Articles
Published 2017-05-15
Keywords
- Adaptive Modelling,
- hierarchical properties,
- mesh enrichment,
- mesh generation
How to Cite
Ródenas, J. J., Albelda, J., TUR, M., & Fuenmayor, F. (2017). A hierarchical h-adaptivity methodology based on element subdivision. Revista UIS Ingenierías, 16(2), 263–280. https://doi.org/10.18273/revuin.v16n2-2017024
Abstract
This paper presents a hierarchical h adaptive methodology for Finite Element Analysis based on the hierarchical relations between parent and child elements that come out if these elements are geometrically similar. Under this similarity condition the terms involved in the evaluation of element stiffness matrices of parent and child elements are related by a constant which is a function of the element sizes ratio (scaling factor). These relations have been the basis for the development of a hierarchical h adaptivity methodology based on element subdivision and the use of multi-point-constraints to ensure C0 continuity. The use of a hierarchical data structure significantly reduces the amount of calculations required for the mesh refinement, the evaluation of the global stiffness matrix, element stresses and element error estimation. The data structure also produces a natural reordering of the global stiffness matrix that improves the behaviour of the Cholesky factorization.
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References
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A. Tabarraei and N. Sukumar, Adaptive computations on conforming quadtree meshes, Finite Elements Anal. Des. 41 (2005) 686-702.
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A. Plaza, M. A. Padrón, J. P. Suárez, Non-degeneracy study of the 8-tetrahedra longest-edge partition, Applied Numerical Mathematics 55 (2005) 458-472.
C. James, D. A. Cavendish, H. William, An approach to automatic three-dimensional finite element mesh generation, Int J Numer Methods Eng 21 (1985) 329-347.
M. C. Rivara, Algorithms for refining triangular grids suitable for adaptive and multigrid techniques, Int J Numer Methods Eng 20 (1984) 745-756.
J. F. Abel and M. S. Shephard, An algorithm for multipoint constraints in finite element analysis, Int J Numer Methods Eng 14 (1979) 464-467.
C. Farhat, C. Lacour, D. Rixen, Incorporation of linear multipoint constraints in substructure based iterative solvers. part 1: A numerically scalable algorithm, Int J Numer Methods Eng 43 (1998) 997-1016.
J. J. Ródenas, M. Tur, J. E. Tarancón, F. J. Fuenmayor, H-adaptatividad de elementos finitos en refinamiento por subdivisión de malla. In XIV Congreso Nacional de Ingeniería Mecánica. Anales de Ingeniería Mecánica, Asociación Española de Ingeniería Mecánica, 2000.
M. Tur, J. Fuenmayor, J. J. Ródenas, E. Giner, 3D analysis of the influence of specimen dimensions on fretting stresses, Finite Elements Anal. Des. 39 (2003) 933-949.
A. Suzuki and M. Tabata, Finite element matrices in congruent subdomains and their effective use for large-scale computations, Int J Numer Methods Eng 62 (2005) 1807-1831.
J. J. Ródenas, J. E. Tarancón, J. Albelda, A. Roda, F. J. Fuenmayor, Hierarquical properties in elements obtained by subdivision: A hierarquical h-adaptivity program. In Adaptive Modeling and Simulation 2005, P. Díez, N.E. Wiberg (Eds.), CIMNE, 2005.
Matlab®,7.9.0.529 (R2009b)The MathWorks Inc, Natick, MA, 2009
O. C. Zienkiewicz and J. Z. Zhu, A simple error estimation and adaptive procedure for practical engineering analysis, Int. J. Numer. Methods Eng. 24 (1987) 337-357.
J. J. Ródenas, Fuenmayor, F.J., A. Vercher, Improvement of the superconvergent patch recovery technique by the use of constraint equations: The SPR-C technique, Int. J. Numer. Methods Eng. 70 (2007) 705-727.
O. C. Zienkiewicz and J. Z. Zhu, The superconvergent patch recovery and a posteriori error estimates. part I: The recovery technique, Int. J. Numer. Methods Eng. 33 (1992) 1331-1364.
P. Ladeveze and D. Leguillon, Error estimate procedure in the finite element method and applications, SIAM Journal on Numerical Analysis 20 (1983) 485-509.
P. Ladeveze, P. Marin, J. P. Pelle, J. L. Gastine, Accuracy and optimal meshes in finite element computation for nearly incompressible materials, Comput. Methods Appl. Mech. Eng. 94 (1992) 303-315.
P. Coorevits, P. Ladeveze, J. P. Pelle, An automatic procedure with a control of accuracy for finite element analysis in 2D elasticity, Comput. Methods Appl. Mech. Eng. 121 (1995) 91-120.
