DOCSLIB.ORG

(Galerkin) Finite Element Approximations

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
(Galerkin) Finite Element Approximations (Galerkin) Finite element approximations The finite element method (FEM): special choice for the shape functions φ. ˜ Ne = 5 Subdivide Ω into elements Ωe: Ne Ω = Ωe e=1 [ Ω1 Ω2 Ω3 Ω4 Ω5 Ωe1 Ωe2 = x = a x = b ∩ ∅ Approximate u on each element separately by a polynomial of some degree p, for example by Lagrangian interpolation (using p + 1 nodal points per element). The end points of an element must be nodal points. Example: linear elements Global shape functions: Ne = 5 n T 1 uh(x) = uiφi(x) = φ (x)u φ3(x) i=1 ˜ ˜ X Ω1 Ω2 Ω3 Ω4 Ω5 n: number of global nodal points. x1 = ax2 x3 x4 x5 x6 = b Local element shape functions: e e e T e u (x) = u φ1(x)+u φ2(x) = φ (x)u h 1 2 ˜ φ1(x) φ2(x) ˜ 1 1 T with φ = [φ1(x), φ2(x)] and ˜ e e x x2 x x1 e e e e φ1(x) = − , φ2(x) = − x1 x2 x1 x2 xe xe xe xe 1 − 2 2 − 1 Example: quadratic elements Local element shape functions: φ1(x) φ2(x) φ3(x) 1 1 1 e e e e e e e e e x1 x2 x3 x1 x2 x3 x1 x2 x3 e e e T e uh(x) = u1φ1(x) + u2φ2(x) = φ (x)u ˜ ˜ T with φ = [φ1(x), φ2(x), φ3(x)] and ˜ (x xe)(x xe) (x xe)(x xe) φ (x) = − 2 − 3 , φ (x) = − 1 − 3 , 1 (xe xe)(xe xe) 2 (xe xe)(xe xe) 1 − 2 1 − 3 2 − 1 2 − 3 (x xe)(x xe) φ (x) = − 1 − 2 3 (xe xe)(xe xe) 3 − 1 3 − 2 Example: higher-order elements General polynomials of order p: p+1 e e T e uh(x) = ui φi(x) = φ (x)u i=1 ˜ ˜ X T with φ = [φ1, φ2, . , φp+1]. ˜ Various expansions possible: B Gauss-Lobatto integration points (includes end points) and Lagrangian interpolation (Spectral elements). B Hierarchical base functions: end points are nodes but internal shape functions have no nodes (similar to Legrende polynomials). (hp-FEM). B Legrendre polynomials in discontinuous Galerkin methods. Global numbering 4 N =3 e uh(x) T uh(x) = uiφi(x) = φ (x)u i=1 ˜ ˜ X u 2 u3 u1 u4 u1 φ1(x) u φ (x) Ω1 Ω2 Ω3 u = 2 , φ(x) = 2 u φ (x) x1 = ax2 x3 x4 = b ˜ 3 ˜ 3 u φ (x) 1 4 4 φ (x) Ne = 3 3 Ω1 Ω2 Ω3 x1 = ax2 x3 x4 = b Local numbering in elements Weak form: Find uh in Sh such that dv du ( h,A h) + v (b)h = (v , f) for all v V dx dx h b h h ∈ h Split: N N e dv du e ( h,A h) + v (b)h = (v , f) for all v V dx dx h b h h ∈ h e=1 e=1 X X Write in each element e: 2 2 e e T e e T uh(x) = ui φi(x) = φ (x)ue, vh(x) = vi φi(x) = φ (x)ve i=1 ˜ ˜ i=1 ˜ ˜ X X T e e T where ue = (u1, u2) and φ (x) = (φ1(x), φ2(x)). ˜ ˜ Element matrix and vector We get: Ne Ne T T (ve Keue) + vnhb = ve f e e=1 ˜ ¯ ˜ e=1 ˜ ˜ X X where dφ dφT dφ dφT Ke = ˜,A ˜ = ˜A ˜ dx ¯ dx dx e Ω dx dx Z e f e = (φ, f)e = φf dx ˜ ˜ ZΩe ˜ are the element matrix Ke and element vector f e. ¯ ˜ Local global (assembling) → The local vectors ue and ve are part of the global vectors: ˜ ˜ ue = P eu, ve = P ev, ˜ ¯ ˜ ˜ ¯ ˜ So we get Ne Ne Ne T T T T ˜ v Keue = v P KeP e u = v Ke u ˜e ¯ ˜ ˜ ¯ e ¯ ¯ ˜ ˜ ¯ ˜ e=1 e=1 ˜ e=1 X X Ke X ¯ Ne Ne | {z } Ne vT f = vT P T f = vT f˜ e e e e e e=1 ˜ ˜ e=1 ˜ ¯ ˜ ˜ e=1 ˜ X X f˜ X e | ˜{z } Weak form Substitution into the weak form: vT Ku = vT f for all v ˜ ¯ ˜ ˜ ˜ ˜ or Ku = f ¯ ˜ with ˜ Ne ˜ K = Ke ¯ e=1 ¯ X 0 Ne 0 f = f˜ + . e ˜ e=1 ˜ 0 X h − b Example (1) For example: u1 u2 0 1 0 0 u2 u2 = = = P 2u u3 0 0 1 0 u3 ¯ ˜ ˜ u4 P 2 ¯ 0| 0 {z } 2 2 ˜ T 1 0 K11 K12 0 1 0 0 K2 = P 2 K2P 2 = 2 2 ¯ ¯ ¯ ¯ 0 1 K21 K22 0 0 1 0 0 0 0 0 0 0 2 2 0 K11 K12 0 = 2 2 0 K21 K22 0 0 0 0 0 Example (2) Assembly of K: ¯ 1 1 K11 K12 0 0 0 0 0 0 0 0 0 0 1 1 2 2 K21 K22 0 0 0 K11 K12 0 0 0 0 0 K = + 2 2 + 3 3 ¯ 0 0 0 0 0 K21 K22 0 0 0 K11 K12 0 0 0 0 0 0 0 0 0 0 K3 K3 21 22 1 1 K11 K12 0 0 1 1 2 2 K21 K22 + K11 K12 0 = 2 2 3 3 0 K21 K22 + K11 K12 0 0 K3 K3 21 22 Similar for assembly of f. ˜ Exercise 5 The ‘bandwidth’ of K for linear shape functions is 3. How large is the bandwidth for K when using polynomial¯ shape functions of order k, k 1? ¯ ≥ Dirichlet conditions Renumber and split vector and matrix (‘u’ unknown, ‘p’ prescribed) uu vu Kuu Kup f u u = ˜ , v = ˜ ,K = ¯ ¯ , f = ˜ ˜ up ˜ vp ¯ Kpu Kpp ˜ f p ˜ ˜ ¯ ¯ ˜ Note: up prescribed values, vp = 0. Weak form ˜ ˜ ˜ vT Ku = vT f for all v ˜ ¯ ˜ ˜ ˜ ˜ with vp = 0 leads to ˜ ˜ T T vu (Kuuuu + Kupup) = vu f u for all vu ˜ ¯ ˜ ¯ ˜ ˜ ˜ ˜ or Kuuuu = f u Kupup ¯ ˜ ˜ − ¯ ˜ 1D convection-diffusion-reaction equation 1D convection-diffusion-reaction Eq.: find u(x) such that for x (a, b) ∈ ∂u ∂u ∂ ∂u + a (A ) + bu = f ∂t ∂x − ∂x ∂x and u = ua(t), at x = a, t > 0 (ΓD) du A = h (t) at x = b, t > 0 (Γ ) − dx b N Notes: B Strong form; Classical (strong) solution u(x, t) 0 2 B f(x, t) C (a, b) (continuous) then u C (a, b) (twice continuously differentiable)∈ ∈ Oldroyd-B/UCM viscoelastic model λ τ5 +τ = 2ηD where τ5= τ˙ L τ τ LT − · − · or ∂τ τ η + ~u τ L τ τ LT + = 2 D ∂t · ∇ − · − · λ λ ∂u ∂u bu f ∂t a∂x |{z} | {z } | {z } | {z } Diffusion is missing (A = 0). Dimensionless form (1) Scaling: