Mixed Finite Elements for Flexoelectric Solids

2017 ◽  
Vol 84 (8) ◽  
Author(s):  
Feng Deng ◽  
Qian Deng ◽  
Wenshan Yu ◽  
Shengping Shen
Keyword(s):  
Finite Elements ◽  
Mixed Finite Element ◽  
Gradient Elasticity ◽  
Mixed Finite Elements ◽  
Mixed Finite Element Method ◽  
Dielectric Materials ◽  
Concentrated Force ◽  
Lagrangian Multipliers ◽  
Strain Gradients ◽  
Mixed Fem

Flexoelectricity (FE) refers to the two-way coupling between strain gradients and the electric field in dielectric materials, and is universal compared to piezoelectricity, which is restricted to dielectrics with noncentralsymmetric crystalline structure. Involving strain gradients makes the phenomenon of flexoelectricity size dependent and more important for nanoscale applications. However, strain gradients involve higher order spatial derivate of displacements and bring difficulties to the solution of flexoelectric problems. This dilemma impedes the application of such universal phenomenon in multiple fields, such as sensors, actuators, and nanogenerators. In this study, we develop a mixed finite element method (FEM) for the study of problems with both strain gradient elasticity (SGE) and flexoelectricity being taken into account. To use C0 continuous elements in mixed FEM, the kinematic relationship between displacement field and its gradient is enforced by Lagrangian multipliers. Besides, four types of 2D mixed finite elements are developed to study the flexoelectric effect. Verification as well as validation of the present mixed FEM is performed through comparing numerical results with analytical solutions for an infinite tube problem. Finally, mixed FEM is used to simulate the electromechanical behavior of a 2D block subjected to concentrated force or voltage. This study proves that the present mixed FEM is an effective tool to explore the electromechanical behaviors of materials with the consideration of flexoelectricity.

A Stabilized Mixed FEM for Viscoplastic Flow Analysis

2000 ◽  
Author(s):  
Yong Liu ◽  
Antoinette M. Maniatty ◽  
Ottmar Klaas ◽  
Mark S. Shephard
Keyword(s):  
Mixed Finite Element ◽  
Flow Analysis ◽  
Mixed Finite Element Method ◽  
Viscoplastic Flow ◽  
Lagrange Equations ◽  
Flow Problems ◽  
Bulk Forming ◽  
Mixed Fem ◽  
Lbb Condition ◽  
Stabilized Mixed Finite Element

Abstract A stabilized, mixed finite element method for viscoplastic flow analysis is presented. Preliminary results show promise for modeling steady-state bulk forming processes. In this work, the Ladyzenskaya-Babuska-Brezzi (LBB) condition is circumvented by adding mesh dependent terms (stabilization terms), which are functions of the residual of the Euler-Lagrange equations, to the usual Galerkin method. The stabilized formulation and applications to plastic flow problems are presented. Numerical experiments using the stabilization method show that the stabilized, mixed FEM is effective and efficient for non-linear steady forming problems.

scholarly journals MIXED FINITE ELEMENT APPROXIMATION OF THE VECTOR LAPLACIAN WITH DIRICHLET BOUNDARY CONDITIONS

2012 ◽  
Vol 22 (09) ◽  
pp. 1250024 ◽  
Author(s):  
DOUGLAS N. ARNOLD ◽  
RICHARD S. FALK ◽  
JAY GOPALAKRISHNAN
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Stokes Equations ◽  
Mixed Finite Element ◽  
Mixed Finite Elements ◽  
Mixed Finite Element Method ◽  
Two Dimensions ◽  
Dirichlet Boundary Conditions ◽  
Element Approximation

We consider the finite element solution of the vector Laplace equation on a domain in two dimensions. For various choices of boundary conditions, it is known that a mixed finite element method, in which the rotation of the solution is introduced as a second unknown, is advantageous, and appropriate choices of mixed finite element spaces lead to a stable, optimally convergent discretization. However, the theory that leads to these conclusions does not apply to the case of Dirichlet boundary conditions, in which both components of the solution vanish on the boundary. We show, by computational example, that indeed such mixed finite elements do not perform optimally in this case, and we analyze the suboptimal convergence that does occur. As we indicate, these results have implications for the solution of the biharmonic equation and of the Stokes equations using a mixed formulation involving the vorticity.

