An efficient threshold dynamics method for topology optimization for fluids

Abstract

We propose an efficient threshold dynamics method for topology optimization for fluids modeled with the Stokes equation. The proposed algorithm is based on minimization of an objective energy function that consists of the dissipation power in the fluid and the perimeter approximated by nonlocal energy, subject to a fluid volume constraint and the incompressibility condition. We show that the minimization problem can be solved with an iterative scheme in which the Stokes equation is approximated by a Brinkman equation. The indicator functions of the fluid-solid regions are then updated according to simple convolutions followed by a thresholding step. We demonstrate mathematically that the iterative algorithm has the total energy decaying property. The proposed algorithm is simple and easy to implement. A simple adaptive time strategy is also used to accelerate the convergence of the iteration. Extensive numerical experiments in both two and three dimensions show that the proposed iteration algorithm converges in much fewer iterations and is more efficient than many existing methods. In addition, the numerical results show that the algorithm is very robust and insensitive to the initial guess and the parameters in the model.

A provably efficient monotonic-decreasing algorithm for shape optimization in Stokes flows by phase-field approaches

“…Assume that D is a designed domain that contains the optimized shape domain Ω. To ensure that the velocity vanishes on the inner boundary in the fluid region D, an additional term is required to add into the objective functional to penalize the velocity field in the void domain, which is expressed as (see, for instant, [12,9,7,43,1,24,41]).…”

“…Diffuser in 3D. We now consider the design of a diffuser in 3D [9]. The computational domain is set to be (D = [0, 1] 3 ) and the maximum flow velocity is ḡ = 1 at the inlet Γ i on the left.…”

“…However, re-initialization is always required to keep the regularity of the level set function. Recently, the threshold dynamics method [9,10,38] has been applied to shape optimization for fluids. The authors proposed an iterative scheme that has a monotonic-decaying property.…”

“…Stimulated by the pioneering work [12,23,9,22], we propose an efficient decoupled monotonic-decaying phase-field scheme for shape optimization and constrain the phase-field values in [0, 1] by a cut-off postprocessing. It is an iterative method via updating the velocity variable, the phase variable, and the volume parameter.…”