Boundary element analysis of dynamic coupled thermoelasticity problems

Some possibility of numerical analysis of coupled dynamic problems of linear elastic heat conductors on classical thermoelasticity theory by using the boundary element method is shown in this paper. The boundary integral equation formulation and its numerical implementation of the two-dimensional problem are developed in the manner by the newly derived fundamental solution for the coupled equations of elliptic type in the transformed space and the numerical inversion of Laplace transformation. The boundary element unsteady solutions of the first and second Danilovskaya problems and the Sternberg and Chakravorty problem in the half-space are demonstrated through comparison with the existing solutions.     

The boundary element method applied to the analysis of two-dimensional nonlinear sloshing problems

This paper deals with an application of the boundary element method to the analysis of nonlinear sloshing problems, namely nonlinear oscillations of a liquid in a container subjected to forced oscillations. First, the problem is formulated mathematically as a nonlinear initial-boundary value problem by the use of a governing differential equation and boundary conditions, assuming the fluid to be inviscid and incompressible and the flow to be irrotational. Next, the governing equation (Laplace equation) and boundary conditions, except the dynamic boundary condition on the free surface, are transformed into an integral equation by employing the Galerkin method. Two dynamic boundary condition is reduced to a weighted residual equation by employing the Galerkin method. Two equations thus obtained are discretized by the use of the finite element method spacewise and the finite difference method timewise. Collocation method is employed for the discretization of the integral equation. Due to the nonlinearity of the problem, the incremental method is used for the numerical analysis. Numerical results obtained by the present boundary element method are compared with those obtained by the conventional finite element method and also with existing analytical solutions of the nonlinear theory. Good agreements are obtained, and this indicates the availability of the boundary element method as a numerical technique for nonlinear free surface fluid problems.     

Three-dimensional dynamic fracture analysis with the dual reciprocity method in Laplace domain

In this paper, the dual reciprocity boundary element method in the Laplace domain has been developed for the analysis of three-dimensional elastodynamic fracture mechanics mixed-mode problems. The boundary element method is used to calculate the unknowns of transformed boundary displacement and traction and the domain integrals in the elastodynamic equation are transformed into boundary integrals by the use of the dual reciprocity method. The transformed dynamic stress intensity factors are determined by the crack opening displacement (COD) directly in the Laplace domain. By using Durbin's inversion technique, the dynamic stress intensity factors in the time domain are obtained. Several numerical examples are presented to demonstrate the good agreement with existing solutions.     

The mixed-type integral equations for transient elastodynamic plane crack problems

In the present paper, by use of the boundary integral equation method and the techniques of Green fundamental solution and singularity analysis, the dynamic infinite plane crack problem is investigated. For the first time, the problem is reduced to solving a system of mixed-typed integral equations in Laplace transform domain. The equations consist of ordinary boundary integral equations along the outer boundary and Cauchy singular integral equations along the crack line. The equations obtained are strictly proved to be equivalent with the dual integral equations obtained by Sih in the special case of dynamic Griffith crack problem. The mixed-type integral equations can be solved by combining the numerical method of singular integral equation with the ordinary boundary element method. Further use the numerical method for Laplace transform, several typical examples are calculated and their dynamic stress intensity factors are obtained. The results show that the method proposed is successful and can be used to solve more complicated problems.     

微极饱和土波动分析中的变分原理

主要给出饱和多孔微极介质波动方程变分所对应的泛函表达式和有限元离散化方程。首先对u-U形式的饱和多孔微极介质波动方程和边界条件进行Laplace 变换,形成力学中的非齐次边值问题,然后构造变分后满足波动方程和边界条件的泛函,最后将有限元插值形式代入泛函表达式得到单元体的有限元离散方程。此方程对微极饱和多孔介质的动力固结问题数值分析具有重要意义。     

