DOCSLIB.ORG

Multigrid Method for This Step

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Multigrid Method for This Step The Pennsylvania State University The Graduate School College of Engineering PARALLEL PARTICLE-IN-CELL PERFORMANCE OPTIMIZATION: A CASE STUDY OF ELECTROSPRAY SIMULATION A Thesis in Computer Science and Engineering by Ramachandran Kodanganallur Narayanan © 2016 Ramachandran Kodanganallur Narayanan Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science May 2016 The thesis of Ramachandran Kodanganallur Narayanan was reviewed and approved∗ by the following: Kamesh Madduri Assistant Professor in Department of Computer Science and Engineering Thesis Advisor Mahmut Taylan Kandemir Professor in Department of Computer Science and Engineering Director of Graduate Studies John Hannan Associate Professor in Department of Computer Science and Engineering Interim Associate Department Head ∗Signatures are on file in the Graduate School. ii Abstract The particle-in-cell (PIC) numerical technique is frequently used in physics and engineering simulations. In this work, we describe ES-PICBench, a new shared- memory parallel implementation of the PIC technique for electrospray simulations. Electrospray simulations are used in aerospace applications, and the goal of an electropray simulation is to understand behavior of an electrospray thruster or a colloid thruster. We discuss performance optimizations for various steps of a PIC- based electrospray simulation. One of the main steps in this simulation is solving the Poisson partial differential equation, and this step can be in turn converted to solving a system of linear equations. We develop a parallel implementation of the Multigrid method for this step. We demonstrate that ES-PICBench is significantly faster than other parallel PIC electrospray simulation implementations on Intel Xeon multicore platforms. Further, ES-PICBench can serve as a real-world scientific computing benchmark for analyzing parallel system performance. iii Table of Contents List of Figures vii List of Tables ix List of Symbolsx Acknowledgments xii Chapter 1 Introduction1 Chapter 2 Problem Formulation3 2.1 Problem Parameters .......................... 4 2.2 Methodology .............................. 6 2.3 Prior Work ............................... 8 Chapter 3 PIC Implementation9 3.1 Creation of Grid ............................ 10 3.2 Calculation of Release Rate ...................... 10 3.3 Preallocation and Storage ....................... 11 3.4 Releasing Particles ........................... 11 3.5 Particles leaving the domain...................... 13 3.6 Weighting charges to nodes ...................... 14 3.7 Resort Particles............................. 17 3.8 Resetting RHS Vector ......................... 18 3.9 Moving Particles ............................ 18 3.10 Parallelization.............................. 19 3.10.1 Releasing Particles ....................... 20 iv 3.10.2 Particles leaving the domain.................. 20 3.10.3 Update Charge Fractions.................... 21 3.10.4 Moving Particles ........................ 22 3.10.5 Performance........................... 22 3.11 Results-Particle Distribution...................... 24 Chapter 4 Poisson Solve 27 4.1 Initial Serial Solver........................... 28 4.1.1 Matrix-free approach...................... 28 4.1.2 Exploration of Solver Options with PETSc.......... 31 4.2 Results.................................. 32 Chapter 5 Multigrid Method 35 5.1 Methodology .............................. 35 5.1.1 Pre/Post Smoother....................... 36 5.1.2 Restriction............................ 36 5.1.3 Recurse/Direct Solve...................... 36 5.1.4 Prolongation .......................... 37 5.1.5 Matrix-Free Approach ..................... 37 5.2 Implementation............................. 39 5.3 Validation................................ 41 5.3.1 Dirichlet conditions....................... 41 5.3.2 Mixed Neumann ........................ 43 5.3.3 Small domain length issue................... 45 5.3.4 Troubleshooting Multigrid................... 45 5.4 Comparison with PETSc........................ 46 5.5 Parallelization.............................. 46 5.5.1 Smoother ............................ 46 5.5.2 Direct Solve........................... 47 5.5.3 Remaining steps......................... 48 5.5.4 Results.............................. 48 5.5.5 SubComponent Timings .................... 