Composite Finite Elements for Elliptic Boundary Value Problems with Discontinuous Coefficients

In this paper, we will introduce composite finite elements for solving elliptic boundary value problems with discontinuous coefficients. The focus is on problems where the geometry of the interfaces between the smooth regions of the coefficients is very complicated. On the other hand, efficient numerical methods such as, e.g., multigrid methods, wavelets, extrapolation, are based on a multi-scale discretization of the problem. In standard finite element methods, the grids have to resolve the structure of the discontinuous coefficients. Thus, straightforward coarse scale discretizations of problems with complicated coefficient jumps are not obvious. In this paper, we define composite finite elements for problems with discontinuous coefficients. These finite elements allow the coarsening of finite element spaces independently of the structure of the discontinuous coefficients. Thus, the multigrid method can be applied to solve the linear system on the fine scale. We focus on the construction of the composite finite elements and the efficient, hierarchical realization of the intergrid transfer operators. Finally, we present some numerical results for the multigrid method based on the composite finite elements (CFE–MG).     

Large natural strains and some special difficulties due to non-linearity and incompressibility in finite elements

The finite element method is now a well established tool for the routine treatment of large linear problems, but the treatment of non-linear problems by the method is yet at the beginning.Section 1 of the present work extends the idea of natural strains and stresses to large strains in simplex finite elements.Section 2 applies some algorithms developed for structural dynamics to the problem of non-linear wave propagation including a case of shock development.Section 3 discusses with numerical examples special difficulties when displacement finite elements are used to solve problems with incompressible or nearly incompressible material.     

A numerical method for a kinetic equation and its application to propagating shock waves

An efficient numerical method for kinetic equations and its application to analyses of moving shock wave problems are presented. The present study aims to give an efficient scheme for two-dimensional unsteady gas flows. An explicit MacCormack difference method is applied to solve a BGK-model equation. The efficiency and accuracy of the scheme are examined in an application to one-dimensional shock structure problems. Furthermore, the scheme is applied to a two-dimensional flow problem: nonstationary reflection of a shock wave at a wedge. The present scheme is found to be useful and efficient for the analyses of two-dimensional unsteady rarefied gas flows.     

Mixed finite elements based upon marguerre theory for the study of geometrically nonlinear behavior of thin shells

Simple triangular and quadrangular finite elements based on Marguerre's theory are proposed and are shown to greatly improve the solution over plane shell elements for a small additional computation cost.Several features of the developments are worth noting, namely: The presentation of a dual approach for the derivation of Marguerre's theory of shallow shells with moderate rotations, based on Fraeijs de Veubeke's variational principle, with a precise statement of hypotheses and applicability; the choice of hybrid connectors for solving the compatibility problem generated by Kirchhoff's hypothesis; a treatment of pressure loading, body forces and inextensional bending modes which employs a ‘static approach’ for the membrane; and discussion about the performance of some algorithms used to solve elastic stability problems. Numerical studies indicate that accurate results may be obtained by the approach advocated.     

四阶高分辨率熵相容算法

针对一维Burgers方程和一维Euler方程组的数值求解问题,提出了一种四阶高分辨率熵相容算法。新算法时间方向采用半离散方式,空间方向应用四阶中心加权基本无振荡(CWENO)重构方法,数值通量引入Ismail通量函数,将新的四阶算法应用于静态激波问题、激波管问题以及强稀疏波问题的数值求解中,并将所得结果同准确解以及已有算法所得结果进行了分析与比较。数值结果表明:新算法计算结果正确、分辨率高,能够准确捕捉激波及稀疏波,并能有效避免膨胀激波的产生。新算法适用于准确解决一维Burgers方程和一维Euler方程组的数值求解问题。     

Dispersive and Dissipative Properties of Discontinuous Galerkin Finite Element Methods for the Second-Order Wave Equation

Discontinuous Galerkin finite element methods (DGFEM) offer certain advantages over standard continuous finite element methods when applied to the spatial discretisation of the acoustic wave equation. For instance, the mass matrix has a block diagonal structure which, used in conjunction with an explicit time stepping scheme, gives an extremely economical scheme for time domain simulation. This feature is ubiquitous and extends to other time-dependent wave problems such as Maxwell’s equations. An important consideration in computational wave propagation is the dispersive and dissipative properties of the discretisation scheme in comparison with those of the original system. We investigate these properties for two popular DGFEM schemes: the interior penalty discontinuous Galerkin finite element method applied to the second-order wave equation and a more general family of schemes applied to the corresponding first order system. We show how the analysis of the multi-dimensional case may be reduced to consideration of one-dimensional problems. We derive the dispersion error for various schemes and conjecture on the generalisation to higher order approximation in space     

