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Polygonal finite element methods for contact-impact problems on non-conformal meshes. (English) Zbl 1296.74105

Summary: A polygonal finite element method is presented for large deformation frictionless dynamic contact-impact problems with non-conformal meshes. The geometry and interfaces of the problem are modeled independent of the background mesh based on the level set method to produce polygonal elements at the intersection of the interface with the regular FE mesh. Various polygonal shape functions are employed to investigate the capability of polygonal-FEM technique in modeling frictionless contact-impact problems. The contact constraints are imposed between polygonal elements produced along the contact surface through the node-to-surface contact algorithm. Several contact-impact problems are modeled using various polygonal interpolation functions, including the Wachspress interpolation functions, the metric coordinate shape functions, the natural neighbor based functions, and the mean value coordinate functions to demonstrate the efficiency of proposed technique in modeling contact-impact problems.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
74M20 Impact in solid mechanics
74M15 Contact in solid mechanics
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[1] Strouboulis, T.; Copps, K.; Babuska, I., The generalized finite element method, Computer Methods in Applied Mechanics and Engineering, 190, 4081-4193 (2001) · Zbl 0997.74069
[2] Duarte, C. A.; Hamzeh, O. N.; Liszka, T. J.; Tworzydlo, W. W., A generalized finite element method for the simulation of three-dimensional dynamic crack propagation, Computer Methods in Applied Mechanics and Engineering, 190, 2227-2262 (2001) · Zbl 1047.74056
[3] Belytschko, T.; Black, T., Elastic crack growth in finite elements with minimal remeshing, International Journal for Numerical Methods in Engineering, 45, 601-620 (1999) · Zbl 0943.74061
[4] Daux, C.; Moës, N.; Dolbow, J.; Sukumar, N.; Belytschko, T., Arbitrary branched and intersecting cracks with the extended finite element method, International Journal for Numerical Methods in Engineering, 48, 1741-1760 (2000) · Zbl 0989.74066
[6] Leveque, R. J.; Li, Z., The immersed interface method for elliptic equations with discontinous coefficients and sinhular sources, SIAM Journal of Numerical Analysis, 31, 1019-1044 (1994) · Zbl 0811.65083
[7] Noble, D. R.; Newren, E. P.; Lechman, J. B., A conformal decomposition finite element method for modeling stationary fluid interface problems, International Journal for Numerical Methods in Fluids, 63, 725-742 (2010) · Zbl 1423.76266
[8] Li, Z.; Lin, T.; Wu, X., New Cartesian grid methods for interface problems using the finite element formulation, Numerische Mathematik, 96, 61-98 (2003) · Zbl 1055.65130
[9] Sanches, R. A.K.; Bornemann, P. B.; Cirak, F., Immersed b-spline (i-spline) finite element method for geometrically complex domains, Computer Methods in Applied Mechanics and Engineering, 200, 1432-1445 (2011) · Zbl 1228.74097
[10] Tabarraei, A.; Sukumar, N., Conforming polygonal finite elements, International Journal for Numerical Methods in Engineering, 61, 2045-2066 (2004) · Zbl 1073.65563
[11] Biabanaki, S. O.R.; Khoei, A. R., A polygonal finite element method for modeling arbitrary interfaces in large deformation problems, Computational Mechanic, 50, 19-33 (2012) · Zbl 1312.74032
[12] Wachspress, E. L., A Rational Finite Element Basis (1975), Academic Press: Academic Press New York · Zbl 0322.65001
[13] Meyer, M.; Lee, H.; Barr, A. H.; Desbrun, M., Generalized barycentric coordinates for irregular \(n\)-gons, Journal of Graphics Tools, 7, 13-22 (2002)
[14] Floater, M. S., Mean value coordinates, Computer Aided Geometric Design, 20, 19-27 (2003) · Zbl 1069.65553
[15] Sibson, R., A vector identity for the Dirichlet tessellation, Mathematical Proceedings of the Cambridge Philosophical Society, 87, 151-155 (1980) · Zbl 0466.52010
[16] Malsch, E. A.; Lin, J. J.; Dasgupta, G., Smooth two dimensional interpolants: a receptor all polygons, Journal of Graphics Tools, 10, 27-39 (2005)
[17] Sukumar, N., Construction of polygonal interpolants: a maximum entropy approach, International Journal for Numerical Methods in Engineering, 61, 2159-2181 (2004) · Zbl 1073.65505
[18] Sukumar, N.; Malsch, E. A., Recent advances in the construction of polygonal finite element interpolants, Archives of Computational Methods in Engineering, 13, 129-163 (2006) · Zbl 1101.65108
[19] Zuppa, C.; Simonetti, G., Building shape functions on convex polyhedra using MLS interpolants and rational weights, Mecanina Computacional, XXI, 1471-1483 (2002)
