DIFFERENCE SCHEME AND NUMERICAL SIMULATION BASED ON MIXED FINITE ELEMENT METHOD FOR NATURAL CONVECTION PROBLEM

The non_stationary natural convection problem is studied. A lowest order finite difference scheme based on mixed finite element method for non_stationary natural convection problem, by the spatial variations discreted with finite element method and time with finite difference scheme was derived, where the numerical solution of velocity, pressure, and temperature can be found together, and a numerical example to simulate the close square cavity is given, which is of practical importance.

An Adaptive Least-Squares Mixed Finite Element Method for Fourth Order Parabolic Problems

A least-squares mixed finite element (LSMFE) method for the numerical solution of fourth order parabolic problems analyzed and developed in this paper. The Ciarlet-Raviart mixed finite element space is used to approximate. The a posteriori error estimator which is needed in the adaptive refinement algorithm is proposed. The local evaluation of the least-squares functional serves as a posteriori error estimator. The posteriori errors are effectively estimated. The convergence of the adaptive least-squares mixed finite element method is proved.

Error estimates of H^1-Galerkin mixed finite element method for Schrdinger equation

An H^1-Galerkin mixed finite element method is discussed for a class of second order SchrSdinger equation. Optimal error estimates of semidiscrete schemes are derived for problems in one space dimension. At the same time, optimal error estimates are derived for fully discrete schemes. And it is showed that the H1-Galerkin mixed finite element approximations have the same rate of convergence as in the classical mixed finite element methods without requiring the LBB consistency condition.

A MIXED FINITE ELEMENT METHOD FOR THE CONTACT PROBLEM IN ELASTICITY

Based on the analysis of [7] and [10], we present the mixed finite element approximation of the variational inequality resulting from the contact problem in elasticity. The convergence rate of the stress and displacement field are both improved from O（h3/4） to quasi-optimal O（h│logh│^1/4）. If stronger but reasonable regularity is available, the convergence rate can be optimal O（h）.

Discrete formulation of mixed finite element methods for vapor deposition chemical reaction equations

The vapor deposition chemical reaction processes, which are of extremely extensive applications, can be classified as a mathematical model by the following governing nonlinear partial differential equations containing velocity vector, temperature field, pressure field, and gas mass field. The mixed finite element （MFE） method is employed to study the system of equations for the vapor deposition chemical reaction processes. The semidiscrete and fully discrete MFE formulations are derived. And the existence and convergence （error estimate） of the semidiscrete and fully discrete MFE solutions are demonstrated. By employing MFE method to treat the system of equations for the vapor deposition chemical reaction processes, the numerical solutions of the velocity vector, the temperature field, the pressure field, and the gas mass field can be found out simultaneously. Thus, these researches are not only of important theoretical means, but also of extremely extensive applied vistas.

A Posteriori Error Estimate of Two Grid Mixed Finite Element Methods for Semilinear Elliptic Equations

In this paper, we present the a posteriori error estimate of two-grid mixed finite element methods by averaging techniques for semilinear elliptic equations. We first propose the two-grid algorithms to linearize the mixed method equations. Then, the averaging technique is used to construct the a posteriori error estimates of the two-grid mixed finite element method and theoretical analysis are given for the error estimators. Finally, we give some numerical examples to verify the reliability and efficiency of the a posteriori error estimator.

A mixed finite element method for the unilateral contact problem in elasticity

In this paper, we provide a new mixed finite element approximation of the variational 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.

Nonlinear simulation of arch dam cracking with mixed finite element method

This paper proposes a new, simple and efficient method for nonlinear simulation of arch dam cracking from the construction period to the operation period, which takes into account the arch dam construction process and temperature loads. In the calculation mesh, the contact surface of pair nodes is located at places on the arch dam where cracking is possible. A new effective iterative method, the mixed finite element method for friction-contact problems, is improved and used for nonlinear simulation of the cracking process. The forces acting on the structure are divided into two parts： external forces and contact forces. The displacement of the structure is chosen as the basic variable and the nodal contact force in the possible contact region of the local coordinate system is chosen as the iterative variable, so that the nonlinear iterative process is only limited within the possible contact surface and is much more economical. This method was used to simulate the cracking process of the Shuanghe Arch Dam in Southwest China. In order to prove the validity and accuracy of this method and to study the effect of thermal stress on arch dam cracking, three schemes were designed for calculation. Numerical results agree with actual measured data, proving that it is feasible to use this method to simulate the entire process of nonlinear arch dam cracking.

