有限元模型位移解与应力解误差特性分析

Analysis on the error characteristics of displacement solution & stress solution in the finite element model

将有限元求解误差分解为每个单元上的残值所导致的误差之和。对于有限元模型中的任一单元小片,将其误差分解为全局误差与局部误差两部分:局部误差是由该单元小片内单元的残值所引起的,而全局误差是由单元小片之外区域的残值所导致。分别探讨计算结果中位移解误差与应力解误差的特性。通过一系列数值算例研究,发现有限元模型中的应力奇异点、高应力梯度区是导致全局误差产生的原因。应力奇异点或高应力梯度区导致任一单元小片所产生的误差与其奇异性强度及二者之间的距离有关。奇异性越强,距离越近,所产生的全局误差就越大。得到的相关结论为建立高精度的有限元模型提供理论指导。

The finite-element solution error serves as the sum of errors caused by the residual values on each element.It can be divided into the local error and the global error,which are the solutions of each element,when the domain is loaded by near-field residuals or far-field residuals respectively.Both the characteristics of displacement error and stress error are analyzed.A series of numerical studies show that both the stress singular point and the high stress gradient domains cause the global error within the domain;both the singularity and the distance in between exert influence on the specific error:the stronger the singularity,the shorter distance in between,the greater the global error is.The results provide theoretical guidance for setting up a high-precision finite element model.