Boundary Particle Method with High-Order Trefftz Functions
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Preprint, 1703.10016, 2017
ASC Report No. 20/2018 Optimal additive Schwarz preconditioning for adaptive 2D IGA boundary element methods T. Fuhrer,¨ G. Gantner, D. Praetorius, and S. Schimanko Institute for Analysis and Scientific Computing Vienna University of Technology | TU Wien www.asc.tuwien.ac.at ISBN 978-3-902627-00-1 Most recent ASC Reports 19/2018 A. Arnold, C. Klein, and B. Ujvari WKB-method for the 1D Schr¨odinger equation in the semi-classical limit: enhanced phase treatment 18/2018 A. Bespalov, T. Betcke, A. Haberl, and D. Praetorius Adaptive BEM with optimal convergence rates for the Helmholtz equation 17/2018 C. Erath and D. Praetorius Optimal adaptivity for the SUPG finite element method 16/2018 M. Fallahpour, S. McKee, and E.B. Weinm¨uller Numerical simulation of flow in smectic liquid crystals 15/2018 A. Bespalov, D. Praetorius, L. Rocchi, and M. Ruggeri Goal-oriented error estimation and adaptivity for elliptic PDEs with parametric or uncertain inputs 14/2018 J. Burkotova, I. Rachunkova, S. Stanek, E.B. Weinm¨uller, S. Wurm On nonsingular BVPs with nonsmooth data. Part 1: Analytical results 13/2018 J. Gambi, M.L. Garcia del Pino, J. Mosser, and E.B. Weinm¨uller Post-Newtonian equations for free-space laser communications between space- based systems 12/2018 T. F¨uhrer, A. Haberl, D. Praetorius, and S. Schimanko Adaptive BEM with inexact PCG solver yields almost optimal computational costs 11/2018 X. Chen and A. J¨ungel Weak-strong uniqueness of renormalized solutions to reaction-cross-diffusion systems 10/2018 C. Erath, G. Gantner, and D. Praetorius Optimal convergence behavior of adaptive FEM driven by simple (h-h/2)-type error estimators Institute for Analysis and Scientific Computing Vienna University of Technology Wiedner Hauptstraße 8{10 1040 Wien, Austria E-Mail: [email protected] WWW: http://www.asc.tuwien.ac.at FAX: +43-1-58801-10196 ISBN 978-3-902627-00-1 c Alle Rechte vorbehalten. -
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. -
Dear Colleagues
Contents Part 1: Plenary Lectures The expanding role of applications in the development and validation of CFD at NASA D. M. Schuster .......…………………………………………….....…………………..………. Thermodynamically consistent systems of hyperbolic equations S. K. Godunov ...................................................................................................................................... Part 2: Keynote Lectures A brief history of shock-fitting M.D. Salas …………………………………………………………………....…..…………. Understanding aerodynamics using computers M.M. Hafez ……….............…………………………………………………………...…..………. Part 3: High-Order Methods A unifying discontinuous CPR formulation for the Navier-Stokes equations on mixed grids Z.J. Wang, H. Gao, T. Haga ………………………………………………………………................. Assessment of the spectral volume method on inviscid and viscous flows O. Chikhaoui, J. Gressier, G. Grondin .........……………………………………..............…..…. .... Runge–Kutta Discontinuous Galerkin method for multi–phase compressible flows V. Perrier, E. Franquet ..........…………………………………………..........………………..……... Energy stable WENO schemes of arbitrary order N.K. Yamaleev, M.H. Carpenter ……………………………………………………….............…… Part 4: Two-Phase Flow A hybrid method for two-phase flow simulations K. Dorogan, J.-M. Hérard, J.-P. Minier ……………………………………………..................…… HLLC-type Riemann solver for the Baer-Nunziato equations of compressible two-phase flow S.A. Tokareva, E.F. Toro ………………………………………………………………...........…… Parallel direct simulation Monte Carlo of two-phase gas-droplet -
Boundary Knot Method for Heat Conduction in Nonlinear Functionally Graded Material
