Gradient based biobjective shape optimization to improve reliability and cost of ceramic components
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Gradient based biobjective shape optimization to improve reliability and cost of ceramic components O. T. Doganay1 · H. Gottschalk1 · C. Hahn1 · K. Klamroth1 · J. Schultes1 · M. Stiglmayr1 Received: 27 May 2019 / Revised: 20 September 2019 / Accepted: 4 December 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract We consider the simultaneous optimization of the reliability and the cost of a ceramic component in a biobjective PDE constrained shape optimization problem. A probabilistic Weibull-type model is used to assess the probability of failure of the component under tensile load, while the cost is assumed to be proportional to the volume of the component. Two different gradient-based optimization methods are suggested and compared at 2D test cases. The numerical implementation is based on a first discretize then optimize strategy and benefits from efficient gradient computations using adjoint equations. The resulting approximations of the Pareto front nicely exhibit the trade-off between reliability and cost and give rise to innovative shapes that compromise between these conflicting objectives. Keywords Biobjective shape optimization · Shape gradients · Probability of failure · Descent algorithms Mathematics Subject Classification 90B50 · 49Q10 · 65C50 · 60G55
* O. T. Doganay [email protected]‑wuppertal.de H. Gottschalk hanno.gottschalk@uni‑wuppertal.de C. Hahn [email protected]‑wuppertal.de K. Klamroth [email protected]‑wuppertal.de J. Schultes [email protected]‑wuppertal.de M. Stiglmayr [email protected]‑wuppertal.de 1
University of Wuppertal, Gaussstraße 20, 42119 Wuppertal, Germany
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1 Introduction The optimization of the design of mechanical structures is a central task in mechanical engineering. If the material for a component is chosen and the use cases are defined, implying in particular the mechanical loads, then the central task of engineering design is to define the shape of the component. Among all possible choices, those shapes are preferred that guarantee the desired functionality at minimal cost. The functionality, however, is only guaranteed if the mechanical integrity of the component is preserved. The fundamental design requirements of functional integrity and cost are almost always in conflict, which makes mechanical engineering an optimization problem with at least two objective functions to consider. Mathematically, the task of choosing the shape of a structure is formulated by the theory of shape optimization, see, e.g., Allaire (2001), Bucur and Buttazzo (2005), Haslinger and Mäkinen (2003) and Sokolovski and Zolesio (1992) for an introduction. We thus consider admissible shapes Ω ⊆ ℝp , p = 2, 3 , along with an objective function f (Ω) which returns lower values for better designs. The task then is to find an admissible shape Ω∗ ∈ arg min f (Ω) . The existence of optimal shapes has been studied in Chenais (1975), Fujii (1988) and Haslinger and Mäkinen (2003)—for the specific objective function f of complian
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