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Generalized Gaussian quadrature rules for discontinuities and crack singularities in the extended finite element method. (English) Zbl 1225.74099
Summary: New Gaussian integration schemes are presented for the efficient and accurate evaluation of weak form integrals in the extended finite element method. For discontinuous functions, we construct Gauss-like quadrature rules over arbitrarily-shaped elements in two dimensions without the need for partitioning the finite element. A point elimination algorithm is used in the construction of the quadratures, which ensures that the final quadratures have minimal number of Gauss points. For weakly singular integrands, we apply a polar transformation that eliminates the singularity so that the integration can be performed efficiently and accurately. Numerical examples in elastic fracture using the extended finite element method are presented to illustrate the performance of the new integration techniques.
MSC:
74S05Finite element methods in solid mechanics
74G70Stress concentrations, singularities
74R10Brittle fracture
Software:
DECUHR; XFEM
References:
[1]Duarte, C. A.; Oden, J. T.: An h – p adaptive method using clouds, Comput. methods appl. Mech. engrg. 139, 237-262 (1996) · Zbl 0918.73328 · doi:10.1016/S0045-7825(96)01085-7
[2]Melenk, J. M.; Babuška, I.: The partition of unity finite element method: basic theory and applications, Comput. methods appl. Mech. engrg. 139, 289-314 (1996) · Zbl 0881.65099 · doi:10.1016/S0045-7825(96)01087-0
[3]Babuška, I.; Melenk, J. M.: The partition of unity method, Int. J. Numer. methods engrg. 40, 727-758 (1997) · Zbl 0949.65117 · doi:10.1002/(SICI)1097-0207(19970228)40:4<727::AID-NME86>3.0.CO;2-N
[4]Belytschko, T.; Black, T.: Elastic crack growth in finite elements with minimal remeshing, Int. J. Numer. methods engrg. 45, No. 5, 601-620 (1999) · Zbl 0943.74061 · doi:10.1002/(SICI)1097-0207(19990620)45:5<601::AID-NME598>3.0.CO;2-S
[5]Moës, N.; Dolbow, J.; Belytschko, T.: A finite element method for crack growth without remeshing, Int. J. Numer. methods engrg. 46, No. 1, 131-150 (1999) · Zbl 0955.74066 · doi:10.1002/(SICI)1097-0207(19990910)46:1<131::AID-NME726>3.0.CO;2-J
[6]Dolbow, J.; Moës, N.; Belytschko, T.: Discontinuous enrichment in finite elements with a partition of unity method, Finite elem. Anal. des. 36, 235-260 (2000) · Zbl 0981.74057 · doi:10.1016/S0168-874X(00)00035-4
[7]Daux, C.; Moës, N.; Dolbow, J.; Sukumar, N.; Belytschko, T.: Arbitrary branched and intersecting cracks with the extended finite element method, Int. J. Numer. methods engrg. 48, No. 12, 1741-1760 (2000) · Zbl 0989.74066 · doi:10.1002/1097-0207(20000830)48:12<1741::AID-NME956>3.0.CO;2-L
[8]Belytschko, T.; Moës, N.; Usui, S.; Parimi, C.: Arbitrary discontinuities in finite elements, Int. J. Numer. methods engrg. 50, 993-1013 (2001) · Zbl 0981.74062 · doi:10.1002/1097-0207(20010210)50:4<993::AID-NME164>3.0.CO;2-M
[9]Hansbo, A.; Hansbo, P.: A finite element method for the simulation of strong and weak discontinuities in solid mechanics, Comput. methods appl. Mech. engrg. 193, No. 33 – 35, 3523-3540 (2004) · Zbl 1068.74076 · doi:10.1016/j.cma.2003.12.041
