Infinite Element in Meshless Approaches
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A High-Order Boris Integrator
A high-order Boris integrator Mathias Winkela,∗, Robert Speckb,a, Daniel Ruprechta aInstitute of Computational Science, University of Lugano, Switzerland. bJ¨ulichSupercomputing Centre, Forschungszentrum J¨ulich,Germany. Abstract This work introduces the high-order Boris-SDC method for integrating the equations of motion for electrically charged particles in an electric and magnetic field. Boris-SDC relies on a combination of the Boris-integrator with spectral deferred corrections (SDC). SDC can be considered as preconditioned Picard iteration to com- pute the stages of a collocation method. In this interpretation, inverting the preconditioner corresponds to a sweep with a low-order method. In Boris-SDC, the Boris method, a second-order Lorentz force integrator based on velocity-Verlet, is used as a sweeper/preconditioner. The presented method provides a generic way to extend the classical Boris integrator, which is widely used in essentially all particle-based plasma physics simulations involving magnetic fields, to a high-order method. Stability, convergence order and conserva- tion properties of the method are demonstrated for different simulation setups. Boris-SDC reproduces the expected high order of convergence for a single particle and for the center-of-mass of a particle cloud in a Penning trap and shows good long-term energy stability. Keywords: Boris integrator, time integration, magnetic field, high-order, spectral deferred corrections (SDC), collocation method 1. Introduction Often when modeling phenomena in plasma physics, for example particle dynamics in fusion vessels or particle accelerators, an externally applied magnetic field is vital to confine the particles in the physical device [1, 2]. In many cases, such as instabilities [3] and high-intensity laser plasma interaction [4], the magnetic field even governs the microscopic evolution and drives the phenomena to be studied. -
(PFEM) for Solid Mechanics
Development and preliminary evaluation of the main features of the Particle Finite Element Method (PFEM) for solid mechanics Vasilis Papakrivopoulos Development and preliminary evaluation of the main features of the Particle Finite Element Method (PFEM) for solid mechanics by Vasilis Papakrivopoulos to obtain the degree of Master of Science at the Delft University of Technology, to be defended publicly on Monday December 10, 2018 at 2:00 PM. Student number: 4632567 Project duration: February 15, 2018 – December 10, 2018 Thesis committee: Dr. P.J.Vardon, TU Delft (chairman) Prof. dr. M.A. Hicks, TU Delft Dr. F.Pisanò, TU Delft J.L. Gonzalez Acosta, MSc, TU Delft (daily supervisor) An electronic version of this thesis is available at http://repository.tudelft.nl/. PREFACE This work concludes my academic career in the Technical University of Delft and marks the end of my stay in this beautiful town. During the two-year period of my master stud- ies I managed to obtain experiences and strengthen the theoretical background in the civil engineering field that was founded through my studies in the National Technical University of Athens. I was given the opportunity to come across various interesting and challenging topics, with this current project being the highlight of this course, and I was also able to slowly but steadily integrate into the Dutch society. In retrospect, I feel that I have made the right choice both personally and career-wise when I decided to move here and study in TU Delft. At this point, I would like to thank the graduation committee of my master thesis. -
Strong Form Meshfree Collocation Method for Higher Order and Nonlinear Pdes in Engineering Applications
University of Colorado, Boulder CU Scholar Civil Engineering Graduate Theses & Dissertations Civil, Environmental, and Architectural Engineering Spring 1-1-2019 Strong Form Meshfree Collocation Method for Higher Order and Nonlinear Pdes in Engineering Applications Andrew Christian Beel University of Colorado at Boulder, [email protected] Follow this and additional works at: https://scholar.colorado.edu/cven_gradetds Part of the Civil Engineering Commons, and the Partial Differential Equations Commons Recommended Citation Beel, Andrew Christian, "Strong Form Meshfree Collocation Method for Higher Order and Nonlinear Pdes in Engineering Applications" (2019). Civil Engineering Graduate Theses & Dissertations. 