DOCSLIB.ORG

A Technique to Combine Meshfree and Finite- Element

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
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. Rubber components often have geometric features 1 that give rise to very steep or even singular gradients. Elements in the neighborhood of these features inevitably become highly distorted, often to the point of material eversion (negative de- terminant of deformation gradient). This event effectively terminates the solution process and renders the model useless for further simulation. For finite element models with coarse elements, it is often possible to achieve the desired degree of loading before element failure occurs. These model, however, will not provide accurate solutions. When fine meshes are used this “element collapse” can occur under very small applied loads. Too much is demanded of an element adja- cent to, for example, a singular point. The analysis must then trade load level for accuracy, since accurate solutions for the necessary load levels can not be obtained. Meshfree methods such as the hp-cloud method [15, 16], the method of finite spheres [9], reproducing kernel particle methods [26–28], the element-free Galerkin method [3–6, 22, 37], the finite point method [34–36], the generalized finite difference method [24, 25, 45, 46], the diffuse element method [32], the modified local Petrov-Galerkin method [1], smooth particle hydrodynamics [19, 43, 44], among others, offer an attractive alternative for the solution of many classes of problems that are difficult or even not feasible to solve using the finite element method. In particular, meshfree methods have shown to be very effective for the solution of problems involving large deformations like those described above [7, 8]. Excellent overviews of meshfree methods and their applications can be found in, for example, [3, 27]. In meshfree methods, the approximation of field variables is constructed in terms of nodes without the aid of a mesh. The actual implementation of some meshfree methods, however, re- quires the partition of the domain through the use of a “background grid” for domain integration. Nevertheless, due to the flexibility in constructing conforming shape functions to meet specific needs for different applications, it has been reported [7, 8, 12, 13, 15, 16, 30] that meshfree meth- ods are particularly suitable for the simulation of crack propagation, hp-adaptivity, and modeling of large deformation problems. The use of smooth shape functions appears to be particularly effective in dealing with large deformation problems. One of the main drawbacks of meshfree methods has been the fact that the computational cost is too high in some applications due to the fact that one has to use a great number of integration points in order to integrate the meshfree functions and their products over a computational domain. One approach to ameliorate the computational cost is to use these methods only in parts of the domain where they are strictly needed and use a finite element discretization elsewhere. Besides reducing the overall computational cost, this approach has several other appealing features. It facilitates, for example, the implementation of Dirichlet boundary conditions [22] and the coupling with classical structural finite elements, like rods, shell, rigid bars, etc. Several techniques to couple meshfree and finite element methods have been proposed. Belytschko et al. [6] proposed a coupling technique in which some finite element nodes are re- placed by meshfree nodes and a ramp function is used to build the transition between the finite element and meshfree discretizations. Linear consistency is attained with this approach. A gener- alization of this idea was proposed by Hegen [20] based on the use of Lagrange multipliers. Huerta and Fernandez-Mendez [21] proposed a technique to couple finite element and 2 meshfree discretizations based on consistency or reproducibility conditions of the resulting shape functions and moving least squares techniques. The support size of the meshfree functions and their distribution must obey some rules in order to be admissible [21]. A related technique was proposed by Liu et al. [29] with the goal of improving a finite element discretization with meshfree shape functions. Many of the meshfree shape functions like moving least squares and Shepard functions, constitute a partition of unity. In other words, these functions add to the unity at any point in the domain. Lagrangian finite element shape functions also possess this property. In this paper, we present a technique to couple meshfree and finite element discretizations that explores the partition of unity property of these functions. The procedure, while simple, is quite flexible and generic. The only requirement on the finite element and meshfree shape functions is that the union of their supports completely covers the computational domain. H and p type approximations can be built in the finite element and meshfree parts of the domain. Exponential convergence of the resulting coupled approximation is demonstrated though a numerical example. Another approach to reduce the cost of numerical integration of meshfree shape functions is to use a finite