Numerical operational methods for time-dependent linear problems

A general and systematic discussion on the use of the operational method of Laplace transform for numerically solving complex time-dependent linear problems is presented. Application of Laplace transform with respect to time on the governing differential equations as well as the boundary and initial conditions of the problem reduces it to one independent of time, which is solved in the transform domain by any convenient numerical technique, such as the finite element method, the finite difference method or the boundary integral equation method. Finally, the time domain solution is obtained by a numerical inversion of the transformed solution. Eight existing methods of numerical inversion of the Laplace transform are systematically discussed with respect to their use, range of applicability, accuracy and computational efficiency on the basis of some framework vibration problems. Other applications of the Laplace transform method in conjunction with the finite element method or the boundary integral equation method in the areas of earthquake dynamic response of frameworks, thermaliy induced beam vibrations, forced vibrations of cylindrical shells, dynamic stress concentrations around holes in plates and viscoelastic stress analysis are also briefly described to demonstrate the generality and advantages of the method against other known methods.     

An alternative collocation boundary element method for static and dynamic problems

A collocation boundary element formulation is presented which is based on a mixed approximation formulation similar to the Galerkin boundary element method presented by Steinbach (SIAM J Numer Anal 38:401–413, 2000) for the solution of Laplace’s equation. The method is also applicable to vector problems such as elasticity. Moreover, dynamic problems of acoustics and elastodynamics are included. The resulting system matrices have an ordered structure and small condition numbers in comparison to the standard collocation approach. Moreover, the employment of Robin boundary conditions is easily included in this formulation. Details on the numerical integration of the occurring regular and singular integrals and on the solution of the arising systems of equations are given. Numerical experiments have been carried out for different reference problems. In these experiments, the presented approach is compared to the common nodal collocation method with respect to accuracy, condition numbers, and stability in the dynamic case.     

Fundamental solutions for dynamic BE analyses of incompressible problems

The dilatational wave velocity theoretically tends to infinity as Poisson's ratio approaches 0·5. This infinite wave velocity can cause serious numerical difficulty in boundary element analyses of dynamic incompressible problems. This paper shows that when Poisson's ratio equals 0·5 Stokes' solutions are independent of the dilatational wave velocity. Consequently, by using the modified fundamental solutions, the boundary element method can effectively analyse dynamic incompressible problems without special difficulty.     

Free vibration analysis of anisotropic solids with the boundary element method

In this paper, the boundary element method is employed for the solution of three-dimensional anisotropic free vibration problems. The formulation is based upon the use of static fundamental solutions in conjunction with the dual reciprocity method. This approach is very advantageous for the solution of free vibration problems and circumvents the problems related to the anisotropic dynamic fundamental solutions. By means of numerical examples, the influence of the internal collocation points on the representation of the mass matrix and the occurrence of complex-valued eigenfrequencies is investigated. The eigenfrequencies and mode shapes obtained with the boundary element method are compared to finite element computations and excellent agreement is observed.     

层合梁的弯曲与自由振动分析

本文在铁摩辛柯梁理论的基础上, 应用迭合刚度的方法和Hamilton原理, 导出了适合于层合梁静力分析和动力分析的控制方程组(在单层情况下, 将退化成Timoshenko梁的方程)及边界条件。而且, 利用所获得的控制方程, 求得了层合梁一些问题的解析解及相应的数值结果。      

Analysis of functionally graded plates by meshless method: A purely analytical formulation

Functionally graded plates under static and dynamic loads are investigated by the local integral equation method (LIEM) in this paper. Plate bending problem is described by the Reissner moderate thick plate theory. The governing equations for the functionally graded material with respect to the neutral plane are presented in the Laplace transform domain and therefore the in-plane and bending problems are uncoupled. Both isotropic and orthotropic material properties are considered. The local integral equation method is developed with the locally supported radial basis function (RBF) interpolation. As the closed forms of the local boundary integrals are obtained, there are no domain or boundary integrals to be calculated numerically in this approach. The solutions of the nodal values for the entire plate are obtained by solving a set of linear algebraic equation system with certain boundary conditions. Details of numerical procedures are presented and the accuracy and convergence characteristics of the method are examined. Several examples are presented for the functionally graded plates under static and dynamic loads and the accuracy for proposed method has been observed compared with 3D analytical solutions.     