471. https://scholar.colorado.edu/cven_gradetds/471 This Thesis is brought to you for free and open access by Civil, Environmental, and Architectural Engineering at CU Scholar. It has been accepted for inclusion in Civil Engineering Graduate Theses & Dissertations by an authorized administrator of CU Scholar. For more information, please contact [email protected]. Strong Form Meshfree Collocation Method for Higher Order and Nonlinear PDEs in Engineering Applications Andrew Christian Beel B.S., University of Colorado Boulder, 2019 A thesis submitted to the Faculty of the Graduate School of the University of Colorado Boulder in partial fulfillment of the requirement for the degree of Master of Science Department of Civil, Environmental and Architectural Engineering 2019 This thesis entitled: Strong Form Meshfree Collocation Method for Higher Order and Nonlinear PDEs in Engineering Applications written by Andrew Christian Beel has been approved for the Department of Civil, Environmental and Architectural Engineering Dr. Jeong-Hoon Song (Committee Chair) Dr. Ronald Pak Dr. Victor Saouma Dr. Richard Regueiro Date The final copy of this thesis has been examined by the signatories, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline. -
Overview of Meshless Methods
Technical Article Overview of Meshless Methods Abstract— This article presents an overview of the main develop- ments of the mesh-free idea. A review of the main publications on the application of the meshless methods in Computational Electromagnetics is also given. I. INTRODUCTION Several meshless methods have been proposed over the last decade. Although these methods usually all bear the generic label “meshless”, not all are truly meshless. Some, such as those based on the Collocation Point technique, have no associated mesh but others, such as those based on the Galerkin method, actually do Fig. 1. EFG background cell structure. require an auxiliary mesh or cell structure. At the time of writing, the authors are not aware of any proposed formal classification of these techniques. This paper is therefore not concerned with any C. The Element-Free Galerkin (EFG) classification of these methods, instead its objective is to present In 1994 Belytschko and colleagues introduced the Element-Free an overview of the main developments of the mesh-free idea, Galerkin Method (EFG) [8], an extended version of Nayroles’s followed by a review of the main publications on the application method. The Element-Free Galerkin introduced a series of im- of meshless methods to Computational Electromagnetics. provements over the Diffuse Element Method formulation, such as II. MESHLESS METHODS -THE HISTORY • Proper determination of the approximation derivatives: A. The Smoothed Particle Hydrodynamics In DEM the derivatives of the approximation function The advent of the mesh free idea dates back from 1977, U h are obtained by considering the coefficients b of the with Monaghan and Gingold [1] and Lucy [2] developing a polynomial basis p as constants, such that Lagrangian method based on the Kernel Estimates method to h T model astrophysics problems. -
Workshop High-Order Numerical Approximation for Partial Differential Equations
Workshop High-Order Numerical Approximation for Partial Differential Equations Hausdorff Center for Mathematics University of Bonn February 6-10, 2012 Organizers Alexey Chernov (University of Bonn) Christoph Schwab (ETH Zurich) Program overview Time Monday Tuesday Wednesday Thursday Friday 8:00 Registration 8:40 Opening 9:00 Stephan Sloan Ainsworth Demkowicz Buffa 10:00 Canuto Nobile Sherwin Houston Beir~ao da Veiga 10:30 Reali 11:00 Coffee break 11:30 Quarteroni Gittelson Sch¨oberl Sch¨otzau Sangalli 12:00 Bespalov 12:30 Lunch break 14:30 Dauge Xiu Wihler D¨uster Excursions: 15:00 Yosibash Arithmeum 15:30 Melenk Litvinenko Hesthaven or Coffee break 16:00 Maischak Rank Botanic 16:30 Coffee break Coffee break Gardens 17:00 Maday Hackbusch Hiptmair {18:00 Free Time 20:00 Dinner 2 Detailed program Monday, 9:00{9:55 Ernst P. Stephan hp-adaptive DG-FEM for Parabolic Obstacle Problems Monday, 10:00{10:55 Claudio Canuto Adaptive Fourier-Galerkin Methods Monday, 11:30{12:25 Alfio Quarteroni Discontinuous approximation of elastodynamics equations Monday, 14:30{15:25 Monique Dauge Weighted analytic regularity in polyhedra Monday, 15:30{16:25 Jens Markus Melenk Stability and convergence of Galerkin discretizations of the Helmholtz equation Monday, 17:00{17:55 Yvon Maday A generalized empirical interpolation method based on moments and application Tuesday, 9:00{9:55 Ian H. Sloan PDE with random coefficients as a problem in high dimensional integration Tuesday, 10:00{10:55 Fabio Nobile Stochastic Polynomial approximation of PDEs with random coefficients Tuesday, -
