Abstract

The objective of this paper is to gain insight into the implementation of 3D finite elements in curvilinear coordinates using the fundamental equations of 3D elasticity and the Principle of Virtual Displacements. After reviewing the mathematical model of the geometry, we present the formulation of hexahedral finite elements that represent a generalization of classical shell finite elements, in which also the thickness direction can be curvilinear. These finite elements are equivalent to the widely-used continuum mechanics based solid finite elements, but the adoption of local curvilinear reference system will allow in future works to apply mixed methods, such as the Mixed Interpolation of Tensorial Components (MITC), to contrast the locking phenomenon in the three spatial directions. For the assessment of the present elements, we consider different examples of curved geometries: cylindrical, conical, toroidal and spherical. A free-vibration analysis of curved 3D components is performed and the results, in terms of natural frequencies, are compared with the convergence solutions computed with the commercial software MSC Patran/Nastran.

摘要

本文的目的是深入了解使用 3D 弹性基本方程和虚拟位移原理在曲线坐标中实现 3D 有限元。在回顾了几何的数学模型之后，我们提出了代表经典壳有限元推广的六面体有限元的公式，其中厚度方向也可以是曲线的。这些有限元等效于广泛使用的基于连续介质力学的固体有限元，但采用局部曲线参考系统将允许在未来的工作中应用混合方法，例如张量分量的混合插值 (MITC)，以对比三个空间方向的锁定现象。为了评估当前要素，我们考虑弯曲几何形状的不同示例：圆柱形、圆锥形、环形和球形。对弯曲的 3D 组件进行自由振动分析，并将结果在自然频率方面与使用商业软件 MSC Patran/Nastran 计算的收敛解进行比较。