Domain Decomposition Algorithm Research Articles

Nonlinear domain decomposition scheme for sequential fully implicit formulation of compositional multiphase flow

New sequential fully implicit (SFI) methods for compositional flow simulation have been recently investigated. These SFI schemes decouple the fully coupled problem into separate pressure and transport problems and have convergence properties comparable with those of the fully implicit (FI) method. The pressure system is a parabolic problem with fixed overall compositions and the transport system is a hyperbolic problem with fixed pressure and total velocity. We discuss some aspects of how to design optimal SFI schemes for compositional flow with general equation-of-states by localizing the computations. The different systems are solved sequentially and the fully implicit solution is recovered by controlling the a posteriori splitting errors due to the choice of decoupling. When the parabolic and the hyperbolic operators are separated, it is possible to design nonlinear domain decomposition schemes taking the advantage of the specific properties of each operator. Usually, for reservoir simulation models, most of the reservoir is converged with SFI methods in one outer iteration. However, in some localized regions with strong coupling between the pressure and the compositions, the SFI algorithms may need several outer iterations. Here, we propose a domain decomposition method based on a predictor-corrector strategy. As a first step, the nonlinear parabolic pressure equation is solved on the whole domain with the Multiscale Restriction-Smooth Basis (MsRSB) method used as a linear domain decomposition solver. In a second step, the compositions system is solved. At the end of this first outer iteration, most of the reservoir is converged. Based on a posteriori splitting errors of the SFI scheme in volume and velocity, we define local regions where additional global outer iterations would be required in the conventional SFI scheme. We then fix Dirichlet boundary conditions for the pressure and the compositions and solve local problems in these non converged regions. After convergence of these smaller nonlinear problems, if the boundary conditions are changed by the updated regions, the global pressure problem is revisited. An additional post-processing of local transport iterations makes sure mass is conserved everywhere. The resulting algorithm converges to the same solution as the FI solver, with all simultaneous updates to composition and pressure in localized regions. We demonstrate the robustness of this nonlinear domain decomposition algorithm across a wide parameter range. Realistic compositional models with gas and water injection are presented and discussed.

Parallel analysis system for free-surface flow using MPS method with explicitly represented polygon wall boundary model

This study develops a parallel solver of free-surface flow based on a mesh-free particle method, the explicit MPS method, with polygon boundary representation. We adopt the explicitly represented polygon (ERP) wall boundary model, which expresses wall boundaries as arbitrarily shaped triangular polygons. A bucket-based domain decomposition algorithm for dynamic load balancing is expanded to the polygon-based computation of the ERP model. The validity and parallel efficiency of the developed solver are tested by analyzing a dam break problem, and the numerical results are compared with experimental results. Our developed solver can simplify the evaluation of the integrity of coastal structures since it can be easily connected to finite element analyses of the structures.

A coercive heterogeneous media Helmholtz model: formulation, wavenumber-explicit analysis, and preconditioned high-order FEM

We consider a frequency-domain heterogeneous wave propagation model governed by the Helmholtz partial differential equation (PDE) and an impedance boundary condition. The celebrated standard (H1) variational formulation of the model is non-coercive. It is an open problem to establish a coercive variational formulation of the heterogeneous model. The main focus of this article is on solving this continuous model formulation and analysis problem, and hence establishing an efficient preconditioned numerical algorithm for simulating our novel coercive variational formulation. We develop the variational formulation for the heterogeneous model (in a Hilbert space V equipped with a stronger norm than the H1-norm) and prove that the associated sesquilinear form is coercive, with a wavenumber-independent coercivity constant. We use this result to derive a wavenumber-independent bound for solutions of the heterogeneous media wave propagation model in the V -norm. Additionally, we prove continuity of the sesquilinear form, with a wavenumber-explicit continuity constant. Using our analysis-supported coercive formulation, we develop a high-order frequency robust-preconditioned finite element method (FEM)-based heterogeneous media discrete wave model. For demonstrating efficiency and convergence of the coercive high-order FEM model, we use non-convex media comprising curved and non-smooth boundaries and low- to high-frequency input incident waves. For the heterogeneous media, with size varying from tens to hundreds of wavelengths, we demonstrate that our new preconditioned-FEM model requires a very low number of GMRES iterations, and the number of iterations is independent of the wavenumber of the model. We also use a class of additive Schwarz domain decomposition (DD) algorithms to implement the preconditioned-FEM model. The DD-based high-order preconditioned-FEM results and comparisons further demonstrate efficiency of the coercive formulation to simulate wave propagation in heterogenous media.