49 5.5.5.1 Smoother....................... 50 5.5.5.2 Restriction and Prolongation ............ 51 5.5.6 Smoother Performance..................... 52 5.6 Code Results .............................. 55 5.6.1 Potential Distribution ..................... 55 5.6.2 Effect of Particles on Potential................. 55 v Chapter 6 Conclusions and Future Work 57 6.1 Future work............................... 57 Bibliography 59 vi List of Figures 2.1 Side View With Boundary Conditions. ................ 4 2.2 View from the Capillary with Boundary Conditions.......... 5 2.3 View from the Extractor with Boundary Conditions. ........ 5 3.1 Outline of Particle class ........................ 9 3.2 Insertion of new particles into gaps .................. 13 3.3 Swap particles at end to cover leftover gaps.............. 14 3.4 Charges in a sample 2D grid. Arrows show the contribution towards a particular node............................. 15 3.5 Area weighted contribution of charge.................. 15 3.6 Laplace Particle Routine Scalability.................. 24 3.7 Particle Distribution at Timestep 50 ................. 25 3.8 Particle Distribution at Timestep 100................. 25 3.9 Particle Distribution at Timestep 150................. 25 3.10 Particle Distribution at Timestep 250................. 25 3.11 Front view of distribution at 250 timesteps.............. 26 4.1 Form of Matrix A with coefficients in case of Laplace equation. 27 4.2 7-pt stencil with Axes convention.................... 28 4.3 Sparsity Pattern of matrix A. ..................... 29 4.4 Evaluating Ax = b............................ 30 4.5 Graph indicating scaling for Poisson Run. .............. 34 5.1 Allocation of hierarchy of grids for a 2D face.............. 40 5.2 Form of the V-Cycle........................... 40 5.3 Iterations taken for varying Problem Size. .............. 42 5.4 Iterations taken for varying Problem Size............... 44 5.5 Red-Black coloring ........................... 47 5.6 Multigrid Dirichlet (9 levels, 3 coarse points) Speedup graph. 49 5.7 Scalability plot of Smoother from the Subcomponent Timings. 51 vii 5.8 Plot of Restriction and Prolongation Scalability............ 52 5.9 Smoother Scalability with 7 Multigrid levels.............. 54 5.10 Potential Distribution at Timestep 50................. 55 5.11 Electric Field at Timestep 50 ..................... 55 5.12 Potential Contours Timestep 50.................... 56 5.13 Potential Contours at Timestep 250.................. 56 viii List of Tables 2.1 Domain Lengths and Voltage Values used............... 4 3.1 Scalability study for the particle routines. .............. 23 3.2 Speedup values for the Moving Particles routine and for the Total Time. .................................. 23 4.1 PETSc-3.4.3 Solver and Preconditioner combinations explored. 31 4.2 PETSc-3.6.2 Solver and Preconditioner combinations explored. 31 4.3 Details of Performance Setup on a Single Node on Ganga Cluster. 32 4.4 Scaling results for the Poisson Run with 500 Timesteps. 33 5.1 Validation of Multigrid Solver with Dirichlet conditions. 42 5.2 Validation of the Multigrid solver with Neumann Boundary Conditions. 43 5.3 Multigrid Dirichlet with 9 levels and 3 points on coarsest level. 48 5.4 Timings for the individual subcomponent steps in the Multigrid Dirichlet case............................... 50 5.5 Scalability study of Smoother from the Subcomponent Timings. 50 5.6 Tabulation of Restriction and Prolongation Scalability. 52 5.7 Details of the Memory Hierarchy for LionXG. ............ 53 5.8 Cachegrind Performance Results.................... 53 5.9 Standalone Smoother Scalability Setup with 7 Multigrid levels and 3 coarse grid points. .......................... 54 ix List of Symbols ∇ Gradient operator, p.3 Φ Electric potential, p.3 ρq Density of Space charge, p.3 o Free space permittivity, p.3 E Electric Field, p.3 NMD Release rate of particles in the Molecular Dynamics code, p. 10 ∆tMD Molecular Dynamics code timestep, p. 10 Np,rel Release rate of particles in a PIC timestep, p. 10 tPIC PIC timestep, p. 10 Ip,rel Integer part of Np,rel, p. 10 Dp,rel Decimal part of Np,rel, p. 10 ryz,MD The magnitude of position vector of released particle from capillary center, p. 11 vyz,MD The magnitude of velocity of released particle from capillary center, p. 11 θ, η Angles used while randomizing particle attributes, p. 11 yPIC , zPIC New