Parallel Solution Methods Research Articles

A matrix-free parallel solution method for the three-dimensional heterogeneous Helmholtz equation

A matrix-free parallel solution method for the three-dimensional heterogeneous Helmholtz equation

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.

Globalization of Nonlinear FETI-DP Domain Decomposition Methods Using an SQP Approach

The globalization of Nonlinear FETI-DP (Dual Primal Finite Element Tearing and Interconnecting) methods is considered using a Sequential Quadratic Programming (SQP) approach. Nonlinear FETI-DP methods are parallel iterative solution methods for nonlinear finite element problems, based on divide and conquer, using Lagrange multipliers. In these methods, nonlinear elimination is an important ingredient to increase the convergence radius of Newton’s method. We prove standard globalization results for SQP-based globalization of Nonlinear FETI-DP, first for the case that the elimination set is empty. We then show how to combine nonlinear elimination and SQP-based globalization. The globalization preserves the block structure of the FETI-DP operator, which is the basis of the computational parallelism.Supporting numerical experiments using homogenous and heterogeneous model problems from nonlinear structural mechanics are provided. In the numerical experiments, we consider four standard choices of different elimination sets and different problem setups including stiff or almost incompressible inclusions in every subdomain. The numerical experiments illustrate that a good elimination set is important. However, the use of the SQP-based globalization approach presented here can improve the convergence of Nonlinear FETI-DP methods further, especially, if combined with a good choice of the elimination set.

An extensive and systematic literature review for hybrid flowshop scheduling problems

Hybrid flowshop scheduling problem (HFSP) is a mixture of two classical scheduling problems as parallel machine scheduling (PMS) and flowshop scheduling (FS). In the HFSP, a series of jobs are processed respectively in a set of stages that at least one of these stages has more than one parallel machine (identical, uniform or unrelated). HFSP is a widespreadly studied subject in the literature and there are various application areas such as transportation, healthcare management, agricultural activities, cloud computing, and the most common manufacturing. Therefore, it will be useful to present a review study including recent papers and developments related to this problem for researchers. For this aim, in this paper, a systematic literature survey has been conducted with respect to HFSPs by means of Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) methodology which enables to realize systematic review and meta-analyses in a specified subject. 172 articles which are published in the 2010-2020 year interval, related to production scheduling and including a mathematical programming model to express scheduling problems have been determined as a result of this methodological review process. These articles have been statistically analyzed according to many features such as year, country, journal, number of stages, type of parallel machines, constraints, objective functions, solution methods, test instances and type of parameters. The results of statistical analyses have been presented through charts so as to provide a visual demonstration to researchers. Furthermore, it has been aimed to answer 14 predetermined research questions by means of analyses realized in the scope of this review study. Consequently, it has revealed the existing literature, recent developments and future research suggestions related to HFSP and therefore it is possible to say that this review paper provides a beneficial road map for researchers studying in this field.

A Simple Parallel Solution Method for the Navier–Stokes Cahn–Hilliard Equations

We present a discretization method of the Navier–Stokes Cahn–Hilliard equations which offers an impressing simplicity, making it easy to implement a scalable parallel code from scratch. The method is based on a special pressure projection scheme with incomplete pressure iterations. The resulting scheme admits solution by an explicit Euler method. Hence, all unknowns decouple, which enables a very simple implementation. This goes along with the opportunity of a straightforward parallelization, for example, by few lines of Open Multi-Processing (OpenMP) or Message Passing Interface (MPI) routines. Using a standard benchmark case of a rising bubble, we show that the method provides accurate results and good parallel scalability.

Accelerating the convergence of AFETI partitioned analysis of heterogeneous structural dynamical systems

Variationally based algorithms for the partitioned solution of structural mechanics problems are presented. Two key features of the present algorithms are the judicious application of the d’Alembert-Lagrange principal equations and the use of dominant substructural deformation modes. The paper includes three developments:1. Variational derivation of AFETI parallel solution methods.2. One-level and two-level AFETI implicit transient analysis algorithms with coarse problem included in the projector and based on free floating rigid body modes.3. A new AFETI implicit transient solution algorithm derived by constraining the interface equilibrium equations with the floating and dominant deformational modes.In addition to variational derivations of solution algorithms, the present paper is strived to offer new physical and/or numerical insight as each of variational derivational steps is succinctly explained. Performance evaluations of the algorithms described herein are presented.