Ambient Vibration Testing of a Pedestrian Bridge Using Low-Cost Accelerometers for SHM Applications

Damage detection and structural health monitoring have always been of great importance to civil engineers and researchers. Vibration-based damage detection has several advantages compared to traditional methods of non-destructive evaluation, such as ground penetrating radar (GPR) or ultrasonic testing, since they give a global response and are feasible for large structures. Damage detection requires a comparison between two systems states, the baseline or “healthy state”, i.e., the initial modal parameters, and the damaged state. In this study, system identification (SI) was carried out on a pedestrian bridge by measuring the dynamic response using six low-cost triaxial accelerometers. These low-cost accelerometers use a micro-electro-mechanical system (MEMS), which is cheaper compared to a piezoelectric sensor. The frequency domain decomposition algorithm, which is an output-only method of modal analysis, was used to obtain the modal properties, i.e., natural frequencies and mode shapes. Three mode shapes and frequencies were found out using system identification and were compared with the finite element model (FEM) of the bridge, developed using the commercial finite element software, Abaqus. A good comparison was found between the FEM and SI results. The frequency difference was nearly 10%, and the modal assurance criterion (MAC) of experimental and analytical mode shapes was greater than 0.80, which proved to be a good comparison despite the small number of accelerometers available and the simplifications and idealizations in FEM.

Time-based dynamic load balancing algorithm for domain decomposition with particle method adopting three-dimensional polygon-wall boundary model

Time-based dynamic load balancing algorithm for domain decomposition with particle method adopting three-dimensional polygon-wall boundary model

Simulation of Challenging Electromagnetic Problems Using a Massively Parallel Finite Element Method Solver

This communication presents an efficient massively parallel finite element method solver for the solution of complex and electrically large electromagnetic problems with arbitrary structures. The solver makes use of a domain decomposition algorithm to decompose the original problem into several non-overlapping sub-domains that may be solved independently in parallel through the application of the corresponding transmission conditions on the interfaces of the adjacent sub-domains. A numerical exact mesh truncation algorithm called finite element-iterative integral equation evaluation, accelerated with multilevel fast multipole algorithm, is implemented to meet the highly accurate requirements of today's challenging simulations. What's more, a hybrid message passing interface and an open multi-processing parallel framework are designed to achieve large-scale parallel performance on supercomputers. Through several numerical examples, the accuracy, effectiveness, and scalability of the proposed solver will be demonstrated, achieving more than 60% parallel efficiency on an eight times CPU core scale (from 1280 to 10 240 cores).

Computation of volume potentials on structured grids with the method of local corrections

We present a new version of the Method of Local Corrections (MLC) \cite{mlc}, a multilevel, low communications, non-iterative, domain decomposition algorithm for the numerical solution of the free space Poisson's equation in 3D on locally-structured grids. In this method, the field is computed as a linear superposition of local fields induced by charges on rectangular patches of size $O(1)$ mesh points, with the global coupling represented by a coarse grid solution using a right-hand side computed from the local solutions. In the present method, the local convolutions are further decomposed into a short-range contribution computed by convolution with the discrete Green's function for an $Q^{th}$-order accurate finite difference approximation to the Laplacian with the full right-hand side on the patch, combined with a longer-range component that is the field induced by the terms up to order $P-1$ of the Legendre expansion of the charge over the patch. This leads to a method with a solution error that has an asymptotic bound of $O(h^P) + O(h^Q) + O(\epsilon h^2) + O(\epsilon)$, where $h$ is the mesh spacing, and $\epsilon$ is the max norm of the charge times a rapidly-decaying function of the radius of the support of the local solutions scaled by $h$. Thus we have eliminated the low-order accuracy of the original method (which corresponds to $P=1$ in the present method) for smooth solutions, while keeping the computational cost per patch nearly the same with that of the original method. Specifically, in addition to the local solves of the original method we only have to compute and communicate the expansion coefficients of local expansions (that is, for instance, 20 scalars per patch for $P=4$). Several numerical examples are presented to illustrate the new method and demonstrate its convergence properties.