E. Stein, M. Rüter, S. Ohnimus, Adaptive finite element analysis and modelling of solids and structures. findings, problems and trends, Int J Numer Methods Eng 60 (2004) 103-138.
M. Rüter and E. Stein, Goal-oriented a posteriori error estimates in linear elastic fracture mechanics, Comput. Methods Appl. Mech. Eng. 195 (2006) 251-278.
J. J. Ródenas, J. Albelda, C. Corral, J. Mas, Efficient implementation of domain decomposition methods using a hierarchical h-adaptive finite element program. In III European Conference on Computational Mechanics. Solids, Structures and Coupled Problems in Engineering. Book of Abstracs, Springer, 2006.
J. J. Ródenas, J. Albelda, C. Corral, J. Mas, A domain decomposition iterative solver based on a hierarchical h-adaptive FE code. In Proceedings of the Fifth International Conference on Engineering Computational Technology, B.H.V. Topping, G. Montero, R. Montenegro (Eds.), Civil-comp Press, 2006.
J. J. Ródenas, C. Corral, J. Albelda, J. Mas, C. Adam, Solver directo de división recursiva en subdominios basado en un programa de refinamiento h-adaptable de estructura jerárquica. In Métodos Numéricos e Computacionais em Engenharia. CMNE CILAMCE 2007, J.César de Sá, Raimundo Delgado, Abel d. Santos, Antonio Rodríguez-Ferrán, Javier Oliver: Paulo R.M. Lyra, José L. D. Alves (Eds.), APMTAC, SEMMNI, ABMEC, 2007.
J. J. Ródenas, C. Corral, J. Albelda, J. Mas, C. Adam, Nested domain decomposition direct and iterative solvers based on a hierarchical h adaptive finite element code. In ADMOS 2007. IV International Conference on Adaptive Modeling and Simulation, K. Runesson, P. Díez (Eds.), Internacional Center for Numerical Methods in Engineering (CIMNE), 2007.
M. S. Shephard, An algorithm for defining a single near‐optimum mesh for multiple‐load‐case problems, Int J Numer Methods Eng 15 (1980) 617-625.
I. Babuška and W. Rheinboldt, Adaptive approaches and reliability estimations in finite element analysis, Comput. Methods Appl. Mech. Eng. 17 (1979) 519-540.
J. F. Thompson, B. K. Soni, N. P. Weatherill, Handbook of Grid Generation, CRC, 1999.
H. Jin and N. E. Wiberg, Two-dimensional mesh generation, adaptive remeshing and refinement, Int J Numer Methods Eng 29 (1990) 1501-1526.
J. Z. Zhu, E. Hinton, O. C. Zienkiewicz, Mesh enrichment against mesh regeneration using quadrilateral elements, Communications in Numerical Methods in Engineering 9 (1993) 547-554.
S. Zhang, On the nested refinement of quadrilateral and hexahedral finite elements and the affine approximation, Numerische Mathematik 98 (2004) 559-579.
M. A. Yerry and M. S. Shephard, Automatic three‐dimensional mesh generation by the modified‐octree technique, Int J Numer Methods Eng 20 (1984) 1965-1990.
M. C. Rivara, A grid generator based on 4‐triangles conforming mesh‐refinement algorithms, Int J Numer Methods Eng 24 (1987) 1343-1354.
M. T. Jones and P. E. Plassmann, Adaptive refinement of unstructured finite-element meshes, Finite Elements Anal. Des. 25 (1997) 41-60.
A. Tabarraei and N. Sukumar, Adaptive computations on conforming quadtree meshes, Finite Elements Anal. Des. 41 (2005) 686-702.
Á Plaza, J. P. Suárez, M. A. Padrón, S. Falcón, D. Amieiro, Mesh quality improvement and other properties in the four-triangles longest-edge partition, Comput. Aided Geom. Des. 21 (2004) 353-369.
A. Plaza, M. A. Padrón, J. P. Suárez, Non-degeneracy study of the 8-tetrahedra longest-edge partition, Applied Numerical Mathematics 55 (2005) 458-472.
C. James, D. A. Cavendish, H. William, An approach to automatic three-dimensional finite element mesh generation, Int J Numer Methods Eng 21 (1985) 329-347.
M. C. Rivara, Algorithms for refining triangular grids suitable for adaptive and multigrid techniques, Int J Numer Methods Eng 20 (1984) 745-756.
J. F. Abel and M. S. Shephard, An algorithm for multipoint constraints in finite element analysis, Int J Numer Methods Eng 14 (1979) 464-467.
