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.     

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.     

求解应力强度因子的样条虚边界元交替法

为求解平面裂纹问题的应力强度因子,提出基于Muskhelishvili基本解和样条虚边界元法的样条虚边界元交替法.该方法将平面内带裂纹有限域问题分解成带裂纹无限域问题与不带裂纹有限域问题的叠加.带裂纹无限域问题利用Muskhelishvili基本解法直接得出,不带裂纹有限域问题采用样条虚边界元法求解.利用该方法对复合型中心裂纹方板和I型偏心裂纹矩形板进行分析.数值结果表明该方法精度高且适用性强.     

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.     

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.     

基于结合扩展精度技术的基本解方法的非线性功能梯度材料热传导问题求解

采用基本解方法结合扩展精度技术和Kirchhoff变换求解功能梯度材料的二维热传导问题.在求解瞬态热传导问题时运用Laplace变换处理时间变量,将时域问题转化为频域问题求解;采用基本解方法计算得到高精度的频域数值解,再分别采用Stehfest和Talbot这2种数值Laplace逆变换恢复原瞬态热传导问题的计算结果.通过3个非线性功能梯度材料的稳态和瞬态热传导基准算例,分析结合扩展精度技术的基本解方法的计算精度与扩展精度位数、边界布点数和虚拟边界参数三者之间的关系.比较Stehfest和Talbot这2种数值Laplace逆变换算法的优劣.采用结合扩展精度技术的基本解方法数值研究热传导系数随位置剧烈变化的功能梯度材料热传导行为.数值结果表明该方法具有求解精度高、适用性好等特点,能高效模拟非线性功能梯度材料的二维稳态与瞬态热传导行为.     

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.     

Dynamic response of frameworks by fast fourier transform

A general numerical method for determining the dynamic response of linear elastic plane frameworks to dynamic shocks, wind forces or earthquake excitations is presented. The method consists of formulating and solving the dynamic problem in the frequency domain by the finite element method and of obtaining the response by a numerical inversion of the transformed solution with the aid of the fast Fourier transform algorithm. 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, and axial 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 advantages over other methods are presented.     

Solution of contaminant transport with equilibrium and non-equilibrium adsorption

In this paper we present a numerical approximation scheme for the solution of contaminant transport problems with diffusion and adsorption in equilibrium and non-equilibrium mode. The method is based on time stepping and operator splitting. The non-linear transport is solved semi-analytically via the multiple Riemann problem, the non-linear diffusion by a finite volume method and by Newton’s type of linearisation, and finally the reaction part, incorporating the non-equilibrium adsorption, is transformed to an integral equation which is solved numerically using time discretization. Various results of numerical experiments are shown, and the method is applied to the practical dual-well problem.     

Dynamic analysis of beams by the boundary element method

Free and forced flexural vibrations of beams are numerically studied with the aid of the direct boundary element method. The free vibration case is treated as an eigenvalue problem, while the forced vibration one is treated with the aid of the Laplace transform. The structural dynamic response is finally obtained by a numerical inversion of the transformed solution. The effects of a constant axial force, external viscous or internal viscoelastic damping, and an elastic foundation on the response are also considered. Various numerical examples serve to illustrate the method and demonstrate its advantages and disadvantages.     

Some Mathematical and Numerical Aspects in Aluminum Production

In this paper, we present a mathematical modeling of some magnetohydrodynamic effects arising in an aluminum production cell as well as its numerical approximation by a finite element method. We put the emphasis on the magnetic effects which live in the whole three dimensional space and which are solved numerically with a domain decomposition method.     

Structural optimization using global stress-deviation objective function via the level-set method

The paper deals with minimum stress design using a novel stress-related objective function based on the global stress-deviation measure. The shape derivative, representing the shape sensitivity analysis of the structure domain, is determined for the generalized form of the global stress-related objective function. The optimization procedure is based on the domain boundary evolution via the level-set method. The elasticity equations are, instead of using the usual ersatz material approach, solved by the extended finite element method. The Hamilton-Jacobi equation is solved using the streamline diffusion finite element method. The use of finite element based methods allows a unified numerical approach with only one numerical framework for the mechanical problem as also for the boundary evolution stage. The numerical examples for the L-beam benchmark and the notched beam are given. The results of the structural optimization problem, in terms of maximum von Mises stress corresponding to the obtained optimal shapes, are compared for the commonly used global stress measure and the novel global stress-deviation measure, used as the stress-related objective functions.     

