Overlapping Domain Decomposition Approach Research Articles

Multilevel domain decomposition-based architectures for physics-informed neural networks

Physics-informed neural networks (PINNs) are a powerful approach for solving problems involving differential equations, yet they often struggle to solve problems with high frequency and/or multi-scale solutions. Finite basis physics-informed neural networks (FBPINNs) improve the performance of PINNs in this regime by combining them with an overlapping domain decomposition approach. In this work, FBPINNs are extended by adding multiple levels of domain decompositions to their solution ansatz, inspired by classical multilevel Schwarz domain decomposition methods (DDMs). Analogous to typical tests for classical DDMs, we assess how the accuracy of PINNs, FBPINNs and multilevel FBPINNs scale with respect to computational effort and solution complexity by carrying out strong and weak scaling tests. Our numerical results show that the proposed multilevel FBPINNs consistently and significantly outperform PINNs across a range of problems with high frequency and multi-scale solutions. Furthermore, as expected in classical DDMs, we show that multilevel FBPINNs improve the accuracy of FBPINNs when using large numbers of subdomains by aiding global communication between subdomains.

A parallel grad‐div stabilized finite element algorithm for the Navier–Stokes equations with a nonlinear damping term

AbstractIn this work, we propose a parallel grad‐div stabilized finite element algorithm for the Navier–Stokes equations attached with a nonlinear damping term, using a fully overlapping domain decomposition approach. In the proposed algorithm, we calculate a local solution in a defined subdomain on a global composite mesh which is fine around the defined subdomain and coarse in other regions. The algorithm is simple to carry out on the basis of available sequential solvers. By a local a priori estimate of the finite element solution, we deduce error bounds of the approximations from our presented algorithm. We perform also some numerical experiments to verify the effectiveness of the proposed algorithm.

A parallel finite element variational multiscale method for the Navier-Stokes equations with nonlinear slip boundary conditions

Based on a fully overlapping domain decomposition approach and a recent variational multiscale method, a parallel finite element variational multiscale method for the Navier-Stokes equations with nonlinear slip boundary conditions is proposed and analyzed. In this parallel method, a global composite grid is used to find a stabilized finite element solution for each subproblem, where a stabilization term based on two local Gauss integrations at the element level is employed to stabilize the system. Using the technical tool of local a priori estimate for the finite element solution, error estimates in H1-norm of velocity and L2-norm of pressure are derived. Numerical results are given to verify the validity of the theoretical predictions and illustrate the high efficiency of the proposed method.

Multidimensional coupling: A variationally consistent approach to fiber-reinforced materials

A novel mathematical model for fiber-reinforced materials is proposed. It is based on a 1-dimensional beam model for the thin fiber structures, a flexible and general 3-dimensional elasticity model for the matrix and an overlapping domain decomposition approach. From a computational point of view, this is motivated by the fact that matrix and fibers can be easily meshed independently. Our main interest is in fiber-reinforced polymers where the Young modulus is quite different. Thus the modeling error from the overlapping approach is of no significance. The coupling conditions acknowledge both, the forces and the moments of the beam model and transfer them to the background material. A suitable static condensation procedure is applied to remove the beam balance equations. The condensed system then forms our starting point for a numerical approximation in terms of isogeometric analysis. The choice of our discrete basis functions of higher regularity is motivated by the fact, that as a result of the static condensation, we obtain second gradient terms in fiber direction. Eventually, a series of benchmark tests demonstrate the flexibility and robustness of the proposed methodology. As a proof-of-concept, we show that our new model is able to capture bending, torsion and shear dominated situations.

A parallel stabilized finite element variational multiscale method based on fully overlapping domain decomposition for the incompressible Navier-Stokes equations

Based on a fully overlapping domain decomposition approach, a parallel stabilized finite element variational multiscale method for the incompressible Navier-Stokes equations is proposed, where the stabilizations both for the velocity and pressure are based on two local Gauss integrations at the element level. The basic idea of the method is to use a locally refined global mesh to compute a stabilized solution in the given subdomain of interest. The proposed method only requires the application of an existing Navier-Stokes sequential solver on the locally refined global mesh associated with each subdomain, and thus can reuse the existing sequential solver without substantial recoding. Error bound of the approximate solutions from the proposed method is estimated with the use of local a priori error estimate for the stabilized solution. Algorithmic parameter scalings of the method are also derived. Some numerical simulations are presented to demonstrate the effectiveness of the method.