t = tct∗ tc : characteristic time u = Uu∗ U : characteristic value solution x = Lx∗ L : characteristic length scale Dimensionless variables: (1). O 2 U ∂u∗ aU ∂u∗ AU ∂ u∗ + 2 2 + bUu∗ = f tc ∂t∗ L ∂x∗ − L ∂x∗ Relative to convection: 2 L ∂u∗ ∂u∗ 1 ∂ u∗ bL L + 2 + u∗ = f atc ∂t∗ ∂x∗ − Pe∂x∗ a aU aL Pe = : P´eclet number, convection/diffusion. A Dimensionless form (2) Time scales: L convection: B a L2 diffusion: B A 1 source: (’relaxation time’) B b B time scales in b.c. B externally or internally generated frequencies (’von Karman vortex’) We choose tc = L/a (convection) and get 2 ∂u∗ ∂u∗ 1 ∂ u∗ + 2 + b∗u∗ = f ∗ ∂t∗ ∂x∗ − Pe∂x∗ bL L with b = and f = f. Pe > 1 : convection dominated ∗ a ∗ aU Exercise 6 L2 Assume b = 0 and f = 0. We take for the typical time scale t = (diffusion c A time scale). Show that we now have the non-dimensional form 2 ∂u∗ ∂u∗ ∂ u∗ + Pe 2 = 0 ∂t∗ ∂x∗ − ∂x∗ When will this non-dimensional form be preferable over the one on the previous slide? Steady state 1D convection-diffusion-reaction equation steady state: ∂u/∂t = 0 du d du (u) = a (A ) + bu = f L dx − dx dx : linear operator, with b.c. L u = ua, at x = a (ΓD) du A = h at x = b (Γ ) − dx b N Weak form Multiply with test function v and integrate: (v, u f) = 0 for all v L − Partial integration of the diffusion term and inserting b.c. we get: Find u S such that ∈ dv du du ( ,A ) + (v, a ) + (v, bu) + v(b)h = (v, f) for all v V dx dx dx b ∈ where S and V are appropriate spaces. Galerkin FEM approximations (1) Approximation spaces Sh and Vh: n T uh(x) = uiφi(x) = φ (x)u i=1 ˜ ˜ X n T vh(x) = viφi(x) = φ (x)v i=1 ˜ ˜ X where φ(x) are global shape functions. Substituting this into the weak form gives: Find u˜ S such that h ∈ h dv du du ( h,A h) + (v , a h) + (v , bu ) + v (b)h = (v , f) for all v V dx dx h dx h h h b h h ∈ h or vT Ku = vT f for all v ˜ ¯ ˜ ˜ ˜ ˜ Galerkin FEM approximations (2) and thus Ku = f ¯ ˜ ˜ with T dφ dφ dφi dφj K = ( ˜,A ˜ ) or Kij = ( ,A ) ¯ dx dx dx dx T dφ dφj + (φ, a ˜ ) + (φi, a ) ˜ dx dx T + (φ, bφ ) + (φi, bφj) ˜ ˜ f = (φ, f) hbφ(b) fi = (φi, f) hbφi(b) ˜ ˜ − ˜ − Galerkin FEM approximations (3) Build from element matrices. For example with constant coefficients and linear element shape functions: dφ dφT 1 1 1 ( , )e = − “stiffness matrix” dx˜ dx˜ h 1 1 − dφT 1 1 1 (φ, )e = − “convection matrix” dx˜ 2 1 1 ˜ − h 2 1 (φ, φT ) = “mass matrix” e 6 1 2 ˜ ˜ where h is the element length.
