微分变换法求解二维非线性Volterra积分微分方程

为了解决二维非线性Volterra积分微分方程的求解问题,本文给出微分变换法.利用该方法将方程中的微分部分和积分部分进行变换,这样简化了原方程,进而得到非线性代数方程组,从而将原问题转换为求解非线性代数方程组的解,使得计算更简便.文中最后数值算例说明了该方法的可行性和有效性.     

求解延迟微分代数方程的多步Runge-Kutta方法的渐近稳定性

延迟微分代数方程(DDAEs)广泛出现于科学与工程应用领域．本文将多步Runge-Kutta方法应用于求解线性常系数延迟微分代数方程，讨论了该方法的渐近稳定性．数值试验表明该方法对求解DDAEs是有效的．     

预测校正公式求解指标2微分代数方程

1．前言微分代数方程（EEES）是经常出现于实际问题中的一类方程．其数值求解已成为常微分方程数值求解领域十分活跃的一个方向．目前微分代数方程求解的数值方法主要是nunge－Kutta型方法及BDF方法．Runge－Kutta型方法在网,问中有详细的介绍．Hairer等人据此编制了软件RADAU,而目前使用最广泛的软件还是PetZold等编制的DASSL．DASSL使用的方法为BDF方法,它在微分代数方程中的应用最早可以追述到Gear的开创性工作问．BDF方法一个很大的优点是刚性稳定．然而对于非刚性的微分代数方程,刚性稳定已不是主要考虑的因素．因此…     

上三角算子矩阵的本征函数展开法在应力形式的二维弹性问题中的应用

深入研究了求解基于应力形式的二维弹性问题的本征函数展开法．根据已有的研究结果,将基于应力形式的二维弹性问题的基本偏微分方程组等价地转化为上三角微分系统,并导出了相应的上三角算子矩阵．通过深入研究,分别获得了该算子矩阵的两个对角块算子更为简洁的正交本征函数系,并证明了它们在相应空间中的完备性,进而应用本征函数展开法给出了该二维弹性问题的更为简洁实用的一般解．此外,对该二维弹性问题,还指出了什么样的边界条件可以应用此方法求解．最后应用具体的算例验证了所得结论的合理性．     

分数阶热弹理论下重力场对二维纤维增强介质的影响

基于Sherief等提出的分数阶广义热弹性耦合理论，研究了在热冲击作用下二维纤维增强弹性体的热弹性问题.考虑了重力对二维纤维增强线性热弹性各向同性介质的影响，建立了控制方程.运用正则模态法，经过数值计算，对控制方程进行求解，得到了不同分数阶参数和不同重力场下无量纲温度、位移和应力分量的表达式，以图形的方式展示了变量的分布规律并对结果展开了讨论.研究结果为:重力场和分数阶参数对纤维增强介质的位移及应力有着重要的影响，但对温度的影响有限.     

在动载荷作用下框架结构大变形分析的微分代数方法

采用弧坐标首先建立了在动载荷作用下,具有不连续性条件和初始位移的框架结构大变形分析的非线性数学模型．其次, 在空间区域内, 采用微分求积单元法（DQEM）来离散非线性数学模型, 并提出了在使用DQEM来求解结构大变形分析中,多个变量具有间断性条件的有效方法,得到了一组非线性DQEM的离散化方程,它是时间域内的一组具有奇异性的非线性微分-代数方程．同时也给出了求解非线性微分-代数方程组的一个解法A·D2作为应用,求解了受集中力和分布力作用的框架和组合框架的大变形静动力学问题,并与现有结果进行了比较．数值算例表明,处理多个变量具有间断性条件的方法和求解代数-微分系统的方法是一个有效的和一般的方法,它具有较少的节点、 较小的计算工作量、 较高的精度、良好的收敛性、 操作简单以及应用广泛等优点．     

考虑记忆效应及尺寸效应窄长薄板的磁-热弹性耦合动态响应

引入记忆依赖微分的双相滞后热弹性理论能较完善地描述非Fourier导热现象,然而迄今尚未发现该理论综合考虑微尺度效应和磁、热、弹等多场耦合效应对材料力学行为的影响。通过考虑记忆依赖效应和非局部效应修正了双相滞后广义热弹性理论,基于改进后的理论研究了受周期性变化热源作用时窄长薄板的磁-热弹性耦合问题。首先建立问题的控制方程;然后结合边界条件与初值条件,利用Laplace变换和反变换技术对该问题进行求解;最后分别考察了磁场、相位滞后、时间延迟因子、核函数、非局部效应、时间对各无量纲量的影响,为微尺度材料的动态响应提供了有力参考依据。     

