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Based on domain decomposition, a semi-analytical method (SAM) is applied to analyze the free vibration of double-curved shells of revolution with a general curvature radius made from graphene nanoplatelet (GPL)-reinforced porous composites. The mechanical properties of the GPL-reinforced composition are assessed with the Halpin–Tsai model. The double-curvature shell of revolution is broken down into segments along its axis in accordance with first-order shear deformation theory (FSDT) and the multi-segment partitioning technique, to derive the shell's functional energy. At the same time, interfacial potential is used to ensure the continuity of the contact surface between neighboring segments. By applying the least-squares weighted residual method (LWRM) and modified variational principle (MVP) to relax and achieve interface compatibility conditions, a theoretical framework for analyzing vibrations is developed. The displacements and rotations are described through Fourier series and Chebyshev polynomials, accordingly, converting a two-dimensional issue into a suite of decoupled one-dimensional problems. The obtained solutions are contrasted with those achieved using the finite element method (FEM) and other existing results, and the current formulation's validity and precision are confirmed. Example cases are presented to demonstrate the free vibration of GPL-reinforced porous composite double-curved paraboloidal, elliptical, and hyperbolical shells of revolution. The findings demonstrate that the natural frequency of the shell is related to pore coefficients, porosity, the mass fraction of the GPLs, and the distribution patterns of the GPLs. [ABSTRACT FROM AUTHOR]

To enhance solution accuracy and training efficiency in neural network approximation to partial differential equations, partitioned neural networks can be used as a solution surrogate instead of a single large and deep neural network defined on the whole problem domain. In such a partitioned neural network approach, suitable interface conditions or subdomain boundary conditions are combined to obtain a convergent approximate solution. However, there has been no rigorous study on the convergence and parallel computing enhancement on the partitioned neural network approach. In this paper, iterative algorithms are proposed to enhance parallel computation performance in the partitioned neural network approximation. Our iterative algorithms are based on classical additive Schwarz domain decomposition methods. For the proposed iterative algorithms, their convergence is analyzed under an error assumption on the local and coarse neural network solutions. Numerical results are also included to show the performance of the proposed iterative algorithms. [ABSTRACT FROM AUTHOR]

The phase field modeling of fracture is able to simulate the nucleation and the propagation of complex crack patterns. However, the relatively small internal lengths that are required usually lead to very fine meshes and high computational costs, especially for three‐dimensional applications. In the present work, additional cost also comes from the implicit dynamics regularization of unstable crack propagations which potentially leads to a large variation of time steps when switching from quasi‐static to dynamic regimes. To reduce the time to solution in this context, this study proposes a domain decomposition framework and acceleration techniques for the phase field fracture staggered solver. The displacement subproblem and the phase field one are solved with parallel domain decomposition solvers. Dual domain decomposition methods provide low cost preconditioner well adapted to the phase field subproblem. For displacement subproblems undergoing unstable crack propagations, primal domain decomposition methods are preferred to be less sensitive to the treatment of floating substructures. Preconditioners performances are assessed and scalability studies over academic test cases, up to 324 subdomains, are presented. Finally, the robustness of the approach is illustrated on two semi‐industrial simulations. [ABSTRACT FROM AUTHOR]

Considering positive H (curl , Ω) problems, we propose a substructured version of the Hiptmair-Xu preconditioner based on a new formula that expresses the inverse of Schur systems in terms of the inverse matrix of the global volume problem. [ABSTRACT FROM AUTHOR]

Copyright of Engineering Mechanics / Gongcheng Lixue is the property of Engineering Mechanics Editorial Department and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This paper presents an electromagnetic finite element analysis method using domain decomposition based on the Finite Element Tearing and Interconnecting (FETI) method. This method specifically addresses the extended FETI techniques of FETI-dual, FETI-DP, and FETI-H. The accuracy of the solutions obtained from each method was validated using the electromagnetic scattering field and the Radar Cross Section (RCS) of a PEC cube. It was demonstrated that the FETI-H method, which employs a First-Order Transmission Condition (FOTC) at the subdomain boundaries, exhibits improved convergence and computation speed. [ABSTRACT FROM AUTHOR]

Although over 300 million protein sequences are registered in a reference sequence database, only 0.2 % have experimentally determined functions. This suggests that many valuable proteins, potentially catalyzing novel enzymatic reactions, remain undiscovered among the vast number of function-unknown proteins. In this study, we developed a method to predict whether two proteins catalyze the same enzymatic reaction by analyzing sequence and structural similarities, utilizing structural models predicted by AlphaFold2. We performed pocket detection and domain decomposition for each structural model. The similarity between protein pairs was assessed using features such as full-length sequence similarity, domain structural similarity, and pocket similarity. We developed several models using conventional machine learning algorithms and found that the LightGBM-based model outperformed the models. Our method also surpassed existing approaches, including those based solely on full-length sequence similarity and state-of-the-art deep learning models. Feature importance analysis revealed that domain sequence identity, calculated through structural alignment, had the greatest influence on the prediction. Therefore, our findings demonstrate that integrating sequence and structural information improves the accuracy of protein function prediction.

