Implementation of B-Spline Based Finite Element Analysis. for the Boundary-Value Problem. Andrew J. Marker. Master of Science

Transcription

1 Implementaton of B-Splne Based Fnte Element Analyss for the Boundary-Value Problem Andrew J. Marker A project submtted to the faculty of Brgham Young Unversty n partal fulfllment of the requrements for the degree of Master of Scence Mchael A. Scott, Char Rchard J. Ballng Norman L. Jones Department of Cvl and Envronmental Engneerng Brgham Young Unversty December 2013 Copyrght 2013 Andrew Marker All Rghts Reserved

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3 ABSTRACT Implementaton of B-Splne Based Fnte Element Analyss for the Boundary-Value Problem Andrew J. Marker Department of Cvl and Envronmental Engneerng Master of Scence Ths project wll eamne the use of B-Splnes as the bass shaped functons n fnte element (FE) analyss. The analyss wll be appled to a boundary-value problem. A program wll be wrtten to verfy that B-Splne fnte element (BSFE) analyss accurately represents the boundary-value problem. Keywords: b-splnes, fnte element

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5 TABLE OF CONTENTS LIST OF TABLES... v LIST OF FIGURES... 1 Introducton Background... 2 Fnte Element Method Basc Concept... 3 B-Splnes Basc Concept Bass Functons Methods... 7 The Boundary Value Problem The Strong Form The Weak Form The Galerkn Form Matr Form Local Element pont of vew... 9 MATLAB Program Gaussan Quadrature Bass Values and Dervatves v

6 3.2.3 Mappng from Local to Global Error Results and Conclulsons Dscusson of Results Sources of Error Conclusons REFERENCES Append A. Plots of Solutons v

7 LIST OF TABLES Table 4-1: Error for f() = Table 4-2: Error for f() = Table 4-3: Error for f() = v

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9 LIST OF FIGURES Fgure 2-1: Bézer and B-Splne Curves... 4 Fgure 2-2: Lnear B-Splne Bass over Fve Elements... 5 Fgure 2-3: Quadratc B-Splne Bass over Four Elements... 5 Fgure 2-4: Cubc B-Splne Bass over S Elements... 6 Fgure 4-1: f() = 1, n = Fgure 4-2: Enlarged pcture of f() = 1, n= Fgure 4-3: f() = 1, n = Fgure 4-4: Error for f() =

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11 1 INTRODUCTION The fnte element method (FEM) has been wdely used for engneerng purposes snce ts ntroducton n the 1940 s and 1950 s. The age of computers accelerated the use and understandng of FEM. FEM requres the use of bass equatons to appromate the functon. Often pece-wse contnuous polynomals, LaGrange or Hermte polynomals, have been used as the bass functons n FEM. B-splnes offer an alternatve bass that s contnuous over the whole model. In nterestng applcaton of B-Splne based FEM s so-geometrc analyss (IGA). B-Splnes are used etensvely n computer modelng whch allows for B-Splnes to be a lnk between modelng and analyss. Ths project wll eamne the use of B-Splnes as the bass shaped functons n FEM analyss. The analyss wll be appled to the boundary-value problem. A program was developed to run FEM on the boundary-value problem wth frst, second, and thrd-order bass functons. The error between the actual soluton and the FEM results are calculated and compared wth FEM usng Lagrange polynomals of the same order. The rate of convergence was also calculated and used to determne the effcency of B-Splne based FEM. Ths paper provdes a bref overvew of the fnte element method and B-Splnes. It dscusses the method of the project. The results wll be presented and dscussed. 1

12 2 BACKGROUND Ths secton provdes a bref ntroducton to the fnte element method and b-splnes. It s meant to provde enough nformaton for the reader to understand the project, and s certanly not ehaustve. For each topc a bref hstory s gven as well as an ntroducton to the basc dea. Fnte Element Method The fnte method s used today for many engneerng applcatons. Much research has and contnues to be devoted to ths topc because of ts mportance n engneerng analyss, although the fnte element method s relatvely new. Its orgns are n the 1940 s and 1950 s when mathematcans and engneers were lookng to solve hgh order problems wth new numercal methods. Frst publcatons on the subject nclude R.L. Courant s Varatonal Methods for the Soluton of Problems of Equlbrum and Vbraton n whch he used pece-wse lnear appromatons on trangles. The method s more clearly outlned by Argyrs and Turner wth others n the 1950 s. The technque was frst appled to the dscplne of structures but found other uses ncludng heat conducton and fluds n the 1960 s. The development of computers has ncreased the amount of research and applcaton of the FEM. (Pepper, Henrch, 2006, ). Along wth the development of hgh-speed dgtal computers, the applcaton of the fnte element method also progressed at a very mpressve rate (Rao, 2001). FEM contnues to be a hghly regarded and used numercal method n engneerng and other dscplnes. Research contnues n order fnd new technques and applcatons. 2