C. Farhat, C. Lacour, D. Rixen, Incorporation of linear multipoint constraints in substructure based iterative solvers. part 1: A numerically scalable algorithm, Int J Numer Methods Eng 43 (1998) 997-1016.
J. J. Ródenas, M. Tur, J. E. Tarancón, F. J. Fuenmayor, H-adaptatividad de elementos finitos en refinamiento por subdivisión de malla. In XIV Congreso Nacional de Ingeniería Mecánica. Anales de Ingeniería Mecánica, Asociación Española de Ingeniería Mecánica, 2000.
M. Tur, J. Fuenmayor, J. J. Ródenas, E. Giner, 3D analysis of the influence of specimen dimensions on fretting stresses, Finite Elements Anal. Des. 39 (2003) 933-949.
A. Suzuki and M. Tabata, Finite element matrices in congruent subdomains and their effective use for large-scale computations, Int J Numer Methods Eng 62 (2005) 1807-1831.
J. J. Ródenas, J. E. Tarancón, J. Albelda, A. Roda, F. J. Fuenmayor, Hierarquical properties in elements obtained by subdivision: A hierarquical h-adaptivity program. In Adaptive Modeling and Simulation 2005, P. Díez, N.E. Wiberg (Eds.), CIMNE, 2005.
Matlab®,7.9.0.529 (R2009b)The MathWorks Inc, Natick, MA, 2009
O. C. Zienkiewicz and J. Z. Zhu, A simple error estimation and adaptive procedure for practical engineering analysis, Int. J. Numer. Methods Eng. 24 (1987) 337-357.
J. J. Ródenas, Fuenmayor, F.J., A. Vercher, Improvement of the superconvergent patch recovery technique by the use of constraint equations: The SPR-C technique, Int. J. Numer. Methods Eng. 70 (2007) 705-727.
O. C. Zienkiewicz and J. Z. Zhu, The superconvergent patch recovery and a posteriori error estimates. part I: The recovery technique, Int. J. Numer. Methods Eng. 33 (1992) 1331-1364.
P. Ladeveze and D. Leguillon, Error estimate procedure in the finite element method and applications, SIAM Journal on Numerical Analysis 20 (1983) 485-509.
P. Ladeveze, P. Marin, J. P. Pelle, J. L. Gastine, Accuracy and optimal meshes in finite element computation for nearly incompressible materials, Comput. Methods Appl. Mech. Eng. 94 (1992) 303-315.
P. Coorevits, P. Ladeveze, J. P. Pelle, An automatic procedure with a control of accuracy for finite element analysis in 2D elasticity, Comput. Methods Appl. Mech. Eng. 121 (1995) 91-120.
E. Stein, M. Rüter, S. Ohnimus, Adaptive finite element analysis and modelling of solids and structures. findings, problems and trends, Int J Numer Methods Eng 60 (2004) 103-138.
M. Rüter and E. Stein, Goal-oriented a posteriori error estimates in linear elastic fracture mechanics, Comput. Methods Appl. Mech. Eng. 195 (2006) 251-278.
J. J. Ródenas, J. Albelda, C. Corral, J. Mas, Efficient implementation of domain decomposition methods using a hierarchical h-adaptive finite element program. In III European Conference on Computational Mechanics. Solids, Structures and Coupled Problems in Engineering. Book of Abstracs, Springer, 2006.
J. J. Ródenas, J. Albelda, C. Corral, J. Mas, A domain decomposition iterative solver based on a hierarchical h-adaptive FE code. In Proceedings of the Fifth International Conference on Engineering Computational Technology, B.H.V. Topping, G. Montero, R. Montenegro (Eds.), Civil-comp Press, 2006.
J. J. Ródenas, C. Corral, J. Albelda, J. Mas, C. Adam, Solver directo de división recursiva en subdominios basado en un programa de refinamiento h-adaptable de estructura jerárquica. In Métodos Numéricos e Computacionais em Engenharia. CMNE CILAMCE 2007, J.César de Sá, Raimundo Delgado, Abel d. Santos, Antonio Rodríguez-Ferrán, Javier Oliver: Paulo R.M. Lyra, José L. D. Alves (Eds.), APMTAC, SEMMNI, ABMEC, 2007.
J. J. Ródenas, C. Corral, J. Albelda, J. Mas, C. Adam, Nested domain decomposition direct and iterative solvers based on a hierarchical h adaptive finite element code. In ADMOS 2007. IV International Conference on Adaptive Modeling and Simulation, K. Runesson, P. Díez (Eds.), Internacional Center for Numerical Methods in Engineering (CIMNE), 2007.