Numerical approximation of phase field based shape and topology optimization for fluids

2014/01/01 by Harald Garcke, Claudia Hecht, Garcke, Harald +5 · 4 citations
Computer Science · Engineering · Materials Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Solidification and crystal growth phenomena #Topology Optimization in Engineering #math.OC #msc:35Q35 #msc:35Q56 #msc:35R35 #msc:49Q10 #msc:65M12 #msc:65M22 #msc:65M60 #msc:76S05

paper · pdf · doi:10.48550/arxiv.1405.3480

arxiv created 2014/05/14 · arxiv updated 2014/05/15

Abstract

We consider the problem of finding optimal shapes of fluid domains. The fluid obeys the Navier--Stokes equations. Inside a holdall container we use a phase field approach using diffuse interfaces to describe the domain of free flow. We formulate a corresponding optimization problem where flow outside the fluid domain is penalized. The resulting formulation of the shape optimization problem is shown to be well-posed, hence there exists a minimizer, and first order optimality conditions are derived. For the numerical realization we introduce a mass conserving gradient flow and obtain a Cahn--Hilliard type system, which is integrated numerically using the finite element method. An adaptive concept using reliable, residual based error estimation is exploited for the resolution of the spatial mesh. The overall concept is numerically investigated and comparison values are provided.

Citations

• Algorithm 832
• An optimal Poincaré inequality for convex domains

Cited by

• Applying a phase field approach for shape optimization of a stationary Navier-Stokes flow
• On the Crouzeix-Raviart Finite Element Approximation of Phase-Field Dependent Topology Optimization in Stokes Flow
• Convergence Analysis of an Adaptive Nonconforming FEM for Phase-Field Dependent Topology Optimization in Stokes Flow
• Shape optimization for surface functionals in Navier--Stokes flow using a phase field approach

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• On the Calculation of the Brinkman Penalization Term in Density-Based Topology Optimization of Fluid-Dependent Problems