Computations for large-amplitude two-dimensional body motions

A numerical method is presented for the time-domain simulation of large-amplitude motions of a 2-D surface-piercing body with arbitrary shape in deep water. Based on potential theory, panels are distributed on the body and desingularized sources are distributed above the calm water surface. The body boundary condition is satisfied on the exact submerged body surface. The free-surface boundary conditions are linearized and satisfied on the calm water level. The solution is stepped forward in time by integrating the free-surface kinematic and dynamic conditions. The numerical solutions for the oscillation problem are compared with experimental results and other numerical results, and found to agree well. The results for the impact problem are compared with similarity solutions. Finally, results for the large-amplitude sinusoidal motion of a 45-degree wedge are presented.     

An accurate solution method for the static and dynamic deflections of orthotropic plates with general boundary conditions

Extending the previous work on isotropic beams and plates by the second author [Li WL, et al. An exact series solution for the transverse vibration of rectangular plates with general elastic boundary supports. J Sound Vib 2009;321:254–69], this paper describes an accurate analytical method for calculating the static deflections and modal characteristics of orthotropic plates with general elastic boundary supports. The displacement function is expressed as a 2-D Fourier cosine series supplemented with several terms in the form of 1-D series. The series expansions for all the relevant derivatives can be directly obtained through term-by-term differentiations of the displacement series. Thus, a classical solution can be derived by letting the series exactly satisfy the governing differential equation at every field point and all the boundary conditions at every boundary point, respectively. Several numerical examples are presented to demonstrate the excellent accuracy and convergence of the current solutions.     

Heat Conduction Analysis of 3-D Axisymmetric and Anisotropic FGM Bodies by Meshless Local Petrov–Galerkin Method

The meshless local Petrov–Galerkin method is used to analyze transient heat conduction in 3-D axisymmetric solids with continuously inhomogeneous and anisotropic material properties. A 3-D axisymmetric body is created by rotation of a cross section around an axis of symmetry. Axial symmetry of geometry and boundary conditions reduces the original 3-D boundary value problem into a 2-D problem. The cross section is covered by small circular subdomains surrounding nodes randomly spread over the analyzed domain. A unit step function is chosen as test function, in order to derive local integral equations on the boundaries of the chosen subdomains, called local boundary integral equations. These integral formulations are either based on the Laplace transform technique or the time difference approach. The local integral equations are nonsingular and take a very simple form, despite of inhomogeneous and anisotropic material behavior across the analyzed structure. Spatial variation of the temperature and heat flux (or of their Laplace transforms) at discrete time instants are approximated on the local boundary and in the interior of the subdomain by means of the moving least-squares method. The Stehfest algorithm is applied for the numerical Laplace inversion, in order to retrieve the time-dependent solutions.     

3-D boundary element method for cracks in a plate

New fundamental solutions which automatically satisfy boundary conditions at the interfaces of an elastic plate perfectly bonded to two elastic halfspaces are implemented in a 3-D boundary element method (BEM) for crack problems. The BEM features a new integration scheme for highly singular kernels. The capability is achieved through a part analytic and part numerical integration procedure, such that the analytic part of the integration is similar for all slip/opening variations, ‘Part-through’ elliptic cracks in an elastic plate with traction-free surfaces are analysed and the stress intensity factor (SIF) values along the crack front are found to compare favourably with widely accepted numerically obtained SIF results by Raju and Newman.¹     

Static response and free vibration of two-dimensional functionally graded metal/ceramic open cylindrical shells under various boundary conditions

The free vibration and static response of a two-dimensional functionally graded (2-D FGM) metal/ceramic open cylindrical shell are analyzed using 2-D generalized differential quadrature method. The open cylindrical shell is assumed to be simply supported at one pair of opposite edges and arbitrary boundary conditions at the other edges such that trigonometric functions expansion can be used to satisfy the boundary conditions precisely at simply supported edges. This paper presents a novel 2-D power-law distribution for ceramic volume fraction of 2-D FGM that gives designers a powerful tool for flexible designing of structures under multifunctional requirements. Various material profiles in two radial and axial directions are illustrated using the 2-D power-law distribution. The effective material properties at a point are determined in terms of the local volume fractions and the material properties by the Mori–Tanaka scheme. The 2-D generalized differential quadrature method as an efficient and accurate numerical tool is used to discretize the governing equations and to implement the boundary conditions. The convergence of the method is demonstrated, and to validate the results, comparisons are made with the available solutions for FGM cylindrical shells. The interesting results indicate that a graded ceramic volume fraction in two directions has a higher capability to reduce the mechanical stresses and natural frequency than conventional 1-D FGM. The achieved results confirm that natural frequency and mechanical stress distribution can be modified to a required manner by selecting an appropriate volume fraction profile in two directions.     

