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.

Self-adaptive one-dimensional nonlinear finite element method based on element energy projection method

The element energy projection （EEP） method for computation of super- convergent resulting in a one-dimensional finite element method （FEM） is successfully used to self-adaptive FEM analysis of various linear problems, based on which this paper presents a substantial extension of the whole set of technology to nonlinear problems. The main idea behind the technology transfer from linear analysis to nonlinear analysis is to use Newton＇s method to linearize nonlinear problems into a series of linear problems so that the EEP formulation and the corresponding adaptive strategy can be directly used without the need for specific super-convergence formulation for nonlinear FEM. As a re- sult, a unified and general self-adaptive algorithm for nonlinear FEM analysis is formed. The proposed algorithm is found to be able to produce satisfactory finite element results with accuracy satisfying the user-preset error tolerances by maximum norm anywhere on the mesh. Taking the nonlinear ordinary differential equation （ODE） of second-order as the model problem, this paper describes the related fundamental idea, the imple- mentation strategy, and the computational algorithm. Representative numerical exam- ples are given to show the efficiency, stability, versatility, and reliability of the proposed approach.

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.

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.

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.

Implementing the Node Based Smoothed Finite Element Method as User Element in Abaqus for Linear and Nonlinear Elasticity

In this paper,the node based smoothed-strain Abaqus user element(UEL)in the framework of finite element method is introduced.The basic idea behind of the node based smoothed finite element(NSFEM)is that finite element cells are divided into subcells and subcells construct the smoothing domain associated with each node of a finite element cell[Liu,Dai and Nguyen-Thoi(2007)].Therefore,the numerical integration is globally performed over smoothing domains.It is demonstrated that the proposed UEL retains all the advantages of the NSFEM,i.e.,upper bound solution,overly soft stiffness and free from locking in compressible and nearly-incompressible media.In this work,the constant strain triangular(CST)elements are used to construct node based smoothing domains,since any complex two dimensional domains can be discretized using CST elements.This additional challenge is successfully addressed in this paper.The efficacy and robustness of the proposed work is obtained by several benchmark problems in both linear and nonlinear elasticity.The developed UEL and the associated files can be downloaded from https://github.com/nsundar/NSFEM.

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.

RANDOM MICROSTRUCTURE FINITE ELEMENT METHOD FOR EFFECTIVE NONLINEAR PROPERTIES OF COMPOSITE MATERIALS

Some theoretical methods have been reported to deal with nonlinear problems of composite materials but the accuracy is not so good. In the meantime, a lot of linear problems are difficult to be managed by the theoretical methods. The present study aims to use the developed method, the random microstructure finite element method, to deal with these nonlinear problems. In this paper, the random microstructure finite element method is used to deal with all three kinds of nonlinear property problems of composite materials. The analyzed results suggest the influences of the nonlinear phenomena on the effective properties of composite materials are significant and the random microstructure finite element method is an effective tool to investigate the nonlinear problems.

NONLINEAR QUASI-CONFORMING FINITE ELEMENT METHOD

The nonlinear quasi-conforming FEM is presented based on the basic concept of the quasi- -conforming finite element. First, the incremental principle of stationary potential energy is discussed, Then, the formulation process of the nonlinear quasi-conforming FEM is given. Lastly, two computational examples of shells are given.

Quasi-optimal complexity of adaptive finite element method for linear elasticity problems in two dimensions

This paper introduces an adaptive finite element method （AFEM） using the newest vertex bisection and marking exclusively according to the error estimator without special treatment of oscillation. By the combination of the global lower bound and the localized upper bound of the posteriori error estimator, perturbation of oscillation, and cardinality of the marked element set, it is proved that the AFEM is quasi-optimal for linear elasticity problems in two dimensions, and this conclusion is verified by the numerical examples.

Finite Element Methods for Coupled Stokes and Darcy Problems

We derived and analyzed a new numerical scheme for the coupled Stokes and Darcy problems by using H（div） conforming elements in the entire domain. The approach employs the mixed finite element method for the Darcy equations and a stabilized H（div） finite element method for the Stokes equations. Optimal error estimates for the fluid velocity and pressure are derived. The finite element solutions from the new scheme not only feature a full satisfaction of the continuity equation, which is highly demanded in scientific computing, but also satisfy the mass conservation.

