Domain Decomposition Research Articles

A 3D frequency domain finite element formulation for solving the wave equation in the presence of rotating obstacles

A frequency domain numerical method aimed at solving the propagation of sound waves in a three-dimensional domain enclosing rotating obstacles is presented. It relies on a domain decomposition whereby rotating parts are all embedded in a cylindrically-shaped domain. Following the classical Arbitrary Lagrangian Eulerian formalism, the governing equation, which is written in the local rotating frame, takes the form of a convected wave equation. The transmission conditions at the interface between the fixed and rotating domains is accomplished via the Frequency Scattering Boundary Conditions which, after classical FEM discretization, give rise to a series of coupled problems associated with a discrete set of frequencies. The performances of the method are demonstrated through several test cases involving rotating sources and/or obstacles inserted in circular and rectangular ducts.

A compatible finite element discretisation for the nonhydrostatic vertical slice equations

We present a compatible finite element discretisation for the vertical slice compressible Euler equations, at next-to-lowest order (i.e., the pressure space is bilinear discontinuous functions). The equations are numerically integrated in time using a fully implicit timestepping scheme which is solved using monolithic GMRES preconditioned by a linesmoother. The linesmoother only involves local operations and is thus suitable for domain decomposition in parallel. It allows for arbitrarily large timesteps but with iteration counts scaling linearly with Courant number in the limit of large Courant number. This solver approach is implemented using Firedrake, and the additive Schwarz preconditioner framework of PETSc. We demonstrate the robustness of the scheme using a standard set of testcases that may be compared with other approaches.

PGRASS-Solver: A Graph Spectral Sparsification-Based Parallel Iterative Solver for Large-Scale Power Grid Analysis

With the increase in the complexity of VLSI chips, power grid analysis has become a challenging task, because linear equations of extremely large size need to be solved. Recent graph sparsification based solvers have shown promising performance for power grid analysis. However, existing graph sparsification algorithms are implemented in serial computing, while factorization and backward/forward substitution of the sparsifier’s Laplacian matrix are hard to parallelize. On the other hand, partition based iterative methods which are inherently parallel lack a direct control of the relative condition number of the preconditioner and consume more memory. In this work, we propose a novel parallel iterative solver called pGRASS-Solver. We first propose a practically-efficient parallel graph sparsification algorithm. Then domain decomposition method (DDM) is utilized to solve the sparsifier’s Laplacian matrix. To further improve the efficiency, a variant of DDM which employs partial Cholesky factorization and Schur complement matrix sparsification is proposed. Thus, we obtain an efficient parallel preconditioner, which not only leads to fast convergence but also enjoys ease of parallelization. Numerous experiments are conducted to illustrate the superior efficiency of the proposed pGRASS-Solver for large-scale power grid analysis, showing an average 6.8X speedup over a recent parallel iterative solver [1]. Moreover, it solves a real-world power grid matrix with 0.36 billion nodes and 8.7 billion nonzeros within 20 minutes on a 16-core machine, which is 10.9X faster than the best result of sequential graph sparsification based solver [2].

ADVANCES IN UNSTRUCTURED WAVEWATCH III AND APPLICATIONS TO NEARSHORE WAVES

The spectral wave generation and propagation model WAVEWATCH III (WW3) (WW3DG 2019) is undergoing rapid community development to extend capability and nearshore modeling applicability. An option for unstructured grids and implicit solution provides WW3 with the flexibility and efficiency to resolve complex shorelines and high-gradient wave zones to drive nearshore circulation, wave setup, and wave-driven sediment transport with multi-scale spatial coverage over approximately three orders of magnitude. A hybrid approach to parallelization uses spectral partitioning for advection in geographical space and domain decomposition for spectral advection and the source term integration. The advection part of wave action equation is integrated fully implicitly, and a new convergent action limiter derived from Komen et al. (1994) and Hersbach and Janssen (1999) is applied. The model is compatible with community-based coupling infrastructure to facilitate two-way coupling with circulation models for simulating hurricane storm surge and waves. Abdolali et al. (2020) evaluated WW3 enhancements on a large-scale numerical domain employing the new parallelization algorithm and implicit solver for unstructured grids and compared to the existing parallelization algorithm, domain decomposition, and robust explicit numerical solver of WW3 on both structured and unstructured grids. These new capabilities in the wave model push the limitations of the model, including minimum resolution (10s of meters), maximum number of model grid points (~2 million), and computational scalability, as well as better accuracy in contrast to the explicit scheme. This paper describes model enhancements and validation.