Inverse heat conduction problems by meshless local Petrov–Galerkin method

The meshless local Petrov–Galerkin (MLPG) method is used to solve stationary and transient heat conduction inverse problems in 2-D and 3-D axisymmetric bodies. A 3-D axisymmetric body is generated by rotating a cross section around an axis of symmetry. Axial symmetry of geometry and boundary conditions reduce the original 3-D boundary value problem to a 2-D problem. The analyzed domain is covered by small circular subdomains surrounding nodes randomly spread over the analyzed domain. A unit step function is chosen as test function in deriving the local integral equations (LIEs) on the boundaries of the chosen subdomains. The time integration schemes are formulated based on the Laplace transform technique and the time difference approach, respectively. The local integral equations are non-singular and take a very simple form. Spatial variation of the temperature and heat flux (or of their Laplace transforms) at discrete time instants are approximated on the local boundary and in the interior of the subdomain by means of the moving least-squares (MLS) method. Singular value decomposition (SVD) is applied to solve the ill-conditioned linear system of algebraic equations obtained from the LIE after MLS approximation. The Stehfest algorithm is applied for the numerical Laplace inversion, in order to retrieve the time-dependent solutions.     

A 3-D boundary element formulation of viscoelastic media with gravity

This paper presents a boundary element formulation for 3-D linear and viscoelastic bodies subjected to the body force of gravity. The Laplace transformation is first used to suppress the time variable, and solutions of displacements and stresses are found in the transformed domain. The time domain solutions are then found by an accurate and efficient numerical inversion method which requires only real calculations for all quantities. Input and output data, and solutions in the transformed and time domains are connected through an Interactive Data Language code written by the authors. While particular solutions of stresses and displacements related to the body force of gravity (which is applied at time t = 0 and is kept constant) are derived, the Green's functions in the Laplace domain are obtained through the correspondence principle. The new formulation has been implemented into an existing 3-D BEM program, and several numerical examples involving 3-D viscoelastic bodies are presented. Although the discussion in this paper focuses on Maxwell viscoelastic and isotropic media, other linear isotropic and even anisotropic viscoelastic models can also be incorporated, without difficulty, into the 3-D viscoelastic BEM program.     

复杂边界条件功能梯度板三维分析的半解析方法

推导出适应功能梯度材料构件分析的半解析方法基本算式,并针对功能梯度构件的材料参数随空间坐标变化的特点,将材料参数纳入到力学方程中进行整体积分计算,从而编制统一程序计算不同边界条件下的板件问题.该法适应性强而又简洁高效,且不同于一般的半解析法,可采用一维离散,给出三维分析结果,是一种解决功能梯度构件力学性能分析的有效数值方法.文中用半解析法分析几种具有不同复杂边界条件的功能梯度板,给出了板件的力学量三维分布形态.     

Nonlinear wave propagation and run-up generated by subaerial landslides modeled using meshless method

The focus of present study is on water waves generated by landslides. Because such problems involve moving boundaries and large deformation of the computational domain, a 2-D numerical model is established with a meshless method and a fully nonlinear Lagrangian time marching scheme. The method chosen in this study is a RBF collocation method developed in the way that the collocations of both the governing equation and boundary conditions are applied at each of the boundary points. This guarantees the accuracies of the partial derivatives of the velocity potential near the free surface, which results in the precise prediction of the free surface. A very effective treatment is proposed for the landslide boundary in this study. Present model is verified by comparing the numerical results of waves generated by a submerged landslide with other numerical solutions, including those obtained using the BIEM and Boussinesq-type models. Fairly good agreements are observed. Present model is then applied to simulate subaerial landslide-induced waves. Various slopes are considered. The landslide-induced wave propagation and shoreline motions are examined. The effects of sliding horizontal distance along a given slope on the induced wave are also discussed.     