Engineering Analysis with Boundary Elements 35 (2011) 729–734 Contents lists available at ScienceDirect Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound Boundary knot method for heat conduction in nonlinear functionally graded material Zhuo-Jia Fu a,b, Wen Chen a,n, Qing-Hua Qin b a Center for Numerical Simulation Software in Engineering and Sciences, Department of Engineering Mechanics, Hohai University, Nanjing, Jiangsu, PR China b School of Engineering, Building 32, Australian National University, Canberra ACT 0200, Australia article info abstract Article history: This paper firstly derives the nonsingular general solution of heat conduction in nonlinear functionally Received 13 August 2010 graded materials (FGMs), and then presents boundary knot method (BKM) in conjunction with Kirchhoff Accepted 11 December 2010 transformation and various variable transformations in the solution of nonlinear FGM problems. The proposed BKM is mathematically simple, easy-to-program, meshless, high accurate and integration-free, Keywords: and avoids the controversial fictitious boundary in the method of fundamental solution (MFS). Numerical Boundary knot method experiments demonstrate the efficiency and accuracy of the present scheme in the solution of heat Kirchhoff transformation conduction in two different types of nonlinear FGMs. Nonlinear functionally graded material & 2010 Elsevier Ltd. All rights reserved. Heat conduction Meshless 1. Introduction MLPG belong to the category of weak-formulation, and MFS belongs to the category of strong-formulation. Functionally graded materials (FGMs) are a new generation of This study focuses on strong-formulation meshless methods composite materials whose microstructure varies from one mate- due to their inherent merits on easy-to-program and integration- rial to another with a specific gradient. -
Numerische Untersuchung Der Akustischen Eigenschaften Von Trennenden Und flankierenden Bauteilen
Braunschweiger Schriften zur Mechanik Nr. 64-2008 Numerische Untersuchung der akustischen Eigenschaften von trennenden und flankierenden Bauteilen von Dirk Clasen Institut für Angewandte Mechanik Technische Universität Braunschweig Herausgegeben vom Mechanik-Zentrum der Technischen Universität Braunschweig Schriftleiter: Prof. Dr. rer. nat. H. Antes Institut für Angewandte Mechanik Postfach 3329 38023 Braunschweig Tel.: 0531 / 391-7101 Fax: 0531 / 391-5843 Von der Fakultät Architektur, Bauingenieurwesen und Umweltwissenschaften der Technischen Universität Carolo-Wilhelmina zu Braunschweig zur Erlangung des Grades eines Doktor-Ingenieur (Dr.-Ing.) genehmigte Dissertation Tag der Einreichung: 27.06.2007 Tag der Prüfung: 14.04.2008 Berichter: Prof. Dr.–Ing. S. Langer Prof. Dr.–Ing. O. von Estorff Copyright 2008 D. Clasen, Braunschweig BSM 64-2008 ISBN 978-3-920395-63-0 Alle Rechte, insbesondere der Übersetzung in fremde Sprachen, vorbehalten. Mit Genehmigung des Autors ist es gestattet, dieses Heft ganz oder teilweise zu vervielfältigen. Zusammenfassung Die Bevölkerung Deutschlands ist dem Lärm durch eine Vielzahl von Schallquellen ausgesetzt. Straßen, Schienenwege, Flugplätze, Gewerbeanlagen, Sportanlagen und nicht zuletzt Nachbarn führen häufig zu Lärmproblemen. Auf Grund dieser immer stärker werdenden Lärmbelastungen gewinnt die Berechnung der Schalldämmung zunehmend an Bedeutung. Dabei spielt die Erfas- sung von beliebigen geometrischen, bauphysikalischen und bauakustischen Randbedingungen ei- ne entscheidende Rolle. Oftmals wird der Nachweis -
Symmetric Boundary Knot Method
Symmetric boundary knot method W. Chen* Simula Research Laboratory, P. O. Box. 134, 1325 Lysaker, Norway E-mail: [email protected] (submitted 28 Sept. 2001, revised 25, Feb. 2002) Abstract The boundary knot method (BKM) is a recent boundary-type radial basis function (RBF) collocation scheme for general PDE’s. Like the method of fundamental solution (MFS), the RBF is employed to approximate the inhomogeneous terms via the dual reciprocity principle. Unlike the MFS, the method uses a non-singular general solution instead of a singular fundamental solution to evaluate the homogeneous solution so as to circumvent the controversial artificial boundary outside the