scholarly journals A Framework for Nonconforming Mixed Finite Element Method for Elliptic Problems in ℝ3

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Gwanghyun Jo ◽  
J. H. Kim
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Experiments ◽  
Mixed Finite Element ◽  
Mixed Finite Elements ◽  
Mixed Finite Element Method ◽  
Optimal Order ◽  
Order Convergence ◽  
New Family ◽  
Optimal Order Convergence

In this paper, we suggest a new patch condition for nonconforming mixed finite elements (MFEs) on parallelepiped and provide a framework for the convergence. Also, we introduce a new family of nonconforming MFE space satisfying the new patch condition. The numerical experiments show that the new MFE shows optimal order convergence in Hdiv and L2-norm for various problems with discontinuous coefficient case.

A mesh-dependent norm analysis of low-order mixed finite element for elasticity with weakly symmetric stress

2014 ◽  
Vol 24 (11) ◽  
pp. 2155-2169 ◽  
Author(s):  
Mika Juntunen ◽  
Jeonghun Lee
Keyword(s):  
Finite Element ◽  
Stability Analysis ◽  
Finite Elements ◽  
Mixed Finite Element ◽  
Three Dimensional ◽  
Mixed Finite Elements ◽  
Complex Approach ◽  
Weakly Symmetric ◽  
Rectangular Element ◽  
Low Order

We consider mixed finite elements for linear elasticity with weakly symmetric stress. We propose a low-order three-dimensional rectangular element with optimal O(h) rate of convergence for all the unknowns. The element is a rectangular analogue of the simplified Arnold–Falk–Winther element. Instead of the elasticity complex approach, our stability analysis is based on new mesh-dependent norms.

scholarly journals Raviart–Thomas finite elements of Petrov–Galerkin type

2019 ◽  
Vol 53 (5) ◽  
pp. 1553-1576
Author(s):  
Francois Dubois ◽  
Isabelle Greff ◽  
Charles Pierre
Keyword(s):  
Finite Elements ◽  
Finite Volume ◽  
Mixed Finite Element ◽  
Mixed Finite Elements ◽  
Finite Volume Methods ◽  
Finite Volume Scheme ◽  
Test Functions ◽  
Shape Functions ◽  
Local Computation ◽  
Necessary And Sufficient

Finite volume methods are widely used, in particular because they allow an explicit and local computation of a discrete gradient. This computation is only based on the values of a given scalar field. In this contribution, we wish to achieve the same goal in a mixed finite element context of Petrov–Galerkin type so as to ensure a local computation of the gradient at the interfaces of the elements. The shape functions are the Raviart–Thomas finite elements. Our purpose is to define test functions that are in duality with these shape functions: precisely, the shape and test functions will be asked to satisfy some orthogonality property. This paradigm is addressed for the discrete solution of the Poisson problem. The general theory of Babuška brings necessary and sufficient stability conditions for a Petrov–Galerkin mixed problem to be convergent. In order to ensure stability, we propose specific constraints for the dual test functions. With this choice, we prove that the mixed Petrov–Galerkin scheme is identical to the four point finite volume scheme of Herbin, and to the mass lumping approach developed by Baranger, Maitre and Oudin. Convergence is proven with the usual techniques of mixed finite elements.

scholarly journals DUAL MATHEMATICAL MODELS OF LIMIT LOAD ANALYSIS PROBLEMS OF STRUCTURES BY MIXED FINITE ELEMENTS

1997 ◽  
Vol 3 (10) ◽  
pp. 43-51
Author(s):  
Stanislovas Kalanta
Keyword(s):  
Finite Elements ◽  
Mathematical Programming ◽  
Mathematical Models ◽  
Optimization Problems ◽  
Mixed Finite Elements ◽  
Limit Load ◽  
Parameter Analysis ◽  
Lagrangian Multipliers ◽  
Multipliers Method ◽  
Load Analysis

