Generalized Gaussian Quadrature Rules in Enriched Finite Element Methods

Transcription

1 Generalized Gaussian Quadrature Rules in Enriched Finite Element Methods Abstract In this paper, we present new Gaussian integration schemes for the efficient and accurate evaluation of weak form integrals that arise in enriched finite element methods. For discontinuous functions we present an algorithm for the construction of Gauss-like quadrature rules over arbitraril-shaped elements without partitioning. In case of singular integrands, we introduce a new polar transformation that eliminates the singularit so that the integration can be performed with fewer integration points while maintaining high accurac. We combine the quadrature construction technique with a point elimination algorithm, which ensures that the final quadratures have as few number of Gauss points as possible without sacrificing accurac. Ke words: Heaviside function, singular function, numerical integration, Duff transformation, crack modeling Introduction In partition of unit finite element methods, special basis functions are added to the solution space to allow the inclusion of a priori knowledge about the local behavior of the solution []. In the etended finite element method [2], which is used for crack modeling without remeshing, the displacement approimation is enriched b incorporating a discontinuous function and the near-tip asmptotic fields via the framework of partition of unit. In these methods, a well-known and outstanding issue is the efficient and accurate evaluation of the entries in the stiffness matri corresponding to the enriched degrees of freedom since the gradients of enriched basis functions contain discontinuities and/or singularities. Traditionall, these elements are partitioned into integration subcells (normall subtriangles) that conform to the crack geometr [2] and higher-order Gauss quadratures is used over them for integration. Recentl, Ventura [3] demonstrated that an equivalent polnomial function eists such that its integral gives the eact value of the discontinuous/nondifferentiable function integrated on subcells. This technique is applicable to triangular and tetrahedral constant strain elements. Holdch et al. [4] propose a similar method for construction of quadrature rules with fied points but variable weights for integration of discontinuous functions over triangles and tetrahedrons. The limitation of the aforementioned methods is that the are dependent on the element topolog and due to the compleit of the algebraic epressions, crack kinks within an element can not be handled. In the Preprint submitted to Elsevier 6 Januar 29

2 method proposed in this paper, generalized quadrature rules are constructed on arbitrar polgons with arbitrar configuration of cracks that can integrate polnomials of an desired order eactl within the cracked domain. Another numerical issue in the etended finite element method is the evaluation of the stiffness matri entries containing the crack tip due to the presence of singular terms such as /r and / r.moës et al. [2] subdivide the element containing the crack tip into conforming triangles and use higher-order Gauss quadratures over each of them. Béchet et al. [5] present a more sophisticated scheme in which first the domain is triangulated with the singularit ling at a verte and then the integration of the singular kernel over the triangle is transformed into a smooth integration over the bi-unit square through a series of complicated transformations. Laborde et al. [6] emplo a rather straightforward transformation from the triangle to the unit square to remove the singularit though not realized, this approach was originall conceived b Duff [7]. In this paper, we introduce a more general transformation that significantl improves the performance of the Duff transformation b reducing the number of Gauss points. The technique used in this paper for construction of quadratures is combined with a point elimination algorithm (originall presented b Xiao and Gimbutas [8] and later generalized to regular polgons b the authors [9]), which is based on Newton s method and ensures that the number of integration points in a quadrature rule are as few as possible without loss of accurac. 2 Algorithm for Construction of Efficient Quadrature Schemes 2. Quadrature Rule for Discontinuous Functions A quadrature is a formula of the form n ω() f() d w i f( i ), () Ω i= with Ω the integration region, f an integrand defined on Ω and ω the weight function. ω() = H() is used to account for the discontinuit with H()being the generalized Heaviside function. i and w i are called quadrature points and weights, respectivel. We can design a quadrature so that Eq. () is eact for all functions in a pre-selected set, e.g., {φ i } m i=,bsolvingeq.(2): Ω ω()φ () d φ ( ) φ ( 2 )... φ ( n ) w Ω ω()φ 2() d φ = 2 ( ) φ 2 ( 2 )... φ 2 ( n ) w 2. (2)... Ω ω()φ m() d φ m ( ) φ m ( 2 )... φ m ( n ) w n The basis functions in our case are polnomials up to a certain degree with respect to and in a 2D problem. Once an initial guess such as a tensorproduct is chosen, Newton s method is used to solve Eq. (2) iterativel. After 2

