A simple, stable, and accurate linear tetrahedral finite element for transient, nearly, and fully incompressible solid dynamics: a dynamic variational multiscale approach. (English) Zbl 1352.74434
Summary: We propose a new approach for the stabilization of linear tetrahedral finite elements in the case of nearly incompressible transient solid dynamics computations. Our method is based on a mixed formulation, in which the momentum equation is complemented by a rate equation for the evolution of the pressure field, approximated with piecewise linear, continuous finite element functions. The pressure equation is stabilized to prevent spurious pressure oscillations in computations. Incidentally, it is also shown that many stabilized methods previously developed for the static case do not generalize easily to transient dynamics. Extensive tests in the context of linear and nonlinear elasticity are used to corroborate the claim that the proposed method is robust, stable, and accurate.
MSC:
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 74B05 | Classical linear elasticity |
| 74B20 | Nonlinear elasticity |
Keywords:
tetrahedral finite element; piecewise linear interpolation; stabilized method; transient dynamics; nearly incompressible elasticityReferences:
| [1] | HughesTJR. The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Prentice‐Hall: Englewood Cliffs, New Jersey, 1987. (Dover reprint, 2000). · Zbl 0634.73056 |
| [2] | HughesTJR. Equivalence of finite elements for nearly incompressible elasticity. Journal of Applied Mechanics, Transaction ASME1977; 44(1):181-183. |
| [3] | HughesTJR. Generalization of selective integration procedure to anisotropic and nonlinear media. International Journal of Numerical Methods in Engineering1980; 15(1):1413-1418. · Zbl 0437.73053 |
| [4] | de Souza NetoEA, Andrade PiresFM, OwenDRJ. F‐bar‐based linear triangles and tetrahedra for finite strain analysis of nearly incompressible solids. Part I: formulation and benchmarking. International Journal of Numerical Methods in Engineering2005; 62(3):353-383. · Zbl 1179.74159 |
| [5] | de Souza NetoEA, PerićD, HuangGC, OwenDRJ. Remarks on the stability of enhanced strain elements in finite elasticity and elastocplasticity. Communications in Numerical Methods in Engineering1995; 11(11):951-961. · Zbl 0836.73066 |
| [6] | De Souza NetoEA, PerićD, DutkoM, OwenDRJ. Design of simple low order finite elements for large strain analysis of nearly incompressible solids. International Journal of Solids and Structures1996; 33(20-22):3277-3296. · Zbl 0929.74102 |
| [7] | NagtegaalJC, ParkDM, RiceJR. On numerically accurate finite element solutions in the fully plastic range. Computer Methods in Applied Mechanics and Engineering1974; 4:53-177. · Zbl 0284.73048 |
| [8] | BabuškaI. Error‐bounds for finite element method. Numerische Mathematik1971; 16(4):322-333. · Zbl 0214.42001 |
| [9] | BrezziF, FortinM. Mixed and Hybrid Finite Element Methods (1st edn), Springer Series in Computational Mathematics, vol. 15. Springer: New York, 1991. · Zbl 0788.73002 |
| [10] | ErnA, GuermondJL. Theory and Practice of Finite Elements, Applied Mathematical Sciences, vol. 159. Springer: New York, 2004. · Zbl 1059.65103 |
| [11] | QianJ, ZhangY. Automatic all‐hexahedral mesh generation from B‐Reps for non‐manifold CAD assemblies. Engineering with Computers2012; 28:345-359. |
| [12] | MalkusDS, HughesTJR. Mixed finite element methods – reduced and selective integration techniques: a unification of concepts. Computer Methods in Applied Mechanics and Engineering1978; 15(1):68-81. · Zbl 0381.73075 |
