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.     

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.     

Meshless Radial Basis Function Method for Transient Electromagnetic Computations

We propose a novel numerical method to simulate transient electromagnetic problems. The time derivatives are still tackled with the customary explicit leapfrog time scheme. But in the space domain, the fields at the collocation points are expanded into a series of radial basis functions and are treated with a meshless method procedure. Our method solves numerically Maxwell's equations with various assigned boundary conditions and current source excitation. Furthermore, the numerical stability condition of our method is obtained through a one-dimensional (1-D) wave equation and thus the relationship between control parameters is accounted for. To verify the accuracy and effectiveness of the new formulation, we compare the results of the proposed method with those of the conventional finite-difference time-domain method through a 1-D case study with different boundary conditions.      

Dynamic response of rigid strip-foundations by a time-domain boundary element method

The dynamic response of rigid strip-foundations placed on or embedded in a homogeneous, isotropic, linear elastic half-space under conditions of plane strain to either external forces or obliquely incident seismic waves of arbitrary time variation is numerically obtained. The above mixed boundary-value problems are treated by the time domain boundary element method which is used in a step-by-step timewise fashion to provide the foundation response to a rectangular impulse. Numerical examples are presented in detail to demonstrate the use and importance of the proposed method. The method appears to be more advantageous than frequency domain techniques, because it provides the transient foundation response in a natural and direct way and can form the basis for extension to the non-linear case.     

Modeling of porous piezoelectric structures by the meshless local Petrov-Galerkin method

A meshless method based on the local Petrov-Galerkin approach is proposed to solve initial-boundary value problems of porous piezoelectric solids. Constitutive equations for porous piezoelectric materials possess a coupling between mechanical displacements and electric intensity vectors in both solid and fluid phases. Stationary and transient 2-D and 3-D axisymmetric problems are considered in this article. Nodal points are spread on the problem domain, and each node is surrounded by a small circle for simplicity. The spatial variation of displacements and electric potentials for both phases is approximated by the moving least-squares scheme. After performing the spatial integration, one obtains a system of ordinary differential equations for certain nodal unknowns. The resulting system is solved numerically by the Houbolt finite-difference scheme as a time stepping method. The proposed method is applied to bending problems associated with a porous piezoelectric 2-D plate and 3-D axisymmetric cylinder under simply supported and clamped boundary conditions.     

Meshless local Petrov-Galerkin method for continuously nonhomogeneous linear viscoelastic solids

A meshless method based on the local Petrov-Galerkin approach is proposed for the solution of quasi-static and transient dynamic problems in two-dimensional (2-D) nonhomogeneous linear viscoelastic media. A unit step function is used as the test functions in the local weak form. It is leading to local boundary integral equations (LBIEs) involving only a domain-integral in the case of transient dynamic problems. The correspondence principle is applied to such nonhomogeneous linear viscoelastic solids where relaxation moduli are separable in space and time variables. Then, the LBIEs are formulated for the Laplace-transformed viscoelastic problem. The analyzed domain is covered by small subdomains with a simple geometry such as circles in 2-D problems. The moving least squares (MLS) method is used for approximation of physical quantities in LBIEs.     

Time-dependent fundamental solutions for homogeneous diffusion problems

This paper describes the applications of the method of fundamental solutions (MFS) for 1-, 2- and 3-D diffusion equations. The time-dependent fundamental solutions for diffusion equations are used directly to obtain the solution as a linear combination of the fundamental solution of the diffusion operator. The proposed scheme is free from the conventionally used Laplace transform or the finite difference scheme to deal with the time derivative of the governing equation. By properly placing the field points and the source points at a given time level, the solution is advanced in time until steady state solutions are reached. Test results obtained for 1-, 2- and 3-D diffusion problems show good comparisons with the analytical solutions and some with the MFS based on the modified Helmholtz fundamental solutions, thus the demonstration present numerical scheme of MFS with the space–time unification has been demonstrated as a promising mesh-free numerical tool to solve homogeneous diffusion problem.     

A least-squares front-tracking finite element method analysis of phase change with natural convection

This paper focuses on the numerical modelling of phase-change processes with natural convection. In particular, two-dimensional solidification and melting problems are studied for pure metals using an energy preserving deforming finite element model. The transient Navier–Stokes equations for incompressible fluid flow are solved simultaneously with the transient heat flow equations and the Stefan condition. A least-squares variational finite element method formulation is implemented for both the heat flow and fluid flow equations. The Boussinesq approximation is used to generate the bulk fluid motion in the melt. The mesh motion and mesh generation schemes are performed dynamically using a transfinite mapping. The consistent penalty method is used for modelling incompressibility. The effect of natural convection on the solid/liquid interface motion, the solidification rate and the temperature gradients is found to be important. The proposed method does not possess some of the false diffusion problems associated with the standard Galerkin formulations and it is shown to produce accurate numerical solutions for convection dominated phase-change problems.     

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.     

