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Introduction

  • Marcus Olavi RüterEmail author
Chapter
Part of the Lecture Notes in Applied and Computational Mechanics book series (LNACM, volume 88)

Abstract

In this introductory chapter, the topics of this monograph are briefly discussed and embedded into the bigger picture of computational validation and verification strategies in Computational Mechanics. More precisely, different types of errors are introduced that appear during the numerical simulation process of a physical phenomenon based on various Galerkin methods. The Galerkin methods dealt with in this monograph are the (conventional) finite element method (FEM) and, in particular, advanced versions of the finite element method, such as the extended finite element method (XFEM).

References

  1. Ainsworth, M.: A posteriori error estimation for fully discrete hierarchic models of elliptic boundary value problems on thin domains. Numer. Math. 80, 325–362 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  2. Ainsworth, M., Oden, J.T.: A unified approach to a posteriori error estimation based on element residual methods. Numer. Math. 65, 23–50 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  3. Ainsworth, M., Oden, J.T.: A Posteriori Error Estimation in Finite Element Analysis. John Wiley & Sons, New York (2000)zbMATHCrossRefGoogle Scholar
  4. Atluri, S.N., Zhu, T.: A new Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics. Comput. Mech. 22, 117–127 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  5. Aubin, J.-P.: Behavior of the error of the approximate solution of boundary value problems for linear elliptic operators by Galerkin’s and finite difference methods. Ann. Scuola Norm. Sup. Pisa 21, 599–637 (1967)zbMATHGoogle Scholar
  6. Auricchio, F., Beirão da Veiga, L., Brezzi, F., Lovadina, C.: Mixed finite element methods. In: Stein, E., de Borst, R., Hughes, T.J.R. (eds.) Encyclopedia of Computational Mechanics, vol. 1, 2nd edn., pp. 149–201. John Wiley & Sons, Chichester (2017)Google Scholar
  7. Babuška, I., Melenk, J.M.: The partition of unity method. Int. J. Numer. Meth. Engng. 40, 727–758 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  8. Babuška, I., Miller, A.: The post-processing approach in the finite element method. Part 3: A posteriori error estimates and adaptive mesh selection. Int. J. Numer. Meth. Engng. 20, 2311–2324 (1984)zbMATHCrossRefGoogle Scholar
  9. Babuška, I., Miller, A.: A feedback finite element method with a posteriori error estimation: Part I. The finite element method and some basic properties of the a posteriori error estimator. Comput. Methods Appl. Mech. Engrg. 61, 1–40 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  10. Babuška, I., Rheinboldt, W.C.: A-posteriori error estimates for the finite element method. Int. J. Numer. Meth. Engng. 12, 1597–1615 (1978a)zbMATHCrossRefGoogle Scholar
  11. Babuška, I., Rheinboldt, W.C.: Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15, 736–754 (1978b)MathSciNetzbMATHCrossRefGoogle Scholar
  12. Babuška, I., Schwab, C.: A posteriori error estimation for hierarchic models of elliptic boundary value problems on thin domains. SIAM J. Numer. Anal. 33, 221–246 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  13. Babuška, I., Strouboulis, T.: The Finite Element Method and its Reliability. Oxford University Press, Oxford (2001)zbMATHGoogle Scholar
  14. Bangerth, W., Rannacher, R.: Adaptive Finite Element Methods for Differential Equations. Birkhäuser, Basel (2003)zbMATHCrossRefGoogle Scholar
