# Mathematical Analysis of Continuum Mechanics and Industrial Applications IIIShape Optimization of Flow Fields Considering Proper Orthogonal Decomposition

Mathematical Analysis of Continuum Mechanics and Industrial Applications III: Shape Optimization... [This paper presents a new shape optimization method suppressing time periodic flows driven only by the non-stationary boundary condition at a sufficiently low Reynolds number using Proper Orthogonal Decomposition (POD). For the shape optimization problem, the eigenvalue in POD is defined as a cost function. The main problems are the non-stationary Navier–Stokes problem and the eigenvalue problem of POD. An objective functional is described using Lagrange multiplier method with finite element method. Two-dimensional cavity flow with a disk-shaped isolated body is adopted, where the non-stationary boundary condition is defined on the top boundary and non-slip boundary condition for the boundaries not only of the side and bottom but also of the disk. For numerical demonstrations, the disk boundary is used as the design boundary. Therefore the disk is reshaped by the shape optimization process as the cost function decreases. Numerical results reveal that the cost function (eigenvalues in POD) is decreased. The eigenvalues in the initial and the optimal domains are compared. Results clarify suppression of the amplitude of the time periodic flow, driven only by the non-stationary boundary condition at a sufficiently low Reynolds number.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

# Mathematical Analysis of Continuum Mechanics and Industrial Applications IIIShape Optimization of Flow Fields Considering Proper Orthogonal Decomposition

Part of the Mathematics for Industry Book Series (volume 34)
Editors: Itou, Hiromichi; Hirano, Shiro; Kimura, Masato; Kovtunenko, Victor A.; Khludnev, Alexandr M.
20 pages

/lp/springer-journals/mathematical-analysis-of-continuum-mechanics-and-industrial-qhestCP6gd
Publisher
Springer Singapore
© Springer Nature Singapore Pte Ltd. 2020
ISBN
978-981-15-6061-3
Pages
125 –145
DOI
10.1007/978-981-15-6062-0_9
Publisher site
See Chapter on Publisher Site

### Abstract

[This paper presents a new shape optimization method suppressing time periodic flows driven only by the non-stationary boundary condition at a sufficiently low Reynolds number using Proper Orthogonal Decomposition (POD). For the shape optimization problem, the eigenvalue in POD is defined as a cost function. The main problems are the non-stationary Navier–Stokes problem and the eigenvalue problem of POD. An objective functional is described using Lagrange multiplier method with finite element method. Two-dimensional cavity flow with a disk-shaped isolated body is adopted, where the non-stationary boundary condition is defined on the top boundary and non-slip boundary condition for the boundaries not only of the side and bottom but also of the disk. For numerical demonstrations, the disk boundary is used as the design boundary. Therefore the disk is reshaped by the shape optimization process as the cost function decreases. Numerical results reveal that the cost function (eigenvalues in POD) is decreased. The eigenvalues in the initial and the optimal domains are compared. Results clarify suppression of the amplitude of the time periodic flow, driven only by the non-stationary boundary condition at a sufficiently low Reynolds number.]

Published: Aug 30, 2020

Keywords: Cavity flow; Shape optimization problem; Adjoint method; Proper orthogonal decomposition