Viscoelastic analysis of nearly incompressible solids

The finite element method is well established for the analysis of structures and other field problems. However, its straightforward application for the analysis of nearly incompressible solids yields erratic results. In the present work, an efficient special purpose code for the Viscoelastic Analysis of Nearly Incompressible Solids (VANIS) is developed using isoparametric elements with selective integration procedure, which is a third order Gauss rule for deviatoric response and second order Gauss rule for volumetric response. The software can be effectively employed for the structures with lower Poisson's ratios. VANIS is based on the direct formulation using linear uncoupled thermoviscoelastic theory for the thermorheologically simple materials. The element library consists of 8-noded plane strain, 8-noded axisymmetric solid and 20-noded three dimensional quadratic isoparametric elements. These elements meet all the possible structural idealisation requirements of the solid continua. Experimentally obtained rigidity modulus can be used directly or expressing it in Prony series. The software is tested on a number of problems and gives very accurate results for all the permissible values of the Poisson's ratio.     

Moving grid method without interpolations

In their method of solving a one-dimensional moving boundary problem Crank and Gupta suggest a grid system which moves with the interface. The method requires some interpolations to be carried out which they perform by using a cubic spline or an ordinary polynomial. In the present paper these interpolations are avoided by employing a Taylor's expansion in space and time dimensions. A practical diffusion problem is solved and the results are compared with those obtained from other methods.     

Superconvergence of Discontinuous Finite Element Solutions for Transient Convection–diffusion Problems

We present a study of the local discontinuous Galerkin method for transient convection–diffusion problems in one dimension. We show that p-degree piecewise polynomial discontinuous finite element solutions of convection-dominated problems are O(Δx ^p+2) superconvergent at Radau points. For diffusion- dominated problems, the solution’s derivative is O(Δx ^p+2) superconvergent at the roots of the derivative of Radau polynomial of degree p+1. Using these results, we construct several asymptotically exact a posteriori finite element error estimates. Computational results reveal that the error estimates are asymptotically exact.This revised version was published online in July 2005 with corrected volume and issue numbers.     

Continuous and discontinuous finite element methods for burgers' equation

We apply two discontinuous finite element methods to the inviscid Burgers' equation and to the full equation with viscosity. In both cases we compare with a continuous space-time finite element method previously studied. For v = 0 discontinuous methods give better results, while the reverse prevails for the viscous equation.     

Accurate determination of asymptotic postbuckling stresses by the finite element method

Koiter's method for initial postbuckling and imperfection sensitivity analysis of elastic structures is conveniently formulated in terms of finite elements. Special care must, however, be taken in the computation of the postbuckling stresses and the postbucklmg constant determining the initial curvature of the postbuckling path. The cause of the problem lies in the different degree of approximation of in-plane and lateral displacements and is inevitable in a standard compatible finite element formulation. It is shown that only a minor change in the computation of the postbuckling stresses is needed. The procedure is extended to cover Byskov and Hutchinson's method for cases with nearly sinultaneous buckling modes.     

Discontinuous Finite Volume Element Method for a Coupled Non-stationary Stokes–Darcy Problem

In this paper, a discontinuous finite volume element method was presented to solve the nonstationary Stokes–Darcy problem for the coupling fluid flow in conduits with porous media flow. The proposed numerical method is constructed on a baseline finite element family of discontinuous linear elements for the approximation of the velocity and hydraulic head, whereas the pressure is approximated by piecewise constant elements. The unique solvability of the approximate solution for the discrete problem is derived. Optimal error estimates of the semi-discretization and full discretization with backward Euler scheme in standard \(L^2\)-norm and broken \(H^1\)-norm are obtained for three discontinuous finite volume element methods (symmetric, non-symmetric and incomplete types). A series of numerical experiments are provided to illustrate the features of the proposed method, such as the optimal accuracy orders, mass conservation, capability to deal with complicated geometries, and applicability to the problems with realistic parameters.     

Mixed finite element models for plate bending analysis: A new element and its applications

With a few exceptions, finite element packages available in today's commercial software environment contain in their libraries displacement-type elements only. The present paper aims to demonstrate the feasibility that properly formulated mixed-type elements compete most favorably with displacement-type elements and should, therefore, be considered as potential candidates for inclusion in general purpose finite element packages. In doing so, the development of a new triangular doubly—curved mixed-hybrid shallow shell element and its extensive testing in carefully chosen example problems are reported on.     

