×

zbMATH — the first resource for mathematics

An arbitrary order variationally consistent integration for Galerkin meshfree methods. (English) Zbl 1352.65481
Summary: Because most approximation functions employed in meshfree methods are rational functions with overlapping supports, sufficiently accurate domain integration becomes costly, whereas insufficient accuracy in the domain integration leads to suboptimal convergence. In this paper, we show that it is possible to achieve optimal convergence by enforcing variational consistency between the domain integration and the test functions, and optimal convergence can be achieved with much less computational cost than using higher-order quadrature rules. In fact, stabilized conforming nodal integration is variationally consistent, whereas Gauss integration and nodal integration are not. In this work the consistency conditions for arbitrary order exactness in the Galerkin approximation are set forth explicitly. The test functions are then constructed to be variationally consistent with the integration scheme up to a desired order. Attempts are also made to correct methods that are variationally inconsistent via modification of test functions, and several variationally consistent methods are derived under a unified framework. It is demonstrated that the solution errors of PDEs due to quadrature inaccuracy can be significantly reduced when the variationally inconsistent methods are corrected with the proposed method, and consequently the optimal convergence rate can be either partially or fully restored.

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Chen, A Lagrangian reproducing kernel particle method for metal forming analysis, Computational Mechanics 22 pp 289– (1998) · Zbl 0928.74115 · doi:10.1007/s004660050361
[2] Belytschko, Dynamic fracture using element-free Galerkin methods, International Journal for Numerical Methods in Engineering 39 pp 923– (1996) · Zbl 0953.74077 · doi:10.1002/(SICI)1097-0207(19960330)39:6<923::AID-NME887>3.0.CO;2-W
[3] Moës, A finite element method for crack growth without remeshing, International Journal for Numerical Methods in Engineering 46 pp 131– (1999) · Zbl 0955.74066 · doi:10.1002/(SICI)1097-0207(19990910)46:1<131::AID-NME726>3.0.CO;2-J
[4] Guan, Semi-Lagrangian reproducing kernel particle method for fragment-impact problems, International Journal of Impact Engineering 38 pp 1033– (2011) · doi:10.1016/j.ijimpeng.2011.08.001
[5] Guan, Semi-Lagrangian reproducing kernel formulation and application to modeling earth moving operations, Mechanics of Materials 41 pp 670– (2009) · doi:10.1016/j.mechmat.2009.01.030
[6] Nayroles, Generalizing the finite element method: diffuse approximation and diffuse elements, Computational Mechanics 10 pp 307– (1992) · Zbl 0764.65068 · doi:10.1007/BF00364252
[7] Belytschko, Element-free Galerkin methods, International Journal for Numerical Methods in Engineering 37 pp 229– (1994) · Zbl 0796.73077 · doi:10.1002/nme.1620370205
[8] Dolbow, Numerical integration of the Galerkin weak form in meshfree methods, Computational Mechanics 23 pp 219– (1999) · Zbl 0963.74076 · doi:10.1007/s004660050403
[9] Atluri, A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics, Computational Mechanics 22 pp 117– (1998) · Zbl 0932.76067 · doi:10.1007/s004660050346
[10] De, The method of finite spheres, Computational Mechanics 25 pp 329– (2000) · Zbl 0952.65091 · doi:10.1007/s004660050481
[11] De, The method of finite spheres with improved numerical integration, Computers and Structures 79 pp 2183– (2001) · doi:10.1016/S0045-7949(01)00124-9
[12] Liu, A new support integration scheme for the weakform in mesh-free methods, International Journal for Numerical Methods in Engineering 82 pp 699– (2010) · Zbl 1188.74084
[13] Carpinteri, The partition of unity quadrature in meshless methods, International Journal for Numerical Methods in Engineering 54 pp 987– (2002) · Zbl 1028.74047 · doi:10.1002/nme.455
