Mathematical Sciences: Direct Relaxation of Optimal Layout Problems

9305840 Lurie这项研究将开发一种直接的方法来松弛优化设计的非自伴问题。利用被积函数的拟鞍性作为极小极大问题存在的充要条件，研究了相应的极小极大问题。我们还发展了另一种方法来解决这个问题，在勒让德变换应用到伴随场之后，将它简化为一个标准的最小化问题。%研究的目标是开发一种方法，用于确定在各种外部条件(载荷、静态和动态响应、失稳)或在组合物理场(应力和温度、电场和磁场等)作用下工作的弹性体的最佳结构特征。分析结果将用一种新的数值技术来实现，以使用先进的复合材料计算最优设计。与现有方法相比，这项技术将提供真正的最佳设计。***

9305840 Lurie This research will develop a direct approach to the relaxation of nonselfadjoint problems of optimal design. The associated min-max problems are investigated applying the notion of quasi-saddle behavior of integrand as the necessary and sufficient condition for the existence of min-max. We also develop an alternative approach to the problem reducing it to a standard minimization after the Legendre transform is applied to the adjoint field. %%% The goal of the research is to develop a methodology for the determination of optimal structural characteristics of elastic bodies designed for work in a variety of external conditions (load, static and dynamic response, loss of stability,) or under the action of combined physical fields (stresses and temperature, electric and magnetic fields, etc.) The analytic results will be implemented in a new numerical technique to compute optimal designs using advanced composites. This technique will give truly optimal designs as compared to existing methods. ***