功能梯度材料热传导问题的仿真

针对功能梯度材料的热传导问题,提出了一种新的无网格数值算法。首先,从功能梯度材料的二维稳态热传导方程出发,通过引入变量代换,系统地推导了指数型、二次型和三角函数型功能梯度材料热传导方程的基本解。然后基于基本解构造了无网格方法求解功能梯度材料的稳态热传导方程,进而对功能梯度材料的温度分布情况进行仿真。最后,分别以指数型、二次型及三角函数型功能梯度材料板为例,对其温度分布进行仿真。结果表明无网络算法是正确、有效的,且无需网格划分,有理论简单、易于程序实现等优点。     

基于自然元法和精细积分法的功能梯度材料瞬态热传导问题求解

为促进无网格法分析技术在热传导分析中的应用,提出空间离散采用自然单元法、时间离散采用精细积分法求解功能梯度材料瞬态热传导问题的数值计算方法.在计算过程中,取高斯点的材料参数模拟功能梯度材料特性的变化.温度场采用自然邻接点插值形函数进行离散插值.数值算例验证该数值算法的正确性和有效性.     

生物组织热传导问题的数值仿真

针对生物组织热传导方程解析解无法获得的问题,提出了将Laplace变换与杂交有限元相结合的数值求解算法.首先,利用Lapace变换对生物组织热传导方程进行处理,将时间域内的边值问题转化为Laplace空间内;接着,利用杂交元方法在Laplace空间对此新的边值问题进行求解;然后,通过Stehfest算法进行Laplace反变换,将结果变换会时间域中,从而得到生物组织的温度分布.最后,以皮肤组织的热传导问题为例,利用Matlab编制程序对其温度场分布进行数值仿真,算例结果表明肿瘤病变会改变皮肤组织的温度分布,即肿瘤处的温度较周围组织明显升高.仿真结果表明LT和HFEM相结合算法的正确性和有效性,为非入侵式医疗诊断提供了依据.     

三维有限元非线性导热计算程序FemHC

介绍采用有限元法求解稳态和瞬态非线性导热方程的计算程序FemHC,明确求解导热问题所依据的控制方程和有限元离散方法,描述FemHC的基本框架和核心类,并通过典型导热问题验证FemHC计算的准确性。采用FemHC计算三维涡轮叶片的热传导,结果表明FemHC可以用于实际涡轮叶片等复杂结构导热问题的计算。     

基于时频域交替法的迟滞非线性振动系统的稳态响应分析

连接界面的黏滑、摩擦行为不仅是引起结构刚度和阻尼非线性的主要原因,而且是结构无源阻尼的主要来源.Iwan模型能够较好地复现连接界面的黏滑、摩擦行为.本文采用时频域交替法(Alternating Frequency/Time Domain Method,AFT)研究含Iwan非线性模型的单自由度振子系统的稳态响应.时频域交替法具有频域法求解线性系统响应的高效性和时域法判断非线性力的便捷性特点,采用离散傅里叶变换和傅里叶逆变换,在频域和时域内分别求解系统响应和对应的非线性恢复力,再反复迭代计算系统的稳态响应.将时频域交替法计算结果和中心差分法计算的结果进行对比,并研究激励幅值对系统非线性特征的影响.结果表明,时频域交替法计算的结果与中心差分计算的结果具有较好的一致性,且求解效率较高,计算耗时减少50%;随着激励幅值的增加,系统的能量耗散增加,刚度降低,固有频率降低.     

基于复化Gauss-Legendre积分的Laplace变换数值反演及其应用

Laplace变换的数值反演是一个病态问题.采用代数精度较高的数值积分近似Laplace变换截断积分,合理选取复平面上样本点以形成离散线性代数方程组是解决这个问题的途径之一.本文采用代数精度较高的复化Gauss-Legendre数值积分近似Laplace变换截断积分,推导了一种Laplace变换数值反演算法.其间,对于所形成的条件数很大的线性方程组采用基于约化奇异值分解的最小二乘法进行求解,以尽可能降低数值解的误差.使用该算法对简单测试算例进行数值反演,并将其结果与精确解进行对比,结果表明,相比经典的Gaver-Stehfest方法和基于GaussLegendre积分的方法,本文推导的反演算法可以达到满意的数值精度.同时,结合该算法采用半解析半数值方法对一个较为复杂的冲击凿岩问题的数值反演结果也表明该数值反演算法具有一定的实用性.     

