Nitsche-type Mortaring for Maxwell s Equations
|
|
- Collin Garrett
- 6 years ago
- Views:
Transcription
1 Nitsche-type Mortaring for Maxwell s Equations Joachim Schöberl Karl Hollaus, Daniel Feldengut Computational Mathematics in Engineering at the Institute for Analysis and Scientific Computing Vienna University of Technology Dzevat Omeragic, Schlumberger-Doll Research Markus Wabro, CST AG Joachim Schöberl Page 1
2 Problem setup Domain decomposition on non-matching meshes: Contents: Method for scalar model problem Nitsche-method for Maxwell s equation Numerical results Joachim Schöberl Page 2
3 A model problem Poisson equation: u = f on Ω 1 Ω 2 u = 0 at (Ω 1 Ω 2 ) Interface conditions on Γ: [u] := u 1 u 2 = 0 n1 u 1 + n2 u 2 = 0 Joachim Schöberl Page 3
4 Mortar method Pose the constraint as additional equation. Find u H 1 0,D (Ω 1) H 1 0,D (Ω 1) and λ H 1/2 (Γ) such that Ω 1 Ω 2 u v + Γ [v]λ = fv v Γ [u]µ = 0 µ The Lagrange parameter λ is the normal flux n u. Requires stability condition for finite element spaces (LBB). Leads to an indefinite system matrix. Joachim Schöberl Page 4
5 Nitsche / Discontinuous Galerkin method Allows discontinuous approximation by keeping extra boundary terms: Find u H 1 0,D (Ω 1) H 1 0,D (Ω 1) such that Ω 1 Ω 2 u v Γ n u [v] n v [u] + α Γ Γ [u] [v] = fv v with soft penalty term α p 2 /h sufficiently large. No extra stability condition is required. Leads to a symmetric positive definite stiffness matrix. Joachim Schöberl Page 5
6 Integration of boundary terms Both methods require to compute integrals of finite element functions from different meshes: Mortar method: Γ u 2µ Nitsche method: Γ v 1 n u 2 Requires the calculation of an intersection mesh Complicated implementation, in particular on curved interfaces in 3D Joachim Schöberl Page 6
7 Hybrid Nitsche method - derivation Introduce a new variable for the primal unknown on the interface: λ := u Γ Multiply by test-functions, and integrate by parts: u v n u v = Ω i Γ i fv v H 1 (Ω 1 ) H 1 (Ω 2 ) Use continuity of n u and introduce single-valued test function µ on interface: u v n u (v µ) = fv v µ i Γ Use u = λ on Γ to symmetrize and stabilize with α p 2 /h. u v n u (v µ) n v (u λ) + α i Γ Γ Γ (u λ)(v µ) = fv v µ For α chosen right, the discrete formulation is stable independent of the choice of fe spaces for u and λ. Joachim Schöberl Page 7
8 Discretizing and numerical integration In general, numerical integration is still difficult. We propose to use smooth B-spline functions for discretizing the hybrid variable λ. This allows evaluation in global coordinates efficient numerical integration by Gauss-rules on the surface elements u u λ Joachim Schöberl Page 8
9 Numerical experiments in 2D f = x in circle, else f = 0. Solution u: Solution u/ x: Finite element order p = 5. Joachim Schöberl Page 9
10 Numerical experiments in 3D f = xz in cylinder, else f = 0. Solution u: Solution u/ x: Finite element order p = 4. Joachim Schöberl Page 10
11 Maxwell s Equations Time harmonic Maxwell s equations curl µ 1 curl u + κu = j in Ω i with κ = iωσ ω 2 ɛ, and Transmission conditions E = iωu, H = µ 1 curl u. u 1 n 1 + u 2 n 2 = 0, µ 1 1 curl u 1 n 1 + µ 1 2 curl u 2 n 2 = 0. Joachim Schöberl Page 11
12 Hybrid Nitsche formulation proceed as in the scalar case: Ω i {µ 1 curl u curl v + κu v} + Ω i µ 1 curl u (v n) = Ω i j v add symmetry and penalty terms: find (u, λ) such that { 2 i=1 Ω i µ 1 {curl u curl v + κu v} + + Ω i µ 1 curl u [(v µ) n] Ωi µ 1 curl v [(u λ) n] + αp2 µh Ω i [(u λ) n] [(v µ) n] } = Ω j v, where u, v H(curl, Ω 1 ) H(curl, Ω 2 ), and λ, µ are tangential vector valued fields on the interface. Joachim Schöberl Page 12
13 Overpenalization of gradient fields The natural energy norm is For gradient fields u = φ, this norms scales as u 2 = µ 1 curl u 2 L 2 + κ u 2 L 2 φ 2 = O(κ) This is small for small frequencies/conductivities. The norm for the Nitsche method is (u, λ) 2 = 2 i=1 { } µ 1 curl u 2 Ω i + κ u 2 Ω i + αµ 1 (u λ) n 2 Γ But, the last term of this norm does not scale with κ for gradient fields. Thus, the penalty term u λ leads to an overpenalization of the jump for gradient fields. Joachim Schöberl Page 13
14 Scalar potential at the boundary Goal: Want to replace the continuity condition (u i λ) n i = 0 i = 1, 2 by (u i φ i λ) n i = 0 φ i φ Γ = 0 with arbitrary scalar fields φ 1 = φ 2 = φ Γ on the boundary. This allows to scale the penalty terms for gradients and rotations differently. Joachim Schöberl Page 14