Reproducing kernel particle methods for structural dynamics

This paper explores a Reproducing Kernel Particle Method (RKPM) which incorporates several attractive features. The emphasis is away from classical mesh generated elements in favour of a mesh free system which only requires a set of nodes or particles in space. Using a Gaussian function or a cubic spline function, flexible window functions are implemented to provide refinement in the solution process. It also creates the ability to analyse a specific frequency range in dynamic problems reducing the computer time required. This advantage is achieved through an increase in the critical time step when the frequency range is low and a large window is used. The stability of the window function as well as the critical time step formula are investigated to provide insight into RKPMs. The predictions of the theories are confirmed through numerical experiments by performing reconstructions of given functions and solving elastic and elastic–plastic one-dimensional (1-D) bar problems for both small and large deformation as well as three 2-D large deformation non-linear elastic problems. Numerical and theoretical results show the proposed reproducing kernel interpolation functions satisfy the consistency conditions and the critical time step prediction; furthermore, the RKPM provides better stability than Smooth Particle Hydrodynamics (SPH) methods. In contrast with what has been reported in SPH literature, we do not find any tensile instability with RKPMs.     

The linear θ method for 2-D elastodynamic BE analysis

A linear θ method is used here to solve 2-D elastodynamic problems. Linear time and space interpolation functions are used, and expressions for the computation of stresses at internal points by means of appropriate integral equations are presented. When compared with the standard point collocation procedure, the linear θ method is more stable for 2-D elastodynamic problems, with an equivalent computer cost. As the linear θ method presents more numerical damping than the standard approach, special attention is required when either θ or the time step length is large. Two examples are presented in order to verify the accuracy of the proposed formulation. Received 15 November 1998     

Numerical damping of spurious oscillations: a comparison between the bulk viscosity method and the explicit dissipative Tchamwa–Wielgosz scheme

The use of Finite Element and Finite Difference methods of spatial and temporal discretization for solving structural dynamics problems gives rise to purely numerical errors. Among the many numerical methods used to damp out the spurious oscillations occurring in the high frequency domain, it is proposed here to analyse and compare the Bulk Viscosity method, which involves calculating the stresses, and a method recently presented by Tchamwa and Wielgosz, which is based on an explicit time integration algorithm. The 1-D study and the 2-D axisymmetric study on a bar subjected to compression and impact loads presented here show that the former method is insensitive to meshing irregularities, whereas the latter method is not. The Bulk Viscosity method was found to be sensitive, however, to the behavior of the material, contrary to the Tchamwa–Wielgosz method. Since comparisons of this kind are rather complex, a specific method of analysis was developed.     

Iterative coupling of FEM and BEM in 3D transient elastodynamics

A domain decomposition approach is presented for the transient analysis of three-dimensional wave propagation problems. The subdomains are modelled using the FEM and/or the BEM, and the coupling of the subdomains is performed in an iterative manner, employing a sequential Neumann–Dirichlet interface relaxation algorithm which also allows for an independent choice of the time step length in each subdomain. The approach has been implemented for general 3D problems. In order to investigate the convergence behaviour of the proposed algorithm, using different combinations of FEM and BEM subdomains, a parametric study is performed with respect to the choice of the relaxation parameters. The validity of the proposed method is shown by means of two numerical examples, indicating the excellent accuracy and applicability of the new formulation.     

Iterative coupling of FEM and BEM in 3D transient elastodynamics

A domain decomposition approach is presented for the transient analysis of three-dimensional wave propagation problems. The subdomains are modelled using the FEM and/or the BEM, and the coupling of the subdomains is performed in an iterative manner, employing a sequential Neumann–Dirichlet interface relaxation algorithm which also allows for an independent choice of the time step length in each subdomain. The approach has been implemented for general 3D problems. In order to investigate the convergence behaviour of the proposed algorithm, using different combinations of FEM and BEM subdomains, a parametric study is performed with respect to the choice of the relaxation parameters. The validity of the proposed method is shown by means of two numerical examples, indicating the excellent accuracy and applicability of the new formulation.     

A faster and accurate explicit algorithm for quasi‐harmonic dynamic problems

It is known that the explicit time integration is conditionally stable. The very small time step leads to increase of computational time dramatically. In this paper, a mass‐redistributed method is formulated in different numerical schemes to simulate transient quasi‐harmonic problems. The essential idea of the mass‐redistributed method is to shift the integration points away from the Gauss locations in the computation of mass matrix for achieving a much larger stable time increment in the explicit method. For the first time, it is found that the stability of explicit method in transient quasi‐harmonic problems is proportional to the softened effect of discretized model with mass‐redistributed method. With adjustment of integration points in the mass matrix, the stability of transient models is improved significantly. Numerical experiments including 1D, 2D and 3D problems with regular and irregular mesh have demonstrated the superior performance of the proposed mass‐redistributed method with the combination of smoothed finite element method in terms of accuracy as well as stability. Copyright © 2016 John Wiley & Sons, Ltd.     