13 2.1.1 Basc Concept The fnte element method (FEM) s a numercal method. Lke all numercal methods, the dea s to fnd an appromaton to a comple problem by replacng t wth a smpler one. The dea behnd FEM s to break up a larger geometry nto smaller sub-geometres, or elements. Analyss of the ndvdual elements can be completed based on the governng equatons of the problem. Appromatons are made by lnear combnatons of the bass functons. The results are than assembled nto a global soluton whch appromates the real soluton. The appromaton mproves as the mesh s refned, that s as the number of elements ncreases. Dfferent technques or greater computatonal effort can also be appled to FEM to create closer appromatons. (Huebner 2001, Reddy, 1984)). B-Splnes Smlar to the fnte element method, the hstory of B-splnes (bass splnes) s relatvely new.. B-splnes were frst ntroduced for use n statstcal data smoothng by Schoenberg n 1946 who coned the phrase. B-splnes were developed for the automotve ndustry n the 1960 s as an easy way to model the comple curves as found on automobles. Two French mathematcans, Bézer wth Renault and De Casteljau s wth Ctroën, smultaneously developed B-splnes for geometrc desgn. There are two mportant mathematcal developments n the hstory of B-splnes. Frst s the development of the recursve relaton for defnng and evaluatng B-splnes bass functons. Ths s often credted to de Boor et al. The second s the creaton of the polar form by Ramshaw whch allows for quck understandng and manpulatng of B-splnes. Today B-splnes 3

14 are used n the felds of statstcs, computer aded desgn (CAD), computer aded geometrc desgn, anmaton, graphc desgn, and fnte element analyss. Even the fonts and curves on a word processor are B-splnes Basc Concept B-splnes are a sequence of Bezer curves constructed n a way to create C n-1 contnuty, where C s the order. B-splnes are defned by the control ponts, P, assocated wth weghts, and the knot vector whch controls the parameterzaton of the splne. The polar form of B-splne mentoned above makes t easy to fnd ntermedate values between control ponts Fgure 1 shows a sngle Bézer curve and a B-splne consstng of three Bézer curves (Sederberg, 2012). P(1, 2, 3) P(2, 3, 3) P(1, 1, 2) a) Bézer Curve P(1, 1, 1) Knot Vector ( ) b) B-Splne Fgure 2-1: Bézer and B-Splne Curves P(3, 3, 3) Bass Functons An mportant aspect of FEM s choosng a bass functon, or a lnear combnaton by whch functons can be appromated. B-splne bass functons do ths and also guarantee C n-1 contnuty whch becomes convenent for hgher order problems. Usng the knot vector and recursve relaton B-splne bass functons are nepensve to evaluate as are the dervatves. AS wth any lnear combnaton, the sum of the values at any pont s one, and the sum of the dervatves at any pont 4

15 s zero. Another nnovatve reason for usng B-splne bass functons wth FEM s for the use n so-geometrc analyss, whch tres to close the gap between the graphcal model and the model used for analyss. There s currently a lot of research n ths area. Fgure 2-2, Y and Z show the lnear, quadratc and cubc bass functons. Notce there are C1 splne over each element, where C agan s the order of the splnes. Fgure 2-2: Lnear B-Splne Bass over Fve Elements Fgure 2-3: Quadratc B-Splne Bass over Four Elements 5

16 6 Fgure 2-4: Cubc B-Splne Bass over S Elements

17 3 METHODS The purpose of ths project was to mplement a B-splnes bass fnte element program to solve the boundary value problem. The man porton of the project conssted of defnng the problem, and solvng the problem wth a program n MATLAB. Also an Ecel spreadsheet was created and programmed to generate values for the bass functon for a gven order and number of elements. The Boundary Value Problem The boundary value problem (BVP) has been wdely studed as an ntroducton to fnte element method. It s a smple one dmensonal frst-order problem. It s ntroduced here brefly as n Hughes (87) and can be studed n greater detal by referrng to that book. The goal of the followng dervaton s to get to a matr form of the equaton whch s famlar and easy to solve The Strong Form The strong or classcal form of the BVP n terms of u on Ω = [0,1] s (SS) Gven ff Ω R and constants gg and h, fnd uu Ω R, ssssssh tthaaaa uu, ff = 0 oooo Ω uu(1) = gg uu, (0) = h 3.1 where a comma represent dfferentaton and f s a functon defned on Ω. The second and thrd equatons represent the boundary condtons. Both g and h are 0 for ths project. 7