15 Variational formulation { 2 i=1 Ω i {µ 1 curl u curl v + κuv}+ µ 1 curl u [(v µ) n] + µ 1 curl v [(u φ λ) n]+ Ω i Ω i α µ 1 [(u φ λ) n][(v ψ µ) n]+ Ω i } α κ(φ φ Γ )(ψ ψ Γ ) Ω i = Ω jv Joachim Schöberl Page 15
16 A boundary identity Testing the weak form with v = ψ gives κu ψ + µ 1 curl u ( ψ n) = Ω i Ω i Ω j ψ Taking the divergence in the strong form, and integrating by parts leads to div(κu) ψ = div j ψ Ω i Ω i κu ψ + κu n ψ = j ψ + j n ψ Ω i Ω i Ω i Ω i Adding up leads to the boundary relation Ω i µ 1 curl u ( ψ n) + Ω i κu n ψ = Ω i j n ψ Joachim Schöberl Page 16
17 { 2 i=1 Final variational formulation {µ 1 curl u curl v + κuv} + µ 1 curl u [(v ψ µ) n] Ω i Ω i + µ 1 curl v [(u φ λ) n] + α µ 1 [(u φ λ) n][(v ψ µ) n] Ω i Ω i } κu n (ψ ψ Γ ) κv n (φ φ Γ ) + α κ(φ φ Γ )(ψ ψ Γ ) = Ω i Ω i Ω i { 2 i=1 jv j n ψ Ω i Ω i } u, v... H(curl) conforming element basis functions on Ω i φ, ψ... H 1 conforming element basis functions on Ω i Γ λ, µ... tangential vector valued spline functions on Γ φ Γ, ψ Γ... scalar spline functions on Γ Joachim Schöberl Page 17
18 Magnetostatics Permanent magnet (red) with two domains (green and blue) Joachim Schöberl Page 18
19 Magnetic flux magnetic flux x-component of magnetic flux Finite element order p = 4. Joachim Schöberl Page 19
20 LWD-Tool borehole with soil tool with antennas Joachim Schöberl Page 20
21 LWD-Tool B-field B-field, field-lines: Joachim Schöberl Page 21
22 LWD-Tool Numerical results for first order elements: frequency [khz] standard, rec volt [nv] Nitsche, rec volt [nv] dofs dofs i i i i i i i i 5255 Numerical results for second order elements: frequency [khz] standard, rec volt [nv] Nitsche, rec volt [nv] dofs dofs i i i i i i i i 5255 Joachim Schöberl Page 22
23 Conclusions and Ongoing Work We have Hybrid Nitsche-type mortaring for scalar equation Stable transmission conditions for low frequency Maxwell s equations Interface fields discretized by smooth B-spline spaces for simple numerical integration We work on Error estimators and adaptivity for spline space and numerical integration Iterative solvers, in particular BDDC domain decomposition methods The methods are implemented within Netgen/NGSolve software [K. Hollaus, D. Feldengut, J. Schberl, M. Wabro, D. Omeragic: Nitsche-type Mortaring for Maxwell s Equations PIERS Proceedings, , July 5-8, Cambridge, USA 2010] Joachim Schöberl Page 23
FEM: Domain Decomposition and Homogenization for Maxwell s Equations of Large Scale Problems
FEM: and Homogenization for Maxwell s Equations of Large Scale Problems Karl Hollaus Vienna University of Technology, Austria Institute for Analysis and Scientific Computing February 13, 2012 Outline 1
More informationDiscontinuous Galerkin Methods
Discontinuous Galerkin Methods Joachim Schöberl May 20, 206 Discontinuous Galerkin (DG) methods approximate the solution with piecewise functions (polynomials), which are discontinuous across element interfaces.
More informationHybrid (DG) Methods for the Helmholtz Equation
Hybrid (DG) Methods for the Helmholtz Equation Joachim Schöberl Computational Mathematics in Engineering Institute for Analysis and Scientific Computing Vienna University of Technology Contributions by
More informationRecovery-Based A Posteriori Error Estimation
Recovery-Based A Posteriori Error Estimation Zhiqiang Cai Purdue University Department of Mathematics, Purdue University Slide 1, March 2, 2011 Outline Introduction Diffusion Problems Higher Order Elements
More informationHybrid Discontinuous Galerkin methods for incompressible flow problems
Hybrid Discontinuous Galerkin methods incompressible flow problems Christoph Lehrenfeld, Joachim Schöberl MathCCES Computational Mathematics in Engineering Workshop Linz, May 31 - June 1, 2010 Contents
More informationHybridized Discontinuous Galerkin Methods
Hybridized Discontinuous Galerkin Methods Theory and Christian Waluga Joint work with Herbert Egger (Uni Graz) 1st DUNE User Meeting, Stuttgart Christian Waluga (AICES) HDG Methods October 6-8, 2010 1
More informationHybridized DG methods
Hybridized DG methods University of Florida (Banff International Research Station, November 2007.) Collaborators: Bernardo Cockburn University of Minnesota Raytcho Lazarov Texas A&M University Thanks:
More informationMixed Finite Elements Method
Mixed Finite Elements Method A. Ratnani 34, E. Sonnendrücker 34 3 Max-Planck Institut für Plasmaphysik, Garching, Germany 4 Technische Universität München, Garching, Germany Contents Introduction 2. Notations.....................................