Computational Fluid and Solid Mechanics
COMPUTATIONAL FLUID AND SOLID MECHANICS Proceedings First MIT Conference on Computational Fluid and Solid Mechanics June 12-15,2001 Editor: K.J. Bathe Massachusetts Institute of Technology, Cambridge, MA, USA VOLUME 2 UNIVERSITATSBIBLIOTHEK HANNOVER TECHKUSCHE INFORMATIONSBIBLIOTHEK 2001 ELSEVIER Amsterdam - London - New York - Oxford - Paris - Shannon - Tokyo Contents Volume 2 Preface v Session Organizers vi Fellowship Awardees vii Sponsors ix Fluids Achdou, Y., Pironneau, O., Valentin, E, Comparison of wall laws for unsteady incompressible Navier-Stokes equations over rough interfaces 762 Allik, H., Dees, R.N., Oppe, T.C., Duffy, D., Dual-level parallelization of structural acoustics computations 764 Altai, W., Chu, V, K-e Model simulation by Lagrangian block method 767 Alves, M.A., Oliveira, P.J., Pinho, F.T., Numerical simulations of viscoelastic flow around sharp corners 772 Badeau,A., CelikJ., A droplet formation model for stratified liquid-liquid shear flows 776 Balage, S., Saghir, M.Z., Buoyancy and Marangoni convections of Te-doped GaSb 779 Bauer, A.C., Patra, A.K., Preconditioners for parallel adaptive hp FEM for incompressible flows 782 Berger, S.A., Stroud, J.S., Flow in sclerotic carotid arteries 786 Bouhairie, S., Chu, V.H., Gehr, R., Heat transfer calculations of high-Reynolds-number flows around a circular cylinder 791 Cabral, E.L.L., Sabundjian, G, Hierarchical expansion method in the solution of the Navier-Stokes equations for incompressible fluids in laminar two-dimensional flow 795 Chaidron, G., Chinesta, E, On the -
ICES REPORT 14-04 Collocation on Hp Finite Element Meshes
ICES REPORT 14-04 February 2014 Collocation On Hp Finite Element Meshes: Reduced Quadrature Perspective, Cost Comparison With Standard Finite Elements, And Explicit Structural Dynamics by Dominik Schillinger, John a. Evans, Felix Frischmann, Rene R. Hiemstra, Ming-Chen Hsu, Thomas J.R. Hughes The Institute for Computational Engineering and Sciences The University of Texas at Austin Austin, Texas 78712 Reference: Dominik Schillinger, John a. Evans, Felix Frischmann, Rene R. Hiemstra, Ming-Chen Hsu, Thomas J.R. Hughes, "Collocation On Hp Finite Element Meshes: Reduced Quadrature Perspective, Cost Comparison With Standard Finite Elements, And Explicit Structural Dynamics," ICES REPORT 14-04, The Institute for Computational Engineering and Sciences, The University of Texas at Austin, February 2014. Collocation on hp finite element meshes: Reduced quadrature perspective, cost comparison with standard finite elements, and explicit structural dynamics Dominik Schillingera,b,∗, John A. Evansc, Felix Frischmannd, Ren´eR. Hiemstrab, Ming-Chen Hsue, Thomas J.R. Hughesb aDepartment of Civil Engineering, University of Minnesota, Twin Cities, USA bInstitute for Computational Engineering and Sciences, The University of Texas at Austin, USA cAerospace Engineering Sciences, University of Colorado Boulder, USA dLehrstuhl f¨ur Computation in Engineering, Technische Universit¨at M¨unchen, Germany eDepartment of Mechanical Engineering, Iowa State University, Ames, USA Abstract We demonstrate the potential of collocation methods for efficient higher-order analysis on standard nodal finite element meshes. We focus on a collocation method that is variation- ally consistent and geometrically flexible, converges optimally, embraces concepts of reduced quadrature, and leads to symmetric stiffness and diagonal consistent mass matrices. At the same time, it minimizes the evaluation cost per quadrature point, thus reducing formation and assembly effort significantly with respect to standard Galerkin finite element methods. -
The Numerical Solution of Helmholtz Equation Using Modified Wavelet Multigrid Method
