我们提出了新的用于线性弹性的矩形混合有限元。该方法基于对Hellinger-Reissner泛函的修改，其中应力场的对称性通过引入拉格朗日乘子被弱施加。这些单元是Arnold、Falk和Winther[弱对称条件下线性弹性的混合有限元方法。Mathematics of Computation 76 (2007), pp. 1699--1723]中描述的最低阶单元的类似物。分片常数用于逼近位移和旋转。一阶BDM元用于逼近应力场的每一行。
英文摘要：
We present new rectangular mixed finite elements for linear elasticity. The approach is based on a modification of the Hellinger-Reissner functional in which the symmetry of the stress field is enforced weakly through the introduction of a Lagrange multiplier. The elements are analogues of the lowest order elements described in Arnold, Falk and Winther [ Mixed finite element methods for linear elasticity with weakly imposed symmetry. Mathematics of Computation 76 (2007), pp. 1699--1723]. Piecewise constants are used to approximate the displacement and the rotation. The first order BDM elements are used to approximate each row of the stress field.
We consider and discretize a mixed formulation for linear elasticity with weakly imposed symmetry in two and three dimensions. Whereas existing methods mainly deal with simplicial or polygonal meshes, we take advantage of isogeometric analysis (IGA) and consequently allow for shapes with curved boundaries. To introduce the discrete spaces we use isogeometric discrete differential forms defined by proper B-spline spaces. For the proposed schemes a proof of well-posedness and an error estimate are given. Further we discuss our ansatz by means of different numerical examples.
We introduce a pure--stress formulation of the elasticity eigenvalue problem with mixed boundary conditions. We propose an H(div)-based discontinuous Galerkin method that imposes strongly the symmetry of the stress for the discretization of the eigenproblem. Under appropriate assumptions on the mesh and the degree of polynomial approximation, we demonstrate the spectral correctness of the discrete scheme and derive optimal rates of convergence for eigenvalues and eigenfunctions. Finally, we provide numerical examples in two and three dimensions.
In this article we consider the linear elasticity problem in an axisymmetric three dimensional domain, with data which are axisymmetric and have zero angular component. The weak formulation of the the three dimensional problem reduces to a two dimensional problem on the meridian domain, involving weighted integrals. The problem is formulated in a mixed method framework with both the stress and displacement treated as unknowns. The symmetry condition for the stress tensor is weakly imposed. Well posedness of the continuous weak formulation and its discretization are shown. Two approximation spa…
Weakly imposed symmetry and robust preconditioners for Biot's consolidation model
University of Oslo（奥斯陆大学）；The Institute for Computational and Engineering Sciences, University of Texas Austin（德克萨斯大学奥斯汀分校计算与工程科学研究所）；Simula Research Laboratory（西幕拉研究实验室）
In this article we suggest two discretization methods based on isogeometric analysis (IGA) for planar linear elasticity. On the one hand, we apply the well-known ansatz of weakly imposed symmetry for the stress tensor and obtain a well-posed mixed formulation. Such modified mixed problems have been already studied by different authors. But we concentrate on the exploitation of IGA results to handle also curved boundary geometries. On the other hand, we consider the more complicated situation of strong symmetry, i.e. we discretize the mixed weak form determined by the so-called Hellinger-Reissn…
We develop a mixed finite element method for the coupled problem arising in the interaction between a free fluid governed by the Stokes equations and flow in deformable porous medium modeled by the Biot system of poroelasticity. Mass conservation, balance of stress, and the Beavers--Joseph--Saffman condition are imposed on the interface. We consider a fully mixed Biot formulation based on a weakly symmetric stress-displacement-rotation elasticity system and Darcy velocity-pressure flow formulation. A velocity-pressure formulation is used for the Stokes equations. The interface conditions are i…
In this work we introduce a dG framework for nonlinear elasticity based on a Bassi-Rebay (BR2) formulation. The framework encompasses compressible and incompressible hyperelastic materials and is capable of dealing with large deformations. In order to achieve stability, we combine higher-order lifting operators for the BR2 stabilization term with an adaptive stabilization strategy which relies on the BR2 Laplace operator stabilization and a penalty parameter based on the spectrum of the fourth-order elasticity tensor. Dirichlet boundary conditions for the displacement can be imposed by means o…
This study presents a generalized multiscale multimesh finite element method ($\text{M}^2$-FEM) that addresses several long-standing challenges in the numerical simulation of integral structural theories, often used to model multiscale and nonlocal effects. The major challenges in the numerical simulation of integral boundary value problems are primarily rooted in the coupling of the spatial discretization of the global (parent) and integral (child) domains which severely restricts the computational efficiency of existing algorithms by imposing an implicit trade-off in the accuracy achieved by…
The thoracic diaphragm is the muscle that drives the respiratory cycle of a human being. Using a system of partial differential equations (PDEs) that models linear elasticity we compute displacements and stresses in a two-dimensional cross section of the diaphragm in its contracted state. The boundary data consists of a mix of displacement and traction conditions. If these are imposed as they are, and the conditions are not compatible, this leads to reduced smoothness of the solution. Therefore, the boundary data is first smoothed using the least-squares radial basis function generated finite …