More informationDivergence-conforming multigrid methods for incompressible flow problems
Divergence-conforming multigrid methods for incompressible flow problems Guido Kanschat IWR, Universität Heidelberg Prague-Heidelberg-Workshop April 28th, 2015 G. Kanschat (IWR, Uni HD) Hdiv-DG Práha,
More informationHybrid Discontinuous Galerkin Methods for Fluid Dynamics and Solid Mechanics
Hybrid Discontinuous Galerkin Methods for Fluid Dynamics and Solid Mechanics Joachim Schöberl Computational Mathematics in Engineering Institute for Analysis and Scientific Computing Vienna University
More informationDomain Decomposition Preconditioners for Spectral Nédélec Elements in Two and Three Dimensions
Domain Decomposition Preconditioners for Spectral Nédélec Elements in Two and Three Dimensions Bernhard Hientzsch Courant Institute of Mathematical Sciences, New York University, 51 Mercer Street, New
More informationLecture 2. Introduction to FEM. What it is? What we are solving? Potential formulation Why? Boundary conditions
Introduction to FEM What it is? What we are solving? Potential formulation Why? Boundary conditions Lecture 2 Notation Typical notation on the course: Bolded quantities = matrices (A) and vectors (a) Unit
More informationDual-Primal Isogeometric Tearing and Interconnecting Solvers for Continuous and Discontinuous Galerkin IgA Equations
Dual-Primal Isogeometric Tearing and Interconnecting Solvers for Continuous and Discontinuous Galerkin IgA Equations Christoph Hofer and Ulrich Langer Doctoral Program Computational Mathematics Numerical
More informationHomogenization of the Eddy Current Problem in 2D
Homogenization of the Eddy Current Problem in 2D K. Hollaus, and J. Schöberl Institute for Analysis and Scientific Computing, Wiedner Hauptstr. 8, A-14 Vienna, Austria E-mail: karl.hollaus@tuwien.ac.at
More informationChapter 5. Magnetostatics
Chapter 5. Magnetostatics 5.4 Magnetic Vector Potential 5.1.1 The Vector Potential In electrostatics, E Scalar potential (V) In magnetostatics, B E B V A Vector potential (A) (Note) The name is potential,
More informationMSFEM with Network Coupling for the Eddy Current Problem of a Toroidal Transformer
MSFEM with Network Coupling for the Eddy Current Problem of a Toroidal Transformer Markus Schöbinger, Simon Steentjes, Joachim Schöberl, Kay Hameyer, Karl Hollaus July 18, 2017 ISTET 2017 Illmenau July
More informationCHAPTER 2. COULOMB S LAW AND ELECTRONIC FIELD INTENSITY. 2.3 Field Due to a Continuous Volume Charge Distribution
CONTENTS CHAPTER 1. VECTOR ANALYSIS 1. Scalars and Vectors 2. Vector Algebra 3. The Cartesian Coordinate System 4. Vector Cartesian Coordinate System 5. The Vector Field 6. The Dot Product 7. The Cross
More informationMultigrid Methods for Maxwell s Equations
Multigrid Methods for Maxwell s Equations Jintao Cui Institute for Mathematics and Its Applications University of Minnesota Outline Nonconforming Finite Element Methods for a Two Dimensional Curl-Curl
More informationAn Extended Finite Element Method for a Two-Phase Stokes problem
XFEM project An Extended Finite Element Method for a Two-Phase Stokes problem P. Lederer, C. Pfeiler, C. Wintersteiger Advisor: Dr. C. Lehrenfeld August 5, 2015 Contents 1 Problem description 2 1.1 Physics.........................................
