Modelling, Analysis and Computation in Viscoelasticity

具有褪色记忆的粘弹性材料表现出介于弹性材料的非线性双曲响应和粘性材料的强扩散抛物线响应之间的行为；它们包含由褪色记忆效应引起的微妙的耗散机制。了解这些影响对于正确设计模拟非定常和定常粘弹性流动的计算方法是至关重要的。通过采用双曲守恒律研究的方法，将发展非定常问题的分析和计算技术。在理解经典解及其分解的基础上，将使用两种方法来研究控制方程的弱解：补偿紧致法和随机选择法。后者需要了解粘弹性流动的黎曼问题；它的解是设计计算算法的基本要素。这些想法将被用来扩大已建立的定常二维粘弹性流动模拟程序的范围，目的是解决在高Weissenberg数情况下的严重计算困难。这项研究代表了科学计算中的一个项目，该项目统一了数学、计算和工程学科的方法。

Viscoelastic materials with fading memory exhibit behavior that is intermediate between the nonlinear hyperbolic response of elastic materials and the strongly diffusive, parabolic response of viscous materials; they incorporate a subtle dissipative mechanism induced by the effects of fading memory. Understanding these effects is crucial to the proper design of computational methods for modeling both unsteady and steady viscoelastic flows. Analytical and computational techniques will be developed for unsteady problems by adapting methods from the study of hyperbolic conservation laws. Building upon the understanding of classical solutions and their breakdown, a study of weak solutions to the governing equations will be conducted using two approaches: the method of compensated compactness and the random choice method. The latter requires an understanding of the Riemann problem for viscoelastic flows; its solution is an essential ingredient for designing computational algorithms. These ideas will be used to broaden the scope of the established program of simulation of steady, two-dimensional viscoelastic flows, with the objective of resolving serious computational difficulties at high Weissenberg numbers. This research represents a program in scientific computation that unites methods from mathematical, computational, and engineering disciplines.