3 4 β = β = 2 β = 3 β = 4 4 β = β = 2 β = 3 β = 4 Ma. absolute error 8 2 Ma. absolute error Number of evaluation points Number of evaluation points (a) α = (b) α = 2 Fig. 2. Error in integration of p(, )/r α. convergence is reached, an n-point quadrature rule is produced that can integrate the selected polnomials multiplied b Heaviside function, specificall designed for a cracked element. Now, an interesting idea is to eliminate one of the points in the n-point quadrature rule and solve Eq. (2) again starting with the n -point quadrature. This process is continued until Newton s method does not converge anmore. For more details of the point elimination algorithm and its implementation, see References [8] and [9]. 2.2 Quadrature Rule for Singular Functions v u (a) (b) Fig.. Duff transformation. (a) Standard triangle and (b) unit square Duff transformation [7]: = u, = v = uv maps the standard triangle (Fig. a) to a unit square (Fig. b) and as a result eliminates singularities of the form /r α. The singular integrand over the standard triangle is transformed into a smooth function over the unit square that can be integrated with fewer integration points [6, 7], e.g., b using a tensorproduct rule. Integration of an singular function over an element with the singularit inside it can be carried out with these steps: triangulate the element so that the singularit occurs at one verte of all triangles, map each one of the triangles to the standard one and evaluate the integration b using Duff transformation and a standard Gauss quadrature rule over the unit square. In this paper, we introduce a new transformation: = u β, = v β = u β v β, which generalizes the Duff transformation and maps the standard triangle to the unit square. The rest of the process of integration is similar to the case of Duff transformation. Fig. 2 shows the maimum absolute error of integrations of the form Ω st p(, )/rα da for different values of β, whereω st stands for the standard triangle shown in Fig. a and p(, ) belongs to the set of bivariate monomials up to order two with respect to and. Theeact integrations are evaluated using the Maple TM smbolic package. As seen in this figure, while β = can give reasonable accurac for the singularit of the form r, it is far from acceptable for a singularit of the form r /2,whereasfor 3

4 2 2 (a) nsp =4 (b) nsp =4 (c) nsp =4 (d) nsp =3 2 2 (e) nsp =6 (f) nsp =9 (g) nsp =7 (h) nsp =6 Fig. 3. Quadrature rules for discontinuous functions. (a-d) Quadratic precision and (e-h) Quartic precision. β = 2 one can get much better results with the same number of integration points over the unit square. For the numerical results presented in this paper, we use β =forr and β =2forr /2 singularities. Now, having established a robust scheme for integration of those singular functions, the same algorithm used for construction and optimization of quadrature rules for discontinuous functions can be used for singular functions b changing the weight function to ω() =r and ω() =r /2 with the slight difference of replacing the eact integrations on the left-hand-side of Eq. (2) with the approimate ones obtained b application of Duff transformation and our new transformation, respectivel. 3 Numerical Eamples (a) (b) (c) Fig. 4. Singular quadratures (quadratic precision). +:r and : r /2 singularit. The algorithm for construction and optimization of quadratures is implemented in MATLAB TM. Fig. 3 shows quadrature rules for discontinuous functions over different element shapes and cracks for quadratic and quartic precision. All these quadratures are eact (within machine precision). Fig. 4 shows quadrature rules for singular integrands of tpes r and r /2 over elements with different topologies and crack configurations. The convergence curves of these quadratures are similar to those shown in Fig. 2 with β =andβ =2forr and r /2 singularities respectivel, therefore one can have strict control of accurac b selecting adequate number of integration points within the unit square. Notice that the number of integration points on the horizontal ais of Fig. 2 are onl for numerical integration within the unit square to evaluate the left-hand-side of Eq. (2), whereas the final quadrature rules produced with this algorithm 4

5 can have as few as three points for a triangle with the singularit at a verte as shown in Fig. 4a. The quadratures in Figures 4b and 4c are constructed b subdividing the domains into triangles with the crack-tip at a verte of all of them and then appling the algorithm to each of the triangles. This algorithm can be applied to other singularities as well, e.g., r 2/3 which arises in crack modeling in bimaterials (see Fig. 5). The average eecution time for a quadrature rule is about 2 seconds on a Linu workstation. We presented a new approach for constructing and optimizing quadrature rules for general classes of functions. The algorithm was used for construction of quadrature rules for eact integration of discontinuous functions in 2D without partitioning. This technique is ver fleible and can be used for arbitrar polgons with arbitrar crack configurations. A new polar transformation was introduced that significantl improves the convergence of the Duff transformation. The transformation combined Fig. 5. Quadrature rule for singularit r 2/3, quadratic precision, β =3. with the point elimination algorithm was used to construct quadrature rules for singular functions over arbitrar conve polgons. Higher-order quadratures (cubic, quartic, etc.) for enrichment functions ψ = rf(θ) canbeconstructed with this technique to account for enrichments that possess angular dependence. The proposed quadrature rules are promising for use in enriched finite element methods to ensure sstematic improvabilit and control of numerical integration errors. References [] I. Babuška, J. M. Melenk, IJNME 4 (997) [2] N. Moës, J. Dolbow, T. Beltschko, IJNME 46 () (999) 3 5. [3] G. Ventura, IJNME 66 (26) [4] D. J. Holdch, D. R. Noble, R. B. Secor, IJNME 73 (28) [5] E. Béchet, H. Minnebo, N. Moës, B. Burgardt, IJNME 64 (25) [6] P. Laborde, J. Pommier, Y. Renard, M. Salaun, IJNME 64 (25) [7] M. G. Duff, SINUM 9 (6) (982) [8] H. Xiao, Z. Gimbutas, A new algorithm for the construction of efficient quadratures in two and higher dimensions, submitted, 28. [9] Authors, Generalized Gaussian quadrature rules on regular polgons, submitted, 28. 5