ON THE USE OF A CONTINUOUS STRONG”“FORM RESIDUUM FIELD FOR ERROR ESTIMATION IN SMOOTH GFEM APPROXIMATIONS
DOI:
https://doi.org/10.26512/ripe.v2i14.21365Keywords:
Subdomain error estimators. Implicit residual methods. Ck”“GFEM. Smoothness. Strong-form residuum field.Abstract
This investigation proposes the use of continuous strong”“form residuum fields, obtained through smooth Generalized Finite Element Method (GFEM) , for error estimation in terms of the energy norm. Aspects on the construction of Ck”“GFEM”“based approximation functions (Duarte, Kim & Quaresma, 2006), using domain triangulation, are addressed. It is shown how the smoothness may be exploited in implicit residual algorithms for error estimation since the approximated Ck”“GFEM stress field can be directly continuously differentiated, to verify the equilibrium equations in strong form, locally, and then leading to a continuous residuum field. The subdomain strategy (Barros et al., 2013; Par´es, D´Ä±ez & Huerta, 2006) for implicit error estimation is employed, in such a way the local error problems are defined on the clouds, the patch of elements around the node, through the weighting provided by the Partition of Unity (PoU) functions. Its implementation fits very well into GFEM routines because such strategy is naturally tailored to the nodal enrichment procedure of the method (Barros, Barcellos & Duarte, 2007), producing nodal error indicators. Two types of weighting for the variational residuum functional (Prudhomme et al., 2004; Strouboulis et al., 2006) are tested in order to verify the performance for the effectivity of the nodal indicators and the global estimators. Numerical examples show that both the indicator and the estimator may be effective for two-dimensional linear elastic problems even in the presence of singularities.
Downloads
References
Ainsworth, M., & Oden, J. T., 2000. A posteriori error estimation in finite element analysis. John Wiley and Sons.
Anuvriev, I., Korneev, V., & Kostylev, V., 2007. A posteriori error estimation by means of the exactly equilibrated fields. Austrian Academy of Sciences, Institutional Report.
Babuˇska, I., & Strouboulis, T., 2001. The finite element method and its reliability. Oxford.
Babuˇska, I., Whiteman, J. R., & Strouboulis, T., 2011. Finite elements: an introduction to the method and error estimation. Oxford.
Barcellos, C. S. de, Mendonc¸a, P. T. R., & Duarte, C. A., 2009. A Ck continuous generalized finite element formulation applied to laminated Kirchhoff plate model. Computational Mechanics, vol. 44, pp. 377”“393.
Barros, F. B., Barcellos, C. S. de, & Duarte, C. A., 2007. p-Adaptive Ck generalized finite element method for arbitrary polygonal clouds. Computational Mechanics, vol. 41, pp. 175”“187.
Barros, F. B., Barcellos, C. S. de, & Duarte, C. A., 2009. Subdomain-based flux-free a posteriori estimator for generalized finite element method. Thirth Ibero-Latin-American Congress on Computational Methods in Engineering (XXX-CILAMCE)
Barros, F. B., Barcellos, C. S. de, Duarte, C. A., & Torres, D. A. F., 2013. Subdomain-based error techniques for generalized finite element approximations of problems with singular stress fields. Computational Mechanics, vol. 52, pp. 1395”“1415.
Barros, F. B., Proenc¸a, S. P. B., & Barcellos, C. S. de, 2004. On error estimator and p-adaptivity in the generalized finite element method. International Journal for Numerical Methods in Engineering, vol. 60, pp. 2373”“2398.
Belytschko, T., & Black, T., 1999. Elastic crack growth in finite elements with minimal remeshing. International Journal for Numerical Methods in Engineering, vol. 45, pp. 601”“620.
Belytschko, T., & Gracie, R., 2007. On XFEM applications to dislocations and interfaces. International Journal of Plasticity, vol. 23, n. 10, pp. 1721”“1738.
