Variational r-adaption in elastodynamics

International Journal for Numerical Methods in Engineering

Volumn 74, Issue 7, 2008, Pages 1162-1197

  Zielonka, M G ^a   Ortiz, M ^a   Marsden, J E ^a  

^a California Institute of Technology   (United States)

Author keywords

Elastodynamics; Mesh adaption; Variational methods

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; FINITE ELEMENT METHOD; GALERKIN METHODS; HAMILTONIANS; INTERPOLATION; VARIATIONAL TECHNIQUES;

ELASTODYNAMICS; GALERKIN DISCRETIZATION; MESH ADAPTION;

DYNAMICS;

CONVERGENCE OF NUMERICAL METHODS; DYNAMICS; FINITE ELEMENT METHOD; GALERKIN METHODS; HAMILTONIANS; INTERPOLATION; VARIATIONAL TECHNIQUES;

EID: 43749118642     PISSN: 00295981     EISSN: 10970207     Source Type: Journal    
DOI: 10.1002/nme.2205     Document Type: Article

References (57)

1. Radovitzki R, Ortiz M. Error estimation and adaptive meshing in strongly non-linear dynamic problems. Computer Methods in Applied Mechanics and Engineering 1999; 172(1-4):203-240.
2. McNiece GM, Marcal PM. Optimization of finite element grids based on minimum potential energy. Transaction of the ASME 1976; 1: 186-190.
3. Felippa C. Optimization of finite element grids by direct energy search. Applied Mathematical Modelling 1976; 1:93-96.
4. Felippa C. Numerical experiments in finite element grid optimization by direct energy search. Applied Mathematical Modelling 1977; 1:239-244.
5. Eshelby JD. The force on an elastic singularity. Philosophical Transactions of the Royal Society of London 1951; 244:87-112.
6. Eshelby JD. The elastic Eshelby momentum tensor. Journal of Elasticity 1975; 5:321-335.
7. Braun M. Configurational forces induced by finite element discretization. Proceedings of the Estonian Academy of Sciences, Physics, and Mathematics 1997; 46:24-31.
8. Mueller R, Maugin GA. On material forces and finite element discretizations. Computational Mechanics 2002; 29:52-60.
9. Mueller R, Kolling S, Gross D. On configurational forces in the context of the finite element method. International Journal for Numerical Methods in Engineering 2002; 53:1557-1574.
10. Thoutireddy P. Variational arbitrary Lagrangian-Eulerian method. Ph.D. Thesis, California Institute of Technology, 2003. Available at: http://etd.caltech.edu/etd/available/etd-05292003-113845/.
11. Thoutireddy P, Ortiz M. A variational r-adaption and shape-optimization method for finite-deformation elasticity. International Journal for Numerical Methods in Engineering 2004; 61:1-21.
12. Mueller R, Gross D, Maugin GA. Use of material forces in adaptive finite element methods. Computational Mechanics 2004; 33:421-434.
13. Kuhl E, Askes H, Steinman P. An ALE formulation based on spatial and material settings of continuum mechanics. Part 1: generic hyperelastic formulation. Computer Methods in Applied Mechanics and Engineering 2004; 193:4203-4222.
14. Kuhl E, Steinmann P. A hyperelastodynamic ALE formulation based on referential, spatial and material settings of continuum mechanics. Acta Mechanica 2005; 174:201-222.
15. Askes H, Kuhl E, Steinman P. An ALE formulation based on spatial and material settings of continuum mechanics. Part 2: classification and applications. Computer Methods in Applied Mechanics and Engineering 2004; 193:4223-4245.
16. Mosler J, Ortiz M. On the numerical implementation of variational arbitrary Lagrangian-Eulerian (VALE) formulations. International Journal for Numerical Methods in Engineering 2006; 67:1272-1289.
17. Lew A. Variational time integrators in computational solid mechanics. Ph.D. Thesis, California Institute of Technology, 2003. Available at: http://etd.caltech.edu/etd/available/etd-05262003-200254/.
18. Lew A, Marsden JE, Ortiz M, West M. Asynchronous variational integrators. Archive for Rational Mechanics and Analysis 2003; 167:85-146.
19. Lew A, Marsden JE, Ortiz M, West M. Variational time integrators. International Journal for Numerical Methods in Engineering 2002; 3:1-10.
20. West M. Variational integrators. Ph.D. Thesis, California institute of technology, 2004. Available at: http://etd.caltech.edu/etd/ available/etd-06072004-161416/0.
21. Leok M. Foundations of computational geometric mechanics. Ph.D. Thesis, California Institute of Technology, 2004. Available at: http://etd.caltech.edu/etd/available/etd-03022004-000251/.
22. Pontryagin LS, Boltyanskii VG, Gamkrelidze RV, Mishchenko EF. The Mathematical Theory of Optimal Processes. Wiley: New York, 1962.
23. Yoshimura H, Marsden JE. Dirac structures in Lagrangian mechanics. Part II: variational structures. Journal of Geometry and Physics 2006; 57:209-250.
24. Kharevych L, Tong WY, Kanso E, Marsden JE, Schröder P, Desbrun M. Geometric, variational integrators for computer animation. Eurographics ACM SIGGRAPH Symposium on Computer Animation, Vienna, Austria, 2006; 1-9.
