Family Of Mixed Finite Elements Research Articles

Taylor-Hood Like Finite Elements for Nearly Incompressible Strain Gradient Elasticity Problems

We propose a family of mixed finite elements that are robust for the nearly incompressible strain gradient model, which is a fourth-order singular perturbed elliptic system. The element is similar to [C. Taylor and P. Hood, Comput. & Fluids, 1(1973), 73–100] in the Stokes flow. Using a uniform discrete B-B inequality for the mixed finite element pairs, we show the optimal rate of convergence that is robust in the incompressible limit. By a new regularity result that is uniform in both the materials parameter and the incompressibility, we prove the method converges with 1/2 order to the solution with strong boundary layer effects. Moreover, we estimate the convergence rate of the numerical solution to the unperturbed second-order elliptic system. Numerical results for both smooth solutions and the solutions with sharp layers confirm the theoretical prediction.

A family of mixed finite elements for the biharmonic equations on triangular and tetrahedral grids

This paper introduces a new family of mixed finite elements for solving a mixed formulation of the biharmonic equations in two and three dimensions. The symmetric stress σ = −∇2u is sought in the Sobolev space H(divdiv, Ω; $$\mathbb{S}$$ S ) simultaneously with the displacement u in L2(Ω). By stemming from the structure of H(div, Ω; $$\mathbb{S}$$ S ) conforming elements for the linear elasticity problems proposed by Hu and Zhang (2014), the H(divdiv, Ω; $$\mathbb{S}$$ S ) conforming finite element spaces are constructed by imposing the normal continuity of divσ on the H (div, Ω; $$\mathbb{S}$$ S ) conforming spaces of Pk symmetric tensors. The inheritance makes the basis functions easy to compute. The discrete spaces for u are composed of the piecewise Pk−2 polynomials without requiring any continuity. Such mixed finite elements are inf-sup stable on both triangular and tetrahedral grids for k ⩾ 3, and the optimal order of convergence is achieved. Besides, the superconvergence and the postprocessing results are displayed. Some numerical experiments are provided to demonstrate the theoretical analysis.

Higher order multipoint flux mixed finite element methods on quadrilaterals and hexahedra

We develop higher order multipoint flux mixed finite element (MFMFE) methods for solving elliptic problems on quadrilateral and hexahedral grids that reduce to cell-based pressure systems. The methods are based on a new family of mixed finite elements, which are enhanced Raviart–Thomas spaces with bubbles that are curls of specially chosen polynomials. The velocity degrees of freedom of the new spaces can be associated with the points of tensor-product Gauss–Lobatto quadrature rules, which allows for local velocity elimination and leads to a symmetric and positive definite cell-based system for the pressures. We prove optimal [Formula: see text]th order convergence for the velocity and pressure in their natural norms, as well as [Formula: see text]st order superconvergence for the pressure at the Gauss points. Moreover, local postprocessing gives a pressure that is superconvergent of order [Formula: see text] in the full [Formula: see text]-norm. Numerical results illustrating the validity of our theoretical results are included.

Compatible-strain mixed finite element methods for incompressible nonlinear elasticity

We introduce a new family of mixed finite elements for incompressible nonlinear elasticity – compatible-strain mixed finite element methods (CSFEMs). Based on a Hu–Washizu-type functional, we write a four-field mixed formulation with the displacement, the displacement gradient, the first Piola–Kirchhoff stress, and a pressure-like field as the four independent unknowns. Using the Hilbert complexes of nonlinear elasticity, which describe the kinematics and the kinetics of motion, we identify the solution spaces of the independent unknown fields. In particular, we define the displacement in H1, the displacement gradient in H(curl), the stress in H(div), and the pressure field in L2. The test spaces of the mixed formulations are chosen to be the same as the corresponding solution spaces. Next, in a conforming setting, we approximate the solution and the test spaces with some piecewise polynomial subspaces of them. Among these approximation spaces are the tensorial analogues of the Nédélec and Raviart–Thomas finite element spaces of vector fields. This approach results in compatible-strain mixed finite element methods that satisfy both the Hadamard compatibility condition and the continuity of traction at the discrete level independently of the refinement level of the mesh. By considering several numerical examples, we demonstrate that CSFEMs have a good performance for bending problems and for bodies with complex geometries. CSFEMs are capable of capturing very large strains and accurately approximating stress and pressure fields. Using CSFEMs, we do not observe any numerical artifacts, e.g., checkerboarding of pressure, hourglass instability, or locking in our numerical examples. Moreover, CSFEMs provide an efficient framework for modeling heterogeneous solids.

