9,688 research outputs found
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
Applying a phase field approach for shape optimization of a stationary Navier-Stokes flow
We apply a phase field approach for a general shape optimization problem of a
stationary Navier-Stokes flow. To be precise we add a multiple of the
Ginzburg--Landau energy as a regularization to the objective functional and
relax the non-permeability of the medium outside the fluid region. The
resulting diffuse interface problem can be shown to be well-posed and
optimality conditions are derived. We state suitable assumptions on the problem
in order to derive a sharp interface limit for the minimizers and the
optimality conditions. Additionally, we can derive a necessary optimality
system for the sharp interface problem by geometric variations without stating
additional regularity assumptions on the minimizing set
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
A Nitsche-based cut finite element method for a fluid--structure interaction problem
We present a new composite mesh finite element method for fluid--structure
interaction problems. The method is based on surrounding the structure by a
boundary-fitted fluid mesh which is embedded into a fixed background fluid
mesh. The embedding allows for an arbitrary overlap of the fluid meshes. The
coupling between the embedded and background fluid meshes is enforced using a
stabilized Nitsche formulation which allows us to establish stability and
optimal order \emph{a priori} error estimates,
see~\cite{MassingLarsonLoggEtAl2013}. We consider here a steady state
fluid--structure interaction problem where a hyperelastic structure interacts
with a viscous fluid modeled by the Stokes equations. We evaluate an iterative
solution procedure based on splitting and present three-dimensional numerical
examples.Comment: Revised version, 18 pages, 7 figures. Accepted for publication in
CAMCo
- …