Optimal Design by Adaptive Mesh Refinement on Shape Optimization of Flow Fields Considering Proper Orthogonal Decomposition

Abstract

This paper presents optimal design using Adaptive Mesh Refinement (AMR) with shape optimization method. The method suppresses time periodic flows driven only by the non-stationary boundary condition at a sufficiently low Reynolds number using Snapshot Proper Orthogonal Decomposition (Snapshot POD). For shape optimization, the eigenvalue in Snapshot POD is defined as a cost function. The main problems are non-stationary Navier-Stokes problems and eigenvalue problems of POD. An objective functional is described using Lagrange multipliers and finite element method. Two-dimensional cavity flow with a disk-shaped isolated body is adopted. The non-stationary boundary condition is defined on the top boundary and non-slip boundary condition respectively for the side and bottom boundaries and for the disk boundary. For numerical demonstration, the disk boundary is used as the design boundary. Using H 1 gradient method for domain deformation, all triangles over a mesh are deformed as the cost function decreases. To avoid decreasing the numerical accuracy based on squeezing triangles, AMR is applied throughout the shape optimization process to maintain numerical accuracy equal to that of a mesh in the initial domain. The combination of eigenvalues that can best suppress the time periodic flow is investigated.

Model order reduction using Karhunen-Loève expansion in shape optimization analysis of hyperelastic body

“…The use of POD in a level-set topology optimization problem can be found in Xia and Breitkopf (2014). The application of POD to a non-parametric shape optimization problem of a flow field was shown by Nakazawa (2019) and Nakazawa and Nakajima (2019), who extracted dominant modes from a numerical solution of the Navier-Stokes problem using POD, applied the eigenvalues to formulate a shape optimization problem of a flow field improving hydrodynamic stability (Nakazawa and Azegami, 2016;Kiriyama et al, 2018), and solved the optimization problem through a gradientbased method known as the H 1 gradient method for a domain variation (Azegami, 2020, Chapter 9), which was also referred to as the traction method (Azegami and Wu, 1996;Azegami and Takeuchi, 2006).…”

Shape optimization for suppressing coherent structure of two-dimensional open cavity flow

“…This paper presents an optimal design obtained using a shape optimization problem in a domain with a singular point. The author earlier reported specific examination of construction of a shape optimization method with Snapshot Proper Orthogonal Decomposition (snapshot POD; Nakazawa, 2019), in which eigenvalues of Snapshot POD are defined as a cost function. Time average and time fluctuation parts of transient flows are suppressed directly in a two-dimensional cavity flow with an isolated disk, where the isolated disk is used as a design boundary.…”

“…Time average and time fluctuation parts of transient flows are suppressed directly in a two-dimensional cavity flow with an isolated disk, where the isolated disk is used as a design boundary. As described herein, the shape optimization problem suggested in work by Nakazawa (2019) is improved for a domain with a boundary with a singular point, which is used as a design boundary. Thereby, a two-dimensional open cavity flow investigated by Sipp et al (2007) and Barbagallo et al (2009) is chosen as an initial domain.…”

“…To address the challenge explained above, the author constructed a shape optimization method using Snapshot POD (Nakazawa, 2019), in which the eigenvalue in Snapshot POD is defined as the cost function. Such the Snapshot POD has the same numerical procedure as Primary Component Analysis (PCA) and play a role as one of Reduced Order Model (ROM) based method as well as Dynamic Mode Decompostion (DMD).…”