physical domain. The BKM is meshfree, super-convergent, integration-free, very easy to learn and program. The original BKM, however, loses symmetricity in the presence of mixed boundary. In this study, by analogy with Fasshauer’s Hermite RBF interpolation, we developed a symmetric BKM scheme. The accuracy and efficiency of the symmetric BKM are also numerically validated in some 2-D and 3-D Helmholtz and diffusion-reaction problems under complicated geometries. Keyword: boundary knot method; radial basis function; meshfree; method of fundamental solution; dual reciprocity BEM; symmetricity * This research is supported by Norwegian Research Council. 1 1.Introduction In recent years much effort has been devoted to developing a variety of meshfree schemes for the numerical solution of partial differential equations (PDE’s). The driving force behind the scene is that the mesh-based methods such as the standard FEM and BEM often require prohibitive computational effort to mesh or remesh in handling high- dimensional, moving boundary, and complex-shaped boundary problems. -
A Self-Singularity-Capturing Scheme for Fractional Differential Equations
A Self-Singularity-Capturing Scheme for Fractional Differential Equations Jorge L. Suzukia,b and Mohsen Zayernouria,c aDepartment of Mechanical Engineering, Michigan State University, East Lansing, MI, USA bDepartment of Computational Mathematics, Science and Engineering, Michigan State University, East Lansing, MI, USA cDepartment of Probability and Statistics, Michigan State University, East Lansing, MI, USA ARTICLE HISTORY Compiled April 30, 2020 ABSTRACT We develop a two-stage computational framework for robust and accurate time- integration of multi-term linear/nonlinear fractional differential equations. In the first stage, we formulate a self-singularity-capturing scheme, given avail- able/observable data for diminutive time, experimentally obtained or sampled from an approximate numerical solution utilizing a fine grid nearby the initial time. The fractional differential equation provides the necessary knowledge/insight on how the hidden singularity can bridge between the initial and the subsequent short-time so- lution data. In the second stage, we utilize the multi-singular behavior of solution in a variety of numerical methods, without resorting to making any ad-hoc/uneducated guesses for the solution singularities. Particularly, we employed an implicit finite- difference method, where the captured singularities, in the first stage, are taken into account through some Lubich-like correction terms, leading to an accuracy of order O(∆t3−α). We show that this novel framework can control the error even in the presence of strong multi-singularities. -
Singular Boundary Method for Heat Conduction in Layered Materials
Copyright © 2011 Tech Science Press CMC, vol.24, no.1, pp.1-14, 2011 Singular Boundary Method for Heat Conduction in Layered Materials H. Htike1;2, W. Chen1;2;3 and Y. Gu1;2 Abstract: In this paper, we investigate the application of the singular boundary method (SBM) to two-dimensional problems of steady-state heat conduction in isotropic bimaterials. A domain decomposition technique is employed where the bimaterial is decomposed into two subdomains, and in each subdomain, the solu- tion is approximated separately by an SBM-type expansion. The proposed method is tested and compared on several benchmark test problems, and its relative merits over the other boundary discretization methods, such as the method of fundamental solution (MFS) and the boundary element method (BEM), are also discussed. Keywords: Singular boundary method, origin intensity factor, heat conduction, bimaterials, meshless. 1 Introduction During machining processes, heat is a source that strongly influences the tool per- formance, such as tool wear, tool life, surface quality as well as process quality. The layers of material provide a barrier for the intensive heat flow from the con- tact area into the substrate material. Thus, a clear understanding of the temperature distribution in layered materials is very useful and important [Atluri (2005); Atluri, Liu, and Han (2006);Chang, Liu, and Chang (2010);Chen and Liu (2001)]. The method of fundamental solutions (MFS) [Karageorghis (1992);Fairweather and Karageorghis (1998);Chen, Golberg, and Hon (1998)] is a successful technique for the solution of some elliptic boundary value problems in engineering applica- tions. It is applicable when a fundamental solution of the operator of the governing equation is known. -