Development of a software platform for bridge modal and damage identification based on ambient excitation

Modal and damage identification based on ambient excitation can greatly improve the efficiency of high-speed railway bridge vibration detection. This paper first describes the basic principles of stochastic subspace identification, peak-picking, and frequency domain decomposition method in modal analysis based on ambient excitation, and the effectiveness of these three methods is verified through finite element calculation and numerical simulation. Then the damage element is added to the finite element model to simulate the crack, and the curvature mode difference and the curvature mode area difference square ratio are calculated by using the stochastic subspace identification results to verify their ability of damage identification and location. Finally, the above modal and damage identification techniques are integrated to develop a bridge modal and damage identification software platform. The final results show that all three modal identification methods can accurately identify the vibration frequency and mode shape, both damage identification methods can accurately identify and locate the damage, and the developed software platform is simple and efficient.

Researches advanced in fine-grained image classification based on deep learning

In the recent past, the task of fine-grained image recognition has become a hot research topic for scholars. Due to its substantial intra-class and little inter-class variance, fine-grained image classification task is more difficult than traditional image classification task. From a deep learning standpoint, this paper first introduces representative algorithms for strongly supervised fine-grained image classification, consisting of the Part-based R-CNN algorithm, the Pose Normalized CNN algorithm, and the Mask-CNN model. Second, weakly supervised process in image classification is expounded from the representative methods of attention mechanism, end-to-end visual coding, zero-shot learning with adaptive domain decomposition. Third, in the experimental part, the most popular databases for precise picture categorization, the evaluation metrics and conclusion of the algorithms are listed. Finally, it summarizes and looks forward, points out the problems and deficiencies of current research, and gives guidance for possible solutions in the future.

3D NUMERICAL STUDY OF HYDROYNAMICS AROUND A MONOPILE FOUNDATION UNDER COMBINED WAVE CURRENT EFFECT

A numerical investigation of hydrodynamics around a monopile foundation is presented in the paper. An open-source CFD program REEF3D is used for the present study. The REEF3D model solves the Reynolds-averaged Naiver-Stokes equation with k-w turbulence closure to obtain the flow hydrodynamics. The code utilizes a staggered cartesian grid system. Also, Message Passing Interface (MPI) based parallelization is used for domain decomposition. In order to ensure the precision of the flow field, the model is first validated for four different combined wave-current scenarios, where the currents and waves are following each other. The results obtained from the validation studies exhibit good agreement with the already reported experimental results. Initially, a grid convergence study has been performed to ensure that the Numerical wave tank is independent of the mesh size. Finally, numerical experiments corresponding to four different combined wave-current cases for different wave steepness are simulated using the code.

A hybrid probabilistic domain decomposition algorithm suited for very large-scale elliptic PDEs

A hybrid probabilistic domain decomposition algorithm suited for very large-scale elliptic PDEs

A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks (PINNs) and their temporal decompositions