| [13] | HughesTJR, FrancaLP, BalestraM. A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuška-Brezzi condition: a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal‐order interpolations. Computer Methods in Applied Mechanics and Engineering1986 November; 59(1):85-99. · Zbl 0622.76077 |
| [14] | HughesThomasJR, FrancaLP. A new finite element formulation for computational fluid dynamics: VII. The Stokes problem with various well‐posed boundary conditions: symmetric formulations that converge for all velocity/pressure spaces. Computer Methods in Applied Mechanics and Engineering1987; 65(1):85-96. · Zbl 0635.76067 |
| [15] | FrancaLP, HughesTJR. Two classes of mixed finite element methods. Computer Methods in Applied Mechanics and Engineering1988; 69(1):89-129. · Zbl 0629.73053 |
| [16] | FrancaLP, HughesTJR, LoulaAFD, MirandaI. A new family of stable elements for nearly incompressible elasticity based on a mixed Petrov-Galerkin finite element formulation. NM1988; 53(1-2):123-141. · Zbl 0656.73036 |
| [17] | KlaasO, ManiattyA, ShephardMS. A stabilized mixed finite element method for finite elasticity. Formulation for linear displacement and pressure interpolation. Computer Methods in Applied Mechanics and Engineering1999; 180(1-2):65-79. · Zbl 0959.74066 |
| [18] | ManiattyAM, LiuY, KlaasO, ShephardMS. Stabilized finite element method for viscoplastic flow: formulation and a simple progressive solution strategy. Computer Methods in Applied Mechanics and Engineering2001; 190(35-36):4609-4625. · Zbl 1059.74056 |
| [19] | ManiattyAM, LiuY, KlaasO, ShephardMS. Higher order stabilized finite element method for hyperelastic finite deformation. Computer Methods in Applied Mechanics and Engineering2002; 191(13-14):1491-1503. · Zbl 1098.74704 |
| [20] | ManiattyAM, LiuY. Stabilized finite element method for viscoplastic flow: formulation with state variable evolution. International Journal of Numerical Methods in Engineering2003; 56(2):185-209. · Zbl 1116.74431 |
| [21] | RameshB, ManiattyAM. Stabilized finite element formulation for elastic-plastic finite deformations. Computer Methods in Applied Mechanics and Engineering2005; 194(6-8):775-800. · Zbl 1112.74536 |
| [22] | MasudA, XiaK. A stabilized mixed finite element method for nearly incompressible elasticity. Journal of Applied Mechanics2005 January; 72(5):711-720. · Zbl 1111.74548 |
| [23] | NakshatralaKB, MasudA, HjelmstadKD. On finite element formulations for nearly incompressible linear elasticity. Computational Mechanics2008; 41(4):547-561. · Zbl 1162.74472 |
| [24] | XiaK, MasudA. A stabilized finite element formulation for finite deformation elastoplasticity in geomechanics. Computers and Geotechnics2009; 36(3):396-405. |
| [25] | MasudA, TrusterTJ. A framework for residual‐based stabilization of incompressible finite elasticity: stabilized formulations and methods for linear triangles and tetrahedra. Computer Methods in Applied Mechanics and Engineering2013; 267:359-399. · Zbl 1286.74018 |
| [26] | HughesTJR. Multiscale phenomena: Green’s functions, the Dirichlet‐to‐Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods. Computer Methods in Applied Mechanics and Engineering1995; 127(1-4):387-401. · Zbl 0866.76044 |
| [27] | HughesTJR, FeijóoGR, MazzeiL, QuincyJB. The variational multiscale method – a paradigm for computational mechanics. Computer Methods in Applied Mechanics and Engineering1998; 166(1-2):3-24. · Zbl 1017.65525 |
| [28] | HughesTJR, ScovazziG, FrancaLP. Multiscale and stabilized methods. In Encyclopedia of Computational Mechanics, SteinE (ed.), de BorstR (ed.), HughesTJR (ed.) (eds).John Wiley & Sons, 2004. |