Infinite boundary elements for dynamic problems of 3-D half space

In this paper, a new procedure for solving 3-D dynamic problems of unbounded foundations in the frequency domain by using BEM is studied. For simulations of wave propagations due to far field effects, a type of infinite boundary element (IBEM) is presented for modelling a 3-D regular or irregular half space. The wave type considered could be compressional, shear or a combination of the two. Through the analysis of the asymptotic behaviour of 3-D fundamental solutions for elasto dynamics, a rather feasible technique for obtaining singular integral coefficients for dynamic problems has been developed. Through the analysis of the dynamic response for a 3-D square foundation under a uniform load distribution, excellent accuracy has been achieved in agreement with previous numerical solutions. Another example–analysis of the dynamic compliance of a rigid square plate on a half space–has also shown very good results. The development of this infinite boundary element provides a powerful tool for dealing with 3-D structure foundation interaction or wave propagation problems for irregular foundations such as arch dam canyons.     

Adaptive time stepping strategy for solidification processes based on modified local time truncation error

Adaptive time step methods provide a computationally efficient means of simulating transient problems with a high degree of numerical accuracy. However, choosing appropriate time steps to model the transient characteristics of solidification processes is difficult. The Gresho–Lee–Sani predictor–corrector strategy, one of the most commonly applied adaptive time step methods, fails to accurately model the latent heat release associated with phase change due to its exaggerated time steps while the apparent heat capacity method is applied. Accordingly, the current study develops a modified local time truncation error‐based strategy designed to adaptively adjust the size of the time step during the simulated solidification procedure in such a way that the effects of latent heat release are more accurately modeled and the precision of the computational solutions correspondingly improved. The computational accuracy and efficiency of the proposed method are demonstrated via the simulation of several one‐dimensional and two‐dimensional thermal problems characterized by different phase change phenomena and boundary conditions. The feasibility of the proposed method for the modeling of solidification processes is further verified via its applications to the enthalpy method. Copyright © 2009 John Wiley & Sons, Ltd.     

二维正交各向异性结构弹塑性问题的边界元分析

给出了二维正交各向异性结构弹塑性问题的边界元分析方法, 包括相应边界积分方程、内点应力公式、边界元求解格式以及弹塑性应力计算方法。在弹塑性分析中, 引入了Hill-Tsai 屈服准则, 采用初应力法和切向预测径向返回法确定实际应力状态。通过具体算例分析了二维正交各向异性结构的弹塑性应力和塑性区分布情况, 部分数值结果与已有结果进行了比较, 两者基本吻合。结果表明, 本文中给出的边界元法可以有效地用于求解二维正交各向异性结构的弹塑性问题。      

Free surface flow through rock-fill dams analyzed by FEM with level set approach

 A stabilized-finite element formulation is coupled with a level set technique for computations of incompressible non-linear flow with interfaces between two immiscible fluids. An interface capturing formulation (ICF) for non-linear, free surface, seepage flow in rock-fill dams is proposed. The formulation is derived for two- and three-dimensional flow within a fixed mesh domain. The resulting formulation is general and applicable for various steady and transient two-phase flow problems. FE-refinement is processed for the entire fixed mesh domains. A general solver is also reviewed for large and non-symmetric non-positive definite linear system of equations with the GMRES-update technique based on a Newton-iterative method. The computational procedure has been implemented in MATLAB. A comparison is performed between the 2-D computed test problem for coarse and refined meshes together with some proposed analytical solutions for nonlinear seepage flow with free surface in rock-fill dams. An expansion of the 2-D program code to a 3-D one for a rectangular rock-fill dam is also developed and simulated in MATLAB. The performance of the computations in 3-D is very promising and its opening the future for possible industrial applications using the same simple technique. Computations for a simple 3-D seepage flow problem with free surface in rock-fill dam are included in present paper. A general mesh generator and solver for large scale and complex 3-D flow problems in a real embankment dam is also under construction in C++.     

柔性框架结构动力非线性分析的刚体准则法

大跨、高层等柔性结构,其动力响应往往表现出大位移、大转动等非线性特征。动力非线性问题的分析关键在于运动方程的高效稳定求解,以及单元大转动产生的结点力增量的有效计算。动力时程分析通常采用直接积分法,但对于强非线性动力问题,直接积分法难以兼顾计算精度与稳定性。该文基于几何非线性分析的刚体准则,针对杆件结构大转动小应变的非线性问题,提出了一种新型空间杆系结构动力非线性分析的刚体准则法。该方法采用满足刚体准则的空间非线性梁单元,结合HHT-α法求解结构运动方程,并将刚体准则植入动力增量方程的迭代求解过程以计算结点力增量。通过典型柔性框架算例结果表明,该文方法可以有效分析柔性框架结构的强动力非线性行为。与高精度单元相比,该文采用的单元刚度矩阵构造简明,计算过程简洁;与商业软件所用方法相比,单元数和迭代步少,精度高,适于工程应用。