Load more

Recommended publications
  • A FETI Method with a Mesh Independent Condition Number for the Iteration Matrix Christine Bernardi, Tomas Chacon-Rebollo, Eliseo Chacòn Vera
    A FETI method with a mesh independent condition number for the iteration matrix Christine Bernardi, Tomas Chacon-Rebollo, Eliseo Chacòn Vera To cite this version: Christine Bernardi, Tomas Chacon-Rebollo, Eliseo Chacòn Vera. A FETI method with a mesh in- dependent condition number for the iteration matrix. Computer Methods in Applied Mechanics and Engineering, Elsevier, 2008, 197 (issues-13-16), pp.1410-1429. 10.1016/j.cma.2007.11.019. hal- 00170169 HAL Id: hal-00170169 https://hal.archives-ouvertes.fr/hal-00170169 Submitted on 6 Sep 2007 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. A FETI method with a mesh independent condition number for the iteration matrix C. Bernardi1, T . Chac´on Rebollo2, E. Chac´on Vera2 August 21, 2007 Abstract. We introduce a framework for FETI methods using ideas from the 1 decomposition via Lagrange multipliers of H0 (Ω) derived by Raviart-Thomas [22] and complemented with the detailed work on polygonal domains devel- oped by Grisvard [17]. We compute the action of the Lagrange multipliers 1/2 using the natural H00 scalar product, therefore no consistency error appears. As a byproduct, we obtain that the condition number for the iteration ma- trix is independent of the mesh size and there is no need for preconditioning.
    [Show full text]
  • An Adaptive Hp-Refinement Strategy with Computable Guaranteed Bound
    An adaptive hp-refinement strategy with computable guaranteed bound on the error reduction factor∗ Patrik Danielyz Alexandre Ernzy Iain Smearsx Martin Vohral´ıkyz April 24, 2018 Abstract We propose a new practical adaptive refinement strategy for hp-finite element approximations of elliptic problems. Following recent theoretical developments in polynomial-degree-robust a posteriori error analysis, we solve two types of discrete local problems on vertex-based patches. The first type involves the solution on each patch of a mixed finite element problem with homogeneous Neumann boundary conditions, which leads to an H(div; Ω)-conforming equilibrated flux. This, in turn, yields a guaranteed upper bound on the error and serves to mark mesh vertices for refinement via D¨orfler’s bulk-chasing criterion. The second type of local problems involves the solution, on patches associated with marked vertices only, of two separate primal finite element problems with homogeneous Dirichlet boundary conditions, which serve to decide between h-, p-, or hp-refinement. Altogether, we show that these ingredients lead to a computable guaranteed bound on the ratio of the errors between successive refinements (error reduction factor). In a series of numerical experiments featuring smooth and singular solutions, we study the performance of the proposed hp-adaptive strategy and observe exponential convergence rates. We also investigate the accuracy of our bound on the reduction factor by evaluating the ratio of the predicted reduction factor relative to the true error reduction, and we find that this ratio is in general quite close to the optimal value of one. Key words: a posteriori error estimate, adaptivity, hp-refinement, finite element method, error reduction, equilibrated flux, residual lifting 1 Introduction Adaptive discretization methods constitute an important tool in computational science and engineering.
    [Show full text]
  • Newton-Krylov-BDDC Solvers for Nonlinear Cardiac Mechanics
    Newton-Krylov-BDDC solvers for nonlinear cardiac mechanics Item Type Article Authors Pavarino, L.F.; Scacchi, S.; Zampini, Stefano Citation Newton-Krylov-BDDC solvers for nonlinear cardiac mechanics 2015 Computer Methods in Applied Mechanics and Engineering Eprint version Post-print DOI 10.1016/j.cma.2015.07.009 Publisher Elsevier BV Journal Computer Methods in Applied Mechanics and Engineering Rights NOTICE: this is the author’s version of a work that was accepted for publication in Computer Methods in Applied Mechanics and Engineering. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Computer Methods in Applied Mechanics and Engineering, 18 July 2015. DOI:10.1016/ j.cma.2015.07.009 Download date 07/10/2021 05:34:35 Link to Item http://hdl.handle.net/10754/561071 Accepted Manuscript Newton-Krylov-BDDC solvers for nonlinear cardiac mechanics L.F. Pavarino, S. Scacchi, S. Zampini PII: S0045-7825(15)00221-2 DOI: http://dx.doi.org/10.1016/j.cma.2015.07.009 Reference: CMA 10661 To appear in: Comput. Methods Appl. Mech. Engrg. Received date: 13 December 2014 Revised date: 3 June 2015 Accepted date: 8 July 2015 Please cite this article as: L.F. Pavarino, S. Scacchi, S. Zampini, Newton-Krylov-BDDC solvers for nonlinear cardiac mechanics, Comput. Methods Appl. Mech. Engrg. (2015), http://dx.doi.org/10.1016/j.cma.2015.07.009 This is a PDF file of an unedited manuscript that has been accepted for publication.
