Free Vibration Analysis of a Liquid in a Circular Cylindrical Rigid Tank Using the Hierarchical Finite Element Method

Abstract A hierarchical finite element is developed for the free vibration analysis of a liquid in a rigid cylindrical tank with or without a free surface. It is a hierarchical quadrilateral element and has the advantage that the hierarchical mode number is allowed to vary independently of direction. Liquid behavior in tanks with large aspect ratios can therefore be solved very accurately by using a higher hierarchical mode number in the longer direction than in the shorter one. Furthermore, it is possible to idealize the liquid by using only one element. The solution can therefore be obtained to any desired degree of accuracy simply by increasing the hierarchical mode number. In this method, the liquid behavior is described by the displacements alone. The pressure and velocity potential are not considered as unknowns. The results are compared with other methods and show good agreement.

摘要：针对带或不带自由液面的刚性圆柱储液罐内液体的自由振动分析问题，本文开发了一种分层有限元（hierarchical finite element）方法。该方法采用分层四边形单元，其优势在于分层模态阶数可沿不同方向独立选取。对于高长宽比储液罐内的液体，可在长边方向采用比短边方向更高的分层模态阶数，从而实现高精度求解。此外，仅需单个单元即可完成液体的理想化建模，仅需提升分层模态阶数即可将求解精度提升至任意所需水平。该方法仅通过位移场描述液体的运动特性，无需将压力和速度势作为未知量。将该方法的求解结果与其他方法进行对比，结果吻合良好。