25. Washizu K. Variational Methods in Elasticity and Plasticity (3rd edn). International Series of Monographs in Aeronautics and Astronautics: Solid and Structural Mechanics, vol. 9. Pergamon Press: Oxford, 1982.
26. Felippa C. On the original publication of the general canonical functional of linear elasticity. Journal of Applied Mechanics 2000; 67:217-219 (Brief notes).
27. Zielonka MG. Configurational forces and variational mesh adaption in solid dynamics. Ph.D. Thesis, California Institute of Technology, 2006. Available at: http://etd.caltech.edu/etd/available/ etd-05112006-162905/.
28. Truesdell CA, Noll W. The non-linear field theories of mechanics. Flügge's Handbuch der Physik, vol. 3, Part 3. Springer: Berlin, 1965 (2nd edn, corrected, 1992).
29. Lodge AS. On the use of convected coordinate systems in the mechanics of continuous media. Proceedings of the Cambridge Philosophical Society 1951; 47:575.
30. Yavari A, Marsden JE, Ortiz M. On spatial sand material covariant balance laws in elasticity. Journal of Mathematical Physics 2006; 47:1-53.
31. Maugin GA. Material Inhomogeneities in Elasticity. Chapman & Hall: London, 1993.
32. Maugin GA. 'Material' mechanics of materials. Theoretical and Applied Mechanics 2002; 27:1-12.
33. 3 (mechanics on the material manifold). Theoretical and Applied Mechanics 2002; 28-29:221-233.
34. Maugin GA. Material forces: concepts and applications. Applied Mechanics Reviews 1995; 48(5):213-245.
35. Noll W. Materially uniform simple bodies with inhomogeneities. Archive for Rational Mechanics and Analysis 1967; 27:1-32.
36. Epstein M, Maugin GA. The energy-momentum tensor and material uniformity in finite elasticity. Acta Mechanica 1990; 83:127-133.
37. Epstein M, Maugin GA. On the geometrical material structure of anelasticity. Acta Mechanica 1996; 115: 119-131.
38. Epstein M. The Eshelby tensor and the theory of continuous distribution of inhomogeneities. Mechanics Research Communications 2002; 29:501-506.
39. Maugin GA, Epstein M. Geometrical material structure of elastoplasticity. International Journal of Plasticity 1999; 14:109-115.
40. Maugin GA. Pseudo-plasticity and pseudo-inhomogeneity effects in materials mechanics. Journal of Elasticity 2003; 71:81-103.
41. Cermelli P, Fried E. The influence of inertia on the configurational forces in a deformable solid. Proceedings of the Royal Society of London, Series A 1997; 453:1915-1927.
42. Gurtin ME. The nature of configurational forces. Archive for Rational Mechanics and Analysis 1995; 131:67-100.
43. Gurtin ME. Configurational Forces as Basic Concepts of Continuum Physics. Applied Mathematical Sciences, vol. 137. Springer, 2000.
44. Kalpakides VK, Dascalu C. On the configurational force balance in thermomechanics. Proceedings of the Royal Society of London, Series A 2002; 458:3023-3039.
45. Knowles JK, Sternberg E. On a class of conservation laws in linearized and finite elastostatics. Archive for Rational Mechanics and Analysis 1971; 44:187-211.
46. Marsden JE, Hughes TJR. Mathematical Foundations of Elasticity. Prentice-Hall: Englewood Cliffs, NJ, 1983.
47. Kienzler R, Herrmann G. Mechanics in Material Space with Applications to Defect and Fracture Mechanics. Springer: Germany, 2000.
48. Maugin G, Trimarco C. Pseudo quantite de mouvernent et milieux elastiques inhomogenes. Comptes Rendus de l'Académie Sciences Serie II 1991; 313:851-856.
49. Podio-Guidugli P. Configurational balances via variational arguments. Interfaces and Free Boundaries 2001; 3:1-10.
50. Steinmann P. On spatial and material settings of thermo-hyperelastodynamics. Journal of Elasticity 2002; 66(2):109-157.
51. Kane C, Marsden JE, Ortiz M. Symplectic-energy-momentum preserving variational integrators. Journal of Mathematical Physics 1999; 40(7):3353-3371.
52. Mosler J, Ortiz M. Variational h-adaption in finite deformation elasticity and plasticity. International Journal for Numerical Methods in Engineering 2007; 72(5):505-523.
53. Green AE, Naghdi PM. A re-examination of the basic postulates of thermomechanics. Proceedings of the Royal Society of London, Series A 1991; 432:171-194.
54. Maugin GA. Towards an analytical mechanics of dissipative materials. Rendiconti del Seminario Matematico, vol. 58(2). Geometry, Continua and Microstuctures. Universita e Politecnico di Torino, Torino, 2000.
55. Maugin GA, Kalpakides VK. A Hamiltonian formulation for elasticity and thermoplasticity. Journal of Physics A - Mathematical and General 2002; 35:10775-10788.
56. Kalpakides VK, Maugin GA. Canonical formulation and conservation laws of theromelasticity without dissipation. Reports on Mathematica Physics 2004; 53(3):371-391.
57. Dascalu C, Maugin GA. The thermoclastic material-momentum equation. Journal of Elasticity 1995; 39:201-212.