283 research outputs found
Relating phase field and sharp interface approaches to structural topology optimization
A phase field approach for structural topology optimization which allows for topology
changes and multiple materials is analyzed. First order optimality conditions are
rigorously derived and it is shown via formally matched asymptotic
expansions that these conditions converge to classical first order conditions obtained in
the context of shape calculus. We also discuss how to deal with triple junctions where
e.g. two materials and the void meet. Finally, we present several
numerical results for mean compliance problems and a cost involving the least square error
to a target displacement
An extension of the projected gradient method to a Banach space setting with application in structural topology optimization
For the minimization of a nonlinear cost functional under convex
constraints the relaxed projected gradient process is
a well known method. The analysis is classically performed in a Hilbert space
. We generalize this method to functionals which are differentiable in a
Banach space. Thus it is possible to perform e.g. an gradient method if
is only differentiable in . We show global convergence using
Armijo backtracking in and allow the inner product and the scaling
to change in every iteration. As application we present a
structural topology optimization problem based on a phase field model, where
the reduced cost functional is differentiable in . The
presented numerical results using the inner product and a pointwise
chosen metric including second order information show the expected mesh
independency in the iteration numbers. The latter yields an additional, drastic
decrease in iteration numbers as well as in computation time. Moreover we
present numerical results using a BFGS update of the inner product for
further optimization problems based on phase field models
A PDE-constrained optimization approach for topology optimization of strained photonic devices
Recent studies have demonstrated the potential of using tensile-strained, doped Germanium
as a means of developing an integrated light source for (amongst other things) future microprocessors.
In this work, a multi-material phase-field approach to determine the optimal material
configuration within a so-called Germanium-on-Silicon microbridge is considered. Here, an ``optimal"
configuration is one in which the strain in a predetermined minimal optical cavity within
the Germanium is maximized according to an appropriately chosen objective functional. Due to
manufacturing requirements, the emphasis here is on the cross-section of the device; i.e. a socalled
aperture design. Here, the optimization is modeled as a non-linear optimization problem
with partial differential equation (PDE) and manufacturing constraints. The resulting problem is
analyzed and solved numerically. The theory portion includes a proof of existence of an optimal
topology, differential sensitivity analysis of the displacement with respect to the topology, and the
derivation of first and second-order optimality conditions. For the numerical experiments, an array
of first and second-order solution algorithms in function-space are adapted to the current setting,
tested, and compared. The numerical examples yield designs for which a significant increase in
strain (as compared to an intuitive empirical design) is observed
Topology optimization subject to additive manufacturing constraints
In topology optimization the goal is to find the ideal material distribution in a domain subject to external forces. The structure is optimal if it has the highest possible stiffness. A volume constraint ensures filigree structures, which are regulated via a Ginzburg–Landau term. During 3D printing overhangs lead to instabilities. As a remedy an additive manufacturing constraint is added to the cost functional. First order optimality conditions are derived using a formal Lagrangian approach. With an Allen-Cahn interface propagation the optimization problem is solved iteratively. At a low computational cost the additive manufacturing constraint brings about support structures, which can be fine tuned according to demands and increase stability during the printing process
Sharp interface limit for a phase field model in structural optimization
We formulate a general shape and topology optimization problem in structural
optimization by using a phase field approach. This problem is considered in
view of well-posedness and we derive optimality conditions. We relate the
diffuse interface problem to a perimeter penalized sharp interface shape
optimization problem in the sense of -convergence of the reduced
objective functional. Additionally, convergence of the equations of the first
variation can be shown. The limit equations can also be derived directly from
the problem in the sharp interface setting. Numerical computations demonstrate
that the approach can be applied for complex structural optimization problems
Relating phase field and sharp interface approaches to structural topology optimization
A phase field approach for structural topology optimization which
allows for topology changes and multiple materials is analyzed. First order
optimality conditions are rigorously derived and it is shown via formally
matched asymptotic expansions that these conditions converge to classical
first order conditions obtained in the context of shape calculus. We also
discuss how to deal with triple junctions where e.g. two materials and the
void meet. Finally, we present several numerical results for mean compliance
problems and a cost involving the least square error to a target
displacement
Topology optimization subject to additive manufacturing constraints
In Topology Optimization the goal is to find the ideal material distribution in a domain subject to external forces. The structure is optimal if it has the highest possible stiffness. A volume constraint ensures filigree structures, which are regulated via a Ginzburg-Landau term. During 3D Printing overhangs lead to instabilities, which have only been tackled unsatisfactorily. The novel idea is to incorporate an Additive Manufacturing Constraint into the phase field method. A rigorous analysis proves the existence of a solution and leads to first order necessary optimality conditions. With an Allen-Cahn interface propagation the optimization problem is solved iteratively. At a low computational cost the Additive Manufacturing Constraint brings about support structures, which can be fine tuned according to engineering demands. Stability during 3D Printing is assured, which solves a common Additive Manufacturing problem
Topology optimization for incremental elastoplasticity: a phase-field approach
We discuss a topology optimization problem for an elastoplastic medium. The
distribution of material in a region is optimized with respect to a given
target functional taking into account compliance. The incremental elastoplastic
problem serves as state constraint. We prove that the topology optimization
problem admits a solution. First-order optimality conditions are obtained by
considering a regularized problem and passing to the limit
Topology optimization subject to additive manufacturing constraints
In Topology Optimization the goal is to find the ideal material distribution in a domain subject to external forces. The structure is optimal if it has the highest possible stiffness. A volume constraint ensures filigree structures, which are regulated via a Ginzburg-Landau term. During 3D Printing overhangs lead to instabilities, which have only been tackled unsatisfactorily. The novel idea is to incorporate an Additive Manufacturing Constraint into the phase field method. A rigorous analysis proves the existence of a solution and leads to first order necessary optimality conditions. With an Allen-Cahn interface propagation the optimization problem is solved iteratively. At a low computational cost the Additive Manufacturing Constraint brings about support structures, which can be fine tuned according to engineering demands. Stability during 3D Printing is assured, which solves a common Additive Manufacturing problem
Phase-Field Methods for Spectral Shape and Topology Optimization
We optimize a selection of eigenvalues of the Laplace operator with Dirichlet
or Neumann boundary conditions by adjusting the shape of the domain on which
the eigenvalue problem is considered. Here, a phase-field function is used to
represent the shapes over which we minimize. The idea behind this method is to
modify the Laplace operator by introducing phase-field dependent coefficients
in order to extend the eigenvalue problem on a fixed design domain containing
all admissible shapes. The resulting shape and topology optimization problem
can then be formulated as an optimal control problem with PDE constraints in
which the phase-field function acts as the control. For this optimal control
problem, we establish first-order necessary optimality conditions and we
rigorously derive its sharp interface limit. Eventually, we present and discuss
several numerical simulations for our optimization problem
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