A BICUBIC B-SPLINE FINITE ELEMENT METHOD FOR FOURTH-ORDER SEMILINEAR PARABOLIC OPTIMAL CONTROL PROBLEMS

A bicubic B-spline finite element method is proposed to solve optimal control problems governed by fourth-order semilinear parabolic partial differential equations.Its key feature is the selection of bicubic B-splines as trial functions to approximate the state and costate variables in two space dimensions.A Crank-Nicolson difference scheme is constructed for time discretization.The resulting numerical solutions belong to C2in space,and the order of the coefficient matrix is low.Moreover,the Bogner-Fox-Schmit element is considered for comparison.Two numerical experiments demonstrate the feasibility and effectiveness of the proposed method.

Neural network method for solving elastoplastic finite element problems

A basic optimization principle of Artificial Neural Network—the Lagrange Programming Neural Network (LPNN) model for solving elastoplastic finite element problems is presented. The nonlinear problems of mechanics are represented as a neural network based optimization problem by adopting the nonlinear function as nerve cell transfer function. Finally, two simple elastoplastic problems are numerically simulated. LPNN optimization results for elastoplastic problem are found to be comparable to traditional Hopfield neural network optimization model.

Gradient Recovery Based Two-Grid Finite Element Method for Parabolic Integro-Differential Optimal Control Problems

In this paper, the optimal control problem of parabolic integro-differential equations is solved by gradient recovery based two-grid finite element method. Piecewise linear functions are used to approximate state and co-state variables, and piecewise constant function is used to approximate control variables. Generally, the optimal conditions for the problem are solved iteratively until the control variable reaches error tolerance. In order to calculate all the variables individually and parallelly, we introduce a gradient recovery based two-grid method. First, we solve the small scaled optimal control problem on coarse grids. Next, we use the gradient recovery technique to recover the gradients of state and co-state variables. Finally, using the recovered variables, we solve the large scaled optimal control problem for all variables independently. Moreover, we estimate priori error for the proposed scheme, and use an example to validate the theoretical results.

Stress and Displacement Distribution of Soft Clay Slope with 2D and 3D Elastoplastic Finite Element Method

Based on elastoplastic model, 2D and 3D finite element method （FEM） are used to calculate the stress and displacement distribution in the soft clay slope under gravity and uniform load at the slope top. Stability analyses indicate that 3D boundary effect varies with the stress level of the slope. When the slope is stable, end effect of 3D space is not remarkable. When the stability decreases, end effect occurs; when the slope is at limit state, end effect reaches maximum. The energy causing slope failure spreads preferentially along y-z section, and when the failure resistance capability reaches the limit state, the energy can extend along x-axis direction. The 3D effect of the slope under uniform load on the top is related to the ratio of load influence width to slope height, and the effect is remarkable with the decrease of the ratio.

A FINITE ELEMENT METHOD FOR STRESS ANALYSIS OR ELASTOPLASTIC BODY WITH POLYGONAL LINE STRAIN——HARDENING

In this paper, the stress-strain curve of material is fitted by polygonal line composed of three lines. According to the theory of proportional loading in elastoplasticity, we simplify the complete stress-strain relations, which are given by the increment theory of elastoplasticity. Thus, the finite element equation with the solution of displacement is derived. The assemblage elastoplastic stiffness matrix can be obtained by adding something to the elastic matrix, hence it will shorten the computing time. The determination of every loading increment follows the von Mises yield criteria. The iterative method is used in computation. It omits the redecomposition of the assemblage stiffness matrix and it will step further to shorten the computing time. Illustrations are given to the high-order element application departure from proportional loading, the computation of unloading fitting to the curve and the problem of load estimation.

