Multidisciplinary topology optimization solved as a Nash game

Abstract

SUMMARYIn the present work, multidisciplinary optimization is formulated in the game theory framework. We choose a coupled heat transfer-thermoelastic system as the case study for which a topology design approach is developed. The multidisciplinary optimization problem is solved as a non-cooperative game and we determine a Nash equilibrium. The game has two players and the parameterization of the design domain is such that the design variables describe the material density and a parameter which influences the heat flow by convection to the surrounding fluid. The first player controls the structure and the second player controls the temperature distribution in the structure. For the second player, we present mathematical proof of existence of a discrete valued optimal solution and it is concluded that no regularization of the suboptimization problem is needed. We present two numerical examples which illustrate the proposed methodology. One of the examples is also solved by weighting the objectives to a scalar valued objective function and the result is compared with the Nash game solution.

Game theory approach to robust topology optimization with uncertain loading

“…In particular the Hessian of L 1 is large and dense and Newton-based methods are therefore expected to be too expensive, unless perhaps (limited memory) quasi-Newton approximations can be used. Périaux et al (2001) and Habbal et al (2004) used a nonlinear Jacobi-type algorithm (Facchinei and Kanzow 2010) where the two players update their strategies simultaneously, i.e. both optimization problems are solved at the same time and independent of each other.…”

“…Game theory approaches for structural optimization have been used in a number of papers, but only two papers (known by the authors) involve TO: Habbal (2005) formulated a TO framework to model growth of cancerous tumors and Habbal et al (2004) optimized a thermoelastic system where a heat source as well as an external load were applied to the design space; one player was aiming to minimize the compliance of the thermoelastic system by varying the topology, the other player was aiming to minimize the temperature in the structure by adding structural elements, such as fins, in order to maximize the heat flow to the surroundings.…”

Multi Objective Aerodynamic Optimization Using Parallel Nash Evolutionary/Deterministic Hybrid Algorithms

“…Since Nash 1,2 introduces Nash Equilibrium (NE) theory in the early 50's, it has become an efficient tool to solve Multi Objective Optimization (MOO) problems in aerodynamics 3,4,5,6 and other relative fields 7,8 . The solution of a MOO problem can be viewed as a NE under the concept of competitive games.…”

Topology optimization for thermal insulation: an application to building engineering

“…to Gao et al (2008), that addresses load effects in conduction-dominated problems with heat generation depending on the material state, or to Bruns (2007) and Iga et al (2009), that propose ad hoc methodologies to handle the effects of evolving structural boundaries also in the case of convection-driven problems. Thermal aspects are taken into account within several multi-physics formulations, as in the case of the optimal design of thermo-elastic components addressed in Cho and Choi (2005), Habbal et al (2004) and Diaz and Benard (2003). Peculiar issues of the optimization for the heat conduction problem are discussed in the multi-material design presented in Zhuang et al (2010) and in the nano-scale investigations of Evgrafov et al (2008).…”