Physics-informed neural networks (PINNs) as a means of solving partial differential equations (PDE) have garnered much attention in the Computational Science and Engineering (CS&E) world. However, a recent topic of interest is exploring various training (i.e., optimization) challenges – in particular, arriving at poor local minima in the optimization landscape results in a PINN approximation giving an inferior, and sometimes trivial, solution when solving forward time-dependent PDEs with no data. This problem is also found in, and in some sense more difficult, with domain decomposition strategies such as temporal decomposition using XPINNs. We furnish examples and explanations for different training challenges, their cause, and how they relate to information propagation and temporal decomposition. We then propose a new stacked-decomposition method that bridges the gap between time-marching PINNs and XPINNs. We also introduce significant computational speed-ups by using transfer learning concepts to initialize subnetworks in the domain and loss tolerance-based propagation for the subdomains. Finally, we formulate a new time-sweeping collocation point algorithm inspired by the previous PINNs causality literature, which our framework can still describe, and provides a significant computational speed-up via reduced-cost collocation point segmentation. The proposed methods form our unified framework, which overcomes training challenges in PINNs and XPINNs for time-dependent PDEs by respecting the causality in multiple forms and improving scalability by limiting the computation required per optimization iteration. Finally, we provide numerical results for these methods on baseline PDE problems for which unmodified PINNs and XPINNs struggle to train.

Performance Evaluation and Optimization of the Weather Research and Forecasting (WRF) Model Based on Kunpeng 920

The Weather Research and Forecasting (WRF) model is a mesoscale numerical weather prediction system, which is widely used in major high-performance server platforms. This study focuses on the performance evaluation and optimization of WRF on Huawei’s self-developed kunpeng 920 processor platform, aiming to improve the operational efficiency of WRF. The results of the study show that the scalability of WRF on kunpeng 920 processor is well performed; the performance of WRF on kunpeng 920 processor is improved by 32.6% after invoking the Fast Math Library and Domain Decomposition Core Tile Division optimization. In terms of IO, the main optimizations are parallel IO and asynchronous IO. Eventually, the single output time of WRF is reduced from 37.28 s in serial IO mode to 0.14 s in asynchronous IO mode, and the overall running time is reduced from 1078.80 s to 807.94 s.

A domain decomposition strategy for mCRE-based model updating in dynamics

In structural dynamics, the modified constitutive relation error is a widely used functional for model updating. The method involves a two-step iterative process to minimize the functional: first, regions with high modeling errors are localized, then their associated parameters are updated. The localization step requires solving large linear systems at each iteration, which makes the method less suitable for industrial models. In this work, a domain decomposition formulation is proposed. It makes the method more flexible, as subdomains can use different models which are independently modified during the updating algorithm without any additional cost. Moreover, this approach is very efficient numerically as it benefits from the same advantages as domain decomposition methods. In its standard form, it lacks a coarse problem which seems problematic; this is however addressed by using a multi-preconditioned solver such as MPGMRES. The performance of the proposed approach is illustrated by means of academic and industrial examples, with practical implementation in a commercial software.

On the space-time decomposition and reconstruction of the pump-jet propulsor flow field

In this numerical study, a modal approach based on the stress blend eddy simulation (SBES) was employed to investigated the morphology and development regime of the vortices in a pump-jet propulsor (PJP). Modal analysis was conducted on the corresponding surfaces to gain insights into the evaluation of the flow field, with a focus on spectral proper orthogonal decomposition (SPOD) and dynamic mode decomposition (DMD). These methods consider both temporal and spatial domain decomposition. By comparing the modes obtained from SOPD and DMD, it was observed that the coherent structure morphology exhibited similarities with only minor differences. The results indicated that the flow field could be effectively reconstructed using a few low-order modes. Both SPOD and DMD accurately captured the flow field, expect in the downstream hub region where chaotic flow was present. This study provides valuable insights into wake flow field of the PJP and sheds light on the understanding of vortex dynamics.

A posteriori error analysis for a space‐time parallel discretization of parabolic partial differential equations

AbstractWe 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.