In order to balance the accuracy and efficiency of neutron transport simulation, performing self-adaptive mesh refinement and domain decomposition parallelization for unstructured meshes are technologies that worth exploring. For the neutron flux distribution of each energy group generally has significant difference with distributions of other groups, using a common mesh in the simulation for all energy groups cannot best match the flux spatial profiles of every group. In that case, the common mesh has to be excessively globally refined, or the accuracy is expected to lose some extent. An energy-group wise local mesh refinement scheme for multi-group neutron transport simulation was adopted in this paper, which was based on independent posterior error estimations for the neutron flux of all energy groups. The obtained unstructured fine meshes were stored separately in multiple containers, they are different on the active top level, but share a common coarse mesh on the base level where the fine meshes are derived from. These multi-sets of hierarchical meshes were named base-derived meshes hereafter. Obviously, these base-derived meshes can best match the flux distribution of each energy group respectively, but the data transfer and interpolation between groups for calculation of inter-group fission and scattering sources come to be the key issues. A couple algorithm for multi-group transport simulation on such group-wise refined base-derived meshes was introduced. It achieves high accuracy and efficiency by successive adaptive refinement cycles, and finally reaches the mesh insensitive state automatically regardless the resolution of initial coarse base mesh. The transport code ENTER-Ⅱ was developed based on this energy-group wise self-adaptive (GWA) refinement algorithm, and the discontinuous finite element method was applied to enable the flexible modeling and meshing of complex geometry. Domain decomposition based parallel computing capability was realized with the help of the open-source library DEAL.Ⅱ. Preliminary verifications to the group-wise self-adaptive refinement algorithm were presented, which show that the simulation accuracy is acceptable, and the time efficiency is remarkably better. The time costs in solving angular flux on the successively refined meshes are quite low, due to the good estimations of initial values of flux that are transferred from the solutions on the coarser meshes and then interpolated on the fine meshes. It concludes that, with the help of energy-group wise self-adaptive refinement, the initial meshing of geometry domain does not need to rely on expertise, and the high-fidelity simulation efficiency raises effectively.

This study revisits the freely vibrating problem of asymmetric L-shaped membranes using a three-segmented domain decomposition (3-SDD) strategy. Motivated by the need for more accurate and flexible methods, the 3-SDD strategy is compared with the previously proposed two-segmented domain decomposition (2-SDD) strategy. The region-matching technique is used to derive an alternative series solution, and the eigenvalues obtained are compared with those in existing research. The convergence behavior and sensitivity analyses reveal that the 3-SDD strategy offers improved accuracy and stability, particularly for higher truncation terms. Detailed comparisons of the first four eigenvalue squares show strong agreement between the 3-SDD and 2-SDD strategies, confirming the reliability of both methods. This research establishes a foundation for the vibration analysis of complex membrane structures, emphasizing the benefits of the 3-SDD approach for upcoming engineering applications and showcasing its potential for broader applicability in practical scenarios. The findings underscore the importance of utilizing multi-segmented decomposition strategies to enhance the accuracy and flexibility of free vibration analysis. [ABSTRACT FROM AUTHOR]

The efficiency of numerical solvers of PDEs depends on the approximation properties of the discretization methods and the conditioning of the resulting linear systems. If applicable, the boundary element methods typically provide better approximation with unknowns limited to the boundary than the Schur complement of the finite element stiffness matrix with respect to the interior variables. Since both matrices correctly approximate the same object, the Steklov-Poincaré operator, it is natural to assume that the matrices corresponding to the same fine boundary discretization are similar. However, this note shows that the distribution of the spectrum of the boundary element stiffness matrix is significantly better conditioned than the finite element Schur complement. The effect of the favorable conditioning of BETI clusters is demonstrated by solving huge problems by H-TBETI-DP and H-TFETI-DP. [ABSTRACT FROM AUTHOR]

This article presents a comprehensive study on the convergence behavior of the Dirichlet-Neumann Waveform Relaxation algorithm applied to solve the time fractional sub-diffusion and diffusion-wave equations in multiple subdomains, considering the presence of some heterogeneous media. Our analysis focuses on estimating the convergence rate of the algorithm and investigates how this estimate varies with different fractional orders. Furthermore, we extend our analysis to encompass the 2D sub-diffusion case. To validate our findings, we conduct numerical experiments to verify the estimated convergence rate. The results confirm the theoretical estimates and provide empirical evidence for the algorithm’s efficiency and reliability. Moreover, we push the boundaries of the algorithm’s applicability by extending it to solve the time fractional Allen-Chan equation, a problem that exceeds our initial theoretical estimates. Remarkably, we observe that the algorithm performs exceptionally well in this extended scenario for both short and long-time windows. [ABSTRACT FROM AUTHOR]