Fast Boundary Knot Method for Solving Axisymmetric Helmholtz Problems with High Wave Number
Copyright © 2013 Tech Science Press CMES, vol.94, no.6, pp.485-505, 2013 Fast Boundary Knot Method for Solving Axisymmetric Helmholtz Problems with High Wave Number J. Lin1, W. Chen 1;2, C. S. Chen3, X. R. Jiang4 Abstract: To alleviate the difficulty of dense matrices resulting from the bound- ary knot method, the concept of the circulant matrix has been introduced to solve axi-symmetric Helmholtz problems. By placing the collocation points in a circular form on the surface of the boundary, the resulting matrix of the BKM has the block structure of a circulant matrix, which can be decomposed into a series of smaller matrices and solved efficiently. In particular, for the Helmholtz equation with high wave number, a large number of collocation points is required to achieve desired accuracy. In this paper, we present an efficient circulant boundary knot method algorithm for solving Helmholtz problems with high wave number. Keywords: boundary knot method, Helmholtz problem, circulant matrix, ax- isymmetric. 1 Introduction The boundary knot method (BKM) [Chen and Tanaka (2002); Chen (2002); Chen and Hon (2003); Wang, Ling, and Chen (2009); Wang, Chen, and Jiang (2010); Zhang and Wang (2012); Zheng, Chen, and Zhang (2013)] has been widely ap- plied for solving certain classes of boundary value problems. Instead of using the singular fundamental solution as the basis function in the method of funda- mental solutions (MFS) [Fairweather and Karageorghis (1998); Fairweather, Kara- georghis, and Martin (2003); Alves and Antunes (2005); Chen, Cho, and Golberg (2009); Drombosky, Meyer, and Ling (2009); Gu, Young, and Fan (2009); Lin, Gu, and Young (2010); Lin, Chen, and Wang (2011)], the BKM uses the non-singular general solution for the approximation of the solution. -
J. Calderon-Sanchez
Universidad Polit´ecnicade Madrid Escuela T´ecnicaSuperior de Ingenieros Navales Programa de Doctorado en Ingenier´ıaNaval y Oce´anica Numerical studies of the sloshing phenomenon using the Smoothed Particle Hydrodynamics (SPH) method. Javier Calderon-Sanchez Supervisor: Daniel Duque Campayo January, 2020 `Nothing in life is to be feared, it is only to be understood. Now is the time to understand more, so that we may fear less.' Marie Curie i ii Tribunal designado por la Comisión de Doctorado de la Universidad Politécnica de Madrid, en su reunión del día...............de.............................de 20..... Presidente: Vocal: Vocal: Vocal: Secretario: Suplente: Suplente: Realizado el acto de defensa y lectura de la Tesis el día..........de........................de 20 ... en la E.T.S.I. /Facultad.................................................... Calificación .................................................. EL PRESIDENTE LOS VOCALES EL SECRETARIO iv Declaration I hereby declare that the contents presented in this dissertation are original and, to the best of my knowledge, they contain no material published by another person, or substantial proportions of material that have been submitted for any degree or other purposes, except where specific reference is made to the work of others. I certify that the intellectual content of this thesis is the product of my own work and that all the assistance received in preparing this thesis and sources have been acknowledged. Javier Calderon-Sanchez January, 2020 v vi Abstract The purpose of the present thesis is to increase the applicability of Smoothed Particle Hydro- dynamics (SPH) method to sloshing problems relevant in engineering. The thesis is structured around three main topics: theoretical improvements on SPH, imple- mentation of tools and physical models within the open-source software AQUAgpusph, and studies on 3-D geometries and application cases. -
