Advances in Computers and Information in Engineering Research, Volume 1
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In most implementations of the Finite Element Method (FEM) [1–3], interpolation functions or shape functions are predefined. On the other hand, in meshless or meshfree methods these functions have to be determined by solving local equations. Therefore, the computational efficiency of the FEM is higher than the computational efficiency of meshless methods. In addition, very complex problem domains can be well represented by finite element meshes that are generated using adaptive algorithms [4, 5], which is very challenging to achieve in current meshless methods. Nevertheless, it is well known that the performance of the FEM is significantly affected by the quality of finite element meshes, which is measured by element shapes in the meshes, or more accurately, by the relative locations of nodes in the elements. The existence of a few distorted or invalid elements in a mesh may ruin the whole finite element solutions, or at the best, compromise the accuracy. As mentioned in , with readily available commercial software used for geometrical modeling and mesh generation, generation of finite element meshes is not a difficult task any more. However, tuning the quality of a finite element mesh to make all elements in the mesh have ideal shapes is very time consuming, especially if the problem domain has a complex geometric shape, as an optimization algorithm is usually required to obtain the optimal mesh. Element distortion during finite element simulation has been a major issue that makes the FEM inconvenient or inefficient in solving engineering problems involving large deformation, propagation of discontinuities, evolving interface of material phases, etc. Local mesh modification and remeshing of the evolved problem domain at every time or load step during simulation have been the two commonly used techniques to correct or remove distorted and invalid elements. Both of them are very time consuming and thus not practical for solving large-scale problems [4, 5].