This article presents a parallel finite element discretization scheme for solving numerically the steady natural convection equations, where a fully overlapping domain decomposition technique is used for parallelization. In this scheme, each processor computes independently a local solution in its subdomain using a mesh that covers the entire domain. It has a small mesh size h around the subdomain and a large mesh size H away from the subdomain. The discretization scheme is easy to implement based on existing serial software. It can yield an optimal convergence rate for the approximate solutions with suitable algorithmic parameters. Compared with the standard finite element method, the scheme is able to obtain an approximate solution of comparable accuracy with considerable reduction in computational time. Theoretical and numerical results show the promise of the scheme, where numerical simulation results for some benchmark problems such as the buoyancy-driven square cavity flow, right-angled triangular cavity flow and sinusoidal hot cylinder flow are provided. [ABSTRACT FROM AUTHOR]

In this paper, we develop a new splitting characteristic finite difference scheme for solving the time-fractional mobile-immobile nonlinear advection–diffusion equation by combining non-overlapping block-divided domain decomposition method, the operator splitting technique and the characteristic finite difference method. Over each sub-domain, the solutions and fluxes along x-direction in the interiors of sub-domains are computed by the implicit characteristic finite difference method while the intermediate fluxes on the interfaces of sub-domains are computed by local multi-point weighted average from the approximate solutions at characteristic tracking points which are solved by the quadratic interpolation. Secondly, the solutions and fluxes along y direction in the interiors of sub-domains are computed lastly by the implicit characteristic difference method while the time fractional derivative is approximated by L1-format and the intermediate fluxes on the interfaces of sub-domains are computed by local multi-point weighted average from the approximate solutions at characteristic tracking points are solved by the quadratic interpolation. Applying Brouwer fixed point theorem, we prove strictly the existence and uniqueness of the proposed scheme. The conditional stability and convergence with O Δ t + Δ t 2 - α + h 2 + H 5 2 of the proposed scheme are given as well. Numerical experiments verify the theoretical results. [ABSTRACT FROM AUTHOR]

We construct and analyze a multiscale finite element method for an elliptic distributed optimal control problem with pointwise control constraints, where the state equation has rough coefficients. We show that the performance of the multiscale finite element method is similar to the performance of standard finite element methods for smooth problems and present corroborating numerical results. [ABSTRACT FROM AUTHOR]

In this work, we address parametric non–stationary fluid dynamics problems within a model order reduction setting based on domain decomposition. Starting from the optimisation–based domain decomposition approach, we derive an optimal control problem, for which we present a convergence analysis in the case of non–stationary incompressible Navier–Stokes equations. We discretise the problem with the finite element method and we compare different model order reduction techniques: POD–Galerkin and a non–intrusive neural network procedures. We show that the classical POD–Galerkin is more robust and accurate also in transient areas, while the neural network can obtain simulations very quickly though being less precise in the presence of discontinuities in time or parameter domain. We test the proposed methodologies on two fluid dynamics benchmarks with physical parameters and time dependency: the non–stationary backward–facing step and lid–driven cavity flow. • Optimisation-based domain-decomposition algorithm for non-stationary computational fluid dynamics. • Convergence analysis for the domain-decomposition optimal control problem. • Comparison of the intrusive POD-Galerkin and a non-intrusive Neural Networks based approaches. [ABSTRACT FROM AUTHOR]

. This work proposes a discretization of the acoustic wave equation with possibly oscillatory coefficients based on a superposition of discrete solutions to spatially localized subproblems computed with an implicit time discretization. Based on exponentially decaying entries of the global system matrices and an appropriate partition of unity, it is proved that the superposition of localized solutions is appropriately close to the solution of the (global) implicit scheme. It is thereby justified that the localized (and especially parallel) computation on multiple overlapping subdomains is reasonable. Moreover, a restart is introduced after a certain number of time steps to maintain a moderate overlap of the subdomains. Overall, the approach may be understood as a domain decomposition strategy in space on successive short time intervals that completely avoids inner iterations. Numerical examples are presented. [ABSTRACT FROM AUTHOR]

For the Poisson equation posed in a domain containing a large number of polygonal perforations, we propose a low-dimensional coarse approximation space based on a coarse polygonal partitioning of the domain. Similarly to other multiscale numerical methods, this coarse space is spanned by locally discrete harmonic basis functions. Along the subdomain boundaries, the basis functions are piecewise polynomial. The main contribution of this article is an error estimate regarding the H 1 -projection over the coarse space; this error estimate depends only on the regularity of the solution over the edges of the coarse partitioning. For a specific edge refinement procedure, the error analysis establishes superconvergence of the method even if the true solution has a low general regularity. Additionally, this contribution numerically explores the combination of the coarse space with domain decomposition (DD) methods. This combination leads to an efficient two-level iterative linear solver which reaches the fine-scale finite element error in few iterations. It also bodes well as a preconditioner for Krylov methods and provides scalability with respect to the number of subdomains. [ABSTRACT FROM AUTHOR]

