TV SM -5 1 0 7 & TV B A -5 0 2 9 F R E D R IK H O L M S T R Ö M S T R U C T U R E -A C O U S T IC A N A L Y S IS U S IN G B E M /F E M ; IM P L E M E STRUCTURE-ACOUSTIC ANALYSIS N T A T IO USING BEM/FEM; N IN M IMPLEMENTATION IN MATLAB A T L A B FREDRIK HOLMSTRÖM Structural Mechanics & Master’s Dissertation Engineering Acoustics Structural Mechanics ISRN LUTVDG/TVSM--01/5107--SE (1-100) ISSN 0281-6679 Engineering Acoustics ISRN LUTVDG/TVBA--01/5029--SE (1-100) ISSN 0281-8477 STRUCTURE-ACOUSTIC ANALYSIS USING BEM/FEM; IMPLEMENTATION IN MATLAB Master’s Dissertation by FREDRIK HOLMSTRÖM Supervisors PETER DAVIDSSON, Structural Mechanics JONAS BRUNSKOG, Engineering Acoustics Copyright © 2001 by Structural Mechanics & Engineering Acoustics, LTH, Sweden. Printed by KFS i Lund AB, Lund, Sweden, May 2001. For information, address: Structural Mechanics, LTH, Lund University, Box 118, SE-221 00 Lund, Sweden. Homepage: http://www.byggmek.lth.se Engineering Acoustics, LTH, Lund University, Box 118, SE-221 00 Lund, Sweden. Homepage: http://www.akustik.lth.se Detta (cid:228)r en tom sida! Abstract Thismasterdissertationdescribestheprocessofimplementingacoupling between the boundary element method (BEM) and the finite element method (FEM) for three dimensional time harmonic structure-acoustic models in CALFEM, which is a finite element toolbox to MATLAB. Since no boundary elements earlier have been represented in CALFEM the development and implementation of constant and linear boundary elements is also described in this thesis. To verify the correctness of the implemented functions and to show how they are used three model ex- amples are performed: one using only BEM, and two structure-acoustics. Keywords: BEM, FEM, Coupled, CALFEM, Vibro-Acoustics, Acous- tics, Implementation. Cover Picture: Sound pressure level on vibrating plate. i ii Preface ThismasterdissertationhasbeenperformedattheDivisionofStructural Mechanics and the Division of Engineering Acoustics, Lund University, during the autumn and spring term of year 2000/2001. Supervisors for this work have been Eng. Lic. Jonas Brunskog and M. Sc. Peter Davidsson. I would like to express my gratitude for their support, during times of both success and frustration. I would also like tothankthepersonalatStructuralMechanicsandatBuildingAcoustics, that have been involved in my work. Finally, Andreas, Bj¨orn, Anders, andPeterdeservesathank, forkeep- ing the spirit up in the ”master’s dissertation room”. Lund, May 2001 Fredrik Holmstro¨m iii iv Summary The purpose of this master dissertation is to implement BEM (Boundary Element Method) and a coupling between BEM and FEM (Finite Ele- ment Method) in CALFEM (a FEM toolbox to MATLAB) for structure- acoustic models. Twodifferentboundaryelements,whichformulationsarebasedonthe Helmholtz integral equation, has been developed: a one node constant, and a four node linear. Both elements are quadrilateral and can take any orientation in the three dimensional space. The linear boundary element can be coupled with two different four node finite elements: a plate element with 12 degrees of freedom, and a shell element with 24 degrees of freedom. Since BEM and coupled BEM/FEM problems, modelled with func- tionsdevelopedinthisdissertation,canbeverytimeconsuming,themost important future improvement is to construct time reducing measures. v vi Contents 1 Introduction 1 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Purpose with the Thesis . . . . . . . . . . . . . . . . . . 2 1.3 Basic Relationships . . . . . . . . . . . . . . . . . . . . . 3 1.4 Essential Features . . . . . . . . . . . . . . . . . . . . . . 4 2 Fundamental Functions and Equations 5 2.1 The Helmholtz Equation . . . . . . . . . . . . . . . . . . 5 2.2 The Free-Space Green’s Function . . . . . . . . . . . . . 8 2.3 The Helmholtz Integral Equation . . . . . . . . . . . . . 10 3 BEM Formulation 13 3.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.1.1 Pre-Processing . . . . . . . . . . . . . . . . . . . 13 3.1.2 Post-Processing . . . . . . . . . . . . . . . . . . . 15 3.2 Constant Element . . . . . . . . . . . . . . . . . . . . . . 15 3.2.1 Pre-Processing . . . . . . . . . . . . . . . . . . . 15 3.2.2 Post Processing . . . . . . . . . . . . . . . . . . . 18 3.3 Four-node Linear Elements . . . . . . . . . . . . . . . . . 18 3.3.1 Pre-Processing . . . . . . . . . . . . . . . . . . . 20 3.3.2 Post-Processing . . . . . . . . . . . . . . . . . . . 21 4 Implemented BEM Functions 23 4.1 Bem infl1q . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.2 Bem infl4q . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.3 Bem spacang . . . . . . . . . . . . . . . . . . . . . . . . 27 4.4 Bem assem . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.5 Bem solveq . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.6 Bem acouspost . . . . . . . . . . . . . . . . . . . . . . . 31 vii