ABSTRACT
One of the most important advances in the field of numerical methods was the development of the finite element method (FEM) in the late 1950s. In FEM, a continuum is divided into discrete elements that are connected together by a topological map and usually are called meshes. FEM is a robust and thoroughly developed method for computational science and engineering. However, FEM has several shortcomings, for example:
• Difficulty in meshing and re-meshing • Low accuracy, especially for computing stresses • Difficulties when dealing with certain class of problems
− Simulating the dynamic crack growth with arbitrary paths that usually do not coincide with the original element mesh lines
− Handling large deformation that leads to an extremely skewed mesh − Simulating the breakage of structures or components with large
numbers of fragments − Solving dynamic contacts with moving boundaries − Solving multi-physics problems
A close examination of these problems associated with FEM will reveal their root causes, namely the use of elements or meshes. As long as the elements or meshes are used with FEM, the problems mentioned above will not have easy solutions. Therefore, the idea of getting rid of the elements and meshes is naturally proposed, such that the meshless or meshfree methods have been developed continuously for a long time.
