A G Space Theory and Weakened Weak (W2) Formulation of Meshfree Methods
This paper briefs first the G space theory using the generalized gradient smoothing technique for a unified formulation of a wide class of meshfree methods of special properties. The G space accommodates discontinuous functions allowing the use of much more types of methods/ techniques for the creation of shape functions for meshfree methods. It is also shown that W2 formulation can be used to construct many meshfree methods, and methods based on the finite element settings can also be formulated in the similar manner. Properties and a set of important inequalities for G spaces are then proven in theory and analyzed in detail. We prove that the numerical methods developed based on the W2 formulation will be spatially stable, and convergent to exact solutions. We then present examples of some of the possible W2 models, and show the major properties of these models: 1) it is variationally consistent in a conventional sense, if the solution is sought in a proper H space (compatible cases); 2) it passes the standard patch test when the solution is sought in a proper G space with discontinuous functions (incompatible cases); 3) the stiffness of the discretized model is reduced 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; 4) the W2 models are less sensitive to the quality of the mesh, and triangular meshes can be used without any accuracy problems. These properties and theories have been confirmed numerically via examples solved using a number of W2 models including compatible and incompatible cases. A number of W2 models, such as NS-PIM, NS-FEM, ES-PIM, ES-FEM, CS-PIM and CS-FEM, are then presented. The NS-models are used for real-time computation based on the reduced basis approximation.
Numerical methods, meshfree methods, G Space, FEM, real-time computation, solution bound, inverse analysis
G. R. Liu
Centre for Advanced Computations in Engineering Science, Department of Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1, Singapore 117576 Singapore-MIT Alliance (SMA), E4-04-10, 4 Engineeri
国际会议
南京
英文
2-3
2009-10-12（万方平台首次上网日期，不代表论文的发表时间）