Copyright of Atomic Energy Science & Technology is the property of Editorial Board of Atomic Energy Science & Technology and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The high-resolution 3D groundwater flow and transport simulation problem requires massive discrete linear systems to be solved, leading to significant computational time and memory requirements. The domain decomposition method is a promising technique that facilitates the parallelization of problems with minimal communication overhead by dividing the computation domain into multiple subdomains. However, directly utilizing a domain decomposition scheme to solve massive linear systems becomes impractical due to the bottleneck in algebraic operations required to coordinate the results of subdomains. In this paper, we propose a two-level domain decomposition method, named dual-domain decomposition, to efficiently solve the massive discrete linear systems in high-resolution 3D groundwater simulations. The first level of domain decomposition partitions the linear system problem into independent linear sub-problems across multiple subdomains, enabling parallel solutions with significantly reduced complexity. The second level introduces a domain decomposition preconditioner to solve the linear system, known as the Schur system, used to coordinate results from subdomains across their boundaries. This additional level of decomposition parallelizes the preconditioning of the Schur system, addressing the bottleneck of the Schur system solution while improving its convergence rates. The dual-domain decomposition method facilitates the partition and distribution of the computation to be solved into independent finely grained computational subdomains, substantially reducing both computational and memory complexity. We demonstrate the scalability of our proposed method through its application to a high-resolution 3D simulation of chromium contaminant transport in groundwater. Our results indicate that our method outperforms both the vanilla domain decomposition method and the algebraic multigrid preconditioned method in terms of runtime, achieving up to 8.617× and 5.515× speedups, respectively, in solving massive problems with approximately 108 million degrees of freedom. Therefore, we recommend its effectiveness and reliability for high-resolution 3D simulations of groundwater flow and transport. [ABSTRACT FROM AUTHOR]

Aluminum honeycomb sandwich structure with panels made of carbon fiber reinforced polymer (CFRP) are widely used in aerospace and other fields. Simulation of the eddy current (EC) testing of the sandwich structure using the finite element (FE) method is challenging as the traditional FE method has difficulties in mesh division and the solution of the algebraic equations. This paper proposes to use the domain decomposition FE method to solve such problems. The top CFRP panel, the aluminum honeycomb core, and the bottom CFRP panel of the sandwich structure and the ferrite core of the coil are placed in different subdomains and the subdomains are meshed independently. This method simplifies the mesh generation and does not require regenerating the meshes when simulating the scanning testing with the ferrite-core coil. In this way, the efficiency of simulation is greatly improved. The EC distributions in the sandwich structure are computed and the influence of defect on EC distribution is analyzed. The C scans of the sandwich structures are simulated. The images of the EC responses to the defects, such as wall fracture, node disconnection, and core wrinkle, are obtained. The simulation results are validated by experiments. [ABSTRACT FROM AUTHOR]

This paper explores the potential of Physics-Informed Neural Networks (PINNs) to serve as Reduced Order Models (ROMs) for simulating the flow field within stirred tank reactors (STRs). We solve the two-dimensional stationary Navier-Stokes equations within a geometrically intricate domain and explore methodologies that allow us to integrate additional physical insights into the model. These approaches include imposing the Dirichlet boundary conditions (BCs) strongly and employing domain decomposition (DD), with both overlapping and non-overlapping subdomains. We adapt the Extended Physics-Informed Neural Network (XPINN) approach to solve different sets of equations in distinct subdomains based on the diverse flow characteristics present in each region. Our exploration results in a hierarchy of models spanning various levels of complexity, where the best models exhibit $ \ell_1 $ prediction errors of less than 1% for both pressure and velocity. To illustrate the reproducibility of our approach, we track the errors over repeated independent training runs of the best identified model and show its reliability. Subsequently, by incorporating the stirring rate as a parametric input, we develop a fast-to-evaluate model of the flow capable of interpolating across a wide range of Reynolds numbers. Although we exclusively restrict ourselves to STRs in this work, we conclude that the steps taken to obtain the presented model hierarchy can be transferred to other applications. [ABSTRACT FROM AUTHOR]

Predicting computational intensity (CI) is essential for domain decomposition and load balance in parallel geoprocessing. However, traditional CI prediction is limited in capturing the heterogeneity of spatial domain, leading to poor accuracy and load imbalance. Leveraging recent advancements in deep learning from Artificial Intelligence (AI), this paper proposes a deep learning-based approach for predicting CI and enhancing domain decomposition, which reduces the dependency on expert knowledge through automatic feature learning. In the approach, Convolutional Neural Networks are employed to capture the heterogeneity of spatial domain, encompassing structural, distribution, and topological characteristics. A fully connected layer is then utilized for CI prediction and optimized domain decomposition. Comparative experiments were implemented between the proposed approach and three traditional methods, using two cases: spatial intersection on vector data and peak perilousness assessment. The results demonstrate that the proposed approach achieves a speedup ratio of 19.8 and a parallel efficiency of 0.82. The findings highlight the advantages of the proposed approach in terms of parallel performance and usability. This study serves as a valuable reference for illustrating how deep learning can enhance parallel geoprocessing, providing a roadmap for applying deep learning techniques to geocomputations and fostering further advancements of AI GIS.

