Space-Time Isogeometric Analysis for linear and non-linear elastodynamics

Abstract In this study we investigate a stabilized space-time formulation for linear and non-linear elastodynamics. We use Isogeometric Analysis (IGA) in order to benefit from its numerical qualities. We focus on two points: the formulation of stabilized weak-forms in a linear and non-linear context and the interest of using continuous Galerkin schemes in space and time with higher order and higher continuity basis functions. We illustrate the numerical performance of these methods through typical impact or vibration problems commonly encountered in the field of the elastodynamics of solids.

[1]  A. Grebénnikov.  Isogeometric approximation of functions of one variable , 1982 .

[2]  Thomas J. R. Hughes,et al.  Stability, convergence, and accuracy of a new finite element method for the circular arch problem , 1987 .

[3]  D. Eyheramendy,et al.  Hybrid free energy approach for nearly incompressible behaviors at finite strain , 2018, Continuum Mechanics and Thermodynamics.

[4]  Giancarlo Sangalli,et al.  Space-time least-squares isogeometric method and efficient solver for parabolic problems , 2018, Math. Comput..

[5]  John C. Bruch,et al.  Transient two‐dimensional heat conduction problems solved by the finite element method , 1974 .

[6]  K. S. Chung.  The fourth‐dimention concept in the finite element analysis of transient heat transfer problems , 1981 .

[7]  Thomas J. R. Hughes,et al.  Improved numerical dissipation for time integration algorithms in structural dynamics , 1977 .

[8]  A. Ammar,et al.  Space–time proper generalized decompositions for the resolution of transient elastodynamic models , 2013 .

[9]  Donald A. French,et al.  A space-time finite element method for the wave equation* , 1993 .

[10]  Christoph Lehrenfeld,et al.  Analysis of a Nitsche XFEM-DG Discretization for a Class of Two-Phase Mass Transport Problems , 2013, SIAM J. Numer. Anal..

[11]  L. Franca,et al.  The Galerkin gradient least-squares method , 1989 .

[12]  Antonio Huerta,et al.  3D NURBS‐enhanced finite element method (NEFEM) , 2008 .

[13]  Alexander V. Idesman,et al.  SOLUTION OF LINEAR ELASTODYNAMICS PROBLEMS WITH SPACE-TIME FINITE ELEMENTS ON STRUCTURED AND UNSTRUCTURED MESHES , 2007 .

[14]  D. Peric,et al.  Subdivision based mixed methods for isogeometric analysis of linear and nonlinear nearly incompressible materials , 2016 .

[15]  David A. Peters,et al.  Application of triangular space-time finite elements to problems of wave propagation , 1994 .

[16]  Gregory M. Hulbert,et al.  Discontinuity-capturing operators for elastodynamics , 1992 .

[17]  Julien Réthoré,et al.  A combined space–time extended finite element method , 2005 .

[18]  Czesław I. Bajer,et al.  Triangular and tetrahedral space-time finite elements in vibration analysis , 1986 .

[19]  Elena Atroshchenko,et al.  Weakening the tight coupling between geometry and simulation in isogeometric analysis: From sub‐ and super‐geometric analysis to Geometry‐Independent Field approximaTion (GIFT) , 2016, 1706.06371.

[20]  Tayfun E. Tezduyar,et al.  Space–time computational analysis of tire aerodynamics with actual geometry, road contact, tire deformation, road roughness and fluid film , 2019, Computational Mechanics.

[21]  Thomas J. R. Hughes,et al.  Mixed Petrov-Galerkin methods for the Timoshenko beam problem , 1987 .

[22]  Tom Lyche,et al.  Polynomial splines over locally refined box-partitions , 2013, Comput. Aided Geom. Des..

[23]  M. Lucchesi,et al.  Space–time elements for the shock wave propagation problem , 1980 .

[24]  Tayfun E. Tezduyar,et al.  Space–time Isogeometric flow analysis with built-in Reynolds-equation limit , 2019, Mathematical Models and Methods in Applied Sciences.

[25]  Vinh Phu Nguyen,et al.  Nitsche’s method for two and three dimensional NURBS patch coupling , 2013, 1308.0802.

[26]  P Jamet,et al.  Numerical solution of the eulerian equations of compressible flow by a finite element method which follows the free boundary and the interfaces , 1975 .

