An adaptive hybrid stress transition quadrilateral finite element method for linear elasticity (original) (raw)
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International Journal for Numerical Methods in Engineering, 1990
A modified global approach to choosing stress terms for hybrid finite elements in plane stress problems is based on the known requirement of minimum number of stress parameters. Let nS be the number of independent B-stress parameters, n, the number of nodal displacements and nR the number of rigid body degrees of freedom, then the satisfaction of the criterion na 2 n, -nR of the assembled structure instead of the individual element enables the reduction of np. New rectangular hybrid transition elements applied in adaptive mesh refinement and a new eight node rectangular invariant element including only 12 / I parameters, based on the modified criterion, are presented here.
Convergence of an adaptive finite element method on quadrilateral meshes
2008
Convergence d'une méthodeéléments finis adaptative pour des maillages quadrilatéraux Résumé : Nous démontrons la convergence d'un algorithme d'éléments finis adaptatifs sur un maillage formés de quadrilatéres. Le raffinement local du maillage consiste en une subdivision réguliére des mailles marquées, faisant ainsi apparaitres des noeuds flottants. De plus, nous interdisons que deux mailles voisines aient deux niveaux de raffinements d'cart, et pour cela nous sommes contraints d'introduire des raffinement supplémentaires. Nous donnons alors une estimation de la complexité de cette technique de raffinement. Par rapport au cas des maillages triangulaires l'estimateur d'erreur contient un terme supplémentaire mesurant la dformation des quadrilatéres par rapportà un parallélogramme. Le résultat classiqe en P 1 sur la borne inférieure de l'estimateur n'est plus vérifié dans ce cas et nous dḿontrons alors une décroissance de l'estimateur pourétablir la convergence et analyser la complexité de la méthode Mots-clés : Eléments finis adaptatifs, convergence d'algorithmes adaptatifs, estimation de la complexité, maillages quadrilatéraux, noeuds flottants Convergence of an adaptive finite element method on quadrilateral meshes 3
Full H(div)-approximation of linear elasticity on quadrilateral meshes based on ABF finite elements
Computer Methods in Applied Mechanics and Engineering, 2019
For meshes of nondegenerate, convex quadrilaterals, we present a family of stable mixed finite element spaces for the mixed formulation of planar linear elasticity. The problem is posed in terms of the stress tensor, the displacement vector and the rotation scalar fields, with the symmetry of the stress tensor weakly imposed. The proposed spaces are based on the Arnold-Boffi-Falk (ABF k , k ≥ 0) elements for the stress and piecewise polynomials for the displacement and the rotation. We prove that these finite elements provide full H(div)−approximation of the stress field, in the sense that it is approximated to order h k+1 , where h is the mesh diameter, in the H(div)−norm. We show that displacement and rotation are also approximated to order h k+1 in the L 2 −norm. The convergence is optimal order for k ≥ 1, while the lowest order case, index k = 0, requires special treatment. The spaces also apply to both compressible and incompressible isotropic problems, i.e., the Poisson ratio may be one-half. The implementation as a hybrid method is discussed, and numerical results are given to illustrate the effectiveness of these finite elements.
Computers & Structures, 1992
Mesh refinement has become a standard tool in practical engineering stress analysis problems. In this paper, an algorithm is presented which combines the process of mesh refinement and redistribution in order to position the new nodes at asymptotically optimal locations. The algorithm is described in the context of a two-dimensional boundary element treatment of the equations of linear elasticity. Two examples are presented which verify the algorithm in the case of problems with and without singularities.
Convergence of adaptive finite element methods in computational mechanics
Applied Numerical Mathematics, 2009
The a priori convergence of finite element methods is based on the density property of the sequence of finite element spaces which essentially assumes a quasi-uniform meshrefining. The advantage is guaranteed convergence for a large class of data and solutions; the disadvantage is a global mesh refinement everywhere accompanied by large computational costs. Adaptive finite element methods (AFEMs) automatically refine exclusively wherever their refinement indication suggests to do so and consequently leave out refinements at other locations. In other words, the density property is violated on purpose and the a priori convergence is not guaranteed automatically and, in fact, crucially depends on algorithmic details. The advantage of AFEMs is a more effective mesh in many practical examples accompanied by smaller computational costs; the disadvantage is that the desirable convergence property is not guaranteed a priori. Efficient error estimators can justify a numerical approximation a posteriori and so achieve reliability. But it is not theoretically justified from the start that the adaptive mesh-refinement will generate an accurate solution at all. In order to foster the development of a convergence theory and improved design of AFEMs in computational engineering and sciences, this paper describes a particular version of an AFEM and analyses convergence results for three model problems in computational mechanics: linear elastic material (A), nonlinear monotone elastic material (B), and Hencky elastoplastic material (C). It establishes conditions sufficient for error-reduction in (A), for energy-reduction in (B), and eventually for strong convergence of the stress field in (C) in the presence of small hardening.