| [29] | ChiumentiM, ValverdeQ, de SaracibarCA, CerveraM. A stabilized formulation for incompressible elasticity using linear displacement and pressure interpolations. Computer Methods in Applied Mechanics and Engineering2002; 191(46):5253-5264. · Zbl 1083.74584 |
| [30] | CerveraM, ChiumentiM, ValverdeQ, Agelet de SaracibarC. Mixed linear/linear simplicial elements for incompressible elasticity and plasticity. Computer Methods in Applied Mechanics and Engineering2003; 192(49-50):5249-5263. · Zbl 1054.74050 |
| [31] | ChiumentiM, ValverdeQ, Agelet de SaracibarC, CerveraM. A stabilized formulation for incompressible plasticity using linear triangles and tetrahedra. International Journal Plasticity2004; 20(8-9):1487-1504. · Zbl 1066.74587 |
| [32] | CodinaR. Stabilization of incompressibility and convection through orthogonal sub‐scales in finite element methods. Computer Methods in Applied Mechanics and Engineering2000; 190(13):1579-1599. · Zbl 0998.76047 |
| [33] | BonetJ, MarriottH, HassanO. Stability and comparison of different linear tetrahedral formulations for nearly incompressible explicit dynamic applications. International Journal of Numerical Methods in Engineering2001; 50(1):119-133. · Zbl 1082.74547 |
| [34] | BonetJ, BurtonAJ. A simple average nodal pressure tetrahedral element for incompressible and nearly incompressible dynamic explicit applications. Communications in Numerical Methods in Engineering1998; 14(5):437-449. · Zbl 0906.73060 |
| [35] | PusoMA, SolbergJ. A stabilized nodally integrated tetrahedral. International Journal of Numerical Methods in Engineering2006; 67(6):841-867. · Zbl 1113.74075 |
| [36] | DohrmannCR, HeinsteinMW, JungJ, KeySW, WitkowskiWR. Node‐based uniform strain elements for three‐node triangular and four‐node tetrahedral meshes. International Journal of Numerical Methods in Engineering2000; 47(9):1549-1568. · Zbl 0989.74067 |
| [37] | GeeMW, DohrmannCR, KeySW, WallWA. A uniform nodal strain tetrahedron with isochoric stabilization. International Journal of Numerical Methods in Engineering2009; 78(4):429-443. · Zbl 1183.74275 |
| [38] | StenbergR. A family of mixed finite elements for the elasticity problem. NM1988; 53(5):513-538. · Zbl 0632.73063 |
| [39] | CerveraM, ChiumentiM, CodinaR. Mixed stabilized finite element methods in nonlinear solid mechanics. Part I: formulation. Computer Methods in Applied Mechanics and Engineering2010; 199(37-40):2559-2570. · Zbl 1231.74404 |
| [40] | CerveraM, ChiumentiM, CodinaR. Mixed stabilized finite element methods in nonlinear solid mechanics. Part II: strain localization.Computer Methods in Applied Mechanics and Engineering2010; 199(37-40):2571-2589. · Zbl 1231.74405 |
| [41] | CerveraM, ChiumentiM, BenedettiL, CodinaR. Mixed stabilized finite element methods in nonlinear solid mechanics. Part III: compressible and incompressible plasticity. Computer Methods in Applied Mechanics and Engineering2015; 285:752-775. · Zbl 1423.74149 |
| [42] | ChiumentiM, CerveraM, CodinaR. A mixed three‐field FE formulation for stress accurate analysis including the incompressible limit. Computer Methods in Applied Mechanics and Engineering2013. |
| [43] | ChiumentiM, CerveraM, CodinaR. A mixed three‐field FE formulation for stress accurate analysis including the incompressible limit. Computer Methods in Applied Mechanics and Engineering2015; 283:1095-1116. · Zbl 1423.74871 |
| [44] | LafontaineNM, RossiR, CerveraM, ChiumentiM. Explicit mixed strain‐displacement finite element for dynamic geometrically non‐linear solid mechanics. Computational Mechanics2015; 55(3):543-559. · Zbl 1311.74028 |