element mesh to build them and enforce that the support of all functions coincide with the support of corresponding global Lagrangian shape functions defined on the same mesh. In this case, the method is no longer strictly meshfree but some of the attractive features of meshfree approximations, like high regularity of the approximation, can still be retained. Edwards [18] has proposed such approach and have shown that it can be used in any dimension and for any kind of finite element (triangular, quadrilateral, tetrahedral, hexahedral, etc) while rendering C 1 finite element shape functions. Edwards’ approach however, has a serious practical limitation– It requires that the support of the functions be convex. It is not possible, in general, to build finite element meshes such that the support of the corresponding finite element shape functions be convex. In this paper, we generalize Edwards’ approach to handle non-convex supports with the aid of the so-called R-functions [38, 39]. In the following sections the formulation of the proposed coupling technique and the gen- eralization of Edwards’ approach [18] to build finite element shape functions with arbitrary regu- larity on non-convex supports is introduced. Illustrative numerical experiments are also presented. 2 Partition of Unity Shape Functions In this section, the construction of partition of unity shape functions is briefly reviewed. Examples of this kind of shape functions are hp-cloud [15,16] and generalized finite element shape functions [11, 12, 41, 42]. Let the functions 'α; α = 1; : : : ; N, denote a partition of unity (PoU) subordinate to the N n open covering TN = f!αgα=1 of a domain Ω ⊂ RI ; n = 1; 2; 3. Here, !α is the support of the partition of unity function 'α and N the number of functions. We call !α a cloud and associate with each one of them a node, denoted by xα. 3 From the above we have that s 'α 2 C0 (!α); s ≥ 0; 1 ≤ α ≤ N 'α(x) = 1 8 x 2 Ω α X Let χα(!α) = spanfLiαgi2I(α) denote local spaces defined on !α; α = 1; : : : ; N, where I(α); α = 1; : : : ; N, are index sets and fLiαgi2I(α) a basis for the space χα(!α). Functions Liα are also denoted by enrichment or local approximation functions. The partition of unity shape functions associated with a node xα are then defined by α φi := 'αLiα; i 2 I(α) (no sum on α) (1) Different choices for the partition of unity functions are possible. Each one of then will lead to a different class of shape functions. Some of the possible choices are discussed below. 2.1 Shepard Partition of Unity Shepard’s formula [23, 40] is a very simple approach to build partition of unity functions and it is n often used in meshfree methods [9, 15, 16]. Let Wα : RI ! RI denote a weighting function with s compact support !α and that belongs to the space C0 (!α); s ≥ 0. Suppose that such weighting function is built at every cloud !α; α = 1; : : : ; N. Then, the partition of unity functions 'α associated with the clouds !α, α = 1; : : : ; N, are defined by Wα(x) 'α(x) = β(x) 2 fγ j Wγ (x) =6 0g α = 1; : : : ; N (2) β(x) Wβ(x) which are known as ShepardP functions [23, 40].
Load more

Recommended publications
  • Finite Difference Methods for Solving Differential
    FINITE DIFFERENCE METHODS FOR SOLVING DIFFERENTIAL EQUATIONS I-Liang Chern Department of Mathematics National Taiwan University May 16, 2013 2 Contents 1 Introduction 3 1.1 Finite Difference Approximation . ........ 3 1.2 Basic Numerical Methods for Ordinary Differential Equations ........... 5 1.3 Runge-Kuttamethods..... ..... ...... ..... ...... ... ... 8 1.4 Multistepmethods ................................ 10 1.5 Linear difference equation . ...... 14 1.6 Stabilityanalysis ............................... 17 1.6.1 ZeroStability................................. 18 2 Finite Difference Methods for Linear Parabolic Equations 23 2.1 Finite Difference Methods for the Heat Equation . ........... 23 2.1.1 Some discretization methods . 23 2.1.2 Stability and Convergence for the Forward Euler method.......... 25 2.2 L2 Stability – von Neumann Analysis . 26 2.3 Energymethod .................................... 28 2.4 Stability Analysis for Montone Operators– Entropy Estimates ........... 29 2.5 Entropy estimate for backward Euler method . ......... 30 2.6 ExistenceTheory ................................. 32 2.6.1 Existence via forward Euler method . ..... 32 2.6.2 A Sharper Energy Estimate for backward Euler method . ......... 33 2.7 Relaxationoferrors.............................. 34 2.8 BoundaryConditions ..... ..... ...... ..... ...... ... 36 2.8.1 Dirichlet boundary condition . ..... 36 2.8.2 Neumann boundary condition . 37 2.9 The discrete Laplacian and its inversion . .......... 38 2.9.1 Dirichlet boundary condition . ..... 38 3 Finite Difference Methods for Linear elliptic Equations 41 3.1 Discrete Laplacian in two dimensions . ........ 41 3.1.1 Discretization methods . 41 3.1.2 The 9-point discrete Laplacian . ..... 42 3.2 Stability of the discrete Laplacian . ......... 43 3 4 CONTENTS 3.2.1 Fouriermethod ................................ 43 3.2.2 Energymethod ................................ 44 4 Finite Difference Theory For Linear Hyperbolic Equations 47 4.1 A review of smooth theory of linear hyperbolic equations .............