基于动态近似边界条件的气动弹性数值模拟

发展了一种利用欧拉方程计算非定常气动力的数值方法,通过在固定物面边界上满足动态近似边界条件计算出非定常气动力,避免了在每个时间步重新生成网格或需用动网格技术进行网格变形处理过程,提高了计算效率。运用这种方法计算了一系列非定常气动力算例,并与非结构动网格准确边界条件下的欧拉方程解和实验数据进行了比较,进一步分析了翼型俯仰角和马赫数对非定常气动力相对误差的影响。将气动力解算器与结构方程耦合进行气动弹性数值模拟,计算了跨音速具有S型颤振边界的二元气动弹性标准算例-Isogaiwing。算例结果表明,利用动态近似边界条件的欧拉方程具有简便、高效的特点,并能在小振幅情况下得到与精确边界条件精度相当的非定常流场解,还可以用于气动弹性分析。     

Fracture analysis of cracks in magneto-electro-elastic solids by the MLPG

A meshless method based on the local Petrov–Galerkin approach is proposed for crack analysis in two-dimensional (2-D) and three-dimensional (3-D) axisymmetric magneto-electric-elastic solids with continuously varying material properties. Axial symmetry of geometry and boundary conditions reduces the original 3-D boundary value problem into a 2-D problem in axial cross section. Stationary and transient dynamic problems are considered in this paper. The local weak formulation is employed on circular subdomains where surrounding nodes randomly spread over the analyzed domain. The test functions are taken as unit step functions in derivation of the local integral equations (LIEs). The moving least-squares (MLS) method is adopted for the approximation of the physical quantities in the LIEs. The accuracy of the present method for computing the stress intensity factors (SIF), electrical displacement intensity factors (EDIF) and magnetic induction intensity factors (MIIF) are discussed by comparison with numerical solutions for homogeneous materials.     

考虑高阶横向剪切正交各向异性板振动的微分求积方法

分析了计及高阶横向剪切变形的正交各向异性弹性板自由振动的问题。采用微分求积方法(DQ方法)导出了控制方程的DQ形式，推广DQWB技巧处理了高阶矩的边界条件。研究了数值结果的收敛性，同时考察了不同的节点分布对收敛速度的影响。并将数值结果与三维精确解进行了比较，表明该方法的可靠性和有效性。     

A boundary point interpolation method for stress analysis of solids

 A boundary point interpolation method (BPIM) is proposed for solving boundary value problems of solid mechanics. In the BPIM, the boundary of a problem domain is represented by properly scattered nodes. The boundary integral equation (BIE) for 2-D elastostatics has been discretized using point interpolants based only on a group of arbitrarily distributed boundary points. In the present BPIM formulation, the shape functions constructed using polynomial basis function in a curvilinear coordinate possess Dirac delta function property. The boundary conditions can be implemented with ease as in the conventional boundary element method (BEM). The BPIM for 2-D elastostatics has been coded in FORTRAN, and used to obtain numerical results for stress analysis of two-dimensional solids. Received 10 January 2000     

On boundary conditions in the element-free Galerkin method

 Accurate imposition of essential boundary conditions in the Element Free Galerkin (EFG) method often presents difficulties because the Moving Least Squares (MLS) interpolants, used in this method, lack the delta function property of the usual finite element or boundary element method shape functions. A simple and logical strategy, for alleviating the above problem, is proposed in this paper. A discrete norm is typically minimized in the EFG method in order to obtain certain variable coefficients. The strategy proposed in this work involves a new definition of this discrete norm. This new strategy works very well in all the numerical examples, for 2-D potential problems, that are presented here. In addition to the discussion of boundary conditions, some recommendations are also made in this paper regarding strategies for refinements in order to improve the accuracy of numerical solutions from the EFG method.     

