21,120 research outputs found
Dynamic programming approach to structural optimization problem – numerical algorithm
In this paper a new shape optimization algorithm is presented. As a model application we consider state problems related to fluid mechanics, namely the Navier-Stokes equations for viscous incompressible fluids. The general approach to the problem is described. Next, transformations to classical optimal control problems are presented. Then, the dynamic programming approach is used and sufficient conditions for the shape optimization problem are given. A new numerical method to find the approximate value function is developed
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
Optimal wavy surface to suppress vortex shedding using second-order sensitivity to shape changes
A method to find optimal 2nd-order perturbations is presented, and applied to
find the optimal spanwise-wavy surface for suppression of cylinder wake
instability. Second-order perturbations are required to capture the stabilizing
effect of spanwise waviness, which is ignored by standard adjoint-based
sensitivity analyses. Here, previous methods are extended so that (i) 2nd-order
sensitivity is formulated for base flow changes satisfying linearised
Navier-Stokes, and (ii) the resulting method is applicable to a 2D global
instability problem. This makes it possible to formulate 2nd-order sensitivity
to shape modifications. Using this formulation, we find the optimal shape to
suppress the a cylinder wake instability. The optimal shape is then perturbed
by random distributions in full 3D stability analysis to confirm that it is a
local optimal at the given amplitude and wavelength. Furthermore, it is shown
that none of the 10 random wavy shapes alone stabilize the wake flow at Re=50,
while the optimal shape does. At Re=100, surface waviness of maximum height 1%
of the cylinder diameter is sufficient to stabilize the flow. The optimal
surface creates streaks by passively extracting energy from the base flow
derivatives and effectively altering the tangential velocity component at the
wall, as opposed to spanwise-wavy suction which inputs energy to the normal
velocity component at the wall. This paper presents a fully two-dimensional and
computationally affordable method to find optimal 2nd-order perturbations of
generic flow instability problems and any boundary control (such as boundary
forcing, shape modulation or suction).Comment: 19 pages, 6 figure
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