FEMBEM NOTES.pdf

FEM/BEM NOTES Professor Peter Hunter p.hunterauckland.ac.nz Associate Professor Andrew Pullan a.pullanauckland.ac.nz Department of Engineering Science The University of Auckland New Zealand June 17, 2003 c Copyright 1997-2003 Department of Engineering Science The University of Auckland Contents 1Finite Element Basis Functions1 1.1Representing a One-Dimensional Field . . . . . . . . . . . . . . . . . . . . . . . .1 1.2Linear Basis Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2 1.3Basis Functions as Weighting Functions . . . . . . . . . . . . . . . . . . . . . . .4 1.4Quadratic Basis Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7 1.5Two- and Three-Dimensional Elements. . . . . . . . . . . . . . . . . . . . . . .7 1.6Higher Order Continuity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10 1.7Triangular Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14 1.8Curvilinear Coordinate Systems. . . . . . . . . . . . . . . . . . . . . . . . . . .16 1.9CMISS Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20 2Steady-State Heat Conduction23 2.1One-Dimensional Steady-State Heat Conduction. . . . . . . . . . . . . . . . . .23 2.1.1Integral equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24 2.1.2Integration by parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24 2.1.3Finite element approximation. . . . . . . . . . . . . . . . . . . . . . . .25 2.1.4Element integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26 2.1.5Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27 2.1.6Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29 2.1.7Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29 2.1.8Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29 2.2An -Dependent Source Term. . . . . . . . . . . . . . . . . . . . . . . . . . . .30 2.3The Galerkin Weight Function Revisited . . . . . . . . . . . . . . . . . . . . . . .31 2.4Two and Three-Dimensional Steady-State Heat Conduction . . . . . . . . . . . . .32 2.5Basis Functions - Element Discretisation . . . . . . . . . . . . . . . . . . . . . . .34 2.6Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .36 2.7Assemble Global Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37 2.8Gaussian Quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39 2.9CMISS Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .42 3The Boundary Element 43 3.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43 3.2The Dirac-Delta Function and Fundamental Solutions . . . . . . . . . . . . . . . .43 3.2.1Dirac-Delta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43 3.2.2Fundamental solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . .45 iiCONTENTS 3.3The Two-Dimensional Boundary Element . . . . . . . . . . . . . . . . . .48 3.4Numerical Solution Procedures for the Boundary Integral Equation . . . . . . . . .53 3.5 Numerical uation of Coeffi cient Integrals . . . . . . . . . . . . . . . . . . . .55 3.6The Three-Dimensional Boundary Element . . . . . . . . . . . . . . . . .57 3.7A Comparison of the FE and BE s . . . . . . . . . . . . . . . . . . . . . .58 3.8More on Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . .60 3.8.1Logarithmic quadrature and other special schemes. . . . . . . . . . . . .60 3.8.2Special solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .61 3.9The Boundary Element Applied to other Elliptic PDEs . . . . . . . . . . .61 3.10 Solution of Matrix Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .61 3.11 Coupling the FE and BE techniques. . . . . . . . . . . . . . . . . . . . . . . . .62 3.12 Other BEM techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .64 3.12.1 Trefftz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .64 3.12.2 Regular BEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .64 3.13 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .65 3.14 Axisymmetric Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .67 3.15 Infi nite Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .69 3.16 Appendix Common Fundamental Solutions . . . . . . . . . . . . . . . . . . . . .72 3.16.1 Two-Dimensional equations . . . . . . . . . . . . . . . . . . . . . . . . .72 3.16.2 Three-Dimensional equations. . . . . . . . . . . . . . . . . . . . . . . .72 3.16.3 Axisymmetric problems. . . . . . . . . . . . . . . . . . . . . . . . . . .73 3.17 CMISS Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .73 4Linear Elasticity75 4.