YZ positions after rotating position vector through angles, p. 11 x vy,P IC , vz,P IC New YZ velocities after rotating velocity vector through angles, p. 11 q Charge on particle, p. 18 F~ Force acting on particle of charge q, p. 18 mp Mass of particle, p. 18 ~v t Velocity of particle at time t, p. 18 xi Acknowledgments This research is supported in part by the US National Science Foundation award #1253881. Firstly, I would like to thank Dr. Kamesh Madduri for the support throughout this research project and his guidance on tackling various issues at different stages of the project. I had learnt valuable lessons prototyping an algorithm and the aspects to look out for, while aiming for performance. The ability to look at problems farther away as well as dive into the details, was particularly instructive. I would
Load more

Recommended publications
  • Multigrid Methods
    Multigrid Methods Volker John Winter Semester 2013/14 Contents 1 Literature 2 2 Model Problems 3 3 Detailed Investigation of Classical Iterative Schemes 8 3.1 General Aspects of Classical Iterative Schemes . .8 3.2 The Jacobi and Damped Jacobi Method . 10 3.3 The Gauss{Seidel Method and the SOR Method . 15 3.4 Summary . 18 4 Grid Transfer 19 4.1 Algorithms with Coarse Grid Systems, the Residual Equation . 19 4.2 Prolongation or Interpolation . 21 4.3 Restriction . 23 5 The Two-Level Method 27 5.1 The Coarse Grid Problem . 27 5.2 General Approach for Proving the Convergence of the Two-Level Method . 30 5.3 The Smoothing Property of the Damped Jacobi Iteration . 32 5.4 The Approximation Property . 33 5.5 Summary . 36 6 The Multigrid Method 37 6.1 Multigrid Cycles . 37 6.2 Convergence of the W-cycle . 39 6.3 Computational Work of the Multigrid γ-Cycle . 45 7 Algebraic Multigrid Methods 49 7.1 Components of an AMG and Definitions . 49 7.2 Algebraic Smoothness . 52 7.3 Coarsening . 56 7.4 Prolongation . 58 7.5 Concluding Remarks . 60 8 Outlook 61 1 Chapter 1 Literature Remark 1.1 Literature. There are several text books about multigrid methods, e.g., • Briggs et al. (2000), easy to read introduction, • Hackbusch (1985), the classical book, sometimes rather hard to read, • Shaidurov (1995), • Wesseling (1992), an introductionary book, • Trottenberg et al. (2001). 2 2 Chapter 2 Model Problems Remark 2.1 Motivation. The basic ideas and properties of multigrid methods will be explained in this course on two model problems.
    [Show full text]
  • Hybrid Particle-Grid Water Simulation Using Multigrid Pressure Solver Per Karlsson
    LiU-ITN-TEK-G--14/006-SE Hybrid Particle-Grid Water Simulation using Multigrid Pressure Solver Per Karlsson 2014-03-13 Department of Science and Technology Institutionen för teknik och naturvetenskap Linköping University Linköpings universitet nedewS ,gnipökrroN 47 106-ES 47 ,gnipökrroN nedewS 106 47 gnipökrroN LiU-ITN-TEK-G--14/006-SE Hybrid Particle-Grid Water Simulation using Multigrid Pressure Solver Examensarbete utfört i Medieteknik vid Tekniska högskolan vid Linköpings universitet Per Karlsson Handledare George Baravdish Examinator Camilla Forsell Norrköping 2014-03-13 Upphovsrätt Detta dokument hålls tillgängligt på Internet – eller dess framtida ersättare – under en längre tid från publiceringsdatum under förutsättning att inga extra- ordinära omständigheter uppstår. Tillgång till dokumentet innebär tillstånd för var och en att läsa, ladda ner, skriva ut enstaka kopior för enskilt bruk och att använda det oförändrat för ickekommersiell forskning och för undervisning. Överföring av upphovsrätten vid en senare tidpunkt kan inte upphäva detta tillstånd. All annan användning av dokumentet kräver upphovsmannens medgivande. För att garantera äktheten, säkerheten och tillgängligheten finns det lösningar av teknisk och administrativ art. Upphovsmannens ideella rätt innefattar rätt att bli nämnd som upphovsman i den omfattning som god sed kräver vid användning av dokumentet på ovan beskrivna sätt samt skydd mot att dokumentet ändras eller presenteras i sådan form eller i sådant sammanhang som är kränkande för upphovsmannens litterära eller konstnärliga anseende eller egenart. För ytterligare information om Linköping University Electronic Press se förlagets hemsida http://www.ep.liu.se/ Copyright The publishers will keep this document online on the Internet - or its possible replacement - for a considerable time from the date of publication barring exceptional circumstances.
