195 research outputs found
Revisiting Non-Convexity in Topology Optimization of Compliance Minimization Problems
Purpose: This is an attempt to better bridge the gap between the mathematical
and the engineering/physical aspects of the topic. We trace the different
sources of non-convexification in the context of topology optimization problems
starting from domain discretization, passing through penalization for
discreteness and effects of filtering methods, and end with a note on
continuation methods. Design/Methodology/Approach: Starting from the global
optimum of the compliance minimization problem, we employ analytical tools to
investigate how intermediate density penalization affects the convexity of the
problem, the potential penalization-like effects of various filtering
techniques, how continuation methods can be used to approach the global
optimum, and how the initial guess has some weight in determining the final
optimum. Findings: The non-convexification effects of the penalization of
intermediate density elements simply overshadows any other type of
non-convexification introduced into the problem, mainly due to its severity and
locality. Continuation methods are strongly recommended to overcome the problem
of local minima, albeit its step and convergence criteria are left to the user
depending on the type of application. Originality/Value: In this article, we
present a comprehensive treatment of the sources of non-convexity in
density-based topology optimization problems, with a focus on linear elastic
compliance minimization. We put special emphasis on the potential
penalization-like effects of various filtering techniques through a detailed
mathematical treatment
Design-dependent loads in topology optimization
We present, analyze, and implement a new method for the design of the stiffest structure subject to a pressure load or a given field of internal forces. Our structure is represented as a subset S of a reference domain, and the complement of S is made of two other phases , the vold and a fictitious liquid that exerts a pressure force on its interface with the solid structure. The problem we consider is to minimize the compliance of the structure S, which is the total work of the pressure and internal forces at the equilibrium displacement. In order to prevent from homogenization we add a penalization on the perimeter of S. We propose an approximation of our problem in the framework of Γ-convergence, based on an approximation of our three phases by a smooth phase-field. We detail the numerical implementation of the approximate energies and show a few experiments. © EDP Sciences, SMAI 2003
Multi-phase structural optimization via a level set method
33 pagesInternational audienceWe consider the optimal distribution of several elastic materials in a fixed working domain. In order to optimize both the geometry and topology of the mixture we rely on the level set method for the description of the interfaces between the different phases. We discuss various approaches, based on Hadamard method of boundary variations, for computing shape derivatives which are the key ingredients for a steepest descent algorithm. The shape gradient obtained for a sharp interface involves jump of discontinuous quantities at the interface which are difficult to numerically evaluate. Therefore we suggest an alternative smoothed interface approach which yields more convenient shape derivatives. We rely on the signed distance function and we enforce a fixed width of the transition layer around the interface (a crucial property in order to avoid increasing "grey" regions of fictitious materials). It turns out that the optimization of a diffuse interface has its own interest in material science, for example to optimize functionally graded materials. Several 2-d examples of compliance minimization are numerically tested which allow us to compare the shape derivatives obtained in the sharp or smoothed interface cases
Solution of the Kirchhoff-Plateau problem
The Kirchhoff-Plateau problem concerns the equilibrium shapes of a system in
which a flexible filament in the form of a closed loop is spanned by a liquid
film, with the filament being modeled as a Kirchhoff rod and the action of the
spanning surface being solely due to surface tension. We establish the
existence of an equilibrium shape that minimizes the total energy of the system
under the physical constraint of non-interpenetration of matter, but allowing
for points on the surface of the bounding loop to come into contact. In our
treatment, the bounding loop retains a finite cross-sectional thickness and a
nonvanishing volume, while the liquid film is represented by a set with finite
two-dimensional Hausdorff measure. Moreover, the region where the liquid film
touches the surface of the bounding loop is not prescribed a priori. Our
mathematical results substantiate the physical relevance of the chosen model.
Indeed, no matter how strong is the competition between surface tension and the
elastic response of the filament, the system is always able to adjust to
achieve a configuration that complies with the physical constraints encountered
in experiments
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