AbstractSpatial heterogeneity is of great importance in social, economic, and environmental science studies. The spatially varying coefficient model is a popular and effective spatial regression technique to address spatial heterogeneity. However, accounting for heterogeneity comes at the cost of reducing model parsimony. To balance flexibility and parsimony, this article develops a class of generalized partially linear spatially varying coefficient models which allow the inclusion of both constant and spatially varying effects of covariates. Another significant challenge in many applications comes from the enormous size of the spatial datasets collected from modern technologies. To tackle this challenge, we design a novel distributed heterogeneity learning (DHL) method based on bivariate spline smoothing over a triangulation of the domain. The proposed DHL algorithm has a simple, scalable, and communication-efficient implementation scheme that can almost achieve linear speedup. In addition, this article provides rigorous theoretical support for the DHL framework. We prove that the DHL constant coefficient estimators are asymptotic normal and the DHL spline estimators reach the same convergence rate as the global spline estimators obtained using the entire dataset. The proposed DHL method is evaluated through extensive simulation studies and analyses of U.S. loan application data. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work. [ABSTRACT FROM AUTHOR]

In this paper, we consider a semiconducting device with an active zone made of a single-layer material. The associated Poisson equation for the electrostatic potential (to be solved in order to perform self-consistent computations) is characterized by a surface particle density and an out-of-plane dielectric permittivity in the region surrounding the single-layer. To avoid mesh refinements in such a region, we propose an interface problem based on the natural domain decomposition suggested by the physical device. Two different interface continuity conditions are discussed. Then, we write the corresponding variational formulations adapting the so called three-fields formulation for domain decomposition and we approximate them using a proper finite element method. Finally, numerical experiments are performed to illustrate some specific features of this interface approach. [ABSTRACT FROM AUTHOR]

This paper proposes an adaptive space-time algorithm based on domain decomposition for the large-scale simulation of a recently developed thermodynamically consistent reservoir problem. In the approach, the bound constraints are represented by means of a minimum-type complementarity function to enforce the positivity of the reservoir model, and a space-time mixed finite element method is applied for the parallel-in-time monolithic discretization. In particular, we propose a time-adaptive strategy using the improved backward differencing formula of second order, to take full advantage of the high degree of space-time parallelism. Moreover, the complicated dynamics with higher nonlinearity of space-time discretization require some innovative nonlinear and linear solution strategies. Therefore, we present a class of modified semismooth Newton algorithms to enhance the convergence rate of nonlinear iterations. Multilevel space-time restricted additive Schwarz algorithms, whose subdomains cover both space and time variables, are also studied for domain decomposition-based preconditioning. Numerical experiments demonstrate the robustness and parallel scalability of the proposed adaptive space-time algorithm on a supercomputer with tens of thousands of processor cores. [ABSTRACT FROM AUTHOR]

Summary: This paper addresses the overwhelming computational resources needed with standard numerical approaches to simulate architected materials. Those multiscale heterogeneous lattice structures gain intensive interest in conjunction with the improvement of additive manufacturing as they offer, among many others, excellent stiffness‐to‐weight ratios. We develop here a dedicated HPC solver that benefits from the specific nature of the underlying problem in order to drastically reduce the computational costs (memory and time) for the full fine‐scale analysis of lattice structures. Our purpose is to take advantage of the natural domain decomposition into cells and, even more importantly, of the geometrical and mechanical similarities among cells. Our solver consists in a so‐called inexact FETI‐DP method where the local, cell‐wise operators and solutions are approximated with reduced order modeling techniques. Instead of considering independently every cell, we end up with only few principal local problems to solve and make use of the corresponding principal cell‐wise operators to approximate all the others. It results in a scalable algorithm that saves numerous local factorizations. Our solver is applied for the isogeometric analysis of lattices built by spline composition, which offers the opportunity to compute the reduced basis with macro‐scale data, thereby making our method also multiscale and matrix‐free. The solver is tested against various 2D and 3D analyses. It shows major gains compared to black‐box solvers; in particular, problems of several millions of degrees of freedom can be solved with a simple computer within few minutes. [ABSTRACT FROM AUTHOR]

AbstractMaCaw is a Multiphysics Object-Oriented Simulation Environment (MOOSE)–based application that enables domain-decomposed neutral particle transport calculations in MOOSE. It leverages MOOSE’s ray-tracing module for unstructured mesh particle tracking and OpenMC for collision physics. Additionally, the OpenMC implementation of several calculation steps (e.g. initialization and normalization) needed in a Monte Carlo particle transport eigenvalue calculation were adapted for domain decomposition. This paper reports on MaCaw’s implementation and several limitations, a single verification case, and early single-node scaling studies. This paper also serves as an announcement of the public release of MaCaw on the Idaho National Laboratory GitHub at https://github.com/idaholab/macaw. [ABSTRACT FROM AUTHOR]

Methods for representing and processing algorithms for dynamically adaptive locally refined meshes are proposed for serial and parallel computing systems, including hybrid ones. Estimates of the complexity of the algorithms are given. New parallel algorithms for the decomposition of locally condensed meshes and dynamic load balancing, which ensure low overhead costs and reduce the overall computation time for two- and three-dimensional numerical modeling, are proposed. Time reduction is achieved by decreasing the number of cells in the computational mesh (relative to the regular mesh) and through parallel processing. [ABSTRACT FROM AUTHOR]