Effective Condition Number for Boundary Knot Method
Copyright © 2009 Tech Science Press CMC, vol.12, no.1, pp.57-70, 2009 Effective Condition Number for Boundary Knot Method F.Z. Wang1, L. Ling2 and W. Chen1;3 Abstract: This study makes the first attempt to apply the effective condition number (ECN) to the stability analysis of the boundary knot method (BKM). We find that the ECN is a superior criterion over the traditional condition number. The main difference between ECN and the traditional condition numbers is in that the ECN takes into account the right hand side vector to estimates system stability. Numerical results show that the ECN is roughly inversely proportional to the nu- merical accuracy. Meanwhile, using the effective condition number as an indicator, one can fine-tune the user-defined parameters (without the knowledge of exact so- lution) to ensure high numerical accuracy from the BKM. Keywords: boundary knot method, effective condition number, traditional con- dition number. 1 Introduction In recent years, a variety of boundary meshless methods have been introduced, such as the method of fundamental solutions (MFS)[Fairweather and Karageorghis (1998);Young, Tsai, Lin, and Chen (2006);Chen, Karageorghis, and Smyrlis (2008)], boundary knot method (BKM) [Chen and Tanaka (2002);Chen and Hon (2003);Wang, Chen, and Jiang (2009)], boundary collocation method [Chen, Chang, Chen, and Lin (2002);Chen, Chen, Chen, and Yeh (2004)], boundary node method [Mukher- jee and Mukherjee (1997);Zhang, Yao, and Li (2002)], and regularized meshless method [Young, Chen, and Lee (2005);Young, Chen, Chen, and Kao (2007);Chen, Kao, and Chen (2009)]. All these methods are of the boundary type numerical technique in which only the boundary knots are required in the numerical solution of homogeneous problems. -
Arxiv:1808.04585V2 [Math.NA]
OPTIMAL ADDITIVE SCHWARZ PRECONDITIONING FOR ADAPTIVE 2D IGA BOUNDARY ELEMENT METHODS THOMAS FUHRER,¨ GREGOR GANTNER, DIRK PRAETORIUS, AND STEFAN SCHIMANKO Abstract. We define and analyze (local) multilevel diagonal preconditioners for isogeo- metric boundary elements on locally refined meshes in two dimensions. Hypersingular and weakly-singular integral equations are considered. We prove that the condition number of the preconditioned systems of linear equations is independent of the mesh-size and the re- finement level. Therefore, the computational complexity, when using appropriate iterative solvers, is optimal. Our analysis is carried out for closed and open boundaries and numerical examples confirm our theoretical results. 1. Introduction In the last decade, the isogeometric analysis (IGA) had a strong impact on the field of scientific computing and numerical analysis. We refer, e.g., to the pioneering work [HCB05] and to [CHB09, BdVBSV14] for an introduction to the field. The basic idea is to utilize the same ansatz functions for approximations as are used for the description of the geometry by some computer aided design (CAD) program. Here, we consider the case, where the geometry is represented by rational splines. For certain problems, where the fundamental solution is known, the boundary element method (BEM) is attractive since CAD programs usually only provide a parametrization of the boundary ∂Ω and not of the volume Ω itself. Isogeometric BEM (IGABEM) has first been considered for 2D BEM in [PGK+09] and for 3D BEM in [SSE+13]. We refer to [SBTR12, PTC13, SBLT13, NZW+17] for numerical experiments, to [HR10, TM12, MZBF15, DHP16, DHK+18, DKSW18] for fast IGABEM based on wavelets, fast multipole, -matrices resp.