Domain Decomposition Research Articles

A hybrid shifted Laplacian multigrid and domain decomposition preconditioner for the elastic Helmholtz equations

In this work we extend the shifted Laplacian approach to the elastic Helmholtz equation. The shifted Laplacian multigrid method is a common preconditioning approach for the discretized acoustic Helmholtz equation. In some cases, like geophysical seismic imaging, one needs to consider the elastic Helmholtz equation, which is harder to solve: it is three times larger and contains a nullity-rich grad-div term. These properties make the solution of the equation more difficult for multigrid solvers. The key idea in this work is combining the shifted Laplacian with approaches for linear elasticity. We provide local Fourier analysis and numerical evidence that the convergence rate of our method is independent of the Poisson's ratio. Moreover, to better handle the problem size, we complement our multigrid method with the domain decomposition approach, which works in synergy with the local nature of the shifted Laplacian, so we enjoy the advantages of both methods without sacrificing performance. We demonstrate the efficiency of our solver on 2D and 3D problems in heterogeneous media.

Improved benders-and-price algorithm for the multi-product assembly routing problem with time windows: A domain decomposition strategy for the benders-master model

To meet the optimization requirements of many-to-one assembly systems comprising a plant and multiple suppliers, the multi-product assembly routing problem with time windows (MPARPTW) is investigated in this study. Besides optimizing the assembly of different types of products in a plant to satisfy the periodic market demand within a specific planning horizon, MPARPTW factors in the vehicle routes for collecting many components from suppliers to meet the inventory requirements that are imposed by the production. The MPARPTW is formulated as a mixed-integer linear programming model to minimize the setup, production, transportation, and inventory costs. The model is solved using an improved Benders-and-Price algorithm that combines Benders decomposition and branch-and-price algorithm. The MPARPTW model is decomposed into a lot-sizing Benders-master (BM) model and a vehicle-routing Benders-sub (BS) model. A domain decomposition strategy is developed in this study to further decompose the BM model, thereby generating Benders cuts efficiently and increasing the convergence speed. Numerical experiments are designed to test the performance of the proposed algorithm. Within a limited maximum computing time (compared with the off-the-shelf solver), the improved algorithm reduces the average cost and computing time by 0.6% and 59.3%, respectively. The significance of the proposed model and algorithm in determining the plant investment directions is demonstrated. Moreover, compared with the conventional algorithm, the improved algorithm exhibits overall improvements. In large-scale instances, the cost is reduced by 1.8% on average and 4.2% maximum. The computing time yields average and maximum reductions of 35.2% and 76.0%, respectively.

Adaptive fail-safe topology optimization using a hierarchical parallelization scheme

This work presents an efficient, flexible, and scalable strategy to implement density-based topology optimization formulation for fail-safe structural design. Such optimized designs can operate after faults modeled as loss of stiffness. However, the need for assessing the physical behavior in all the fault cases can make this problem unfeasible computationally. We combine many ingredients to exploit the different levels of parallelism of the formulation to deal successfully with this problem. We use a non-overlapping domain decomposition method to solve the failure cases using multi-core computing with distributed memory computation. These subdomains use adaptive mesh refinement (AMR) techniques to focus computing efforts on the regions of interest. A key point to evaluate the fault cases is the organization of the computing threads, especially in clusters of computers. We group the computing threads by physical computing nodes to reduce the inter-node communications, maximizing intra-node communications to mitigate bandwidth problems. Another ingredient of paramount importance is the use of computing buffers to adapt the evaluation of the fault cases to the computing resources. We test the scalability of the computational framework using a large cluster of computers.

Scalability analysis of a two level domain decomposition approach in space and time solving data assimilation models

SummaryWe are concerned with the mapping on high performance hybrid architectures of a parallel software implementing a two level overlapping domain decomposition, that is, along space and time directions, of the four dimensional variational data assimilation model. The reference architecture belongs to the SCoPE (Sistema Cooperativo Per Elaborazioni scientifiche multidisciplinari) data center, located at University of Naples Federico II. We consider the initial boundary problem of the shallow water equation and analyse both strong and weak scaling. Keeping the efficiency always greater than and about in most cases, we experimentally find that the isoefficiency function grows a little more than linearly with respect to the number of processes. Results, obtained by using the parallel computing toolbox of MATLABR2013a, are in agreement with the algorithm's performance prevision based on the scale up factor, confirming the appropriate mapping of the algorithm on the hybrid architecture.