Dynamic analysis of large-scale SSI systems for layered unbounded media via a parallelized coupled finite-element/boundary-element/scaled boundary finite-element model

An algorithm for a parallelized coupled model based on finite element method (FEM), boundary element method (BEM), and scaled boundary FEM (SBFEM) for harmonic and transient dynamic response of large-scale 2D structures embedded in or on layered soil media is presented. The BEM and SBFEM are used for modelling the dynamic response of the unbounded media. The standard FEM is used for modelling the finite region and the embedded structure. The objective of the development of this parallelized coupled model is to use the power of high performance computing, and to take into account the advantages and evade the disadvantages of the above mentioned numerical methods for modelling of the unbounded media in soil-structure interaction (SSI) systems. The development of the parallel algorithm for this model is essential for solving arbitrarily shaped large-scale SSI problems, which cannot be solved within reasonable elapsed times by a serial algorithm. The efficiency of the proposed parallel algorithm and the validity of the coupled model are shown by means of three numerical examples, indicating the excellent accuracy and applicability of the parallel algorithm with considerable time-savings in large-scale problems.     

A combined scheme of generalized finite difference method and Krylov deferred correction technique for highly accurate solution of transient heat conduction problems

This paper presents a numerical framework for the highly accurate solutions of transient heat conduction problems. The numerical framework discretizes the temporal direction of the problems by introducing the Krylov deferred correction (KDC) approach, which is arbitrarily high order of accuracy while remaining the computational complexity same as in the time-marching of first-order methods. The discretization by employing the KDC method yields a boundary value problem of the inhomogeneous modified Helmholtz equation at each time step. The meshless generalized finite difference method (GFDM) or meshless finite difference method (MFDM), a meshless method, is then applied to the solution of resulting boundary value problems at each time step. Six numerical experiments in one-, two-, and three-dimensional cases show that the proposed hybrid KDC-GFDM scheme allows big time step size for a long-time dynamic simulation and has a great potential for the problems with complex boundaries. In addition, some comparisons are also presented between the present method, the COMSOL software, and the GFDM with implicit Euler method.     

Comparative study on orthotropic solid waves by time-domain BEM and dynamic photoelasticity

Experimental research and numerical analysis are two basic tools in the study of wave propagation problems in orthotropic media. In this paper, an experimental method, namely dynamic orthotropic photoelasticity, which studies the dynamic behavior of orthotropic materials on a macroscopic scale by employing orthotropic birefringent materials, is established. Meanwhile, a numerical method, namely time domain boundary element method (BEM) for wave propagation in orthotropic media, is also presented. The two methods are used together in the analysis of semi-infinite orthotropic plates with and without a circular hole modeled by a unidirectional fiber-reinforced composite under impact loading. The propagation, reflection and diffraction of stress waves in the orthotropic media are recorded experimentally and investigated. Time histories of birefringent fringe orders or stresses for specific points of the plates are obtained, respectively, from the two methods and compared with each other. The comparative study demonstrates the applicability and accuracy of the two methods for wave propagation problems in orthotropic media.     

地震作用下水-轴对称柱体相互作用的子结构分析方法

针对水-轴对称柱体动力相互作用问题,提出了一种地震作用下水-结构相互作用的时域子结构分析方法.基于三维不可压缩水体的波动方程和边界条件,利用分离变量法将其转换为环向解析、竖向和径向数值的二维模型;基于比例边界有限元推导了截断边界处无限域水体的动力刚度方程,并将水体内域有限元方程和人工边界处的动水压力进行耦合,从而得到结...     

