Convergence of Diffusion Generated Motion to Motion by Mean Curvature

Swartz, Drew; Yip, Nung Kwan

doi:10.48550/arxiv.1703.06519

Cited by 1 publication

(1 citation statement)

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“…The MBO method has been shown to converge to continuous motion by mean curvature [3,9,18,47]. Esedoglu and Otto gave a variational formulation for the original MBO scheme and successfully generalized this type of method to multiphase problems with arbitrary surface tensions [15].…”

Section: Introductionmentioning

confidence: 99%

An efficient threshold dynamics method for topology optimization for fluids

Chen¹,

Leng²,

Wang³

et al. 2018

Preprint

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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.

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