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Multi-field spacetime discontinuous Galerkin methods for linearized elastodynamics. (English) Zbl 1231.74430
Summary: We extend the single-field spacetime discontinuous Galerkin (SDG) method for linearized elastodynamics of R. Abedi [Comput. Methods Appl. Mech. Eng. 195, No. 25–28, 3247–3273 (2006; Zbl 1130.74044)] to multi-field versions. A three-field method, in displacement, velocity and strain, is derived by invoking a Bubnov-Galerkin weighted residuals procedure on the system of spacetime field equations and the corresponding jump conditions. A two-field formulation, in displacement and velocity, and the one-field displacement formulation of [loc. cit.] are obtained from the three-field model through strong enforcement of kinematic compatibility relations. All of these formulations balance linear and angular momentum at the element level, and we prove that they are energy-dissipative and unconditionally stable. As in [loc. cit.], we implement the SDG models using a causal, advancing-front meshing procedure that enables a patch-by-patch solution procedure with linear complexity in the number of spacetime elements. Numerical results show that the three-field formulation is most efficient, wherein all interpolated fields converge at the optimal, O(h p+1 ), rate. For a given mesh size, the three-field model delivers error values that are more than an order of magnitude smaller than those of the one- andtwo-field models. The three-field formulation’s efficiency is also superior, independent of whether the comparison is based on matching polynomial orders or matching convergence rates.
MSC:
74S05Finite element methods in solid mechanics
74B15Equations linearized about a deformed state (small deformations superposed on large)
74H15Numerical approximation of solutions for dynamical problems in solid mechanics
References:
[1]Abedi, R.; Petracovici, B.; Haber, R. B.: A spacetime discontinuous Galerkin method for linearized elastodynamics with element-wise momentum balance, Comput. methods appl. Mech. engrg. 195, 3247-3273 (2006) · Zbl 1130.74044 · doi:10.1016/j.cma.2005.06.013
[2]Abedi, R.; Haber, R. B.; Thite, S.; Erickson, J.: An h-adaptive spacetime-discontinuous Galerkin method for linear elastodynamics, Revue européenne de mécanique numérique 15, 619-642 (2006) · Zbl 1208.74143 · doi:10.3166/remn.15.619-642
[3]Palaniappan, J.; Haber, R. B.; Jerrard, R. J.: A spacetime discontinuous Galerkin method for scalar conservation laws, Comput. methods appl. Mech. engrg. 193, 3607-3631 (2004) · Zbl 1077.65108 · doi:10.1016/j.cma.2004.01.028
[4]Miller, S. T.; Haber, R. B.: A spacetime discontinuous Galerkin method for hyperbolic heat conduction, Comput. methods appl. Mech. engrg. 198, 194-209 (2008) · Zbl 1194.80118 · doi:10.1016/j.cma.2008.07.016
[5]Üngör, A.; Sheffer, A.: Pitching tents in spacetime: mesh generation for discontinuous Galerkin method, Int. J. Foundations comput. Sci. 13, 201-221 (2002) · Zbl 1066.65138 · doi:10.1142/S0129054102001059
[6]Abedi, R.; Chung, S. -H.; Erickson, J.; Fan, Y.; Garland, M.; Guoy, D.; Haber, R.; Sullivan, J.; Thite, S.; Zhou, Y.: Space – time meshing with adaptive refinement and coarsening, Proceedings 20th annual ACM symposium on computational geometry, 300-309 (2004)
[7]B. Rivière, M. Wheeler, Discontinuous finite element methods for acoustic and elastic wave problems: Part I: semidiscrete error estimates, Tech. Rep. 01-02, TICAM, University of Texas, Austin, TX, 2001.
