Shape optimization problem based on the generalized J integral considering RANS and Snapshot POD

Abstract

For suppression of time periodic flow and normal stress using optimal design techniques, this paper presents an optimal shape by sensitivity based on the Generalized J Integral, and makes a comparison to results of sensitivity evaluated by boundary and volume integrations of a type that is widely used for shape optimization problems. To date, J integral has been used to evaluate the energy release rate of stress concentrated near cracks or corners in the field of fracture mechanics. The Generalized J integration type used for this study is sensitive to avoid singularity and to engender the suppression of stress concentration in the domain and at the boundary. For such shape optimization studied here, the cost function is defined as eigenvalues with modes of the time fluctuation component in Snapshot POD. The main problems are the Reynolds Average Navier–Stokes problem and eigenvalue problem of Snapshot POD. An objective functional is described using Lagrange multipliers and finite element method. The sensitivity is obtained in three kinds of ways, the boundary and volume, the Generalized J integration types. Numerical results reveal that the cost function is minimized as the time periodic flow is suppressed in such the three ways. Especially, normal stress on the boundary in the sensitivity evaluated by the Generalized J integration type is suppressed most of them.

Keywords

• adjoint method
• cavity flow
• shape optimization problem
• snapshot proper orthogonal decomposition