Probabilistic domain decomposition (PDD) is an alternative paradigm for solving boundary value problems (BVPs) in parallel with excellent scalability properties, thanks to its reliance on stochastic representations of the BVP. However, there are cases when the latter is less numerically convenient, or unknown. Semilinear elliptic BVPs and the Helmholtz equation are prominent examples of either class. In this paper, we overcome this issue by designing suitable iterative schemes for either problem. These schemes not only retain the desirable properties of PDD but also are optimally suited for pathwise variance reduction, resulting in a systematic, nearly cost-free reduction of the statistical error through the iterations. Numerical tests carried out on the supercomputer Marconi100 are presented. [ABSTRACT FROM AUTHOR]

We consider a non-overlapping domain decomposition method for optimal control problems of the tracking type governed by time-fractional diffusion equations in one space dimension, where the fractional time derivative is considered in the Caputo sense. We concentrate on a transmission problem defined on two adjacent intervals, where at the interface we introduce an iterative non-overlapping domain decomposition in the spirit of P.L. Lions for the corresponding first-order optimality system, such that the optimality system corresponding to the optimal control problem on the entire domain is iteratively decomposed into two systems on the respective sub-domains; this approach can be framed as first optimize, then decompose. We show that the iteration involving the states and adjoint states converges in the appropriate spaces. Moreover, we show that the decomposed systems on the sub-domain can in turn be interpreted as optimality systems of so-called virtual control problems on the sub-domains. Using this property, we are able to solve the original optimal control problem by an iterative solution of optimal control problems on the sub-domains. This approach can be framed as first decompose, then optimize. We provide a mathematical analysis of the problems as well as a numerical finite difference discretization using the L1-method with respect to the Caputo derivative, along with two examples in order to verify the method. [ABSTRACT FROM AUTHOR]

This work presents a deep neural network (DNN)-based approach for identifying the modal field distributions of closed non-radiating waveguides. Specifically, physics-informed neural networks (PINNs) are used to solve the Helmholtz partial differential equation. The PINN architecture includes incorporation of boundary conditions and selection of initial conditions to obtain required modes inside the waveguides. In this paper, furthermore, the use of this method is illustrated for waveguides consisting of inhomogeneous and anisotropic media, where we apply a domain decomposition-based deep learning method. Our approach successfully identifies all eigenmode distributions with an error of less than −12 dB as compared to analytical and full-wave simulation results. Notably, we further enhance the efficiency of our approach by utilizing transfer learning, achieving a 23 times reduction in solution time. Our results demonstrate PINNs as an alternative to traditional methods in accurately calculating waveguide modal field distributions and its applicability to other partial differential equation based EM problems.

Based on domain decomposition, a semi-analytical method (SAM) is applied to analyze the free vibration of double-curved shells of revolution with a general curvature radius made from graphene nanoplatelet (GPL)-reinforced porous composites. The mechanical properties of the GPL-reinforced composition are assessed with the Halpin–Tsai model. The double-curvature shell of revolution is broken down into segments along its axis in accordance with first-order shear deformation theory (FSDT) and the multi-segment partitioning technique, to derive the shell’s functional energy. At the same time, interfacial potential is used to ensure the continuity of the contact surface between neighboring segments. By applying the least-squares weighted residual method (LWRM) and modified variational principle (MVP) to relax and achieve interface compatibility conditions, a theoretical framework for analyzing vibrations is developed. The displacements and rotations are described through Fourier series and Chebyshev polynomials, accordingly, converting a two-dimensional issue into a suite of decoupled one-dimensional problems. The obtained solutions are contrasted with those achieved using the finite element method (FEM) and other existing results, and the current formulation’s validity and precision are confirmed. Example cases are presented to demonstrate the free vibration of GPL-reinforced porous composite double-curved paraboloidal, elliptical, and hyperbolical shells of revolution. The findings demonstrate that the natural frequency of the shell is related to pore coefficients, porosity, the mass fraction of the GPLs, and the distribution patterns of the GPLs.

The paper presents a fast and stable solver algorithm for structural problems. The point is the distance between the eigenvector of the constrained stiffness matrix and the unconstrained matrix. The coarse motions are close to the kernel of the unconstrained matrix. We use lower-frequency deformation modes to construct an iterative solver algorithm through domain decomposition expressing near-rigid-body motions, deflation algorithms, and two-level algorithms. We remove the coarse space from the solution space and hand over the iteration space to the fine space. Our solver is parallelized, and the solver thus has two sets of domain decomposition. One decomposition generates the coarse space, and the other is for parallelization. The basic framework of the solver is the parallel conjugate gradient (CG) method on the fine space. We use the CG method for the basic framework instead of the (simplest) domain decomposition method. We conducted benchmark tests using elastic static analysis for thin plate models. A comparison with the standard CG solver results shows the new solver's high-speed performance and remarkable stability. [ABSTRACT FROM AUTHOR]