  15. Bank, R.E., Smith, R.K.: A posteriori error estimates based on hierarchical bases. SIAM J. Numer. Anal. 30, 921–935 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  16. Bank, R.E., Weiser, A.: Some a posteriori error estimators for elliptic partial differential equations. Math. Comp. 44, 283–301 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  17. Barsoum, R.S.: On the use of isoparametric finite elements in linear fracture mechanics. Int. J. Numer. Meth. Engng. 10, 25–37 (1976)zbMATHCrossRefGoogle Scholar
  18. Becker, R., Rannacher, R.: A feed-back approach to error control in finite element methods: Basic analysis and examples. East-West J. Numer. Math. 4, 237–264 (1996)MathSciNetzbMATHGoogle Scholar
  19. Belytschko, T., Black, T.: Elastic crack growth in finite elements with minimal remeshing. Int. J. Numer. Meth. Engng. 45, 601–620 (1999)zbMATHCrossRefGoogle Scholar
  20. Belytschko, T., Lu, Y.Y., Gu, L.: Element-free Galerkin methods. Int. J. Numer. Meth. Engng. 37, 229–256 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  21. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, New York (1991)zbMATHCrossRefGoogle Scholar
  22. Carstensen, C., Funken, S.A.: Averaging technique for FE - a posteriori error control in elasticity. Part I: Conforming FEM. Comput. Methods Appl. Mech. Engrg. 190, 2483–2498 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  23. Chamoin, L., Díez, P. (eds.): Verifying Calculations—Forty Years On—An Overview of Classical Verification Techniques for FEM Simulations. Springer, New York (2016)Google Scholar
  24. Chen, J.S., Hillman, M., Rüter, M., Hu, H.Y., Chi, S.W.: The role of quadrature in meshfree methods: Variational consistency in Galerkin weak form and collocation in strong form. IACM Expr. 34, 11–16 (2013)Google Scholar
  25. Chen, J.S., Liu, W.K., Hillman, M.C., Chi, S.W., Lian, Y., Bessa, M.A.: Reproducing kernel particle method for solving partial differential equations. In: Stein, E., de Borst, R., Hughes, T.J.R. (eds.) Encyclopedia of Computational Mechanics, vol. 2, 2nd edn., pp. 691–734. John Wiley & Sons, Chichester (2017)Google Scholar
  26. Chen, J.S., Pan, C., Wu, C.T., Liu, W.K.: Reproducing Kernel Particle Methods for large deformation analysis of non-linear structures. Comput. Methods Appl. Mech. Engrg. 139, 195–227 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  27. Chen, J.S., Wu, C.T., Yoon, S., You, Y.: A stabilized conforming nodal integration for Galerkin mesh-free methods. Int. J. Numer. Meth. Engng. 50, 435–466 (2001)zbMATHCrossRefGoogle Scholar
  28. Cherepanov, G.P.: Crack propagation in continuous media. J. Appl. Math. Mech. 31, 503–512 (1967)zbMATHCrossRefGoogle Scholar
  29. Cirak, F., Ramm, E.: A posteriori error estimation and adaptivity for linear elasticity using the reciprocal theorem. Comput. Methods Appl. Mech. Engrg. 156, 351–362 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  30. Courant, R.: Variational methods for the solution of problems of equilibrium and vibrations. Bull. Amer. Math. Soc. 49, 1–23 (1943)MathSciNetzbMATHCrossRefGoogle Scholar
  31. Eriksson, K., Estep, D., Hansbo, P., Johnson, C.: Introduction to adaptive methods for differential equations. Acta Numer. 106–158 (1995)Google Scholar
  32. Eshelby, J.D.: The force on an elastic singularity. Philos. Trans. Roy. Soc. London Ser. A 244, 87–112 (1951)Google Scholar
  33. Galerkin, B.G.: Series solutions of some cases of equilibrium of elastic beams and plates (in Russian). Vestn. Inshenernov. 1, 897–908 (1915)Google Scholar
  34. Gingold, R.A., Monaghan, J.J.: Smoothed particle hydrodynamics: theory and application to non-spherical stars. Mon. Not. R. astr. Soc. 181, 375–389 (1977)zbMATHCrossRefGoogle Scholar
  35. Heintz, P., Larsson, F., Hansbo, P., Runesson, K.: Adaptive strategies and error control for computing material forces in fracture mechanics. Int. J. Numer. Meth. Engng. 60, 1287–1299 (2004)zbMATHCrossRefGoogle Scholar