| [45] | AguirreM, GilAJ, BonetJ, CarreñoAA. A vertex centred finite volume Jameson-Schmidt-Turkel (JST) algorithm for a mixed conservation formulation in solid dynamics. Journal of Computational Physics2014; 259:672-699. · Zbl 1349.74341 |
| [46] | GilAJ, OrtigosaR, BonetJ. A computational framework for polyconvex large strain electro‐mechanics. Submitted to Journal of the Mechanics and Physics of Solids2015. |
| [47] | LeeCH, GilAJ, BonetJ. Development of a stabilised Petrov-Galerkin formulation for conservation laws in Lagrangian fast solid dynamics. Computer Methods in Applied Mechanics and Engineering2014; 268:40-64. · Zbl 1295.74099 |
| [48] | LeeCH, GilAJ, BonetJ. Development of a cell centred upwind finite volume algorithm for a new conservation law formulation in structural dynamics. Computers and Structures2013; 118:13-38. |
| [49] | TrangensteinJA. Second‐order Godunov algorithm for two‐dimensional solid mechanics. Computational Mechanics1994; 13:343-359. · Zbl 0793.73102 |
| [50] | PemberRB, TrangensteinJA. Numerical algorithms for strong discontinuities in elastic-plastic solids. Journal of Computational Physics1992; 103(1):63-89. · Zbl 0779.73080 |
| [51] | ColellaP, TrangensteinJ. A higher‐order Godunov method for modeling finite deformation in elastic-plastic solids. Communications on Pure and Applied Mathematics1991; 44:41-100. · Zbl 0714.73027 |
| [52] | TrangensteinJA. The Riemann problem for longitudinal motion in an elastic-plastic bar. SJSSC1991; 12:180-207. · Zbl 0714.73028 |
| [53] | BonetJ, GilAJ, OrtigosaR. A computational framework for polyconvex large strain elasticity. Computer Methods in Applied Mechanics and Engineering2015; 283:1061-1094. · Zbl 1423.74122 |
| [54] | BonetJ, GilAJ, LeeCH, AguirreM, OrtigosaR. A first order hyperbolic framework for large strain computational solid dynamics. Part I: total Lagrangian isothermal elasticity. Computer Methods in Applied Mechanics and Engineering2015; 283:689-732. · Zbl 1423.74010 |
| [55] | GilAJ, LeeCH, BonetJ, AguirreM. A stabilised Petrov-Galerkin formulation for linear tetrahedral elements in compressible, nearly incompressible and truly incompressible fast dynamics. Computer Methods in Applied Mechanics and Engineering2014; 276:659-690. · Zbl 1423.74883 |
| [56] | ShakibF, HughesTJR. A new finite element formulation for computational fluid dynamics: IX. Fourier analysis of space-time Galerkin/least‐squares algorithms. Computer Methods in Applied Mechanics and Engineering1991; 87:35-58. · Zbl 0760.76051 |
| [57] | ShakibF, HughesTJR, JohanZ. A new finite element formulation for computational fluid dynamics: X. The compressible Euler and Navier-Stokes equations. Computer Methods in Applied Mechanics and Engineering1991; 89:141-219. |
| [58] | HughesTJR, FrancaLP, MalletM. A new finite element formulation for computational fluid dynamics: I. Symmetric forms of the compressible Euler and Navier-Stokes equations and the second law of thermodynamics. Computer Methods in Applied Mechanics and Engineering1986; 54:223-234. · Zbl 0572.76068 |
| [59] | HughesTJR, MalletM, MizukamiA. A new finite element formulation for computational fluid dynamics: II. Beyond SUPG. Computer Methods in Applied Mechanics and Engineering1986; 54:341-355. · Zbl 0622.76074 |
| [60] | HughesTJR, MalletM. A new finite element formulation for computational fluid dynamics: III. The generalized streamline operator for multidimensional advective-diffusive systems. Computer Methods in Applied Mechanics and Engineering1986; 58:305-328. · Zbl 0622.76075 |