Multiscale sampling for the inverse modeling of partial differential equations

We are concerned with a novel Bayesian statistical framework for the characterization of natural subsurface formations, a very challenging task. Because of the large dimension of the stochastic space of the prior distribution in the framework, typically a dimensional reduction method, such as a Karhunen-Leove expansion (KLE), needs to be applied to the prior distribution to make the characterization computationally tractable. Due to the large variability of properties of subsurface formations (such as permeability and porosity) it may be of value to localize the sampling strategy so that it can better adapt to large local variability of rock properties.In this paper, we introduce the concept of multiscale sampling to localize the search in the stochastic space. We combine the simplicity of a preconditioned Markov Chain Monte Carlo method with a new algorithm to decompose the stochastic space into orthogonal subspaces, through a one-to-one mapping of the subspaces to subdomains of a non-overlapping domain decomposition of the region of interest. The localization of the search is performed by a multiscale blocking strategy within Gibbs sampling: we apply a KL expansion locally, at the subdomain level. Within each subdomain, blocking is applied again, for the sampling of the KLE random coefficients.The effectiveness of the proposed framework is tested in the solution of inverse problems related to elliptic partial differential equations arising in porous media flows. We use multi-chain studies in a multi-GPU cluster to show that the new algorithm clearly improves the convergence rate of the preconditioned MCMC method. Moreover, we illustrate the importance of a few conditioning points to further improve the convergence of the proposed method.

Seismo-VLAB: An Open-Source Software for Soil–Structure Interaction Analyses

In the fields of structural and geotechnical engineering, improving the understanding of soil–structure interaction (SSI) effects is critical for earthquake-resistant design. Engineers and practitioners often resort to finite element (FE) software to advance this objective. Unfortunately, the availability of software equipped with boundary representation for absorbing scattered waves and ensuring consistent input ground motion prescriptions, which is necessary for accurately representing SSI effects, is currently limited. To address such limitations, the authors developed Seismo-VLAB (SVL v1.0-stable) an open-source software designed to perform SSI simulations. The methodology considers the integration of advanced techniques, including the domain decomposition method (DDM), perfectly matched layers (PMLs), and domain reduction method (DRM), in addition to parallel computing capabilities to accelerate the solution of large-scale problems. In this work, the authors provide a detailed description of the implementation for addressing SSI modeling, validate some of the SVL’s features needed for such purpose, and demonstrate that the coupled DRM–PML technique is a necessary condition for accurately solving SSI problems. It is expected that SVL provides a significant contribution to the SSI research community, offering a self-contained and versatile alternative. The software’s practical application in analyzing SSI and directionality effects on 3D structures under seismic loading demonstrates its capability to model real-world earthquake responses in structural engineering.

Fast Non-overlapping Domain Decomposition Methods for Continuous Multi-phase Labeling Problem

Fast Non-overlapping Domain Decomposition Methods for Continuous Multi-phase Labeling Problem

Identification of damage in steel beam by natural frequency using ANNS models

Beams have played a significant role in engineering applications and they have been commonly used for modelling civil problems. In fact, different models and methods have been developed to identify the damage to the beams. In this paper, the artificial neural network (ANN) model was developed to predict the location, width and depth of the saw-cut of steel beams by the change of natural frequencies. The natural frequencies of a steel beam in different scenarios were identified by the Finite Element Method (FEM). In order to validate the accuracy of FEM model, the natural frequencies of the steel beam in the case of no saw-cut determined by this model were compared with those determined by the Frequency Domain Decomposition (FDD) method. The results indicated that, the combination of FEM method, FDD method and ANN model would have great significance in structure health monitoring.