  36. Henshell, R.D., Shaw, K.G.: Crack tip finite elements are unnecessary. Int. J. Numer. Meth. Engng. 9, 495–507 (1975)zbMATHCrossRefGoogle Scholar
  37. Huerta, A., Belytschko, T., Fernández-Méndez, S., Rabczuk, T., Zhuang, X., Arroyo, M.: Meshfree methods. In: Stein, E., de Borst, R., Hughes, T.J.R. (eds.) Encyclopedia of Computational Mechanics, vol. 2, 2nd edn., pp. 653–690. John Wiley & Sons, Chichester (2017)Google Scholar
  38. Koenke, C., Harte, R., Krätzig, W.B., Rosenstein, O.: On adaptive remeshing techniques for crack simulation problems. Engrg. Comput. 15, 74–88 (1998)zbMATHCrossRefGoogle Scholar
  39. Ladevèze, P., Leguillon, D.: Error estimate procedure in the finite element method and applications. SIAM J. Numer. Anal. 20, 485–509 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  40. Ladevèze, P., Pelle, J.-P.: Mastering Calculations in Linear and Nonlinear Mechanics. Springer, New York (2005)zbMATHGoogle Scholar
  41. Lancaster, P., Salkauskas, K.: Surfaces generated by moving least squares methods. Math. Comp. 37, 141–158 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  42. Larsson, F., Hansbo, P., Runesson, K.: Strategies for computing goal-oriented a posteriori error measures in non-linear elasticity. Int. J. Numer. Meth. Engng. 55, 879–894 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  43. Larsson, F., Runesson, K.: Modeling and discretization errors in hyperelasto-(visco-)plasticity with a view to hierarchical modeling. Comput. Methods Appl. Mech. Engrg. 193, 5283–5300 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  44. Liu, G.R., Dai, K.Y., Nguyen, T.T.: A smoothed finite element method for mechanics problems. Comput. Mech. 39, 859–877 (2007)zbMATHCrossRefGoogle Scholar
  45. Liu, W.K., Jun, S., Li, S., Adee, J., Belytschko, T.: Reproducing kernel particle methods for structural dynamics. Int. J. Numer. Meth. Engng. 38, 1655–1679 (1995a)MathSciNetzbMATHCrossRefGoogle Scholar
  46. Liu, W.K., Jun, S., Zhang, Y.F.: Reproducing kernel particle methods. Int. J. Numer. Meth. Fluids 20, 1081–1106 (1995b)MathSciNetzbMATHCrossRefGoogle Scholar
  47. Lu, Y.Y., Belytschko, T., Gu, L.: A new implementation of the element free Galerkin method. Comput. Methods Appl. Mech. Engrg. 113, 397–414 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  48. Lucy, L.B.: A numerical approach to the testing of the fission hypothesis. Astron. J. 82, 1013–1024 (1977)CrossRefGoogle Scholar
  49. Maugin, G.A.: Material Inhomogeneities in Elasticity. Chapman & Hall, London (1993)zbMATHCrossRefGoogle Scholar
  50. Melenk, J.M., Babuška, I.: The partition of unity finite element method: Basic theory and applications. Comput. Methods Appl. Mech. Engrg. 139, 289–314 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  51. Moës, N., Dolbow, J., Belytschko, T.: A finite element method for crack growth without remeshing. Int. J. Numer. Meth. Engng. 46, 131–150 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  52. Moës, N., Dolbow, J.E., Sukumar, N.: Extended finite element methods. In: Stein, E., de Borst, R., Hughes, T.J.R. (eds.) Encyclopedia of Computational Mechanics, vol. 3, 2nd edn., pp 173–193, John Wiley & Sons, Chichester (2017)Google Scholar
  53. Murthy, K.S.R.K., Mukhopadhyay, M.: Adaptive finite element analysis of mixed-mode crack problems with automatic mesh generator. Int. J. Numer. Meth. Engng. 49, 1087–1100 (2000)zbMATHCrossRefGoogle Scholar