Finite element simulation of elastoplastic fieldnear crack tips and results for a central crackedplate of le-lhp material under tension

The elastoplastic field near crack tips is investigated through finite element simulation.A refined mesh model near the crack tip is proposed. In the mesh refining area, element size continuously varies from the nanometer scale to themicrometer scale and the millimeter scale. Graphics of the plastic zone, the crack tip blunting, and the deformed crack tip elements are given in the paper.Based on the curves of stress and plastic strain, closely near the crack tip, the stresssingularity index and the stress intensity factor,as well as the plastic strain singularity index and the plastic strain intensity factor are determined.Thestress and plastic strainsingular index vary with the load, while the dimensions of the stress and the plastic strain intensity factorsdependon the stress and the plastic strain singularity index, respectively. The singular field near the elastoplastic crack tip is characterized by the stress singularity index and the stress intensity factor, or alternativelythe plastic strain singularity index and the plastic strain intensityfactor.At the end of the paper, following Irwin’s concept of fracture mechanics,σδKσδKcriterion andεδQεδQcriterion are proposed.Besides, crack tip angle criterion is also presented.

HIGH ACCURACY FINITE VOLUME ELEMENT METHOD FOR TWO-POINT BOUNDARY VALUE PROBLEM OF SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS

In this paper, a high accuracy finite volume element method is presented for two-point boundary value problem of second order ordinary differential equation, which differs from the high order generalized difference methods. It is proved that the method has optimal order error estimate O(h3) in H1 norm. Finally, two examples show that the method is effective.

Two-grid methods for semi-linear elliptic interface problems by immersed finite element methods

In this paper,two-grid immersed finite element (IFE) algorithms are proposed and analyzed for semi-linear interface problems with discontinuous diffusion coefficients in two dimension.Because of the advantages of finite element (FE) formulation and the simple structure of Cartesian grids,the IFE discretization is used in this paper.Two-grid schemes are formulated to linearize the FE equations.It is theoretically and numerically illustrated that the coarse space can be selected as coarse as H =O(h^1/4)(or H =O(h^1/8)),and the asymptotically optimal approximation can be achieved as the nonlinear schemes.As a result,we can settle a great majority of nonlinear equations as easy as linearized problems.In order to estimate the present two-grid algorithms,we derive the optimal error estimates of the IFE solution in the L^p norm.Numerical experiments are given to verify the theorems and indicate that the present two-grid algorithms can greatly improve the computing efficiency.

TWO-LEVEL MULTISCALE FINITE ELEMENT METHODS FOR THE STEADY NAVIER-STOKES PROBLEM

In this article, on the basis of two-level discretizations and multiscale finite element method, two kinds of finite element algorithms for steady Navier-Stokes problem are presented and discussed. The main technique is first to use a standard finite element discretization on a coarse mesh to approximate low frequencies, then to apply the simple and Newton scheme to linearize discretizations on a fine grid. At this process, multiscale finite element method as a stabilized method deals with the lowest equal-order finite element pairs not satisfying the inf-sup condition. Under the uniqueness condition, error analyses for both algorithms are given. Numerical results are reported to demonstrate the effectiveness of the simple and Newton scheme.

A Priori and A Posteriori Error Estimates of Streamline Diffusion Finite Element Method for Optimal Control Problem Governed by Convection Dominated Diffusion Equation

In this paper,we investigate a streamline diffusion finite element approxi- mation scheme for the constrained optimal control problem governed by linear con- vection dominated diffusion equations.We prove the existence and uniqueness of the discretized scheme.Then a priori and a posteriori error estimates are derived for the state,the co-state and the control.Three numerical examples are presented to illustrate our theoretical results.

A POSTERIORI ERROR ESTIMATES OF FINITEELEMENT METHOD FOR PARABOLIC PROBLEMS

This paper deals with a-posteriori error estimates for piecewise linear finite element approximations of parabolic problems in two space dimensions. The analysis extends previous results for elliptic problems to the parabolic context.

Immersed Interface Finite Element Methods for Elasticity Interface Problems with Non-Homogeneous Jump Conditions

In this paper,a class of new immersed interface finite element methods (IIFEM) is developed to solve elasticity interface problems with homogeneous and non-homogeneous jump conditions in two dimensions.Simple non-body-fitted meshes are used.For homogeneous jump conditions,both non-conforming and conforming basis functions are constructed in such a way that they satisfy the natural jump conditions. For non-homogeneous jump conditions,a pair of functions that satisfy the same non-homogeneous jump conditions are constructed using a level-set representation of the interface.With such a pair of functions,the discontinuities across the interface in the solution and flux are removed;and an equivalent elasticity interface problem with homogeneous jump conditions is formulated.Numerical examples are presented to demonstrate that such methods have second order convergence.

