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 www.nasatech.com/tsp under the Mechanics category.
NPO-21133
This Brief includes a Technical Support Package (TSP).
PVP-MP Method for Wrinkling Analysis of Space Membrane Structures
(reference NPO-21133) is currently available for download from the TSP library.
Don't have an account?
Overview
The document discusses the Parametric-Variational-Principle based Mathematical Programming (PVP-MP) method, a computational technique developed for analyzing wrinkles in membranes, particularly in lightweight structures used in space. This method is significant for its ability to address the complexities of membrane behavior under various conditions—taut, slack, and wrinkled—by transforming the traditional wrinkling problem into a mathematical programming challenge.
The PVP-MP method is unique because it employs a parametric variational principle that allows for a unified description of membrane states. This principle is particularly effective for wrinkled membranes, as it guarantees numerical convergence, which is often a challenge in traditional methods. Unlike conventional approaches that rely on iterative techniques, the PVP-MP method utilizes optimization to find the minimum of the variational principle, ensuring that the global minimum is reached without the risk of settling for a local minimum.
The document highlights the limitations of ordinary static analysis and finite element methods in predicting wrinkles, which are a result of local buckling phenomena in membranes subjected to in-plane compressions. Traditional numerical methods often face convergence issues or yield incorrect solutions due to the lack of a minimum energy principle. The PVP-MP method was developed in response to the need for a more accurate and efficient wrinkling analysis technique.
The application of the PVP-MP method extends beyond space structures; it can also be utilized in various industrial products, including textiles and paper. This versatility makes it a valuable tool for industries that rely on membrane structures. The document emphasizes the importance of disseminating this technology through platforms like NASA Tech Briefs, which can connect the innovation with interested companies and industries.
In summary, the PVP-MP method represents a significant advancement in the analysis of membrane structures, providing a reliable and efficient means to understand and predict wrinkling behavior. Its innovative approach to optimization and convergence makes it a promising solution for both aerospace applications and broader industrial uses.