A Brief Summary of Finite Element Method Applications to Nonlinear Wave-structure Interactions

We review recent advances in the finite element method (FEM) simulations of interactions between waves and structures. Our focus is on the potential theory with the fully nonlinear or second-order boundary condition. The present paper has six sections. A review of previous work on interactions between waves and ocean structures is presented in Section one. Section two gives the mathematical formulation. In Section three, the finite element discretization, mesh generation and the finite element linear system solution methods are described. Section four presents numerical methods including time marching schemes, computation of velocity, remeshing and smoothing techniques and numerical radiation conditions. The application of the FEM to the wave-structure interactions are presented in Section five followed by the concluding remarks in Section six.

An Adaptive Finite Element Method Based on Optimal Error Estimates for Linear Elliptic Problems

The subject of the work is to propose a series of papers about adaptive finite element methods based on optimal error control estimate. This paper is the third part in a series of papers on adaptive finite element methods based on optimal error estimates for linear elliptic problems on the concave corner domains. In the preceding two papers (part 1:Adaptive finite element method based on optimal error estimate for linear elliptic problems on concave corner domain; part 2:Adaptive finite element method based on optimal error estimate for linear elliptic problems on nonconvex polygonal domains), we presented adaptive finite element methods based on the energy norm and the maximum norm. In this paper, an important result is presented and analyzed. The algorithm for error control in the energy norm and maximum norm in part 1 and part 2 in this series of papers is based on this result.

A Posteriori Error Estimates for Finite Element Methods for Systems of Nonlinear,Dispersive Equations

The present study regards the numerical approximation of solutions of systems of Korteweg-de Vries type,coupled through their nonlinear terms.In our previous work[9],we constructed conservative and dissipative finite element methods for these systems and presented a priori error estimates for the semidiscrete schemes.In this sequel,we present a posteriori error estimates for the semidiscrete and fully discrete approximations introduced in[9].The key tool employed to effect our analysis is the dispersive reconstruction devel-oped by Karakashian and Makridakis[20]for related discontinuous Galerkin methods.We conclude by providing a set of numerical experiments designed to validate the a posteriori theory and explore the effectivity of the resulting error indicators.

Finite Element Method Based on Equivalent Magnetic Energy Method for Computation of 2D Nonlinear Eddy Current Field

In this paper, the finite element method using vector potential in applications to 2D nonlinear eddy current field is discussed. The authors use the equivalent magnetic energy method to deal with magnetization curve of ferromagnetic material,and present the formulation of 2D nonlinear eddy current field.With this method the authors analyze the eddy current field in an induction ladle furnace and the force distribution in the charge (molten metal),and plot the corresponding curves.

A WEIGHTED PENALTY FINITE ELEMENT METHOD FOR THE ANALYSIS OF POWER-LAW FLUID FLOW PROBLEMS

In this paper, a new finite element method for the flow analysis of the viscous incompressible power-law fluid is proposed by the use of penalty-hybrid/mixed finite element formulation and by the introduction of an alternative perturbation, which is weighted by viscosity, of the continuity equation. A numerical example is presented to exhibit the efficiency of the method.

Analytical Model of Nonlinear Semi-rigid Frames Using Finite Element Method

Performance-based design for a constructional steel frame in nonlinear-plastic region requires an improvement in order to achieve a reliable structural analysis.The need to explicitly consider the nonlinear behaviour of structures makes the numerical modelling approach much more favourable than expensive and potentially dangerous experimental work.The parameters considered in the analysis are not limited to the linear change of geometry and material yielding,but also include the effect of large deformations,geometrical imperfections,load eccentricities,residual stresses,strain-unloading,and the nonlinear boundary conditions.Such analysis requires the use of accurate mathematical modelling and effective numerical procedures for solving equations of equilibrium.With that in mind,this paper presents the mathematical formulations and finite element procedures of nonlinear inelastic steel frame analysis with quasi-static semi-rigid connections.Verification and validation of the developed analytical procedures are conducted and good agreements are obtained.It is an approach that enables the structural behaviour of constructional steel frames to be traced throughout the entire range of loading until failure.It also provides information on the derivation of the structural analysis by using finite element method.