Gurtin变分原理及其应用的时间有限元法

动力学Ｇｕｒｔｉｎ变分原理完整地表征了动力学的全部特征。由于在Ｇｕｒｔｉｎ变分原理的泛函中含有双重卷积，给时间域的离散带来很大困难。本文通过在Ｌａｐｌａｃｅ空间构造泛函，获得了几个具有单重卷积的Ｇｕｒｔｉｎ原理。由于卷积降阶，所给出的泛函更加便于应用。本文还通过在时间域采用适当的插值多项式逼近广义节点坐标，进一步讨论了时间有限元法实施的基本原理和步骤。     

求解交变载荷下异形板动力响应的Gurtin半解析法

该方法以Ｇｕｒｔｉｎ变分原理为基础，在空间域作有限元离散，时间域取级数。本文研究了在交变载荷ｑ（ｘ，ｙ，ｔ）＝Ｐ（ｘ，ｙ）ｓｉｎωｔ作用下任意形状板时域级数的取法，使得各种支承条件的动力响应问题均可由本计算模式得到具有相当精度和效率的解。     

部分应力杂交动力分析模型的混合变分原理

本文通过对具有单重卷积积分的二类变量的Ｇｕｒｔｉｎ型混合变分原理进行修正，采用“部分应力杂交”的概念，建立了线弹性动力分析 部分应力杂交变分格式，并在此基础上构造出时间有限元模型，这种模型对于分层复合板的动力分析尤其适用。     

基于位移型Gurtin变分原理计算动力响应的逐步积分法

本文利用位移型Ｇｕｒｔｉｎ变分原理,在时间域上采用三次Ｈｅｒｍｉｔｅ插值函数进行离散,给出了一种计算结构动力响应的逐步积分方法。通过稳定性分析研究了该方法的稳定区情况表明,当１．６４≤θ≤２．０８时,该方法的数值计算精度很高,但是条件稳定积分格式。当θ≥４．１时,该方法是无条件稳定的积分格式,精度较高。     

耦合热弹性动力学的统一变分原理族

对耦合热弹性动力学问题,迄今文献中只建立了Ｇｕｒｔｉｎ型合卷积的统一变分原理,其缺点是只适用于常系数的线性问题,且因含卷积而使实际数值离散和求解复杂化．文中首次成功地建立了耦合热弹性动力学问题经典型（不含卷积）统一变分原理族,其关键是建议了动态差分变换和初终值条件的新处理法．该方法可以推广到各向异性材料以及非线性问题上去,同时在应用有限元法离散和求解上都比较简便．     

卷积型加权残值法求解薄壳的动力学问题

板壳等弹性体,受到动荷载作用时,其动力分析问题是比较重要且难于解决的．利用卷积型加权残值法,推导出Gurtin变分原理,并应用卷积型加权残值法计算了薄壳的动力学问题,为计算薄壳的结构动力学问题提供了一种有效的方法．     

热压电变分原理

热释电介的许多传感器及机敏结构或中的关键材料，本文系统地讨论了在基础理论和数值计算中极具重要地位的各类变分原理，包括准静态变分原理、动态变分原理和关于固有频率的变分原理。最后建立了关于静态压电要板的变分原理，并由此导出了各向异性压电板的控制方程及边界条件。     

变水深环境下中厚度浮板耦振问题的一个变分解

本文从Reissner 板的理论出发,提出并论证了变水深环境下中厚度浮板耦振问题的一个局部变分原理,它是建立中厚度浮板各种近似分析方法的一个有力工具,借助这个变分原理,文中还导得了一个计算中厚度浮板固有频率的一个变分式,数值算例表明本文方法具有精度高和省机时的优点。     

结构物在水下运行过程中的自振特性及动力响应研究

本文运用流体力学、结构动力学及变分原理推导了结构－流体耦合系统的动力学方程，并对水下结构物在运行时的自振特征及动力响应给出了数值计算分析。     

耦合热弹性动力学中各类Hamilton型拟变分原理

根据古典阴阳互补和现代对偶互补的基本思想,通过作者早已提出的一条简单而统一的途径,系统地建立了耦合热弹性动力学的各类Hamilton型拟变分原理。这种以单一泛函的变分式表示的Hamilton型拟变分原理,能精确反映耦合热弹力动力学初值－边值问题的全部特征。文中首先给出一个在力学上可以认为是广义拟虚功原理的表式。然后从该式出发,通过所给出的一系列广义Legendre变换,系统地推导出耦合热弹性动力学的8类变量、6类变量、4类变量和2类变量Hamilton型拟变分原理。同时,通过这条途径还能阐明这些原理的内在联系。     

微孔压电弹性动力学的能量原理

根据古典阴阳互补和现代对偶互补的基本思想,通过作者早已提出一条简单而统一的新途径,系统地建立了微孔压电弹性动力学的能量原理,给出一个重要的以卷积表示的积分关系式,可以认为,在力学上它是广义虚功原理的表式,从该式出发,不仅能得到微孔压电弹性动力学的虚功原理和互等定理,而且通过作者所给出的一系列广义Legendre变换,能系统地导出成互补关系的11类变量、9类变量、6类变量和3类变量简化Gurtin型变分原理的泛函,同时,通过这条新途径,还能清楚地阐明这些原理之间的内在联系。     

