A novel Galerkin-like weakform and a superconvergent alpha finite element method (S αFEM) for mechanics problems using triangular meshes
A carefully designed procedure is presented to modify the piecewise constant strain field of linear triangular FEM models, and to reconstruct a strain field with an adjustable parameter α. A novel Galerkin-like weakform derived from the Hellinger–Reissner variational principle is proposed for establ...
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| Veröffentlicht in: | Journal of computational physics Jg. 228; H. 11; S. 4055 - 4087 |
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| Format: | Journal Article |
| Sprache: | Englisch |
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20.06.2009
Elsevier |
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| ISSN: | 0021-9991, 1090-2716 |
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| Abstract | A carefully designed procedure is presented to modify the piecewise constant strain field of linear triangular FEM models, and to reconstruct a strain field with an adjustable parameter
α. A novel Galerkin-like weakform derived from the Hellinger–Reissner variational principle is proposed for establishing the discretized system equations. The new weak form is very simple, possesses the same good properties of the standard Galerkin weakform, and works particularly well for strain construction methods. A superconvergent alpha finite element method (S
αFEM) is then formulated by using the constructed strain field and the Galerkin-like weakform for solid mechanics problems. The implementation of the S
αFEM is straightforward and no additional parameters are used. We prove theoretically and show numerically that the S
αFEM always achieves more accurate and higher convergence rate than the standard FEM of triangular elements (T3) and even more accurate than the four-node quadrilateral elements (Q4) when the same sets of nodes are used. The S
αFEM can always produce both lower and upper bounds to the exact solution in the energy norm for all elasticity problems by properly choosing an
α. In addition, a preferable-
α approach has also been devised to produce very accurate solutions for both displacement and energy norms and a superconvergent rate in the energy error norm. Furthermore, a model-based selective scheme is proposed to formulate a combined S
αFEM/NS-FEM model that handily overcomes the volumetric locking problems. Intensive numerical studies including singularity problems have been conducted to confirm the theory and properties of the S
αFEM. |
|---|---|
| AbstractList | A carefully designed procedure is presented to modify the piecewise constant strain field of linear triangular FEM models, and to reconstruct a strain field with an adjustable parameter alpha. A novel Galerkin-like weakform derived from the Hellinger-Reissner variational principle is proposed for establishing the discretized system equations. The new weak form is very simple, possesses the same good properties of the standard Galerkin weakform, and works particularly well for strain construction methods. A superconvergent alpha finite element method (SalphaFEM) is then formulated by using the constructed strain field and the Galerkin-like weakform for solid mechanics problems. The implementation of the SalphaFEM is straightforward and no additional parameters are used. We prove theoretically and show numerically that the SalphaFEM always achieves more accurate and higher convergence rate than the standard FEM of triangular elements (T3) and even more accurate than the four-node quadrilateral elements (Q4) when the same sets of nodes are used. The SalphaFEM can always produce both lower and upper bounds to the exact solution in the energy norm for all elasticity problems by properly choosing an alpha. In addition, a preferable-alpha approach has also been devised to produce very accurate solutions for both displacement and energy norms and a superconvergent rate in the energy error norm. Furthermore, a model-based selective scheme is proposed to formulate a combined SalphaFEM/NS-FEM model that handily overcomes the volumetric locking problems. Intensive numerical studies including singularity problems have been conducted to confirm the theory and properties of the SalphaFEM. A carefully designed procedure is presented to modify the piecewise constant strain field of linear triangular FEM models, and to reconstruct a strain field with an adjustable parameter α. A novel Galerkin-like weakform derived from the Hellinger–Reissner variational principle is proposed for establishing the discretized system equations. The new weak form is very simple, possesses the same good properties of the standard Galerkin weakform, and works particularly well for strain construction methods. A superconvergent alpha finite element method (S αFEM) is then formulated by using the constructed strain field and the Galerkin-like weakform for solid mechanics problems. The implementation of the S αFEM is straightforward and no additional parameters are used. We prove theoretically and show numerically that the S αFEM always achieves more accurate and higher convergence rate than the standard FEM of triangular elements (T3) and even more accurate than the four-node quadrilateral elements (Q4) when the same sets of nodes are used. The S αFEM can always produce both lower and upper bounds to the exact solution in the energy norm for all elasticity problems by properly choosing an α. In addition, a preferable- α approach has also been devised to produce very accurate solutions for both displacement and energy norms and a superconvergent rate in the energy error norm. Furthermore, a model-based selective scheme is proposed to formulate a combined S αFEM/NS-FEM model that handily overcomes the volumetric locking problems. Intensive numerical studies including singularity problems have been conducted to confirm the theory and properties of the S αFEM. |
| Author | Xu, X. Liu, G.R. Nguyen-Xuan, H. Nguyen-Thoi, T. |
| Author_xml | – sequence: 1 givenname: G.R. surname: Liu fullname: Liu, G.R. organization: Singapore-MIT Alliance (SMA), E4-04-10, 4 Engineering Drive 3, Singapore 117576, Singapore – sequence: 2 givenname: H. surname: Nguyen-Xuan fullname: Nguyen-Xuan, H. email: smanxh@nus.edu.sg, nxhung@hcmuns.edu.vn organization: Singapore-MIT Alliance (SMA), E4-04-10, 4 Engineering Drive 3, Singapore 117576, Singapore – sequence: 3 givenname: T. surname: Nguyen-Thoi fullname: Nguyen-Thoi, T. organization: Center for Advanced Computations in Engineering Science (ACES), Department of Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1, Singapore 117576, Singapore – sequence: 4 givenname: X. surname: Xu fullname: Xu, X. organization: Center for Advanced Computations in Engineering Science (ACES), Department of Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1, Singapore 117576, Singapore |
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| Keywords | Meshfree methods Node-based smoothed finite element method (NS-FEM) Solution bounds Alpha finite element method ( αFEM) Numerical methods Strain construction methods Finite element method (FEM) Superconvergence Lower bound Singularity Digital simulation Alpha finite element method (αFEM) Elasticity Numerical method Linear model Bounded solution Calculation methods Exact solution Finite element method Upper bound Variational principle Convergence rate Calculation |
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| SubjectTerms | Alpha finite element method ( αFEM) Computational techniques Exact sciences and technology Finite element method (FEM) Mathematical methods in physics Meshfree methods Node-based smoothed finite element method (NS-FEM) Numerical methods Physics Solution bounds Strain construction methods Superconvergence |
| Title | A novel Galerkin-like weakform and a superconvergent alpha finite element method (S αFEM) for mechanics problems using triangular meshes |
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