A Multiscale Domain Decomposition Algorithm for Boundary Value Problems for Eikonal Equations

In this paper, we present a new multiscale domain decomposition algorithm for computing solutions of static Eikonal equations. In our new method, the decomposition of the domain does not depend on the slowness function in the Eikonal equation or the boundary conditions. The novelty of our new method is a coupling of coarse grid and fine grid solvers to propagate information along the characteristics of the equation efficiently. The method involves an iterative parareal-like update scheme in order to stabilize the method and speed up convergence. One can view the new method as a general framework where an effective coarse grid solver is computed “on the fly” from coarse and fine grid solutions that are computed in previous iterations. We study the optimal weights used to define the effective coarse grid solver and the stable update scheme via a model problem. To demonstrate the framework, we develop a specific scheme using Cartesian grids and the fast sweeping method for solving Eikonal equations. Numerical examples are given to show the method's effectiveness on Eikonal equations involving a variety of multiscale slowness functions.

A parallel domain decomposition algorithm for large scale image denoising

Total variation denoising (TVD) is an effective technique for image denoising, in particular, for recovering blocky, discontinuous images from noisy background. The problem is formulated as an optimization problem in the space of bounded variation functions, and the solution is obtained by solving the associated Euler–Lagrange equation defined on the domain occupied by the entire image. The method offers high quality results, but is computationally expensive for large images, especially for three-dimensional problems. In this paper, we introduce a highly parallel version of the algorithm which formulates the problem as multiple overlapping, but independent, optimization problems, and each is defined on a portion of the image domain. This approach is similar to the overlapping Schwarz type domain decomposition method, but is non-iterative, for solving partial differential equations, and is highly scalable, without using any coarse grids, for parallel computers with a large number of processors. We show by a theory and also by some two- and three-dimensional numerical experiments that the new approach has similar numerical accuracy as the classical TVD approach, but is much more efficient on parallel computers.

Explicit/implicit and Crank–Nicolson domain decomposition methods for parabolic partial differential equations

Two parallel non-overlapping domain decomposition algorithms for solving parabolic partial differential equations are proposed. The algorithms combine Crank–Nicolson scheme with implicit Galerkin finite element methods in sub-domains and explicit flux approximation along inner boundaries at each time step. Thus, parallelism can be easily achieved. L2-norm error estimates for these explicit/implicit procedures are presented, in which time step constraints are proved to be less severe than that of fully explicit schemes. Numerical experiments are also performed to verify the theoretical analysis.

A parallel implicit domain decomposition algorithm for the large eddy simulation of incompressible turbulent flows on 3D unstructured meshes

SummaryWe present a parallel fully implicit algorithm for the large eddy simulation (LES) of incompressible turbulent flows on unstructured meshes in three dimensions. The LES governing equations are discretized by a stabilized Galerkin finite element method in space and an implicit second‐order backward differentiation scheme in time. To efficiently solve the resulting large nonlinear systems, we present a highly parallel Newton‐Krylov‐Schwarz algorithm based on domain decomposition techniques. Analytic Jacobian is applied in order to obtain the best achievable performance. Two benchmark problems of lid‐driven cavity and flow passing a square cylinder are employed to validate the proposed algorithm. We then apply the algorithm to the LES of turbulent flows passing a full‐size high‐speed train with realistic geometry and operating conditions. The numerical results show that the algorithm is both accurate and efficient and exhibits a good scalability and parallel efficiency with tens of millions of degrees of freedom on a computer with up to 4096 processors. To understand the numerical behavior of the proposed fully implicit scheme, we study several important issues, including the choices of linear solvers, the overlapping size of the subdomains, and, especially, the accuracy of the Jacobian matrix. The results show that an exact Jacobian is necessary for the efficiency and the robustness of the proposed LES solver.