热传导方程的无网格Galerkin方法数值模拟研究

为了更好地数值模拟热传导方程,将无网格Galerkin( EFG)方法引入热传导问题的求解中,时间导数采用θ加权方法离散,同时与有限元(FE)方法的数值结果进行了比较,并研究了EFG方法中若干参数的选取对数值结果的影响.计算结果表明:相对于有限元方法,EFG方法能更好地吻合微分方程的解析解,EFG方法在节点布置较稀疏时,也可以获得很高的计算精度;θ≥1/2,EFG方法无条件稳定,且θ=1时数值解精度最高;算例中影响半径取为1.2h≤r＜2.8h,EFG方法可获得较为理想的计算结果.     

基于扩展FEAST的大规模特征值求解问题研究

为了求解非线性特征值问题,在线性FEAST特征值算法的基础上,提出一种非线性FEAST扩展算法.通过将复平面分割为不相交的区域集合,计算每个区域的特征对.扩展算法使用与线性FEAST算法相同的一系列运算,通过修改围道积分来支持非线性特征值求解的固定移位集合和固定子空间维数.与线性FEAST算法相似,扩展算法可以通过并行求解额外的线性系统,改进数值围道积分或提升近似特征向量子空间的维数,从而提高非线性FEAST的收敛速度.通过三个计算模型问题验证了非线性FEAST算法的多项式特征值行为.     

功能梯度材料圆板的非线性热振动及屈曲

采用弹性理论建立了功能梯度材料板的静力平衡方程,利用静力平衡方程确定了功能梯度材料板的中性面位置,在此基础上推导出了功能梯度材料板在均匀温度场中的非线性振动及屈曲微分方程组,求得了功能梯度材料圆板的非线性振动及屈曲的近似解,讨论分析了中性面位置、梯度指数、温度等因素对功能梯度材料圆板非线性振动及屈曲的影响.把该方法计算结果与有限元计算结果进行了比较,验证了该方法的计算结果是可靠的.算例分析表明,中性面位置对均匀温度场中功能梯度材料圆板的非线性振动及屈曲有一定影响.     

功能梯度材料椭圆板的非线性热振动及屈曲

采用弹性理论建立了功能梯度材料板的静力平衡方程,利用静力平衡方程确定了功能梯度材料板的中性面位置,在此基础上推导出了功能梯度材料板在均匀温度场中的非线性振动及屈曲微分方程组,求得了功能梯度材料椭圆板的非线性振动及屈曲的近似解,讨论分析了中性面位置、梯度指数、温度等因素对功能梯度材料椭圆板非线性振动及屈曲的影响.把该方法计算结果与有限元计算结果进行了比较,验证了该方法的计算结果是可靠的.算例分析表明,中性面位置对均匀温度场中功能梯度材料椭圆板的非线性振动及屈曲有一定影响.     

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.     

Sampling Archimedean copulas

The challenge of efficiently sampling exchangeable and nested Archimedean copulas is addressed. Specific focus is put on large dimensions, where methods involving generator derivatives are not applicable. Additionally, new conditions under which Archimedean copulas can be mixed to construct nested Archimedean copulas are presented. Moreover, for some Archimedean families, direct sampling algorithms are given. For other families, sampling algorithms based on numerical inversion of Laplace transforms are suggested. For this purpose, the Fixed Talbot, Gaver Stehfest, Gaver Wynn rho, and Laguerre series algorithm are compared in terms of precision and runtime. Examples are given, including both exchangeable and nested Archimedean copulas.     