Frequency response analyses in vibroacoustics using the method of fundamental solutions

The method of fundamental solutions, one of the promising boundary-type meshless methods, is proposed as a direct procedure to formulate and analyze the vibroacoustic problem. The coupled system discussed in this study is composed of an acoustic-cavity and excited by an external force or an internal sound source harmonically. The wall of cavity consists of the beam or the plate components, respectively, in two- and three-dimensional problems. The two independent sub-systems interact at the interface simultaneously by satisfying the necessary equilibrium and compatibility conditions. The mathematical formulations described by the presented meshless method demonstrate straightforwardly the frequency responses of the vibroacoustic problems with no boundary integrals. General characteristics of the dynamic coupling effect are displayed, based on the systematic natural frequencies and mode shapes. Feasible results simulated by the presented numerical scheme are validated through meshless numerical experiments including the acoustic-wave propagation problems and the vibroacoustic problems.     

A numerical method for elastic and viscoelastic dynamic fracture problems in homogeneous and bimaterial systems

We present a summary of recent advances in the development of an efficient numerical scheme to be used in the investigation of a wide range of 2D and 3D dynamic fracture problems. The numerical scheme, which is based on a spectral representation of the boundary integral relations, can be applied to homogeneous and interfacial dynamic fracture problems involving planar cracks and faults of arbitrary shapes buried in elastic and viscoelastic media. Spontaneous propagation of the crack is achieved by combining the elastodynamic integral relations with a stress-based cohesive failure model. The objective of this paper is to present some of the major differences existing between the various formulations within the simpler 2D scalar framework of anti-plane shear (mode III) loading conditions. Examples are presented to illustrate some capabilities of the method.     

Modal analysis of anisotropic plates using the boundary element method

A numerical formulation for analysis of dynamic problems of thin anisotropic plates bending is presented. The bending behavior follows Kirchhoff's hypothesis. The formulation is based on the direct boundary element method. The problem is simplified by using the elastostatic fundamental solution of an infinite plate. Domain integrals arising from inertial terms are transformed into boundary integrals using the dual reciprocity technique. Boundary integrals are discretized and evaluated numerically. Natural frequencies for free vibration are obtained and the respective mode shapes are shown. The accuracy of numerical results obtained is assured by comparison with analytical or finite element results.     

Least squares element method for boundary eigenvalue problems

Linear and non-linear boundary eigenvalue problems are discretized by a new finite element like method. The reason for the new construction principle is the non-linear dependence of the dynamic stiffness element matrix on an eigenparameter. The dynamic stiffness element matrix is evaluated for a fixed number of parameters and is then elementwise replaced by a polynomial in the eigenparameter by solving least squares problems. A fast solver is introduced for the resulting non-linear matrix eigenvalue problem. It consists of a combination of bisection method and inverse iteration. The superiority of the newconstructionprinciple in comparison with the finite or dynamic element method is demonstrated finally for some numerical examples.     

Boundary element analysis of stress intensity factor KI in some two-dimensional dynamic thermoelastic problems

This paper presents a numerical technique for the calculation of stress intensity factor as a function of time for coupled thermoelastic problems. In this task, effect of inertia term considering coupled theory of thermoelasticity is investigated and its importance is shown.A boundary element method using Laplace transform in time-domain is developed for the analysis of fracture mechanic considering dynamic coupled thermoelasticity problems in two-dimensional finite domain. The Laplace transform method is applied to the time-domain and the resulting equations in the transformed field are discretized using boundary element method. Actual physical quantities in time-domain is obtained, using the numerical inversion of the Laplace transform method.The singular behavior of the temperature and stress fields in the vicinity of the crack tip is modeled by quarter-point elements. Thermal dynamic stress intensity factor for mode I is evaluated using J-integral method. By using J-integral method effects of inertia term and other terms such as strain energy on stress intensity factor may be investigated separately and their importance may be shown. The accuracy of the method is investigated through comparison of the results with the available data in literature.