[8]Grote, M. J.; Schneebeli, A.; Schötzau, D.: Discontinuous Galerkin finite element method for the wave equations, SIAM J. Numer. anal. 44, No. 6, 2408-2431 (2006) · Zbl 1129.65065 · doi:10.1137/05063194X
[9]Romero, I.: On the stability and convergence of fully discrete solutions in linear elastodynamics, Comput. methods appl. Mech. engrg. 191, 3857-3882 (2002) · Zbl 1039.74050 · doi:10.1016/S0045-7825(02)00320-1
[10]Ainsworth, M.; Monk, P.; Muniz, W.: Dispersive and dissipative properties of discontinuous Galerkin finite element methods for the second-order wave equation, J. sci. Comput. 27, No. 1 – 3, 5-40 (2006) · Zbl 1102.76032 · doi:10.1007/s10915-005-9044-x
[11]Hu, F. Q.; Hussaini, M. Y.; Rasetarinera, P.: An analysis of the discontinuous Galerkin method for wave propagation problems, J. comput. Phys. 151, 921-946 (1999) · Zbl 0933.65113 · doi:10.1006/jcph.1999.6227
[12]Dumbser, M.; Käser, M.: An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes – ii. The three-dimensional isotropic case, Geophys. J. Int. 167, 319-336 (2006)
[13]Borri, M.; Bottasso, C.: A general framework for interpreting time finite element formulations, Comput. mech. 13, 133-142 (1993) · Zbl 0789.70003 · doi:10.1007/BF00370131
[14]Li, X.; Yao, D.; Lewis, R. W.: A discontinuous Galerkin finite element method for dynamic and wave propagation problems in non-linear solids and saturated posous media, Int. J. Numer. methods engrg. 57, 1775-1800 (2000) · Zbl 1062.74621 · doi:10.1002/nme.741
[15]Chien, C. -C.; Wu, T. -Y.: An improved predictor/multi-corrector algorithm for a time-discontinuous Galerkin finite element method in structural dynamics, Comput. mech. 25, 430-437 (2000) · Zbl 0976.74063 · doi:10.1007/s004660050490
[16]Chien, C. C.; Wang, C. S.; Tang, J. H.: Three-dimensional transient elastodynamic analysis by a space and time-discontinuous Galerkin finite element method, Finite elem. Anal. des. 39, 561-580 (2003)
[17]Hughes, T. J. R.; Hulbert, G. M.: Space – time finite element methods for elastodynamics: formulations and error estimates, Comput. methods appl. Mech. engrg. 66, 339-363 (1988) · Zbl 0616.73063 · doi:10.1016/0045-7825(88)90006-0
[18]Hulbert, G. M.; Hughes, T. J. R.: Space – time finite element methods for second-order hyperbolic equations, Comput. methods appl. Mech. engrg. 84, 327-348 (1990) · Zbl 0754.73085 · doi:10.1016/0045-7825(90)90082-W
[19]Johnson, C.: Discontinuous Galerkin finite element methods for second order hyperbolic problems, Comput. methods appl. Mech. engrg. 107, 117-129 (1993) · Zbl 0787.65070 · doi:10.1016/0045-7825(93)90170-3
[20]French, D. A.: A space – time finite element method for the wave equation, Comput. methods appl. Mech. engrg. 107, 145-157 (1993) · Zbl 0787.65069 · doi:10.1016/0045-7825(93)90172-T
[21]Huang, H.; Costanzo, F.: On the use of space – time finite elements in the solution of elasto-dynamic problems with strain discontinuities, Comput. methods appl. Mech. engrg. 191, 5315-5343 (2002) · Zbl 1083.74587 · doi:10.1016/S0045-7825(02)00460-7
[22]Costanzo, F.; Huang, H.: Proof of unconditional stability for a single-field discontinuous Galerkin finite element formulation for linear elasto-dynamics, Comput. methods appl. Mech. engrg. 194, 2059-2076 (2005) · Zbl 1095.74032 · doi:10.1016/j.cma.2004.07.011