    [Show full text]
  • Exercises from Finite Difference Methods for Ordinary and Partial
    Exercises from Finite Difference Methods for Ordinary and Partial Differential Equations by Randall J. LeVeque SIAM, Philadelphia, 2007 http://www.amath.washington.edu/ rjl/fdmbook ∼ Under construction | more to appear. Contents Chapter 1 4 Exercise 1.1 (derivation of finite difference formula) . 4 Exercise 1.2 (use of fdstencil) . 4 Chapter 2 5 Exercise 2.1 (inverse matrix and Green's functions) . 5 Exercise 2.2 (Green's function with Neumann boundary conditions) . 5 Exercise 2.3 (solvability condition for Neumann problem) . 5 Exercise 2.4 (boundary conditions in bvp codes) . 5 Exercise 2.5 (accuracy on nonuniform grids) . 6 Exercise 2.6 (ill-posed boundary value problem) . 6 Exercise 2.7 (nonlinear pendulum) . 7 Chapter 3 8 Exercise 3.1 (code for Poisson problem) . 8 Exercise 3.2 (9-point Laplacian) . 8 Chapter 4 9 Exercise 4.1 (Convergence of SOR) . 9 Exercise 4.2 (Forward vs. backward Gauss-Seidel) . 9 Chapter 5 11 Exercise 5.1 (Uniqueness for an ODE) . 11 Exercise 5.2 (Lipschitz constant for an ODE) . 11 Exercise 5.3 (Lipschitz constant for a system of ODEs) . 11 Exercise 5.4 (Duhamel's principle) . 11 Exercise 5.5 (matrix exponential form of solution) . 11 Exercise 5.6 (matrix exponential form of solution) . 12 Exercise 5.7 (matrix exponential for a defective matrix) . 12 Exercise 5.8 (Use of ode113 and ode45) . 12 Exercise 5.9 (truncation errors) . 13 Exercise 5.10 (Derivation of Adams-Moulton) . 13 Exercise 5.11 (Characteristic polynomials) . 14 Exercise 5.12 (predictor-corrector methods) . 14 Exercise 5.13 (Order of accuracy of Runge-Kutta methods) .