| [61] | HughesTJR, MalletM. A new finite element formulation for computational fluid dynamics: IV. A discontinuity‐capturing operator for multidimensional advective-diffusive systems. Computer Methods in Applied Mechanics and Engineering1986; 58:329-336. · Zbl 0587.76120 |
| [62] | HughesTJR, FrancaLP, MalletM. A new finite element formulation for computational fluid dynamics: VI. Convergence analysis of the generalized SUPG formulation for linear time‐dependent multidimensional advective-diffusive systems. Computer Methods in Applied Mechanics and Engineering1987; 63:97-112. · Zbl 0635.76066 |
| [63] | HughesTJR, FrancaLP, HulbertGM. A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/least‐squares method for advective-diffusive equations. Computer Methods in Applied Mechanics and Engineering1989; 73:173-189. · Zbl 0697.76100 |
| [64] | HaukeG. Simple stabilizing matrices for the computation of compressible flows in primitive variables. Computer Methods in Applied Mechanics and Engineering2001; 190:6881-6893. · Zbl 0996.76047 |
| [65] | HaukeG, HughesTJR. A unified approach to compressible and incompressible flows. Computer Methods in Applied Mechanics and Engineering1994; 113:389-396. · Zbl 0845.76040 |
| [66] | HaukeG, HughesTJR. A comparative study of different sets of variables for solving compressible and incompressible flows. Computer Methods in Applied Mechanics and Engineering1998; 153:1-44. · Zbl 0957.76028 |
| [67] | AguirreM, GilAJ, BonetJ, LeeCH. An upwind vertex centred finite volume solver for Lagrangian solid dynamics, 2015. (to appear). · Zbl 1349.74342 |
| [68] | ZienkiewiczOC, RojekJ, TaylorRL, PastorM. Triangles and tetrahedra in explicit dynamic codes for solids. International Journal of Numerical Methods in Engineering1998; 43(3):565-583. · Zbl 0939.74073 |
| [69] | MiraP, PastorM, LiT, LiuX. A new stabilized enhanced strain element with equal order of interpolation for soil consolidation problems. Computer Methods in Applied Mechanics and Engineering2003; 192(37-38):4257-4277. · Zbl 1181.74138 |
| [70] | LiX, HanX, PastorM. An iterative stabilized fractional step algorithm for finite element analysis in saturated soil dynamics. Computer Methods in Applied Mechanics and Engineering2003; 192(35-36):3845-3859. · Zbl 1054.74732 |
| [71] | PastorM, QuecedoM, ZienkiewiczOC. A mixed displacement-pressure formulation for numerical analysis of plastic failure. Computers and Structures1997; 62(1):13-23. · Zbl 0910.73063 |
| [72] | PastorM, ZienkiewiczOC, LiT, XiaoqingL, HuangM. Stabilized finite elements with equal order of interpolation for soil dynamics problems. Archives of Computational Methods in Engineering1999; 6(1):3-33. |
| [73] | GuermondJL, QuartapelleL. On stability and convergence of projection methods based on pressure Poisson equation. International Journal for Numerical Methods in Fluids1998; 26(9):1039-1053. · Zbl 0912.76054 |
| [74] | RojekJ, OñateE, TaylorRL. CBS‐based stabilization in explicit solid dynamics. International Journal of Numerical Methods in Engineering2006; 66(10):1547-1568. · Zbl 1110.74856 |
| [75] | OñateE, RojekJ, TaylorRL, ZienkiewiczOC. Finite calculus formulation for incompressible solids using linear triangles and tetrahedra. International Journal of Numerical Methods in Engineering2004; 59(11):1473-1500. · Zbl 1041.74546 |
| [76] | GuoY, OrtizM, BelytschkoT, RepettoEA. Triangular composite finite elements. International Journal of Numerical Methods in Engineering2000; 47(1-3):287-316. · Zbl 0985.74068 |