The general and discrete dual mathematical models of the limit load analysis and optimization problems of rigid-plastic body are created in the article. The discrete models are formulated by mixed finite elements and presented in terms of kinematic and static formulation. In these models the velocity of the energy dissipation is estimated not only within the volume of finite elements, but also at the plastic surfaces between elements, where the discontinuities of displacement velocities functions appear. The theory of plastic flow, the theory of duality and mathematical programming are applied. The mixed energy functional (1) and (3) of both problems are formulated using the general static formulations of these problems, presented in the article [10], and Lagrangian multipliers method. The mixed finite elements are used for their discretization. The discrete expressions (8), (9) and (13) of mixed functionals are given choosing the interpolation functions (7) for the stress, displacement velocities, plastic multipliers and external load. Stationary conditions are created by static variables (stress and load vectors) of theses functionals. The discrete expressions of the geometric compatibility equations and constraint of load power are received from them. Using them as preliminary conditions for the functionals (8) and (9), the mathematical models (14), (15) and (17) of kinematic formulation of limit load analysis and optimization problems are formulated. The model (20) with a smaller number of unknowns is formed by elimination the displacement velocities. Using Lagrangian multipliers method, the mathematical models (21)-(23) of static formulation for the limit load parameter analysis problem and the models (24)-(26) for the load optimization problem are derived. All of them are the problems of mathematical programming. The mathematical models of static formulation for engineering purposes are more important and fit better. They are easier solved (a smaller quantity of unknowns), besides, they allow to determine the optimum distribution of the load. The formulated mathematical models allow to determine upper values of limit load, stresses, displacement and plastic multipliers velocities. Together with equilibrium models of these problems, presented in the article [10], they allow to determine the lower and upper values of aforementioned parameters. So, a good possibility is created to check reliability and exactness of numerical calculation results and to establish, if the computing net density of finite elements is sufficient.

A NOTE ON UNIFORM SUPERCONVERGENCE FOR THE TIMOSHENKO BEAM USING MIXED FINITE ELEMENTS

1994 ◽  
Vol 04 (06) ◽  
pp. 795-806 ◽  
Author(s):  
JAN H. BRANDTS
Keyword(s):  
Finite Elements ◽  
Timoshenko Beam ◽  
Mixed Finite Element ◽  
Plate Thickness ◽  
Mixed Finite Elements ◽  
Posteriori Error ◽  
Mindlin Plate ◽  
Plate Bending ◽  
The One ◽  
Posteriori Error Estimators

In this paper we present some results on the discretization by mixed finite elements of the Timoshenko beam, i.e. the one-dimensional Reissner-Mindlin plate bending problem. The results concern superconvergence. Superconvergence (of the displacement at nodal points and of the gradient at Gaussian points) for plate bending problems was considered before, but these earlier results degenerate for small values of the plate thickness d. Here, we prove superconvergence of the mixed finite element solutions to projections of the real solutions on the approximating spaces in the global H1(I)-norm uniform in d. These facts can be used to obtain asymptotically exact a posteriori error estimators, uniform in d, by means of an easy implementable and cheap post-processing. Numerical experiments illustrate the conclusions.

MIXED FINITE ELEMENTS WITH MASS-LUMPING FOR THE TRANSIENT WAVE EQUATION

2000 ◽  
Vol 08 (01) ◽  
pp. 171-188 ◽  
Author(s):  
GARY COHEN ◽  
SANDRINE FAUQUEUX
Keyword(s):  
Finite Elements ◽  
Stiffness Matrix ◽  
Mixed Finite Element ◽  
Mass Matrix ◽  
Mixed Finite Elements ◽  
Computation Time ◽  
Spectral Elements ◽  
Transient Wave ◽  
Time Step ◽  
Mass Lumping

Solving the acoustics equation by finite elements with mass-lumping requires the use of spectral elements. Although avoiding the inversion of a mass-matrix at each time-step, these elements remain expensive from the point of view of the stiffness-matrix. In this paper, we give a mixed finite element method which provides a factorization of the stiffness-matrix which leads to a gain of storage and computation time which grows with the order of the method and the dimension in space. After proving the equivalence between classical spectral elements and this method, we give a dispersion analysis on nonregular periodic meshes. Then, we analyze the accuracy and the stability of Q3 and Q5 approximations on numerical tests in 2D.

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