18 3.1.2 The Weak Form To get the weak form of the BVP equaton we frst defne the tral soluton space, δ, and weghtng functon space, υ, as the followng δδ = {uu uu HH 1, uu(1) = gg} 3.2 Υ = {ww ww HH 1, ww(1) = 0} 3.3 where H 1 are functons who satsfy the followng equaton. 1 (uu, ) 2 dddd < Then the weak form can be wrtten as Gven ff, gg, and h, as before. Fnd uu δδ such that for all ww Υ (WW) = 1 1 ww, uu, dddd = wwww dddd ww(0)h Let 1 aa(ww, uu) = ww, uu, dddd (ww, ff) = wwww dddd Then the weak form can be wrtten as aa(ww, uu) = (ww, ff) ww(0)h The Galerkn Form Dscretzng usng Galerkn s appromaton we create subsets of the tral and watng spaces such that δδ h δδ aaaaaa Υ h Υ. Then u h s constructed such that u h = υ h g h generatng the Galerkn form: Gven ff, gg, aand h, as before, fnd uu h = υυ h gg h wwheeeeee υυ h Υ h, (GG) such that for all ww h Υ h aa(ww h, υυ h ) = (ww h, ff) ww h (0)h aa(ww h, gg h ) 3.9 8

19 3.1.4 Matr Form The Galerkn form results n a system of lnear equatons whch can be nterpolated wth the bass functons and wrtten n a matr form famlar to engneers, Kd = F, where KK = [KK AAAA ], where KK AAAA = aa(nn AA, NN BB ) 3.10 FF = {FF AA }, where FF AA = (NN AA, ff) NN AA (0)h aa(nn AA, NN nn1 )gg 3.11 Note that NA represents the bass functons, or n ths case the B-splne bass functons. Recall that g and h are both zero for ths project smplfyng the equaton for FA to FA = (NA, f). Once F and K are assembled d can be solved for whch s our soluton u Local Element pont of vew In FEM each element s analyzed and then element components are assembled nto the global problem. For each element the k e matr and f e vector s calculated wth the followng equatons. However the local lmts of ntegraton are now -1 to 1 whch requres some adjustment when assemblng K. ee kk aaaa = aa(nnnn, NNNN) 3.12 ff aa ee = (ff, NNNN) 3.13 MATLAB Program Key portons of the program most pertnent to the results wll be hghlghted here. Append A contans an outlne vew of the entre program as well as the code. The code was modfed from prevous code whch used Lagrange polynomals as the bass. The code the Lagrange bass was kept for comparson. 9

20 3.2.1 Gaussan Quadrature Gaussan quadrature was used to evaluate the ntegrals n the analyss and n fndng the error. Gaussan quadrature works by takng a weghted sum of values at f at key ntegratons ponts. 1 nn ff() ff(ξξ ll ) WW ll 1 ll= It s an effectve way to evaluate complcated ntegrals. Hughes reports that accuracy of order 2nnt s ganed by nnt ponts of ntegraton (Hughes, 1987). A nnt of 3 was used for ths project requrng that ξξ 1 = 3/5, ξξ 2 = 0, ξξ 3 = 3/5 W1 = W3 = 5/9, W2 = 8/ Bass Values and Dervatves A large porton of the new code was to evaluate the new bass. Because Gaussan quadrature was used to appromate the ntegrals, so the bass had to be evaluated at three ponts n each element. Wth Lagrange polynomals the functons are the same over each element, so only three ponts had to be evaluated. Wth the B-splnes, three ponts for each element needed to be calculated. Usng a unform knot nterval k:[ n-1 n n n] for n elements, and the recursve relaton, the equatons for the bass functons, and ther dervatves can easly be evaluated at any pont. For eample the quadratc formulas are 10

21 = elsewhere B 0, ), [, ) )( ( ) ( ), [, ) )( ( ) )( ( 1 ), [, ) )( ( ) ( ) ( Where represent the knots. Functons were created that returned values and dervatves for each of the bass equatons Mappng from Local to Global The element space s dfferent than the global space due to the way the local element s defned on [-1 1] and the global element s defned on [a b] where b a = 1. To handle ths apply a change of varables to the local equatons whch result n a dervatve term n the ntegral. In one dmenson ths s related to the frst dervatve. Pror to ntegratng, the dervatve term at each quadrature pont s determned as cc aa NN aa (ξξ) CC1 aa= Where ca are the control ponts of the B-Splnes. For convenence the control pont were chosen at the grevlle abscssae so the geometrc space and parametrc space are equvalent. 11

