Finite element analysis of a compressible dynamic viscoelastic sphere using FFT

An extension to a compressible dynamic viscoelastic hollow sphere problem with both finite and infinite outer radius is performed. The governing viscoelastic equations of motion are transformed into the Laplace domain via the elastic–viscoelastic correspondence principle. Real and imaginary parts of the nodal displacements are obtained by solving a non-symmetric matrix equation in the complex Laplace domain. Inversion into the time domain is performed using the discrete inverse Fourier transform. Use is made of an infinite element in the infinite sphere problem. Numerical solutions are compared to both the exact Laplace and time domain solutions wherever possible.     

Boundary element analysis of linear viscoelastic problems using realistic relaxation functions

This paper presents an efficient method for the stress analysis of realistic viscoelastic solids by the time-domain boundary element method. The fundamental solutions and stress kernels are obtained using the elastic-viscoelastic correspondence principle. Since it is inconvenient to obtain the Laplace transform of the relaxation functions of realistic viscoelastic solids, the method of collocation has been employed and the relaxation function has been expanded in a sum of exponentials. Numerical results of example problems show the effectiveness and applicability of the proposed method.     

Application of fractional order theory of magneto-thermoelasticity to an infinite perfect conducting body with a cylindrical cavity

A fractional model of the equations of generalized magneto-thermoelasticity for a perfect conducting isotropic thermoelastic media is given. This model is applied to solve a problem of an infinite body with a cylindrical cavity in the presence of an axial uniform magnetic field. The boundary of the cavity is subjected to a combination of thermal and mechanical shock acting for a finite period of time. The solution is obtained by a direct approach by using the thermoelastic potential function. Laplace transform techniques are used to derive the solution in the Laplace transform domain. The inversion process is carried out using a numerical method based on Fourier series expansions. Numerical computations for the temperature, the displacement and the stress distributions as well as for the induced magnetic and electric fields are carried out and represented graphically. Comparisons are made with the results predicted by the generalizations, Lord–Shulman theory, and Green–Lindsay theory as well as to the coupled theory.

     

Dynamic response of frameworks by numerical laplace transform

A general method for determining the dynamic response of complex three-dimensional frameworks to dynamic shocks, wind forces or earthquake excitations is presented. The method consists of formulating and solving the dynamic problem in the Laplace transform domain by the finite element method and of obtaining the response by a numerical inversion of the transformed solution. The formulation is based on the exact solution of the transformed governing equation of motion of a beam element, and it consequently leads to the exact solution of the problem. Flexural, axial and torsional motion of the framework members are considered. The effects of damping (external viscous or internal viscoelastic), axial forces on bending, rotatory inertia and shear deformation on the dynamic response are also taken into account. Numerical examples to illustrate the method and demonstrate its merits are presented.     

Fracture analysis of plane piezoelectric/piezomagnetic multiphase composites under transient loading

The transient response of cracked composite materials made of piezoelectric and piezomagnetic phases, when subjected to in-plane magneto-electro-mechanical dynamic loads, is addressed in this paper by means of a mixed boundary element method (BEM) approach. Both the displacement and traction boundary integral equations (BIEs) are used to develop a single-domain formulation. The convolution integrals arising in the time-domain BEM are numerically computed by Lubich’s quadrature, which determines the integration weights from the Laplace transformed fundamental solution and a linear multistep method. The required Laplace-domain fundamental solution is derived by means of the Radon transform in the form of line integrals over a unit circumference. The singular and hypersingular BIEs are numerically evaluated in a precise and efficient manner by a regularization procedure based on a simple change of variable, as previously proposed by the authors for statics. Discontinuous quarter-point elements are used to properly capture the behavior of the extended crack opening displacements (ECOD) around the crack-tip and directly evaluate the field intensity factors (stress, electric displacement and magnetic induction intensity factors) from the computed nodal data. Numerical results are obtained to validate the formulation and illustrate its capabilities. The effect of the combined application of electric, magnetic and mechanical loads on the dynamic field intensity factors is analyzed in detail for several crack configurations under impact loading.     

Boundary identification for a 3D Laplace equation using a genetic algorithm

In this paper we consider the identification of the geometric structure of the boundary of the solution domain for the three-dimensional Laplace equation. We investigate the determination of the shape of an unknown portion of the boundary of a solution domain from Cauchy data on the remaining portion of the boundary. This problem arises in the study of quantitative non-destructive evaluation of corrosion in materials in which boundary measurements of currents and voltages are used to determine the material loss caused by corrosion. The domain identification problem is considered as a variational problem to minimize a defect functional, which utilises some additional data on certain known parts of the boundary. A real coded genetic algorithm (RCGA) is used in order to minimise the objective functional. The unknown boundary is parameterized using B-splines. The Laplace equation is discretised using the method of fundamental solutions (MFS). Numerical results are presented and discussed for several test examples.     

