抄録
This paper provides a brief overview on computational methods with emphases on numerical methods that combines the finite element method (FEM) with the meshfree techniques. We introduces first a general fundamental theory for computational methods, the G space theory, that accommodates much more types of techniques for creating shape functions for numerical methods. Weakened weak (W2) formulations are then used to construct methods with meshfree technique in finite element settings, which ensures spatially stability and convergent to exact solutions. We next present examples of some of the possible W2 models, and show the major properties of these models. It is shown that the stiffness W2 models is "softer" compared to the FEM model and even the exact model, allowing us to obtain upper bound solutions with respect to both the FEM and the exact solutions. W2 models are also found less sensitive to the mesh quality, and triangular meshes can be used with excellent accuracy, which opens widely a window of opportunity for adaptive or automatic analysis of various types of problems, including fracture mechanics, plates, shells and membranes, contact problems, acoustics, heat transfer, non-linear problems, bio-mechanics, shakedown analysis, real-time computation, inverse and optimization problems.
