Numerical Modeling of Material Discontinuity Using Mixed MLPG Collocation Method

Transcription

1 umerical odeling of aterial Dicontinuit Uing ied LPG Collocation ethod B. alušić 1,. Sorić 1 and T. arak 1 Abtract A mied LPG collocation method i applied for the modeling of material dicontinuit in heterogeneou material compoing of homogeneou domain. Two homogeneou iotropic material with different linear elatic propertie are conidered. The olution for the entire domain i obtained b enforcing the correponding continuit condition along the interface of homogeneou material. For the approimation of the unknown field variable LS function with interpolator condition are applied. The accurac and numerical efficienc of the mied approach i compared with a tandard primal mehle approach and demontrated b a repreentative numerical eample. Keword: ied mehle approach, LS approimation, heterogeneou material 1 ntroduction n recent decade, a new group of numerical approache known a mehle method ha attracted tremendou interet due to it potential to overcome certain hortcoming of meh-baed method uch a element ditortion problem and time-demanding meh generation proce. everthele, the calculation of mehle approimation function due to it high computational cot i till a major drawback. Thi deficienc can be alleviated to a certain etent b uing the mied ehle Local Petrov-Galerkin (LPG) ethod paradigm [Atluri, Liu, Han (2006)]. n the preent contribution, the LPG formulation baed on the mied approach i adapted for the modeling of deformation repone of heterogeneou material. A heterogeneou tructure conit of two homogeneou material which are dicretized b grid point in which equilibrium equation are impoed. The linear elatic boundar value problem for each homogeneou material i dicretized b uing the independent approimation of diplacement and tre component. ndependent variable are approimated uing mehle function in uch a wa that each material i treated a a eparate problem [Chen, Wang, Hu, Chi (2009)]. The global olution for the entire heterogeneou tructure i acquired b enforcing appropriate diplacement and traction condition along the interface of two homogeneou material. A collocation mehle method i ued, which ma be conidered a a pecial cae of the LPG approach, where the Dirac delta function i utilized a the tet function. Since the collocation method i 1 Facult of echanical Engineering and aval Architecture, Univerit of Zagreb, vana Lučića 5, Zagreb, Croatia.

2 emploed, the trong form of equilibrium equation i ielded and time-conuming numerical integration proce i avoided. The moving leat quare (LS) approimation function [Atluri (2004)] with interpolator propertie are applied. Thi enable imple impoition of diplacement boundar condition a in FE. Traction boundar condition on outer edge are enforced via the direct collocation approach. n order to derive the final cloed tem of dicretized governing equation with the diplacement a unknown variable, the nodal tre value are epreed in term of the diplacement component uing the kinematic and contitutive relation. The mied LPG collocation method for modeling of material dicontinuit i preented and eplained at large in chapter 2. Efficienc of the propoed mied method i analzed in detail on a wellknown numerical eample of a plate with circular incluion with different material propertie. n chapter 3 the accurac of the utilized approach i alo compared to a tandard mehle primal approach. 2 ied LPG collocation method for heterogeneou material The two-dimenional heterogeneou material which occupie the global computational domain urrounded b the global outer boundar i conidered, a hown in Figure 1. The boundar repreent the interface between two ubdomain i with different homogeneou material propertie. eparate the global domain in uch a manner that and. Figure 1: Two-dimenional heterogeneou material The governing equation for the preented eample are the trong form 2D equilibrium equation which have to be atified within the global computational domain divided into and σ b 0, within, σ b 0, within. (1) ij, j i ij, j i Equation (1) have to atif the boundar condition precribed on the local ubdomain boundarie u u, on, u u, on, (2) i i u i i u

3 t n t, on, t n t, on (3) + i ij j i t i ij j i t and interface condition on the boundar. For the dicretization of, the double node concept in emploed. At each node on, the interface condition of diplacement continuit and traction equilibrium are enforced u u 0, t t 0. (4) i i i i The eternal boundar of the local ubdomain i compoed of everal part, + + +, where denote the part of with precribed u u t t diplacement boundar condition u u u u, while i denote the part, where the + t t t traction boundar condition t i are applied. The two-dimenional heterogeneou continuum i dicretized b two different et of node 1,2,..., and 1,2,..., P, where and P indicate the total number of node within and, repectivel. According to the mied collocation procedure [Atluri, Liu, Han (2006)], the unknown field variable are the diplacement and tre component. All unknown field variable are approimated eparatel within ubdomain and, where the ame approimation function are emploed for all the diplacement and tre component. For the ubdomain it can be written u i i 1 h u, (5) ij ij 1 h, (6) where repreent the nodal value of two-dimenional hape function for node, tand for the number of node within the approimation domain, while ui denote the nodal value of diplacement and tre component. The diplacement ij and tre component are analogoul approimated over the ubdomain. For the hape function contruction the well-known LS approimation cheme [Atluri (2004)] i emploed. The interpolator propertie of the LS approimation function are achieved b utilizing the weight function according to [ot, Bucher (2008)]. Appling the mied LPG trateg the equilibrium equation (1) are dicretized b the tre approimation (6), leading to T T 1 1 B σ b 0, within, B σ b 0, within, (7) where B B and and B B indicate the matrice coniting of the firtorder derivative of hape function. t can be verified that the number of equation at the global level obtained b (7) i le than the total number of tre unknown. Therefore,