Boresi, A. P., Chong, K. P., & Lee, J. D., 2011. Elasticity in engineering mechanics. 3rd. ed., John Wiley and Sons.
Brebbia, C. A., Telles, J. C. F., & Wrobel, L. C., 1984. Boundary element techniques: theory and applications in engineering. Springer-Verlag.
D´Ä±ez, P., Par´es, N., & Huerta, A., 2004. Accurate upper and lower bounds by solving flux-free local problems in stars. Revue Europ´eenne des El´ements Finis, vol. 13, pp. 497”“507
Duarte, C. A., 1996. The hp-cloud method. PhD thesis, The University of Texas at Austin.
Duarte, C. A., Babuˇska, I., & Oden, J. T., 2000. Generalized finite element method for threedimensional structural mechanics problems. Computers and Structures, vol. 77, pp. 215”“ 232.
Duarte, C. A., Kim, D. J., & Quaresma, D. M., 2006. Arbitrarily smooth generalized finite element approximations. Computer Methods in Applied Mechanics and Structures, vol. 196, pp. 33”“56.
Duarte, C. A., & Oden, J. T., 1996. An hô€€€ p adaptive method using cloud. Computer Methods in Applied Mechanics and Engineering, vol. 139, pp. 237”“262.
Freitas, A., Torres, D. A. F., Mendonc¸a, P. T. R., & Barcellos, C. S. de, 2015. Comparative analysis of Ck- and C0-GFEM applied to two-dimensional problems of confined plasticity. Latin American Journal of Solids and Structures, vol. 12, n. 5, pp. 861”“882.
Fries, T. P., & Belytschko, T., 2010. The extended/generalized finite element method: an overview of the method and its applications. International Journal for Numerical Methods in Engineering, vol. 84, pp. 253”“304.
Kreyszig, E., 1989. Introductory Functional Analysis with Applications. Wiley.
Ladev`eze, P., & Leguillon, D., 1983. Error estimate procedure in the finite element method and applications. SIAM Journal on Numerical Analysis, vol. 20, pp. 485”“509.
Liu, G. R., 2003. Mesh free methods: moving beyond the finite element method. CRC Press.
Marin´e, N. P., 2005. Error assessment for functional outputs of PDE’s: bounds and goal-oriented adaptivity. PhD thesis, Universitat Polit`ecnica de Catalunya.
Mendonc¸a, P. T. R., Barcellos, C. S. de, & Torres, D. A. F., 2011. Analysis of anisotropic Mindlin plate model by continuous and non-continuous GFEM. Finite Elements in Analysis and Design, vol. 47, pp. 698”“717.
Mendonc¸a, P. T. R., Barcellos, C. S. de, & Torres, D. A. F., 2013. Robust Ck=C0 generalized FEM approximations for higher-order conformity requirements: application to Reddy’s HSDT model for anisotropic laminated plates. Composite Structures, vol. 96, pp. 332”“345.
Morin, P., Nochetto, R. H., & Siebert, K. G., 2003. Local problems on stars: a posteriori error
estimators, convergence, and performance. Mathematics of Computations, vol. 72, pp. 1067”“1097.
Oden, J. T., Demkowicz, L., Rachowicz, W., & Westermann, T. A., 1989. Toward a universal
hô€€€ p adaptive finite element strategy. Part 2: A posteriori error estimation. Computer
Methods in Applied Mechanics and Engineering, vol. 77, pp. 113”“180.
Oden, J. T., Duarte, C. A., & Zienkiewicz, O. C., 1998. A new cloud-based hp finite element method. Computer Methods in Applied Mechanics and Engineering, vol. 153, pp. 117”“ 126.
Oden, J. T., & Reddy, J. N., 1976. An introduction to the mathematical theory of finite elements. Wiley.
Par´es, N., D´Ä±ez, P., & Huerta, A., 2006. Subdomain-based flux-free a posteriori error estimators. Computer Methods in Applied Mechanics and Engineering, vol. 195, pp. 297”“323.
