Solving Partial Differential Equations by Taylor Meshless Method Jie Yang
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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. -
Notes on Partial Differential Equations John K. Hunter
Notes on Partial Differential Equations John K. Hunter Department of Mathematics, University of California at Davis Contents Chapter 1. Preliminaries 1 1.1. Euclidean space 1 1.2. Spaces of continuous functions 1 1.3. H¨olderspaces 2 1.4. Lp spaces 3 1.5. Compactness 6 1.6. Averages 7 1.7. Convolutions 7 1.8. Derivatives and multi-index notation 8 1.9. Mollifiers 10 1.10. Boundaries of open sets 12 1.11. Change of variables 16 1.12. Divergence theorem 16 Chapter 2. Laplace's equation 19 2.1. Mean value theorem 20 2.2. Derivative estimates and analyticity 23 2.3. Maximum principle 26 2.4. Harnack's inequality 31 2.5. Green's identities 32 2.6. Fundamental solution 33 2.7. The Newtonian potential 34 2.8. Singular integral operators 43 Chapter 3. Sobolev spaces 47 3.1. Weak derivatives 47 3.2. Examples 47 3.3. Distributions 50 3.4. Properties of weak derivatives 53 3.5. Sobolev spaces 56 3.6. Approximation of Sobolev functions 57 3.7. Sobolev embedding: p < n 57 3.8. Sobolev embedding: p > n 66 3.9. Boundary values of Sobolev functions 69 3.10. Compactness results 71 3.11. Sobolev functions on Ω ⊂ Rn 73 3.A. Lipschitz functions 75 3.B. Absolutely continuous functions 76 3.C. Functions of bounded variation 78 3.D. Borel measures on R 80 v vi CONTENTS 3.E. Radon measures on R 82 3.F. Lebesgue-Stieltjes measures 83 3.G. Integration 84 3.H. Summary 86 Chapter 4. -
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 -
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. -
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. -
Industrial and Systems Engineering (ISE)
Lehigh University 2021-22 1 Industrial and Systems Engineering (ISE) Courses ISE 224 Information Systems Analysis and Design 3 Credits ISE 100 Industrial Employment 0 Credits An introduction to the technological as well as methodological Usually following the junior year, students in the industrial engineering aspects of computer information systems. Content of the course curriculum are required to do a minimum of eight weeks of practical stresses basic knowledge in database systems. Database design and work, preferably in the field they plan to follow after graduation. A evaluation, query languages and software implementation. Students report is required. Must have sophomore standing. that take CSE 241 cannot receive credit for this course. ISE 111 Engineering Probability 3 Credits ISE 226 Engineering Economy and Decision Analysis 3 Credits Random variables, probability models and distributions. Poisson Economic analysis of engineering projects; interest rate factors, processes. Expected values and variance. Joint distributions, methods of evaluation, depreciation, replacement, breakeven covariance and correlation. analysis, aftertax analysis. decision-making under certainty and risk. Prerequisites: MATH 022 or MATH 096 or MATH 032 or MATH 052 Prerequisites: ISE 111 or MATH 231 or IE 111 Can be taken Concurrently: ISE 111, MATH 231, IE 111 ISE 112 Computer Graphics 1 Credit Introduction to interactive graphics and construction of multiview ISE 230 Introduction to Stochastic Models in Operations representations in two and three dimensional space. Applications in Research 3 Credits industrial engineering. Must have sophomore standing in industrial Formulating, analyzing, and solving mathematical models of real- engineering. world problems in systems exhibiting stochastic (random) behavior. Discrete and continuous Markov chains, queueing theory, inventory ISE 121 Applied Engineering Statistics 3 Credits control, Markov decision process. -
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 -
Major Factors That Influence the Employment Decisions of Generation X Consulting Engineers
Old Dominion University ODU Digital Commons Engineering Management & Systems Engineering Management & Systems Engineering Theses & Dissertations Engineering Spring 2002 Major Factors That Influence the Employment Decisions of Generation X Consulting Engineers Robert William Mayfield Old Dominion University Follow this and additional works at: https://digitalcommons.odu.edu/emse_etds Part of the Engineering Commons, Organizational Behavior and Theory Commons, and the Work, Economy and Organizations Commons Recommended Citation Mayfield, Robert W.. "Major Factors That Influence the Employment Decisions of Generation X Consulting Engineers" (2002). Master of Science (MS), Thesis, Engineering Management & Systems Engineering, Old Dominion University, DOI: 10.25777/t41p-rd52 https://digitalcommons.odu.edu/emse_etds/102 This Thesis is brought to you for free and open access by the Engineering Management & Systems Engineering at ODU Digital Commons. It has been accepted for inclusion in Engineering Management & Systems Engineering Theses & Dissertations by an authorized administrator of ODU Digital Commons. For more information, please contact [email protected]. MAJOR FACTORS THAT INFLUENCE THE EMPLOYMENT DECISIONS OF GENERATION X CONSULTING ENGINEERS bv Robert William Mayfield B.S. March 1994, The Ohio State University A Thesis Submitted to the Faculty o f Old Dominion University in Partial Fulfillment o f the Requirement for the Degree of MASTER OF SCIENCE ENGINEERING MANAGEMENT OLD DOMINION UNIVERSITY May 2002 Approved by: Charles Keating (Direct Paul Kauffmai ember) Andres Sousa-Poza (Member) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABSTRACT MAJOR FACTORS THAT INFLUENCE THE EMPLOYMENT DECISIONS OF GENERATION X CONSULTING ENGINEERS Robert William Mayfield Old Dominion University. 2002 Director: Dr. Charles Keating The purpose of this research was to study Generation X consulting engineers (those bom between the years 1964 and 1980) in Lynchburg. -
