A combined finite and infinite element approach for modeling spherically symmetric transient subsurface flow

This paper presents a finite element-infinite element coupling approach for modeling a spherically symmetric transient flow problem in a porous medium of infinite extent. A finite element model is used to examine the flow potential distribution in a truncated bounded region close to the spherical cavity. In order to give an appropriate artificial boundary condition at the truncated boundary, a transient infinite element, that is developed to describe transient flow in the exterior unbounded domain, is coupled with the finite element model. The coupling procedure of the finite and infinite elements at their interface is described by means of the boundary integro-differential equation rather than through a matrix approach. Consequently, a Neumann boundary condition can be applied at the truncated boundary to ensure the C¹-continuity of the solution at the truncated boundary. Numerical analyses indicate that the proposed finite element-infinite element coupling approach can generate a correct artificial truncated boundary condition to the finite element model for the unbounded flow transport problem.     

Composite infinite element analysis of unbounded two-phase media

This paper discusses the development and application of a mapped type composite infinite element for modelling the response of unbounded two-phase media. The coordinate ascent mapping technique which uses conventional shape functions and the Gauss-Legendre integration method has been used for the formulation. The element is constructed such that it preserves compatibility between variations in effective stress and pore pressure. This compatibility is a basic requirement for the composite element to accurately model the two-phase media. Numerical analyses of problems with unbounded saturated media using these elements have shown considerably improved predictions compared to conventional analyses. A subroutine listing is presented to demonstrate the implementation of the element in finite element programs.     

The coupling of boundary integral and finite element methods for the exterior steady Oseen problem

In this article, we present a new numerical method for solving the steady Oseen equations in an unbounded plane domain. The technique consists in coupling the boundary integral and the finite element methods. An artificial smooth boundary is introduced separating an interior inhomogeneous region from an exterior homogeneous one. The solution in exterior region is represented by an integral equation over the artificial boundary. This integral equation is incorporated into a velocity-pressure formulation for the interior region, and a finite element method is used to approximate the resulting variational problem. Finally, the optimal error estimates of the numerical solution are derived.Computer results will be discussed in a forthcoming paper.     

Moments-method analysis of a finite array of arbitrary shaped microstrip patch radiating elements

A full-wave model for the analysis of a finite array of arbitrary shaped, probe fed, microstrip elements is presented. The method of moments is used first to obtain a space-domain solution to the mixed potential integral equation (MPIE) for the isolated element in terms of linear ramp basis functions over triangular subdomains. A second order matrix equation for the array problem is then set up using characteristic current modes as basis functions. These converge rapidly and have orthogonality properties such that only the intermode coupling between separated elements need be computed—a relatively simple operation. Predicted and measured performance characteristics for a 37-element 8-GHz array of dual-mode patches are given which demonstrate the accuracy obtainable with this approach.     

Generation of finite element mesh with variable size over an unbounded 2D domain

With the advance of the finite element, general fluid dynamic and traffic flow problems with arbitrary boundary definition over an unbounded domain are tackled. This paper describes an algorithm for the generation of finite element mesh of variable element size over an unbounded 2D domain by using the advancing front circle packing technique. Unlike the conventional frontal method, the procedure does not start from the object boundary but starts from a convenient point within an open domain. The sequence of construction of the packing circles is determined by the shortest distance from the fictitious centre in such a way that the generation front is more or less a circular loop with occasional minor concave parts due to element size variation. As soon as a circle is added to the generation front, finite elements are directly generated by properly connecting frontal segments with the centre of the new circle. In contrast to other mesh generation schemes, the domain boundary is not considered in the process of circle packing, this reduces a lot of geometrical checks for intersection with frontal segments, and a linear time complexity for mesh generation can be achieved. In case the boundary of the domain is needed, simply generate an unbounded mesh to cover the entire object. As the element adjacency relationship of the mesh has already been established in the circle packing process, insertion of boundary segments by neighbour tracing is fast and robust. Details of such a boundary recovery procedure are described, and practical meshing problems are given to demonstrate how physical objects are meshed by the unbounded meshing scheme followed by the insertion of domain boundaries.     

An improved formulation of space stiffeners

This paper presents a displacement based finite element model for predicting the constraint torsion effect of stiffeners. In structural modelling, the plate/shell and the stiffeners are treated as separate elements where the displacement compatibility transformation between these two types of elements takes into account the constraint torsional warping effect in the stiffeners. The development is based on a general beam theory which includes flexural-torsion coupling, constrained torsion warping, and shear-centre location. The virtual work principle includes the second order terms of finite beam rotations. For finite element analysis, cubic Hermitian polynomials are used as shape functions of the straight space frame element with two nodes. Elastic stiffness and geometric stiffness matrices for an arbitrary cross-section are evaluated in a closed form, and load correction stiffness for eccentric stiffener loads are considered. To demonstrate the importance of torsion warping constraints and to illustrate the accuracy of this formulation, finite element solutions are presented and compared with available solutions.     