High-Resolution Modeling and Decentralized Dispatch of Heat and Electricity Integrated Energy System

The heat and electricity integrated energy system (HE-IES) has high energy efficiency and bright application prospects. In the HE-IES, the heat transmission in the heating network and the indoor temperature variation in the buildings are governed by (partial) differential equations, which bring the heat dynamics problem in the system operation. In this paper, we propose a comprehensive approach for the operation of the HE-IES to coordinate the electricity and heat dynamics, including modeling, evaluation, and dispatch procedure. First, we formulate a high-resolution model for the heating network and buildings to describe the heat dynamics; two indices are proposed to measure the impact of heat dynamics, which are then used to select the time resolution. Second, an optimal dispatch model with high resolution for the heat dynamics is formulated for the HE-IES; a decentralized and parallel solution method is proposed based on the alternating direction method of multipliers. Third, a two-stage procedure for the time resolution selection is proposed, which consists of a dispatch decision stage and a time resolution evaluation stage. A small illustrative case is studied to verify the effectiveness of the proposed method. A big case based on a real heating network in Jilin Province, China, is also simulated.

GPU-based model predictive control for continuous casting spray cooling control system using particle swarm optimization

Model predictive control (MPC) for spray cooling control system requires a repeated online solution of an optimization problem that includes partial differential equations (PDEs). To simulate the future temperature behavior of steel billets, 3D dynamic heat transfer model is used. The special solution domain of PDEs has led to large computation cost, which is the main challenge in the real-time practical application of spray cooling control system. Meanwhile, the heat transfer coefficients need to be identified using the measured surface temperature. This work presents a two-level parallel solution method implemented on a Graphics processing unit (GPU) for MPC of spray cooling control systems and a weighted least squares modified conjugate gradient method (WLS–MCG) for identification of heat transfer coefficients. Two-level parallel solution method consists of parallel-based heat transfer model and stream parallel particle swarm optimization (PSO). PSO is used to solve the optimization problem. WLS–MCG consists of the weighted least squares (WLS) and modified conjugate gradient method (MCG). The experimental results show that the two-level parallel solution method has good computational performance and achieves satisfactory control performance.

The Parallel Solution Method of Sparse Triangular System Based on Correlation Decomposition

The Parallel Solution Method of Sparse Triangular System Based on Correlation Decomposition

Fast uncertainty quantification of activation sequences in patient-specific cardiac electrophysiology meeting clinical time constraints.

We present a fast, patient-specific methodology for uncertainty quantification in electrophysiology, aimed at meeting the time constraints of clinical practitioners. We focus on computing the statistics of the activation map, given the uncertainties associated with the conductivity tensor modeling the fiber orientation in the heart. We use a fast parallel solution method implemented on a graphics processing unit for the eikonal approximation, in order to compute the activation map and to sample the random fiber field with correlation on the basis of geodesic distances. While this enables to perform uncertainty quantification studies with a manageable computational effort, the required time frame still exceeds clinically suitable time expectations. In order to reduce it further by 2 orders of magnitude, we rely on Bayesian multifidelity methods. In particular, we propose a low-fidelity model that is patient-specific and free from the additional training cost associated with reduced models. This is achieved by a sound physics-based simplification of the full eikonal model. The low-fidelity output is then corrected by the standard multifidelity framework. In practice, the complete procedure only requires approximately 100 new runs of our eikonal graphics processing unit solver for producing the sought estimates and their associated credible intervals, enabling a full online analysis in less than 5minutes.

PARALLEL SOLUTION METHODS AND PRECONDITIONERS FOR EVOLUTION EQUATIONS

The recent development of the high performance computer platforms shows a clear trend towards heterogeneity and hierarchy. In order to utilize the computational power, particular attention must be paid to finding new algorithms or adjust existing ones so that they better match the HPC computer architecture. In this work we consider an alternative to classical time-stepping methods based on use of time-harmonic properties and discuss solution approaches that allow efficient utilization of modern HPC resources. The method in focus is based on a truncated Fourier expansion of the solution of an evolutionary problem. The analysis is done for linear equations and it is remarked on the possibility to use two- or multilevel mesh methods for nonlinear problems, which can enable further, even higher degree of parallelization. The arising block matrix system to be solved admits a two-by-two block form with square blocks, for which a very efficient preconditioner exists. It leads to tight eigenvalue bounds for the preconditioned matrix and, hence, to a very fast convergence of a preconditioned Krylov subspace or iterative refinement method. The analytical background is shown as well as some illustrating numerical examples.

