流形元法固定边界约束处理研究

在数值流形方法中,对于材料的固定边界,一般采用罚函数的方法进行处理,即在固定边界上设置刚性弹簧约束其位移来实现固定约束条件的近似满足。罚函数法在理论上不是严格的固定约束处理方法,罚弹簧的布置与弹簧刚度的大小对模拟的效果都会产生影响。基于流形单元上位移函数的组成提出了流形方法固定边界约束处理的新方法,在组成流形单元的物理覆盖上,通过取消相应的覆盖函数在流形单元位移函数中的组成来实现双向固定的约束条件,通过使用只包含单方向位移的覆盖函数使x向固定约束条件和y向固定约束条件得到实现,推导了相应固定约束条件下的流形单元刚度矩阵的数值计算格式。该方法严格满足固定约束的物理意义,简化了固定边界的处理,并经算例证明是有效和准确的,有利于数值流形方法的程序实现和工程应用。

Research on constraints treatment of fixed boundary in numerical manifold method

In numerical manifold method,generally the fixed boundary of material is treated by penalty function method.Theoretically penalty function method does not strictly meet the condition of fixed constraints,the arrangement of penalty spring and the spring's stiffness value will influence the effect of the numerical simulation.Based on the composition of displacement function on manifold element,a new method of constraints treatment of fixed boundary is proposed,on the physical cover made up of the manifold element,the bi-directional fixed constraint terms is realized by canceling the cover function's composition of the displacement function of manifold element,the x-direction or y-direction fixaed constraint terms is realized by using the cover function which only include the directional displacement.The corresponding numerical calculation format of manifold element's stiffness matrix with fixed constraint terms is deduced.This method strictly meets the physical meaning of fixed constraint and simplifies the treatment of the fixed boundary,the compution example shows that it is effective and accurate to program of numerical manifold method and engineering application.