More informationNumerical approximation of output functionals for Maxwell equations
Numerical approximation of output functionals for Maxwell equations Ferenc Izsák ELTE, Budapest University of Twente, Enschede 11 September 2004 MAXWELL EQUATIONS Assumption: electric field ( electromagnetic
More informationTrefftz-discontinuous Galerkin methods for time-harmonic wave problems
Trefftz-discontinuous Galerkin methods for time-harmonic wave problems Ilaria Perugia Dipartimento di Matematica - Università di Pavia (Italy) http://www-dimat.unipv.it/perugia Joint work with Ralf Hiptmair,
More informationChapter 1 Mathematical Foundations
Computational Electromagnetics; Chapter 1 1 Chapter 1 Mathematical Foundations 1.1 Maxwell s Equations Electromagnetic phenomena can be described by the electric field E, the electric induction D, the
More informationRadiation Integrals and Auxiliary Potential Functions
Radiation Integrals and Auxiliary Potential Functions Ranga Rodrigo June 23, 2010 Lecture notes are fully based on Balanis [?]. Some diagrams and text are directly from the books. Contents 1 The Vector
More informationIntroduction. J.M. Burgers Center Graduate Course CFD I January Least-Squares Spectral Element Methods
Introduction In this workshop we will introduce you to the least-squares spectral element method. As you can see from the lecture notes, this method is a combination of the weak formulation derived from
More informationThe Mortar Boundary Element Method
The Mortar Boundary Element Method A Thesis submitted for the degree of Doctor of Philosophy by Martin Healey School of Information Systems, Computing and Mathematics Brunel University March 2010 Abstract
More informationVarious Two Dimensional Multiscale Finite Element Formulations for the Eddy Current Problem in Iron Laminates
Various Two Dimensional Multiscale Finite Element Formulations for the Eddy Current Problem in Iron Laminates Karl Hollaus and Joachim Schöberl Institute for Analysis and Scientific Computing Vienna University
More informationInverse source estimation problems in magnetostatics
Inverse source estimation problems in magnetostatics ERNSI Workshop, Lyon, 2017 Juliette Leblond INRIA Sophia Antipolis (France), Team APICS Joint work with: Laurent Baratchart, Sylvain Chevillard, Doug
More informationElectromagnetic wave propagation. ELEC 041-Modeling and design of electromagnetic systems
Electromagnetic wave propagation ELEC 041-Modeling and design of electromagnetic systems EM wave propagation In general, open problems with a computation domain extending (in theory) to infinity not bounded
More informationChapter 5. Magnetostatics
Chapter 5. Magnetostatics 5.1 The Lorentz Force Law 5.1.1 Magnetic Fields Consider the forces between charges in motion Attraction of parallel currents and Repulsion of antiparallel ones: How do you explain
More informationA SADDLE POINT LEAST SQUARES METHOD FOR SYSTEMS OF LINEAR PDES. Klajdi Qirko
A SADDLE POINT LEAST SQUARES METHOD FOR SYSTEMS OF LINEAR PDES by Klajdi Qirko A dissertation submitted to the Faculty of the University of Delaware in partial fulfillment of the requirements for the degree
More informationNumerical Analysis of Electromagnetic Fields
Pei-bai Zhou Numerical Analysis of Electromagnetic Fields With 157 Figures Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest Contents Part 1 Universal Concepts
More informationAntennas and Propagation. Chapter 2: Basic Electromagnetic Analysis
Antennas and Propagation : Basic Electromagnetic Analysis Outline Vector Potentials, Wave Equation Far-field Radiation Duality/Reciprocity Transmission Lines Antennas and Propagation Slide 2 Antenna Theory
More informationScientific Computing WS 2017/2018. Lecture 18. Jürgen Fuhrmann Lecture 18 Slide 1
Scientific Computing WS 2017/2018 Lecture 18 Jürgen Fuhrmann juergen.fuhrmann@wias-berlin.de Lecture 18 Slide 1 Lecture 18 Slide 2 Weak formulation of homogeneous Dirichlet problem Search u H0 1 (Ω) (here,
More information2 FORMULATIONS 2.1 The H-J formulation Let Ω be a domain consisting of a conducting region R and a non-conducting region S. Here assume that Ω, R and
Annual Report of ADVENTURE Project ADV-99-1(1999) LARGE-SCALE MAGNETIC FIELD ANALYSES Hiroshi KANAYAMA Department of Intelligent Machinery and Systems Graduate School of Engineering, Kyushu University
More informationINTRODUCTION TO FINITE ELEMENT METHODS ON ELLIPTIC EQUATIONS LONG CHEN
INTROUCTION TO FINITE ELEMENT METHOS ON ELLIPTIC EQUATIONS LONG CHEN CONTENTS 1. Poisson Equation 1 2. Outline of Topics 3 2.1. Finite ifference Method 3 2.2. Finite Element Method 3 2.3. Finite Volume
More informationAMS 529: Finite Element Methods: Fundamentals, Applications, and New Trends
AMS 529: Finite Element Methods: Fundamentals, Applications, and New Trends Lecture 25: Introduction to Discontinuous Galerkin Methods Xiangmin Jiao SUNY Stony Brook Xiangmin Jiao Finite Element Methods
More informationChapter 2. Electrostatics. Introduction to Electrodynamics, 3 rd or 4 rd Edition, David J. Griffiths
Chapter 2. Electrostatics Introduction to Electrodynamics, 3 rd or 4 rd Edition, David J. Griffiths 2.3 Electric Potential 2.3.1 Introduction to Potential E 0 We're going to reduce a vector problem (finding
More informationA LAGRANGE MULTIPLIER METHOD FOR ELLIPTIC INTERFACE PROBLEMS USING NON-MATCHING MESHES
A LAGRANGE MULTIPLIER METHOD FOR ELLIPTIC INTERFACE PROBLEMS USING NON-MATCHING MESHES P. HANSBO Department of Applied Mechanics, Chalmers University of Technology, S-4 96 Göteborg, Sweden E-mail: hansbo@solid.chalmers.se
More informationNumerical problems in 3D magnetostatic FEM analysis
Numerical problems in 3D magnetostatic FEM analysis CAZACU DUMITRU Department of electronics, computers and electrical engineering University of Pitesti Str.Tirgu din Vale nr.1 Pitesti ROMANIA cazacu_dumitru@yahoo.com
More informationFinite Element Method (FEM)
Finite Element Method (FEM) The finite element method (FEM) is the oldest numerical technique applied to engineering problems. FEM itself is not rigorous, but when combined with integral equation techniques
More informationScientific Computing WS 2018/2019. Lecture 15. Jürgen Fuhrmann Lecture 15 Slide 1
Scientific Computing WS 2018/2019 Lecture 15 Jürgen Fuhrmann juergen.fuhrmann@wias-berlin.de Lecture 15 Slide 1 Lecture 15 Slide 2 Problems with strong formulation Writing the PDE with divergence and gradient
More informationMaxwell s Equations. In the previous chapters we saw the four fundamental equations governging electrostatics and magnetostatics. They are.