  54. Nayroles, B., Touzot, G., Villon, P.: Generalizing the finite element method: diffuse approximation and diffuse elements. Comput. Mech. 10, 307–318 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  55. Neittaanmäki, P., Repin, S.: Reliable Methods for Computer Simulation—Error Control and A Posteriori Estimates. Elsevier B.V., Amsterdam (2004)zbMATHCrossRefGoogle Scholar
  56. Nitsche, J.A.: Ein Kriterium für die Quasioptimalität des Ritzschen Verfahrens. Numer. Math. 11, 346–348 (1968)MathSciNetzbMATHCrossRefGoogle Scholar
  57. Oden, J.T., Prudhomme, S.: Estimation of modeling error in computational mechanics. J. Comput. Phys. 182, 496–515 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  58. Oden, J.T., Vemaganti, K.: Estimation of local modeling error and goal-oriented modeling of heterogeneous materials; Part I: error estimates and adaptive algorithms. J. Comput. Phys. 164, 22–47 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  59. Ohnimus, S., Stein, E., Walhorn, E.: Local error estimates of FEM for displacements and stresses in linear elasticity by solving local Neumann problems. Int. J. Numer. Meth. Engng. 52, 727–746 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  60. Organ, D., Fleming, M., Terry, T., Belytschko, T.: Continuous meshless approximations for nonconvex bodies by diffraction and transparency. Comput. Mech. 18, 225–235 (1996)zbMATHCrossRefGoogle Scholar
  61. Osher, S., Fedkiw, R.: Level Set Methods and Dynamic Implicit Surfaces. Springer, New York (2002)zbMATHGoogle Scholar
  62. Osher, S., Sethian, J.A.: Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79, 12–49 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  63. Paraschivoiu, M., Patera, A.T.: A hierarchical duality approach to bounds for the outputs of partial differential equations. Comput. Methods Appl. Mech. Engrg. 158, 389–407 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  64. Parés, N., Díez, P., Huerta, A.: Subdomain-based flux-free a posteriori error estimators. Comput. Methods Appl. Mech. Engrg. 195, 297–323 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  65. Prasad, M.V.K.V., Krishnamoorthy, C.S.: Adaptive finite element analysis of mode I fracture in cement-based materials. Int. J. Numer. Anal. Meth. Geomech. 25, 1131–1147 (2001)zbMATHCrossRefGoogle Scholar
  66. Prudhomme, S., Oden, J.T.: On goal-oriented error estimation for local elliptic problems: application to the control of pointwise errors. Comput. Methods Appl. Mech. Engrg. 176, 313–331 (1999)Google Scholar
  67. Randles, P.W., Libersky, L.D.: Smoothed Particle Hydrodynamics: Some recent improvements and applications. Comput. Methods Appl. Mech. Engrg. 139, 375–408 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  68. Rannacher, R., Suttmeier, F.-T.: A posteriori error estimation and mesh adaptation for finite element models in elasto-plasticity. Comput. Methods Appl. Mech. Engrg. 176, 333–361 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  69. Rice, J.R.: A path independent integral and the approximate analysis of strain concentration by notches and cracks. J. Appl. Mech. 35, 379–386 (1968)CrossRefGoogle Scholar
  70. Roache, P.J.: Quantification of uncertainty in computational fluid dynamics. Annu. Rev. Fluid. Mech. 29, 123–160 (1997)MathSciNetCrossRefGoogle Scholar
  71. Ródenas, J.J., Tur, M., Fuenmayor, F.J., Vercher, A.: Improvement of the superconvergent patch recovery technique by the use of constraint equations: The SPR-C technique. Int. J. Numer. Meth. Engng. 70, 705–727 (2007)zbMATHCrossRefGoogle Scholar