基于应力形式的二维弹性问题的本征展开法

给出求解基于应力形式的二维弹性问题的本征函数展开方法．通过引入适当的状态函数,将该问题的基本偏微分方程等价地转化为上三角微分系统,导出相应的上三角算子矩阵．证明了该矩阵的两个对角块算子均具有规范的正交本征函数系,并得到它们在相应空间中的完备性．此外,基于本征函数系的完备性,应用本征函数展开法给出了二维弹性问题的一般解．     

病态代数方程求解的一种改进精细积分法

通过在病态代数方程精细积分法的基础上增加一个迭代改善算法,建立了病态代数方程求解的改进精细积分法．该方法进一步提高了病态代数方程精细积分法的精度和效率,具有良好的应用前景．算例证明了该方法在病态代数方程求解中的有效性．     

非一致性界面热流固耦合作用的整体求解

提出了非一致性界面热流固耦合作用整体求解的一种方法．热流体求解基于Boussinesq假设和不可压缩的Navier-Stokes方程．流体区域的运动采用任意Lagrange-Euler（ALE）方法．拟固体元方法实现流体区域的变形．使用几何非线性的热弹性动力学描述固体运动．为了保证界面处应力和传热的平衡,采用了基于Gauss积分点的数据交换方法,对热流固耦合最终形成的强非线性方程实现整体求解．数值实例分析表明该方法的健壮性和有效性．     

Numerical resolution of an exact heat conduction model with a delay term

In this paper we analyze, from the numerical point of view, a dynamic thermoelastic problem. Here, the so-called exact heat conduction model with a delay term is used to obtain the heat evolution. Thus, the thermomechanical problem is written as a coupled system of partial differential equations, and its variational formulation leads to a system written in terms of the velocity and the temperature fields. An existence and uniqueness result is recalled. Then, fully discrete approximations are introduced by using the classical finite element method to approximate the spatial variable and the implicit Euler scheme to discretize the time derivatives. A priori error estimates are proved, from which the linear convergence of the algorithm could be derived under suitable additional regularity conditions. Finally, a two-dimensional numerical example is solved to show the accuracy of the approximation and the decay of the discrete energy.     

Numerical Methods for Stiff Index-3 DAEs

Space semidiscretization of PDAEs, i.e. coupled systems of PDEs and algebraic equations, give raise to stiff DAEs and thus the standard theory of numerical methods for DAEs is not valid. As the study of numerical methods for stiff ODEs is done in terms of logarithmic norms, it seems natural to use also logarithmic norms for stiff DAEs. In this paper we show how the standard conditions imposed on the PDAE and the semidiscretized problem are formally the same if they are expressed in terms of logarithmic norms. To study the mathematical problem and their numerical approximations, this link between the standard conditions and logarithmic norms allow us to use for stiff DAEs techniques similar to the ones used for stiff ODEs. The analysis is done for problems which appear in the context of elastic multibody systems, but once the tools, i.e., logarithmic norms, are developed, they can also be used for the analysis of other PDAEs/DAEs.     

Structural analysis of electrical circuits including magnetoquasistatic devices

Modeling electric circuits that contain magnetoquasistatic (MQS) devices leads to a coupled system of differential-algebraic equations (DAEs). In our case, the MQS device is described by the eddy current problem being already discretized in space (via edge-elements). This yields a DAE with a properly stated leading term, which has to be solved in the time domain. We are interested in structural properties of this system, which are important for numerical integration. Applying a standard projection technique, we are able to deduce topological conditions such that the tractability index of the coupled problem does not exceed two. Although index-2, we can conclude that the numerical difficulties for this problem are not severe due to a linear dependency on index-2 variables.     