[23]Wiberg, N. -E.; Zeng, L.; Li, X.: Error estimation and adaptivity in elastodynamics, Comput. methods appl. Mech. engrg. 101, 369-395 (1992) · Zbl 0802.73076 · doi:10.1016/0045-7825(92)90030-N
[24]Li, X. D.; Wiberg, N. E.: Structural dynamic analysis by a time-discontinuous Galerkin finite element method, Comput. methods appl. Mech. engrg. 39, No. 12, 2131-2152 (1996) · Zbl 0885.73081 · doi:10.1002/(SICI)1097-0207(19960630)39:12<2131::AID-NME947>3.0.CO;2-Z
[25]Li, X. D.; Wiberg, N. E.: Implementation and adaptivity of a space – time finite element method for structural dynamics, Comput. methods appl. Mech. engrg. 156, 211-229 (1998) · Zbl 0960.74065 · doi:10.1016/S0045-7825(97)00207-7
[26]Idesman, A. V.; Schmidt, M.; Sierakowski, R. L.: A new explicit predictor – multicorrector high-order accurate method for linear elastodynamics, J. sound vib. 310, 217-229 (2008)
[27]Richter, G. R.: An explicit finite element method for the wave equation, Appl. numer. Math. 16, 65-80 (1994) · Zbl 0816.65062 · doi:10.1016/0168-9274(94)00048-4
[28]Falk, R. S.; Richter, G. R.: Explicit finite element methods for symmetric hyperbolic equations, SIAM J. Numer. anal. 36, 935-952 (1999) · Zbl 0923.65065 · doi:10.1137/S0036142997329463
[29]Yin, L.; Acharia, A.; Sobh, N.; Haber, R. B.; Tortorelli, D. A.: A spacetime discontinuous Galerkin method for elastodynamics analysis, Discontinuous Galerkin methods: theory, computation and applications, 459-464 (2000) · Zbl 1041.74553
[30]L. Yin, A new spacetime discontinuous Galerkin finite element method for elastodynamics analysis., Ph.D. thesis, University of Illinois at Urbana-Champaign, 2002.
[31]Monk, P.; Richter, G. R.: A discontinuous Galerkin method for linear symmetric hyperbolic systems in inhomogeneous media, J. sci. Comput. 22 – 23, 443-447 (2005) · Zbl 1082.65099 · doi:10.1007/s10915-004-4132-5
[32]Petersen, S.; Farhat, C.; Tezaur, R.: A space – time discontinuous Galerkin method for the solution of the wave equation in the time domain, Int. J. Numer. methods engrg. (2008)
[33]Spivak, M.: Calculus on manifolds, (1965) · Zbl 0141.05403
[34]Fleming, W.: Functions of several variables, (1977)
[35]Abraham, R.; Marsden, J. E.; Ratiu, T.: Manifolds, tensor analysis and applications, (1988) · Zbl 0875.58002
[36]Arnold, V. I.: Mathematical methods of classical mechanics, (1989)
[37]Ambrosio, L.; Fusco, N.; Pallara, D.: Functions of bounded variation and free discontinuity problems, (2000) · Zbl 0957.49001
[38]B. Petracovici, Analysis of a spacetime discontinuous Galerkin method for elastodynamics., Ph.D. thesis, University of Illinois at Urbana-Champaign, 2004.
[39]Palaniappan, J.; Miller, S. T.; Haber, R. B.: Sub-cell shock capturing and spacetime discontinuity tracking for nonlinear conservation laws, Int. J. Numer. methods fluids 57, 1115-1135 (2008)
[40]Erickson, J.; Guoy, D.; Sullivan, J.; Üngör, A.: Building spacetime meshes over arbitrary spatial domains, Proceedings of the 11th international meshing roundtable (Sandia national laboratories), 391-402 (2002)
[41]Üngör, A.; Sheffer, A.: Pitching tents in spacetime: mesh generation for discontinuous Galerkin method, Int. J. Foundations comput. Sci. 13, No. 2, 201-221 (2002) · Zbl 1066.65138 · doi:10.1142/S0129054102001059
[42]Shewchuk, J. R.: Triangle: engineering a 2d quality mesh generator and Delaunay triangulator, Lecture notes in computer science 1148, 203-222 (1996)