14 research outputs found
Phase-field approaches to structural topology optimization
The mean compliance minimization in structural topology optimization is solved with the help of a phase field approach. Two steepest descent approaches based on L2- and H-1 gradient flow dynamics are discussed.
The resulting flows are given by Allen-Cahn and Cahn-Hilliard type dynamics coupled to a linear elasticity system. We finally compare numerical results obtained from the two different approaches
Allen-Cahn and Cahn-Hilliard variational inequalities solved with Optimization Techniques
Parabolic variational inequalities of Allen-Cahn and Cahn-
Hilliard type are solved using methods involving constrained optimization. Time discrete variants are formulated with the help of Lagrange multipliers for local and non-local equality and inequality constraints. Fully discrete problems resulting from finite element discretizations in space are solved with the help of a primal-dual active set approach. We
show several numerical computations also involving systems of parabolic variational inequalities
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
Truncated nonsmooth Newton multigrid methods for simplex-constrained minimization problems
We present a multigrid method for the minimization of strongly convex functionals defined on a finite product of simplices. Such problems result, for example, from the discretization of multi-component phase-field problems. Our algorithm is globally convergent, requires no regularization parameters, and achieves multigrid convergence rates. We present numerical results for the vector-valued Allen--Cahn equation and observe that the convergence rate is independent from the temperature parameter and the number of components
Topology optimization of multiple anisotropic materials, with application to self-assembling diblock copolymers
We propose a solution strategy for a multimaterial minimum compliance
topology optimization problem, which consists in finding the optimal allocation
of a finite number of candidate (possibly anisotropic) materials inside a
reference domain, with the aim of maximizing the stiffness of the body. As a
relevant and novel application we consider the optimization of self-assembled
structures obtained by means of diblock copolymers. Such polymers are a class
of self-assembling materials which spontaneously synthesize periodic
microstructures at the nanoscale, whose anisotropic features can be exploited
to build structures with optimal elastic response, resembling biological
tissues exhibiting microstructures, such as bones and wood. For this purpose we
present a new generalization of the classical Optimality Criteria algorithm to
encompass a wider class of problems, where multiple candidate materials are
considered, the orientation of the anisotropic materials is optimized, and the
elastic properties of the materials are assumed to depend on a scalar
parameter, which is optimized simultaneously to the material allocation and
orientation. Well-posedness of the optimization problem and well-definition of
the presented algorithm are narrowly treated and proved. The capabilities of
the proposed method are assessed through several numerical tests
A semismooth Newton method with analytical path-following for the -projection onto the Gibbs simplex
An efficient, function-space-based second-order method for the -projection onto the Gibbs-simplex is presented. The method makes use of the theory of semismooth Newton methods in function spaces as well as Moreau-Yosida regularization and techniques from parametric optimization. A path-following technique is considered for the regularization parameter updates. A rigorous first and second-order sensitivity analysis of the value function for the regularized problem is provided to justify the update scheme. The viability of the algorithm is then demonstrated for two applications found in the literature: binary image inpainting and labeled data classification. In both cases, the algorithm exhibits mesh-independent behavior
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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 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
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Mini-Workshop: Geometries, Shapes and Topologies in PDE-based Applications
The aim of the workshop was to study geometrical objects and their sensitivities in applications based on partial differential equations or differential variational inequalities. Focus topics comprised analytical investigations, numerical developments, issues in applications as well as new and future directions. Particular emphasis was put on: (i) combined shape and topological sensitivity; (ii) extended topological expansions and their numerical realization; (iii) level set based shape and topology optimization
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 T-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