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The vertex solution theorem and its coupled framework for static analysis of structures with interval parameters
Author(s) -
Qiu Zhiping,
Lv Zheng
Publication year - 2017
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.5523
Subject(s) - vertex (graph theory) , interval (graph theory) , mathematics , interval arithmetic , finite element method , regular polygon , upper and lower bounds , discrete mathematics , mathematical analysis , combinatorics , geometry , graph , engineering , bounded function , structural engineering
Summary This work gives new statement of the vertex solution theorem for exact bounds of the solution to linear interval equations and its novel proof by virtue of the convex set theory. The core idea of the theorem is to transform linear interval equations into a series of equivalent deterministic linear equations. Then, the important theorem is extended to find the upper and lower bounds of static displacements of structures with interval parameters. Following discussions about the computational efforts, a coupled framework based on vertex method (VM) is established, which allows us to solve many large‐scale engineering problems with uncertainties using deterministic finite element software. Compared with the previous works, the contribution of this work is not only to obtain the exact bounds of static displacements but also lay the foundation for development of an easy‐to‐use interval finite element software. Numerical examples demonstrate the good accuracy of VM. Meanwhile, the implementation of VM and availability of the coupled framework are demonstrated by engineering example. Copyright © 2017 John Wiley & Sons, Ltd.