    [Show full text]
  • Combining Perturbation Theory and Transformation Electromagnetics for Finite Element Solution of Helmholtz-Type Scattering Probl
    Journal of Computational Physics 274 (2014) 883–897 Contents lists available at ScienceDirect Journal of Computational Physics www.elsevier.com/locate/jcp Combining perturbation theory and transformation electromagnetics for finite element solution of Helmholtz-type scattering problems ∗ Mustafa Kuzuoglu a, Ozlem Ozgun b, a Dept. of Electrical and Electronics Engineering, Middle East Technical University, 06531 Ankara, Turkey b Dept. of Electrical and Electronics Engineering, TED University, 06420 Ankara, Turkey a r t i c l e i n f o a b s t r a c t Article history: A numerical method is proposed for efficient solution of scattering from objects with Received 31 August 2013 weakly perturbed surfaces by combining the perturbation theory, transformation electro- Received in revised form 13 June 2014 magnetics and the finite element method. A transformation medium layer is designed over Accepted 15 June 2014 the smooth surface, and the material parameters of the medium are determined by means Available online 5 July 2014 of a coordinate transformation that maps the smooth surface to the perturbed surface. The Keywords: perturbed fields within the domain are computed by employing the material parameters Perturbation theory and the fields of the smooth surface as source terms in the Helmholtz equation. The main Transformation electromagnetics advantage of the proposed approach is that if repeated solutions are needed (such as in Transformation medium Monte Carlo technique or in optimization problems requiring multiple solutions for a set of Metamaterials perturbed surfaces), computational resources considerably decrease because a single mesh Coordinate transformation is used and the global matrix is formed only once.
    [Show full text]
  • Robust Finite Element Solution
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Computational and Applied Mathematics 212 (2008) 374–389 www.elsevier.com/locate/cam An elliptic singularly perturbed problem with two parameters II: Robust finite element solutionଁ Lj. Teofanova,∗, H.-G. Roosb aDepartment for Fundamental Disciplines, Faculty of Technical Sciences, University of Novi Sad, Trg D.Obradovi´ca 6, 21000 Novi Sad, Serbia bInstitute of Numerical Mathematics, Technical University of Dresden, Dresden D-01062, Germany Received 8 November 2005; received in revised form 8 December 2006 Abstract In this paper we consider a singularly perturbed elliptic model problem with two small parameters posed on the unit square. The problem is solved numerically by the finite element method using piecewise linear or bilinear elements on a layer-adapted Shishkin mesh. We prove that method with bilinear elements is uniformly convergent in an energy norm. Numerical results confirm our theoretical analysis. © 2007 Elsevier B.V. All rights reserved. MSC: 35J25; 65N30; 65N15; 76R99 Keywords: Singular perturbation; Convection–diffusion problems; Finite element method; Two small parameters; Shishkin mesh 1. Introduction In this paper, we consider a finite element method for the following singularly perturbed elliptic two-parameter problem posed on the unit square = (0, 1) × (0, 1) Lu := −1u + 2b(x)ux + c(x)u = f(x,y) in , u = 0onj, (1) b(x) > 0, c(x) > 0 for x ∈[0, 1], (2) where 0 < 1, 2>1 and and are constants. We assume that b, c and f are sufficiently smooth functions and that f satisfies the compatibility conditions f(0, 0) = f(0, 1) = f(1, 0) = f(1, 1) = 0.
