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1.
A LOCKING-FREE ANISOTROPIC NONCONFORMING FINITE ELEMENT FOR PLANAR LINEAR ELASTICITY PROBLEM 总被引:1,自引:1,他引:1
The main aim of this article is to study the approximation of a locking-free anisotropic nonconforming finite element for the pure displacement boundary value prob-lem of planar linear elasticity. The optimal error estimates are obtained by using some novel approaches and techniques. The method proposed in this article is robust in the sense that the convergence estimates in the energy and L2-norms are independent of the Lame parameterλ. 相似文献
2.
In this paper, a four-parameter quadrilateral nonconforming finite element with DSP (double set parameters) is presented.
Then we discuss the quadrilateral nonconforming finite element approximation to the linear elastic equations with pure displacement
boundary. The optimal convergence rate of the method is established in the broken energy and -norms, and in particular, the convergence is uniform with respect to the Lamé parameter . Also the performance of the scheme does not deteriorate as the material becomes nearly incompressible. Lastly, a numerical
test is carried out, which coincides with our theoretical analysis.
The research is supported by NSF of China (No. 10471133) and the project of Creative Engineering of Henan Province of China. 相似文献
3.
Son‐Young Yi 《Numerical Methods for Partial Differential Equations》2013,29(5):1749-1777
In this article, we develop a nonconforming mixed finite element method to solve Biot's consolidation model. In particular, this work has been motivated to overcome nonphysical oscillations in the pressure variable, which is known as locking in poroelasticity. The method is based on a coupling of a nonconforming finite element method for the displacement of the solid phase with a standard mixed finite element method for the pressure and velocity of the fluid phase. The discrete Korn's inequality has been achieved by adding a jump term to the discrete variational formulation. We prove a rigorous proof of a‐priori error estimates for both semidiscrete and fully‐discrete schemes. Optimal error estimates have been derived. In particular, optimality in the pressure, measured in different norms, has been proved for both cases when the constrained specific storage coefficient c0 is strictly positive and when c0 is nonnegative. Numerical results illustrate the accuracy of the method and also show the effectiveness of the method to overcome the nonphysical pressure oscillations. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013 相似文献
4.
Ruishu WANG Xiaoshen WANG Kai ZHANG Qian ZHOU 《Frontiers of Mathematics in China》2018,13(5):1121-1140
A hybridization technique is applied to the weak Galerkin finite element method (WGFEM) for solving the linear elasticity problem in mixed form. An auxiliary function, the Lagrange multiplier defined on the boundary of elements, is introduced in this method. Consequently, the computational costs are much lower than the standard WGFEM. Optimal order error estimates are presented for the approximation scheme. Numerical results are provided to verify the theoretical results. 相似文献
5.
This paper introduces a new family of nonconforming mixed finite elements for solving the linear elasticity equations on simplicial grids. Besides, this paper describes the construction of the lowest order basis functions. The construction only involves simple computations due to the new explicit stress shape function spaces and the procedure applies for high order cases. Numerical experiments for four benchmark problems in mechanics indicate the robust locking‐free behavior and show that the lowest order nonconforming mixed method leads to smaller stress errors than the first and second order standard Galerkin methods for the nearly incompressible case. 相似文献
6.
7.
Douglas N. Arnold Richard S. Falk Ragnar Winther. 《Mathematics of Computation》2007,76(260):1699-1723
In this paper, we construct new finite element methods for the approximation of the equations of linear elasticity in three space dimensions that produce direct approximations to both stresses and displacements. The methods are based on a modified form of the Hellinger-Reissner variational principle that only weakly imposes the symmetry condition on the stresses. Although this approach has been previously used by a number of authors, a key new ingredient here is a constructive derivation of the elasticity complex starting from the de Rham complex. By mimicking this construction in the discrete case, we derive new mixed finite elements for elasticity in a systematic manner from known discretizations of the de Rham complex. These elements appear to be simpler than the ones previously derived. For example, we construct stable discretizations which use only piecewise linear elements to approximate the stress field and piecewise constant functions to approximate the displacement field.
8.
HUA Dongying & WANG Lieheng The First Fundamental Department Beijing Information Technology Institute Beijing China Institute of Computational Mathematics Scientific/Engineering Computing Academy of Mathematics System Sciences Chinese Academy of Sciences Beijing China 《中国科学A辑(英文版)》2006,49(4):513-524
In this paper, we provide a new mixed finite element approximation of the varia-tional inequality resulting from the unilateral contact problem in elasticity. We use the continuous piecewise P2-P1 finite element to approximate the displacement field and the normal stress component on the contact region. Optimal convergence rates are obtained under the reasonable regularity hypotheses. Numerical example verifies our results. 相似文献
9.
Mohamed Farhloul And Michel Fortin 《Numerical Methods for Partial Differential Equations》1997,13(5):445-457
In a recent work, Hiptmair [Mathematisches Institut, M9404, 1994] has constructed and analyzed a family of nonconforming mixed finite elements for second-order elliptic problems. However, his analysis does not work on the lowest order elements. In this article, we show that it is possible to construct a nonconforming mixed finite element for the lowest order case. We prove the convergence and give estimates of optimal order for this finite element. Our proof is based on the use of the properties of the so-called nonconforming bubble function to control the consistency terms introduced by the nonconforming approximation. We further establish an equivalence between this mixed finite element and the nonconforming piecewise quadratic finite element of Fortin and Soulie [J. Numer. Methods Eng., 19, 505–520, 1983]. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 445–457, 1997 相似文献
10.