[10]Fries, T. P.; Belytschko, T.: The intrinsic XFEM: a method for arbitrary discontinuities without additional unknowns, Int. J. Numer. methods engrg. 68, 1358-1385 (2006) · Zbl 1129.74045 · doi:10.1002/nme.1761
[11]Chahine, E.; Laborde, P.; Renard, Y.: Crack tip enrichment in the XFEM using a cutoff function, Int. J. Numer. methods engrg. 75, 629-646 (2008) · Zbl 1195.74167 · doi:10.1002/nme.2265
[12]Shen, Y.; Lew, A.: An optimally convergent discontinuous-Galerkin-based extended finite element method for fracture mechanics, Int. J. Numer. methods engrg. 82, No. 6, 716-755 (2010) · Zbl 1188.74070 · doi:10.1002/nme.2781
[13]Mousavi, S. E.; Sukumar, N.: Generalized duffy transformation for integrating vertex singularities, Comput. mech. 45, No. 2 – 3, 127-140 (2010)
[14]Belytschko, T.; Chen, H.; Xu, J.; Zi, G.: Dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment, Int. J. Numer. methods engrg. 58, 1873-1905 (2003) · Zbl 1032.74662 · doi:10.1002/nme.941
[15]Réthoré, J.; Gravouil, A.; Combescure, A.: An energy-conserving scheme for dynamic crack growth using the extended finite element method, Int. J. Numer. methods engrg. 63, 631-659 (2005) · Zbl 1122.74519 · doi:10.1002/nme.1283
[16]Prabel, B.; Combescure, A.; Gravouil, A.; Marie, S.: Level set X-FEM non-matching meshes: application to dynamic crack propagation in elastic – plastic media, Int. J. Numer. methods engrg. 69, 1553-1569 (2007) · Zbl 1194.74465 · doi:10.1002/nme.1819
[17]Wells, G. N.; Sluys, L. J.: A new method for modelling cohesive cracks using finite elements, Int. J. Numer. methods engrg. 50, 2667-2682 (2001) · Zbl 1013.74074 · doi:10.1002/nme.143
[18]Moës, N.; Belytschko, T.: Extended finite element method for cohesive crack growth, Engrg. fract. Mech. 69, 813-833 (2002)
[19]Mergheim, J.; Kuhl, E.; Steinmann, P.: A finite element method for the computational modelling of cohesive cracks, Int. J. Numer. methods engrg. 63, 276-289 (2005) · Zbl 1118.74349 · doi:10.1002/nme.1286
[20]Gross, S.; Reusken, A.: An extended pressure finite element space for two-phase incompressible flows with surface tension, J. comput. Phys. 224, 40-58 (2007)
[21]Zilian, A.; Legay, A.: The enriched space – time finite element method (EST) for simultaneous solution of fluid – structure interaction, Int. J. Numer. methods engrg. 75, 305-334 (2008) · Zbl 1195.74212 · doi:10.1002/nme.2258
[22]Gerstenberger, A.; Wall, W. A.: An extended finite element method/Lagrange multiplier based approach for fluid – structure interaction, Comput. methods appl. Mech. engrg. 197, 1699-1714 (2008) · Zbl 1194.76117 · doi:10.1016/j.cma.2007.07.002
[23]Fries, T. P.: The intrinsic XFEM for two-fluid flows, Int. J. Numer. methods engrg. 60, 437-471 (2009) · Zbl 1161.76026 · doi:10.1002/fld.1901
[24]L.M. Vigneron, J.G. Verly, S.K. Warfield, Modelling surgical cuts, retractions, and resections via extended finite element method, in: C. Barillot, D.R. Haynor, P. Hellier (Eds.), Proceedings of the Medical Image Computing and Computer Assisted Intervention – MICCAI 2004, PT 2, vol. 3217 (Part 2), Lecture Notes in Computer Science, Springer-Verlag, Berlin, Heidelberger Platz 3, D-14197 Berlin, Germany, 2004, pp. 311 – 318.