| [77] | ThoutireddyP, MolinariJF, RepettoEA, OrtizM. Tetrahedral composite finite elements. International Journal of Numerical Methods in Engineering2002; 53(6):1337-1351. · Zbl 1112.74544 |
| [78] | PiresFMA, de Souza NetoEA, de la Cuesta PadillaJL. An assessment of the average nodal volume formulation for the analysis of nearly incompressible solids under finite strains. Communications in Numerical Methods in Engineering2004; 20(7):569-583. · Zbl 1302.74173 |
| [79] | EyckAT, LewA. Discontinuous Galerkin methods for non‐linear elasticity. International Journal of Numerical Methods in Engineering2006; 67:1204-1243. · Zbl 1113.74068 |
| [80] | EyckAT, CelikerF, LewA. Adaptive stabilization of discontinuous Galerkin methods for nonlinear elasticity: analytical estimates. Computer Methods in Applied Mechanics and Engineering2008; 197(33-40):2989-3000. · Zbl 1194.74390 |
| [81] | EyckAT, CelikerF, LewA. Adaptive stabilization of discontinuous Galerkin methods for nonlinear elasticity: motivation, formulation, and numerical examples. Computer Methods in Applied Mechanics and Engineering2008; 197(45-48):3605-3622. · Zbl 1194.74389 |
| [82] | EyckAT, LewA. An adaptive stabilization strategy for enhanced strain methods in non‐linear elasticity. International Journal of Numerical Methods in Engineering2010; 81(11):1387-1416. · Zbl 1183.74270 |
| [83] | KabariaH, LewAJ, CockburnB. A hybridizable discontinuous Galerkin formulation for non‐linear elasticity. Computer Methods in Applied Mechanics and Engineering2015; 283:303-329. · Zbl 1423.74895 |
| [84] | TaylorRL. A mixed‐enhanced formulation tetrahedral finite elements. International Journal of Numerical Methods in Engineering2000; 47(1-3):205-227. · Zbl 0985.74074 |
| [85] | KasperEP, TaylorRL. A mixed‐enhanced strain method. Part I: geometrically linear problems. Computers and Structures2000; 75:237-250. |
| [86] | KasperEP, TaylorRL. A mixed‐enhanced strain method. Part II: geometrically nonlinear problems. Computers and Structures2000; 75:252-260. |
| [87] | AuricchioF, da VeigaLB, LovadinaC, RealiA. An analysis of some mixed‐enhanced finite element for plane linear elasticity. Computer Methods in Applied Mechanics and Engineering2005; 194(27):2947-2968. · Zbl 1091.74046 |
| [88] | ArnoldDN, WintherR. Mixed finite elements for elasticity. Numerische Mathematik2002; 92(3):401-419. · Zbl 1090.74051 |
| [89] | ArnoldDN, WintherR. Nonconforming mixed elements for elasticity. Mathematical Models and Methods in Applied Sciences2003; 13(3):295-307. · Zbl 1057.74036 |
| [90] | ArnoldDN, FalkRS, WintherR. Differential complexes and stability of finite element methods II: the elasticity complex. Compatible Spatial Discretizations: The IMA Volumes in Mathematics and its Applications2006; 142:47-67. · Zbl 1119.65399 |
| [91] | ArnoldDN, FalkRS, WintherR. Mixed finite element methods for linear elasticity with weakly imposed symmetry. Mathematics of Computation2007; 76:1699-1723. · Zbl 1118.74046 |
| [92] | ArnoldDN, FalkRS, WintherR. Differential complexes and stability of finite element methods I. The de Rham complex. Compatible Spatial Discretizations: The IMA Volumes in Mathematics and its Applications2006; 142:23-46. · Zbl 1119.65398 |
| [93] | ChiH, TalischiC, Lopez‐PamiesO, PaulinoGH. Polygonal finite elements for finite elasticity. International Journal of Numerical Methods in Engineering2015; 101:305-328. · Zbl 1352.74044 |
| [94] | ScovazziG. Lagrangian shock hydrodynamics on tetrahedral meshes: a stable and accurate variational multiscale approach. Journal of Computational Physics2012; 231(24):8029-8069. |
