Mathematical Sciences: Numerical Solution of Differential Equations in Mechanics

点击翻译按钮获取中文摘要

9500672 Arnold This project will involve devising, improving, and analyzing methods for the numerical simulation of complex physical phenomena modeled by partial differential equations. A major emphasis will be on the modeling of thin structures (plates and shells). Specifically, an innovative new finite element method recently proposed by a team of computational engineers for plate bending will be analyzed and a new approach the investigator introduced in collaboration with Brezzi to develop and analyze new finite element methods for modeling the deformation of elastic shells will be pursued. This latter approach has produced the best theoretical results to date for avoiding the phenomenon known as membrane locking which has plagued shell computations. In addition to studying numerical methods and the errors they introduce, the investigator will consider the errors introduced by mathematical models of thin structures. Specifically, a new technique of analysis to study the convergence of two-dimensional models of thin plates will be used. This technique gives the possibility of obtaining sharp convergence estimates for plate modeling errors both globally and away from the lateral boundary (where convergence may be higher). It is hoped that this research will clarify in what situations the Reissner-Mindlin plate model affords better approximation than the classical Kirchhoff model, an important question that eluded resolution for decades. A second area of emphasis will be on the development, implementation, validation, and dissemination of a finite element code to simulate the collision of black holes. This problem involves the solution of the Einstein field equations on an unbounded domain in four dimensional space-time and is extremely demanding computationally. Work will focus, at least initially, on the solution of the initial data problem for black hole collisions, i.e., the determination of physical ly consistent and meaningful initial data. The initial data problem is a nonlinear elliptic system posed on an unbounded three-dimensional domain. For this problem the computer code to be produced will involve a variety of sophisticated modern techniques that have not yet been applied in computational relativity, including adaptive tetrahedral mesh generation and multigrid preconditioning. The investigator's work is directly related to several critical technological issues. It is generally agreed by practitioners that the current state of the art of numerical modeling of shells is inadequate, and this has been implicated in the failure of engineering structures such as stadium roofs and airplane fuselages. The investigator's work in both the area of the development and analysis of finite element methods for plates and shells, and on plate modeling errors is directed at overcoming these problems. The work on the simulation of black hole collisions is directly motivated by the ongoing construction of the LIGO gravitational wave observatory. LIGO is expected to be able to detect radiation from black hole collisions early in the next decade. Massive numerical simulations are needed to identify the cosmological events which are the sources for the detected radiation. It is clear that these simulations push the limits of available hardware and software technology. The investigator's work is directed at improving the speed and accuracy of these simulations by employing and adapting recent advances in numerical algorithms. The resulting computer codes will be adapted to parallel computer architectures and will provide an excellent opportunity for the graduate students and postdoctoral fellows working under the supervision of the investigator to learn high performance computing techniques.