XIE XiaoPing & XU JinChao School of Mathematics Sichuan University Chengdu China 《中国科学 数学(英文版)》2011,(7)
We consider mixed finite elements for the plane elasticity system and the Stokes equation. For the unmodified Hellinger-Reissner formulation of elasticity in which the stress and displacement fields are the primary unknowns, we derive two new nonconforming mixed finite elements of triangle type. Both elements use piecewise rigid motions to approximate the displacement and piecewise polynomial functions to approximate the stress, where no vertex degrees of freedom are involved. The two stress finite element ... 相似文献
11.
The purpose of this article is to study a mixed formulation of the elasticity problem in plane polygonal domains and its numerical approximation. In this mixed formulation the strain tensor is introduced as a new unknown and its symmetry is relaxed by a Lagrange multiplier, which is nothing else than the rotation. Because of the corner points, the displacement field is not regular in general in the vicinity of the vertices but belongs to some weighted Sobolev space. Using this information, appropriate refinement rules are imposed on the family of triangulations in order to recapture optimal error estimates. Moreover, uniform error estimates in the Lamé coefficient λ are obtained for λ large. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 323–339, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10009 相似文献
12.
Two locking-free nonconforming finite elements are presented for three-dimensional elasticity problem with pure displacement boundary condition. Convergence rate of the elements are uniformly optimal with respect to λ. The energy norm and L2 norm errors are O(h2) and O(h3), respectively. Lastly, a numerical experiment is carried out, which coincides with the theoretical analysis. 相似文献
13.
In this paper, a new splitting positive definite nonconforming mixed finite element method is proposed for pseudo-hyperbolic equations, in which a quasi-Wilson quadrilateral element is used for the flux p, and the bilinear element is used for u. Superconvergence results in ||·||div,h norm for p and optimal error estimates in L2 norm for u are derived for both semi-discrete and fully discrete schemes under almost uniform meshes. 相似文献
14.
A superconvergent nonconforming mixed finite element method for the Navier–Stokes equations 下载免费PDF全文
The superconvergence for a nonconforming mixed finite element approximation of the Navier–Stokes equations is analyzed in this article. The velocity field is approximated by the constrained nonconforming rotated Q1 (CNRQ1) element, and the pressure is approximated by the piecewise constant functions. Under some regularity assumptions, the superconvergence estimates for both the velocity in broken H1‐norm and the pressure in L2‐norm are obtained. Some numerical examples are presented to demonstrate our theoretical results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 646–660, 2016 相似文献
15.
A nonconforming mixed finite element scheme is proposed for Sobolev equations based on a new mixed variational form under semi-discrete and Euler fully-discrete schemes. The corresponding optimal convergence error estimates and superclose property are obtained without using Ritz projection, which are the same as the traditional mixed finite elements. Furthemore, the global superconvergence is obtained through interpolation postprocessing technique. The numerical results show the validity of the theoretical analysis. 相似文献
16.
SHI Dong-yang~ WANG Cai-xia~ Dept.of Math. Zhengzhou Univ. Zhengzhou China. Faculty of Math.and Inform.Sci. North China Univ.of Water Conservancy Electric Power Zhengzhou China. 《高校应用数学学报(英文版)》2008,23(1):9-18
This paper deals with a new nonconforming anisotropic rectangular finite element approximation for the planar elasticity problem with pure displacement boundary condition. By use of the special properties of this element, and by introducing the complementary space and a series of novel techniques, the optimal error estimates of the energy norm and the L^2-norm are obtained. The restrictions of regularity assumption and quasi-uniform assumption or the inverse assumption on the meshes required in the conventional finite element methods analysis are to be got rid of and the applicable scope of the nonconforming finite elements is extended. 相似文献
17.
Jan H. Brandts 《Applications of Mathematics》2009,54(3):225-235
We show that a non-standard mixed finite element method proposed by Barrios and Gatica in 2007, is a higher order perturbation
of the least-squares mixed finite element method. Therefore, it is also superconvergent whenever the least-squares mixed finite
element method is superconvergent. Superconvergence of the latter was earlier investigated by Brandts, Chen and Yang between
2004 and 2006. Since the new method leads to a non-symmetric system matrix, its application seems however more expensive than
applying the least-squares mixed finite element method.
Dedicated to Ivan Hlaváček on the occasion of his 75th birthday 相似文献
18.
A combined method consisting of mixed finite element method (MFEM) for the pressure equation and expanded mixed finite element method with characteristics(CEMFEM) for the concentration equation is presented to solve the coupled system of incompressible miscible displacement problem. To solve the resulting nonlinear system of equations efficiently, the two‐grid algorithm relegates all of the Newton‐like iterations to grids much coarser than the original one, with no loss in order of accuracy. It is shown that coarse space can be extremely coarse and our algorithm achieve asymptotically optimal approximation when the mesh sizes satisfy . Numerical experiment is provided to confirm our theoretical results. 相似文献
19.
Ruchi Guo Tao Lin Yanping Lin 《Numerical Methods for Partial Differential Equations》2019,35(3):1243-1268
We construct and analyze a group of immersed finite element (IFE) spaces formed by linear, bilinear, and rotated Q1 polynomials for solving planar elasticity equation involving interface. The shape functions in these IFE spaces are constructed through a group of approximate jump conditions such that the unisolvence of the bilinear and rotated Q1 IFE shape functions are always guaranteed regardless of the Lamé parameters and the interface location. The boundedness property and a group of identities of the proposed IFE shape functions are established. A multi‐point Taylor expansion is utilized to show the optimal approximation capabilities for the proposed IFE spaces through the Lagrange type interpolation operators. 相似文献
20.
Shaochun Chen 《Journal of Computational and Applied Mathematics》2010,233(10):2534-2548
In this paper, we present two nonconforming finite elements for the pure displacement planar elasticity problem. Both of them are locking-free and have two order of convergence. Some numerical results attest the validity of our theoretical analysis. 相似文献