Accelerating explicit ODE methods on GPUs by kernel fusion

SummaryGraphics processing units (GPUs) have a promising architecture for implementing highly parallel solution methods for systems of ordinary differential equations (ODEs). However, their high performance comes at the price of caveats such as small caches or wide SIMD. For ODE methods, optimizing the memory access pattern is often crucial. In this article, instead of considering only one specific method, we generalize the description of explicit ODE methods by using data flow graphs consisting of basic operations that are suitable to cover the types of computations occurring in all common explicit methods. After showing that the straightforward approach for processing the data flow graph by calling one kernel per basic operation is memory bound, we explain how the number of memory accesses can be reduced by the kernel fusion technique, which fuses several basic operations into one kernel. Moreover, we will present enabling transformations that allow additional fusions and thus can reduce the number of memory accesses even further. We apply these optimizations to three different classes of explicit ODE methods: embedded Runge–Kutta (RK) methods, parallel iterated RK (PIRK) methods, and peer methods. A detailed experimental evaluation on three modern GPUs showed speedups between 1.86 and 3.51 compared to unfused implementations.

Decomposition-Coordinating Method for Parallel Solution of a Multi-area Combined Economic Emission Dispatch Problem

<p>Multi-area Combined Economic Emission Dispatch (MACEED) problem is an optimization task in power system operation for allocating the amount of generation to the committed units within the system areas. Its objective is to minimize the fuel cost and the quantity of emissions subject to the power balance, generator limits, transmission line and tie-line constraints. The solutions of the MACEED problem in the conditions of deregulation are difficult, due to the model size, nonlinearities, and the big number of interconnections, and require intensive computations in real-time. High-Performance Computing (HPC) gives possibilities for the reduction of the problem complexity and the time for calculation by the use of parallel processing techniques for running advanced application programs efficiently, reliably and quickly. These applications are considered as very new in the power system control centers because there are not available optimization methods and software based on them that can solve the MACEED problem in parallel, paying attention to the existence of the power system areas and the tie-lines between them. A decomposition-coordinating method based on Lagrange’s function is developed in this paper. Investigations of the performance of the method are done using IEEE benchmark power system models.</p>

Toward Extremely Scalable Nonlinear Domain Decomposition Methods for Elliptic Partial Differential Equations

The solution of nonlinear problems, e.g., in material science, requires fast and highly scalable parallel solvers. Finite element tearing and interconnecting dual primal (FETI-DP) domain decomposition methods are parallel solution methods for implicit problems discretized by finite elements. Recently, nonlinear versions of the well-known FETI-DP methods for linear problems have been introduced. In these methods, the nonlinear problem is decomposed before linearization. This approach can be viewed as a strategy to further localize computational work and to extend the parallel scalability of FETI-DP methods toward extreme-scale supercomputers. Here, a recent nonlinear FETI-DP method is combined with an approach that allows an inexact solution of the FETI-DP coarse problem. We combine the nonlinear FETI-DP domain decomposition method with an algebraic multigrid (AMG) method and thus obtain a hybrid nonlinear domain decomposition/multigrid method. We consider scalar nonlinear problems as well as nonlinear hyp...

Simulation of MQL Deep Hole Drilling for Predicting Thermally Induced Workpiece Deformations

Abstract The resulting thermomechanical load on the workpiece in deep hole drilling operations using minimum quantity lubrication (MQL) induces a strong in-process deflection of the machined component and can cause an insufficient accuracy of the produced hole. Also subsequent machining operations can be affected by the thermoelastic component of this deformation, which remains within the workpiece after the drilling process. Due to the comparatively long main time of typical deep hole drilling operations the thermomechanical simulation of commonly complex machined parts is challenging. In this paper, a fast finite-element approach using massive parallel solution methods is presented and validated for different wall thickness situations.

Interaction of the Zymne monastery buildings with the soil base under seismic loads

The work presents numerous research findings on the stressstrain state of a building within the “base–basement–construction” system under a seismic load. The soil array was considered as volumetric nonlinear elastic viscose and plastic environment with a new computed behavioral pattern. The latter accounts for peculiarities of viscose and elastic deformation of marginally stiff soil under a dynamic load, elastic and plastic deformation regarding structural strength and possibilities of strengthening, and energy loss in interaction with the environment. We have revealed the underlying principles of the model. Problems can be solved effectively by the FEM-based variety of the central difference method for heterogeneous systems, which allows to increase the average time integration step and to make effective use of parallel solution methods. As an example, there was created a three-dimensional model of the Assumption (Uspensky) Cathedral of the Zymne Monastery. The modeling was based on the automated system for scientific research “VESNA-DYN” and a modified (explicit) central difference method. According to the findings, seismic loads result in irreversible plastic deformations that significantly impact the stress-strain state (SSS) of constructions. The analysis has exposed the areas of stress concentration that lead to possible damages of constructions. The research has traced emergence and evolution of a tilt resulting from irreversible soil deformations. We have estimated the overall carrying capacity of the church structures.

