A new method for solving Biot＇s consolidation of a finite soil layer in the cylindrical coordinate system

A new method is developed to solve Biot＇s consolidation of a finite soil layer in the cylindrical coordinate system. Based on the governing equations of Biot＇s consolidation and the technique of Laplace transform, Fourier expansions and Hankel transform with respect to time t, coordinate θ and coordinate r, respectively, a relationship of displacements, stresses, excess pore water pressure and flux is established between the ground surface （z = 0） and an arbitrary depth z in the Laplace and Hankel transform domain. By referring to proper boundary conditions of the finite soil layer, the solutions for displacements, stresses, excess pore water pressure and flux of any point in the transform domain can be obtained. The actual solutions in the physical domain can be acquired by inverting the Laplace and the Hankel transforms.     

An explicit finite volume element method for solving characteristic level set equation on triangular grids

Level set methods are widely used for predicting evolutions of complex free surface topologies,such as the crystal and crack growth,bubbles and droplets deformation,spilling and breaking waves,and two-phase flow phenomena.This paper presents a characteristic level set equation which is derived from the two-dimensional level set equation by using the characteristic-based scheme.An explicit finite volume element method is developed to discretize the equation on triangular grids.Several examples are presented to demonstrate the performance of the proposed method for calculating interface evolutions in time.The proposed level set method is also coupled with the Navier-Stokes equations for two-phase immiscible incompressible flow analysis with surface tension.The Rayleigh-Taylor instability problem is used to test and evaluate the effectiveness of the proposed scheme.     

用状态方程直接积分法求解饱和土Biot固结方程

试采用状态方程直接积分法求解饱和土Biot固结问题,首次得到以位移未知量表示的状态方程、位移和孔隙流体压力的计算公式。算例结果表明,本方法的计算效率较高,同时精确可靠。     

The perturbation finite element method for solving problems with nonlinear materials

The perturbation method is one of the effective methods for so-lving problems in nonlinear continuum mechanics.It has been de-veloped on the basis of the linear analytical solutions for the o-riginal problems.If a simple analytical solution cannot be ob-tained.we would encounter difficulties in applying this method tosolving certain complicated nonlinear problems.The finite ele-ment method appears to be in its turn a very useful means for sol-ving nonlinear problems,but generally it takes too much time incomputation.In the present paper a mixed approach,namely,theperturbation finite element method,is introduced,which incorpo-rates the advantages of the two above-mentioned methods and enablesus to solve more complicated nonlinear problems with great savingin computing time.Problems in the elastoplastic region have been discussed anda numerical solution for a plate with a central hole under tensionis given in this paper.     

A compact finite difference method for solving Burgers' equation

In this paper, a high‐order accurate compact finite difference method using the Hopf–Cole transformation is introduced for solving one‐dimensional Burgers' equation numerically. The stability and convergence analyses for the proposed method are given, and this method is shown to be unconditionally stable. To demonstrate efficiency, numerical results obtained by the proposed scheme are compared with the exact solutions and the results obtained by some other methods. The proposed method is second‐ and fourth‐order accurate in time and space, respectively. Copyright © 2009 John Wiley & Sons, Ltd.     

The solution of non-linear hyperbolic equation systems by the finite element method

The difficulties experienced in the treatment of hyperbolic systems of equations by the finite element method (or other) spatial discretization procedures are well known. In this paper a temporal discretization precedes the spatial one which in principle is considered along the characteristics to achieve a self adjoint form. By a suitable expansion, the original co-ordinates are preserved and combined with the use of a standard Galerkin process to achieve an accurate discretization. It is shown that the process is equivalent to the Taylor-Galerkin methods of Donea.¹⁷ Several examples illustrate the accuracy and efficiency attainable in such problems as transport, shallow water equations, transonic flow etc.     

Numerical analysis of one-dimensional nonlinear large-strain consolidation by the finite element method

The nonlinear partial differential equation model of Gibson et al. which governs one-dimensional large-strain consolidation is solved numerically using a semi-discrete formulation involving a Galerkin weighted residual approach. The use of quadratic Lagrange basis functions usually complicates the task of solving the system of time-dependent ordinary differential equations that are obtained with the semi-discrete Galerkin procedure. However, an efficient algorithm has been discovered yielding the advantages of quadratic interpolation without undue computational burden.Although considerable effort has already been made to solve the PDE of large-strain consolidation by numerical methods, a satisfactory set of benchmarks is still needed to assess accuracy. To fill this need, three procedures are reported which allow numerical solutions of the large-strain model to be reliably evaluated. One involves the use of perturbation methodology to provide a solution when only self-weight effects are present. A second utilizes an analytical solution developed by Philip when self-weight effects are absent and the third involves the exact calculation of the discharge flux through the upper boundary of a deposit consolidating through self-weight effects alone. All three are restricted to early-time consolidation and are illustrated in the context of the finite element method.     

