Research Initiation: Structural Design Optimization as a Linear Programming Problem

研究人员将研究结构设计优化问题在没有近似的情况下映射到线性规划（LP）问题的条件，并创建利用这些特殊性质的算法。传统的方法是将这类问题视为非线性规划（NLP）问题。对于大型系统，NLP问题的精确解通常是棘手的。这表明，这类结构设计问题可以更有效地解决。该方法要求解一个LP，然后解一个线性方程组。LP的解确定了一个与最优结构具有相同位移场的典型结构。该位移场使平衡方程在未知区域表现为线性系统，求解得到最优设计。因此，土木和航空航天工程师现在可以解决的实际问题比以前想象的要大几个数量级。本研究的目标是扩展LP方法，以涵盖更一般的结构设计优化问题。初步数值证据表明，该方法可推广到位移和频率约束。除了这些约束外，还将考虑屈曲和多重载荷，并将该方法扩展到由连续统有限元离散的结构。如果LP方法成功地解决了这类更广泛的问题，将大大减少解决这些优化问题所需的计算量，因为需要解决LP而不是NLP。这种效率的显著提高将使优化对结构设计师更具吸引力。

The researcher will investigate conditions under which problems in structural design optimization map, without approximation, to linear programming (LP) problems, and to create algorithms which exploit these special properties. The conventional approach is to treat such problems as ones in nonlinear programming (NLP). For large systems, exact solution of the resulting NLP problem is often intractable. It has been shown that a class of structural design problems can be solved much more efficiently. The method requires solution of an LP followed by a system of linear equations. Solution of the LP identifies a canonical structure which possesses the same displacement field as the optimal structure. This displacement field renders the equilibrium equations as a linear system in unknown areas, which can be solved to yield the optimum design. As a consequence, problems can now be solved of practical interest to civil and aerospace engineers several orders of magnitude larger than previously imagined. The goal of this research is to extend the LP methodology to encompass a more general class of structural design optimization problems. Initial numerical evidence indicates that the method is extendable to displacement and frequency constraints. In addition to these constraints, buckling and multiple loads will be considered, and the researchers will define extensions of the method to structures discretized by continuum finite elements. If successful for this broader class of problem, the LP method will greatly reduce the computational effort required for solving these optimization problems, as a consequence of the need to solve an LP rather than a NLP. Such a significant enhancement of efficiency will make optimization more attractive to structural designers.