Maxwell s Equations Introduction In the previous chapters we saw the four fundamental equations governging electrostatics and magnetostatics. They are D = ρ () E = 0 (2) B = 0 (3) H = J (4) In the integral
More informationTrefftz-Discontinuous Galerkin Methods for Maxwell s Equations
Trefftz-Discontinuous Galerkin Methods for Maxwell s Equations Ilaria Perugia Faculty of Mathematics, University of Vienna Vienna P D E Ralf Hiptmair (ETH Zürich), Andrea Moiola (University of Reading)
More informationA NEW APPROXIMATION TECHNIQUE FOR DIV-CURL SYSTEMS
MATHEMATICS OF COMPUTATION Volume 73, Number 248, Pages 1739 1762 S 0025-5718(03)01616-8 Article electronically published on August 26, 2003 A NEW APPROXIMATION TECHNIQUE FOR DIV-CURL SYSTEMS JAMES H.
More informationTrefftz-Discontinuous Galerkin Methods for the Time-Harmonic Maxwell Equations
Trefftz-Discontinuous Galerkin Methods for the Time-Harmonic Maxwell Equations Ilaria Perugia Dipartimento di Matematica - Università di Pavia (Italy) http://www-dimat.unipv.it/perugia Joint work with
More informationngsxfem: Geometrically unfitted discretizations with Netgen/NGSolve (https://github.com/ngsxfem/ngsxfem)
ngsxfem: Geometrically unfitted discretizations with Netgen/NGSolve (https://github.com/ngsxfem/ngsxfem) Christoph Lehrenfeld (with contributions from F. Heimann, J. Preuß, M. Hochsteger,...) lehrenfeld@math.uni-goettingen.de
More informationRecovery-Based a Posteriori Error Estimators for Interface Problems: Mixed and Nonconforming Elements
Recovery-Based a Posteriori Error Estimators for Interface Problems: Mixed and Nonconforming Elements Zhiqiang Cai Shun Zhang Department of Mathematics Purdue University Finite Element Circus, Fall 2008,
More informationA posteriori error estimates for Maxwell Equations
www.oeaw.ac.at A posteriori error estimates for Maxwell Equations J. Schöberl RICAM-Report 2005-10 www.ricam.oeaw.ac.at A POSTERIORI ERROR ESTIMATES FOR MAXWELL EQUATIONS JOACHIM SCHÖBERL Abstract. Maxwell
More informationSpectral finite elements for a mixed formulation in computational acoustics taking flow effects into account
Spectral finite elements for a mixed formulation in computational acoustics taking flow effects into account Manfred Kaltenbacher in cooperation with A. Hüppe, I. Sim (University of Klagenfurt), G. Cohen
More informationCoupling of eddy-current and circuit problems
Coupling of eddy-current and circuit problems Ana Alonso Rodríguez*, ALBERTO VALLI*, Rafael Vázquez Hernández** * Department of Mathematics, University of Trento ** Department of Applied Mathematics, University
More informationFrom the Boundary Element DDM to local Trefftz Finite Element Methods on Polyhedral Meshes
www.oeaw.ac.at From the Boundary Element DDM to local Trefftz Finite Element Methods on Polyhedral Meshes D. Copeland, U. Langer, D. Pusch RICAM-Report 2008-10 www.ricam.oeaw.ac.at From the Boundary Element
More informationPREPRINT 2010:25. Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method ERIK BURMAN PETER HANSBO
PREPRINT 2010:25 Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method ERIK BURMAN PETER HANSBO Department of Mathematical Sciences Division of Mathematics CHALMERS
More informationThe Discontinuous Galerkin Finite Element Method
The Discontinuous Galerkin Finite Element Method Michael A. Saum msaum@math.utk.edu Department of Mathematics University of Tennessee, Knoxville The Discontinuous Galerkin Finite Element Method p.1/41
More informationMATH 332: Vector Analysis Summer 2005 Homework
MATH 332, (Vector Analysis), Summer 2005: Homework 1 Instructor: Ivan Avramidi MATH 332: Vector Analysis Summer 2005 Homework Set 1. (Scalar Product, Equation of a Plane, Vector Product) Sections: 1.9,
More informationA Generalization for Stable Mixed Finite Elements
A Generalization for Stable Mixed Finite Elements Andrew Gillette joint work with Chandrajit Bajaj Department of Mathematics Institute of Computational Engineering and Sciences University of Texas at Austin,
More informationLeast-Squares Finite Element Methods
Pavel В. Bochev Max D. Gunzburger Least-Squares Finite Element Methods Spri ringer Contents Part I Survey of Variational Principles and Associated Finite Element Methods 1 Classical Variational Methods
More informationInverse Problems and Optimal Design in Electricity and Magnetism