    [Show full text]
  • Multigrid Algorithms for High Order Discontinuous Galerkin Methods
    Multigrid algorithms for high order discontinuous Galerkin methods Paola F. Antonietti1, Marco Sarti2, and Marco Verani3 Abstract I n this paper we study the performance of h- and p-multigrid algorithms for high order Discontinuous Galerkin discretizations of elliptic problems. We test the performance of the multigrid schemes employing a wide class of smoothers and considering both two- and three-dimensional test cases. 1 Introduction In the framework of multigrid solvers for Discontinuous Galerkin (DG) schemes, the first contributions are due to Gopalakrishnan and Kanschat [16] and Brenner and Zhao [10]. In [16] a V-cycle preconditioner for a Sym- metric Interior Penalty (SIP) discretization of an elliptic problem is analyzed. They prove that the condition number of the preconditioned system is uni- formly bounded with respect to the mesh size and the number of levels. In [10] V-cycle, F-cycle and W-cycle multigrid schemes for SIP discretizations are presented and analyzed, employing the additive theory developed in [8, 9]. A uniform bound for the error propagation operator is shown provided the number of smoothing steps is large enough. All the previously cited works focus on low order, i.e., linear, DG approximations. With regard to high order DG discretizations, h- and p-multigrid schemes are successfully em- ployed for the numerical solution of many different kinds of problems, see e.g. [14, 20, 22, 21, 24, 6], even if only few theoretical results are available that show the convergence properties of the underlying algorithms. In the MOX, Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano ITALY.
    [Show full text]
  • Hybrid Multigrid/Schwarz Algorithms for the Spectral Element Method
    Hybrid Multigrid/Schwarz Algorithms for the Spectral Element Method James W. Lottes¤ and Paul F. Fischery February 4, 2004 Abstract We study the performance of the multigrid method applied to spectral element (SE) discretizations of the Poisson and Helmholtz equations. Smoothers based on finite element (FE) discretizations, overlapping Schwarz methods, and point-Jacobi are con- sidered in conjunction with conjugate gradient and GMRES acceleration techniques. It is found that Schwarz methods based on restrictions of the originating SE matrices converge faster than FE-based methods and that weighting the Schwarz matrices by the inverse of the diagonal counting matrix is essential to effective Schwarz smoothing. Sev- eral of the methods considered achieve convergence rates comparable to those attained by classic multigrid on regular grids. 1 Introduction The availability of fast elliptic solvers is essential to many areas of scientific computing. For unstructured discretizations in three dimensions, iterative solvers are generally optimal from both work and storage standpoints. Ideally, one would like to have computational complexity that scales as O(n) for an n-point grid problem in lRd, implying that the it- eration count should be bounded as the mesh is refined. Modern iterative methods such as multigrid and Schwarz-based domain decomposition achieve bounded iteration counts through the introduction of multiple representations of the solution (or the residual) that allow efficient elimination of the error at each scale. The theory for these methods is well established for classical finite difference (FD) and finite element (FE) discretizations, and order-independent convergence rates are often attained in practice. For spectral element (SE) methods, there has been significant work on the development of Schwarz-based methods that employ a combination of local subdomain solves and sparse global solves to precondition conjugate gradient iteration.