关于线粘弹性动力学中各种变分原理

本文提出一条简单而统一的新途径,系统地建立了线粘弹性动力学中各种简化Gurtin型变分原理,文中首先给出一个很有用的以卷积表示的积分关系式,然后从该式出发,系统地导出成互补关系的五类变量、四类变量、三类变量、二类变量及一类变量简化Gurtin型变分原理,并清楚地阐明它们之间的内在联系,而且,还发现当前在国际上有广泛影响的力学变分原理方面的名著[1]及文[2]中,所给出的四个变分原理的泛函式均有误.本文除给出这四个正确的泛函式外,还建立了一些新的更一般广义变分原理。     

Simplified Gurtin-type generalized variational principles for fully dynamic magneto-electro-elasticity with geometrical nonlinearity

The fundamental equations, governing all the variables of the initial boundary value problem in fully dynamic magneto-electro-elasticity with geometrical nonlinearity, are expressed in covariant differential form. The generalized principle of virtual work is given in terms of convolutions for the present problem. Two simplified Gurtin-type generalized variational principles, directly leading to all the fundamental equations, are deduced by using He’s semi-inverse method instead of Laplace transforms. By enforcing some fundamental equations as constraint conditions, one of various constrained variational principles is given as an example. By simply dropping out selected field functions, several reduced variational principles are obtained as special forms for piezoelectricity, elastodynamics, and electromagnetics, respectively. This paper aims at providing a more complete theoretical foundation for the finite element applications for the discussed problem.     

有限变形弹性动力学的非传统Gurtin型变分原理

根据古典阴阳互补和现代对偶互补的基本思想，通过作者早已提出的一条简单而统一的途径，系统地建立了有限变形弹性动力学的各类非传统Gurtin型变分原理，给出一个以卷积表示的重要关系式，可以认为，该式是有限变形动力学的广义虚功原理的表式。从该式出发，不仅能得到有限变形动力学的虚功原理，而且通过给出的一系列广义Legendre变换，还能系统地成对导出5类变量，3类变量，2类变量和1类变量非传统Gurtin型变分原理的互补泛函。通过这条途径还能清楚地阐明这些原理之间的内在联系。     

New way to construct high order Hamiltonian variational integrators

This paper develops a new approach to construct variational integrators. A simplified unconventional Hamilton’s variational principle corresponding to initial value problems is proposed, which is convenient for applications. The displacement and momentum are approximated with the same Lagrange interpolation. After the numerical integration and variational operation, the original problems are expressed as algebraic equations with the displacement and momentum at the interpolation points as unknown variables. Some particular variational integrators are derived. An optimal scheme of choosing initial values for the Newton-Raphson method is presented for the nonlinear dynamic system. In addition, specific examples show that the proposed integrators are symplectic when the interpolation point coincides with the numerical integration point, and both are Gaussian quadrature points. Meanwhile, compared with the same order symplectic Runge-Kutta methods, although the accuracy of the two methods is almost the same, the proposed integrators are much simpler and less computationally expensive.     

On some basic principles in dynamic theory of elastic materials with voids

According to the basic idea of dual-complementarity, in a simple and unified way proposed by the author^[1], some basic principles in dynamic theory of elastic materials with voids can be established systematically. In this paper, an important integral relation in terms of convolutions is given, which can be considered as the generalized principle of virtual work in mechanics. Based on this relation, it is possible not only to obtain the principle of virtual work and the reciprocal theorem in dynamic theory of elastic materials with voids, but also to derive systematically the complementary functionals for the eight-field, six-field, four-field and two-field simplified Gurtin-type variational principles. Furthermore, with this approach, the intrinsic relationship among various principles can be explained clearly. The project supported by the Foundation of Zhongshan University Advanced Research Center     

有限变形弹性动力学的非传统简化Gurtin型变分原理--文献[1]的续篇

根据古典阴阳互补和现代对偶互补的基本思想,通过作者提出的一条简单而统一的新途径,建立了有限变形弹性动力学的另一种单卷积形式的变分原理一各类非传统简化Gurtin型变分原理．首先给出一个以卷积表示的关系式,在力学上它是有限变形动力学的广义虚功原理的另一种表式．然后从该式出发,不仅能得到有限变形动力学另一种形式的虚功原理,而且通过文中所给出的一系列广义Legendre变换,还能成对导出5类变量、3类变量、2类变量和1类变量非传统简化Gurtin型变分原理的互补泛函．同时,通过这条新途径还能阐明这些原理之间的内在联系。     

A generalized variational principle in micromorphic thermoelasticity

Recently Nappa obtains a Gurtin-type variational principle for micromorphic thermoelasticity. However, the use of convolutions is of course restricted to the linear case, which sets a limit to the rang of applicability. This paper establishes a classic variational principles for the discussed problem by the semi-inverse method.     

Generalized variational principles for thermopiezoelectricity

Summary  Based on the semi-inverse method of establishing variational principles proposed in [10], a family of variational principles (non Gurtin-type and not involving convolutions) for thermopiezoelectricity is deduced directly from the field equations and boundary/initial conditions. Present theory provides a more complete theoretical basis for the finite element applications and other direct variational methods such as Ritz's, Trefftz's and Kantorovitch's methods. Received 4 November 1999; accepted for publication 6 June 2000     

Dual variational formulation for Trefftz finite element method of elastic materials

A dual variational principle is presented for Trefftz finite element analysis. The proof of the stationary conditions of the variational functional and the theorem on the existence of extremum are provided in this paper. They are boundary displacement condition, surface traction condition and interelement continuity condition. Based on the assumed intraelement and frame fields, element stiffness matrix equation is obtained which can easily be implemented into computer programs for numerical analysis with Trefftz finite element method. Two numerical examples are considered to illustrate the effectiveness and applicability of the proposed element model.