arXiv subjects

Christian Kahle

Publications and source records attributed to Christian Kahle.

21 records · Page 2Linked to original sources

arXiv2015

Shape optimization for surface functionals in Navier--Stokes flow using a phase field approach

We consider shape and topology optimization for fluids which are governed by the Navier--Stokes equations. Shapes are modelled with the help of a phase field approach and the solid body is relaxed to be a porous medium. The phase field method uses a Ginzburg--Landau functional in order to approximate a perimeter penalization. We focus on surface functionals and carefully introduce a new modelling variant, show existence of minimizers and derive first order necessary conditions. These conditions are related to classical shape derivatives by identifying the sharp interface limit with the help of formally matched asymptotic expansions. Finally, we present numerical computations based on a Cahn--Hilliard type gradient descent which demonstrate that the method can be used to solve shape optimization problems for fluids with the help of the new approach.

math.OC↗

arXiv2014

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

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.

math.OC↗

arXiv2014

A stable and linear time discretization for a thermodynamically consistent model for two-phase incompressible flow

A new time discretization scheme for the numerical simulation of two-phase flow governed by a thermodynamically consistent diffuse interface model is presented. The scheme is consistent in the sense that it allows for a discrete in time energy inequality. An adaptive spatial discretization is proposed that conserves the energy inequality in the fully discrete setting by applying a suitable post processing step to the adaptive cycle. For the fully discrete scheme a quasi-reliable error estimator is derived which estimates the error both of the flow velocity, and of the phase field. The validity of the energy inequality in the fully discrete setting is numerically investigated.

math.AP↗