Linear Time Electromigration Analysis Based on Physics-Informed Sparse Regression

In this work, we propose a novel physics-informed sparse regression (PISR) framework to solve stress evolution (described by Korhonen’s equations) in general multi-segment wires using an unsupervised learning scheme. Unlike the existing physics-informed neural network framework (PINN), the PISR method trains the trainable weights through the Moore-Penrose generalized inverse algorithm used in extreme learning machine (ELM), which is extremely faster than the back-propagation algorithm. To improve the accuracy of PISR for complex multi-segment interconnects, we employ domain decomposition schemes in both space and time. For each subdomain, we use different trainable weights but the same shared neural network to represent each subsolution, which leads to more efficient memory usage. Furthermore, we propose to use sparse matrix techniques to accelerate the training speed of the PISR method and prove that the resulting PISR has linear time complexity for analyzing tree-structured interconnects. Finally, we divide the time into many time internals and apply an autoregressive model to simulate each time interval in sequence to further improve scalability and reduce memory cost so that the PISR method can perform EM analysis for large-scale multi-segment interconnects. Experimental results on different kinds of interconnect structures show that the proposed PISR method has the same accuracy level as the numerical methods. The results on NT T-junctions interconnect trees show that the proposed PISR method indeed demonstrates true linear time complexity. Furthermore, PISR can deliver 8.9×, 20.6×, and 1284× speedups over the recently proposed semi-analytic method (ASOV), finite difference method accelerated with model order reduction (FDM-MOR), FDM for the interconnect with NT= 5000, respectively. Furthermore, we show that PISR also achieves an 818× speedup in training over the plain PINN method based on the traditional back-propagation algorithm.

Mutual Interference Mitigation for Automotive FMCW Radar With Time and Frequency Domain Decomposition

Currently, the frequency-modulated continuous-wave (FMCW) millimeter-wave (MMW) radar is a typical choice for automotive and transportation radar systems. As the number of FMCW radars explodes in the current vehicle market and the working frequency is limited in an open window of 76–81 GHz, FMCW radars on the road easily mutually interfere with each other, especially due to their wide bandwidth. Hence, in this article, we rigorously analyze the target echo, i.e., the beat signal, especially the sparsity of the interference signal in the time domain, and the row sparsity of the useful echo signal in the frequency domain. By taking advantage of this feature, we design an interference mitigation optimization problem to extract the target echoes with a row-sparse constraint. A closed-form solution is given in each iteration with specific derivations. Finally, numerical simulations and multiple practical scenes are provided to demonstrate the effectiveness of the proposed method.

Non-overlapping domain decomposition algorithms with only primal velocity unknowns for the discontinuous viscosity Stokes problem

In this paper, non-overlapping domain decomposition algorithms for the Stokes problem with a discontinuous viscosity coefficient are proposed and analyzed. A pair of inf–sup stable finite element spaces with discontinuous pressure finite element functions is employed to obtain a discrete Stokes system. By introducing a non-overlapping subdomain partition for the given finite element spaces, the discrete Stokes system is reformulated into an algebraic system on dual unknowns, where the continuity on the velocity unknowns at the subdomain vertices is enforced strongly and the continuity on the remaining interface velocity unknowns is enforced weakly using the dual unknowns. The resulting algebraic system on the dual unknowns is then solved by an iterative method. The coarse problem in the resulting algebraic system is only related to the strongly coupled primal velocity unknowns and it is thus obtained as a symmetric positive definite matrix equation. To accelerate the iteration convergence, practical versions of lumped and Dirichlet preconditioners are proposed and their condition numbers are also analyzed for the resulting dual algebraic system. Numerical results are also included to confirm our condition number estimates.

Learning adaptive coarse basis functions of FETI-DP

Domain decomposition methods are successful and highly parallel scalable iterative solution methods for discretized partial differential equations. Nevertheless, for many classes of problems, for example, elliptic partial differential equations with arbitrary coefficient distributions, adaptive coarse spaces are necessary to obtain robustness or, in other words, to guarantee a reliable and fast convergence. Adaptive coarse spaces are usually computed by solving many localized eigenvalue problems related to edges or faces of the domain decomposition. This results in a computationally expensive setup of the domain decomposition preconditioner or system operator. In earlier work, to which the authors have contributed, a deep learning based classification model was used to classify the critical edges or faces where eigenvalue problems have to be solved. In the present paper, we suggest to directly learn the adaptive constraints using a deep feedforward neural network regression model and thus completely skip the computationally most expensive part of the setup, i.e., the solution of local eigenvalue problems. We consider a specific adaptive FETI-DP (Finite Tearing and Interconnecting – Dual Primal) approach and concentrate on stationary diffusion problems in two dimensions with arbitrary coefficient functions with large jumps. As an input for the neural network, we use an image representation of the coefficient function which resolves the structure of the coefficient distribution but is not necessarily identical to the discretization of the partial differential equation. Therefore, our approach is independent of the finite element mesh and can, in principle, be easily extended to other adaptive coarse spaces, problems, and domain decomposition methods. We show the robustness of our method for different problems and the generalization property of our trained neural networks by considering different coefficient distributions not contained in the training set. We also combine the learned constraints with computationally cheap frugal constraints to further improve our approach.

