A numerical treatment to the solution of quasiparabolic partial differential equations

This paper presents results obtained by the implementation of a hybrid Laplace transform finite element method to the solution of quasiparabolic problem. The present method removes the time derivatives from the quasiparabolic partial differential equation using the Laplace transform and then solves the associated equation with the finite element method. The numerical inverse of the Laplace transform is realized by solving linear overdetermined systems and a polynomial equation of the kth order. Test examples are used to show that the numerical solution is comparable to the exact solution of the initial-boundary value problem at the given grid points.     

New hybrid Laplace transform/finite element method for three-dimensional transient heat conduction problem

The paper presents results obtained by the implementation of a new hybrid Laplace transform/finite element method developed by the authors. The present method removes the time derivatives from the governing differential equation using the Laplace transform and then solves the associated equation with the finite element method. Previously reported hybrid Laplace transform/finite element methods¹ have been confined to one nodal solution at a time. When applied to many nodes it takes an excessive amount of computer time. By using a similarity transform method on the matrix of the complex number coefficients this restriction is removed and the reported new method provides a more useful tool for the solution of linear transient problems. Test examples are used to show that the basic accuracy is comparable to that obtainable by analytical, finite difference and finite element methods.     

On the Laplace-transformed-based local meshless method for fractional-order diffusion equation

In this work, we formulate a local meshless method based on Laplace transform to estimate the solution of a time-fractional diffusion equation. The collocation is constructed over small subdomains and combined with Laplace transform for a temporal variable. In this approach, the differentiation matrices are constructed by solving small systems over small local domains instead of a large global collocation matrix. The application of Laplace transform avoids the classical time-stepping procedure. This method is capable of solving fractional differential equations in multidimensions with higher accuracy.     

粘弹性基体中环绕纤维的环形裂纹的型和型应力强度因子

分析了粘弹性基体中环绕纤维的环形裂纹的 型和 型应力强度因子及其时间相关性。根据文献 [1 ]、文献 [2 ]中的弹性解 ,求出了粘弹性基体中环绕纤维的环形裂纹的 型和 型应力强度因子在 L aplace变换域内的解。对其进行 Laplace数值反演后 ,得到了相应的 型和 型应力强度因子在时间域内的变化曲线。结果表明 ,给定长度的环形裂纹在尚未接触界面时 ,其两端正则化的 型和 型应力强度因子均随时间增大而减小。     

粘弹性基体中环绕纤维的环形裂纹的&Iota|型和ΙΙ&Iota|型应力强度因子

分析了粘弹性基体中环绕纤维的环形裂纹的é 型和? 型应力强度因子及其时间相关性。根据文献[ 1 ]、文献[2 ]中的弹性解, 求出了粘弹性基体中环绕纤维的环形裂纹的é 型和? 型应力强度因子在Laplace 变换域内的解。对其进行Laplace 数值反演后, 得到了相应的é 型和? 型应力强度因子在时间域内的变化曲线。结果表明, 给定长度的环形裂纹在尚未接触界面时, 其两端正则化的é 型和? 型应力强度因子均随时间增大而减小。     

On a functional approximation for inversion of Laplace transforms

Abstract

A functional representation for inversion of the Laplace transform of a function is considered. The function is given in Laguerre polynomials expansion. The coefficients of the polynomials are in terms of weighted moments which are directly determined from the Laplace transform. The applications to rational and irrational Laplace transforms are presented to illustrate the satisfactory results that the method provides.     

直角坐标系下层状地基力学计算中的传递矩阵技术

给出了在直角坐标系下计算层状地基力学问题的传递矩阵技术,改变了过去只能在柱坐标系下求解层状地基力学问题的状况,大大地简化了任意荷载作用下层状地基力学问题的计算。从直角坐标系下弹性问题的基本方程出发,通过改变坐标系的常规位置,成功地利用Laplace变换及其微分性质,首次推导出直角坐标系下的状态控制方程和对应的传递矩阵,并给出了利用该方法的求解过程。     

A formulation and solution for boundary element analysis of inhomogeneous-nonlinear problem

A new formulation is presented in this paper for the boundary element analysis of a nonlinear potential-type problem wherein the linear term is governed by the Laplace operator, and the nonlinear term is a function of the spatial coordinates as well as the unknown solution function. The formulation aims to transform the domain integral relevant to the inhomogeneous-nonlinear term to a corresponding boundary integral. The proposed approach is different from the more popular schemes for the purpose, such as the Dual Reciprocity and Multiple Reciprocity Methods. The inhomogeneous-nonlinear term is first approximated by a polynomial in terms of the space coordinates with unknown coefficients. Integral equations on the selected points (referred to “computing points”) on the boundary as well as inside domain are employed to determine the above-mentioned unknown coefficients using the least square method. The number of computing points affects the accuracy of the result, which is discussed through some numerical examples in two-dimensional space.     