    [Show full text]
  • FETI-DP Domain Decomposition Method
    ECOLE´ POLYTECHNIQUE FED´ ERALE´ DE LAUSANNE Master Project FETI-DP Domain Decomposition Method Supervisors: Author: Davide Forti Christoph Jaggli¨ Dr. Simone Deparis Prof. Alfio Quarteroni A thesis submitted in fulfilment of the requirements for the degree of Master in Mathematical Engineering in the Chair of Modelling and Scientific Computing Mathematics Section June 2014 Declaration of Authorship I, Christoph Jaggli¨ , declare that this thesis titled, 'FETI-DP Domain Decomposition Method' and the work presented in it are my own. I confirm that: This work was done wholly or mainly while in candidature for a research degree at this University. Where any part of this thesis has previously been submitted for a degree or any other qualification at this University or any other institution, this has been clearly stated. Where I have consulted the published work of others, this is always clearly at- tributed. Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work. I have acknowledged all main sources of help. Where the thesis is based on work done by myself jointly with others, I have made clear exactly what was done by others and what I have contributed myself. Signed: Date: ii ECOLE´ POLYTECHNIQUE FED´ ERALE´ DE LAUSANNE Abstract School of Basic Science Mathematics Section Master in Mathematical Engineering FETI-DP Domain Decomposition Method by Christoph Jaggli¨ FETI-DP is a dual iterative, nonoverlapping domain decomposition method. By a Schur complement procedure, the solution of a boundary value problem is reduced to solving a symmetric and positive definite dual problem in which the variables are directly related to the continuity of the solution across the interface between the subdomains.
    [Show full text]
  • Tr2005-865 a Bddc Algorithm for Flow in Porous Media
    TR2005-865 A BDDC ALGORITHM FOR FLOW IN POROUS MEDIA WITH A HYBRID FINITE ELEMENT DISCRETIZATION XUEMIN TU∗ Abstract. The BDDC (balancing domain decomposition by constraints) methods have been applied successfully to solve the large sparse linear algebraic systems arising from conforming finite element discretizations of elliptic boundary value problems. In this paper, the scalar elliptic problems for flow in porous media are discretized by a hybrid finite element method which is equivalent to a nonconforming finite element method. The BDDC algorithm is extended to these problems which originate as saddle point problems. Edge/face average constraints are enforced across the interface and the same rate of convergence is obtained as in conforming cases. The condition number of the preconditioned system is estimated and numerical experiments are discussed. Key words. BDDC, domain decomposition, saddle point problem, condition number, hybrid finite element method AMS subject classifications 65N30, 65N55 65F10 1. Introduction. Mixed formulations of elliptic problems, see [3], have many applications, e.g., for flow in porous media, for which a good approximation to the velocity, which involves derivatives of the solution of the differential equations, is required. These discretizations lead to large, sparse, symmetric, indefinite linear systems. In our recent paper [24], we extended the BDDC algorithm to this mixed for- mulation of elliptic problems. The BDDC algorithms are nonoverlapping domain de- composition methods, introduced by Dohrmann [6] and further analyzed in [15, 16], are similar to the balancing Neumann-Neumann algorithms, see [14, 7]. However, the BDDC methods have different coarse components which are formed by a small num- ber of continuity constraints enforced across the interface throughout the iterations.