On a fixed point study of an inverse problem governed by Stokes equation

This work deals with domain decomposition like methods for solving an inverse Cauchy problem governed by Stokes equation. As it is well known, this problem is one of highly ill-posed problems in the Hadamard’s sense (Hadamard 1953 Lectures on Cauchy’s Problem in Linear Partial Differential Equations (New York: Dover)). To solve this problem, we develop a technique based on its reformulation into a fixed point one involving a Steklov like operator. Firstly, we show the existence of its fixed point using the topological degrees of Leray–Schauder tools. Then, we propose a fixed point algorithm allowing us to reproduce the alternating one of Kozlov Mazya (1991 Comput. Math. Math. Phys. 31 45–52). So the proposed approach can be considered as a generalization of Kozlov–Mazya algorithm, since it offers the opportunities to exploit other domain decomposition algorithms for solving this inverse problem. Finally, we investigate the numerical approximation of this problem, using Robin–Robin domain decomposition algorithm and the finite element method. The obtained numerical results show the efficiency of the proposed approach.

Parallel implicit matrix-free CFD solver using AMR grids

A novel parallel algorithm for the LU-SGS method with adaptive mesh refinement (AMR) is proposed. Domain decomposition and dynamic load balancing algorithms for spatial discretizations with AMR are described. For improving execution efficiency on targeted GPU-accelerated systems, corresponding coarsening/refining and memory defragmentation parallel algorithms are developed.

A space–time parallel framework for fine-scale visualization of pollen levels across the Eastern United States

ABSTRACTAllergic rhinitis (hay fever) resulting from seasonal pollen affects 15–30% of the population in the United States, and can exacerbate several related conditions, including asthma, atopic eczema, and allergic conjunctivitis. Timely monitoring, accurate prediction, and visualization of pollen levels are critical for public health prevention purposes, such as limiting outdoor exposure or physical activity. The low density of pollen detecting stations and complex movement of pollen represent a challenge for accurate prediction and modeling. In this paper, we reconstruct the dynamics of pollen variation across the Eastern United States for 2016 using space–time interpolation. Pollen levels were extracted according to a stratified spatial sampling design, augmented by additional samples in densely populated areas. These measurements were then used to estimate the space–time cross-correlation, inferring optimal spatial and temporal ranges to calibrate the space–time interpolation. Given the computational requirements of the interpolation algorithm, we implement a spatiotemporal domain decomposition algorithm, and use parallel computing to reduce the computational burden. We visualize our results in a 3D environment to identify the seasonal dynamics of pollen levels. Our approach is also portable to analyze other large space–time explicit datasets, such as air pollution, ash clouds, and precipitation.

Enhanced frequency domain decomposition algorithm: a review of a recent development for unbiased damping ratio estimates

Enhanced frequency domain decomposition (EFDD) is one of OMA methods and has received significant interest from the engineering community involved in the identification of the modal structure. The great attention towards this method is driven by its capability as a user-friendly and fast processing algorithm. However, this method has drawbacks in providing accurate identification of damping ratios, despite natural frequencies and mode shapes can be computed through assuredly and reasonably accurate estimates. The exact practical computation of modal damping is still an open issue, often leading to biased estimates since the errors are coming from every step in EFDD procedures and mainly due to signal processing. Thus, the computation of modal damping becomes tremendously vital in structural dynamics because modal damping is one of the critical parameters of resonance. This review aims to provide relevant essential information on modal damping for a reliable estimation, reduce uncertainties and define error bounds. A literature review has been carried out to find the best practice criteria for modal parameter identification, in particular, modal damping ratio.

OpenMP Parallelization Strategies for a Discontinuous Galerkin Solver

This paper aims to report on the open multi-processing (OpenMP) parallel implementation of a fully unstructured high-order discontinuous Galerkin (DG) solver for computational fluid dynamics and computational aeroacoustics applications. Even if the use of OpenMP paradigm is confined to shared memory systems, it has some advantages over the use of the message passing interface (MPI) library, and getting the best of this approach potentially improves the parallel efficiency of codes running on clusters of multi-core nodes. While with MPI the use of a domain decomposition algorithm is almost unavoidable, the OpenMP shared memory context offers several opportunities. Three strategies, here optimised for a DG solver, are presented and compared: the first refers to a customization of a colouring approach, the second mimics an MPI implementation in the OpenMP context, while the third method is somehow half way between the previous two. The numerical tests performed on both inviscid and viscous test cases indicate that, thanks to the compactness of the DG discretization, all the code versions perform quite satisfactory. In particular, the domain decomposition algorithm reaches the highest level of parallel efficiency at low computational loads while the colouring approach excels at larger computational loads and it can be easily implemented within an existing MPI code. Moreover, colouring is very well suited to deal with hardware accelerators, an opportunity given by the OpenMP 4.0 standard. Finally, the performance gain observed in using a hybrid MPI/OpenMP version of the DG code on high performance computing facilities is demonstrated.