Inverse Problems and Optimal Design in Electricity and Magnetism P. Neittaanmäki Department of Mathematics, University of Jyväskylä M. Rudnicki Institute of Electrical Engineering, Warsaw and A. Savini
More informationModel Reduction via Discretization : Elasticity and Boltzmann
Model Reduction via Discretization : Elasticity and Boltzmann Joachim Schöberl Computational Mathematics in Engineering Institute for Analysis and Scientific Computing Vienna University of Technology IMA
More informationChapter 6. Finite Element Method. Literature: (tiny selection from an enormous number of publications)
Chapter 6 Finite Element Method Literature: (tiny selection from an enormous number of publications) K.J. Bathe, Finite Element procedures, 2nd edition, Pearson 2014 (1043 pages, comprehensive). Available
More informationMulti-Scale FEM and Magnetic Vector Potential A for 3D Eddy Currents in Laminated Media
Multi-Scale FEM and Magnetic Vector Potential A for 3D Eddy Currents in Laminated Media K. Hollaus, and J. Schöberl Institute for Analysis and Scientific Computing, Wiedner Hauptstr. 8, A-14 Vienna, Austria
More informationA DELTA-REGULARIZATION FINITE ELEMENT METHOD FOR A DOUBLE CURL PROBLEM WITH DIVERGENCE-FREE CONSTRAINT
A DELTA-REGULARIZATION FINITE ELEMENT METHOD FOR A DOUBLE CURL PROBLEM WITH DIVERGENCE-FREE CONSTRAINT HUOYUAN DUAN, SHA LI, ROGER C. E. TAN, AND WEIYING ZHENG Abstract. To deal with the divergence-free
More informationFinal Ph.D. Progress Report. Integration of hp-adaptivity with a Two Grid Solver: Applications to Electromagnetics. David Pardo
Final Ph.D. Progress Report Integration of hp-adaptivity with a Two Grid Solver: Applications to Electromagnetics. David Pardo Dissertation Committee: I. Babuska, L. Demkowicz, C. Torres-Verdin, R. Van
More informationSolving the curl-div system using divergence-free or curl-free finite elements
Solving the curl-div system using divergence-free or curl-free finite elements Alberto Valli Dipartimento di Matematica, Università di Trento, Italy or: Why I say to my students that divergence-free finite
More informationTheory of Electromagnetic Nondestructive Evaluation
EE 6XX Theory of Electromagnetic NDE: Theoretical Methods for Electromagnetic Nondestructive Evaluation 1915 Scholl Road CNDE Ames IA 50011 Graduate Tutorial Notes 2004 Theory of Electromagnetic Nondestructive
More informationA primer on Numerical methods for elasticity
A primer on Numerical methods for elasticity Douglas N. Arnold, University of Minnesota Complex materials: Mathematical models and numerical methods Oslo, June 10 12, 2015 One has to resort to the indignity
More informationA Nested Dissection Parallel Direct Solver. for Simulations of 3D DC/AC Resistivity. Measurements. Maciej Paszyński (1,2)
A Nested Dissection Parallel Direct Solver for Simulations of 3D DC/AC Resistivity Measurements Maciej Paszyński (1,2) David Pardo (2), Carlos Torres-Verdín (2) (1) Department of Computer Science, AGH
More informationWeighted Regularization of Maxwell Equations Computations in Curvilinear Polygons
Weighted Regularization of Maxwell Equations Computations in Curvilinear Polygons Martin Costabel, Monique Dauge, Daniel Martin and Gregory Vial IRMAR, Université de Rennes, Campus de Beaulieu, Rennes,
More informationDivergence-free or curl-free finite elements for solving the curl-div system
Divergence-free or curl-free finite elements for solving the curl-div system Alberto Valli Dipartimento di Matematica, Università di Trento, Italy Joint papers with: Ana Alonso Rodríguez Dipartimento di
More informationA Posteriori Error Estimation Techniques for Finite Element Methods. Zhiqiang Cai Purdue University
A Posteriori Error Estimation Techniques for Finite Element Methods Zhiqiang Cai Purdue University Department of Mathematics, Purdue University Slide 1, March 16, 2017 Books Ainsworth & Oden, A posteriori
More informationDG discretization of optimized Schwarz methods for Maxwell s equations
DG discretization of optimized Schwarz methods for Maxwell s equations Mohamed El Bouajaji, Victorita Dolean,3, Martin J. Gander 3, Stéphane Lanteri, and Ronan Perrussel 4 Introduction In the last decades,
More informationKey words. Incompressible magnetohydrodynamics, mixed finite element methods, discontinuous Galerkin methods
A MIXED DG METHOD FOR LINEARIZED INCOMPRESSIBLE MAGNETOHYDRODYNAMICS PAUL HOUSTON, DOMINIK SCHÖTZAU, AND XIAOXI WEI Journal of Scientific Computing, vol. 40, pp. 8 34, 009 Abstract. We introduce and analyze
