Numerical solution to shape optimization problems for non-stationary Navier-Stokes problems

Iwata, Yutaro; Azegami, Hideyuki; Aoyama, Taiki; Katamine, Eiji

doi:10.14495/jsiaml.2.37

Cited by 8 publications

(3 citation statements)

References 13 publications

(15 reference statements)

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“…Subsequently, J. Haslinger and R. A. Makinen [4], B. Mohammadi and O. Pironneau [5], and M. Moubachir and J. P. Zolesio [6] constructed fundamental frameworks of flow-field shape-optimization problems. Recently, many researchers are examining the topic [7][8][9][10]. Efficient but accurate numerical methods must be used for such flow computations within an optimization process.…”

Section: Introductionmentioning

confidence: 99%

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

Nakazawa

Nakajima

2019

IIS

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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.

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Section: Introductionmentioning

confidence: 99%

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

Nakazawa

Nakajima

2019

IIS

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“…In 1973, Pironneau [1] pioneered a structural optimization method for fluid dynamics problems, and obtained minimum drag wing profiles under Stokes flow. Considerable research has been carried out since then and a number of structural optimization methods applicable to fluid dynamics problems have been proposed [2][3][4][5][6][7][8][9][10][11]. However, since the above research was based on shape optimization, the feasible design modifications only consisted in the adjustment of the boundaries of selected parts to the fluid domain.…”

Section: Introductionmentioning

confidence: 99%

Topology optimization using the lattice Boltzmann method incorporating level set boundary expressions

Yaji

Yamada

Yoshino

et al. 2014

Journal of Computational Physics

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This paper presents a topology optimization method for fluid dynamics problems, based on the level set method and using the lattice Boltzmann method (LBM). In this optimization method, the optimization problems are formulated based on the original Boltzmann equation, and the design sensitivities are precisely obtained without the time-consuming numerical operations encountered when dealing with a large-scale asymmetric matrix, in contrast to previous research in which the LBM uses the lattice Boltzmann equation (LBE) for the formulations of optimization problems and the derivation of their adjoint equations. That is, we newly derive sensitivity formulations from the original Boltzmann equation, not the LBE that can be said to be an approximated equation, and these formulations yield strictly correct sensitivities that are error free. Based on the above formulations, we construct a level set-based topology optimization method incorporating a fictitious interface energy for the design of a fluid channel that minimizes flow friction. Furthermore, two-and three-dimensional numerical examples are provided to confirm the validity and utility of the presented method.

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“…The remaining boundaries of the channel are wall boundaries denoted by Γ w , upon which -together with the obstacle boundary Γ f -a no-slip boundary condition is imposed on the fluid. We refer the reader to [11,15,21,25,33] for shape optimization problems involving fluids among others. We also point out that usually, the curl ∇ × u is enough to quantify the vorticity of the fluid.…”

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confidence: 99%

A shape optimization problem constrained with the Stokes equations to address maximization of vortices

Simon

Notsu

2022

EECT

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<p style='text-indent:20px;'>We study an optimization problem that aims to determine the shape of an obstacle that is submerged in a fluid governed by the Stokes equations. The mentioned flow takes place in a channel, which motivated the imposition of a Poiseuille-like input function on one end and a do-nothing boundary condition on the other. The maximization of the vorticity is addressed by the <inline-formula><tex-math id="M1">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-norm of the curl and the <i>det-grad</i> measure of the fluid. We impose a Tikhonov regularization in the form of a perimeter functional and a volume constraint to address the possibility of topological change. Having been able to establish the existence of an optimal shape, the first order necessary condition was formulated by utilizing the so-called rearrangement method. Finally, numerical examples are presented by utilizing a finite element method on the governing states, and a gradient descent method for the deformation of the domain. On the said gradient descent method, we use two approaches to address the volume constraint: one is by utilizing the augmented Lagrangian method; and the other one is by utilizing a class of divergence-free deformation fields.</p>

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Numerical solution to shape optimization problems for non-stationary Navier-Stokes problems

Cited by 8 publications

References 13 publications

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

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

Topology optimization using the lattice Boltzmann method incorporating level set boundary expressions

A shape optimization problem constrained with the Stokes equations to address maximization of vortices

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