COMPUTATIONAL METHODS AND ALGORITHMS Vol. II - Finite Element Method - Jacques-Hervé SAIAC

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1 COMPUTATIONAL METHODS AND ALGORITHMS Vol. II - Fnte Element Method - Jacques-Hervé SAIAC FINITE ELEMENT METHOD Jacques-Hervé SAIAC Départment de Mathématques, Conservatore Natonal des Arts et Méters, Pars, France Keywords: Varatonal equatons, energy mnmzaton, Galerkn method, LaxMlgram s theorem, fnte element meshes, Lagrangan nterpolaton, trangular P k and quadrangular Q k fnte elements, soparametrc elements, quadrature formulas, error analyss. Contents. Introducton.. A Smple One-dmensonal Problem.2. Approxmaton Process wth Lnear Elements.3. Computaton of Matrx Coeffcents 2. Other one-dmensonal boundary problems 2.. The Non-homogeneous Drchlet Problem 2.2. The Neumann Problem 2.3. The Problem wth Robn Boundary Condtons 3. Hgher order elements n one dmenson 3.. P 2 Elements 3.2. Hermte Cubcs 4. Two or three-dmensonal ellptc problems 4.. A Model Problem 4.2. Varatonal Formulaton 4.3. Doman Dscretzaton: The Mesh 4.4. P -Lagrange Trangular Element P Elements The Approxmate Problem wth 5. Two-dmensonal P k Lagrange elements 5.. P Trangle 5.2. P2 Trangle 5.3. P3 Trangle 6. Three-dmensonal P k elements 6.. P Tetrahedron 6.2. P2 Tetrahedron 6.3. P3 Tetrahedron 7. Isoparametrc elements 7.. Quadrlateral Elements Q Elements 7.3. Q Shape Functons 7.4. Q 2 Elements 7.5. Curved P 2 Trangle Encyclopeda of Lfe Support Systems (EOLSS)

2 COMPUTATIONAL METHODS AND ALGORITHMS Vol. II - Fnte Element Method - Jacques-Hervé SAIAC 7.6. Three-dmensonal Isoparametrc Elements 8. Error analyss wth exact ntegraton 8.. Internal Approxmaton 8.2. A General Error Bound for Ellptc Problems 8.3. General Error Bound wth P k or Q k Elements and Exact Integraton 9. Numercal quadrature formulas 9.. One Dmensonal Formulas 9.2. Two-dmensonal Formulas 9.3. Three-dmensonal Formulas. Error analyss wth numercal ntegraton.. Ellptcty Condtons..2. Error Bound wth Numercal Quadrature. Some practcal conclusons.. One-dmensonal Ellptc Problems.2. Two-dmensonal Ellptc Problems.3. Counter-examples: Non-ellptc Approxmate Blnear Forms Bblography Bographcal Sketch Summary The ntroducton to fnte element method may be very trcky. On the one hand, the rgorous foundaton of the mathematcal concepts requres dffcult theorems of functonal analyss, on the other hand, applcatons (to fluds for example) need sometmes complex fnte element spaces. In the 3d case, wth a great number of unknowns, t s very dffcult to produce effcent codes. In ths presentaton, we prefer to concentrate on the fundamental concepts and to llustrate them by smple applcatons. We frst present the fnte element method n the one dmensonal case, even f t s, of course, more useful n hgher dmensons and complex geometry. Ths allows us to thoroughly explan the computatons of element stffness and mass matrces and complete lnear systems wth dfferent boundary condtons. Then, we present twodmensonal trangular and so-parametrc quadrlateral elements and we gve a general dea of the trdmensonal case. Numercal ntegraton formulas are detaled n the trangular and quadrlatal cases. A quck survey on error estmaton and counter examples of bad choces of quadrature formulas are fnally gven.. Introducton The Fnte Element Method (F.E.M.) was developed n the late 95 s to numercally solve equatons of elastcty and structural mechancs. It was ntroduced by engneers as a generalzaton of earler methods used to solve dscrete systems. The fnte element method was based on an analogy between real dscrete elements of a structure and small parts of a contnuum doman, so-called fnte elements. In the feld of sold mechancs, t s nowadays the standard method. Outsde ths feld, the range of applcatons of the fnte method has extended to all engneerng dscplnes. Now ths method s appled wdely. It has been developed by mathematcans and numercal analysts and has become a Encyclopeda of Lfe Support Systems (EOLSS)

