×

A new non-intrusive technique for the construction of admissible stress fields in model verification. (English) Zbl 1227.74078

Summary: In this paper, we investigate a new procedure for constructing admissible stress tensors that are needed in some methods for robust global/goal-oriented error estimation. This procedure is based on properties of the approximate finite element solution which are applied through prolongation conditions. It involves the calculation of equilibrated tractions along element boundaries, leading to the determination of admissible stress tensors at the element level. The main idea of the proposed approach is to use the partition of unity method for the calculation of the equilibrated tractions, thus defining local problems on patches of elements which can be solved in an automatic and non-intrusive manner. Therefore, this hybrid procedure leads to reasonable computational costs and can be easily implemented into existing finite element codes. Two-dimensional experiments illustrate the capabilities of the method as compared to alternative approaches.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
74A10 Stress
74K99 Thin bodies, structures
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] P. Ladevèze, Comparison of models for continuum media, Thèse d’état, Université P. et M. Curie, Paris, 1975 (in French).
[2] Babus˘ka, I.; Rheinboldt, W.C., A posteriori error estimates for the finite element method, Int. J. numer. method engrg., 12, 1597, (1978) · Zbl 0396.65068
[3] Ladevèze, P.; Leguillon, D., Error estimate procedure in the finite element method and application, SIAM J. numer. anal., 20, 3, 485-509, (1983) · Zbl 0582.65078
[4] Verfürth, R., A review of a posteriori error estimation and adaptive mesh-refinement techniques, (1996), Wiley-Teubner Stuttgart · Zbl 0853.65108
[5] Babus˘ka, I.; Strouboulis, T., The finite element method and its reliability, (2001), Oxford University Press · Zbl 0997.74069
[6] ()
[7] Ladevèze, P.; Pelle, J.-P., Mastering calculations in linear and nonlinear mechanics, (2004), Springer NY
[8] Paraschivoiu, M.; Peraire, J.; Patera, A.T., A posteriori finite element bounds for linear functional outputs of elliptic partial differential equations, Comput. method appl. mech. engrg., 150, 289-312, (1997) · Zbl 0907.65102
[9] Rannacher, R.; Suttmeier, F.T., A feedback approach to error control in finite element methods: application to linear elasticity, Comput. mech., 19, 434-446, (1997) · Zbl 0894.73170
[10] Cirak, F.; Ramm, E., A posteriori error estimation and adaptivity for linear elasticity using the reciprocal theorem, Comput. method appl. mech. engrg., 156, 351-362, (1998) · Zbl 0947.74062
[11] Peraire, J.; Patera, A.T., Bounds for linear-functional outputs of coercive partial differential equations; local indicators and adaptive refinements, (), 199-216
[12] Ladevèze, P.; Rougeot, P.; Blanchard, P.; Moreau, J.P., Local error estimators for finite element linear analysis, Computer methods in applied mechanics and engineering, 176, 231-246, (1999) · Zbl 0967.74066
[13] Prudhomme, S.; Oden, J.T., On goal-oriented error estimation for elliptic problems: application to the control of pointwise errors, Comput. method appl. mech. engrg., 176, 313-331, (1999) · Zbl 0945.65123
[14] Strouboulis, T.; Babus˘ka, I.; Datta, D.; Copps, K.; Gangaraj, S.K., A posteriori estimation and adaptive control of the error in the quantity of interest – part 1: A posteriori estimation of the error in the von Mises stress and the stress intensity factors, Comput. method appl. mech. engrg., 181, 261-294, (2000) · Zbl 0973.74083
[15] Becker, R.; Rannacher, R., An optimal control approach to shape a posteriori error estimation in finite element methods, (), 1-120 · Zbl 1105.65349
[16] Wiberg, N.E.; Diez, P., Special issue, Comput. method appl. mech. engrg., 195, 4-6, (2006)
[17] Beckers, P.; Dufeu, E., 3-D error estimation and mesh adaptation using improved R.E.P. method, Stud. appl. mech., 47, 413-426, (1998)
[18] Moitinho de Almeida, J.P.; Almeida Pereira, O.J.B., Upper bounds of the error in local quantities using equilibrated and compatible finite element solutions for linear elastic problems, Comput. method appl. mech. engrg., 195, 4-6, 279-296, (2006) · Zbl 1086.74041
[19] Chamoin, L.; Ladevèze, P., A non-intrusive method for the calculation of strict and efficient bounds of calculated outputs of interest in linear viscoelasticity problems, Comput. method appl. mech. engrg., 197, 9-12, 994-1014, (2008) · Zbl 1169.74357