Numerical solution of the two-dimensional Helmholtz equation with variable coefficients by the radial integration boundary integral and integro-differential equation methods

This paper presents new formulations of the boundary–domain integral equation (BDIE) and the boundary–domain integro-differential equation (BDIDE) methods for the numerical solution of the two-dimensional Helmholtz equation with variable coefficients. When the material parameters are variable (with constant or variable wave number), a parametrix is adopted to reduce the Helmholtz equation to a BDIE or BDIDE. However, when material parameters are constant (with variable wave number), the standard fundamental solution for the Laplace equation is used in the formulation. The radial integration method is then employed to convert the domain integrals arising in both BDIE and BDIDE methods into equivalent boundary integrals. The resulting formulations lead to pure boundary integral and integro-differential equations with no domain integrals. Numerical examples are presented for several simple problems, for which exact solutions are available, to demonstrate the efficiency of the proposed methods.     

Frequency domain analyses of a thin liquid film surface by repetitively applied stress (dynamic response analyses by the long wave equation)

Numerical results were calculated for the dynamic behavior of an ultrathin liquid (lubricant) surface resulting from repetitively applied pressure and shear stress using the frequency domain equation and compared with those obtained using the time domain equation. Frequency domain analyses of the dynamic behavior of the ultrathin liquid (lubricant) surface produced by sinusoidally applied pressure and shear stress clarified the dependence of the liquid surface deformation on the frequency of the stresses and the disk speed. The dynamic behavior resulting from sinusoidally applied pressure and shear stress calculated using the time domain equation were found to gradually coincide with those obtained using the frequency domain equation.     

Dynamic response of framed underground structures

The dynamic response of framed underground structures under conditions of plane strain is numerically determined in this work. The soil deposit surrounding such structures is assumed to be horizontally layered and resting on a rigid base from which shear waves originate, and to exhibit linear elastic or viscoelastic material behavior. The methodology consists of applying the Laplace transform with respect to time to the governing equations of motion of the soil and the structure and subsequently constructing dynamic stiffness influence coefficients for typical soil and structure elements. A numerical inversion of the solution obtained by the finite element methodology employing these influence coefficients in the transformed domain yields the response as a function of time. Numerical examples to illustrate the method and demonstrate its advantages are presented.     

Thermally induced vibrations of beam structures

A general numerical method is developed for determining the dynamic response of beam structures to rapidly applied thermal loads. The method consists of formulating and solving the dynamic problem in the Laplace transform domain with the aid of dynamic stiffness influence coefficients defined for a beam element in that domain and of obtaining the response by a numerical inversion of the transformed solution. Thus, the solution of the associated heat conduction problem, usually obtained by Laplace transform and needed for the computation of the thermal load, can be used in its transformed form. The effects of damping and of axial compressive forces on the structural response are also studied. Three examples are presented in detail to illustrate the proposed method and demonstrate its advantages.     

Computing the principal eigenvalue of the Laplace operator by a stochastic method

We describe a Monte Carlo method for the numerical computation of the principal eigenvalue of the Laplace operator in a bounded domain with Dirichlet conditions. It is based on the estimation of the speed of absorption of the Brownian motion by the boundary of the domain. Various tools of statistical estimation and different simulation schemes are developed to optimize the method. Numerical examples are studied to check the accuracy and the robustness of our approach.     

A new filtering method for the Cauchy problem of the Laplace equation

In the present paper, we consider a Cauchy problem for the Laplace equation in a rectangle domain. A new filtering method is presented for approximating the solution of this problem, and the Hölder-type error estimates are obtained by the different parameter choice rules. Numerical illustration shows that the proposed method works effectively.     

The estimation of parameters in time-dependent transport problems: Dynamic programming and associative memories

Time-dependent radiation and energy transport problems are important in atmospheric science, medicine, biochemistry, and other areas. To determine external energy fields, direct problems (in which parameters are known) can be solved computationally by numerical integration followed by the numerical inversion of Laplace transforms. On the other hand, this paper treats inverse problems of estimating transport parameters on the basis of external observations of radiant intensity. These problems are approached using associative memory neural networks whose associated least squares problem is solved using a new dynamic programming algorithm. The quality of the estimates in the presence of noise in measurements is studied.     

Dynamic response of plate systems by combining finite differences, finite elements and laplace transform

A numerical method for the determination of the dynamic response of large rectangular plates or plate systems to lateral loads is proposed. The method is a combination of the finite difference method, the finite element method and the Laplace transform with respect to time. The plate system is considered as an assemblage of a small number of big rectangular superelements whose stiffness matrices are derived with the aid of the finite difference method in the Laplace transform domain. These superelements are then used to formulate and solve the problem by the finite element method in the transformed domain. The dynamic response is finally obtained by a numerical inversion of the transformed solution. External viscous or internal viscoelastic damping as well as the elastic foundation interaction effect can easily be taken into account. The method is illustrated and its merits demonstrated by means of numerical examples.     

