The optimal design task of this paper seeks the distribution of two materials of prescribed amounts for maximal torsion stiffness of an infinite bar of a given cross section. This example of relaxation in topology optimization leads to a degenerate convex minimization problem $E(v): = \int_\Omega {{\varphi _0}} (|\nabla v|)dx - \int_\Omega {fvdx} $ for $v \in V: = H_0^1\left( \Omega \right)$ with possibly multiple primal solutions u, but with unique stress $\sigma : = \varphi _0^1\left( {|\nabla u|} \right)$ sign ∇u. The mixed finite element method is motivated by the smoothness of the stress variable $\sigma \in H_{loc}^1\left( {\Omega ;{\mathbb{R}^2}} \right)$ , while the primal variables are uncontrollable and possibly nonunique. The corresponding nonlinear mixed finite element method is introduced, analyzed, and implemented. The striking result of this paper is a sharp a posteriori error estimation in the dual formulation, while the a posteriori error analysis in the primal problem suffers from the reliability-efficiency gap. An empirical comparison of that primal formulation with the new mixed discretization schemes is intended for uniform and adaptive mesh refinements.
SIAM Journal on Numerical Analysis contains research articles on the development and analysis of numerical methods including their convergence, stability, and error analysis as well as related results in functional analysis and approximation theory. Computational experiments and new types of numerical applications are also included.
"The Society for Industrial and Applied Mathematics is a leading international association for applied mathematics, and its publications could be the nucleus of an adequate collection in mathematics. One of the objectives of this organization is to make the flow of information between the university and industry smoother. It fulfills this objective admirably and many of the leading academic institutions in the world are members." --Magazines for Libraries, Eighth Edition, 1995, R. R. Bowker, New Providence, New Jersey The Society for Industrial and Applied Mathematics (SIAM), headquartered in Philadelphia, was founded in 1951 to advance the application of mathematics to science and industry, promote mathematical research, and provide media for the exchange of information and ideas among mathematicians, engineers, and scientists. SIAM has a comprehensive publishing program in applied and computational mathematics, including 11 prestigious research journals. For a complete description of our journals and our newly announced SIAM Journals Online, access http://www.siam.org/.
This item is part of a JSTOR Collection.
For terms and use, please refer to our Terms and Conditions
SIAM Journal on Numerical Analysis
© 2012 Society for Industrial and Applied Mathematics
Request Permissions