A massively parallel implicit 3D unstructured grid solver for computing turbulent flows on latest distributed memory computational architectures

An implicit unstructured grid density-based solver–PRAVAHA based on a parallel variant of the lower upper symmetric Gauss-Seidel (LUSGS) method is developed to compute large-scale engineering problems. A four-layered parallel algorithm is designed to efficiently compute three-dimensional turbulent flows on massively parallel modern multiple instruction multiple data-stream (MIMD) computational hardware. This data parallel approach achieves multiple layers of parallelism including continuity of flow solution, transfer of solution gradients, and calculation of drag/lift/solution residuals, right up to the innermost implicit LUSGS solver sub-routine, which is relatively less explored in the literature. Domain decomposition is performed using the METIS software based on multi-level graph partitioning algorithms. Non-blocking message-passing interface library functions are used to manage inter-processor communication through explicit message passing, efficiently. Super-linear scalability of the parallel solver is established on the current state-of-the-art supercomputing facility, the 838 teraflops PARAM seva on up to 6144 cores. Linear or even super-linear speedup on problems of significant size is observed even on ad-hoc parallel computing platforms like workstations and multi-node clusters, for turbulent flow simulations.

A comparative study of scalable multilevel preconditioners for cardiac mechanics

In this work, we provide a performance comparison between the Balancing Domain Decomposition by Constraints (BDDC) and the Algebraic Multigrid (AMG) preconditioners for cardiac mechanics on both structured and unstructured finite element meshes. The mechanical behavior of myocardium can be described by the equations of three-dimensional finite elasticity, which are discretized by finite elements in space and yield the solution of a large scale nonlinear algebraic system. This problem is solved by a Newton-Krylov method, where the solution of the Jacobian linear system is accelerated by BDDC/AMG preconditioners. We thoroughly explore the main parameters of the BDDC preconditioner in order to make the comparison fair. We focus on: the performance of different direct solvers for the local and coarse problems of the BDDC algorithm; the impact of the different choices of BDDC primal degrees of freedom; and the influence of the finite element degree. Scalability tests are performed on Linux clusters up to 1024 processors, and we conclude with a performance study on a realistic electromechanical simulation.

Extended Conductor-like Polarizable Continuum Solvation Model (CPCM-X) for Semiempirical Methods.

We have developed a new method to accurately account for solvation effects in semiempirical quantum mechanics based on a polarizable continuum model (PCM). The extended conductor-like polarizable continuum model (CPCM-X) incorporates a computationally efficient domain decomposition conductor-like screening model (ddCOSMO) for extended tight binding (xTB) methods and uses a post-processing approach based on established solvation models, like the conductor-like screening model for real solvents (COSMO-RS) and the universal solvent model based on solute electron density (SMD). According to various benchmarks, the approach performs well across a broad range of systems and applications, including hydration free energies, non-aqueous solvation free energies, and large supramolecular association reactions of neutral and charged species. Our method for computing solvation free energies is much more accurate than the current methods in the xtb program package. It improves the accuracy of solvation free energies by up to 40% for larger supramolecular association reactions to match even the accuracy of higher-level DFT-based solvation models like COSMO-RS and SMD while being computationally more than 2 orders of magnitude faster. The proposed method and the underlying ddCOSMO model are readily available for a wide variety of solvents and are accessible in xtb for use in various computational applications.

Hyper-fidelity depletion with discrete motion for pebble bed reactors

Hyper-fidelity (HxF) depletion of pebble bed reactors (PBRs) is the capability to model depletion for every pebble while accounting for motion through the core. Previous HxF work demonstrated feasibility to deplete hundreds of thousands of stationary pebbles concurrently within reasonable timeframes. This work illustrates the second step towards HxF, coupling depletion with a discrete motion scheme. The model assumes an ordered bed with pebbles occupying fixed positions. Motion is simplified as discrete since pebbles move in straight lines from one set position to another. The methodology was implemented in Serpent 2, combined with its transport and depletion capabilities. Ad-hoc routines were developed ensuring compatibility with domain decomposition and pebble recirculation after each pass based on discharge criteria and fresh pebble insertion. Capabilities of HxF with discrete motion are demonstrated using a full-scale high-temperature gas-cooled reactor model. Specifically, an approach to equilibrium is performed, and example results are shown for in-core and discarded pebbles. The data illustrates how HxF provides unique insights into PBR fuel, producing information on statistical distributions rather than average values only, as obtained by traditional methods that rely on spectral zoning for depletion. Knowledge of these distributions can greatly improve analysis and assessment of PBRs.