3 COMPUTATIONAL METHODS AND ALGORITHMS Vol. II - Fnte Element Method - Jacques-Hervé SAIAC general tool for numercal soluton of partal dfferental equatons. The fnte element method s partcularly sutable to solve equlbrum or, equvalently, energy mnmzaton problems. Ths method has the bg advantage that complcated geometres, general boundary condtons and varable materal propertes can be handed easly and n a natural way, whereas fnte dfferences or spectral methods ntroduce artfcal complcatons. In flud mechancs problems, the fnte element method competes wth the fnte volume method whch presents the same geometrcal flexblty and naturally preserves the conservaton laws. Modern codes combne both methods, fnte volume dealng wth the conservatve part of the equatons and fnte element wth the dsspatve one. For a mechancal engneerng approach and a revew of many applcatons of the fnte element method, refer to [8]. For a mathematcal study and a detaled error analyss, refer to []. For teachng textbooks, refer to [5] and [6]. At last for fnte element applcatons to Naver-Stokes equatons and flud mechancs, the reader wll proftably read [4], [7] and [3] for extensons to non-lnear problems..4. A Smple One-dmensonal Problem The varatonal formulaton of ellptc dfferental problems and the equvalent energy mnmzaton problem (n the symmetrc case) are the key bass of fnte element methods ( see Varatonal statements of problems. Varatonal methods). Once the contnuous problem s wrtten n varatonal form, thanks to the Lax Mlgram theorem, we obtan the exstence and unqueness of the exact soluton n Hlbert space H. Then, the approxmaton process begns. The fnte element method enables us to buld fnte dmensonal spaces (subspaces of H for the so-called conformng methods).the approxmate soluton belongs to ths fnte dmensonal space. It s, n a certan sense, the best approxmaton n ths subspace of the exact soluton. In contrast wth fnte dfference methods where the approxmated soluton s the vector of dscrete nodal approxmate values, n the fnte element method, the approxmate soluton s a contnuous (n most cases) functon (even f t s, fnally, defned by ts nodal values). Ths s one of the bg advantages of fnte element methods. We can apply to the approxmate soluton the same operators as we appled to the exact soluton. Let us begn by a smple one-dmensonal problem to ntroduce the basc concepts of the fnte element method (F.E.M). An elastc cord fxed at both ends s subject to a vertcal load per unt length f. The vertcal dsplacement u s real functon, the soluton of u ( x) f ( x) x, [ L] ( ) = u( L) = = u (.) where L s the length of the cord. Ths very smple homogenous Drchlet problem can be wrtten n varatonal form. We proceed as n Varatonal statements of problems. Varatonal methods. to obtan Fnd u V such that v V A ( u,v ) = L ( v ) (.) Encyclopeda of Lfe Support Systems (EOLSS)

4 COMPUTATIONAL METHODS AND ALGORITHMS Vol. II - Fnte Element Method - Jacques-Hervé SAIAC where the space V = H ( Ω), form L are defned by: wth = [ L] Ω,, and the blnear form A and the lnear Ldu dv L A ( u,v ) = ( x ) ( x ) dx, L ( v) = f ( x ) v ( x ) dx (.2) dx dx As the blnear form s symmetrc, the problem s equvalent to the followng energy mnmzaton: Fnd u V whch mnmzes 2 L dv L ( ) = ( ) ( ) ( ) E v x dx f x v x dx 2 dx.5. Approxmaton Process wth Lnear Elements (.3) Ths s the startng pont of the approxmaton process. Then F.E.M. can be smply descrbed as a process of constructng fnte dmensonal subspaces V,h of V. The approxmate soluton wll be found n fve steps.. Buld a mesh on the domanω. The doman Ω s subdvded nto a fnte number of smple subsets, called fnte elements. In ths frst one-dmensonal case, elements are the N subntervals x, x for =,... N. Fgure:. One dmensonal mesh 2. Choose a smple functon space to represent the approxmate soluton locally on each element. Here we wll choose lnear polynomals. Then defne the global fnte dmensonal subspace V,h based on the mesh by consderng the contnuous functons satsfyng the Drchlet boundary condtons and whose restrctons on each element are lnear polynomals. Approxmate solutons of the followng knd are obtaned: Fgure: 2. A functon of the dscrete space V,h Encyclopeda of Lfe Support Systems (EOLSS)

