Applications of Gauss-Radau and Gauss-Lobatto Numerical Integrations Over a Four Node Quadrilateral Finite Element

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1 Avaiable online at angladesh J. Sci. Ind. Res. (), 77-86, 008 ANGLADESH JOURNAL OF SCIENTIFIC AND INDUSTRIAL RESEARCH CSIR bsir07gmail.com Abstract Applications of Gauss-Radau and Gauss-Lobatto Numerical Integrations Over a Four Node Quadrilateral Finite Element Md. Shafiqul Islam* and Goutam Saha Department of Mathematics, Universit of Dhaa, Dhaa-000, angladesh In this paper Gauss-Radau and Gauss-Lobatto quadrature rules are presented to evaluate the rational integrals of the element matri for a general quadrilateral. These integrals arise in finite element formulation for second order Partial Differential Equation via Galerin weighted residual method in closed form. Convergence to the analtical solutions and efficienc are depicted b numerical eample. Ke words: Gauss-Radau, Gauss-Lobatto, Rational Integrals, Quadrilateral Finite Element. Introduction Various integrals are determined numericall in the evaluation of the element/stiffness matri, mass matri, bod force vector, initial stress vector, the surface load vector etc., arising in Finite Element Method (FEM). Since the integrals in practical situations are not alwas simple but rational epressions in which the lower order quadrature scheme cannot evaluate eactl. Also there is no order of Gauss quadrature that will evaluate these integrals eactl (Zieniewicz 977, icford 990, Yagawa et al. 990). The integration points have to be increased in order to improve the integration accurac and it is desirable to mae these evaluations b using as few Gauss points (Stroud 97, *Corresponding authour, mdshafiqul@ahoo.com urden and Faires 00, abolian et al. 006) as possible, from the point of view of the computational efficienc. In this aspect, a large number of articles are available in the literature using the quadrilateral element with straight sides, which is one of the most popular elements in FEM. The complications arise from two main sources (arrett 999): the large number of integrations that need to be performed and the presence of the determinant of the Jacobian matri (which will be referred to simpl as the Jacobian throughout this paper) in the denominator of the element matrices. For this some researchers have been attempted to

2 78 Applications of Gauss-Radau () 008 develop analtical integration formula (Rathod and Islam 00, Hacer and Schreer 989) for limited finite elements, but a huge amount of computing time and memor space are involved. Thus numerical integration is an essential tas to strie a proper balance between accurac and efficienc commonl applied for science and engineering problems. However, this paper describes the finite element matri b Galerin weighted residual method using four node quadrilateral elements. Then a brief introduction of Gauss- Radau (Mased-Jamei et al. 005) and Gauss-Lobatto (Eslahchi et al. 005) quadrature rules, which are previousl applied to evaluate for single integrals, is presented in the net section. These numerical methods are eploited to evaluate the double integrals of rational functions belonging to the finite element matri in this paper. A smbolic algebra pacage, Mathematica is used to generate this element matri. Numerical accurac and efficienc are demonstrated b comparing it with the conventional Gaussian quadrature as well as analtical method through numerical eample. Eplicit formulation of element matri Let us consider an arbitrar four noded linear quadrilateral element in the global sstem (,)which is mapped into a -Square in the local parametric sstem (ξ, η) as shown in the following figure. Then the isoparametric coordinate transformation from (,)plane to (ξ, η) plane is given b, = i= N ( ξ, η i i ) and = N i i ( ξ, η) () where ( i, i ), i =-, are the vertices of the element in (,)-plane and denotes the D bilinear basis functions (Zieniewicz 977, icford 990) with (ξ i, η i ) as the natural coordinates in (ξ,η)-plane such that N i Now from equation () we have = [ a ξ η ξη] and + b + c + d = [ a + bξ + cη + d ξη] where, Fig.. () (a) Original -node quadrilateral element and its configuration in ξ η plane. (b) Hence the Jacobian can be epressed as: (, ) J = = = α0 + αξ + α η ( ξ, η) ξ η η ξ (a) i= = ( + ξξ ) ( + ηη ), i = a = b = + + c = + + d = + i i η (-,) (,) (-,-) (,-) a = b = + + c = + + d = + ξ