Prudhomme, S., Nobile, F., Chamoin, L., & Oden, J. T., 2004. Analysis of a subdomain-based error estimator for finite element approximations of elliptic problems. Numerical Methods for Partial Differential Equations, vol. 20, pp. 165”“192.
Rvachev, V. L., & Sheiko, T. I., 1995. R-functions in boundary value problems in mechanics. Applied Mechanics Reviews, vol. 48, pp. 151”“188.
Shepard, D., 1968. A two-dimensional interpolation function for irregularly-spaced data. 23rd ACM National Conference, pp. 517”“524.
Simone, A., Duarte, C. A., & Giessen, E. V. der, 2006. A generalized finite element method for polycrystals with discontinuous grain boundaries. International Journal for Numerical Methods in Engineering, vol. 67, pp. 1122”“1145.
Stazi, F. L., Budyn, E., Chessa, J., & Belytschko, T., 2003. An extended finite element method with higher-order elements for curved cracks. Computational Mechanics, vol. 31, pp. 38”“ 48.
Strouboulis, T., Babuˇska, I., & Copps, K., 2000. The design and analysis of the generalized finite element method. Computer Methods in Applied Mechanics and Engineering, vol. 181, pp. 43”“69.
Strouboulis, T., Copps, K., & Babuˇska, I., 2001. The generalized finite element method. Computer Methods in Applied Mechanics and Engineering, vol. 190, pp. 4081”“4193.
Strouboulis, T., Zhang, L., & Babuˇska, I., 2003. Generalized finite element method using meshbased handbooks: application to problems in domains with many voids. Computer Methods in Applied Mechanics and Engineering, vol. 192, pp. 3109”“3161.
Strouboulis, T., Zhang, L., Wang, D., & Babuˇska, I., 2006. A posteriori error estimation for generalized finite element methods. Computer Methods in Applied Mechanics and Engineering, vol. 195, pp. 852”“879.
Szab´o, B., &Babuˇska, I., 2011. Introduction to finite element analysis: formulation, verification and validation. Wiley.
Torres, D. A. F., 2012. Contributions on the use of continuous approximation functions in the generalized finite element method: evaluation in fracture mechanics. PhD thesis (in portuguese), Federal University of Santa Catarina.
Torres, D. A. F., Barcellos, C. S. de, & Mendonc¸a, P. T. R., 2015. Effects of the smoothness of partitions of unity on the quality of representation of singular enrichments for GFEM/XFEM stress approximations around brittle cracks. Computer Methods in Applied Mechanics and Engineering, vol. 283, pp. 243”“279.
Torres, D. A. F., Mendonc¸a, P. T. R., & Barcellos, C. S. de, 2011. Evaluation and verification of an HSDT-layerwise generalized finite element formulation for adaptive piezoelectric laminated plates. Computer Methods in Applied Mechanics and Engineering, vol. 200, pp. 675”“691.
Wandzura, S., & Xiao, H., 2003. Symmetric quadrature rules on a triangle. Computers and Mathematics with Applications, vol. 45, pp. 1829”“1840.
Downloads
Published
How to Cite
Issue
Section
License
Given the public access policy of the journal, the use of the published texts is free, with the obligation of recognizing the original authorship and the first publication in this journal. The authors of the published contributions are entirely and exclusively responsible for their contents.
1. The authors authorize the publication of the article in this journal.
2. The authors guarantee that the contribution is original, and take full responsibility for its content in case of impugnation by third parties.
3. The authors guarantee that the contribution is not under evaluation in another journal.
4. The authors keep the copyright and convey to the journal the right of first publication, the work being licensed under a Creative Commons Attribution License-BY.
5. The authors are allowed and stimulated to publicize and distribute their work on-line after the publication in the journal.
6. The authors of the approved works authorize the journal to distribute their content, after publication, for reproduction in content indexes, virtual libraries and similars.
7. The editors reserve the right to make adjustments to the text and to adequate the article to the editorial rules of the journal.