A Study of Two Modes of Locking in Poroelasticity

In this paper, we study two modes of locking phenomena in poroelasticity: Possion locking and pressure oscillations. We first study the regularity of the solution of the Biot model to gain some insight into the cause of Poisson locking and show that the displacement gets into a divergence-free state as the Lamé constant $\lambda \to \infty$. We also examine the cause of pressure oscillations from an algebraic point of view when a three-field mixed finite element method is used. Based on the results of our study on the causes of the two modes of locking, we propose a new family of mixed finite elements that are free of both pressure oscillations and Poisson locking. Some numerical results are presented to validate our theoretical studies.

Topology optimization with mixed finite elements on regular grids

Recently, new families of mixed finite elements have been proposed to address the analysis of linear elastic bodies on regular grids adopting a limited number of degrees of freedom per element. A two-dimensional mixed discretization is implemented to formulate an alternative topology optimization problem where stresses play the role of main variables and both compressible and incompressible materials can be dealt with. The structural compliance is computed through the evaluation of the complementary energy, whereas the enforcement of stress constraints is straightforward. Numerical simulations investigate the features of the proposed approach: comparisons with a conventional displacement-based scheme are provided for compressible materials; stress-constrained solutions for structures made of incompressible media are introduced.

The simplest nonconforming mixed finite element method for linear elasticity in the symmetric formulation on [formula omitted]-rectangular grids

A family of mixed finite elements is proposed for solving the first order system of linear elasticity equations in any space dimension, where the stress field is approximated by symmetric finite element tensors. This family of elements has a perfect matching between the stress and the displacement. The discrete spaces for the normal stress τii, the shear stress τij and the displacement ui are span{1,xi}, span{1,xi,xj} and span{1}, respectively, on rectangular grids. In particular, the definition remains the same for all space dimensions. As a result of these choices, the theoretical analysis is independent of the spatial dimension as well. In 1D, the element is nothing else but the 1D Raviart–Thomas element, which is the only conforming element in this family. In 2D and higher dimensions, they are new elements but of the minimal degrees of freedom. The total degrees of freedom per element are 2 plus 1 in 1D, 7 plus 2 in 2D, and 15 plus 3 in 3D. These elements are the simplest element for any space dimension.The well-posedness condition and the optimal a priori error estimate of the family of finite elements are proved. Numerical tests in 2D and 3D are presented to show a superiority of the new elements over others, as a superconvergence is exhibited and proved.

Two Families of $H$(div) Mixed Finite Elements on Quadrilaterals of Minimal Dimension

We develop two families of mixed finite elements on quadrilateral meshes for approximating $({\mathbf u},p)$ solving a second order elliptic equation in mixed form. Standard Raviart--Thomas (RT) and Brezzi--Douglas--Marini (BDM) elements are defined on rectangles and extended to quadrilaterals using the Piola transform, which are well-known to lose optimal approximation of $\nabla\cdot{\mathbf u}$. Arnold--Boffi--Falk spaces rectify the problem by increasing the dimension of RT, so that approximation is maintained after Piola mapping. Our two families of finite elements are uniformly inf-sup stable, achieve optimal rates of convergence, and have minimal dimension. The elements for ${\mathbf u}$ are constructed from vector polynomials defined directly on the quadrilaterals, rather than being transformed from a reference rectangle by the Piola mapping, and then supplemented by two (one for the lowest order) basis functions that are Piola mapped. One family has full $H(div)$-approximation (${\mathbf u}$, $p$...