Hybrid Laplace transform/finite difference method for transient heat conduction problems

The new method involving the combined use of the Laplace transform and the finite difference method is applicable to the problem of time-dependent heat flow systems. The present method removes the time derivatives from the governing differential equation using the Laplace transform and then solves the associated equation with the finite difference method. The transformed temperature is inverted numerically by the method of Honig and Hirdes to obtain the result in the physical quantities. The present results are compared in tables with exact solutions and those obtained from the combined use of the Laplace transform and the finite element method. It is found that the present method is reliable and efficient.     

State space approach to one-dimensional thermal shock problem for a semi-infinite piezoelectric rod

The theory of generalized thermoelasticity, based on the theory of Lord and Shulman with one relaxation time, is used to solve a boundary value problem of one-dimensional semi-infinite piezoelectric rod with its left boundary subjected to a sudden heat. The governing partial differential equations are solved in the Laplace transform domain by the state space approach of the modern control theory. Approximate small-time analytical solutions to stress, displacement and temperature are obtained by means of the Laplace transform and inverse transform. It is found that there are two discontinuous points in both stress and temperature solutions. Numerical calculation for stress, displacement and temperature is carried out and displayed graphically.     

Non-linear analysis of wave propagation using transform methods

Non-linear wave propagation/transient dynamics in lattice structures is modeled using a technique which combines the Laplace transform and the Finite element method. The first step in the technique is to apply the Laplace transform to the governing differential equations and boundary conditions of the structural model. The non-linear terms present in these equations are represented in the transform domain by making use of the complex convolution theorem. Then, a weak formulation of the transformed equations yields a set of element level matrix equations. The trial and test functions used in the weak formulation correspond to the exact solutions of the linear parts of the transformed governing differential equations. Numerical results are presented for a viscoelastic rod and von Karman type beam.     

The mixed finite element method for the quasi‐static and dynamic analysis of viscoelastic timoshenko beams

The quasi‐static and dynamic responses of a linear viscoelastic Timoshenko beam on Winkler foundation are studied numerically by using the hybrid Laplace–Carson and finite element method. In this analysis the field equation for viscoelastic material is used. In the transformed Laplace–Carson space two new functionals have been constructed for viscoelastic Timoshenko beams through a systematic procedure based on the Gâteaux differential. These functionals have six and two independent variables respectively. Two mixed finite element formulations are obtained; TB12 and TB4. For the inverse transform Schapery and Fourier methods are used. The numerical results for quasi‐static and dynamic responses of several visco‐elastic models are presented. Copyright © 1999 John Wiley & Sons, Ltd.     

On the vibration analysis of laminated composite parabolic arches with variable cross-section of various ply stacking sequences

Abstract

The main objective of this study is to perform the free and forced vibration analysis of transversely isotropic and laminated composite parabolic arches with a continuous cross-section variation. The anisotropy of the material of the arch, effects of the rotary inertia, and shear deformations are considered. An efficient unified numerical procedure of the Complementary Functions Method and Laplace transform is applied to solve the strong form of the differential equations that govern the dynamic response of the above structures. The validity and the accuracy of the presented scheme are tested by means of several comparisons with available literature and results of ANSYS. The presented approach has proven to be an accurate and stable numerical method. It is believed that derived results can be used as benchmark solutions for validation of related works in the future.     

Hybrid method for transient response of circular pins

The combined application of the Laplace transform and the finite element method is used to analyse the transient response of circular pins. The present method removes the time-dependence terms from the governing differential equations and boundary conditions using the Laplace transform and then the eigenfunction expansion method is applied to reduce the two-dimensional boundary value problem to that of one dimension. Accordingly, the final transformed equation can easily be solved by the finite element method. The transformed temperature is inverted to the physical quantity numerically. The present results agree well with analytical solutions. In addition, it is seen that the results of axisymmetric transient heat conduction problems with the central node at r = 0 can accurately be obtained using the present method.     