Physics-informed neural network-based petroleum reservoir simulation with sparse data using domain decomposition

Recent advances in deep learning have expanded new possibilities for fluid flow simulation in petroleum reservoirs. However, the predominant approach in existing research is to train neural networks using high-fidelity numerical simulation data. This presents a significant challenge because the sole source of authentic wellbore production data for training is sparse. In response to this challenge, this work introduces a novel architecture called physics-informed neural network based on domain decomposition (PINN-DD), aiming to effectively utilize the sparse production data of wells for reservoir simulation with large-scale systems. To harness the capabilities of physics-informed neural networks (PINNs) in handling small-scale spatial-temporal domain while addressing the challenges of large-scale systems with sparse labeled data, the computational domain is divided into two distinct sub-domains: the well-containing and the well-free sub-domain. Moreover, the two sub-domains and the interface are rigorously constrained by the governing equations, data matching, and boundary conditions. The accuracy of the proposed method is evaluated on two problems, and its performance is compared against state-of-the-art PINNs through numerical analysis as a benchmark. The results demonstrate the superiority of PINN-DD in handling large-scale reservoir simulation with limited data and show its potential to outperform conventional PINNs in such scenarios.

Solution of Multi-Dimensional Non-linear Fractional Differential Equations of Higher Orders

In our paper, we are used here two methods to solve non-linear differential equations from a higher order: the first-one is domain decomposition method is used to estimate the Maxi. Abso. Trunc. Error of Adomain series and the second-one proposed numerical (PNM), these types of equations are studied. When we use these methods, an exclusive solution will be provided, and the approximate analyses of this method applied to these types of equations will be overlooked, and the maximum error that has been informed to solve the ADOMIANS series will be classified. A digital example is prepared clarify the impact method provided and significant following of these equations in our paper is Bagley-Torvik equation.

A scalable domain decomposition method for FEM discretizations of nonlocal equations of integrable and fractional type

Nonlocal models allow for the description of phenomena which cannot be captured by classical partial differential equations. The availability of efficient solvers is one of the main concerns for the use of nonlocal models in real world engineering applications. We present a domain decomposition solver that is inspired by substructuring methods for classical local equations. In numerical experiments involving finite element discretizations of scalar and vectorial nonlocal equations of integrable and fractional type, we observe improvements in solution time of up to 14.6x compared to commonly used solver strategies.

A domain decomposition technique for acoustic wave interactions and scattering

In this work, a two-dimensional domain decomposition technique (DDT) is developed based on an overset method that couples a nonlinear Euler solver with a linearized Euler solver. The nonlinear domain accounts for complex physics in the near field, such as acoustic wave interaction with shock waves and acoustic wave scattering from solid bodies. The linear domain allows the small perturbation waves induced by the nonlinear effects to propagate with minimal numerical dissipation and dispersion while also significantly reducing computational resources. To this end, two scenarios have been considered: (i) wave amplification and modulation in a converging-diverging duct with a standing shock wave and (ii) wave scattering from an incident acoustic wave over a supercritical airfoil. With DDT, these cases show a reduction in computational time by more than 70% compared to their respective fully-nonlinear simulations. The accuracy of the simulations with DDT is found to depend on the extent of overlap between the linear and nonlinear zones. In general, an overlap ratio of at least two is necessary to maintain high order of convergence rate, though still limited by the nonlinear scheme. Furthermore, the implementation of DDT in the external flow case can eliminate numerical noise observed at the far field of the fully-nonlinear counterpart, which do not have any effective non-reflecting boundary treatment.