Novel Substructural Health Monitoring Method Without Interface Measurements Using State Space Analysis and Extended Kalman Filtering
Novel Substructural Health Monitoring Method without Interface Measurements using State Space Analysis and Extended Kalman Filtering by Arianne Layard Muelhausen Under the Supervision of Prof. Brock Hedegaard A thesis submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE (Civil and Environmental Engineering) at the University of Wisconsin-Madison 2018 Contents 1 Introduction 2 2 Literature Review 4 3 Methodology 8 3.1 State Space Analysis . .8 3.2 Kalman Filter . .9 3.3 Extended Kalman Filter . 11 3.4 Substructural Identification using Extended Kalman Filtering . 12 3.4.1 Global Equation of Motion . 13 3.4.2 Substructural Equation of Motion . 14 3.4.3 State Space Formulation: State Transition Equation . 17 3.4.4 State Space Formulation: Measurement Equation . 18 3.4.5 Implementation of Extended Kalman Filter . 19 3.5 Methodology Step-by-Step Summary . 24 4 Results 25 4.1 Model Description . 25 4.2 Case One: All DOFs Measured . 26 4.3 Case Two: All Interface and Some Interior DOFs Measured . 29 4.4 Case Three: Some Interface and Some Interior DOFs Measured . 31 4.5 Case Four: No Interface and Some Interior DOFs Measured . 38 4.6 Discussion on Limitations of Method . 40 5 Conclusion 41 6 Work Cited 42 1 1 Introduction Structural Health Monitoring (SHM) at its core is a process to identify damage, defined as either material or geometric change, of a system that negatively affects the systems performance[7]. SHM seeks to address four questions: (Level 1) Detection: Is damage present? (Level 2) Localization: What is the probable location of damage? (Level 3) Assessment: What is the severity of the damage? (Level 4) Prognosis: What is the remaining service life of the damaged system? SHM can be used to monitor structures affected by external stimuli, long-term movement, ma- terial degradation, or demolition. -
Multivariate Statistical Process Monitoring Using Classical
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Newcastle University eTheses Multivariate Statistical Process Monitoring Using Classical Multidimensional Scaling Prepared by: Mohd Yusri Mohd Yunus A Thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy School of Chemical Engineering and Advanced Materials Newcastle University UK May 2012 ABSTRACT A new Multivariate Statistical Process Monitoring (MSPM) system, which comprises of three main frameworks, is proposed where the system utilizes Classical Multidimensional Scaling (CMDS) as the main multivariate data compression technique instead of using the linear- based Principal Component Analysis (PCA). The conventional method which usually applies variance-covariance or correlation measure in developing the multivariate scores is found to be inappropriately used especially in modelling nonlinear processes, where a high number of principal components will be typically required. Alternatively, the proposed method utilizes the inter-dissimilarity scales in describing the relationships among the monitored variables instead of variance-covariance measure for the multivariate scores development. However, the scores are plotted in terms of variable structure, thus providing different formulation of statistics for monitoring. Nonetheless, the proposed statistics still correspond to the conceptual objective of Hotelling’s T2 and Squared Prediction Errors (SPE). The first framework corresponds to the original -
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. -
Boundary Particle Method with High-Order Trefftz Functions
Copyright © 2010 Tech Science Press CMC, vol.13, no.3, pp.201-217, 2010 Boundary Particle Method with High-Order Trefftz Functions Wen Chen1;2, Zhuo-Jia Fu1;3 and Qing-Hua Qin3 Abstract: This paper presents high-order Trefftz functions for some commonly used differential operators. These Trefftz functions are then used to construct boundary particle method for solving inhomogeneous problems with the boundary discretization only, i.e., no inner nodes and mesh are required in forming the final linear equation system. It should be mentioned that the presented Trefftz functions are nonsingular and avoids the singularity occurred in the fundamental solution and, in particular, have no problem-dependent parameter. Numerical experiments demonstrate the efficiency and accuracy of the present scheme in the solution of inhomogeneous problems. Keywords: High-order Trefftz functions, boundary particle method, inhomoge- neous problems, meshfree 1 Introduction Since the first paper on Trefftz method was presented by Trefftz (1926), its math- ematical theory was extensively studied by Herrera (1980) and many other re- searchers. In 1995 a special issue on Trefftz method, was published in the jour- nal of Advances in Engineering Software for celebrating its 70 years of develop- ment [Kamiya and Kita (1995)]. Qin (2000, 2005) presented an overview of the Trefftz finite element and its application in various engineering problems. The Trefftz method employs T-complete functions, which satisfies the governing dif- ferential operators and is widely applied to potential problems [Cheung, Jin and Zienkiewicz (1989)], two-dimensional elastic problems [Zielinski and Zienkiewicz (1985)], transient heat conduction [Jirousek and Qin (1996)], viscoelasticity prob- 1 Center for Numerical Simulation Software in Engineering and Sciences, Department of Engineer- ing Mechanics, Hohai University, Nanjing, Jiangsu, P.R.China 2 Corresponding author.