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

2014/01/01 by Harald Garcke, Claudia Hecht, Garcke, Harald +5
Computer Science · Materials Science · Engineering · #Advanced Mathematical Modeling in Engineering #Solidification and crystal growth phenomena #Topology Optimization in Engineering

paper · pdf · doi:10.48550/arxiv.1405.3480

Abstract

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

Citations

• Algorithm 832
• An optimal Poincaré inequality for convex domains

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