Numerical analysis of Biot's consolidation process by radial point interpolation method

An algorithm is proposed to solve Biot's consolidation problem using meshless method called a radial point interpolation method (radial PIM). The radial PIM is advantageous over the meshless methods based on moving least-square (MLS) method in implementation of essential boundary condition and over the original PIM with polynomial basis in avoiding singularity when shape functions are constructed. Two variables in Biot's consolidation theory, displacement and excess pore water pressure, are spatially approximated by the same shape functions through the radial PIM technique. Fully implicit integration scheme is proposed in time domain to avoid spurious ripple effect. Some examples with structured and unstructured nodes are studied and compared with closed-form solution or finite element method solutions.     

A moving finite element method for the population balance equation

A moving finite element algorithm has been compared against the upwind-differencing and Smolarkiewicz methods for the population balance equation of multicomponent particle growth processes. Analytical solutions and an error function have been used to test the numerical methods. The moving finite elements technique is much more accurate than other methods for a wide range of parameters. Since this method uses moving grids, it is able to model very narrow particle size distributions. It is also shown that the method can be extended to solve condensational growth problems which include particle curvature and non-continuum mass transfer effects.     

New nonconforming finite element method for solving transient Naiver-Stokes equations

For transient Naiver-Stokes problems, a stabilized nonconforming finite element method is presented, focusing on two pairs inf-sup unstable finite element spaces, i.e., pNC/pNC triangular and pNQ/pNQ quadrilateral finite element spaces. The semi- and full-discrete schemes of the stabilized method are studied based on the pressure projection and a variational multi-scale method. It has some attractive features： avoiding higher-order derivatives and edge-based data structures, adding a discrete velocity term only on the fine scale, being effective for high Reynolds number fluid flows, and avoiding increased computation cost. For the full-discrete scheme, it has second-order estimations of time and is unconditionally stable. The presented numerical results agree well with the theoretical results.     

The perturbation finite element method for solving the plane problem in consideration of linear creep

In this paper, a method (PFMC) for solving plane problem of linear creep is presented by using perturbation finite element. It can be used in plane problem in consideration of creep, such as reinforced concrete beam, presiressed concrete beam, reinforced concrete cylinder and reinforced concrete tunnel in elastic or visco-elastic medium, as well as underground building and so on.In the presented method, the assumption made in the general increment method that variables remain constant in a divided time interval is not taken. The accuracy is improved and the length of time step becomes larger. The computer storage can be reduced and the calculating efficiency can be increased.Perturbation finite element formulae for four-node quadrilateral isoparametric element including reinforcement are established and five numerical examples are given. As contrasted with the analytical solution, the accuracy is satisfactory.     

不可压缩粘性流动的CBS有限元解法

对于二维不可压缩粘性流动,首先通过坐标变换的方式得到了的不含对流项的NS方程,并给出了CBS有限元方法求解的一般过程。结合一类同时含有压力和速度的出口边界条件,对方腔顶盖驱动流、后向台阶绕流和圆柱绕流进行了计算。所得结果与基准解符合良好,验证了CBS算法对于定常、非定常粘性不可压缩流动问题的可行性和所用出口边界条件的无反射特性。特别的,对于圆柱绕流,Re=100时非定常升、阻力系数及漩涡脱落等非定常都得到了较好地模拟,为一进步研究自激振动等更加复杂的非定常流动问题奠定了基础。     

A method of solving the fuzzy finite element equations in monosource fuzzy numbers

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｒｅｈａｖｅｂｅｅｎｍａｎｙｒｅｓｅａｒｃｈｐａｐｅｒｓａｂｏｕｔｔｈｅｆｕｚｚｙｓｔｏｃｈａｓｔｉｃｓｔｒｕｃｔｕｒｅ[1～ 3],ａｎｄｆｕｚｚｉｎｅｓｓａｎｄｒａｎｄｏｍｎｅｓｓａｒｅｔｗｏｉｍｐｏｒｔａｎｔｆａｃｔｏｒｓｉｎｅｎｇｉｎｅｅｒｉｎｇ .Ｂａｓｅｄｏｎｔｈｅｏｐｅｒａｔｉｏｎｒｕｌｅｓｏｆｆｕｚｚｙｎｕｍｂｅｒｓａｎｄｉｎｔｅｒｖａｌｎｕｍｂｅｒｓ,ｔｈｅｆｕｎｃｔｉｏｎｕｎｄｅｒｆｕｚｚｙａｎｄｒａｎｄｏｍｆａｃｔｏｒｓｃａｎｂｅｔｒａｎｓｐｏｓｅｄｉ…     