A three-dimensional nodal-based implementation of a family of discontinuous Galerkin methods for elasticity problems

Four discontinuous Galerkin (DG) methods are proposed to enrich the resource of modeling elasticity problems as they are volume locking-free and allow hanging nodes in meshing. A detailed finite element formulation of these DG methods is presented. For implementation, we coded a three-dimensional nodal-based DG program in which the conventional nodal-based pure displacement finite element codes are fully exploited. The robustness and accuracy of each DG method are demonstrated and compared with mixed methods through solving a rubber beam problem. The coupled use of DG with continuous elements is proposed for some practical applications.     

几种复合型数值方法的算法模拟与性能比较

浅水波问题的数值模拟一直是计算数学、计算流体力学的研究热点之一,采用低阶方法和高阶方法相复合的数值方法引起了人们的注意,并在水力学的数值模拟中取得了很大的成功。文中对三种复合型的数值方法,即Lax-Wendroff(LW)格式与Lax-Friedrichs(LF)格式的复合算法,Upwind格式与Lax-Wendroff(LW)格式的复合算法,WENO格式与LW格式的复合算法,进行了分析比较和改进,并就计算流体力学中的一维浅水波方程的两个算例分别做了数值对比试验,在解的光滑性、锐利性,计算速度等几个方面做了比较,模拟结果表明三种方法均能准确捕捉激波又不产生非物理震荡。     

Saint venant's principle in sandwich strip

The Saint Venant's principle is generally accepted to be valid in most problems in structural mechanics. But some earlier studies have drawn attention to the fact that routine application of Saint Venant's principle in the solution of elasticity problems of sandwich type structures is not justified in general. The aim of the investigation reported herein is to study the applicability of Saint Venant's principle to the problem of a sandwich strip quantitatively in detail. A simple finite element model, with linear strain triangles representing the core and bar elements representing the thin faces, is used for the analysis. Four different load cases are studied, and representative numerical results are presented. In all cases it is observed that significant edge disturbances do exist at relatively large distances from the load point, compared to homogeneous type of construction. Strength of Materials solutions are valid only beyond a distance of “4h” from the loaded end in the case of very soft cores, the distance being less for more rigid cores. Also very high values of face to core modulus ratio do not influence the solution significantly.     

Analysis of shell structures under transient loading using adaptivity in time and space

The adaptive analysis of structures under transient loading leads to the question, which time integration scheme, finite elements or finite differences, is most favorably combined with an adaptive spatial FE discretization. In order to judge this, the properties of different discontinuous Galerkin (DG) and the standard Newmark method are investigated first, also concerning efficiency. In particular, the damping and dispersion effects are discussed in detail for various types of problems. It must be noted that the type of problem has to be carefully checked in order to apply the most appropriate and efficient time integration scheme. It is shown that e.g. for the wave propagation problems the DG method with linear approximations (DG P1-P1) has to be favored when adaptivity in space is applied. Finally, an adaptive time step modification scheme is presented and applied to various problems.     

Application of the Petrov-Galerkin method to chemical-flooding reservoir simulation in one dimension

A one-dimensional, chemical-flooding simulator is described. This program models the effect that polymers, surfactants and salts have on the enhanced recovery of oil. The flow equations are convection dominated and often shock-like profiles develop which traverse the domain. The simulator incorporates, with a few modifications, most of the physical and chemical package which is set forth by Pope and Nelson. The spacial discretization is performed via the Petrov-Galerkin, finite element method. This technique is mainly derived from the work of Hughes and Brooks and is modified for the case of chemical flooding. We compare this finite element method to the upstream, finite difference method and to Galerkin's method. Generally speaking, the Petrov-Galerkin results are significantly less sensitive to the number of grid points than are those of the upstream method. Its stability properties are nearly the same as those of the upstream techniques, which in turn, are superior to those of Galerkin's method.     

Finite element analysis of transient heat conduction application of the weighted residual process

The transient heat conduction problem can be solved by application of Galerkin's method to space as well as time discretization. The formulation corresponds to the procedure known as finite elements in time and space. A linear time expansion leads to a step by step technique which is convergent, consistent and absolutely stable. Several numerical examples are presented using two-dimensional isoparametric elements.     

A lumped Galerkin method for the numerical solution of the modified equal-width wave equation using quadratic B-splines

The numerical solution of the one-dimensional modified equal width wave (MEW) equation is obtained by using a lumped Galerkin method based on quadratic B-spline finite elements. The motion of a single solitary wave and the interaction of two solitary waves are studied. The numerical results obtained show that the present method is a remarkably successful numerical technique for solving the MEW equation. A linear stability analysis of the scheme is also investigated.