We address the challenge of acoustic simulations in three-dimensional (3D) virtual rooms with parametric source positions, which have applications in virtual/augmented reality, game audio, and spatial computing. The wave equation can fully describe wave phenomena such as diffraction and interference. However, conventional numerical discretization methods are computationally expensive when simulating hundreds of source and receiver positions, making simulations with parametric source positions impractical. To overcome this limitation, we propose using deep operator networks to approximate linear wave-equation operators. This enables the rapid prediction of sound propagation in realistic 3D acoustic scenes with parametric source positions, achieving millisecond-scale computations. By learning a compact surrogate model, we avoid the offline calculation and storage of impulse responses for all relevant source/listener pairs. Our experiments, including various complex scene geometries, show good agreement with reference solutions, with root mean squared errors ranging from 0.02 to 0.10 Pa. Notably, our method signifies a paradigm shift as—to our knowledge—no prior machine learning approach has achieved precise predictions of complete wave fields within realistic domains. [ABSTRACT FROM AUTHOR]

This manuscript presents an efficient solver for the linear system that arises from the hierarchical Poincar\'e--Steklov (HPS) discretization of three dimensional variable coefficient Helmholtz problems. Previous work on the HPS method has tied it with a direct solver. This work is the first efficient iterative solver for the linear system that results from the HPS discretization. The solution technique utilizes GMRES coupled with a locally homogenized block-Jacobi preconditioner. The local nature of the discretization and preconditioner naturally yield the matrix-free application of the linear system. Numerical results illustrate the performance of the solution technique. This includes an experiment where a problem approximately 100 wavelengths in each direction that requires more than a billion unknowns to achieve approximately 4 digits of accuracy takes less than 20 minutes to solve. [ABSTRACT FROM AUTHOR]

A parallelizable iterative procedure based on domain decomposition is presented and analyzed for weak Galerkin finite element methods for second order elliptic equations. The convergence analysis is established for the decomposition of the domain into individual elements associated to the weak Galerkin methods or into larger subdomains. A series of numerical tests are illustrated to verify the theory developed in this paper. [ABSTRACT FROM AUTHOR]

We construct a space‐time parallel method for solving parabolic partial differential equations by coupling the parareal algorithm in time with overlapping domain decomposition in space. The goal is to obtain a discretization consisting of "local" problems that can be solved on parallel computers efficiently. However, this introduces significant sources of error that must be evaluated. Reformulating the original parareal algorithm as a variational method and implementing a finite element discretization in space enables an adjoint‐based a posteriori error analysis to be performed. Through an appropriate choice of adjoint problems and residuals the error analysis distinguishes between errors arising due to the temporal and spatial discretizations, as well as between the errors arising due to incomplete parareal iterations and incomplete iterations of the domain decomposition solver. We first develop an error analysis for the parareal method applied to parabolic partial differential equations, and then refine this analysis to the case where the associated spatial problems are solved using overlapping domain decomposition. These constitute our time parallel algorithm and space‐time parallel algorithm respectively. Numerical experiments demonstrate the accuracy of the estimator for both algorithms and the iterations between distinct components of the error. [ABSTRACT FROM AUTHOR]

Microbes are able to deploy different strategies in response to, and depending upon, local environmental conditions. In the setting of a microbial community, this property induces a Nash equilibrium problem because access to environmental resources is bounded. If microbes are also distributed in space, then those resources are subject to transport limitations (encoded in a PDE) and so microbial strategies at one location influence resources and, hence, microbial strategies, at another. Here we formulate the resulting PDE-coupled generalized Nash equilibrium problem for a multispecies biofilm community, and propose a domain-decomposition-based method for its solution. An example consisting of a model with two microbial species biofilm with 33 externally transported chemical concentrations is presented. [ABSTRACT FROM AUTHOR]

We present an overlapping domain decomposition iterative solver for linear systems resulting from the discretization of compressible viscous flows with the Discontinuous Petrov–Galerkin (DPG) method in three dimensions. It is a two-grid solver utilizing the solution on the auxiliary coarse grid and the standard block-Jacobi iteration on patches of elements defined by supports of the coarse mesh base shape functions. The simple iteration defined in this way is used as a preconditioner for the conjugate gradient procedure. Theoretical analysis indicates that the condition number of the preconditioned system should be independent of the actual finite element mesh and the auxiliary coarse mesh, provided that they are quasiuniform. Numerical tests confirm this result. Moreover, they show that presence of strongly flattened or elongated elements does not slow the convergence. The finite element mesh is subject to adaptivity, i.e. dividing the elements with large errors until a required accuracy is reached. The auxiliary coarse mesh is adjusting to the nonuniform actual mesh. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze a non-overlapping substructuring type algorithm for the Cahn–Hilliard equation. Being a nonlinear equation, it is of great importance to develop a robust numerical method for investigating the solution behaviour of the CH equation. We present the formulation of the Neumann–Neumann method applied to the CH equation and study its convergence behaviour in one and two spatial dimension for two subdomains and also extend the method for logarithmic nonlinearity. We also present the nonlinear NN method for the CH equation. We illustrate the theoretical results by providing numerical examples. [ABSTRACT FROM AUTHOR]