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .75 4.2Truss Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .76 4.3Beam Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .79 4.4Plane Stress Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .81 4.4.1Notes on calculating nodal loads . . . . . . . . . . . . . . . . . . . . . . .83 4.5Three-Dimensional Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .84 4.5.1Weighted Residual Integral Equation. . . . . . . . . . . . . . . . . . . .85 4.5.2The Principle of Virtual Work . . . . . . . . . . . . . . . . . . . . . . . .86 4.5.3The Finite Element Approximation. . . . . . . . . . . . . . . . . . . . .87 4.6Linear Elasticity with Boundary Elements . . . . . . . . . . . . . . . . . . . . . .89 4.7Fundamental Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .91 4.8Boundary Integral Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .93 4.9Body Forces and Domain Integrals in General . . . . . . . . . . . . . . . . . . .96 4.10 CMISS Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .98 5Transient Heat Conduction99 5.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .99 5.2Finite Differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .99 5.2.1Explicit Transient Finite Differences . . . . . . . . . . . . . . . . . . . . .99 5.2.2Von Neumann Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . 101 5.2.3Higher Order Approximations . . . . . . . . . . . . . . . . . . . . . . . . 102 C ONTENTSiii 5.3The Transient Advection-Diffusion Equation. . . . . . . . . . . . . . . . . . . . 103 5.4Mass lumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.5CMISS Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 6Modal Analysis111 6.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.2Free Vibration Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.3An Analytic Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.4Proportional Damping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 6.5CMISS Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 7Domain Integrals in the BEM117 7.1Achieving a Boundary Integral ulation . . . . . . . . . . . . . . . . . . . . . 117 7.2Removing Domain Integrals due to Inhomogeneous Terms . . . . . . . . . . . . . 118 7.2.1The Galerkin Vector technique . . . . . . . . . . . . . . . . . . . . . . . . 118 7.2.2The Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 7.2.3Complementary Function-Particular Integral . . . . . . . . . . . . 120 7.3Domain Integrals Involving the Dependent Variable . . . . . . . . . . . . . . . . . 120 7.3.1The Perturbation Boundary Element . . . . . . . . . . . . . . . . 121 7.3.2The Multiple Reciprocity . . . . . . . . . . . . . . . . . . . . . . 122 7.3.3The Dual Reciprocity Boundary Element . . . . . . . . . . . . . . 124 8The BEM for Parabolic PDES135 8.1Time-Stepping s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 8.1.1Coupled Finite Difference - Boundary Element . . . . . . . . . . . 135 8.1.2Direct Time-Integration . . . . . . . . . . . . . . . . . . . . . . . 137 8.2Laplace Trans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 8.3The DR-BEM For Transient Problems . . . . . . . . . . . . . . . . . . . . . . . . 139 8.4The MRM for Transient Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 140 Bibliography143 Index147 Chapter 1 Finite Element Basis Functions 1.1Representing a One-Dimensional Field Consider the problem of fi nding a mathematical expressionto represent a one-dimensional fi eld e.g., measurements of temperature against distance along a bar, as shown in Figure 1.1a. a b FIGURE1.1 a Temperature distribution along a bar. The points are the measured temperatures. b A least-squares polynomial fi t to the data, showing the unacceptable oscillation between data points. One approach would be to use a polynomial expression  “ and to estimate the values of the parameters  ,  ,  and from a least-squares fi t to the data. As the degree of the polynomial is increased the data points are fi