[14] Duflot, A truly meshless Galerkin method based on a moving least squares quadrature, Communications in Numerical Methods in Engineering 18 pp 441– (2002) · Zbl 1008.74082 · doi:10.1002/cnm.503
[15] Beissel, Nodal integration of the element-free Galerkin method, Communications in Numerical Methods in Engineering 139 pp 49– (1996) · Zbl 0918.73329
[16] Chen, A stabilized conforming nodal integration for Galerkin mesh-free methods, International Journal for Numerical Methods in Engineering 50 pp 435– (2001) · Zbl 1011.74081 · doi:10.1002/1097-0207(20010120)50:2<435::AID-NME32>3.0.CO;2-A
[17] Puso, Meshfree and finite element nodal integration methods, International Journal for Numerical Methods in Engineering 74 pp 416– (2008) · Zbl 1159.74456 · doi:10.1002/nme.2181
[18] Chen, Meshfree Methods for Partial Differential Equations III pp 57– (2007) · doi:10.1007/978-3-540-46222-4_4
[19] Liu, A smoothed finite element method for mechanics problems, Computational Mechanics 39 pp 859– (2007) · Zbl 1169.74047 · doi:10.1007/s00466-006-0075-4
[20] Dyka, Stress points for tension instability in SPH, International Journal for Numerical Methods in Engineering 40 pp 2325– (1997) · Zbl 0890.73077 · doi:10.1002/(SICI)1097-0207(19970715)40:13<2325::AID-NME161>3.0.CO;2-8
[21] Krongauz, Consistent pseudo-derivatives in meshless methods, Computer Methods in Applied Mechanics and Engineering 146 pp 371– (1997) · Zbl 0894.73156 · doi:10.1016/S0045-7825(96)01234-0
[22] Bonet, Correction and stabilization of smooth particle hydrodynamics methods with applications in metal forming simulations, International Journal for Numerical Methods in Engineering 47 pp 1189– (2000) · Zbl 0964.76071 · doi:10.1002/(SICI)1097-0207(20000228)47:6<1189::AID-NME830>3.0.CO;2-I
[23] Wang, Locking-free stabilized conforming nodal integration for meshfree Mindlin-Reissner plate formulation, Computer Methods in Applied Mechanics and Engineering 193 pp 1065– (2004) · Zbl 1060.74675 · doi:10.1016/j.cma.2003.12.006
[24] Chen, A constrained reproducing kernel particle formulation for shear deformable shell in Cartesian coordinates, International Journal for Numerical Methods in Engineering 68 pp 151– (2006) · Zbl 1130.74055 · doi:10.1002/nme.1701
[25] Wang, A Hermite reproducing kernel approximation for thin-plate analysis with sub-domain stabilized conforming integration, International Journal for Numerical Methods in Engineering 74 pp 368– (2008) · Zbl 1159.74460 · doi:10.1002/nme.2175
[26] Duan, Second-order accurate derivatives and integration schemes for meshfree methods, International Journal for Numerical Methods in Engineering 92 pp 399– (2012) · Zbl 1352.65390 · doi:10.1002/nme.4359
[27] Babuška, Quadrature for meshless methods, International Journal for Numerical Methods in Engineering 76 pp 1434– (2008) · Zbl 1195.65165 · doi:10.1002/nme.2367
[28] Babuška, Effect of numerical integration on meshless methods, Computer Methods in Applied Mechanics and Engineering 198 pp 2886– (2009) · Zbl 1229.65204 · doi:10.1016/j.cma.2009.04.008
[29] Liu, Reproducing kernel particle methods, International Journal for Numerical Methods in Fluids 20 pp 1081– (1995) · Zbl 0881.76072 · doi:10.1002/fld.1650200824
[30] Chen, Reproducing kernel particle methods for large deformation analysis of non-linear structures, Computer Methods in Applied Mechanics and Engineering 139 pp 195– (1996) · Zbl 0918.73330 · doi:10.1016/S0045-7825(96)01083-3
[31] Chen, Non-linear version of stabilized conforming nodal integration for Galerkin mesh-free methods, International Journal for Numerical Methods in Engineering 53 pp 2587– (2002) · Zbl 1098.74732 · doi:10.1002/nme.338
[32] Yoo, Stabilized conforming nodal integration in the natural-element method, International Journal for Numerical Methods in Engineering 60 pp 861– (2004) · Zbl 1060.74677 · doi:10.1002/nme.972
[33] Lu, A new implementation of the element free Galerkin method, Computer Methods in Applied Mechanics and Engineering 113 pp 397– (1994) · Zbl 0847.73064 · doi:10.1016/0045-7825(94)90056-6
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.