Abstract
In this paper, a new type of finite elements, called as Cross-Line Elements (CLEs), are constructed, which have fewest nodes to interpolate physical variables in both two-dimensional (2D) and three-dimensional (3D) problems. These CLEs are then used in a new mesh free method, the Free Element Method (FREM), for solving general 2D and 3D boundary value problems of partial differential equations. FREM is a strong-form element collocation method, combining the advantages of the finite element method and mesh free method in the aspects of setting up shape functions and generating computational meshes through node by node. The distinct feature of FREM is that only one independent element is needed for each collocation node and the element can be freely formed by the nodes surrounding the collocation node. A few numerical examples for 2D and 3D heat conduction and solid mechanics problems will be given to validate the correctness and demonstrate the potential of the constructed elements and proposed numerical method.
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