[27]  T. Hughes,et al.  Isogeometric analysis : CAD, finite elements, NURBS, exact geometry and mesh refinement , 2005 .

[28]  J. H. Tang,et al.  Three-dimensional transient elastodynamic analysis by a space and time-discontinuous Galerkin finite element method , 2003 .

[29]  Serge Dumont,et al.  4D Remeshing Using a Space-Time Finite Element Method for Elastodynamics Problems , 2018 .

[30]  Nils-Erik Wiberg,et al.  Adaptive FEA of Wave Propagation Induced by High-Speed Trains , 2001 .

[31]  T. Hughes,et al.  Local refinement of analysis-suitable T-splines , 2012 .

[32]  Rolf Krause,et al.  Variational space–time elements for large-scale systems , 2017 .

[33]  Thomas J. R. Hughes,et al.  Isogeometric Analysis: Toward Integration of CAD and FEA , 2009 .

[34]  Ulrich Langer,et al.  Space–time isogeometric analysis of parabolic evolution problems , 2015, 1509.02008.

[35]  Santiago Badia,et al.  Maximum-principle preserving space-time isogeometric analysis , 2018, ArXiv.

[36]  Francesco Costanzo,et al.  A space–time discontinuous Galerkin finite element method for fully coupled linear thermo-elasto-dynamic problems with strain and heat flux discontinuities☆ , 2008 .

[37]  A. Zilian,et al.  Hybridized enriched space-time finite element method for analysis of thin-walled structures immersed in generalized Newtonian fluids , 2010 .

[38]  J. Tinsley Oden,et al.  A GENERAL THEORY OF FINITE ELEMENTS II. APPLICATIONS , 1969 .

[39]  Francesco Costanzo,et al.  On the use of space-time finite elements in the solution of elasto-dynamic problems with strain discontinuities , 2002 .

[40]  Anthony Gravouil,et al.  Heterogeneous asynchronous time integrators for computational structural dynamics , 2015 .

[41]  T. Hughes,et al.  Space-time finite element methods for elastodynamics: formulations and error estimates , 1988 .

[42]  John Argyris,et al.  Finite Elements in Time and Space , 1969, The Aeronautical Journal (1968).

[43]  P Jamet,et al.  Numerical computation of the free boundary for the two-dimensional Stefan problem by space-time finite elements , 1977 .

[44]  J. Reynen,et al.  A space-time least-square finite element scheme for advection-diffusion equations , 1984 .

[45]  Ernst Rank,et al.  Weak coupling for isogeometric analysis of non-matching and trimmed multi-patch geometries , 2014 .

[46]  Andreas Zilian,et al.  The enriched space–time finite element method (EST) for simultaneous solution of fluid–structure interaction , 2008 .

[47]  Thomas J. R. Hughes,et al.  Space-time finite element methods for second-order hyperbolic equations , 1990 .

[48]  T. Hughes,et al.  A Simple Algorithm for Obtaining Nearly Optimal Quadrature Rules for NURBS-based Isogeometric Analysis , 2012 .

[49]  The viscoelastic space-time element , 1986 .

[50]  Giancarlo Sangalli,et al.  An efficient solver for space-time isogeometric Galerkin methods for parabolic problems , 2020, Comput. Math. Appl..

[51]  Nils-Erik Wiberg,et al.  Implementation and adaptivity of a space-time finite element method for structural dynamics , 1998 .

[52]  Yuri Bazilevs,et al.  Rotation free isogeometric thin shell analysis using PHT-splines , 2011 .

[53]  Lonny L. Thompson,et al.  Adaptive space–time finite element methods for the wave equation on unbounded domains , 2005 .

[54]  Czesław I. Bajer,et al.  Notes on the stability of non‐rectangular space–time finite elements , 1987 .

[55]  J. Nitsche.  Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind , 1971 .

[56]  Charbel Farhat,et al.  A space–time discontinuous Galerkin method for the solution of the wave equation in the time domain , 2009 .

[57]  Marek Behr,et al.  Simplex space‐time meshes in two‐phase flow simulations , 2018, ArXiv.

[58]  Francesco Costanzo,et al.  Proof of unconditional stability for a single-field discontinuous Galerkin finite element formulation for linear elasto-dynamics , 2005 .