Solution of plane transient elastodynamic problems by finite elements and laplace transform

The general transient linear elastodynamic problem under conditions of plane stress or plane strain is numerically solved by a special finite element method combined with numerical Laplace transform. A rectangular finite element with eight degrees of freedom is constructed on the basis of the governing equations of motion in the Laplace transformed domain. Thus the problem is formulated and numerically solved in the transformed domain and the time domain response is obtained by a numerical inversion of the transformed solution. Viscoelastic material behavior is easily taken into account by invoking the correspondence principle. The method appears to have certain advantages over conventional finite element techniques.     

The method of fundamental solutions for Helmholtz-type equations in composite materials

In this paper, we introduce and develop the method of fundamental solutions (MFS) for solving Helmholtz-type elliptic partial differential equations in composite materials. This study builds upon the previous developments and applications of the MFS to linear and nonlinear heat conduction, elasticity, and functionally graded composite layered materials. Numerical results are presented and discussed for four examples involving both the modified Helmholtz and the Helmholtz equations in two-dimensional or three-dimensional, bounded or unbounded, smooth or non-smooth composite domains. It was found that the method produces numerical results which are in good agreement with the analytical solutions, where available.     

Dempster–Shafer evidential theory for the automated selection of parameters for Talbot’s method contours and application to matrix exponentiation

In this paper, the Dempster–Shafer theory of evidential reasoning is applied to the problem of optimal contour parameters selection in Talbot’s method for the numerical inversion of the Laplace transform. The fundamental concept is the discrimination between rules for the parameters that define the shape of the contour based on the features of the function to invert. To demonstrate the approach, it is applied to the computation of the matrix exponential via numerical inversion of the corresponding resolvent matrix. Training for the Dempster–Shafer approach is performed on random matrices. The algorithms presented have been implemented in MATLAB. The approximated exponentials from the algorithm are compared with those from the rational approximation for the matrix exponential returned by the MATLAB expm function.     

Boundary element analysis of nonlinear transient heat conduction problems

This paper is concerned with a boundary element formulation and its numerical implementation for the nonlinear transient heat conduction problems with temperature-dependent material properties. By using the Kirchhoff transformation for the material properties a set of pseudo-linear integral equations is obtained in space and time for the fully three-dimensional nonlinear problems under consideration. The resulting boundary integral equations are solved by means of the usual boundary element method. Emphasis is placed on the numerical solution procedure employing constant elements with respect to time. It is shown that in this case there is no need to evaluate the domain integrals resulting from the nonlinearity of the problem. Finally, the powerful usefulness of the proposed method is demonstrated through the numerical computation of several sample problems.     

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.     

Analysis of nonlinear, dynamic coupled thermoviscoelasticity problems by the finite element method

This investigation deals with the numerical solution of a class of nonlinear problem in transient, coupled, thermoviscoelastidty. Equations of motion and heat conduction are derived for finite elements of thermomechanically simple materials and these are adapted to special classes of thermorheologically simple materials. The analysis involves the solution of large systems of nonlinear integrodifferential equations in the nodal displacements and temperatures and their histories. As a representative example, the general equations are applied to the problem of transient response of a thick-walled hollow cylinder subjected to time-varying internal and external pressures, temperatures, and heat fluxes. The integration scheme used to solve the nonlinear equations employs a linear acceleration assumption, representation of nonlinear integral terms by Simpson's rule, and the iterative solution of large systems of nonlinear algebraic equations at each reduced time step by the Newton-Raphson method. Various numerical results are given and are compared with the linearized, isothermal, and quasi-static solutions.     

Monte Carlo solution of Cauchy problem for a nonlinear parabolic equation

In this paper we consider the Monte Carlo solution of the Cauchy problem for a nonlinear parabolic equation. Using the fundamental solution of the heat equation, we obtain a nonlinear integral equation with solution the same as the original partial differential equation. On the basis of this integral representation, we construct a probabilistic representation of the solution to our original Cauchy problem. This representation is based on a branching stochastic process that allows one to directly sample the solution to the full nonlinear problem. Along a trajectory of these branching stochastic processes we build an unbiased estimator for the solution of original Cauchy problem. We then provide results of numerical experiments to validate the numerical method and the underlying stochastic representation.