Collaborative Research: Meshless Methods: Mathematical Foundations and Applications

Syracuse University, Syracuse NY

Investigators

Abstract

Recently Meshless Methods, and the closely related Generalized Finite Element Methods, for the numerical solution of partial differential equations came into the forefront of interest, especially in the engineering community. For example, in the last two years several engineering books on the subject appeared, an international conference was organized, and many engineering papers were published. These methods generalize the classical Finite Element Method in that they are neither based on a mesh nor on classical polynomial approximation. The theoretical underpinning and general state of the art for these methods is similar to that of the Finite Element Method in the 1960s, when already engineering computations based on heuristic principles were done, but the theory had not been developed. This research will address the theoretical and implementational basis of these methods in conjunction with applications. The most important feature of Meshless Methods is their flexibility and potential. It is a challenge to employ this flexibility to achieve their potential: the solution of today's complex problems, with the consideration of the rapidly decreasing ratio of cost of computer arithmetical operations to the human effort. These issues will be investigated in this research project. The broader impact of the proposal lies in the mathematical and computational science aspects of Meshless Methods. The broader mathematical impact includes the understanding of these methods and their potential, which will be directly reflected in the performance of the methods on complex problems of solid and fluid mechanics, for example, multiscale problems. It is likely that the development of the mathematical foundation of Meshless Methods will have similar or larger impact than the mathematical development of the 1970s and 1980s had on the Finite Element Method. The broader impact of this proposal is in its simultaneous consideration of mathematical theory, modeling of engineering and scientific problems, computational issues, and engineering computational experience.