Summary This work presents variational concepts associated with reduced Trefftz type approaches and discusses the interrelationship between various concepts of the displacement, hybrid and Trefftz methods. The basic concept of the displacement version of the reduced Trefftz method operates on the natural boundary conditions enforced in an integral form whereas the stress version of the reduced Trefftz type approach operates on the essential boundary conditions enforced in an integral sense. The application of the method proposed in the framework of the finite element method is briefly outlined. The methods used by the reduced Trefftz type approach for enforcing conformity and interelement continuity between neighboured elements are also discussed. Comparisons with other known methods for the same purpose are performed. General strategy for developing finite elements of general geometric form such as quadrilateral elements with invariance properties is presented. The basic idea of this strategy consists in using the natural coordinate system only for defining the element geometry and performing the element integration in the biunit interval. For defining the approximation functions a local coordinate system defined from the directions of the covariant base vectors and the perpendicular contravariant base vectors computed in the geometric centre of the element is used. This strategy can also be used to implement other versions of finite elements and other forms of finite elements. Different numerical calculations and comparisons in the linear statics and kinetics are performed in order to assess the convergence and the numerical performance of finite elements developed by applying the reduced Trefftz type approach.
翻译:这项工作摘要介绍了与减少特雷夫茨类型方法相关的不同概念,并讨论了与迁移、混合和特雷夫茨方法等不同概念之间的相互关系。减少特雷夫茨方法的迁移版本的基本概念以自然边界条件为整体执行,而减少特雷夫茨类型方法的压力部分则以整体性执行的基本边界条件为主;在有限要素方法框架内提议的方法的应用情况作了简要概述;还讨论了减少特雷夫茨类型方法用于执行相邻要素之间的兼容性和相互连续性的方法;与其他已知的同一目的方法进行了比较;开发一般几何形式的有限要素,如具有变异特性的四边形要素,其基本概念以自然边界条件为主;减少特雷夫茨类型方法的压力部分在整体性条件下执行;在确定要素的几边间隔范围内采用有限要素的拟议方法;在界定近似性功能时,也讨论了减少的特雷夫茨类型方法所使用的方法;为同一目的与其他已知方法中心为同一目的进行的比较进行了比较; 开发了通用几边矩阵形式的一般几度要素的有限性要素的一般战略;在确定性定数的数值模型中,还使用了其他定式的定数性参数的比较。