Parameter-Variation-Principle (PVP) based Mathematical Programming (MP) is the basis of a computational method of analyzing wrinkles in membranes. Devised for original application to lightweight membrane structures in outer space, the method can also be applied on Earth to similar structures, to diverse industrial products that include paper and textiles, and to structures made from these products.

PVP is a variational principle, for which some of membrane strain components, unlike in a traditional variational principle, do not participate in functional variation. PVP is suitable for analyzing wrinkled membranes because it is valid for all three general membrane conditions — taut, slack, and wrinkled. With PVP, the traditional problem of membrane wrinkling is transformed to a mathematical programming problem, which can be efficiently solved by numerical methods. As a result, the present PVP-MP method guarantees numerical convergence for all three conditions. In this method, one uses an optimization technique instead of traditional iteration to search for the minimum of this principle. This search guarantees convergent numerical solutions with finite steps in computation.

A membrane by itself usually has very little resistance to in-plane compression and very little stiffness against out-of-plane bending. Out-of-plane stiffness is usually imparted to a membrane through pre-tensioning. Therefore, out-of-plane stiffness is a function of the distribution of in-plane stress. Wrinkles appear when some areas of a membrane are subjected to in-plane compression to a certain level; indeed, the formation of wrinkles is a membrane local-buckling phenomenon.

Ordinary stress analysis procedures are limited in predicting wrinkles. Numerical iteration methods for wrinkling analysis used heretofore to analyze wrinkles apply different values of membrane material properties, depending on whether it is taut, slack, or wrinkled. These methods often present difficulties that prevent or impede convergence or that lead to incorrect solutions.

The present PVP-MP method guarantees accurate results with much less (relative to prior methods) computational effort. The method involves two main steps. In the first step, one develops a PVP principle, including a controlling parameter vector. With the help of the controlling parameter vector, taut, slack, and wrinkled states of the membrane can be represented by one variational principle. In the second step, one searches for the minimum of the variational principle by use of the applicable optimization technique. Because the search can reach the minimum of the variational principle at the exact solution, this method can predict the distribution of stress throughout the membrane, including any taut, slack, and/or wrinkled areas.

This work was done by Houfei Fang, Michael Lou, and Bingen Yang of Caltech for NASA's Jet Propulsion Laboratory. For further information, access the Technical Support Package (TSP) free on-line at under the Mechanics category.