[25]Jeřábková, L.; Kuhlen, T.: Stable cutting of deformable objects in virtual environments using the XFEM, IEEE comput. Graph. appl. 29, No. 2, 61-71 (2009)
[26]Kaufmann, P.; Martin, S.; Botsch, M.; Grinspun, E.; Gross, M.: Enrichment textures for detailed cutting of shells, ACM trans. Graph. 28, No. 3, 50:1-50:10 (2009)
[27]Ventura, G.: On the elimination of quadrature subcells for discontinuous functions in the extended finite-element method, Int. J. Numer. methods engrg. 66, 761-795 (2006) · Zbl 1110.74858 · doi:10.1002/nme.1570
[28]Tornberg, A. -K.: Multi-dimensional quadrature of singular and discontinuous functions, BIT numer. Math. 42, No. 3, 644-669 (2002) · Zbl 1021.65010 · doi:10.1023/A:1021988001059
[29]Patzák, B.; Jirásek, M.: Process zone resolution by extended finite elements, Eng. fract. Mech. 70, 957-977 (2003)
[30]Oh, H.; Kim, J. G.; Jeong, J. W.: The smooth piecewise polynomial particle shape functions corresponding to patch-wise non-uniformly spaced particles for meshfree particle methods, Comput. mech. 40, 569-594 (2007) · Zbl 1165.74353 · doi:10.1007/s00466-006-0126-x
[31]Benvenuti, E.; Tralli, A.; Ventura, G.: A regularized XFEM model for the transition from continuous to discontinuous displacements, Int. J. Numer. methods engrg. 74, 911-944 (2008) · Zbl 1158.74479 · doi:10.1002/nme.2196
[32]Bordas, S.; Nguyen, P. V.; Guidoum, C. D. A.; Nguyen-Dang, H.: An extended finite element library, Int. J. Numer. methods engrg. 71, 703-732 (2007) · Zbl 1194.74367 · doi:10.1002/nme.1966
[33]Holdych, D.; Noble, D.; Secor, R.: Quadrature rules for triangular and tetrahedral elements with generalized functions, Int. J. Numer. methods engrg. 73, 1310-1327 (2008) · Zbl 1167.74043 · doi:10.1002/nme.2123
[34]Belytschko, T.; Gracie, R.; Ventura, G.: A review of extended/generalized finite element methods for material modeling, Model. simul. Mater. sci. Engrg. 17, 043001 (2009)
[35]Xiao, H.; Gimbutas, Z.: A numerical algorithm for the construction of efficient quadratures in two and higher dimensions, Comput. math. Appl. 59, 663-676 (2010) · Zbl 1189.65047 · doi:10.1016/j.camwa.2009.10.027
[36]Mousavi, S. E.; Xiao, H.; Sukumar, N.: Generalized Gaussian quadrature rules on arbitrary polygons, Int. J. Numer. methods engrg. 82, 99-113 (2010) · Zbl 1183.65026 · doi:10.1002/nme.2759
[37]Barsoum, R. S.: Application of quadratic isoparametric elements in linear fracture mechanics, Int. J. Fract. 10, 603-605 (1974)
[38]Stern, M.; Becker, E. B.: A conforming crack tip element with quadratic variation in the singular fields, Int. J. Numer. methods engrg. 12, 279-288 (1978) · Zbl 0384.65056 · doi:10.1002/nme.1620120209
[39]Tracey, D. M.: Finite elements for determination of crack tip elastic stress intensity factors, Engrg. fract. Mech. 3, 255-265 (1971)
[40]Solecki, J. S.; Swedlow, J. L.: On quadrature and singular finite-elements, Int. J. Numer. methods engrg. 20, 395-408 (1984) · Zbl 0528.73068 · doi:10.1002/nme.1620200302
[41]Akin, J. E.: The generation of elements with singularities, Int. J. Numer. methods engrg. 10, 1249-1259 (1976) · Zbl 0341.65078 · doi:10.1002/nme.1620100605
[42]Tracey, D. M.; Cook, T. S.: Analysis of power type singularities using finite elements, Int. J. Numer. methods engrg. 11, 1225-1233 (1977) · Zbl 0364.65090 · doi:10.1002/nme.1620110804
[43]Lim, I. L.; Johnston, I. W.; Choi, S. K.: Application of singular quadratic distorted isoparametric elements in linear fracture mechanics, Int. J. Numer. methods engrg. 36, 2473-2499 (1993)
[44]Dunham, R. S.: A quadrature rule for conforming quadratic crack tip elements, Int. J. Numer. methods engrg. 14, 287-312 (1979) · Zbl 0393.65012 · doi:10.1002/nme.1620140211