Study of adaptive finite element techniques for contact problem in elastic bodies

The adaptive element techniques of contact problem are studied by means of penalty method, and the error estimators are discussed. Based on error estimators, algorithm of the adaptive element techniques is developed, then the Gauss - Newton iterations are used which allow the nonlinear problem to be transformed into a sequence of linear sub- problems then easily solved. In addition, the algorithm can be applied into the simulation of de -bonding of fiber - reinforced composites.

LEAST-SQUARES MIXED FINITE ELEMENT METHOD FOR A CLASS OF STOKES EQUATION

A least-squares mixed finite element method was formulated for a class of Stokes equations in two dimensional domains. The steady state and the time-dependent Stokes' equations were considered. For the stationary equation, optimal H-t and L-2-error estimates are derived under the standard regularity assumption on the finite element partition ( the LBB-condition is not required). Far the evolutionary equation, optimal L-2 estimates are derived under the conventional Raviart-Thomas spaces.

A Second Order Characteristic Mixed Finite Element Method for Convection Diffusion Reaction Equations

A combined approximate scheme is defined for convection-diffusion-reaction equations. This scheme is constructed by two methods. Standard mixed finite element method is used for diffusion term. A second order characteristic finite element method is presented to handle the material derivative term, that is, the time derivative term plus the convection term. The stability is proved and the L2-norm error estimates are derived for both the scalar unknown variable and its flux. The scheme is of second order accuracy in time increment, symmetric, and unconditionally stable.

Mathematical Description and Finite Element Equation of 3D Coupled Thermo-elastic Contact Problem

Through defining slide yield function and floating potential function of thermo-contact surface, the complementary equation of thermo-contact boundary has been reached, the fundamental equations to solve 3D thermo-contact coupled problem have been listed. On this foundation, the finite element equation and definite solution condition of contact heat transfer have been given out. Based on virtual work principle and contact element technology, the finite element equation of 3D elastic contact system has been deduced under the effect of thermal stress. The pseudo load brought by contact gap have been introduced into this equation in order to reflect the contact state change. During iteration, once contact rigidity matrix is formed, it won’t change, which will make calculation reduce greatly.

EFFECTIVE NUMERICAL METHODS FOR ELASTO-PLASTIC CONTACT PROBLEMS WITH FRICTION

Several effective numerical methods for solving the elasto-plastic contact problems with friction are pres- ented.First,a direct substitution method is employed to impose the contact constraint conditions on condensed finite ele- ment equations,thus resulting in a reduction by half in the dimension of final governing equations.Second,an algorithm composed of contact condition probes and elasto-plastic iterations is utilized to solve the governing equation,which distinguishes two kinds of nonlinearities,and makes the solution unique.In addition,Positive-Negative Sequence Modifica- tion Method is used to condense the finite element equations of each substructure and an analytical integration is intro- duced to determine the elasto-plastic status after each time step or each iteration,hence the computational efficiency is en- hanced to a great extent.Finally,several test and practical examples are pressented showing the validity and versatility of these methods and algorithms.

BOUNDARY ELEMENT ANALYSIS OF CONTACT PROBLEMS USING ARTIFICIAL BOUNDARY NODE APPROACH

An improved version of the regular boundary element method, the artificial boundary node approach, is derived. A simple contact algorithm is designed and implemented into the direct boundary element, regular boundary element and artificial boundary node approaches. The exisiting and derived approaches are tested using some case studies. The results of the artificial boundary node approach are compared with those of the existing boundary element program, the regular element approach, ANSYS and analytical solution whenever possible. The results show the effectiveness of the artificial boundary node approach for a wider range of boundary offsets.