tted with increasing accuracy and polynomials provide a very convenient of expression because they can be differentiated and integrated readily. For low degree polynomials this is a satisfactory approach, but if the polynomial order is increased further to improve the accuracy of fi t a problem arises the polynomial can be made to fi t the data accurately, but it oscillates unacceptably between the data points, as shown in Figure 1.1b. To circumvent this, while retaining the advantages of low degree polynomials, we divide the bar into three subregions and use low order polynomials over each subregion - called elements. For later generality we also introduce a parameter /*,-/13*  *,-A* such that *,- B*,C  B*,C  and refer to these expressions as the basis functions associated with the nodal parameters and  . The basis functions *,and  B*Dare straight lines varying between 7 and / as shown in Figure 1.3. It is convenient always to associate the nodal quantityFEwith element node G and to map the temperatureHJI defi ned at global node K onto local node G of element L by using a connectivity matrixKM8GONLPi.e., EQMH IORSEUTVXW whereKMGJNYLP global node number of local node G of element L . This has the advantage that the 1.2 LINEARBASISFUNCTIONS3 / * / / 7  B*, B*, * / 7 1 FIGURE1.3 Linear basis functions Z [P]\-_[and Z  [U\a[. interpolation *D- B*D4  B*D4  holdsforany elementprovidedthatand  arecorrectlyidentifi ed withtheirglobal counterparts, as shown in Figure 1.4. Thus, in the fi rst element b b b b b b b c c c c c c c d d d d d d d e e e e e e e f f f f f f f f g g g g g g g node h  * node i element / element j element i node / node j 101010  *  * Hk nodes global nodes element H “ HOl H  FIGURE1.4 The relationship between global nodes and element nodes. *D- B*D4  B*D4  1.1 withmHkand  ;H  . In the second element is interpolated by *D- B*D4  B*D4  1.2 withkMH  and  ;H “ , since the parameter H  is shared between the fi rst and second elements 4FINITEELEMENTBASISFUNCTIONS the temperature fi eld is implicitly continuous. Similarly, in the third element is interpolated by *,- B*,C  B*,C  1.3 with noH “ and  pH]l, with the parameter H “ being shared between the second and third elements. Figure 1.6 shows the temperature fi eld defi ned by the three interpolations 1.1–1.3. node / node j If is cubic in*U and cubic in *  , then *u is quadratic in*Uand cubic in *  , and *  is cubic in*uand quadratic in *  . Now consider the side 1–3 in Figure 1.13. The cubic variation of with *  is specifi ed by the four nodal parameters, w *  x , “ and w *  x “ . But since *u the normal derivative is also cubic in *  along that side and is entirely independent of these four parameters, four additional parameters are required to specify this cubic. Two of these are specifi ed by w *Uux and w *uux “ , and the remaining two by w  *U *  x and w  *u *  x “ . 1.6 HIGHERORDERCONTINUITY13 node h node i *u node j *  node / w *uux w *uux “ FIGURE1.13 Interpolation of nodal derivative [ along side 1–3. The bicubic interpolation of these nodal parameters is given by *uN*  k B*UƒF *  †t  *uƒF B*  †   B*UƒF  B*  † “ t  *uƒF  *  †Fl  B*UƒF B*  w *uux t  *uƒF *  w *Uux   B*UƒF  B*  w *uux “ t  *uƒF  *  w *Uux l  B*UƒF B*  w *  x t  *uƒF *  w *  x   B*UƒF  B*  w *  x “ t  *uƒF  *  w *  x l  B*UƒF B*  w  *u *  x   B*us *  w  *u *  x   B*UƒF  B*  w  *u *  x “   B*us  *  w  *u *  x l 1.16 14FINITEELEMENTBASISFUNCTIONS where B*D  /215iu*  jU* “ B*D  **Ž1/P   B*D  *  BiŒ1jU*,  B*D  *  *Ž1A/P 1.17 are the one-dimensional cubic Hermite basis functions see Figure 1.12. Asintheone-dimensionalcaseabove,topreservederivativecontinuityinphysicalx-coordinate space as well as in * -coordinate space the global node derivatives need to be specifi ed with respect to physical arclength. There are now two arclengths to consideriand the Jacobian is   *  “ . The term multiplying the nodal parametersFEis called the element stiffness matrix,   E   EQ w  * * E * D*   E x  * w  D* i E D* i  E x / i D* where the indices  and G are / or j . To uate   E we substitute the basis functions B*D-/213*or * r1Q/  B*D-{*or  * r/ 2 .1 ONE-DIMENSIONALSTEADY-STATEHEATCONDUCTION27 XX X0X 0 X XX X X 0 00 Node 4 X X U “ U