无穷凹角区域椭圆边值问题的重叠型区域分解算法

§1.引言许多科学和工程计算问题都可归结为无界区域上的偏微分方程边值问题,数值求解无界     

Multidimensional exponentially fitted simplicial finite elements for convection-diffusion equations with tensor-valued diffusion

Abstract In this paper we propose and analyze an exponentially fitted simplicial finite element method for the numerical approximation of solutions to diffusion-convection equations with tensor-valued diffusion coefficients. The finite element method is first formulated using exponentially fitted finite element basis functions constructed on simplicial elements in arbitrary dimensions. Stability of the method is then proved by showing that the corresponding bilinear form is coercive. Upper error bounds for the approximate solution and the associated flux are established.     

Acoustic infinite elements for non-separable geometries

We present new infinite element formulations for solving acoustic scattering and radiation problems in the exterior of long, slender bodies. The new infinite elements are geometrically constructed from a prolate spheroid inscribed by the scatterer. These elements need not begin on a level surface of the prolate spheroidal coordinate system. Instead, they may be attached to any convex surface, including that of the scatterer itself. This scheme reduces, or even completely eliminates, finite element modeling of the exterior medium. The formulations may easily be extended to the cases of an interior oblate spheroid or ellipsoid. We present both conjugated and unconjugated formulations without any weighting factors, although it would be simple to include them. We describe a fast numerical scheme for computing the element integrals based on Chebychev approximation. We include numerical results for scattering from spheres and capped cylinders. These results demonstrate the accuracy and the dramatic reduction in computational expense of our new formulations compared to other coupled finite element/infinite element methods.     

A finite element alternative to infinite elements

In this paper, a simple idea based on midpoint integration rule is utilized to solve a particular class of mechanics problems; namely static problems defined on unbounded domains where the solution is required to be accurate only in an interior (and not in the far field). By developing a finite element mesh that approximates the stiffness of an unbounded domain directly (without approximating the far-field displacement profile first), the current formulation provides a superior alternative to infinite elements (IEs) that have long been used to incorporate unbounded domains into the finite element method (FEM). In contrast to most IEs, the present formulation (a) requires no new shape functions or special integration rules, (b) is proved to be both accurate and efficient, and (c) is versatile enough to handle a large variety of domains including those with anisotropic, stratified media and convex polygonal corners. In addition to this, the proposed model leads to the derivation of a simple error expression that provides an explicit correlation between the mesh parameters and the accuracy achieved. This error expression can be used to calculate the accuracy of a given mesh a-priori. This in-turn, allows one to generate the most efficient mesh capable of achieving a desired accuracy by solving a mesh optimization problem. We formulate such an optimization problem, solve it and use the results to develop a practical mesh generation methodology. This methodology does not require any additional computation on the part of the user, and can hence be used in practical situations to quickly generate an efficient and near optimal finite element mesh that models an unbounded domain to the required accuracy. Numerical examples involving practical problems are presented at the end to illustrate the effectiveness of this method.     

Nonlinear soil-structure interaction analysis using coupled finite-infinite elements

Soil-structure interaction analysis for a strip footing is presented using a coupled finite-infinite element formulation. Two-dimensional mapped infinite elements with 1/r type of decay are used. The shape functions for mapped infinite elements with 1/√r type of decay are also presented. In the interactive analysis of footing, non-linear stress-strain behaviour of the soil using the hyperbolic fit is taken into account. The results are presented for both rigid and flexible footings. It is found that the vertical stress distribution for a flexible footing matches very well with the Boussinesq solution. The infinite elements in combination with the conventional finite elements yield a very attractive coupled formulation in terms of far field physical representation, computer use and data preparation.     

A finite-element series-expansion technique for linear surface water waves

A numerical method is presented to deal with the propagation of surface water waves in the framework of the linear theory for an inviscid fluid. For particular geometrical configurations of the region in which wave propagation occurs, refraction, diffraction and reflection phenomena can arise simultaneously, so that the solution of the original Berkhoff equation with appropriate boundary conditions becomes essential to achieve an adequate picture of the resulting field. The method is based on a finite element scheme, in which the element matrices are computed by a series expansion technique. The elements are of arbitrary shape, although of constant depth, and two independent numerical approximations are given for the surface-elevation and velocity fields. An application of the method to the propagation of short water waves in a channel connecting two basins of larger dimensions shows that the method can deal with very large domains, at least when compared to the possibilities of the usual finite element approaches.     

A goal-oriented hp-adaptive finite element approach to radar scattering problems

The paper presents application of an hp-adaptive finite element method for scattering of electromagnetic waves. The main objective of the numerical analysis is to determine the characteristics of the scattered waves indicating the power being scattered at a given direction––i.e. the radar cross-section (RCS). This is achieved considering the scattered far-field which defines RCS and which is expressed as a linear functional of the solution. Techniques of error estimation for the far-field are considered and an h-adaptive strategy leading to the fast reduction of the error of the far-field is presented. The simulations are performed with a three-dimensional version of an hp-adaptive finite element method for electromagnetics based on the hexahedral edge elements combined with infinite elements for modeling the unbounded space surrounding the scattering object.     