4 in order to obtain the cloed tem of equation, the compatibilit between the approimated tree and diplacement i utilized at collocation node K K K K K1 K1 σ D B u, σ D B u, (8) where D and D denote material tenor for each homogeneou domain. nerting the dicretized contitutive relation (8) into the dicretized equilibrium equation (7), a olvable tem of linear algebraic equation with onl the nodal diplacement a unknown i attained 1 1 K u R, within, K u R, within. (9) n equation (9), the nodal tiffne matrice K and K are defined a T T K K K K K1 K1 K B D B, K B D B, (10) while the nodal force vector R and R are impl equal to R b, R b. (11) All approimation function in thi contribution poe the interpolation propert at the node. Conequentl, the diplacement boundar condition are enforced traightforward, analogoul to the procedure in FE. Therefore, b dicretizing the diplacement boundar condition (2) with the approimation (5) we obtain u u 1 1 u u, on, u u, on. (12) Appling the tre approimation (6) and the compatibilit between the approimated tree and diplacement (8) in the boundar equation (3), the dicretized traction boundar condition are derived a K K t K K t K1 K1 t D B u, on, t D B u, on. (13) n a imilar wa the applied interface boundar condition (4) can be dicretized a 1 1 u u, on, (14) K K K K K1 K1 D B u D B u, on. (15)

5 3 umerical Eample 3.1 Plate with circular incluion A rectangular plate of in ize with the radiu of embedded circular incluion equal to 1 ubjected to unit horizontal traction i conidered. Owing to the mmetr, onl one quarter of the plate i modeled. The dicretization of a quarter of the plate with circular incluion, geometr and enforced boundar condition are hown in Figure 3. Thi eample i ued a a benchmark for teting the computational trateg preented. The material propertie of the plate are E 1000, 0.25, while the material propertie of the incluion are E 10000, 0.3. Figure 2: Plate with circular incluion with boundar condition The mehle interpolation cheme uing the econd- and fourth-order bai (LS2, LS4) are applied and compared. Both primal (P) and mied () approache are utilized. Figure 3 and 4 how the comparion of u diplacement and tre component ditribution along 0 for model dicretized with 346 node. Figure 3: Diplacement u at 0 Figure 4: Stre at 0

6 The ditribution attained are compared to referent olution obtained b CPS4 finite element from the finite element oftware Abaqu. The convergence tud of both primal and mied approache emploing the error of normalized diplacement u and normalized tre component at point A are hown in Figure 5 and 6. Figure 5: Convergence of normalized diplacement u at point A Figure 6: Convergence of normalized tre at point A 4 Concluion From the numerical reult it can be perceived that the mied approach i more robut and uperior to the primal formulation. Therefore, a more accurate and numericall efficient modeling of heterogeneou material i achieved if the mied mehle collocation formulation i ued ince it reduce the needed continuit order of the mehle approimation function to onl C 1 continuit for the preented problem. Acknowledgement Thi work ha been full upported b Croatian Science Foundation under the project ulticale umerical odeling of aterial Deformation Repone from acro- to anolevel (2516). Reference: Atluri, S..; Liu, H. T.; Han, Z. D. (2006): ehle Local Petrov-Galerkin (LPG) ied Collocation ethod for Elaticit Problem. CES: Computer odeling in Engineering & Science, vol. 14, no. 3, pp Chen,.-S.; Wang, L.; Hu, H.-Y.; Chi, S.-W. (2009): Subdomain radial bai collocation method for heterogeneou media. nternational ournal for umerical ethod in Engineering, vol. 80, pp Atluri, S.. (2004): The ehle Local Petrov-Galerkin (LPG) ethod. Tech. Science Pre. ot, T.; Bucher, C. (2008): ew Concept for oving Leat Square: An nterpolating oningular Weighting Function and Weighted odal Leat Square. Engineering Anali with Boundar Element, vol. 32, no. 6, pp