3 Islam and Saha 79 where, α = [( α = [( α = [( )( )( )( ) ( ) + ( ) + ( )] )] )] (b) In order to obtain the finite element matri using quadrilateral elements due to second order linear Partial Differential Equation via Galerin weighted residual formulation the integrals of the smmetric matri (Zieniewicz 977, icford 990) is K = [ i, ], = smm (5) where each i,, after a minor simplification, is of the form i = 6 [ J 00 i, (6) Here each coefficient, im n, (0 m, n ), of each i, ( i, ) depends on the four vertices of the phsical quadrilateral, can be written eplicitl. For eample, the coefficients of are:,, + 0 i, )( )( )(,,, ξ + 0 i,,,,, η i, η + i, ξ + i, ξη + i, η ] dξ dη i, = 00, 0, 0, = ( = {( = {( ) + ( )(, )( ) ) + ( ) + ( )( )( )} )} 0,, = ( = {( ) 0, = ( ) + ( ) others can be written similarl Numerical integration )( + ( ) The most common numerical integration is given b (Stroud 97, urden and Faires 00) b n w ( ) f ( ) d = w f ( ) + E[ f ] (7) a = nown as the Gaussian quadrature, where, w ( =,, L, n) and E[ f ] are called nodes (abscissas), weights, and error approimation, respectivel. Since w(), called weight function, and f () are integrable functions over the interval [a,b], and lies in [a,b], the interval of integration. This gives us n parameters to choose. If the coefficients of a polnomial are considered parameters, the class of polnomials of degree at most n - also contains parameters. This, then, is the largest class of polnomials for which it is reasonable to epect the formula to be eact (i.e. E[f]=0). With the proper choice of values and constants, eactness on this set can be obtained. Assuming the eactness (i.e. E[f]=0) and for our convenient (b change of variable), and set w()=, the eqn. (7) the can be written as, f ( ) d ) + ( m n m = w f ( ) + w f ( ) m n = = m+ (8) )( )}

4 80 Applications of Gauss-Radau () 008 which is nown as generalized Gauss quadrature rules. Particular quadratures are as follows:. If we put m=0, then the concluded numerical integration rule is called Gauss-Legendre (urden and Faires 00). Set f()= i, i = 0,,,..., n - in (8), which gives us n parameters (n weights and n nodes), and solving the nonlinear sstem to obtain the required nodes,, and the corresponding weights w for each =,,..., n. since the nodes are smmetric.. If we put m =, = - (or ) then the concluded numerical integration rule is called Gauss-Radau (Mased-Jamei et al. 005). f()= i,i= 0,,,...,n - Setting in (8) to obtain n - parameters (n weights and n - nodes). Then solving the sstem, we can obtain required nodes, =,,...n and the corresponding weights w, =,,...n. Observe that the nodes are unsmmetric.. Rearrange the eqn. (8) as n f ( ) d = w f ( ) + w f ( ) + wn f ( n ) = (9) and putting = - and n = in (9), then the numerical integration rule is called Gauss- Lobatto (Eslahchi et al. 005). If we set f ()= i, i =0,,,...n - in (9), we get n- parameters (n weights and n- nodes). Then solving the sstem, we can obtain required nodes, =,,...n -, and the corresponding weights w, =,,...n. In this case the nodes are smmetric. We ma summarize the nodes and corresponding weights for n =,,5 in Table - I, for this paper. Table I. Nodes and weights for Gauss-Legendre, Gauss-Radau, and Gauss-Lobatto methods Gauss-Legendre (G) Gauss-Radau (GR) Gauss-Lobatto (GL) n Nodes Weights w Nodes Weights w Nodes Weights w ± ± ± ± ± ± ± ± ± ±

5 Islam and Saha 8 In general, the integrals defined in eqn. (6) cannot be evaluated easil. Then each term, i, of (5) gives us rational integral. In fact, these rational integrals are evaluated b Gauss quadratures. Since the denominator (the Jacobian defined in eqn. ()) is a function of two variables ξ and η, the numerator is also a function of two variables ξ and η, so we assume that the eqn.(6) can be epressed as I = f ( ξ, η) dξ dη (0) In particular, for integration over the standard square, we can use the two dimensional Gaussian quadratures, which taes the form: n q= n I = f ( ξ, η ) w p= () where (ξ p, η q ) are the Gauss-Legendre, Gauss-Radau, and Gauss-Lobatto integration points and w p, w q, are the corresponding weighting factors independent of f. Now compute the complete element matri for a general four noded quadrilateral, described in Fig., on the basis of the above information using Mathematica program. Test Eample p q In this section, we wish to compute the element matri using closed form integration formula presented in this paper to compare p w q with the eisting solutions. For this, we consider a simple one-element eample to evaluate the element matri for two-dimensional Laplace's equation. The finite element formulation (Zieniewicz 977, icford 990) of the Laplace's equation is i, = R N N i,, + Ni +, i, (,) (,) (,) R N dd = P i P i, =,,, () (5,) (,) Fig.. Element geometr for Laplacian matrices where R is the tpical four-node isoparametric element shown in Fig. [icford 990, pp. -]. In order to compare the accurac, the complete element matri (smmetric) for the geometr shown in Fig. is given in Table II, and the error compare to the analtical method (Rathod and Islam 00) obtained b Mathematica is also given in Table III obtained b the presented numerical integration methods.