Local and parallel stabilized finite element algorithms based on the lowest equal-order elements for the steady Navier–Stokes equations

Based on a fully overlapping domain decomposition approach, local and parallel stabilized finite element algorithms are proposed and investigated for the steady incompressible Navier–Stokes equations, where the inf−sup unstable lowest equal-order P1−P1 finite element pairs are used and the stabilized term is based on two local Gauss integrations defined by the difference between a consistent and under-integrated matrix of pressure interpolants. In these algorithms, each processor computes a local stabilized solution in its own subdomain using a global grid that is locally refined around its own subdomain, making the algorithms have low communication cost and easy to implement based on a sequential solver. Using the technical tool of the local a priori estimate for the stabilized solution, error bounds of the proposed algorithms are derived. Theoretical and numerical results show that, the algorithms can yield an approximate solution with an accuracy comparable to that of the standard stabilized finite element solution with a substantial decrease in CPU time.

A Two-Level Additive Schwarz Preconditioner for Local $$C^0$$ C 0 Discontinuous Galerkin Methods of Kirchhoff Plates

A two-level additive Schwarz preconditioner based on the overlapping domain decomposition approach is proposed for the local $$C^0$$ discontinuous Galerkin (LCDG) method of Kirchhoff plates. Then with the help of an intergrid transfer operator and its error estimates, it is proved that the condition number is bounded by $$O(1+(H^4/\delta ^4))$$ , where H is the diameter of the subdomains and $$\delta $$ measures the overlap among subdomains. And for some special cases of small overlap, the estimate can be improved as $$O(1+(H^3/\delta ^3))$$ . At last, some numerical results are reported to demonstrate the high efficiency of the two-level additive Schwarz preconditioner.

Algebraic Multigrid Domain and Range Decomposition (AMG-DD/AMG-RD)

In modern large-scale supercomputing applications, algebraic multigrid (AMG) is a leading choice for solving matrix equations. However, the high cost of communication relative to that of computation is a concern for the scalability of traditional implementations of AMG on emerging architectures. This paper introduces two new algebraic multilevel algorithms, algebraic multigrid domain decomposition (AMG-DD) and algebraic multigrid range decomposition (AMG-RD), that replace traditional AMG V-cycles with a fully overlapping domain decomposition approach. While the methods introduced here are similar in spirit to the geometric methods developed by Brandt and Diskin [Multigrid solvers on decomposed domains, in Domain Decomposition Methods in Science and Engineering, Contemp. Math. 157, AMS, Providence, RI, 1994, pp. 135--155], Mitchell [Electron. Trans. Numer. Anal., 6 (1997), pp. 224--233], and Bank and Holst [SIAM J. Sci. Comput., 22 (2000), pp. 1411--1443], they differ primarily in that they are purely algebraic: AMG-RD and AMG-DD trade communication for computation by forming global composite “grids” based only on the matrix, not the geometry. (As is the usual AMG convention, “grids” here should be taken only in the algebraic sense, regardless of whether or not it corresponds to any geometry.) Another important distinguishing feature of AMG-RD and AMG-DD is their novel residual communication process that enables effective parallel computation on composite grids, avoiding the all-to-all communication costs of the geometric methods. The main purpose of this paper is to study the potential of these two algebraic methods as possible alternatives to existing AMG approaches for future parallel machines. To this end, this paper develops some theoretical properties of these methods and reports on serial numerical tests of their convergence properties over a spectrum of problem parameters.

OpenGeoSys-Gem: A numerical tool for calculating geochemical and porosity changes in saturated and partially saturated media

Reactive transport codes that use a Gibbs Energy Minimization (GEM) to solve chemical equilibria are uncommon. We present a new coupling of the Richards flow module of the Finite Element (FE) based OpenGeoSys code with the GEM based chemical solver GEMS3K. The coupled code is highly parallelized using an overlapping domain decomposition approach in combination with execution of multiple threads that solve chemical equilibria in parallel. FE reactive transport schemes are often affected by spurious concentration oscillations. We effectively suppress these oscillations with a linearized algebraic flux corrected transport (FCT) algorithm. An application example is presented which investigates the evolution of material interfaces in a deep geological repository for nuclear waste. The example uses all features of the new coupled code: flow and multi-component transport in variably saturated media, and a very complex chemical setup which makes extensive use of (non-linear) solid solution formulations for mineral phases.