Optimized Schwarz methods for the time-dependent Stokes–Darcy coupling

Abstract This paper derives optimized coefficients for optimized Schwarz iterations for the time-dependent Stokes–Darcy problem using an innovative strategy to solve a nonstandard min-max problem. The coefficients take into account both physical and discretization parameters that characterize the coupled problem, and they guarantee the robustness of the associated domain decomposition method. Numerical results validate the proposed approach in several test cases with physically relevant parameters.

$H$-$\phi$ Formulation in Sparselizard Combined With Domain Decomposition Methods for Modeling Superconducting Tapes, Stacks, and Twisted Wires

The growing interest in the modeling of superconductors has led to the development of effective numerical methods and software. One of the most utilized approaches for magnetoquasistatic simulations in applied superconductivity is the $H$ formulation. However, due to the large number of degrees of freedom (DOFs) present when modeling large and complex systems (e.g. large coils for fusion applications, electrical machines, and medical applications) using the standard $H$ formulation on a desktop machine becomes infeasible. The $H$ formulation solves the Faraday's law formulated in terms of the magnetic field intensity $\mathbf{H}$ using edge elements in the whole modeling domain. For this reason, a very high resistivity is assumed for the non-conducting domains, leading to an ill-conditioned system matrix and therefore long computation times. In contrast, the $H$-$\phi$ formulation uses the $H$ formulation in the conducting region, and the $\phi$ formulation (magnetic scalar potential) in the surrounding non-conducting domains, drastically reducing DOFs and computation time. In this work, we use the $H$-$\phi$ formulation in 2D for the magnetothermal (AC losses and quench) analysis of stacks of REBCO tapes. The same approach is extended to a 3D case for the AC loss analysis of a twisted superconducting wire. All the results obtained by simulations in Sparselizard are compared with results obtained with COMSOL. Our custom tool allows us to distribute the simulations over hundreds of CPUs using domain decomposition methods, considerably reducing the simulation times without compromising accuracy.

A Scalable Multiple-Time-Step Efficient Numerical Algorithm for Wind–Road Vehicle–Train–Bridge Dynamic Simulations

During the past two decades, more and more rail-cum-road bridges have been built to meet the ever-increasing demand of highway and railway transportation in China. The rail-cum-road bridges are expected to withstand multiple types of loadings during normal operation, including wind, road vehicle, and train. Accurate quantifications of the coupling dynamics among bridge, road vehicle, and train are essential to ensure the safety and reliability of rail-cum-road bridges during their lifetime. The present study develops an analytical framework to simulate the complex wind–road vehicle–train–bridge (WRVTB) system. In addition, a scalable multiple-time-step (SMTS) algorithm is proposed to overcome the computational efficiency issue of traditional single-time-step (STS) algorithms for solving the WRVTB interaction. The SMTS algorithm utilizes domain decomposition techniques to enable a fine and a coarse time-step to be assigned for a road vehicle–train subsystem and bridge subsystem, which greatly improves the computational efficiency for solving the WRVTB interaction. A numerical simulation of coupled WRVTB analysis is conducted based on a prototype rail-cum-road bridge to demonstrate the computational efficiency and accuracy of the proposed method. The results indicate that both the MTS and the SMTS algorithms are able to achieve better accuracy and efficiency simultaneously in comparison with the STS algorithm. Although the SMTS algorithm does not exhibit obvious superiority over the MTS algorithm in terms of the overall accuracy and computational efficiency, the major advantage of the proposed SMTS algorithm over the MTS algorithm is the flexibility. When the SMTS algorithm is adopted, a refined time-step can be assigned to a subsystem of interest, while relatively coarse time-steps can be assigned to the remaining subsystems. As such, the desired accuracy of the subsystem of interest is guaranteed at a low computational cost.