5 COMPUTATIONAL METHODS AND ALGORITHMS Vol. II - Fnte Element Method - Jacques-Hervé SAIAC A Lagrangan bass { w } of ths space s the set of hat functons defned by ( j) w x = δ =, N and j =, N (.4) j Fgure:3. A bass functon Any functon vh V,h can be represented by: = N ( ) ( ) vh x = vw x = where v = v h ( x ). The coeffcents v are nothng but the nodal values ofv h. (.5) 3. Project the contnuous problem on the fnte dmensonal subspace V,h followng dscrete problem wll be solved: Fnd u V such that v V A ( uh, vh) = L( vh) h,h h,h Ths problem s equvalent to the lnear system: (.6). The Fnd u, u2,... un such that =, N j= N A ( wj, w) uj = L( w) j= (.7) 4. Compute the matrx and the rght hand sde coeffcents of ths lnear system. Ths s done by splttng the ntegrals nto contrbutons from each element. On each element (here ntervals x, x ), the bass functons w reduce to lnear polynomals. That makes the computatons very easy. After elementary calculatons are performed, the resultng lnear system s obtaned by an assemblng process. Encyclopeda of Lfe Support Systems (EOLSS)

6 COMPUTATIONAL METHODS AND ALGORITHMS Vol. II - Fnte Element Method - Jacques-Hervé SAIAC 5. Fnally the lnear system s solved usng one of the methods descrbed n Soluton of systems of lnear algebrac equatons..6. Computaton of Matrx Coeffcents Let us gve some detals on the practcal computaton of matrx coeffcents n ths smple case. The computaton s done by assemblng the contrbutons of each element T = x, x for =,... N. The coeffcents (, ) ( ) ( ) A A w w w x w x dx = = L j j j of the global matrx A are obtaned by addng elementary contrbutons: k= N x A = k j x j k k= ( ) ( ) w x w x dx. (.8) Let us consder, for example the element T, = x, x. There are only two non zero bass functons on T, namely w and w w T = x x = x x w T x x x x w = w = T x x x x T (.9) (.) Thus the contrbuton of T wll affect only the four followng global matrx coeffcents: A,, A,, A,and. A, It s easy to compute these contrbutons. They are usually wrtten n a matrx form, leadng to the so called element matrx : e, e,2 Elem = (.) e2, e2,2 where e x 2 = w ( x ) dx x = = x 2 dx, x x x ( x x ) (.2) Encyclopeda of Lfe Support Systems (EOLSS)

7 COMPUTATIONAL METHODS AND ALGORITHMS Vol. II - Fnte Element Method - Jacques-Hervé SAIAC e = e x = w ( x ) w ( x ) dx x = dx,2 2, = x x x x ( x x ) 2 ( x x ) 2 e x = w ( x 2 ) dx x = dx 2,2 = x x x x (.3) (.4) and then Elem x x x x = = x x xx xx (.5) The same knd of computatonal technque s used to obtan the rght hand sde of the resultng lnear system. 2. Other One-dmensonal Boundary Problems We can apply the same process to more general one-dmensonal boundary value problems. 2.. The Non-homogeneous Drchlet Problem Let us consder the followng Drchlet problem: u ( x) f ( x) x, [ L] ( ) = α u( L) = β = u We ntroduce an auxlary functon u such that ( ) ( ) u = α, u L = β, (2.) and we consder the new unknown functon u = uu. Problem (2.) s then equvalent to a homogeneous problem for u. A practcal choce for u conssts n expandng t n the fnte element bass u α w β w, = + N where w and wn are the P-Lagrangan bass functons assocated wth the boundary ponts x = and x = L. Thus, u s soluton of N Fnd u V such that v V A ( u,v ) = L ( v ) (2.2) Encyclopeda of Lfe Support Systems (EOLSS)