| [95] | AinsworthM, OdenJT. A unified approach to a posteriori error estimation using residual methods. Numerische Mathematik1993; 65:23-50. · Zbl 0797.65080 |
| [96] | AinsworthM, OdenJT. A posteriori error estimation in finite element analysis. Computer Methods in Applied Mechanics and Engineering1997; 142:1-88. · Zbl 0895.76040 |
| [97] | AinsworthM, OdenJT. An a posteriori error estimate for finite‐element approximations of the Navier-Stokes equations. Computer Methods in Applied Mechanics and Engineering1994; 111:185-202. · Zbl 0844.76056 |
| [98] | CookRD. Improved two‐dimensional finite element. Journal of Structural Division‐ASCE1974; 100(9):1851-1863. |
| [99] | ScovazziG, ChristonMA, HughesTJR, ShadidJN. Stabilized shock hydrodynamics: I. A Lagrangian method. Computer Methods in Applied Mechanics and Engineering2007; 196(4-6):923-966. · Zbl 1120.76334 |
| [100] | ScovazziG. Stabilized shock hydrodynamics: II. Design and physical interpretation of the SUPG operator for Lagrangian computations. Computer Methods in Applied Mechanics and Engineering2007; 196(4-6):966-978. · Zbl 1120.76332 |
| [101] | ScovazziG, ShadidJN, LoveE, RiderWJ. A conservative nodal variational multiscale method for Lagrangian shock hydrodynamics. Computer Methods in Applied Mechanics and Engineering2010; 199(49-52):3059-3100. · Zbl 1225.76204 |
| [102] | JansenK, WhitingC, CollisSS, ShakibF. A better consistency for low‐order stabilized finite‐element methods. Computer Methods in Applied Mechanics and Engineering1999; 174:153-170. · Zbl 0956.76044 |
| [103] | LoveE, ScovazziG. On the angular momentum conservation and incremental objectivity properties of a predictor/multi‐corrector method for Lagrangian shock hydrodynamics. Computer Methods in Applied Mechanics and Engineering2009 September; 198(41-44):3207-3213. · Zbl 1230.76036 |
| [104] | ScovazziG. A discourse on Galilean invariance and SUPG‐type stabilization. Computer Methods in Applied Mechanics and Engineering2007; 196(4-6):1108-1132. · Zbl 1120.76333 |
| [105] | ScovazziG. Galilean invariance and stabilized methods for compressible flows. International Journal of Numerical Methods Fl.2007; 54(6-8):757-778. · Zbl 1207.76094 |
| [106] | ScovazziG, LoveE. A generalized view on Galilean invariance in stabilized compressible flow computations. International Journal of Numerical Methods Fl.2010; 64(10-12):1065-1083. · Zbl 1427.76235 |
| [107] | SongT, ScovazziG. A Nitsche method for wave propagation problems in time domain. Computer Methods in Applied Mechanics and Engineering2015; 293:481-521. · Zbl 1423.35072 |
| [108] | SimoJC, HughesTJR. Computational Inelasticity, Interdisciplinary Applied Mathematics, vol. 7. Springer: New York, 2000. |
| [109] | HartmannS, NeffP. Polyconvexity of generalized polynomial‐type hyperelastic strain energy functions for near‐incompressibility. International Journal of Solids and Structures2003; 40:2767-2791. · Zbl 1051.74539 |
| [110] | GurtinME. An Introduction to Continuum Mechanics. Academic Press: London, 1981. · Zbl 0559.73001 |
| [111] | BonetJ, WoodRD. Nonlinear Continuum Mechanics for Finite Element Analysis (2nd edn.) Cambridge University Press: Cambridge (UK), 2008. · Zbl 1142.74002 |
| [112] | BelytschkoT, LiuWK, MoranB. Nonlinear Finite Elements for Continua and Structures. J. Wiley & Sons: New York, 2000. · Zbl 0959.74001 |
| [113] | EisenstatSC, WalkerHF. Choosing the forcing terms in an inexact Newton method.SIAM Journal of Scientific Computing1996; 17(1):16-32. · Zbl 0845.65021 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