[45]Natarajan, S.; Bordas, S.; Mahapatra, D. R.: Numerical integration over arbitrary polygonal domains based on Schwarz – Christoffel conformal mapping, Int. J. Numer. methods engrg. 80, 103-134 (2009) · Zbl 1176.74190 · doi:10.1002/nme.2589
[46]Sukumar, N.; Tabarraei, A.: Conforming polygonal finite elements, Int. J. Numer. methods engrg. 61, No. 12, 2045-2066 (2004) · Zbl 1073.65563 · doi:10.1002/nme.1141
[47]Tabarraei, A.; Sukumar, N.: Extended finite element method on polygonal and quadtree meshes, Comput. methods appl. Mech. engrg. 197, No. 5, 425-438 (2008) · Zbl 1169.74634 · doi:10.1016/j.cma.2007.08.013
[48]Gander, W.; Gautschi, W.: Adaptive quadrature revisited, BIT numer. Math. 40, No. 1, 84-101 (2004)
[49]Strouboulis, T.; Babuška, I.; Copps, K.: The design and analysis of the generalized finite element method, Comput. methods appl. Mech. engrg. 181, No. 1 – 3, 43-69 (2000) · Zbl 0983.65127 · doi:10.1016/S0045-7825(99)00072-9
[50]Xiao, Q. Z.; Karihaloo, B. L.: Improving the accuracy of XFEM crack tip fields using higher order quadrature and statically admissible stress recovery, Int. J. Numer. methods engrg. 66, 1378-1410 (2006) · Zbl 1122.74529 · doi:10.1002/nme.1601
[51]Espelid, T. O.; Genz, A.: DECUHR: an algorithm for automatic integration of singular functions over a hyperrectangular region, Numer. algorithms 8, 201-220 (1994) · Zbl 0811.65018 · doi:10.1007/BF02142691
[52]Schweitzer, M. A.: An adaptive hp-version of the multilevel particle-partition of unity method, Comput. methods appl. Mech. engrg. 198, 1260-1272 (2009) · Zbl 1157.65494 · doi:10.1016/j.cma.2008.01.009
[53]Griebel, M.; Schweitzer, M. A.: A particle-partition of unity method — part II: Efficient cover construction and reliable integration, SIAM J. Sci. comput. 23, No. 5, 1655-1682 (2002) · Zbl 1011.65069 · doi:10.1137/S1064827501391588
[54]Béchet, E.; Minnebo, H.; Moës, N.; Burgardt, B.: Improved implementation and robustness study of the X-FEM for stress analysis around cracks, Int. J. Numer. methods engrg. 64, No. 8, 1033-1056 (2005) · Zbl 1122.74499 · doi:10.1002/nme.1386
[55]Park, K.; Pereira, J. P.; Duarte, C. A.; Paulino, G. H.: Integration of singular enrichment functions in the generalized/extended finite element method for three-dimensional problems, Int. J. Numer. methods engrg. 78, No. 10, 1220-1257 (2009) · Zbl 1183.74305 · doi:10.1002/nme.2530
[56]Nagarajan, A.; Mukherjee, S.: A mapping method for numerical evaluation of two-dimensional integrals with 1/r singularity, Comput. mech. 12, 19-26 (1993) · Zbl 0776.73073 · doi:10.1007/BF00370482
[57]Ventura, G.; Gracie, R.; Belytschko, T.: Fast integration and weight function blending in the extended finite element method, Int. J. Numer. methods engrg. 77, No. 1, 1-29 (2009) · Zbl 1195.74201 · doi:10.1002/nme.2387
[58]Laborde, P.; Pommier, J.; Renard, Y.; Salaün, M.: High-order extended finite element method for cracked domains, Int. J. Numer. methods engrg. 64, 354-381 (2005) · Zbl 1181.74136 · doi:10.1002/nme.1370
[59]Fairweather, G.; Rizzo, F. J.; Shippy, D. J.: Computation of double integrals in the boundary integral equation method, Advances in computer methods for partial differential equations – III, 331-334 (1979)
[60]Duffy, M. G.: Quadrature over a pyramid or cube of integrands with a singularity at a vertex, SIAM J. Numer. anal. 19, No. 6, 1260-1262 (1982) · Zbl 0493.65011 · doi:10.1137/0719090
[61]Lyness, J. N.; Jespersen, D.: Moderate degree symmetric quadrature rules for the triangle, J. inst. Math. appl. 15, 19-32 (1975) · Zbl 0297.65018 · doi:10.1093/imamat/15.1.19
[62]Nooijen, M.; Velde, G. T.; Baerends, E. J.: Symmetric numerical integration formulas for regular polygons, SIAM J. Numer. anal. 27, No. 1, 198-218 (1990) · Zbl 0691.65007 · doi:10.1137/0727014