A New Family of Efficient Conforming Mixed Finite Elements on Both Rectangular and Cuboid Meshes for Linear Elasticity in the Symmetric Formulation

A new family of mixed finite elements is proposed for solving the classical Hellinger--Reissner mixed problem of the elasticity equations. For two dimensions, the normal stress of the matrix-valued stress field is approximated by an enriched Brezzi--Douglas--Fortin--Marini element of order $k$, the shear stress by the serendipity element of order $k$, and the displacement field by an enriched discontinuous vector-valued $P_{k-1}$ element. The degrees of freedom on each element of the lowest order element, which is of first order, are 10 plus 4. For three dimensions, the normal stress is approximated by an enriched Raviart--Thomas element of order $k$, each component of the shear stress by a product space of the serendipity element space of two variables and the space of polynomials of degree $\leq k-1$ with respect to the rest variable, and the displacement field by an enriched discontinuous vector-valued $Q_{k-1}$ element. The degrees of freedom on each element of the lowest order element, which is of first order, are 21 plus 6. A family of reduced elements is also proposed by dropping some interior bubble functions of the stress and employing the discontinuous vector-valued $P_{k-1}$ (resp., $Q_{k-1}$) element for the displacement field on each element. As a result the lowest order elements have 8 plus 2 and 18 plus 3 degrees of freedom on each element for two and three dimensions, respectively. The well-posedness condition and the optimal a priori error estimate are proved for this family of finite elements. Numerical tests are presented to confirm the theoretical results.

An augmented mixed finite element method for the vorticity–velocity–pressure formulation of the Stokes equations

This paper deals with the numerical approximation of the stationary two-dimensional Stokes equations, formulated in terms of vorticity, velocity and pressure, with non-standard boundary conditions. Here, by introducing a Galerkin least-squares term, we end up with a stabilized variational formulation that can be recast as a twofold saddle point problem. We propose two families of mixed finite elements to solve the discrete problem, in the first family, the unknowns are approximated by piecewise continuous and quadratic elements, Brezzi–Douglas–Marini, and piecewise constant finite elements, respectively, while in the second family, the unknowns are approximated by piecewise linear and continuous, Raviart–Thomas, and piecewise constant finite elements, respectively. The wellposedness of the resulting continuous and discrete variational problems are studied employing an extension of the Babuška–Brezzi theory. We establish a priori error estimates in the natural norms, and we finally report some numerical experiments illustrating the behavior of the proposed schemes and confirming our theoretical findings on structured and unstructured meshes. Additional examples of cases not covered by our theory are also presented.

Anisotropic mixed finite elements for elasticity

SUMMARYIn this paper, we present a family of mixed finite elements, which are suitable for the discretization of slim domains. The displacement space is chosen as Nédélec's space of tangential continuous elements, whereas the stress is approximated by normal–normal continuous symmetric tensor‐valued finite elements. We show stability of the system on a slim domain discretized by a tensor product mesh, where the constant of stability does not depend on the aspect ratio of the discretization. We give interpolation operators for the finite element spaces, and thereby obtain optimal order a priori error estimates for the approximate solution. All estimates are independent of the aspect ratio of the finite elements. Copyright © 2011 John Wiley & Sons, Ltd.

Convergence results of the fictitious domain method for a mixed formulation of the wave equation with a Neumann boundary condition

The problem of modeling acoustic waves scattered by an object with Neumann boundary condition is considered. The boundary condition is taken into account by means of the fictitious domain method, yielding a first order in time mixed variational formulation for the problem. The resulting system is discretized with two families of mixed finite elements that are compatible with mass lumping. We present numerical results illustrating that the Neumann boundary condition on the object is not always correctly taken into account when the first family of mixed finite elements is used. We, therefore, introduce the second family of mixed finite elements for which a theoretical convergence analysis is presented and error estimates are obtained. A numerical study of the convergence is also considered for a particular object geometry which shows that our theoretical error estimates are optimal.