    [Show full text]
  • A Numerical Study of Feti Algorithms for Mortar Finite Element Methods∗
    SIAM J. SCI. COMPUT. c 2001 Society for Industrial and Applied Mathematics Vol. 23, No. 4, pp. 1135–1160 A NUMERICAL STUDY OF FETI ALGORITHMS FOR MORTAR FINITE ELEMENT METHODS∗ DAN STEFANICA† Abstract. The finite element tearing and interconnecting (FETI) method is an iterative sub- structuring method using Lagrange multipliers to enforce the continuity of the finite element solution across the subdomain interface. Mortar finite elements are nonconforming finite elements that al- low for a geometrically nonconforming decomposition of the computational domain into subregions and, at the same time, for the optimal coupling of different variational approximations in different subregions. We present a numerical study of FETI algorithms for elliptic self-adjoint equations discretized by mortar finite elements. Several preconditioners which have been successful for the case of conforming finite elements are considered. We compare the performance of our algorithms when applied to classical mortar elements and to a new family of biorthogonal mortar elements, and we discuss the differences between enforcing mortar conditions instead of continuity conditions for the case of matching nodes across the interface. Our experiments are carried out for both two and three dimensional problems, and include a study of the relative costs of applying different preconditioners for mortar elements. Key words. FETI algorithms, mortar finite elements, Lagrange multipliers, domain decompo- sition AMS subject classifications. 65F10, 65N30, 65N55 PII. S1064827500378829 1. Introduction. The finite element tearing and interconnecting (FETI) method is an iterative substructuring method using Lagrange multipliers which is actively used in industrial-size parallel codes for solving difficult computational mechanics problems. This method was introduced by Farhat and Roux [26]; a detailed presentation is given in [27], a monograph by the same authors.
    [Show full text]
  • Finite Element Tearing and Interconnect for Maxwell's Equations Using
    Finite Element Tearing and Interconnect for Maxwell’s equations using IGA BSc-thesis, MSc-thesis or project/internship work Electrical engineering / Computational engineering / Accelerator physics / Mathematics 1. Context Isogeometric analysis (IGA) is a finite element method (FEM) Computational Electromagnetics using splines for geometry description and basis functions such Prof. Dr. Sebastian Schöps that the geometry can be exactly represented. Recently, an [email protected] isogeometric mortar coupling [1] for electromagnetic problems Office: S4|10 317 was proposed. It is particularly well suited for the eigenfrequency prediction of superconducting accelerator cavities. Each cell, see Numerik und Fig. 1, can be represented by a different subdomain but may still Wissenschaftliches Rechnen share the same discretization. The approach leads to a (stable and Prof. Dr. Herbert Egger spectral correct) saddle-point problem. However, its numerical [email protected] solution is cumbersome and iterative substructuring methods Office: S4|10 101a become attractive. The resulting system is available from a Matlab/Octave code. In this project the finite element tearing and interconnect method (FETI) shall be investigated and standard, possibly low-rank, preconditioners implemented and compared. 2. Task First, familiarize yourself with Maxwell’s eigenvalue problem, the basics of Isogeometric Analysis and the software GeoPDEs [2]. Then understand the ideas behind FETI [3] and implement an iterative solver within the existing software. Finally, experiment with preconditioners to speed up the solution procedure. Fig. 1: 9-cell TESLA cavity 3. Prerequisites decomposed into 9 subdomains Strong background in FEM, basic knowledge of Maxwell’s equations, experience with programming in Matlab/Octave, interest in electromagnetic field simulation.
    [Show full text]
  • A Finite Element Method for Approximating Electromagnetic Scattering from a Conducting Object
    A FINITE ELEMENT METHOD FOR APPROXIMATING ELECTROMAGNETIC SCATTERING FROM A CONDUCTING OBJECT ANDREAS KIRSCH∗ AND PETER MONK† Abstract. We provide an error analysis of a fully discrete finite element – Fourier series method for approximating Maxwell’s equations. The problem is to approximate the electromagnetic field scattered by a bounded, inhomogeneous and anisotropic body. The method is to truncate the domain of the calculation using a series solution of the field away from this domain. We first prove a decomposition for the Poincar´e-Steklov operator on this boundary into an isomorphism and a compact perturbation. This is proved using a novel argument in which the scattering problem is viewed as a perturbation of the free space problem. Using this decomposition, and edge elements to discretize the interior problem, we prove an optimal error estimate for the overall problem. 1. Introduction. Motivated by the problem of computing the interaction of microwave radiation with biological tissue, we shall analyze a finite element method for approximating Maxwell’s equations in an infinite domain. We suppose that there is a bounded anisotropic, conducting object, called the scatterer, illuminated by a time-harmonic microwave source. The microwave source produces an incident electromagnetic field that interacts with the scatterer and produces a scattered time harmonic electromagnetic field. It is the scattered field that we wish to approximate. From the mathematical point of view, the problem can be reduced to that of approximating the total electric field E = E(x), x ∈ R3, which satisfies Maxwell’s equations in all space: −1 2 3 (1.1) ∇ × µr ∇ × E − k rE = 0 , in R where µr and r are respectively the relative permeability tensor (real 3 × 3 matrix) and the relative permit- tivity tensor (complex 3 × 3 matrix) describing the electromagnetic properties of the scatterer.
