Applied Research Of Two-Step Taylor-Galerkin Smoothed Finite Element Method For Wave Shock And Large Deformation Problems

The dynamic problems analyses are of great importance for science, technology and economy. As these problems are usually nonlinear or have extremely complex boundary conditions, the analytical solutions are not always available and numerical methods are employed. There are mainly two types of numerical methods, the finite element method and the meshless method. Due to low accuracy of lower order elements and poor capacity in dealing with elements distortion, the traditional finite element method has difficulty in solving complicated problems. The meshless methods can work perfectly to overcome disadvantages existed in finite element method, but they have problems of complex computational process and lower efficiency as well. In order to obtain high accuracy and high efficiency when using lower order elements such as the triangular elements or tetrahedral elements, a new type of algorithms-- two-step Taylor-Galerkin smoothed finite element methods are proposed based on the gradient smoothing technique and traditional Taylor-Galerkin method. It is proven that these methods can be used to solve typical dynamics problems, because they have great advantages of momentum and energy conservation.There are two algorithms proposed in this paper, based on two different gradient smoothed methods, to solve different types of dynamic problems. Plenty of numerical examples are given to test the properties of these methods, including accuracy,stability and conservativeness. The main work includes the following details:(1)The edge-based smoothed finite element method using two-step TaylorGalerkin algorithm is formulated for the analyses of two-dimension shock wave propagation problems. The smoothed domains are based on the edges of elements,which is used for performing the numerical integration. The method possesses a close-to-exact result because it is much softer than the traditional finite element method. What's more, the use of low order linear triangular elements, without any increase of internal variables or degree of freedoms, can guarantee the high precision and high efficiency at the same time.(2)The two-step Taylor-Galerkin element-based smoothed finite element method is extended for hyperelastic dynamic large deformation and related problems. In this method, field gradients are computed directly only using shape function itself and no derivative of shape function is needed. Unlike the conventional FEM usingisoparametric elements, as no coordinate transformation or mapping is performed, no extra limitation is imposed on the shape function. So it has great advantages in handling the problem of mesh distortion. Moreover, this method, based on two-step Taylor-Galerkin algorithm, keeps the properties of momentum and energy conservation. As a result, it can avoid the influence of shock vibrations and finally reduce the error of results.(3)The properties and features of the two proposed algorithms are demonstrated based on the results of numerical examples. It can be found that the two-step Taylor-Galerkin edge-based smoothed finite element method is suitable for the propagation of shock waves problems because of the formation of smoothing sub-domains and the property of stability. However, the two-step Taylor-Galerkin element-based smoothed finite element method is more available for the large deformation problems, as it avoids the shortcoming of the traditional finite element method when the isoparametric elements are used.