More informationFinite Element Exterior Calculus. Douglas N. Arnold, University of Minnesota The 41st Woudschoten Conference 5 7 October 2016
Finite Element Exterior Calculus Douglas N. Arnold, University of Minnesota The 41st Woudschoten Conference 5 7 October 2016 The Hilbert complex framework Discretization of Hilbert complexes Finite element
More informationCreated by T. Madas VECTOR OPERATORS. Created by T. Madas
VECTOR OPERATORS GRADIENT gradϕ ϕ Question 1 A surface S is given by the Cartesian equation x 2 2 + y = 25. a) Draw a sketch of S, and describe it geometrically. b) Determine an equation of the tangent
More informationDiscontinuous Petrov-Galerkin Methods
Discontinuous Petrov-Galerkin Methods Friederike Hellwig 1st CENTRAL School on Analysis and Numerics for Partial Differential Equations, November 12, 2015 Motivation discontinuous Petrov-Galerkin (dpg)
More informationSimulation and Visualization of Safing Sensor
American Journal of Applied Sciences 2 (8): 1261-1265, 2005 ISSN 1546-9239 2005 Science Publications Simulation and Visualization of Safing Sensor haled M. Furati, Hattan Tawfiq and Abul Hasan Siddiqi
More informationfield using second order edge elements in 3D
The current issue and full text archive of this journal is available at http://www.emerald-library.com using second order edge elements in 3D Z. Ren Laboratoire de GeÂnie Electrique de Paris, UniversiteÂs
More informationHp-Adaptivity on Anisotropic Meshes for Hybridized Discontinuous Galerkin Scheme
Hp-Adaptivity on Anisotropic Meshes for Hybridized Discontinuous Galerkin Scheme Aravind Balan, Michael Woopen and Georg May AICES Graduate School, RWTH Aachen University, Germany 22nd AIAA Computational
More informationAposteriorierrorestimatesinFEEC for the de Rham complex
AposteriorierrorestimatesinFEEC for the de Rham complex Alan Demlow Texas A&M University joint work with Anil Hirani University of Illinois Urbana-Champaign Partially supported by NSF DMS-1016094 and a
More informationarxiv: v2 [math.na] 23 Apr 2016
Improved ZZ A Posteriori Error Estimators for Diffusion Problems: Conforming Linear Elements arxiv:508.009v2 [math.na] 23 Apr 206 Zhiqiang Cai Cuiyu He Shun Zhang May 2, 208 Abstract. In [8], we introduced
More informationAntenna Theory (Engineering 9816) Course Notes. Winter 2016
Antenna Theory (Engineering 9816) Course Notes Winter 2016 by E.W. Gill, Ph.D., P.Eng. Unit 1 Electromagnetics Review (Mostly) 1.1 Introduction Antennas act as transducers associated with the region of
More informationJasmin Smajic1, Christian Hafner2, Jürg Leuthold2, March 23, 2015
Jasmin Smajic, Christian Hafner 2, Jürg Leuthold 2, March 23, 205 Time Domain Finite Element Method (TD FEM): Continuous and Discontinuous Galerkin (DG-FEM) HSR - University of Applied Sciences of Eastern
More informationFull Wave Analysis of RF Signal Attenuation in a Lossy Rough Surface Cave Using a High Order Time Domain Vector Finite Element Method
Progress In Electromagnetics Research Symposium 2006, Cambridge, USA, March 26-29 425 Full Wave Analysis of RF Signal Attenuation in a Lossy Rough Surface Cave Using a High Order Time Domain Vector Finite
More informationNumerical Analysis of Nonlinear Multiharmonic Eddy Current Problems
Numerical Analysis of Nonlinear Multiharmonic Eddy Current Problems F. Bachinger U. Langer J. Schöberl April 2004 Abstract This work provides a complete analysis of eddy current problems, ranging from
More informationThe Divergence Theorem Stokes Theorem Applications of Vector Calculus. Calculus. Vector Calculus (III)
Calculus Vector Calculus (III) Outline 1 The Divergence Theorem 2 Stokes Theorem 3 Applications of Vector Calculus The Divergence Theorem (I) Recall that at the end of section 12.5, we had rewritten Green
More informationFinite Element Methods for Optical Device Design
DFG Research Center Matheon mathematics for key technologies and Zuse Institute Berlin Finite Element Methods for Optical Device Design Frank Schmidt Sven Burger, Roland Klose, Achim Schädle, Lin Zschiedrich
More informationA Balancing Algorithm for Mortar Methods
A Balancing Algorithm for Mortar Methods Dan Stefanica Baruch College, City University of New York, NY 11, USA Dan Stefanica@baruch.cuny.edu Summary. The balancing methods are hybrid nonoverlapping Schwarz
More informationEECS 117 Lecture 19: Faraday s Law and Maxwell s Eq.