A new boundary-type finite element for 2-D- and 3-D-elastic structures

In this paper we discuss the theoretical and numerical formulation of 3-D Trefitz elements. Starting from the variational principle with the so-called hybrid stress method, the trial functions for the stresses have to fulfil the Beltrami equations, which means also the compatibility equations for the strains. The divergence theorem can be applied, and one arrives at a pure boundary formulation in the sense of the Trefftz method. Besides the resulting variational formulation, different regularizations of the interelement conditions are investigated by numerical tests. Two examples show the numerical efficiency of the derived elements. First, a geometric linear 3-D example is presented to show the effects on distorted element meshes. The third example shows the geometrically non-linear analysis of a shallow cylindrical shell segment under a singe load.     

A numerical wave tank model for cavitating hydrofoils

In this paper, an iterative boundary element method (IBEM) for both 2-D and 3-D cavitating hydrofoils moving steadily inside a numerical wave tank (NWT) is presented and some extensive numerical results are given. The cavitating hydrofoil part, the free surface part and the wall parts of NWT are solved separately, with the effects of one on the others being accounted for in an iterative manner. The cavitating hydrofoil surface, the free surface, the bottom surface and the side walls are modelled by a low-order potential based panel method using constant strength dipole and source panels. Second-order correction on the free surface in 2-D are included into the calculations by the method of small perturbation expansion both for potential and for wave elevation. The source strengths on the free surface are expressed in terms of perturbation potential by applying first-order (linearized) and second-order free surface conditions. The IBEM is applied to a 2-D (NACA16006 and NACA0012) and a 3-D rectangular cavitating hydrofoil and the effect of number of iterations, the effect of the depth of the hydrofoil from finite bottom and the effect of the walls of NWT, on the results are discussed.     

The hypersingular boundary contour method for two-dimensional linear elasticity

Summary This paper presents a novel method called the Hypersingular Boundary Contour Method (HBCM) for two-dimensional (2-D) linear elastostatics. This new method can be considered to be a variant of the standard Boundary Element Method (BEM) and the Boundary Contour Method (BCM) because: (a) a regularized form of the hypersingular boundary integral equation (HBIE) is employed as the starting point, and (b) the above regularized form is then converted to a boundary contour version based on the divergence free property of its integrand. Therefore, as in the 2-D BCM, numerical integrations are totally eliminated in the 2-D HBCM. Furthermore, the regularized HBIE can be collocated at any boundary point on a body where stresses are physically continuous. A full theoretical development for this new method is addressed in the present work. Selected examples are also included and the numerical results obtained are uniformly accurate.     

Solution of three dimensional eddy current problems by integral anddifferential methods

A differential T, Ω formulation using nonconformal elements is presented. It is an extension of the T integral formulation. The method has been used as the basis of a finite-element code that handles the open boundary problem of eddy-current analysis in fully 3-D conductors. The results obtained with this formulation for different treatments of the boundary conditions at infinity are shown and compared among themselves and with respect to a T integral formulation. The efficiencies are compared in terms of storage occupation, CPU time, and accuracy     

Heat transfer during fluidized-bed coating

A fully implicit numerical method for linear parabolic free boundary problems with coupled and integral boundary conditions is described. The partial differential equation and the boundary conditions are time discretized with the method of lines. An auxiliary function is introduced to remove the coupled and integral boundary conditions from the resulting free boundary problem for ordinary differential equations. Once separated boundary conditions are obtained, invariant imbedding is used to solve the free boundary problem numerically. The method is illustrated by solving the heat transfer equations for the fluidized-bed coating of a thin-walled cylinder.     

水平层状饱和场地地震响应分析的等效线性化方法

采用饱和土层和半空间精确的动力刚度矩阵,建立了求解P波、SV波斜入射情况下水平层状饱和场地地震响应的一维等效线性化分析方法。将饱和场地退化至干土场地进行方法验证,并将该文结果与二维有限元程序FLUSH结果进行对比,验证了该文方法的准确性。以天津市区某典型饱和场地为例,分别比较了P波和SV波入射情况下饱和场地线性、非线性地震响应之间的差别,以及饱和场地、干土场地非线性地震响应之间的差别,并讨论了入射角度和透水条件等因素对地面运动加速度峰值的影响,总结了饱和场地非线性地震响应的一些基本规律。