离散时间分数阶多自主体系统的时延一致性

复杂工作环境中,许多自然现象的个体动力学特性用整数阶方程不能描述,只能用非整数阶（分数阶）动力学来描述个体的运动行为. 本文假设多自主体系统内部连接组成有向加权网络,个体的动态特性应用分数阶动力学方程描述,个体之间数据传输存在通信时延. 应用分数阶系统的Laplace变换和频域理论,研究了离散时间的分数阶多自主体系统的渐近一致性. 应用Hermit-Biehler 定理,研究了具有样本时延的分数阶多自主体系统的运动一致性,得到保证系统稳定的时延的上界阈值. 最后应用一个实例对结论进行了验证.     

On dual-phase-lag magneto-thermo-viscoelasticity theory with memory-dependent derivative

A new mathematical model of generalized magneto-thermo-viscoelasticity theories with memory-dependent derivatives (MDD) of dual-phase-lag heat conduction law is developed. The equations for one-dimensional problems including heat sources are cast into matrix form using the state space and Laplace transform techniques. The resulting formulation is applied to a problem for the whole space with a plane distribution of heat sources. It is also applied to a perfect conducting semi-space problem with a traction-free surface and plane distribution of heat sources located inside the medium. The inversion of the Laplace transforms is carried out using a numerical approach. Numerical results for the temperature, displacement, stress and heat flux distributions as well as the induced magnetic and electric fields are given and illustrated graphically. A comparison is made with the results obtained in the coupled theory. The impacts of the MDD heat transfer parameter and Alfven velocity on a viscoelastic material, for example, poly (methyl methacrylate) (Perspex) are discussed.

     

Richardson extrapolation applied to boundary element method results in a Dirichlet problem for the Laplace equation

Richardson extrapolation is used to improve the accuracy of the numerical solutions for the normal boundary flux and for the interior potential resulting from the boundary element method. The boundary integral equations arise from a direct boundary integral formulation for solving a Dirichlet problem for the Laplace equation. The Richardson extrapolation is used in two different applications: (i) to improve the accuracy of the collocation solution for the normal boundary flux and, separately, (ii) to improve the solution for the potential in the domain interior. The main innovative aspects of this work are that the orders of dominant error terms are estimated numerically, and that these estimates are then used to develop an a posteriori technique that predicts if the Richardson extrapolation results for applications (i) and (ii) are reliable. Numerical results from test problems are presented to demonstrate the technique.     

Comparative analysis of hybrid-trefftz stress and displacement elements

Summary The alternative stress and displacement models of the hybrid-Trefftz finite element formulation for the analysis of linear boundary value problems are derived in parallel form to emphasise the complementary nature of the fundamental concepts they develop from. In the stress model the stresses in the structural domain and the boundary displacements are independently approximated and inter-element stress continuity is enforced explicitly. Conversely, in the displacement model the displacements in the structural domain and the boundary tractions are independently approximated and inter-element linkage is enforced in the form of displacement continuity. In both models the approximation in the domain is constrained to satisfy locally all field equations, a feature typical of the Trefftz method. Duality is used to interpret physically the finite element equations, which are derived from the fundamental relations of elastostatics. Numerical tests are presented to compare the relative performance of the alternative stress and displacement models.     

A unified numerical approach for thermal stress waves

A unified numerical approach is introduced for the analysis of thermal stress waves. The algorithm is derived from the concept of heat displacement and a variational formulation in Lagrangian form. The objective of the paper is to demonstrate that by using the unified approach an existing computer code for isothermal finite element stress analysis can easily be modified to extend its capability to solve thermal stress problems. Numerical examples are given for the Danilovskaya's problems in dynamic thermoelasticity using a plane analysis computer code. It shows that the unified approach is particularly suitable for the study of thermally-induced waves including thermomechanical coupling effects.     

Elastodynamic fields near running cracks by finite elements

A finite element method is developed for the computation of elastodynamic stress intensity factors at a rapidly moving crack tip. The method is restricted to bodies whose surfaces and two-material interfaces are either parallel to the direction of propagation or are sufficiently remote. The crack tip starts to move at the instant that it is struck by an incident wave. The finite element grid moves undeformed with the crack tip. The main result consists in the fact that the method of non-singular calibrated crack tip elements, in which the stress-intensity factor is determined from its statically calibrated ratio to the crack opening displacement in an adjacent node, is extended to dynamic problems with moving cracks, for both in-plane and anti-plane motions. The dependence of the calibration ratio on the crack tip velocity is established from previously developed analytical solutions for the near-tip displacement fields. Numerical results compare favorably with known analytical solutions for cracks moving in an infinite solid. The grid motion causes an apparent asymmetric additional damping matrix.