Crack tip stress singularities and dislocation interactions with anisotropic viscoelastic bimaterial interfaces

The time evolution of the force on a dislocation and the near tip stress field of a semi-infinite crack, lying at an angle towards the welded interface of a bimaterial and just touching it, is studied. The bimaterial consists of viscoelastic media with the most general anisotropy. Applying the Laplace transform to the field equations gives similar relations to those considered in [3], provided that appropriate elastic coefficients are replaced by their Laplace-transformed viscoelastic counterparts. Numerical methods are then used for the inversion of both the Laplace and Mellin transforms and numerical results for near crack tip stress fields are presented along with asymptotic ones for small and large times. Corresponding results are presented for the force on a dislocation. For simple anisotropies a relation between the force on a dislocation and the time dependence of appropriate crack tip stresses is established.     

Weight Function for a Crack in a Two-dimensional Orthotropic Medium Under Impact Shear Loading

The paper examines the elastodynamic response of an infinite two-dimensional orthotr- opic medium containing a central crack under impact shear loading. Laplace and Fourier integral transforms are employed to reduce the problem to a pair of dual integral equations in the Laplace transformed plane. These equations are reduced to integral differential equations, which have been solved in the low frequency domain by iterations. To determine time dependence, these equations are inverted to yield the dynamic stress intensity factor (SIF) for shear point force loading that corresponds to the weight function for the crack under shear loading. It is used to derive SIF for polynomial loading.     

Fourier differential quadrature method for irregular thin plate bending problems on Winkler foundation

This paper describes Fourier differential quadrature method (FDQM). It is the combination of the Fourier spectral method and differential quadrature method (DQM) in barycentric form as a numerical method for solving problems for thin plates resting on Winkler foundations with irregular domains. The solution is decomposed into a polynomial particular solution for the inhomogeneous equation and the general solution for the homogeneous equation. In the solution procedure, the arbitrary distributed loading is first approximated by the Chebyshev polynomials and thus, the desired polynomial particular solution is obtained. For the latter, we use Fourier series expansion and determine the Fourier coefficients from the boundary conditions. Furthermore, the complex boundary conditions on irregular domains can be solved with DQM directly. Finally, numerical experiments are carried out to demonstrate the flexibility, high efficiency and accuracy of our method for irregular domains.     

Numerical modeling of micro- and macro-behavior of viscoelastic porous materials

This paper presents a numerical method for solving the two-dimensional problem of a polygonal linear viscoelastic domain containing an arbitrary number of non-overlapping circular holes of arbitrary sizes. The solution of the problem is based on the use of the correspondence principle. The governing equation for the problem in the Laplace domain is a complex hypersingular boundary integral equation written in terms of the unknown transformed displacements on the boundaries of the holes and the exterior boundaries of the finite body. No specific physical model is involved in the governing equation, which means that the method is capable of handling a variety of viscoelastic models. A truncated complex Fourier series with coefficients dependent on the transform parameter is used to approximate the unknown transformed displacements on the boundaries of the holes. A truncated complex series of Chebyshev polynomials with coefficients dependent on the transform parameter is used to approximate the unknown transformed displacements on the straight boundaries of the finite body. A system of linear algebraic equations is formed using the overspecification method. The viscoelastic stresses and displacements are calculated through the viscoelastic analogs of the Kolosov–Muskhelishvili potentials, and an analytical inverse Laplace transform is used to provide the time domain solution. Using the concept of representative volume, the effective viscoelastic properties of an equivalent homogeneous material are then found directly from the corresponding constitutive equations for the average field values. Several examples are given to demonstrate the accuracy of the method. The results for the stresses and displacements are compared with the numerical solutions obtained by commercial finite element software (ANSYS). The results for the effective properties are compared with those obtained with the self-consistent and Mori–Tanaka schemes.     

An exact dynamic stiffness matrix for axially loaded double-beam systems

An exact dynamic stiffness method is presented in this paper to determine the natural frequencies and mode shapes of the axially loaded double-beam systems, which consist of two homogeneous and prismatic beams with a distributed spring in parallel between them. The effects of the axial force, shear deformation and rotary inertia are considered, as shown in the theoretical formulation. The dynamic stiffness influence coefficients are formulated from the governing differential equations of the axially loaded double-beam system in free vibration by using the Laplace transform method. An example is given to demonstrate the effectiveness of this method, in which ten boundary conditions are investigated and the effect of the axial force on the natural frequencies and mode shapes of the double-beam system are further discussed.