodatabase. The proposed decomposition framework is a critical step in implementing high quality, multiscale, multiresolution, temporally adaptive and nested grids with least computational burden. We also discuss the algorithms for generating the framework using existing GIS feature objects. The framework is successfully being used in a finite volume based integrated hydrologic model. The framework is generic and can be used in other finite element/volume based hydrologic models.

Parallel discrete element simulation of poly-dispersed granular material

The paper presents parallel 3D DEM simulation of poly-dispersed material described by the normal size distribution. Static domain decomposition and message passing inter-processor communication have been implemented in the DEM code. A novel algorithm for moving particles that exchange processors has been incorporated in the domain decomposition framework. Parallel performance of the developed algorithm and software has been investigated by a series of benchmark tests conducting tri-axial compaction of material with different numbers of particles, heterogeneity ratios and compaction durations. The speed-up equal to 8.81 has been obtained on 10 processors of the distributed memory PC cluster. It has been shown that a drastic increase of computational expenses of simulation for the poly-dispersed material in terms of CPU time is associated with the increase of its heterogeneity. A contribution of the temporal evolution of microscopic behaviour has also been illustrated.

Parallel algebraic hybrid solvers for large 3D convection-diffusion problems

In this paper we study the parallel scalability of variants of an algebraic additive Schwarz preconditioner for the solution of large three dimensional convection diffusion problems in a non-overlapping domain decomposition framework. To alleviate the computational cost, both in terms of memory and floating-point complexity, we investigate variants based on a sparse approximation or on mixed 32- and 64-bit calculation. The robustness and the scalability of the preconditioners are investigated through extensive parallel experiments on up to 2,000 processors. Their efficiency from a numerical and parallel performance view point are reported.

A numerical approximation of the thermal coupling of fluids and solids

AbstractIn this article we analyze the problem of the thermal coupling of fluids and solids through a common interface. We state the global thermal problem in the whole domain, including the fluid part and the solid part. This global thermal problem presents discontinuous physical properties that depend on the solution of auxiliary problems on each part of the domain (a fluid flow problem and a solid state problem). We present a domain decomposition strategy to iteratively solve problems posed in both subdomains and discuss some implementation aspects of the algorithm. This domain decomposition framework is also used to revisit the use of wall function approaches used in this context. Copyright © 2008 John Wiley & Sons, Ltd.

On some fluid–structure iterative algorithms using pressure segregation methods. Application to aeroelasticity

AbstractIn this paper we suggest some algorithms for the fluid–structure interaction problem stated using a domain decomposition framework. These methods involve stabilized pressure segregation methods for the solution of the fluid problem and fixed point iterative algorithms for the fluid–structure coupling. With one single loop the solution of the coupled system tends to one of the strongly coupled monolithic systems. These coupling algorithms are applied to the aeroelastic simulation of suspension bridges. We assess flexural and torsional frequencies for a given inflow velocity. Increasing this velocity we reach the value for which the flutter phenomenon appears. Copyright © 2007 John Wiley & Sons, Ltd.

Local boundary estimates for the Lagrange multiplier discretization of a Dirichlet boundary value problem with application to domain decomposition

We give an estimate on the error resulting from approximating the outer normal derivative of the solution of a second-order partial differential equation with the Lagrange multiplier obtained in using the Lagrange multiplier method for imposing the Dirichlet boundary conditions. We consider both the case of smooth domains and, in view of an application in the framework of domain decomposition, the case of polygonal domains. The estimate given, supported by numerical results, shows that the mesh in the interior of the (sub)domain can be chosen more coarsely than near the boundary, when only a good approximation of the outer normal derivative is needed, as in the case of the evaluation of the Steklov-Poincare operator, or when solving with a Schur complement approach the linear system arising from the three-fields domain decomposition method. Keywords: Lagrange multipliers, error estimate, normal derivative, Steklov-Poincare operator

Nitsche Type Mortaring for Singularly Perturbed Reaction-diffusion Problems

The paper is concerned with the Nitsche mortaring in the framework of domain decomposition where non-matching meshes and weak continuity of the finite element approximation at the interface are admitted. The approach is applied to singularly perturbed reaction-diffusion problems in 2D. Non-matching meshes of triangles being anisotropic in the boundary layers are applied. Some properties as well as error estimates of the Nitsche mortar finite element schemes are proved. In particular, using a suitable degree of anisotropy of triangles in the boundary layers of a rectangle, convergence rates as known for the conforming finite element method are derived. Numerical examples illustrate the approach and the results.

A parallel mixed time integration algorithm for nonlinear dynamic analysis

This paper presents a parallel mixed time integration algorithm formulated by synthesising the implicit and explicit time integration techniques. The proposed algorithm is an extension of the mixed time integration algorithms [Comput. Meth. Appl. Mech. Engng 17/18 (1979) 259; Int. J. Numer. Meth. Engng 12 (1978) 1575] being successfully employed for solving media-structure interaction problems. The parallel algorithm for nonlinear dynamic response of structures employing mixed time integration technique has been devised within the broad framework of domain decomposition. Concurrency is introduced into this algorithm, by integrating interface nodes with explicit time integration technique and later solving the local submeshes with implicit algorithm. A flexible parallel data structure has been devised to implement the parallel mixed time integration algorithm. Parallel finite element code has been developed using portable Message Passing Interface software development environment. Numerical studies have been conducted on PARAM-10000 (Indian parallel supercomputer) to test the accuracy and also the performance of the proposed algorithm. Numerical studies indicate that the proposed algorithm is highly adaptive for parallel processing.

Multidomain finite elements for advection-diffusion equations

In this paper we investigate some domain decomposition methods for the solution of advection-diffusion (AD) equations. We aim at reviewing some well known methods and we introduce and analyze adaptive domain decomposition algorithms for nonoverlapping multidomain partitions. Two of the methods we will deal with were introduced by Cartenzoli and Quarteroni (1993) (Adaptive Dirichlet/Neumann (ADN) and Adaptive Robin/Neumann (ARN)) and are based on an iterative procedure among subdomains in which the transmission conditions take into account the direction of the streamlines. We introduce, also, a variant of ARN, d-ARN (damped-ARN). From the numerical point of view, these methods are applied using a finite element discretization. In particular, ARN and d-ARN perform very well, compared with the classical Dirichlet/Neumann method, when the equation is dominated by convection. We also present some studies related to the use of consistent stabilizations of the Galerkin method in a domain decomposition framework, and show how to enforce the transmission conditions in the presence of cross points.