Adaptive Semi-Structured Mesh Refinement Techniques for the Finite Element Method
Applied Sciences, 2021
The adaptive mesh techniques applied to the Finite Element Method have continuously been an active research line. However, these techniques are usually applied to tetrahedra. Here, we use the triangular prismatic element as the discretization shape for a Finite Element Method code with adaptivity. The adaptive process consists of three steps: error estimation, marking, and refinement. We adapt techniques already applied for other shapes to the triangular prisms, showing the differences here in detail. We use five different marking strategies, comparing the results obtained with different parameters. We adapt these strategies to a conformation process necessary to avoid hanging nodes in the resulting mesh. We have also applied two special rules to ensure the quality of the refined mesh. We show the effect of these rules with the Method of Manufactured Solutions and numerical results to validate the implementation introduced.
Transition finite element families for adaptive analysis of axisymmetric elasticity problems
Finite Elements in Analysis and Design, 2011
In this paper, four transition element families which comprise five-to seven-node quadrilateral elements are developed based on the hybrid-stress and enhanced assumed strain (EAS) formulations for adaptive analyses of axisymmetric elasticity problems. For members in the first hybrid-stress family, a stress field with ten equilibrating stress modes is derived and employed by all members of the family. To study the effect of including more stress modes, another family with two additional stress modes is implemented. On the other hand, two EAS element families are constructed with respect to the incompatible displacement modes of two existing incompatible displacement transition element families. Several numerical examples are exercised. It can be seen that the first hybrid-stress family is the most accurate one among the proposed families. Moreover, the EAS families are close to the respective incompatible families in accuracy yet the former families are not only more efficient in computation but also more concise in formulation.
A convergent finite element method with adaptive√ 3 refinement
We develop an adaptive finite element method (AFEM) using piecewise linears on a sequence of triangulations obtained by adaptive √ 3 refinement. The motivation to consider √ 3 refinement stems from the fact that it is a slower topological refinement than the usual red or red-green refinement, and that it alternates the orientation of the refined triangles, such that certain features or singularities that are not aligned with the initial triangulation might be detected more quickly. On the other hand, the use of √ 3 refinement introduces the additional difficulty that the corresponding finite element spaces are nonnested. This makes the setting nonconforming. First we derive a BPX-type preconditioner for piecewise linears on the adaptively refined triangulations and we show that it gives rise to uniformly bounded condition numbers, so that we can solve the linear systems arising from the AFEM in an efficient way. Then we introduce the AFEM of Morin, Nochetto, and Siebert adapted to our special case for solving the Poisson equation. We prove that this adaptive strategy converges to the solution within any prescribed error tolerance in a finite number of steps. Finally we present some numerical experiments that show the optimality of both the BPX preconditioner and the AFEM.
E3S Web of Conferences
Finite element analyses of irregular structures require adaptive mesh refinement to achieve more accurate results in an efficient manner. This is also true for a non-conventional finite element method with Kriging interpolation, called the Kriging-based finite element method (K-FEM). This paper presents a study of automatic adaptive meshing procedures for analyses of two-dimensional linear elasticity problems using the K-FEM. The Matlab Partial Differential Equation Toolbox was utilized for generating meshes with Delaunay triangulation. Three error indicators, namely, the strain energy error, the gradient of effective stresses, and the element-free Galerkin strain energy error, were employed for estimating the element errors. To find the most effective error indicator, the resulting total number of elements and configurations of the final meshes were compared. The results show that the resulting final meshes were affected by the initial mesh configurations, the refinement criteria, ...
Adaptive refinement of quadrilateral finite element shell meshes
19th AIAA Applied Aerodynamics Conference, 2001
This paper describes and demonstrates a process for adaptive refinement of quadrilateral curved shell meshes using error estimates from MSC.Nastran ® . The meshes have been generated originally using Unigraphics'(UG) Scenario application.