8 COMPUTATIONAL METHODS AND ALGORITHMS Vol. II - Fnte Element Method - Jacques-Hervé SAIAC where the blnear form A s the same as n the homogeneous case, but the lnear form L s now equal to ( ) = ( ) ( ) (, ) L Lv f xvxdx A u v. (2.3) Then, the approxmaton process follows the same lnes The Neumann Problem ( ) We consder now the Neumann problem c L ( L) u ( x) + c( x) u( x) = f ( x), x, [ L], u ( ) = α u ( L) = β The varatonal formulaton of ths problem s: Fnd u V such that v V A ( u,v ) = L ( v) where, ths tme, the space V = H ( Ω), the lnear form L are defned by:, : wth = [ L] (2.4) (2.5) Ω,, and the blnear form A and Ldu dv L A ( u,v ) = ( x ) ( x ) dx, L ( v ) = f ( x ) v ( x ) dx + βv ( L) α v ( ) (2.6) dx dx Remark 2. (see Varatonal statements of problems. Varatonal methods). If c, problem (2.4) s the Neumann problem for the Laplace operator: t s an ll-posed problem. Followng the same approach as before, the fnte dmensonal space V h V s the space of contnuous, pecewse lnear, functons. But, ths tme, there s no restrcton on boundary values. The dmenson of the dscrete space V h s equal to N +: the number of nodes of the mesh. The approxmate soluton reads as follows = N uh ( x) = uw ( x ). (2.7) = The dscrete problem to solve s: Fnd u V such that v V A ( u,v h h) = Lv ( h) h h h h. (2.8) Practcal computatons follow the same lnes as above. Encyclopeda of Lfe Support Systems (EOLSS)

9 COMPUTATIONAL METHODS AND ALGORITHMS Vol. II - Fnte Element Method - Jacques-Hervé SAIAC TO ACCESS ALL THE 35 PAGES OF THIS CHAPTER, Vst: Bblography [] Carlet P.G., The Fnte Element Method for Ellptc Problems, North Holland (979) [A reference on fnte element analyss.]. [2] George P.L., Automatc mesh generaton. Applcatons to fnte element methods, J. Wley and Sons (99). [A key reference on automatc meshng.]. [3] Glownsk R., Numercal Methods for Nonlnear Varatonal Problems, Sprnger Verlag (984) [A very useful book on nonlnear problems and ndustral applcatons. The appendx on lnear varatonal problems s one of the best ntroducton of varatonal method and fnte element approxmaton ever wrtten.]. [4] Gralut V., Ravart P.A., Fnte Element Methods for Naver-Stokes Equatons, Theory and Applcatons, Sprnger Verlag (986) [The reference on fnte element analyss for Naver-Stokes equatons.]. [5] Johnson C., Numercal solutons of Partal Dfferental Equatons by the Fnte Element Method, Cambrdge Unversty Press (987) [A very pedagogcal textbook on fnte element concepts]. [6] Lucqun B., Pronneau O. Introducton to scentfc computng, J. Wley and Sons (998) [An easy readng but rgorous and exhaustve textbook on numercal methods for solvng partal dfferental equatons]. [7] Pronneau O., Fnte Elements for Fluds, J. Wley and Sons (989) [The reference on fnte element applcaton to fluds]. [8] Zenkewcz O.C., Taylor R.L. The Fnte Element Method, fourth edton, Mac Graw Hll (987) [A well-known reference on fnte element method and applcatons]. Bographcal Sketch Jacques-Hervé Saac s Professor at the Conservatore Natonal des Arts et Méters (CNAM) n Pars, France. He graduated from the Ecole Centrale de Pars n 968 and receved hs Ph. D. from UPMC Pars VI Unversty n 97. In 99, under the drecton of Professor Olver Pronneau, he receved the Habltaton à drger des recherches (HDR) dploma n Appled Mathematcs from UPMC Pars VI Unversty. Hs research felds nclude fnte element analyss, optmzaton methods, numercal smulatons n Computatonal Flud Dynamcs and adaptve mesh analyss. Encyclopeda of Lfe Support Systems (EOLSS)