    [Show full text]
  • Multigrid and Saddle-Point Preconditioners for Unfitted Finite
    Multigrid and saddle-point preconditioners for unfitted finite element modelling of inclusions Hardik Kothari∗and Rolf Krause† Institute of Computational Science, Universit`adella Svizzera italiana, Lugano, Switzerland February 17, 2021 Abstract In this work, we consider the modeling of inclusions in the material using an unfitted finite element method. In the unfitted methods, structured background meshes are used and only the underlying finite element space is modified to incorporate the discontinuities, such as inclusions. Hence, the unfitted methods provide a more flexible framework for modeling the materials with multiple inclusions. We employ the method of Lagrange multipliers for enforcing the interface conditions between the inclusions and matrix, this gives rise to the linear system of equations of saddle point type. We utilize the Uzawa method for solving the saddle point system and propose preconditioning strategies for primal and dual systems. For the dual systems, we review and compare the preconditioning strategies that are developed for FETI and SIMPLE methods. While for the primal system, we employ a tailored multigrid method specifically developed for the unfitted meshes. Lastly, the comparison between the proposed preconditioners is made through several numerical experiments. Keywords: Unfitted finite element method, multigrid method, saddle-point problem 1 Introduction In the modeling of many real-world engineering problems, we encounter material discontinuities, such as inclusions. The inclusions are found naturally in the materials, or can be artificially introduced to produce desired mechanical behavior. The finite element (FE) modeling of such inclusions requires to generate meshes that can resolve the interface between the matrix and inclusions, which can be a computationally cumbersome and expensive task.
    [Show full text]
  • Multigrid Method Based on a Space-Time Approach with Standard Coarsening for Parabolic Problems
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CWI's Institutional Repository Applied Mathematics and Computation 317 (2018) 25–34 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc Multigrid method based on a space-time approach with standard coarsening for parabolic problems Sebastião Romero Franco a,b, Francisco José Gaspar c, Marcio Augusto Villela ∗ Pinto d, Carmen Rodrigo e, a Department of Mathematics, State University of Centro-Oeste, Irati, PR, Brazil b Graduate Program in Numerical Methods in Engineering, Federal University of Paraná, Curitiba, PR, Brazil c Centrum Wiskunde & Informatica (CWI), Science Park 123, P.O. Box 94079, Amsterdam 1090, The Netherlands d Department of Mechanical Engineering, Federal University of Paraná, Curitiba, PR, Brazil e IUMA and Applied Mathematics Department, University of Zaragoza, María de Luna, 3, Zaragoza 50018, Spain a r t i c l e i n f o a b s t r a c t MSC: In this work, a space-time multigrid method which uses standard coarsening in both tem- 00-01 poral and spatial domains and combines the use of different smoothers is proposed for the 99-00 solution of the heat equation in one and two space dimensions. In particular, an adaptive Keywords: smoothing strategy, based on the degree of anisotropy of the discrete operator on each Space-time multigrid grid-level, is the basis of the proposed multigrid algorithm. Local Fourier analysis is used Local Fourier analysis for the selection of the crucial parameter defining such an adaptive smoothing approach.
    [Show full text]
  • Electrostatic PIC with Adaptive Cartesian Mesh
    Electrostatic PIC with adaptive Cartesian mesh Vladimir Kolobov1,2 and Robert Arslanbekov1 1 CFD Research Corporation, Huntsville, AL, USA 2 The University of Alabama in Huntsville, Huntsville, AL 35899, USA E-mail: [email protected] Abstract. We describe an initial implementation of an electrostatic Particle-in-Cell (ES-PIC) module with adaptive Cartesian mesh in our Unified Flow Solver framework. Challenges of PIC method with cell-based adaptive mesh refinement (AMR) are related to a decrease of the particle-per-cell number in the refined cells with a corresponding increase of the numerical noise. The developed ES-PIC solver is validated for capacitively coupled plasma, its AMR capabilities are demonstrated for simulations of streamer development during high-pressure gas breakdown. It is shown that cell-based AMR provides a convenient particle management algorithm for exponential multiplications of electrons and ions in the ionization events. 1. Introduction Particle-in-Cell (PIC) is a mature technology widely used for plasma simulations. There are three versions of PIC: an electrostatic (EC) PIC uses Poisson equation for calculating electric field, an electromagnetic (EM) PIC employs full-wave Maxwell solver for calculations of electric and magnetic fields, and a quasi-static (QS) PIC uses a quasi-static version of the Maxwell equations (such as Darwin model) for calculation of the EM fields. Surprisingly, PIC solvers with adaptive mesh refinement (AMR) are relatively rare. It is worth mentioning the WARP code for accelerator applications [1], and an explicit EM-PIC code of Fujimoto [2] for astrophysics applications, both using block-structured AMR. In this paper, we describe an initial implementation of an ES-PIC module in our Unified Flow Solver (UFS) framework [3].
