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COMPUTATIONAL TECHNIQUES FOR CIVIL AND STRUCTURAL ENGINEERING
Edited by: J. Kruis, Y. Tsompanakis and B.H.V. Topping
Stress Constraints and Incompressible Materials in Topology Optimization: State of the Art and New Perspectives
Department of Civil Engineering and Architecture, University of Pavia, Italy
P. Venini, "Stress Constraints and Incompressible Materials in Topology Optimization: State of the Art and New Perspectives", in J. Kruis, Y. Tsompanakis and B.H.V. Topping, (Editors), "Computational Techniques for Civil and Structural Engineering", Saxe-Coburg Publications, Stirlingshire, UK, Chapter 19, pp 439-465, 2015. doi:10.4203/csets.38.19
Keywords: topology optimization, stress constraints, incompressible materials, mixed finite elements.
Among the many theoretical and applicative issues linked to topology optimization this paper focuses on stress constraints and incompressible media design. Both these issues are well known as being difficult for different reasons: stress–constrained topology optimization is in fact an ill-posed problem whose optimal solution may not be found unless suitable remedies are taken; as for incompressible materials, the locking phenomenon that affects standard displacement-based approaches calls for the adoption of ad-hoc strategies that allow the incompressibility constraint to relax. The state–of–the art concerning these two crucial topics is outlined first. In this respect the goal is to describe the approaches that the community has proposed so far and relevant numerical results. There is no intension to provide a complete list of available contributions and further reference is made to the cited papers on some specific subjects for the interested reader. In the second part of the paper a unified approach for stress-constrained and incompressible material topology optimization is presented that is based on the application of a Hu-Washizu variational principle. The adoption of a Hu-Washizu variational principle coupled to the biorthogonal mixed finite element proposed in  is shown to represent an excellent balance between accuracy of stress evaluation, capability to avoid the locking phenomenon, ease of implementation and speed of computation. A few representative numerical examples are proposed to support fully the theoretical framework.
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