Éléments finis mixtes et condensation de masse en élastodynamique linéaire. (I) Construction

We introduce in this Note a new family of mixed finite elements for the linear elastodynamic problem. We present the construction of the lowest order element in the 2D case together with the dispersion analysis for an isotropic homogeneous medium. This construction can be extended to higher orders as well as to the 3D case.

Analysis of the Kleiser‐Schumann method for a family of mixed finite elements for the Stokes equations

AbstractWe study the Stokes problem by using a mixed finite element method introduced by Stenberg [Numer. Math. 56, 827 (1990)] and adapt it to the splitting introduced by Canuto and Sacchi Landriani [Numer. Math. 50, 217 (1986)] and Chinosi and Comodi [Numer. Methods Partial Different. Equations 7, 363 (1991)], by using the incompressibility condition to implicitly define a boundary condition for the pressure. © 1994 John Wiley & Sons, Inc.

Two‐dimensional elastoplasticity: Approximation by mixed finite elements

AbstractTwo‐dimensional elastoplastic problems are used to assess the performance of a family of mixed finite elements in the non‐linear regime. In particular, four‐node bilinear quadrilateral elements that exhibit high accuracy in coarse meshes are used to simulate plane stress, plane strain, axisymmetric and (shear‐deformable) plate bending problems. The excellent performance of these elements is demonstrated with a number of discriminating examples selected from the literature. In addition, agreement is observed in the limit load predicted by the axisymmetric solid and plate bending elements.

Finite-rotation shell elements via mixed formulation

Starting from a tensorial five-parametric finite-rotation shell theory a family of mixed finite elements is developed on the basis of a Reissner-Mindlin type functional. The family developed contains 4-node and 9-node quadrilateral shell elements. In each of them the displacement approximation is combined with various force variable interpolations in order to improve flexibility for numerical applications. The so-called difference vector occurring in the shell theory is expressed in terms of new rotational degrees of freedom which permit a unique determination of this variable in every deformed position. The corresponding constraints are then satisfied at the element level numerically. Due to the underlying theory the numerical models developed are able to predict the physical 2D force variables accurately. Their capability to deal with strongly nonlinear situations is demonstrated by several examples where numerical results due to Kirchhoff-Love type elements are also included for a systematical comparison.

Mixed finite element method applied to a magnetostatic problem

A new method for approximating the magnetostatic problem is given. The charateristics of this method are surface and volume integral formulation and discretization with a family of mixed finite elements. A mathematical analysis is discussed and numerical results are compared with experimental values.

A family of mixed finite elements for linear elasticity

A family of finite elements for use in mixed formulations of linear elasticity is developed. The stresses are not required to be symmetric, but only to satisfy a weaker condition based upon Lagrange multipliers. This is based on the same formulation used in the PEERS finite element spaces. Elements for both two and three dimensional problems are given. Error analysis on these elements is done, and some superconvergence results are proved.

Prismatic mixed finite elements for second order elliptic problems

In this paper, three families of mixed finite elements based on prisms are introduced. These spaces are analogues to those based on simplices and cubes in three space variables. Error estimates in L2 and H−5 are given.

A family of mixed finite elements for the elasticity problem

A new mixed finite element formulation for the equations of linear elasticity is considered. In the formulation the variables approximated are the displacement, the unsymmetric stress tensor and the rotation. The rotation act as a Lagrange multiplier introduced in order to enforce the symmetry of the stress tensor. Based on this formulation a new family of both two-and three-dimensional mixed methods is defined. Optimal error estimates, which are valid uniformly with respect to the Poisson ratio, are derived. Finally, a new postprocessing scheme for improving the displacement is introduced and analyzed.