International Journal of Modern Mathematical Sciences, 2020, 18(1): 76-91 International Journal of Modern Mathematical Sciences ISSN:2166-286X Journal homepage:www.ModernScientificPress.com/Journals/ijmms.aspx Florida, USA Article The Numerical Solution of Helmholtz Equation Using Modified Wavelet Multigrid Method S. C. Shiralashetti1, A. B. Deshi2,*, M. H. Kantli3 1Department of Mathematics, Karnatak University, Dharwad, India-580003 2Department of Mathematics, KLECET, Chikodi, India-591201 3Department of Mathematics, BVV’s BGMIT, Mudhol, India-587313 *Author to whom correspondence should be addressed; E-Mail: [email protected] Article history: Received 11 June 2020, Revised 23 July 2020, Accepted 1 September 2020, Published 14 September 2020. Abstract: This paper presents a modified wavelet multigrid technique for solving elliptic type partial differential equations namely Helmholtz equation. The solution is first obtained on the coarser grid points, and then it is refined by obtaining higher accuracy by increasing the level of resolution. The implementation of the classical numerical methods has been found to involve some difficulty to observe fast convergence in low computational time. To overcome this, we have proposed modified wavelet multigrid method using wavelet intergrid operators similar to classical intergrid operators. Some of the numerical test problems are presented to demonstrate the applicability and attractiveness of the present implementation. Keywords: Wavelet multigrid; Helmholtz equation; Intergrid operators; Numerical methods. Mathematics Subject Classification: 65T60, 97N40, 35J25 1. Introduction The mathematical modelling of engineering problems usually leads to sets of partial differential equations and their boundary conditions. There are several applications of elliptic partial differential equations (EPDEs) in science and engineering. Many physical processes can be modeled using EPDEs. -
Partial Differential Equations
Next: Using Matlab Up: Numerical Analysis for Chemical Previous: Ordinary Differential Equations Subsections z Finite Difference: Elliptic Equations { The Laplace Equations { Solution Techniques { Boundary Conditions { The Control Volume Approach z Finite Difference: Parabolic Equations { The Heat Conduction Equation { Explicit Methods { A Simple Implicit Method { The Crank-Nicholson Method z Finite Element Method { Calculus of variation { Example: The shortest distance between two points { The Rayleigh-Ritz Method { The Collocation and Galerkin Method { Finite elements for ordinary-differential equations z Engineering Applications: Partial Differential Equations Partial Differential Equations An equation involving partial derivatives of an unknown function of two or more independent variables is called a partial differential equation, PDE. The order of a PDE is that of the highest-order partial derivative appearing in the equation. A general linear second-order differential equation is (8.1) Depending on the values of the coefficients of the second-derivative terms eq. (8.1) can be classified int one of three categories. z : Elliptic Laplace equation(steady state with two spatial dimensions) z : Parabolic Heat conduction equation(time variable with one spatial dimension) z : Hyperbolic Wave equation(time variable with one spatial dimension) Finite Difference: Elliptic Equations Elliptic equations in engineering are typically used to characterize steady-state, boundary-value problems. The Laplace Equations z The Laplace equation z The Poisson equation Solution Techniques z The Laplacian Difference Equation : use central difference based on the grid scheme and Substituting these equations into the Laplace equation gives For the square grid, , and by collecting terms This relationship, which holds for all interior point on the plate, is referred to as the Laplacian difference equation. -
65 Numerical Analysis
e Q (e t o 5 M SectionsSet 1Q (Section 65)MR September 2012 65 NUMERICAL ANALYSIS MR2918625 65-06 FRecent advances in scientific computing and matrix analysis. Proceedings of the International Workshop held at the University of Macau, Macau, December 28{30, 2009. Edited by Xiao-Qing Jin, Hai-Wei Sun and Seak-Weng Vong. International Press, Somerville, MA; Higher Education Press, Beijing, 2011. xii+126 pp. ISBN 978-1-57146-202-2 Contents: Zheng-jian Bai and Xiao-qing Jin [Xiao Qing Jin1], A note on the Ulm-like method for inverse eigenvalue problems (1{7) MR2908437; Che-man Cheng [Che-Man Cheng], Kin-sio Fong [Kin-Sio Fong] and Io-kei Lok [Io-Kei Lok], Another proof for commutators with maximal Frobenius norm (9{14) MR2908438; Wai-ki Ching [Wai-Ki Ching] and Dong-mei Zhu [Dong Mei Zhu1], On high-dimensional Markov chain models for categorical data sequences with applications (15{34) MR2908439; Yan-nei Law [Yan Nei Law], Hwee-kuan Lee [Hwee Kuan Lee], Chao-qiang Liu [Chaoqiang Liu] and Andy M. Yip, An additive variational model for image segmentation (35{48) MR2908440; Hai-yong Liao [Haiyong Liao] and Michael K. Ng, Total variation image restoration