Energy functionals for microstructured multi-phase materials

For describing the influence of multi-phase materials with microstructure on different length scales as well as the evolution of phase changes under thermomechanical loading, an energetic model is developed. Relying on the incremental formulation of the energetic model, a new solution procedure for the coupled thermomechanical problem is proposed. This model can be applied to describe e.g. the macroscopic response of carbon fibre reinforced carbon. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Element-based index reduction in electrical circuit simulation

The numerical simulation of very large scale integrated circuits is an important tool in the development of new industrial circuits. In the course of the last years, this topic has received increasing attention. Common modeling approaches for circuits lead to differential-algebraic systems (DAEs). In circuit simulation, these DAEs are known to have index 2, given some topological properties of the network. This higher index leads to several undesirable effects in the numerical solution of the DAEs. Recent approaches try to lower the index of DAEs to improve the numerical behaviour. These methods usually involve costly algebraic transformations of the equations. Especially, for large scale circuit equations, these transformations become too costly to be efficient. We will present methods that change the topology of the network itself, while replacing certain elements in oder to obtain a network that leads to a DAE of index 1, while not altering the analytical solution of the DAE. This procedure can be performed prior to the actual numerical simulation. The decreasing of the index usually leads to significantly improved numerical behaviour. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical analysis and simulations of a quasistatic frictional contact problem with damage in viscoelasticity

We consider a quasistatic problem which models the bilateral contact between a viscoelastic body and a foundation, taking into account the damage and the friction. The damage which results from tension or compression is then involved in the constitutive law and it is modelled using a nonlinear parabolic inclusion. The variational problem is formulated as a coupled system of evolutionary equations for which we state the existence of a unique solution. Then, we introduce a fully discrete scheme using the finite element method to approximate the spatial variable and the Euler scheme to discretize the time derivatives. Error estimates are derived and, under suitable regularity hypotheses, the convergence of the numerical scheme obtained. Finally, a numerical algorithm and results are presented for some two-dimensional examples.     

Non–Isothermal Shear Band Localization in Crystal Plasticity

We analyze the formation of shear bands in plastically deforming single crystals under non-isothermal quasi-static loading conditions. A key feature of the present work is to motivate the constitutive coupled thermomechanical equations for the slip resistance and the slip evolution by micromechanical investigations of defects in crystals. Here, we put particular emphasis on comparative investigations of hardening–softening effects of crystals under non-isothermal conditions, including a quantification of the latent energy storage. Furthermore, we analyze a numerical solution algorithm for the coupled thermomechanical problem and we test the performance on a typical benchmark example of finite-strain single-crystal thermoplasticity: the localization of deformation into a shear band for the case of a rectangular strip. Experimental evidence for this problem is well documented. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Convex and Concave Relaxations for the Parametric Solutions of Semi-explicit Index-One Differential-Algebraic Equations

A method is presented for computing convex and concave relaxations of the parametric solutions of nonlinear, semi-explicit, index-one differential-algebraic equations (DAEs). These relaxations are central to the development of a deterministic global optimization algorithm for problems with DAEs embedded. The proposed method uses relaxations of the DAE equations to derive an auxiliary system of DAEs, the solutions of which are proven to provide the desired relaxations. The entire procedure is fully automatable.     

Numerical Treatment of Second Order Differential-Algebraic Systems

We consider the numerical treatment of systems of second order differential-algebraic equations (DAEs). The classical approach of transforming a second order system to first order by introducing new variables can lead to difficulties such as an increase in the index or the loss of structure. We show how we can compute an equivalent strangeness-free second order system using the derivative array approach and we present Runge-Kutta methods for the direct numerical solution of second order DAEs. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Interval bounds on the solutions of semi-explicit index-one DAEs. Part 2: computation

This article presents two methods for computing interval bounds on the solutions of nonlinear, semi-explicit, index-one differential-algebraic equations (DAEs). Part 1 presents theoretical developments, while Part 2 discusses implementation and numerical examples. The primary theoretical contributions are (1) an interval inclusion test for existence and uniqueness of a solution, and (2) sufficient conditions, in terms of differential inequalities, for two functions to describe componentwise upper and lower bounds on this solution, point-wise in the independent variable. The first proposed method applies these results sequentially in a two-phase algorithm analogous to validated integration methods for ordinary differential equations (ODEs). The second method unifies these steps to characterize bounds as the solutions of an auxiliary system of DAEs. Efficient implementations of both are described using interval computations and demonstrated on numerical examples.