[63]Wandzura, S.; Xiao, H.: Symmetric quadrature rules on a triangle, Comput. math. Appl. 45, 1829-1840 (2003) · Zbl 1050.65022 · doi:10.1016/S0898-1221(03)90004-6
[64]Sukumar, N.; Prévost, J. -H.: Modeling quasi-static crack growth with the extended finite element method. Part I: Computer implementation, Int. J. Solids struct. 40, No. 26, 7513-7537 (2003) · Zbl 1063.74102 · doi:10.1016/j.ijsolstr.2003.08.002
[65]Lasserre, J. B.: Integration on a convex polytope, Proc. am. Math. soc. 126, No. 8, 2433-2441 (1998) · Zbl 0901.65012 · doi:10.1090/S0002-9939-98-04454-2
[66]Lasserre, J. B.: Integration and homogeneous functions, Proc. am. Math. soc. 127, No. 3, 813-818 (1999) · Zbl 0913.65015 · doi:10.1090/S0002-9939-99-04930-8
[67]Sukumar, N.; Srolovitz, D. J.; Baker, T. J.; Prévost, J. -H.: Brittle fracture in polycrystalline microstructures with the extended finite element method, Int. J. Numer. methods engrg. 56, No. 14, 2015-2037 (2003) · Zbl 1038.74652 · doi:10.1002/nme.653
[68]Simone, A.; Duarte, C. A.; Van Der Giessen, E.: A generalized finite element method for polycrystals with discontinuous grain boundaries, Int. J. Numer. methods engrg. 67, No. 8, 1122-1145 (2006) · Zbl 1113.74076 · doi:10.1002/nme.1658
[69]Dunavant, D.: High degree efficient symmetrical Gaussian quadrature rules for the triangle, Int. J. Numer. methods engrg. 21, 1129-1148 (1985) · Zbl 0589.65021 · doi:10.1002/nme.1620210612
[70]A. Haegemans, Cubature formulas for triangles and squares with a 1/r singularity, Tech. Rep. TW 192, Department of Computer Science, K.U. Leuven, Belgium, 1993.
[71]Lyness, J. N.: On handling singularities in finite elements, Numerical integration, recent developments, software and applications, NATO ASI series C: Mathematical and physical sciences 357, 219-233 (1992) · Zbl 0762.65010
[72]Sukumar, N.; Srolovitz, D. J.: Finite element-based model for crack propagation in polycrystalline materials, Comput. appl. Math. 23, 363-380 (2004) · Zbl 1213.74269 · doi:10.1590/S0101-82052004000200014 · doi:http://www.scielo.br/scielo.php?script=sci_abstract&pid=S1807-03022004000200014&lng=en&nrm=iso&tlng=en
[73]Aliabadi, M. H.; Rooke, D. P.; Cartwright, D. J.: Mixed-mode bueckner weight functions using boundary element analysis, Int. J. Fract. 34, 131-147 (1987)
[74]Stazi, F. L.; Budyn, E.; Chessa, J.; Belytschko, T.: An extended finite element method with higher-order elements for curved cracks, Comput. mech. 31, 38-48 (2003) · Zbl 1038.74651 · doi:10.1007/s00466-002-0391-2
[75]Sutradhar, A.; Paulino, G. H.; Gray, L. J.: Symmetric Galerkin boundary element method, (2008)
[76]Sukumar, N.; Chopp, D. L.; Moës, N.; Belytschko, T.: Modeling holes and inclusions by level sets in the extended finite-element method, Comput. methods appl. Mech. engrg. 190, No. 46 – 47, 6183-6200 (2001) · Zbl 1029.74049 · doi:10.1016/S0045-7825(01)00215-8
[77]Moës, N.; Cloirec, M.; Cartraud, P.; Remacle, J. -F.: A computational approach to handle complex microstructure geometries, Comput. methods appl. Mech. engrg. 192, 3163-3177 (2003)
[78]Chessa, J.; Belytschko, T.: An enriched finite element method and level sets for axisymmetric two-phase flow with surface tension, Int. J. Numer. methods engrg. 58, 2041-2064 (2003) · Zbl 1032.76591 · doi:10.1002/nme.946
[79]Legay, A.; Wang, H. W.; Belytschko, T.: Strong and weak arbitrary discontinuities in spectral finite elements, Int. J. Numer. methods engrg. 64, 991-1008 (2005) · Zbl 1167.74045 · doi:10.1002/nme.1388
[80]Zlotnik, S.; Díez, P.: Hierarchical X-FEM for n-phase flow (n>2), Comput. methods appl. Mech. engrg. 198, 2329-2338 (2009) · Zbl 1229.76060 · doi:10.1016/j.cma.2009.02.025