6 8 Applications of Gauss-Radau () 008 Table II. A: Element matri for the geometr in Fig. using order Integration Methods i Legendre (G) Radau (GR) Lobatto (GL) Analtical G GR GL Analtical G Smmetric GR GL Analtical G GR GL Analtical Table: II: Element matri using order Integration Methods i G GR GL Analtical G GR GL Analtical G Smmetric GR GL Analtical G GR GL Analtical

7 Islam and Saha 8 Table. IIC: Element matri using 55 order Integration methods i G GR GL Analtical G GR GL Analtical G Smmetric GR GL Analtical G GR GL Analtical Table III. Error estimation for the results in Table II Order n n Integration methods Legendre Radau Lobatto Errors

8 8 Applications of Gauss-Radau () 008 Table III to be contd. 5 5 Legendre Radau Lobatto Legendre Radau Lobatto In this paper, we use the notations G, GR, and GL for Gauss-Legendre, Gauss-Radau, and Gauss-Lobatto numerical integration methods, respectivel. The error is calculated as follows Error = (Analtical - Approimate) Table IV. We observe that the CPU time of the eact method is 00 times comparing to the numerical methods: G, GR and GL, even if we use higher order quadrature points. The errors given in Table III are ver negligible and the accurac of the presented methods in this paper is ecellent. The computational CPU time for each matri obtained in Table II, is also given in

9 Islam and Saha 85 Table IV. Order Methods and CPU time (in s) n n G GR GL Eact Conclusion Comparison of CPU time for element matri in Table II The integrals arising in finite element analsis usuall involve a huge amount of computing time and memor space. In this paper, a simple and efficient method is focused on the reduction of computing time and space in the stiffness matri for a general four node quadrilateral element through pre- and postintegration process, and b smbolic computation. For this, three tpes of numerical Gaussian quadrature rules are used. Here, the numerical integration methods are used for the comparison purpose but not for the integration itself. The procedures are rather simple and it ma be carried out to optimize the eplicit integration formulas also for the other finite elements. The comparison of the computing time also confirmed one of the main advantages of the smbolic integration approach. Thus the authours' concluding remar is that not onl Gauss-Legendre quadrature rule is efficient but also Gauss- Radau as well as Gauss-Lobatto quadrature rules ma be used for the evaluation of element matrices for a general four-node linear conve quadrilateral element. References abolian, E., Dehghan, M. M.R. Eslahchi (006) Application of Gauss quadrature rule in finding bounds for solution of linear sstem of equations, Appl. Math. Computation, 79: arrett, K.E. (999) Eplicit eight-noded quadrilateral elements, Finite Elements in Analsis and Design, : icford, W.. (990) A First Course in the Finite Element Method, Irwin, Illinois. urden, R.L. Faires, J.D. (00) - Numerical Analsis, roos/cole, 7th Ed. Eslahchi, M.R., Mased-Jamei, M. abolian, E. (005) On numerical improvement of Gauss-Lobatto quadrature rules, Appl. Math. Computation, 6: Hacer, W.L. Schreer, H.L. (989) Eigenvalue analsis of compatible and incompatible rectangular four-node quadrilateral elements, Int. J. Numer. Methods Engrg., 8 :

10 86 Applications of Gauss-Radau () 008 Mased-Jamei, M., Eslahchi, M.R. Dehghan, M. (005) On numerical improvement of Gauss-Radau quadrature rules, Appl. Math. Computation, 68: 5-6. Rathod, H.T. Islam, M. Shafiqul (00) - Reduction of rational integrals related to linear and conve quadrilateral finite elements, Numer. Methods Partial Differ. Eqns, 8: Stroud, A.H. (97) Numerical Quadrature and Solution of Ordinar Differential Equations, Springer-Verlag, New Yor erlin Heidelberg. Yagawa, G., Ye, G.W. Yoshimura, S. (990) A numerical integration scheme for finite element method based on smbolic manipulation, Int. J. Numer. Methods Engrg., 9: Zieniewicz, O.C. (977) The Finite Element Method, rd Edn., Mc Graw Hill Inc Received : December, 05, 007; Accepted : March,, 008.

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