    [Show full text]
  • Meshfree Methods Chapter 1 — Part 1: Introduction and a Historical Overview
    MATH 590: Meshfree Methods Chapter 1 — Part 1: Introduction and a Historical Overview Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2014 [email protected] MATH 590 – Chapter 1 1 Outline 1 Introduction 2 Some Historical Remarks [email protected] MATH 590 – Chapter 1 2 Introduction General Meshfree Methods Meshfree Methods have gained much attention in recent years interdisciplinary field many traditional numerical methods (finite differences, finite elements or finite volumes) have trouble with high-dimensional problems meshfree methods can often handle changes in the geometry of the domain of interest (e.g., free surfaces, moving particles and large deformations) better independence from a mesh is a great advantage since mesh generation is one of the most time consuming parts of any mesh-based numerical simulation new generation of numerical tools [email protected] MATH 590 – Chapter 1 4 Introduction General Meshfree Methods Applications Original applications were in geodesy, geophysics, mapping, or meteorology Later, many other application areas numerical solution of PDEs in many engineering applications, computer graphics, optics, artificial intelligence, machine learning or statistical learning (neural networks or SVMs), signal and image processing, sampling theory, statistics (kriging), response surface or surrogate modeling, finance, optimization. [email protected] MATH 590 – Chapter 1 5 Introduction General Meshfree Methods Complicated Domains Recent paradigm shift in numerical simulation of fluid
    [Show full text]
  • A Meshless Approach to Solving Partial Differential Equations Using the Finite Cloud Method for the Purposes of Computer Aided Design
    A Meshless Approach to Solving Partial Differential Equations Using the Finite Cloud Method for the Purposes of Computer Aided Design by Daniel Rutherford Burke, B.Eng A Thesis submitted to the Faculty of Graduate and Post Doctoral Affairs in partial fulfilment of the requirements for the degree of Doctor of Philosophy Ottawa Carleton Institute for Electrical and Computer Engineering Department of Electronics Carleton University Ottawa, Ontario, Canada January 2013 Library and Archives Bibliotheque et Canada Archives Canada Published Heritage Direction du 1+1 Branch Patrimoine de I'edition 395 Wellington Street 395, rue Wellington Ottawa ON K1A0N4 Ottawa ON K1A 0N4 Canada Canada Your file Votre reference ISBN: 978-0-494-94524-7 Our file Notre reference ISBN: 978-0-494-94524-7 NOTICE: AVIS: The author has granted a non­ L'auteur a accorde une licence non exclusive exclusive license allowing Library and permettant a la Bibliotheque et Archives Archives Canada to reproduce, Canada de reproduire, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par telecommunication ou par I'lnternet, preter, telecommunication or on the Internet, distribuer et vendre des theses partout dans le loan, distrbute and sell theses monde, a des fins commerciales ou autres, sur worldwide, for commercial or non­ support microforme, papier, electronique et/ou commercial purposes, in microform, autres formats. paper, electronic and/or any other formats. The author retains copyright L'auteur conserve la propriete du droit d'auteur ownership and moral rights in this et des droits moraux qui protege cette these. Ni thesis. Neither the thesis nor la these ni des extraits substantiels de celle-ci substantial extracts from it may be ne doivent etre imprimes ou autrement printed or otherwise reproduced reproduits sans son autorisation.
    [Show full text]
  • A Plane Wave Method Based on Approximate Wave Directions for Two
    A PLANE WAVE METHOD BASED ON APPROXIMATE WAVE DIRECTIONS FOR TWO DIMENSIONAL HELMHOLTZ EQUATIONS WITH LARGE WAVE NUMBERS QIYA HU AND ZEZHONG WANG Abstract. In this paper we present and analyse a high accuracy method for computing wave directions defined in the geometrical optics ansatz of Helmholtz equation with variable wave number. Then we define an “adaptive” plane wave space with small dimensions, in which each plane wave basis function is determined by such an approximate wave direction. We establish a best L2 approximation of the plane wave space for the analytic solutions of homogeneous Helmholtz equa- tions with large wave numbers and report some numerical results to illustrate the efficiency of the proposed method. Key words. Helmholtz equations, variable wave numbers, geometrical optics ansatz, approximate wave direction, plane wave space, best approximation AMS subject classifications. 65N30, 65N55. 1. Introduction In this paper we consider the following Helmholtz equation with impedance boundary condition u = (∆ + κ2(r))u(ω, r)= f(ω, r), r = (x, y) Ω, L − ∈ (1.1) ((∂n + iκ(r))u(ω, r)= g(ω, r), r ∂Ω, ∈ where Ω R2 is a bounded Lipchitz domain, n is the out normal vector on ∂Ω, ⊂ f L2(Ω) is the source term and κ(r) = ω , g L2(∂Ω). In applications, ω ∈ c(r) ∈ denotes the frequency and may be large, c(r) > 0 denotes the light speed, which is arXiv:2107.09797v1 [math.NA] 20 Jul 2021 usually a variable positive function. The number κ(r) is called the wave number. Helmholtz equation is the basic model in sound propagation.