A DEM‐based parallel computing hydrodynamic and transport model

ABSTRACTThe sudden and accidental water pollution response system (SAWPRS) for Yangtze River in central China required to develop a hydrodynamic and transport model, which is readily available and capable of simulating a large river system within GIS environment. This study facilitates such effort by developing a parallel computing method based on digital elevation model (DEM) using overlapping domain decomposition approach (ODDA) and message passing interface (MPI) protocol. The hydrodynamic and transport model was redesigned using finite volume method for hydrodynamic and transport model dispersion, the SIMPLEC method for solving the flow field, and the pressure weighted interpolating method for the flow field modification. This modelling approach was verified in two experiments using different sets of computer clusters. The model output was evaluated against the measured data collected for the year 1998 for Wanzhou, an upstream river segment of Yangtze River. The relative error was found to be less than 10%. The performance of parallel computation was found excellent as evident from the cost efficiency values greater than 0.81 in both experiments and increased computation speed while increasing the number of computer clusters. Overall, the parallel computing modelling system developed here was found to meet all requirements of SAWPRS. Copyright © 2010 John Wiley & Sons, Ltd.

A continuum‐to‐atomistic bridging domain method for composite lattices

AbstractThe bridging domain method is an overlapping domain decomposition approach for coupling finite element continuum models and molecular mechanics models. In this method, the total energy is decomposed into atomistic and continuum parts by complementary weight functions applied to each part of the energy in the coupling domain. To enforce compatibility, the motions of the coupled atoms are constrained by the continuum displacement field using Lagrange multipliers. For composite lattices, this approach is suboptimal because the internal modes of the lattice are suppressed by the homogeneous continuum displacement field in the coupling region. To overcome this difficulty, we present a relaxed bridging domain method. In this method, the atom set is divided into primary and secondary atoms; the relative motions between them are often called the internal modes. Only the primary atoms are constrained in the coupling region, which succeed in allowing these internal modes to fully relax. Several one‐ and two‐dimensional examples are presented, which demonstrate improved accuracy over the standard bridging domain method. Copyright © 2009 John Wiley & Sons, Ltd.

Computation of tip-leakage flow in a linear compressor cascade with a second-moment turbulence closure

Strong interaction among tip-leakage vortex, mainstream, passage vortex and turbulence in and behind the blade tip and casing in compressors is one of the major causes of energy losses, affecting the overall gas turbine performance. A number of experiments and recent large-eddy simulations of simplified generic tip-clearance configurations have substantially advanced the understanding of the complex vortical and turbulence structures and energy conversion in and behind a very narrow clearance space. However, the prediction of phenomena in real turbomachinery still relies on the one-point (Reynolds-averaged Navier–Stokes, RANS) Computational Fluid Dynamics which is still burdened by uncertainties related primarily to inadequate turbulence modelling, and also to insufficient accuracy of the numerical algorithms. In this paper we address both these issues by testing an advanced turbulence model and a numerical method in the computations of a tip-clearance flow in a three-dimensional linear compressor cascade. For closing the mean momentum equations we adopted a Second Moment Turbulence Closure (SMC) incorporating low- Re-number and wall proximity effects, which has earlier proved to reproduce accurately a series of generic two- and three-dimensional anisotropic and non-equilibrium flows relevant to turbomachinery. In order to diminish numerical uncertainties, the computations are performed using Petrov-Galerkin stabilized finite element code featuring equal order Q1–Q1 pairs. The solution scheme is based on an in-house parallel Overlapping Domain Decomposition approach featuring a FGMRes linear solver. The results, compared with experiments and the conventional linear low- Re-number eddy viscosity model demonstrate that the SMC here adopted is capable to reproduce all the phenomenological features related to tip-leakage flows, especially the evolution of the strong three-dimensionality and turbulence anisotropy, and can be considered a prospective model for predicting real turbomachinery flows.

An overlapping domain decomposition approach for coupling the finite and boundary element methods

An overlapping iterative domain decomposition approach for the coupling of the finite element method (FEM) and the boundary element method (BEM) is presented in this paper. In this proposed method, the domain of the original problem is subdivided into a FEM sub-domain and a BEM sub-domain, such that the two sub-domains partially overlap over a common region. The common region is modeled by both methods. A brief discussion on the existing iterative coupling methods and their limitations are given in the first part of this paper. In the second part, the proposed overlapping method is described and the convergence conditions are presented. Two numerical examples are given to demonstrate the capability of the proposed method for handling cases where the Neumann boundary conditions are specified on the entire external boundary of the FEM or BEM sub-domains.

Efficient computation of interconnect capacitances using the domain decomposition approach

In this paper, we present a novel technique for efficient computation of capacitance matrices of complex interconnect configurations. It applies the finite difference (FD) method in conjunction with the perfectly matched layer (PML) and impedance boundary condition for mesh truncation, and combines these with the overlapping domain decomposition approach to handle complex configurations that are too large to handle in one step. Convergence and efficiency issues of the proposed algorithm are examined and numerical examples are presented to illustrate the usefulness of the proposed scheme.