[20] Wicke, M.; Botsch, M.; Gross, M., A finite element method on convex polyhedra, Computer Graphics Forum, 26, 355-364 (2007)
[21] Martin, S.; Kaufmann, P.; Botsch, M.; Wicke, M.; Gross, M., Polyhedral finite elements using harmonic basis functions, Eurographics Symposium on Geometry Processing, 27, 1521-1529 (2008)
[24] Gillette, A.; Rand, A.; Bajaj, C., Error estimates for generalized barycentric interpolation, Advances in Computational Mathematics, 37, 417-439 (2012) · Zbl 1259.65013
[25] Rand, A.; Gillette, A.; Bajaj, C., Quadratic serendipity finite elements on polygons using generalized barycentric coordinates, Mathematics of computations (2011), http://www.ams.org/cgi-bin/mstrack/accepted_papers/mcom · Zbl 1300.65091
[26] Cueto, E.; Sukumar, N.; Calvo, B.; Martínez, M. A.; Cegonino, J.; Doblaré, M., Overview and recent advances in natural neighbor Galerkin methods, Archives of Computational Methods in Engineering, 10, 307-384 (2003) · Zbl 1050.74001
[27] Dasgupta, G., Interpolants within convex polygons: Wachspress shape functions, Journal of Aerospace Engineering, 16, 1-8 (2003)
[28] Dasgupta, G.; Wachspress, E., The adjoint for an algebraic finite element, Computers and Mathematics with Application, 55, 1988-1997 (2008) · Zbl 1183.65143
[29] Wachspress, E., Rational bases for convex polyhedral, Computers and Mathematics with Applications, 59, 1953-1956 (2010) · Zbl 1189.52014
[30] Sukumar, N.; Moran, B.; Belytschko, T., The natural element method in solid mechanics, International Journal for Numerical Methods in Engineering, 43, 839-887 (1998) · Zbl 0940.74078
[31] Hiyoshi, H.; Sugihara, K., Two generalizations of an interpolant based on Voronoi diagrams, International Journal of Shape Modeling, 2, 219-231 (1999)
[32] Pinkall, U.; Polthier, K., Computing discrete minimal surfaces and their conjugates, Experimental Mathematics, 2, 15-36 (1993) · Zbl 0799.53008
[33] Rashid, M. M.; Gullett, P. M., On a finite element method with variable element topology, Computer Methods in Applied Mechanics and Engineering, 190, 1509-1527 (2000) · Zbl 1005.74071
[34] Dasgupta, G., Integration within polygonal finite elements, ASCE Journal of Aerospace Engineering, 16, 9-18 (2003)
[35] Lyness, J. N.; Monegato, G., Quadrature rules for regions having regular hexagonal symmetry, SIAM Journal on Numerical Analysis, 14, 283-295 (1977) · Zbl 0365.65014
[36] Nooijen, M.; Velde, G. T.; Baerends, E. J., Symmetric numerical integration formulas for regular polygons, SIAM Journal on Numerical Analysis, 27, 198-218 (1990) · Zbl 0691.65007
[37] Xiao, H.; Gimbutas, Z. A., Numerical algorithm for the construction of efficient quadrature rules in two and higher dimensions, Computers and Mathematics with Applications, 59, 663-676 (2010) · Zbl 1189.65047
[38] Mousavi, S. E.; Xiao, H.; Sukumar, N., Generalized Gaussian quadrature rules on arbitrary polygons, International Journal for Numerical Methods in Engineering, 82, 99-113 (2010) · Zbl 1183.65026
[39] Betsch, P.; Uhlar, S., Energy-momentum conserving integration of multibody dynamics, Multibody System Dynamics, 17, 243-289 (2007) · Zbl 1112.70003
[40] Hesch, C.; Betsch, P., Transient three-dimensional domain decomposition problems: frame-indifferent mortar constraints and conserving integration, International Journal for Numerical Methods in Engineering, 82, 329-358 (2010) · Zbl 1188.74058
[41] Hesch, C.; Betsch, P., Transient three-dimensional contact problems - mortar method: mixed methods and conserving integration, Computational Mechanics, 48, 461-475 (2011) · Zbl 1398.74335
[42] Hesch, C.; Betsch, P., Transient three-dimensional contact problems - NTS method: mixed methods and conserving integration, Computational Mechanics, 48, 437-449 (2011) · Zbl 1352.74481
[43] Crisfield, M. A., Non-Linear Finite Element Analysis of Solids and Structures. Non-Linear Finite Element Analysis of Solids and Structures, Essentials, vol 1 (1991), Wiley · Zbl 0809.73005
[44] Crisfield, M. A., Non-Linear Finite Element Analysis of Solids and Structures. Non-Linear Finite Element Analysis of Solids and Structures, Advanced Topics, vol 2 (1997), Wiley · Zbl 0890.73001
[45] Laursen, T. A., Computational Contact and Impact Mechanics (2002), Springer: Springer Berlin · Zbl 0996.74003
[46] Wriggers, P., Computational Contact Mechanics (2006), Springer · Zbl 1104.74002
[47] Wriggers, P.; Simo, J. C., A note on tangent stiffnesses for fully nonlinear contact problems, Communications in Applied Numerical Methods, 1, 199-203 (1985) · Zbl 0582.73110
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