[20] Cottereau, R.; Diez, P.; Huerta, A., Strict error bounds for linear solid mechanics problems using a subdomain-based flux-free approach, Comput. mech., 44, 4, 533-547, (2009) · Zbl 1297.74116
[21] J. Panetier, P. Ladevèze, L. Chamoin, Strict and effective bounds in goal-oriented error estimation applied to fracture mechanics problems solved with the XFEM, Int. J. Numer. Method Engrg., doi:10.1002/nme.2705. · Zbl 1183.74303
[22] Ladevèze, P., Strict upper error bounds for calculated outputs of interest in computational structural mechanics, Comput. mech., 42, 2, 271-286, (2008) · Zbl 1144.74041
[23] Ladevèze, P.; Waeytens, J., Model verification in dynamics through strict upper error bounds, Comput. method appl. mech. engrg., 198, 21-26, 1775-1784, (2009) · Zbl 1227.74027
[24] Fraeijs de Veubeke, B.M., Displacement and equilibrium models in the finite element method, (), (Chap. 9) · Zbl 0359.73007
[25] Kempeneers, M.; Beckers, P.; Moitinho de Almeida, J.P.; Pereira, O.J.B.A., Modèles équilibre pour l’analyse duale, Revue européenne des éléments finis, 12, 6, 737-760, (2003) · Zbl 1171.74444
[26] Bank, R.E.; Weiser, A., Some a posteriori error estimators for elliptic partial differential equations, Math. comput., 44, 283, (1985) · Zbl 0569.65079
[27] Ainsworth, M.; Oden, J.T., A priori error estimators for second-order elliptic systems. part 2: an optimal order process for calculating self-equilibrating fluxes, Comput. math. appl., 26, 75-87, (1993) · Zbl 0789.65083
[28] Ladevèze, P.; Maunder, E.A.W., A general method for recovering equilibrating element tractions, Comput. method appl. mech. engrg., 137, 111-151, (1996) · Zbl 0886.73065
[29] Ladevèze, P.; Maunder, E.A.W., A general method for recovering equilibrating finite element tractions and stress fields for plates and solid elements, Comput. assist. mech. engrg. sci., 4, 533-548, (1997) · Zbl 0974.74060
[30] Ladevèze, P.; Rougeot, P., New advances on a posteriori error on constitutive relation in finite element analysis, Comput. method appl. mech. engrg., 150, 239-249, (1997) · Zbl 0906.73064
[31] Florentin, E.; Gallimard, L.; Pelle, J.P., Evaluation of the local quality of stresses in 3D finite element analysis, Comput. method appl. mech. engrg., 191, 4441-4457, (2002) · Zbl 1040.74045
[32] Machiels, L.; Maday, Y.; Patera, A.T., A flux-free nodal Neumann subproblem approach to output bounds for partial differential equations, Comptes rendus académie des sciences - Mécanique, Paris, 330, 1, 249-254, (2000) · Zbl 0946.65101
[33] Carstensen, C.; Funken, S.A., Fully reliable localized error control in the FEM, SIAM J. sci. comput., 21, 4, 1465-1484, (2000) · Zbl 0956.65099
[34] Morin, P.; Nochetto, R.H.; Siebert, K.G., Local problems on stars: a posteriori error estimators, convergence, and performance, Math. comput., 72, 243, 1067-1097, (2003) · Zbl 1019.65083
[35] Prudhomme, S.; Nobile, F.; Chamoin, L.; Oden, J.-T., Analysis of a subdomain-based error estimator for finite element approximations of elliptic problems, Numer. method partial differ. eq., 20, 2, 165-192, (2004) · Zbl 1049.65123
[36] Pares, N.; Diez, P.; Huerta, A., Subdomain-based flux-free a posteriori error estimators, Comput. method appl. mech. engrg., 195, 4-6, 297-323, (2006) · Zbl 1193.65191
[37] Pares, N.; Santos, H.; Diez, P., Guaranteed energy error bounds for the Poisson equation using a flux-free approach: solving the local problems in subdomains, Int. J. numer. method engrg., 79, 10, 1203-1244, (2009) · Zbl 1176.65127
[38] J.P. Moitinho de Almeida, E.A.W. Maunder, Recovery of equilibrium on star patches using a partition of unity technique, Comput. Method Appl. Mech. Engrg., in preparation. · Zbl 1176.74201
[39] P. Ladevèze, Verification of finite element calculations: a new non-intrusive technique for the computation of admissible stress fields, Internal Report No. 270, LMT-Cachan, 2008 (in French).
[40] Babus˘ka, I.; Strouboulis, T.; Upadhyay, C.S.; Gangaraj, S.K.; Copps, K., Validation of a posteriori error estimators by numerical approach, Int. J. numer. method engrg., 37, 1073-1123, (1994) · Zbl 0811.65088
[41] www.samtech.com.
[42] Choi, H.W.; Paraschivoiu, M., Adaptive computations of a posteriori finite element output bounds: a comparison of the hybrid-flux approach and the flux-free approach, Comput. method appl. mech. engrg., 193, 36-38, 4001-4033, (2004) · Zbl 1079.74635
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.