P_1-nonconforming triangular finite element method for elliptic and parabolic interface problems

The lowest order Pl-nonconforming triangular finite element method （FEM） for elliptic and parabolic interface problems is investigated. Under some reasonable regularity assumptions on the exact solutions, the optimal order error estimates are obtained in the broken energy norm. Finally, some numerical results are provided to verify the theoretical analysis.

Distributed Lagrange Multiplier/Fictitious Domain Finite Element Method for a Transient Stokes Interface Problem with Jump Coefficients

The distributed Lagrange multiplier/fictitious domain(DLM/FD)-mixed finite element method is developed and analyzed in this paper for a transient Stokes interface problem with jump coefficients.The semi-and fully discrete DLM/FD-mixed finite element scheme are developed for the first time for this problem with a moving interface,where the arbitrary Lagrangian-Eulerian(ALE)technique is employed to deal with the moving and immersed subdomain.Stability and optimal convergence properties are obtained for both schemes.Numerical experiments are carried out for different scenarios of jump coefficients,and all theoretical results are validated.

Finite volume element method for analysis of unsteady reaction-diffusion problems

A finite volume element method is developed for analyzing unsteady scalar reaction-diffusion problems in two dimensions. The method combines the concepts that are employed in the finite volume and the finite element method together. The finite volume method is used to discretize the unsteady reaction-diffusion equation, while the finite element method is applied to estimate the gradient quantities at cell faces. Robustness and efficiency of the combined method have been evaluated on uniform rectangular grids by using available numerical solutions of the two-dimensional reaction-diffusion problems. The numerical solutions demonstrate that the combined method is stable and can provide accurate solution without spurious oscillation along the high-gradient boundary layers.

THE TWO-LEVEL STABILIZED FINITE ELEMENT METHOD BASED ON MULTISCALE ENRICHMENT FOR THE STOKES EIGENVALUE PROBLEM

In this paper,we first propose a new stabilized finite element method for the Stokes eigenvalue problem.This new method is based on multiscale enrichment,and is derived from the Stokes eigenvalue problem itself.The convergence of this new stabilized method is proved and the optimal priori error estimates for the eigenfunctions and eigenvalues are also obtained.Moreover,we combine this new stabilized finite element method with the two-level method to give a new two-level stabilized finite element method for the Stokes eigenvalue problem.Furthermore,we have proved a priori error estimates for this new two-level stabilized method.Finally,numerical examples confirm our theoretical analysis and validate the high effectiveness of the new methods.

ACCURACY ENHANCEMENT FOR THE SIGNORINI PROBLEM WITH FINITE ELEMENT METHOD

In this paper,we study the accuracy enhancement for the frictionless Signorini problem on a polygonal domain with linear finite elements.Numerical test is given to verify our result.

Streamline upwind finite element method using 6-node triangular element with adaptive remeshing technique for convective-diffusion problems

A streamline upwind finite element method using 6-node triangular element is presented. The method is applied to the convection term of the governing transport equation directly along local streamlines. Several convective-diffusion examples are used to evaluate efficiency of the method. Results show that the method is monotonic and does not produce any oscillation. In addition, an adaptive meshing technique is combined with the method to further increase accuracy of the solution, and at the same time, to minimize computational time and computer memory requirement.

A FINITE VOLUME ELEMENT METHOD FOR THERMAL CONVECTION PROBLEMS

Consider the finite volume element method for the thermal convection problem with the infinite Prandtl number. The author uses a conforming piecewise linear function on a fine triangulation for velocity and temperature, and a piecewise constant function on a coarse triangulation for pressure. For general triangulation the optimal order H1 norm error estimates are given.

Application for Superconvergence of Finite Element Approximations for the Elliptic Problem by Global and Local L²-Projection Methods

Numerical experiments are given to verify the theoretical results for superconvergence of the elliptic problem by global and local L2-Projection methods.