Least-squares mixed finite element method for a class of stokes equation

ＩｎｔｒｏｄｕｃｔｉｏｎＡｇｅｎｅｒａｌｔｈｅｏｒｙｏｆｔｈｅｌｅａｓｔ＿ｓｑｕａｒｅｓｍｅｔｈｏｄｈａｓｂｅｅｎｄｅｖｅｌｏｐｅｄｂｙＡＫＡｚｉｚ,ＲＢＫｅｌｌｏｇｇａｎｄＡＢＳｔｅｐｈｅｎｓｉｎ[1].Ｔｈｅｍｏｓｔｉｍｐｏｒｔａｎｔａｄｖａｎｔａｇｅｌｅａｄｓｔｏａｓｙｍｍｅｔｒｉｃｐｏｓｉｔｉｖｅｄｅｆｉｎｉｔｅｐｒｏｂｌｅｍ.ＪＨＢｒａｍｂｌｅａｎｄＪＡＮｉｔｓｈｅｐｒｅｓｅｎｔｅｄａｌｅａｓｔ＿ｓｑｕａｒｅｓｍｅｔｈｏｄｆｏｒＤｉｒｉｃｈｌｅｔｐｒｏｂｌｅｍｓｉｎ[2].Ｔｈｅｍｅｔｈｏｄｇｅ…     

FUZZY ARITHMETIC AND SOLVING OF THE STATIC GOVERNING EQUATIONS OF FUZZY FINITE ELEMENT METHOD

The key component of finite element analysis of structures with fuzzy parameters, which is associated with handling of some fuzzy information and arithmetic relation of fuzzy variables , was the solving of the governing equations of fuzzy finite element method. Based on a given interval representation of fuzzy numbers, some arithmetic rules of fuzzy numbers and fuzzy variables were developed in terms of the properties of interval arithmetic. According to the rules and by the theory of interval finite element method, procedures for solving the static governing equations of fuzzy finite element method of structures were presented. By the proposed procedure, the possibility distributions of responses of fuzzy structures can be generated in terms of the membership functions of the input fuzzy numbers. It is shown by a numerical example that the computational burden of the presented procedures is low and easy to implement. The effectiveness and usefulness of the presented procedures are also illustrated.     

Refining the submesh strategy in the two‐level finite element method: application to the advection–diffusion equation

We introduce a new submesh strategy for the two‐level finite element method. The numerical results show that the new submesh is able to better capture the boundary layer which is caused by the choice of bubble functions. The effect of an improved approximation of the residual free bubbles is studied for the advective–diffusive equation. Copyright © 2002 John Wiley & Sons, Ltd.     

极坐标系下的Legendre谱元方法求解Poisson-型方程

r=0处的坐标奇异性是求解极坐标下Poisson-型方程的关键。本文提出一种极坐标系下基于Galerkin变分的Legendre谱元方法用于求解圆形区域内的Poisson-型方程,物理区域的径向和周向划分若干单元,计算单元均采用Legendre多项式展开;圆心所在单元的径向使用LGR(Legendre Gauss Radau)积分点,其他单元径向使用LGL(Legendre Gauss Lobatto)积分点,从而避免了极点处1/r坐标奇异性,周向单元均采用LGL积分点。利用区域分解技术,可以避免节点在极点附近聚集;最后求解了多个Dirichlet或Neumann边界条件下的Poisson-型方程算例。数值结果表明,谱元方法具有很高的精度。     

Development and application of an adaptive finite element method to reaction-diffusion equations

An adaptive finite element method is developed and applied to study the ozone decomposition laminar flame. The method uses a semidiscrete, linear Galerkin approximation in which the size of the elements is controlled by an integral which minimizes the changes in mesh spacing. The sizes and locations of the elements are controlled by the location and magnitude of the largest temperature gradient. The numerical results obtained with this adaptive finite element method are compared with those obtained using fixed-node finite-difference schemes and an adaptive finite-difference method. It is shown that the adaptive finite element method developed here using 36 elements can yield as accurate flame speeds as fourth-order accurate, fixed-node, finite-difference methods when 272 collocation points are employed in the calculations.     

A total linearization method for solving viscous free boundary flow problems by the finite element method

In this paper a total linearization method is derived for solving steady viscous free boundary flow problems (including capillary effects) by the finite element method. It is shown that the influence of the geometrical unknown in the totally linearized weak formulation can be expressed in terms of boundary integrals. This means that the implementation of the method is simple. Numerical experiments show that the iterative method gives accurate results and converges very fast.     

Reduced-order finite element method based on POD for fractional Tricomi-type equation

The reduced-order finite element method (FEM) based on a proper orthogonal decomposition (POD) theory is applied to the time fractional Tricomi-type equation. The present method is an improvement on the general FEM. It can significantly save memory space and effectively relieve the computing load due to its reconstruction of POD basis functions. Furthermore, the reduced-order finite element (FE) scheme is shown to be unconditionally stable, and error estimation is derived in detail. Two numerical examples are presented to show the feasibility and effectiveness of the method for time fractional differential equations (FDEs).