A Hybrid Large Eddy Simulation Algorithm Based on the Implicit Domain Decomposition

The resolution of small near-wall eddies encountered in high-Reynolds number flows using large eddy simulation (LES) requires very fine meshes that may be computationally prohibitive. As a result, the use of wall-modeled LES as an alternative is becoming more popular. In this paper, the near-wall domain decomposition (NDD) approach that was originally developed for Reynolds-averaged Navier–Stokes simulations (RANSs) is extended to the hybrid RANS/LES zonal decomposition. The algorithm is implemented in two stages. First, the solution is computed everywhere with LES on a coarse grid using a new non-local slip boundary condition for the instantaneous velocity at the wall. The solution is then recomputed in the near-wall region with RANS. The slip boundary conditions used in the first stage guarantee that the composite solution is smooth at the inner/outer region interface. Another advantage of the model is that the turbulent viscosity in the inner region is computed based on the corresponding RANS velocity. This shows improvement over those hybrid models that have only one velocity field in the whole domain obtained from LES. The model is realized in the open source code OpenFOAM with different approximations of turbulent viscosity and is applied to the planar channel flow at frictional Reynolds numbers of Reτ=950, 2000, and 4200. Mean streamwise velocity and Reynolds stress intensities are predicted reasonably well in comparison to the solutions obtained with unresolved LES and available DNS benchmarks. No additional forcing at the interface is required, while the log–layer mismatch is essentially reduced in all cases.

Overlapping multiplicative Schwarz preconditioning for linear and nonlinear systems

For linear and nonlinear systems arising from the discretization of PDEs, multiplicative Schwarz preconditioners can be defined based on subsets of the unknowns that derive from domain decomposition, field splitting, or other collections of conveniently solved subproblems, and are well established theoretically for nonoverlapping subsets. For overlapping subsets, establishing the equivalence of the preconditioned and original iterations is less trivial. We derive herein an explicit formulation for a variety of multiplicative Schwarz preconditioners including overlaps representative of interfacial and bulk coupling in multiphysics systems, thus extending theoretical support for the nonlinear multiplicative Schwarz preconditioned inexact Newton (MSPIN) algorithm to these classes. For nonlinear multiplicative Schwarz preconditioners with overlaps, we illustrate the performance through numerical experiments involving applications such as a shocked duct flow and a natural convection cavity flow. We begin with a broad introduction to nonlinear preconditioning to set the context for those new to the technique.

Stochastic domain decomposition based on variable-separation method

This work proposes a stochastic domain decomposition method for solving steady-state partial differential equations (PDEs) with random inputs. Specifically, based on the efficiency of the variable-separation (VS) method in simulating stochastic partial differential equations (SPDEs), we extend it to stochastic algebraic systems and employ it in the context of stochastic domain decomposition. The resulting method, termed stochastic domain decomposition based on variable-separation (SDD-VS) alleviates the challenge commonly known as the “curse of dimensionality” notably by leveraging explicit representations of stochastic functions derived from physical systems. The primary objective of the proposed SDD-VS method is to obtain a separate representation of the solution for the stochastic interface problem. To enhance computational efficiency, we introduce a two-phase approach consisting of offline and online computation. In the offline phase, we establish an affine representation of stochastic algebraic systems by systematically applying the VS method. During the online phase, we estimate the interface unknowns of SPDEs using a quasi-optimal separated representation, facilitating the construction of efficient surrogate models for subproblems. We substantiate the effectiveness of our proposed approach through numerical experiments involving three specific instances, demonstrating its capability to provide accurate solutions.

Asynchronous time integration while achieving zero interface energy

This contribution deals with an asynchronous direct time integration of the finite-element model. The proposed method is applied to the phenomenon of wave propagation through an elastic linear continuum. The numerical model is partitioned into individual subdomains using the domain decomposition method by means of localized Lagrange multipliers. For each subdomain, different time discretizations are used. No restrictions for relation between subdomain’s time steps are imposed. The coupling of the subdomains is forced by an acceleration continuity condition. Additionally, we use the a posteriori technique to also provide the displacement and velocity continuity at the interfaces, and hence we obtain exact continuity of all three kinematic fields. The proposed method is experimentally validated using the modified SHPB (split Hopkinson pressure bar) setup.