  72. Rodríguez, R.: Some remarks on Zienkiewicz-Zhu estimator. Numer. Methods Partial Differential Eq. 10, 625–635 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  73. Rüter, M., Stein, E.: Goal-oriented a posteriori error estimates in linear elastic fracture mechanics. Comput. Methods Appl. Mech. Engrg. 195, 251–278 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  74. Rüter, M.O., Chen, J.S.: An enhanced-strain error estimator for Galerkin meshfree methods based on stabilized conforming nodal integration. Comput. Math. Appl. 74, 2144–2171 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  75. Simo, J.C., Rifai, M.S.: A class of assumed strain methods and the method of incompatible modes. Int. J. Numer. Meth. Engng. 29, 1595–1638 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  76. Stein, E. (ed.): Error-controlled Adaptive Finite Elements in Solid Mechanics. John Wiley & Sons, Chichester (2003)Google Scholar
  77. Stein, E., Ohnimus, S.: Coupled model- and solution-adaptivity in the finite-element method. Comput. Methods Appl. Mech. Engrg. 150, 327–350 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  78. Stein, E., Rüter, M.: Finite element methods for elasticity with error-controlled discretization and model adaptivity. In: Stein, E., de Borst, R., Hughes, T.J.R. (eds.) Encyclopedia of Computational Mechanics, vol. 3, 2nd edn., pp. 5–100. John Wiley & Sons, Chichester (2017)Google Scholar
  79. Stein, E., Rüter, M., Ohnimus, S.: Adaptive finite element analysis and modelling of solids and structures. Findings, problems and trends. Int. J. Numer. Meth. Engng. 60, 103–138 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  80. Stein, E., Rüter, M., Ohnimus, S.: Error-controlled adaptive goal-oriented modeling and finite element approximations in elasticity. Comput. Methods Appl. Mech. Engrg. 196, 3598–3613 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  81. Steinmann, P.: Application of material forces to hyperelastostatic fracture mechanics. I. Continuum mechanical setting. Int. J. Solids Structures 37, 7371–7391 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  82. Stern, M., Becker, E.B.: A conforming crack tip element with quadratic variation in the singular fields. Int. J. Numer. Meth. Engng. 12, 279–288 (1978)zbMATHCrossRefGoogle Scholar
  83. Stone, T.J., Babuška, I.: A numerical method with a posteriori error estimation for determining the path taken by a propagating crack. Comput. Methods Appl. Mech. Engrg. 160, 245–271 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  84. Strouboulis, T., Babuška, I., Copps, K.: The design and analysis of the Generalized Finite Element Method. Comput. Methods Appl. Mech. Engrg. 181, 43–69 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  85. Strouboulis, T., Copps, K., Babuška, I.: The generalized finite element method. Comput. Methods Appl. Mech. Engrg. 190, 4081–4193 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  86. Turner, M.J., Clough, R.W., Martin, H.C., Topp, L.J.: Stiffness and deflection analysis of complex structures. Journal Aero. Sci. 23, 805–823 (1956)zbMATHCrossRefGoogle Scholar
  87. Verfürth, R.: A Posteriori Error Estimation Techniques for Finite Element Methods. Oxford University Press, Oxford (2013)zbMATHCrossRefGoogle Scholar
  88. Wiberg, N.E., Abdulwahab, F.: Patch recovery based on superconvergent derivatives and equilibrium. Int. J. Numer. Meth. Engng. 36, 2703–2724 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  89. Zienkiewicz, O.C., Zhu, J.Z.: A simple error estimator and adaptive procedure for practical engineering analysis. Int. J. Numer. Meth. Engng. 24, 337–357 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  90. Zienkiewicz, O.C., Zhu, J.Z.: The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique. Int. J. Numer. Meth. Engng. 33, 1331–1364 (1992)MathSciNetzbMATHCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Civil and Environmental EngineeringUniversity of CaliforniaLos AngelesUSA

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