    [Show full text]
  • Hp-Multigrid As Smoother Algorithm for Higher Order Discontinuous Galerkin Discretizations of Advection Dominated Flows Part I
    HP-MULTIGRID AS SMOOTHER ALGORITHM FOR HIGHER ORDER DISCONTINUOUS GALERKIN DISCRETIZATIONS OF ADVECTION DOMINATED FLOWS PART I. MULTILEVEL ANALYSIS J.J.W. VAN DER VEGT∗ AND S. RHEBERGENy AMS subject classifications. Primary 65M55; Secondary 65M60, 76M10. Abstract. The hp-Multigrid as Smoother algorithm (hp-MGS) for the solution of higher order accurate space-(time) discontinuous Galerkin discretizations of advection dominated flows is pre- sented. This algorithm combines p-multigrid with h-multigrid at all p-levels, where the h-multigrid acts as smoother in the p-multigrid. The performance of the hp-MGS algorithm is further improved using semi-coarsening in combination with a new semi-implicit Runge-Kutta method as smoother. A detailed multilevel analysis of the hp-MGS algorithm is presented to obtain more insight into the theoretical performance of the algorithm. As model problem a fourth order accurate space-time dis- continuous Galerkin discretization of the advection-diffusion equation is considered. The multilevel analysis shows that the hp-MGS algorithm has excellent convergence rates, both for low and high cell Reynolds numbers and on highly stretched meshes. Key words. multigrid algorithms, discontinuous Galerkin methods, higher order accurate dis- cretizations, space-time methods, Runge-Kutta methods, Fourier analysis, multilevel analysis 1. Introduction. Discontinuous Galerkin finite element methods are well suited to obtain higher order accurate discretizations on unstructured meshes. The use of basis functions which are only weakly coupled to neighboring elements results in a local discretization which allows, in combination with hp-mesh adaptation, the efficient capturing of detailed structures in the solution, and is also beneficial for parallel computing.
    [Show full text]
  • CONVERGENCE of ALGEBRAIC MULTIGRID BASED on SMOOTHED AGGREGATION∗ ˇ PETR VANEK† , MARIAN BREZINA‡ , and JAN MANDEL§ February 1998 Revised February 2000 Abstract
    CONVERGENCE OF ALGEBRAIC MULTIGRID BASED ON SMOOTHED AGGREGATION∗ ˇ PETR VANEKy , MARIAN BREZINAz , AND JAN MANDELx February 1998 Revised February 2000 Abstract. We prove an abstract convergence estimate for the Algebraic Multigrid Method with prolongator defined by a disaggregation followed by a smoothing. The method input is the problem matrix and a matrix of the zero energy modes of the same problem but with natural boundary conditions. The construction is described in the case of a general elliptic system. The condition number bound increases only as a polynomial of the number of levels, and requires only a uniform weak approximation property for the aggregation operators. This property can be a-priori verified computationally once the aggregates are known. For illustration, it is also verified here for a uniformly elliptic diffusion equations discretized by linear conforming quasiuniform finite elements. Only very weak and natural assumptions on the hierarchy of aggregates are needed. Key words. Algebraic multigrid, zero energy modes, convergence theory, computational mechanics, finite elements, iterative solvers. 1. Introduction. This paper establishes abstract convergence bounds for an Algebraic Multigrid Method (AMG) based on smoothed aggregation. The bounds are obtained by invoking the general convergence theory of [6]. Our main result is a bound on the condition number that grows only as a power of the number of levels, and requires only a weak approximation property for the aggregates, similar to the weak approximation condition in classical AMG investigations [9, 22, 28, 37]. Our weak approximation condition can be easily verified computationally, and we show that it holds for general unstructured meshes and under natural assumptions on aggregates used to construct the coarse levels.
