Abstract

The algorithm for the formation of the stiffness matrix of a four-sided finite element, which is a fragment of an ellipsoidal shell, the middle surface of which was represented by two variants of parametrization, is presented. In the first variant of parametrization of the middle surface of the ellipsoidal shell, the axial coordinate and the angle measured from the applicate axis to the radius vector of the cross-section of the shell are used. In the second version of the representation of the middle surface, the ellipse parameter of the cross section of the ellipsoidal shell was used instead of the angle. The components of the displacement vector and their first derivatives were taken as nodal unknowns of the four-sided finite element. The approximation of the required quantities was carried out in a vector formulation using Hermite polynomials of the third degree. The approximating relations for individual components were obtained using matrix expressions between the basis vectors of the nodal points of a finite element and the basis vectors of its arbitrary point. On the example of calculation of the elliptical cylinder the advantage of the second variant of parametrization of the middle surface of the ellipsoidal shell is shown, as well as the efficiency of the vector formulation of obtaining approximating expressions of the required quantities by the finite element method in curvilinear coordinate systems is demonstrated.

中文翻译：

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中间面可变参数化时基于位移矢量插值的基于椭球壳的有限元计算。

摘要

提出了形成四边形有限元刚度矩阵的算法，该有限元是椭圆形壳的一部分，其椭圆形壳的中间表面由两个参数化形式表示。在椭圆形壳体的中间表面的参数化的第一变型中，使用轴向坐标和从施加轴到壳体横截面的半径矢量的角度。在第二种形式的中间表面表示中，使用椭圆形壳体横截面的椭圆参数代替角度。位移矢量的分量及其一阶导数被视为四边有限元的节点未知量。使用三阶Hermite多项式在向量公式中对所需数量进行近似计算。使用有限元节点点的基础向量与其任意点的基础向量之间的矩阵表达式，可以获取各个组件的近似关系。在椭圆圆柱体的计算示例中，显示了椭圆形壳中间表面第二参数化的优点，以及通过有限元方法获得所需量的近似表达式的矢量公式化的效率在曲线坐标系中进行了演示。使用有限元节点点的基础向量与其任意点的基础向量之间的矩阵表达式，可以获取各个组件的近似关系。在椭圆圆柱体的计算示例中，显示了椭圆形壳中间表面第二参数化的优点，以及通过有限元方法获得所需量的近似表达式的矢量公式化的效率在曲线坐标系中进行了演示。使用有限元节点点的基础向量与其任意点的基础向量之间的矩阵表达式，可以获取各个组件的近似关系。在椭圆圆柱体的计算示例中，显示了椭圆形壳中间表面第二参数化的优点，以及通过有限元方法获得所需量的近似表达式的矢量公式化的效率在曲线坐标系中进行了演示。