| With the development of computer technology and the improvement of commercial finite element simulation software,the finite element method has become one of the most famous numerical calculation methods,and is widely used to solve various engineering problems.However,the finite element method also has some shortcomings,such as the isoparametric element constructed by the principle of minimum potential energy,which has high requirements on the shape of the element.When the element shape is irregular(the initial element is irregular or the deformation causes the mesh distortion),the calculation accuracy of the element will drop significantly.In addition,when analyzing the deformation problems of nearly incompressible materials,distorted elements can cause locking problems,which seriously affect the calculation accuracy.In view of the shortcomings of traditional finite element method such as poor calculation accuracy on deformed elements,scholars have devoted themselves to developing high-performance finite element theory and anti-distortion element models to improve the calculation accuracy of finite element method.The unsymmetric finite element method has shown excellent performance resistance to mesh distortion in recent years.Based on the Petrov-Galerkin formulation,the unsymmetric element US-QUAD8 is developed that can be generalized to elastic-plastic and hyperelastic problems for large deformation analysis.The test function and trial function in Petrov-Galerkin is constructed by classical shape function,and the higher-order Lagrangian basis function in global coordinate system,respectively.The new unsymmetric element eliminates the influence of the negative Jacobian determinant in the stiffness matrix of the distorted element,and the high-order interpolation basis function ensures the high-order completeness of the displacement field of the element under severely distorted meshes.The skew coordinate system eliminates the coordinate system dependence of the original US-QUAD8,and has excellent resistant to mesh distortion.The specific research contents of this paper are as follows:· Under the framework of Galerkin and Petrov-Galerkin theories,symmetric and unsymmetric quadrilateral 8-node elastic and geometric nonlinear finite element equations are deduced respectively.The defects of traditional symmetric quadrilateral 8-node element in computing mesh distortion is analyzed,and a new unsymmetric quadrilateral 8-node element US-QUAD8 is deduced.Corresponding numerical examples show that the US-QUAD8 unsymmetric element has good resistance to mesh distortion in elastic and geometrically nonlinear problems.· For the large elastic-plastic deformation problem,an elastic-plastic unsymmetric finite element equation is established based on the Total Lagrange formulation.Mises criterion is used to judge the plastic state of materials,and the tangent stiffness method is used to solve the problem.Numerical examples show that for the linear isotropic hardening material,the US-QUAD8 unsymmetric element has good resistance to various mesh distortions in both the quadratic displacement field and the cubic displacement field problem.· For the hyperelastic large deformation problem,an unsymmetric finite element equation for the hyperelasticity problem is established based on the Total Lagrangian formulation.The incompressible Neo-Hookean and Mooney Rivlin models are used,and the Newton-Raphson iteration method is used to solve the hyperelasticity problem.Numerical examples show that the US-QUAD8 unsymmetric element proposed in this paper also has good computational accuracy in hyperelastic problems under different load application and different types of mesh distortions.The unsymmetric element developed in this paper is not affected by the geometric shape of the analytical model,and can be easily extended to three-dimensional elastic-plastic and hyperelastic problems.This method providing a new solution to the complex engineering problem of mesh distortion caused by large deformation. |
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