A state vector approach to the one dimensional consolidation of multi-layer soils

A method for determining the behaviour of a one-dimensional multi-layer soil under consolidation is described. A state vector whose components are the excess pore water pressure and discharge velocity is defined. It is shown that problems associated with the enforcement of continuity conditions at the inter-layer boundaries are thereby substantially removed. The time dependent behaviour of the settlement and excess pore water pressure is determined by numerically inverting the Laplace transform (with respect to the time) of the state vector. Examples of the numerical performance of the method are given and numerical comparisons are made with a finite element solution to the problem.     

Enclosure of solutions of weakly nonlinear elliptic boundary value problems and their computation

An algorithm is presented to compute approximations as well as continuous bounds for solutions of weakly nonlinear elliptic boundary value problems. The given problem is majorized in some sense and the obtained new problem is solved by a finite element method. The finite element solution is computed by a monotone iteration process and at last transformed to a continuous (lower) bound for a solution. Convergence is proved and mesh refinement effects are discussed. Two numerical examples are given.     

基于分层模型的复合材料耦合Lamb波频率—波数域特性研究

针对复合材料层合板中耦合Lamb波的传播问题,基于分层模型提出解析建模与有限元数值模拟相结合的方法对其进行预测和评估。利用Legendre正交多项式展开法推导多层各向异性复合材料层合板中耦合Lamb波的控制方程,并对频率-波数域频散特性曲线实现数值求解。基于平面壳单元构建复合材料层合板的有限元模型,采用波结构加载法生成单一Lamb波基本模态,设计复合材料层合板的不同纤维取向、边界和界面约束条件,并经二维傅里叶变换获得有限元模拟数据的频率-波数域频散特性曲线。通过对比验证,结果表明两种方法均有较好的吻合性。     

A parallel method for the numerical solution of integro-differential equation with positive memory

An efficient parallel numerical method is proposed for an integro-differential equation with positive memory. Instead of solving the equation in classical time-marching methods which require massive storage of solutions of previous time steps in order to advance to a next time step, the Fourier–Laplace transformation in time is applied to obtain a set of complex-valued, elliptic problems parameterized by points on a contour in the complex plane. Using the independence of an elliptic problem corresponding to one contour point is independent of those elliptic problems corresponding to other contour points, all elliptic problems can be solved in parallel essentially without data communications. Then the time domain solution can be obtained by the Fourier–Laplace inversion formula. An error analysis and the numerical implementation of this parallel method is presented.     

The Local Artificial Boundary Conditions for Numerical Simulations of the Flow Around a Submerged Body

We consider in this work the numerical approximations of the two-dimensional steady potential flow around a body moving in a liquid of finite constant depth at constant speed and distance below a free surface. Several vertical segments are introduced as the upstream and the downstream artificial boundaries, where a sequence of high-order local artificial boundary conditions are proposed. Then the original problem is solved in a finite computational domain, which is equivalent to a variational problem. The numerical approximations for the original problem are obtained by solving the variational problem with the finite element method. The numerical examples show that the artificial boundary conditions given in this work are very effective.     

Mathematical modeling of diffraction on an inhomogeneity in a waveguide using mixed finite elements

The hybrid finite element method is applied to solve the vector problem of the normal mode diffraction on an inhomogeneity in a waveguide. The algorithm for numerically solving this problem by means of the finite element method with the elements of mixed type is developed and implemented. Partial radiation conditions are used to make the domain bounded.     

Finite element method and laplace transform—Comparative solutions of transient heat conduction problems

The time dependence of temperatures as solutions of transient heat conduction problems, may be obtained by different numerical techniques. Three procedures are presented. The step-by-step methods, based (i) on finite-elements and (ii) finite-differences in time are briefly reviewed, (iii) The application of the numerical Laplace transform is extensively discussed and its introduction in a finite element program is presented. The accuracy and convergence of the numerical results are discussed and a practical engineering problem is solved for which the computer expenses are compared.     

Direct limit analysis via rigid-plastic finite elements

This work concerns the limit analysis of rigid-perfectly plastic plane structures. The continuum is discretized into finite elements, thereby permitting the theoretical nonlinear problem to be transformed to an equivalent linear elastic problem for which the corresponding collapse load may be found in a reasonable number of analysis iterations; as such, the analysis may be termed a quasi-direct approach. The duality concepts for linear elastic-analysis by the finite element method are extended to the plasticity problem using classical variational principles; in this way, both primal and dual quasi-direct approaches to the limit analysis problem are identified. Generalization of the variational principles permits the analysis to be formulated for not only compatible elements or equilibrium finite elements, but also for hybrid elements having two independent fields (i.e. stress and strain). Some numerical examples concerning classical academic problems for plane structures illustrate the efficiency of the method. Finally, a new hybrid element with an arbitrary stress field in the interior and a quadratic displacement field on the boundary is described in detail in an appendix.