Bi-directional coupling between a PDE-domain and an adjacent Data-domain equipped with multi-fidelity sensors

We consider a new prototype problem in domain decomposition with the solution in one domain governed by a known partial differential equation (PDE) whereas the solution in an adjacent domain is reconstructed by information gathered from distributed sensors (data) of variable fidelity. The PDE-domain and the Data-domain are tightly coupled, as the PDE solution is driven by the collected data, while the information gathered from its associated sensors is influenced by the PDE solution. Our overall methodology is based on the Schwarz alternating method and on recent advances in Gaussian process regression (GPR) using multi-fidelity data. The effectiveness of the proposed domain decomposition algorithm is demonstrated by examples of Helmholtz equations in both one-dimensional (1D) and two-dimensional (2D) domains.

A Posteriori Stopping Criteria for Optimized Schwarz Domain Decomposition Algorithms in Mixed Formulations

Abstract This paper develops a posteriori estimates for domain decomposition methods with optimized Robin transmission conditions on the interface between subdomains. We choose to demonstrate the methodology for mixed formulations, with a lowest-order Raviart–Thomas–Nédélec discretization, often used for heterogeneous and anisotropic porous media diffusion problems. Our estimators allow to distinguish the spatial discretization and the domain decomposition error components. We propose an adaptive domain decomposition algorithm wherein the iterations are stopped when the domain decomposition error does not affect significantly the overall error. Two main goals are thus achieved. First, a guaranteed bound on the overall error is obtained at each step of the domain decomposition algorithm. Second, important savings in terms of the number of domain decomposition iterations can be realized. Numerical experiments illustrate the efficiency of our estimates and the performance of the adaptive stopping criteria.

A domain decomposition method to couple nonisothermal compositional gas liquid Darcy and free gas flows

A domain decomposition algorithm is introduced to couple nonisothermal compositional gas liquid Darcy and free gas flow and transport. At each time step, our algorithm solves iteratively the nonlinear system coupling the nonisothermal compositional Darcy flow in the porous medium, the RANS gas flow in the free-flow domain, and the transport of the species and of energy in the free-flow domain. In order to speed up the convergence of the algorithm, the transmission conditions at the interface are replaced by Robin type boundary conditions. The Robin coefficients are obtained from a diagonal approximation of the Dirichlet to Neumann operator related to a simplified model in the neighboring subdomain. The efficiency of our domain decomposition algorithm is assessed on several test cases focusing on the modeling of the mass and energy exchanges at the interface between the geological formation and the ventilation galleries of geological radioactive waste disposal.

A Relaxation-Based Network Decomposition Algorithm for Parallel Transient Stability Simulation with Improved Convergence

Transient stability simulation of a large-scale and interconnected electric power system involves solving a large set of differential algebraic equations (DAEs) at every simulation time-step. With the ever-growing size and complexity of power grids, dynamic simulation becomes more time-consuming and computationally difficult using conventional sequential simulation techniques. To cope with this challenge, this paper aims to develop a fully distributed approach intended for implementation on High Performance Computer (HPC) clusters. A novel, relaxation-based domain decomposition algorithm known as Parallel-General-Norton with Multiple-port Equivalent (PGNME) is proposed as the core technique of a two-stage decomposition approach to divide the overall dynamic simulation problem into a set of subproblems that can be solved concurrently to exploit parallelism and scalability. While the convergence property has traditionally been a concern for relaxation-based decomposition, an estimation mechanism based on multiple-port network equivalent is adopted as the preconditioner to enhance the convergence of the proposed algorithm. The proposed algorithm is illustrated using rigorous mathematics and validated both in terms of speed-up and capability. Moreover, a complexity analysis is performed to support the observation that PGNME scales well when the size of the subproblems are sufficiently large.