with automatic selection of regularization parameters (49{59) MR2908441; Franklin T. Luk and San-zheng Qiao [San Zheng Qiao], Matrices and the LLL algorithm (61{69) MR2908442; Mila Nikolova, Michael K. Ng and Chi-pan Tam [Chi-Pan Tam], A fast nonconvex nonsmooth minimization method for image restoration and reconstruction (71{83) MR2908443; Gang Wu [Gang Wu1], Eigenvalues of certain augmented complex stochastic matrices with applications to PageRank (85{92) MR2908444; Yan Xuan and Fu-rong Lin, Clenshaw-Curtis-rational quadrature rule for Wiener-Hopf equations of the second kind (93{110) MR2908445; Man-chung Yeung [Man-Chung Yeung], On the solution of singular systems by Krylov subspace methods (111{116) MR2908446; Qi- fang Yu, San-zheng Qiao [San Zheng Qiao] and Yi-min Wei, A comparative study of the LLL algorithm (117{126) MR2908447. -
A Technique to Combine Meshfree and Finite- Element
A Technique to Combine Meshfree- and Finite Element-Based Partition of Unity Approximations C. A. Duartea;∗, D. Q. Miglianob and E. B. Beckerb a Department of Civil and Environmental Eng. University of Illinois at Urbana-Champaign Newmark Laboratory, 205 North Mathews Avenue Urbana, Illinois 61801, USA ∗Corresponding author: [email protected] b ICES - Institute for Computational Engineering and Science The University of Texas at Austin, Austin, TX, 78712, USA Abstract A technique to couple finite element discretizations with any partition of unity based approximation is presented. Emphasis is given to the combination of finite element and meshfree shape functions like those from the hp cloud method. H and p type approximations of any polynomial degree can be built. The procedure is essentially the same in any dimension and can be used with any Lagrangian finite element dis- cretization. Another contribution of this paper is a procedure to built generalized finite element shape functions with any degree of regularity using the so-called R-functions. The technique can also be used in any dimension and for any type of element. Numer- ical experiments demonstrating the coupling technique and the use of the proposed generalized finite element shape functions are presented. Keywords: Meshfree methods; Generalized finite element method; Partition of unity method; Hp-cloud method; Adaptivity; P-method; P-enrichment; 1 Introduction One of the major difficulties encountered in the finite element analysis of tires, elastomeric bear- ings, seals, gaskets, vibration isolators and a variety of other of products made of rubbery mate- rials, is the excessive element distortion. Distortion of elements is inherent to Lagrangian formu- lations used to analyze this class of problems. -
ANALYSIS of a TIME MULTIGRID ALGORITHM for DG-DISCRETIZATIONS in TIME 1. Introduction. the Parallelization of Algorithms For
ANALYSIS OF A TIME MULTIGRID ALGORITHM FOR DG-DISCRETIZATIONS IN TIME MARTIN J. GANDER ∗ AND MARTIN NEUMULLER¨ † Abstract. We present and analyze for a scalar linear evolution model problem a time multigrid algorithm for DG-discretizations in time. We derive asymptotically optimized parameters for the smoother, and also an asymptotically sharp convergence estimate for the two grid cycle. Our results hold for any A-stable time stepping scheme and represent the core component for space-time multigrid methods for parabolic partial differential equations. Our time multigrid method has excellent strong and weak scaling properties for parallelization in time, which we show with numerical experiments. Key words. Time parallel methods, multigrid in time, DG-discretizations, RADAU IA AMS subject classifications. 65N55, 65L60, 65F10 1. Introduction. The parallelization of algorithms for evolution problems in the time direction is currently an active area of research, because today’s supercomputers with their millions of cores can not be effectively used any more when only paralleliz- ing the spatial directions. In addition to multiple shooting and parareal [21, 15, 10], domain decomposition and waveform relaxation [14, 13, 11], and direct time parallel methods [23, 9, 12], multigrid methods in time are the fourth main approach that can be used to this effect, see the overview [8] and references therein. The parabolic multigrid method proposed by Hackbusch in [16] was the first multigrid method in space-time. It uses a smoothing iteration (e.g. Gauss-Seidel) over many time levels, but coarsening is in general only possible in space, since time coarsening might lead to divergence of the algorithm.