Landslides are common geological hazards that often result in significant casualties and economic losses. Due to their occurrence in complex terrain areas, conventional geophysical techniques face challenges in early detection and warning of landslides. Semi-airborne electromagnetic (SAEM) technology, utilizing unmanned aerial platforms for rapid unmanned remote sensing, can overcome the influence of complex terrain and serve as an effective approach for landslide detection and monitoring. In response to the low computational efficiency of conventional semi-airborne EM 3D forward modeling, this study proposes an efficient forward modeling method. To handle arbitrarily complex topography and geologic structures, the unstructured tetrahedron mesh is adopted to discretize the earth. Based on the vector finite element formula, the Dual–Primal Finite Element Tearing and Interconnecting (FETI-DP) method is further applied to enhance modeling efficiency. It obtains a reduced order subsystem and avoids directly solving huge overall linear equations by converting the entirety problem into the interface problem. We check our algorithm by comparing it with 1D semi-analytical solutions and the conventional finite element method. The numerical experiments reveal that the FETI-DP method utilities less memory and exhibits higher computation efficiency than the conventional methods. Additionally, we compare the electromagnetic responses with the topography model and flat earth model. The comparison results indicate that the effect of topography cannot be ignored. Further, we analyze the characteristic of electromagnetic responses when the thickness of the aquifer changes in a landslide area. We demonstrate the effectiveness of the 3D SAEM method for landslide detection and monitoring. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The aim of this work is to present a model reduction technique in the framework of optimal control problems for partial differential equations. We combine two approaches used for reducing the computational cost of the mathematical numerical models: domain–decomposition (DD) methods and reduced–order modelling (ROM). In particular, we consider an optimisation–based domain–decomposition algorithm for the parameter–dependent stationary incompressible Navier–Stokes equations. Firstly, the problem is described on the subdomains coupled at the interface and solved through an optimal control problem, which leads to the complete separation of the subdomain problems in the DD method. On top of that, a reduced model for the obtained optimal–control problem is built; the procedure is based on the Proper Orthogonal Decomposition technique and a further Galerkin projection. The presented methodology is tested on two fluid dynamics benchmarks: the stationary backward–facing step and lid-driven cavity flow. The numerical tests show a significant reduction of the computational costs in terms of both the problem dimensions and the number of optimisation iterations in the domain–decomposition algorithm. • Optimisation–based domain–decomposition algorithm for computational fluid dynamics. • Gradient–based optimisation algorithm for decoupling of the subdomains solves. • Projection–based reduced order model for parameterised Navier–Stokes problem. • Significant reduction in space dimension and optimisation iterations. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we propose and present a non-overlapping substructuring-type iterative algorithm for the Cahn–Hilliard (CH) equation, which is a prototype for phase-field models. It is of great importance to develop efficient numerical methods for the CH equation, given the range of applicability of CH equation has. Here we present a formulation for the linear and non-linear Dirichlet–Neumann (DN) methods applied to the CH equation and study the convergence behaviour in one and two spatial dimensions in multiple subdomains. We show numerical experiments to illustrate our theoretical findings and effectiveness of the method. [ABSTRACT FROM AUTHOR]

Aiming at the long-term and high-precision simulation of the magnetohydrodynamic (MHD) instabilities in the tokamak model, we developed a parallelized solver based on a fully implicit difference scheme. A 4th-order precision difference scheme and the Newton–Krylov method are employed in the proposed solver for both the flow and the electromagnetic field. To achieve high parallel efficiency, we adopt a strategy based on the spatial domain decomposition to partition the large Jacobian matrices in the iteration, and a buffer area based on the grid density is utilized to minimize the memory and time consumption. The accuracy of the methodology is verified, and the numerical results are validated by comparison with recognized results. The numerical results of the tearing mode instability in the tokamak model have demonstrated the precision and reliability of the algorithm, and the high parallel efficiency has been proven by the scalability test on the platform with up to 1280 threads, showing significant potential in the large-scale simulation of MHD problems. [ABSTRACT FROM AUTHOR]

In this study, a parallel three-scale homogenization analysis simulator that accounts for macro-, meso-, and micro-scale structures was developed for the carbon-fiber-reinforced plastics (CFRPs) used in high-pressure hydrogen storage vessels manufactured using the filament winding (FW) method. The developed simulator enables detailed analysis that accounts for fiber irregularities in the fiber bundle tapes of FW-CFRP. Because numerical simulations that consider fiber irregularities increase computational time and memory usage, we developed a parallel computation system for three-scale homogenization using the domain decomposition method. Numerical examples using large-scale computers verified that our parallel three-scale homogenization analysis has parallel computing performance close to the ideal acceleration ratio. We then investigated the effects of fiber waviness in fiber bundle tapes on the macro-scale properties of FW–CFRP by considering waviness as an initial irregularity in the carbon fiber arrangement. Our analysis of fiber irregularities revealed that fiber waviness has a significant effect on macroscopic stiffness and stress. Within the scope of this study, the macroscopic stiffness and stress were reduced by 40% and 57%, respectively, compared to the results without irregularities.