平面弹性方程外问题的非重叠型区域分解算法

１．引言 区域分解算法是八十年代兴起的偏微分方程求解新技术．基于有限元法的区域分解算法对求解有界区域问题行之有效［２,４,９］．边界元方法则是处理无界区域问题的强有力的工具［１,１０,１７］,有限元与边界元耦合法得到广泛应用 ［３,５,７］．近年又发展了基于自然边界归化的区域分解算法,特别适用于无界区域问题［８,１１,１２］．迄今这方面的文章主要是针对二维Ｐｏｉｓｓｏｎ方程及双调和方程的［１３－１６］． 本文讨论平面弹性方程的Ｄｉｒｉｃｈｌｅｔ外边值问题其中Ω是充分光滑闭曲线Г０之外的无界区域,ｕ…     

A new algorithm for finite spectrum assignment of single-inputsystems with time delay

Finite spectrum assignment for time-delay systems is the elimination of delay operators from the characteristic function of the closed-loop system and the arbitrary assignment of poles. The control consists of polynomials in the delay operator and finite Laplace transforms. An algorithm for computing the control matrix is presented. In particular, the control matrix over rational functions of a delay operator is computed and expanded to partial fractions. Partial fractions are systematically transformed to finite Laplace transforms     

三维有限元模型的任意剖切及其等值线与彩色云图生成的方法

对三维有限元模型快速有效地生成任意剖面上等值线及彩色云图,是有限元计算后处理中的一个重要技术。该文在建立单元信息描述表的基础上,提出了一种适合于任意三维实体单元类型的通用剖切算法,和在剖切面及外表面生成等值线或高质量彩色云图的方法。     

Local artificial boundary conditions for Schrödinger and heat equations by using high-order azimuth derivatives on circular artificial boundary

The aim of the paper is to design high-order artificial boundary conditions for the Schrödinger equation on unbounded domains in parallel with a treatment of the heat equation. We first introduce a circular artificial boundary to divide the unbounded definition domain into a bounded computational domain and an unbounded exterior domain. On the exterior domain, the Laplace transformation in time and Fourier series in space are applied to achieve the relation of special functions. Then the rational functions are used to approximate the relation of the special functions. Applying the inverse Laplace transformation to a series of simple rational function, we finally obtain the corresponding high-order artificial boundary conditions, where a sequence of auxiliary variables are utilized to avoid the high-order derivatives in respect to time and space. Furthermore, the finite difference method is formulated to discretize the reduced initial–boundary value problem with high-order artificial boundary conditions on a bounded computational domain. Numerical experiments are presented to illustrate the performance of our method.     

Frame element with lateral deformable supports: Formulations and numerical validation

This paper presents the theory and the numerical validation of three different formulations of nonlinear frame elements with nonlinear lateral deformable supports. The governing differential equations of the problem are derived first and the three different finite element formulations are then presented. The first model follows a displacement-based formulation, which is based on the virtual displacement principle. The second one follows the force-based formulation, which is based on the virtual force principle. The third model follows the Hellinger–Reissner mixed formulation, which is based on the two-field mixed variational principle. The selection of the displacement and force interpolation functions for the different formulations is discussed. Tonti’s diagrams are used to conveniently represent the equations governing both the strong and the weak forms of the problem. The general matrix equations of the three formulations are presented, with some details on the issues regarding the elements’ implementations in a general-purpose finite element program. The convergence, accuracy, and computational times of the three elements are studied through a numerical example. The distinctive element characteristics in terms of force and deformation discontinuities between adjacent elements are discussed. The capability of the proposed frame models to trace the softening response due to softening of the foundation is also investigated. Overall, the force-based and the mixed models are much more accurate than the displacement-based model and require very few elements to reach the converged solution. The force-based element is slightly more accurate than the mixed model, but it is more prone to numerical instabilities as it involves inverting the element flexibility matrix.     

An equilibrium method for stress calculation using finite element displacement models

The stress computation concept described in [1]is extended here to arbitrary meshes and elements — in particular to triangular elements. After calculating the nodal displacements — using complete and conforming displacement models — we assume linear stress distributions and corresponding virtual displacements at element peripheries. The nodal stress values are then determined by the principle of virtual work. The right-hand side of the resulting system of algebraic equations consists of the work done by the known nodal stress resultants acting along the virtual displacements. In general, the system of equations is nonquadratic on the structural level. Gauss's transformation produces a symmetric, positive definite band matrix. This kind of stress calculation is called the equilibrium method.A dual node method is also given. It involves the inversion of element matrices instead of the master matrix.Various examples of plane stress, plate bending and shell problems show much better accuracy of stresses in comparison with conventional methods. Furthermore, these techniques improve the computational efficiency considerably. There is also a special advantage in the possibility of choosing arbitrary subregions of the structure for stress calculation.