Optimal Design of Fibered Structures.
Abstract
The problem of shape optimization is the minimize the area that must be filled by expensive material in order to satisfy design constraints. Those constraints require the structure to withstand a given external load (or family of loads) without exceeding a permissible deformation, or a permissible compliance, or the yield limit of a plastic material. The design problem is mathematically subtle, because the underlying optimization problem is not convex. Structural equilibrium is governed by a partial differential equation of the form div(c(x)grad u) = f, but in contrast to the analysis problem (which solves for u), the design problem is to choose c. The control variable is the coefficients, and frequently a conventional solution does not exist. The minimum weight design is approached by coefficients c which jump more and more frequently between alternative states. The design develops a complicated microstructure, and the challenge is to see within that structures a simple and computable pattern. The limit is a composite material, in which the original materials have well-defined densities and orientations. The composite is achieved by homogenization of the original materials. Mathematically this is expressed by a relaxation, or convexification, of the original minimization problem. The project has seen a successful integration of these fundamental theories (previously developed along separate lines), and the explicit computation of optimal designs in a series of significant engineering problems.
Document Details
- Document Type
- Technical Report
- Publication Date
- Feb 01, 1988
- Accession Number
- ADA194165
Entities
People
- Gilbert Strang
Organizations
- Massachusetts Institute of Technology