    [Show full text]
  • New Implementations for the Simultaneous-FETI Method Roberto Molina, François-Xavier Roux
    New implementations for the Simultaneous-FETI method Roberto Molina, François-xavier Roux To cite this version: Roberto Molina, François-xavier Roux. New implementations for the Simultaneous-FETI method. International Journal for Numerical Methods in Engineering, Wiley, 2019, 118 (9), pp.519-535. 10.1002/nme.6024. hal-02294320 HAL Id: hal-02294320 https://hal.archives-ouvertes.fr/hal-02294320 Submitted on 23 Sep 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. ORIG I NAL AR TI CLE New IMPLEMENTATIONS FOR THE Simultaneous-FETI method. Roberto Molina1∗ | FRançois-XaVIER Roux2∗ 1LaborATOIRE Jacques LOUIS Lions, Sorbonne In THIS work, WE PRESENT ALTERNATIVE IMPLEMENTATIONS FOR THE Université, Paris, 75005, FRANCE 2DTIS/ONERA, Université PARIS SaclaY, Simultaneous-FETI (S-FETI) method. DeVELOPED IN RECENT F-91123 Palaiseau, FRANCE Years, THIS METHOD HAS SHOWN TO BE VERY ROBUST FOR HIGHLY Correspondence HETEROGENEOUS problems. HoweVER, THE MEMORY COST IN S- Roberto Molina, Applied Mathematics FETI IS GREATLY INCREASED AND CAN BE A LIMITATION TO ITS use. Department, National LaborATORY FOR ScientifiC Computing - LNCC, Petrópolis, RJ, Our MAIN OBJECTIVE IS TO REDUCE THIS MEMORY USAGE with- 25651-070, BrAZIL OUT LOSING SIGNIfiCANT TIME consumption.
    [Show full text]
  • A Perturbation Method for the 3D Finite Element Modeling of Electrostatically Driven MEMS
    Sensors 2008, 8, 994-1003 sensors ISSN 1424-8220 c 2008 by MDPI www.mdpi.org/sensors Full Paper A Perturbation Method for the 3D Finite Element Modeling of Electrostatically Driven MEMS Mohamed Boutaayamou1,⋆, Ruth V. Sabariego1 and Patrick Dular1,2 1 Department of Electrical Engineering and Computer Science, Applied and Computational Electromagnetics (ACE), 1 University of Li`ege, 1,2 FNRS Sart Tilman Campus, Building B28, B-4000, Li`ege, Belgium E-mails: [email protected]; [email protected]; [email protected] ⋆ Author to whom correspondence should be addressed. Received: 14 September 2007 / Accepted: 8 February 2008 / Published: 19 February 2008 Abstract: In this paper, a finite element (FE) procedure for modeling electrostatically actu- ated MEMS is presented. It concerns a perturbation method for computing electrostatic field distortions due to moving conductors. The computation is split in two steps. First, an un- perturbed problem (in the absence of certain conductors) is solved with the conventional FE method in the complete domain. Second, a perturbation problem is solved in a reduced re- gion with an additional conductor using the solution of the unperturbed problem as a source. When the perturbing region is close to the original source field, an iterative computation may be required. The developed procedure offers the advantage of solving sub-problems in re- duced domains and consequently of benefiting from different problem-adapted meshes. This approach allows for computational efficiency by decreasing the size of the problem. Keywords: Electrostatic field distortions, finite element method, perturbation method, MEMS. 1. Introduction Increased functionality of MEMS has lead to the development of micro-structures that are more and more complex.
    [Show full text]