A parallel finite-element framework for large-scale gradient-based design optimization of high-performance structures

Structural optimization using gradient-based methods is a powerful design technique that is well suited for the design of high-performance structures. However, the ever-increasing complexity of finite-element models and design formulations results in a bottleneck in the computation of the gradients required for the design optimization. Furthermore, in light of current high-performance computing trends, any methods intended to address this bottleneck must efficiently utilize parallel computing resources. Therefore, there is a need for solution and gradient evaluation methods that scale well with the number of design variables, constraints, and processors. We address this need by developing an integrated parallel finite-element analysis tool for gradient-based design optimization that is designed to use specialized parallel solution methods to solve large-scale high-fidelity structural optimization problems with thousands of design variables, millions of state variables, and hundreds of load cases. We describe the most relevant details of the parallel algorithms used within the tool. We present consistent constraint formulations and aggregation techniques for both material failure and buckling constraints. To demonstrate both the solution and functional accuracy, we compare our results to an exact solution of a pressure-loaded cylinder made with either isotropic or orthotropic material. To demonstrate the parallel solution and gradient evaluation performance, we perform a structural analysis and gradient evaluation for a large transport aircraft wing with over 5.44 million unknowns. The results show near-ideal scalability of the structural solution and gradient computation with the number of design variables, constraints, and processors, which makes this framework well suited for large-scale high-fidelity structural design optimization.

Online auto-tuning for the time-step-based parallel solution of ODEs on shared-memory systems

This article considers automatic performance tuning of time-step-based parallel solution methods for initial value problems (IVPs) of systems of ordinary differential equations (ODEs). We apply auto-tuning to the parallel execution of a class of explicit predictor–corrector (PC) methods of Runge–Kutta (RK) type on shared-memory architectures. The performance of parallel multi-threaded implementation variants of these methods depends on various factors only known at runtime, for example, the coupling structure of the ODE system to be solved, the memory access pattern resulting from this coupling structure, and the number of threads executing the program.We propose an online auto-tuning approach that exploits the time-stepping nature of ODE methods by selecting the best parallel implementation variant from a set of candidate implementations at runtime during the first time steps. Thus, the auto-tuning process is not isolated from the computation, but rather contributes to the progress of the solution process. The search space of candidate implementations is a priori reduced by estimating the synchronization overhead of each implementation variant. For implementation variants containing tiled loops, suitable tile sizes are selected using a heuristic empirical search guided by an analytical model. Runtime experiments with two different test problems show the efficiency of the online auto-tuning approach on two different shared-memory systems equipped with 48 and 1040 cores.

A Matrix-Free Newton–Krylov Parallel Implicit Implementation of the Absolute Nodal Coordinate Formulation

This paper sets out to demonstrate three things: (i) implicit integration with absolute nodal coordinate formulation (ANCF) is effective in handling very stiff systems when an accurate computation of the sensitivity matrix is part of the solution sequence, (ii) parallel computing can provide a vehicle for ANCF to tackle very large kinematically constrained problems with millions of degrees of freedom and produce results in a matter of seconds, and (iii) large systems of equations associated with implicit integration can be solved in parallel by relying on an iterative approach that avoids costly matrix factorizations, which would be prohibitively expensive and memory intensive. For (iii), the approach adopted relies on a Krylov–subspace method that is invoked in the Newton stage at each time step of the numerical solution process. The proposed approach is validated against a commercial package and several simple systems for which analytical solutions are available. A set of numerical experiments demonstrates the scaling of the parallel solution method and provides insights in relation to the size of ANCF problems that are tractable using graphics processing unit (GPU) parallel computing and implicit numerical integration.

Finite Element Tearing and Interconnecting Method and its Algorithms for Parallel Solution of Magnetic Field Problems

Abstract Because of the exponential increase of computational resource requirement for numerical field simulations of more and more complex physical phenomena and more and more complex large problems in science and engineering practice, parallel processing appears to be an essential tool to handle the resulting large-scale numerical problems. One way of parallelization of sequential (singleprocessor) finite element simulations is the use of domain decomposition methods. Domain decomposition methods (DDMs) for parallel solution of linear systems of equations are based on the partitioning of the analyzed domain into sub-domains which are calculated in parallel while doing appropriate data exchange between those sub-domains. In this case, the non-overlapping domain decomposition method is the Lagrange multiplier based Finite Element Tearing and Interconnecting (FETI) method. This paper describes one direct solver and two parallel solution algorithms of FETI method. Finally, comparative numerical tests demonstrate the differences in the parallel running performance of the solvers of FETI method. We use a single-phase transformer and a three-phase induction motor as twodimensional static magnetic field test problems to compare the solvers