University of California, Berkeley EECS 117 Lecture 19 p. 1/2 EECS 117 Lecture 19: Faraday s Law and Maxwell s Eq. Prof. Niknejad University of California, Berkeley University of California, Berkeley EECS
More informationAlgebraic Multigrid as Solvers and as Preconditioner
Ò Algebraic Multigrid as Solvers and as Preconditioner Domenico Lahaye domenico.lahaye@cs.kuleuven.ac.be http://www.cs.kuleuven.ac.be/ domenico/ Department of Computer Science Katholieke Universiteit Leuven
More informationMultispace and Multilevel BDDC. Jan Mandel University of Colorado at Denver and Health Sciences Center
Multispace and Multilevel BDDC Jan Mandel University of Colorado at Denver and Health Sciences Center Based on joint work with Bedřich Sousedík, UCDHSC and Czech Technical University, and Clark R. Dohrmann,
More informationDownloaded 11/21/16 to Redistribution subject to SIAM license or copyright; see
SIAM J. SCI. COMPUT. Vol. 38, No. 6, pp. A349 A3514 c 016 Society for Industrial and Applied Mathematics ROBUST PRECONDITIONING FOR XFEM APPLIED TO TIME-DEPENDENT STOKES PROBLEMS SVEN GROSS, THOMAS LUDESCHER,
More informationFourier Series Expansion in a Non-Orthogonal System of Coordinates for the Simulation of 3D AC Borehole Resistivity Measurements
Fourier Series Expansion in a Non-Orthogonal System of Coordinates for the Simulation of 3D AC Borehole Resistivity Measurements D. Pardo[a], C. Torres-Verdín[a], M. J. Nam[a], M. Paszynski[b], and V.
More informationFinite Elements for Magnetohydrodynamics and its Optimal Control
Finite Elements for Magnetohydrodynamics and its Karl Kunisch Marco Discacciati (RICAM) FEM Symposium Chemnitz September 25 27, 2006 Overview 1 2 3 What is Magnetohydrodynamics? Magnetohydrodynamics (MHD)
More informationHIGH VOLTAGE TECHNIQUES REVİEW: Electrostatics & Magnetostatics
HIGH VOLTAGE TECHNIQUES REVİEW: Electrostatics & Magnetostatics Zap You walk across the rug, reach for the doorknob and...zap!!! In the winter, when you change your pullover you hear and/or see sparks...
More informationFinite Difference Solution of Maxwell s Equations
Chapter 1 Finite Difference Solution of Maxwell s Equations 1.1 Maxwell s Equations The principles of electromagnetism have been deduced from experimental observations. These principles are Faraday s law,
More informationAMS 529: Finite Element Methods: Fundamentals, Applications, and New Trends
AMS 529: Finite Element Methods: Fundamentals, Applications, and New Trends Lecture 3: Finite Elements in 2-D Xiangmin Jiao SUNY Stony Brook Xiangmin Jiao Finite Element Methods 1 / 18 Outline 1 Boundary
More informationTECHNO INDIA BATANAGAR
TECHNO INDIA BATANAGAR ( DEPARTMENT OF ELECTRONICS & COMMUNICATION ENGINEERING) QUESTION BANK- 2018 1.Vector Calculus Assistant Professor 9432183958.mukherjee@tib.edu.in 1. When the operator operates on
More informationSome Two-Dimensional Multiscale Finite Element Formulations for the Eddy Current Problem in Iron Laminates
Some Two-Dimensional Multiscale Finite Element Formulations for the Eddy Current Problem in Iron Laminates K. Hollaus, and J. Schöberl Institute for Analysis and Scientific Computing, Vienna University
More informationAn inverse problem for eddy current equations
An inverse problem for eddy current equations Alberto Valli Department of Mathematics, University of Trento, Italy In collaboration with: Ana Alonso Rodríguez, University of Trento, Italy Jessika Camaño,
More informationA posteriori error estimates for a Maxwell type problem
Russ. J. Numer. Anal. Math. Modelling, Vol. 24, No. 5, pp. 395 408 (2009) DOI 0.55/ RJNAMM.2009.025 c de Gruyter 2009 A posteriori error estimates for a Maxwell type problem I. ANJAM, O. MALI, A. MUZALEVSKY,
More informationA Multigrid Method for Two Dimensional Maxwell Interface Problems
A Multigrid Method for Two Dimensional Maxwell Interface Problems Susanne C. Brenner Department of Mathematics and Center for Computation & Technology Louisiana State University USA JSA 2013 Outline A
More information