    [Show full text]
  • Multigrid Solvers for Immersed Finite Element Methods and Immersed Isogeometric Analysis
    Computational Mechanics https://doi.org/10.1007/s00466-019-01796-y ORIGINAL PAPER Multigrid solvers for immersed finite element methods and immersed isogeometric analysis F. de Prenter1,3 · C. V. Verhoosel1 · E. H. van Brummelen1 · J. A. Evans2 · C. Messe2,4 · J. Benzaken2,5 · K. Maute2 Received: 26 March 2019 / Accepted: 10 November 2019 © The Author(s) 2019 Abstract Ill-conditioning of the system matrix is a well-known complication in immersed finite element methods and trimmed isogeo- metric analysis. Elements with small intersections with the physical domain yield problematic eigenvalues in the system matrix, which generally degrades efficiency and robustness of iterative solvers. In this contribution we investigate the spectral prop- erties of immersed finite element systems treated by Schwarz-type methods, to establish the suitability of these as smoothers in a multigrid method. Based on this investigation we develop a geometric multigrid preconditioner for immersed finite element methods, which provides mesh-independent and cut-element-independent convergence rates. This preconditioning technique is applicable to higher-order discretizations, and enables solving large-scale immersed systems at a computational cost that scales linearly with the number of degrees of freedom. The performance of the preconditioner is demonstrated for conventional Lagrange basis functions and for isogeometric discretizations with both uniform B-splines and locally refined approximations based on truncated hierarchical B-splines. Keywords Immersed finite element method · Fictitious domain method · Iterative solver · Preconditioner · Multigrid 1 Introduction [14–21], scan based analysis [22–27] and topology optimiza- tion, e.g., [28–34]. Immersed methods are useful tools to avoid laborious and An essential aspect of finite element methods and iso- computationally expensive procedures for the generation of geometric analysis is the computation of the solution to a body-fitted finite element discretizations or analysis-suitable system of equations.
    [Show full text]
  • HMG--Homogeneous Multigrid For
    HMG — HOMOGENEOUS MULTIGRID FOR HDG PEIPEI LU, ANDREAS RUPP, AND GUIDO KANSCHAT Abstract. We introduce a homogeneous multigrid method in the sense that it uses the same HDG discretization scheme for Poisson’s equation on all levels. In particular, we construct a stable injection operator and prove optimal convergence of the method under the assumption of elliptic regularity. Numerical experiments underline our analytical findings. 1. Introduction While hybridizable discontinuous Galerkin (HDG) methods have be- come popular in mathematics and applications over the last years, lit- erature on efficient solution of the resulting discrete systems is still scarce. In this article, we propose to our knowledge the first multigrid preconditioner for such methods which is homogeneous in the sense that it uses the same discretization scheme on all levels. Such methods are important, since they have the same mathematical properties on all levels. They can also be advantageous from a computational point of view, since their data structures and execution patterns are more regular. HDG methods have been gaining popularity in the last decade. Orig- inally, they were analyzed for the Laplacian, see for instance [12] for an overview. Meanwhile, they have been applied to stationary [11, 14, 15, 17, 28, 29] and instationary [25] Stokes systems, to the locking-free discretization of problems in elasticity [19], as well as to plates [9, 24] and beams [7]. However, only few results have been achieved for solv- ing the large systems of equations arising from this type of numerical method, while multigrid [21] and domain decomposition [18] solvers arXiv:2011.14018v1 [math.NA] 27 Nov 2020 for earlier discontinuous Galerkin (DG) schemes have been available for many years.
    [Show full text]
  • An Efficient Finite Element Based Multigrid Method for Simulations Of
    An efficient finite element based multigrid method for simulations of the mechanical behavior of heterogeneous materials using CT images Xiaodong Liu, Julien Réthoré, Marie-Christine Baietto, Philippe Sainsot, Antonius Lubrecht To cite this version: Xiaodong Liu, Julien Réthoré, Marie-Christine Baietto, Philippe Sainsot, Antonius Lubrecht. An effi- cient finite element based multigrid method for simulations of the mechanical behavior of heterogeneous materials using CT images. Computational Mechanics, Springer Verlag, 2020, 10.1007/s00466-020- 01909-y. hal-02919095 HAL Id: hal-02919095 https://hal.archives-ouvertes.fr/hal-02919095 Submitted on 21 Aug 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution| 4.0 International License An efficient finite element based multigrid method for simulations of the mechanical behavior of heterogeneous materials using CT images Xiaodong Liu1, Julien R´ethor´e∗1, Marie-Christine Baietto2, Philippe Sainsot2, and Antonius Adrianus Lubrecht2 1GeM, CNRS UMR 6183, Centrale Nantes, F-44321, France 2Univ Lyon, INSA-Lyon, CNRS UMR5259, LaMCoS, F-69621, France 19 August, 2020 Abstract X-ray tomography techniques give researchers the full access to material inner structures. With such ample information, employing numerical simulation on real material images becomes more and more common.
    [Show full text]