    [Show full text]
  • Family Name Given Name Presentation Title Session Code
    Family Name Given Name Presentation Title Session Code Abdoulaev Gassan Solving Optical Tomography Problem Using PDE-Constrained Optimization Method Poster P Acebron Juan Domain Decomposition Solution of Elliptic Boundary Value Problems via Monte Carlo and Quasi-Monte Carlo Methods Formulations2 C10 Adams Mark Ultrascalable Algebraic Multigrid Methods with Applications to Whole Bone Micro-Mechanics Problems Multigrid C7 Aitbayev Rakhim Convergence Analysis and Multilevel Preconditioners for a Quadrature Galerkin Approximation of a Biharmonic Problem Fourth-order & ElasticityC8 Anthonissen Martijn Convergence Analysis of the Local Defect Correction Method for 2D Convection-diffusion Equations Flows C3 Bacuta Constantin Partition of Unity Method on Nonmatching Grids for the Stokes Equations Applications1 C9 Bal Guillaume Some Convergence Results for the Parareal Algorithm Space-Time ParallelM5 Bank Randolph A Domain Decomposition Solver for a Parallel Adaptive Meshing Paradigm Plenary I6 Barbateu Mikael Construction of the Balancing Domain Decomposition Preconditioner for Nonlinear Elastodynamic Problems Balancing & FETIC4 Bavestrello Henri On Two Extensions of the FETI-DP Method to Constrained Linear Problems FETI & Neumann-NeumannM7 Berninger Heiko On Nonlinear Domain Decomposition Methods for Jumping Nonlinearities Heterogeneities C2 Bertoluzza Silvia The Fully Discrete Fat Boundary Method: Optimal Error Estimates Formulations2 C10 Biros George A Survey of Multilevel and Domain Decomposition Preconditioners for Inverse Problems in Time-dependent
    [Show full text]
  • Finite Difference and Discontinuous Galerkin Methods for Wave Equations
    Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1522 Finite Difference and Discontinuous Galerkin Methods for Wave Equations SIYANG WANG ACTA UNIVERSITATIS UPSALIENSIS ISSN 1651-6214 ISBN 978-91-554-9927-3 UPPSALA urn:nbn:se:uu:diva-320614 2017 Dissertation presented at Uppsala University to be publicly examined in Room 2446, Polacksbacken, Lägerhyddsvägen 2, Uppsala, Tuesday, 13 June 2017 at 10:15 for the degree of Doctor of Philosophy. The examination will be conducted in English. Faculty examiner: Professor Thomas Hagstrom (Department of Mathematics, Southern Methodist University). Abstract Wang, S. 2017. Finite Difference and Discontinuous Galerkin Methods for Wave Equations. Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1522. 53 pp. Uppsala: Acta Universitatis Upsaliensis. ISBN 978-91-554-9927-3. Wave propagation problems can be modeled by partial differential equations. In this thesis, we study wave propagation in fluids and in solids, modeled by the acoustic wave equation and the elastic wave equation, respectively. In real-world applications, waves often propagate in heterogeneous media with complex geometries, which makes it impossible to derive exact solutions to the governing equations. Alternatively, we seek approximated solutions by constructing numerical methods and implementing on modern computers. An efficient numerical method produces accurate approximations at low computational cost. There are many choices of numerical methods for solving partial differential equations. Which method is more efficient than the others depends on the particular problem we consider. In this thesis, we study two numerical methods: the finite difference method and the discontinuous Galerkin method.
    [Show full text]
  • A Quasi-Static Particle-In-Cell Algorithm Based on an Azimuthal Fourier Decomposition for Highly Efficient Simulations of Plasma-Based Acceleration
    A quasi-static particle-in-cell algorithm based on an azimuthal Fourier decomposition for highly efficient simulations of plasma-based acceleration: QPAD Fei Lia,c, Weiming And,∗, Viktor K. Decykb, Xinlu Xuc, Mark J. Hoganc, Warren B. Moria,b aDepartment of Electrical Engineering, University of California Los Angeles, Los Angeles, CA 90095, USA bDepartment of Physics and Astronomy, University of California Los Angeles, Los Angeles, CA 90095, USA cSLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA dDepartment of Astronomy, Beijing Normal University, Beijing 100875, China Abstract The three-dimensional (3D) quasi-static particle-in-cell (PIC) algorithm is a very effi- cient method for modeling short-pulse laser or relativistic charged particle beam-plasma interactions. In this algorithm, the plasma response, i.e., plasma wave wake, to a non- evolving laser or particle beam is calculated using a set of Maxwell's equations based on the quasi-static approximate equations that exclude radiation. The plasma fields are then used to advance the laser or beam forward using a large time step. The algorithm is many orders of magnitude faster than a 3D fully explicit relativistic electromagnetic PIC algorithm. It has been shown to be capable to accurately model the evolution of lasers and particle beams in a variety of scenarios. At the same time, an algorithm in which the fields, currents and Maxwell equations are decomposed into azimuthal harmonics has been shown to reduce the complexity of a 3D explicit PIC algorithm to that of a 2D algorithm when the expansion is truncated while maintaining accuracy for problems with near azimuthal symmetry.