22 3.2.4 Error Error was calculated usng a least squared approach. Agan quadrature was appled for the followng ntegral. The rate of convergence, change n error per change n the number of elements s a good ndcator of how well a FEM s workng. 1 1/2 EE = (uu aaaatttttttt uu aaaaaaaaaaaa dddd)

23 4 RESULTS AND CONCLULSIONS Dscusson of Results In the results are ncluded numbers for both the Lagrange solutons (uh solutons) and B- Splne solutons (uhb solutons) where P s the bass order. Fgure 4-1 shows the solutons for f() =1 and n = 5. Fgure 4-2 shows s an enlarged vew of the same soluton. Fgure 4-3 s for f() = 1 wth n = Smlar plots other forcng functon and n values are n Append B. There are several thngs to pont out from these plots. Frst, the lnear functons for both Lagrange and B- splnes are the same. On Fgure 4-2, t s notceable that they are ndeed dentcal. Also, t s clear at low numbers of elements that the hgher order B-splnes are not a good appromaton. Also note that the cubc soluton s erroneously less eact than the quadratcs, and smlarly wth the quadratcs and lnears. Fnally t can be seen that the B-Splne solutons do eventually converge. Fgure 4-1: f() = 1, n = 5 13

24 Fgure 4-2: Enlarged pcture of f() = 1, n=5 Fgure 4-3: f() = 1, n =

25 The net few tables show the error and the rate of convergence for dfferent forcng functons. They confrm that the frst-order bass have the same result and that the hgh order converge at about a 1/10 of the speed of the Lagrange bass. That s, the Lagrange bass decreases error by 100 tmes wth an ncrease of n by 10. The hgher order B-Splne only ncrease at rato of 10 to 10 and the cubc B-splne soluton dverge between 500 and 5000 elements. Fgure 4-4 s the graphcal representaton of table 4-3. Table 4-1: Error for f() = 1 F() = 1 N L1 L2 L3 B1 B2 B E E E E E E E E E E E E E E E E E E E E-04 Convergence 2.00E E E E E E-01 Table 4-2: Error for f() = F() = N L1 L2 L3 B1 B2 B E E E E E E E E E E E E E E E E E E E E-04 Convergence 2.00E E E E E E-01 15

26 Table 4-3: Error for f() = 2 F() = 2 N L1 L2 L3 B1 B2 B E E E E E E E E E E E E E E E E E E E E-04 Convergence 2.00E E E E E E Error vs 1/N Error E-08 1E-10 L1 L2 L3 B1 B2 B3 1E-12 1E-14 1/n Fgure 4-4: Error for f() = 2 16

27 Sources of Error Whle there are small errors that come from the appromatons of the FEM, t s clear that the results acqured here has more error than t should. It was brefly mentoned that there s a dfference between the local and global spaces. It s most lkely that ths large error s due to the fact that the code does not properly account for mappng between the two spaces. The dervatve factor used to get the local nto global get beng ncorrect or ths mappng could be otherwse mssed n a dfferent area. Conclusons The program wrtten to mplement a B-splne bass FEM was successful n the fact that t created solutons that converged to the answer wth mesh refnement (as the number of elements ncreased). It was not successful as there s a large source of error, probably due to an ssue mappng between the two spaces, and because t converges too slowly. The program dd not successfully mplement the B-splne bass. In order to make the program a success, the source of error would need to be dentfed. Once the program worked for the boundary value problem, t could than smlarly be appled to hgher order problems. 17

28 REFERENCES Huebner, Kenneth H The Fnte Element Method for Engneers. 4th ed. New York: Wley. Hughes, Thomas J.R The Fnte Element Method: Lnear Statc and Dynamc Fnte Element Analyss. Mneola, NY: Dover Publcatons. Pepper, Darrell W., and Juan C. Henrch The Fnte Element Method: Basc Concepts and Applcatons. 2nd ed. New York: Taylor & Francs. Rao, Sngresu S., The Fnte Element Method n Engneerng. 5th ed. Burlngton, MA: Butterworth-Henemann. Reddy, J. N An Introducton to the Fnte Element Method. New York: McGraw-Hll. Sederberg, Thomas W Computer Aded Geometrc Desgn. 18

29 APPENDIX A. PLOTS OF SOLUTIONS Result plots for n = 5, 50,500, 5000 for each functon f() =1,, ^2. 19

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