Existence and numerical approximation of a solution to frictional contact problem for electro-elastic materials

In this paper,a frictional contact problem between an electro-elastic body and an electrically conductive foundation is studied.The contact is modeled by normal compliance with finite penetration and a version of Coulomb’s law of dry friction in which the coefficient of friction depends on the slip.In addition,the effects of the electrical conductivity of the foundation are taken into account.This model leads to a coupled system of the quasi-variational inequality of the elliptic type for the displacement and the nonlinear variational equation for the electric potential.The existence of a weak solution is proved by using an abstract result for elliptic variational inequalities and a fixed point argument.Then,a finite element approximation of the problem is presented.Under some regularity conditions,an optimal order error estimate of the approximate solution is derived.Finally,a successive iteration technique is used to solve the problem numerically and a convergence result is established.

A NEW METHOD FOR SOLUTION OF 3D ELASTIC-PLASTIC FRICTIONAL CONTACT PROBLEMS

The solution of 3 D elastic-plastic frictional contact problems belongs to the un specified boundary problems where the interaction between two kinds of nonlinearities should occur. Considering the difficulties for the solution of 3 D frictional contact problems, the key part is the determination of the tangential slip states at the contact points, and a great amount of computing work is needed for a high accuracy result. A new method based on a combination of programming and iteration methods, which are respectively known as two main kinds of methods for contact analysis, was put forward to deal with 3 D elastic-plastic contact problems. Numerical results demonstrate the efficiency of the algorithm illustrated here.

UNIFORM SUPERCONVERGENCE ANALYSIS OF A TWO-GRID MIXED FINITE ELEMENT METHOD FOR THE TIME-DEPENDENT BI-WAVE PROBLEM MODELING D-WAVE SUPERCONDUCTORS

In this paper,a two-grid mixed finite element method(MFEM)of implicit Backward Euler(BE)formula is presented for the fourth order time-dependent singularly perturbed Bi-wave problem for d-wave superconductors by the nonconforming EQ_(1)^(rot) element.In this approach,the original nonlinear system is solved on the coarse mesh through the Newton iteration method,and then the linear system is computed on the fine mesh with Taylor’s expansion.Based on the high accuracy results of the chosen element,the uniform superclose and superconvergent estimates in the broken H^(1)-norm are derived,which are independent of the negative powers of the perturbation parameter appeared in the considered problem.Numerical results illustrate that the computing cost of the proposed two-grid method is much less than that of the conventional Galerkin MFEM without loss of accuracy.

Approximation of thermoelasticity contact problem with nonmonotone friction

The paper presents the formulation and approximation of a static thermoelasticity problem that describes bilateral frictional contact between a deformable body and a rigid foundation. The friction is in the form of a nonmonotone and multivalued law. The coupling effect of the problem is neglected. Therefore, the thermic part of the problem is considered independently on the elasticity problem. For the displacement vector, we formulate one substationary problem for a non-convex, locally Lipschitz continuous functional representing the total potential energy of the body. All problems formulated in the paper are approximated with the finite element method.

Error estimates of triangular mixed finite element methods for quasilinear optimal control problems

The goal of this paper is to study a mixed finite element approximation of the general convex optimal control problems governed by quasilinear elliptic partial differential equations. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive a priori error estimates both for the state variables and the control variable. Finally, some numerical examples are given to demonstrate the theoretical results.

Fully Discrete H^(1) -Galerkin Mixed Finite Element Methods for Parabolic Optimal Control Problems

In this paper,we investigate a priori and a posteriori error estimates of fully discrete H^(1)-Galerkin mixed finite element methods for parabolic optimal control prob-lems.The state variables and co-state variables are approximated by the lowest order Raviart-Thomas mixed finite element and linear finite element,and the control vari-able is approximated by piecewise constant functions.The time discretization of the state and co-state are based on finite difference methods.First,we derive a priori error estimates for the control variable,the state variables and the adjoint state variables.Second,by use of energy approach,we derive a posteriori error estimates for optimal control problems,assuming that only the underlying mesh is static.A numerical example is presented to verify the theoretical results on a priori error estimates.