    [Show full text]
  • (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.
    [Show full text]
  • Combined FEM/Meshfree SPH Method for Impact Damage Prediction of Composite Sandwich Panels
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Institute of Transport Research:Publications ECCOMAS Thematic Conference on Meshless Methods 2005 1 Combined FEM/Meshfree SPH Method for Impact Damage Prediction of Composite Sandwich Panels L.Aktay(1), A.F.Johnson(2) and B.-H.Kroplin¨ (3) Abstract: In this work, impact simulations using both meshfree Smoothed Particle Hy- drodynamics (SPH) and combined FEM/SPH Method were carried out for a sandwich composite panel with carbon fibre fabric/epoxy face skins and polyetherimide (PEI) foam core. A numerical model was developed using the dynamic explicit finite element (FE) structure analysis program PAM-CRASH. The carbon fibre/epoxy facings were modelled with standard layered shell elements, whilst SPH particles were positioned for the PEI core. We demonstrate the efficiency and the advantages of pure meshfree SPH and com- bined FEM/SPH Method by comparing the core deformation modes and impact force pulses measured in the experiments to predict structural impact response. Keywords: Impact damage, composite material, sandwich structure concept, Finite Ele- ment Method (FEM), meshfree method, Smoothed Particle Hydrodynamics (SPH) 1 Introduction Modelling of high velocity impact (HVI) and crash scenarios involving material failure and large deformation using classical FEM is complex. Although the most popular numer- ical method FEM is still an effective tool in predicting the structural behaviour in different loading conditions, FEM suffers from large deformation leading problems causing con- siderable accuracy lost. Additionally it is very difficult to simulate the structural behaviour containing the breakage of material into large number of fragments since FEM is initially based on continuum mechanics requiring critical element connectivity.
    [Show full text]
  • A Numerical Method of Characteristics for Solving Hyperbolic Partial Differential Equations David Lenz Simpson Iowa State University
    Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1967 A numerical method of characteristics for solving hyperbolic partial differential equations David Lenz Simpson Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Mathematics Commons Recommended Citation Simpson, David Lenz, "A numerical method of characteristics for solving hyperbolic partial differential equations " (1967). Retrospective Theses and Dissertations. 3430. https://lib.dr.iastate.edu/rtd/3430 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. This dissertation has been microfilmed exactly as received 68-2862 SIMPSON, David Lenz, 1938- A NUMERICAL METHOD OF CHARACTERISTICS FOR SOLVING HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS. Iowa State University, PluD., 1967 Mathematics University Microfilms, Inc., Ann Arbor, Michigan A NUMERICAL METHOD OF CHARACTERISTICS FOR SOLVING HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS by David Lenz Simpson A Dissertation Submitted to the Graduate Faculty in Partial Fulfillment of The Requirements for the Degree of DOCTOR OF PHILOSOPHY Major Subject: Mathematics Approved: Signature was redacted for privacy. In Cijarge of Major Work Signature was redacted for privacy. Headead of^ajorof Major DepartmentDepartmen Signature was redacted for privacy. Dean Graduate Co11^^ Iowa State University of Science and Technology Ames, Iowa 1967 ii TABLE OF CONTENTS Page I. INTRODUCTION 1 II. DISCUSSION OF THE ALGORITHM 3 III. DISCUSSION OF LOCAL ERROR, EXISTENCE AND UNIQUENESS THEOREMS 10 IV.
    [Show full text]
  • 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.
    [Show full text]