17,794 research outputs found
Some flows in shape optimization
Geometric flows related to shape optimization problems of Bernoulli type are
investigated. The evolution law is the sum of a curvature term and a nonlocal
term of Hele-Shaw type. We introduce generalized set solutions, the definition
of which is widely inspired by viscosity solutions. The main result is an
inclusion preservation principle for generalized solutions. As a consequence,
we obtain existence, uniqueness and stability of solutions. Asymptotic behavior
for the flow is discussed: we prove that the solutions converge to a
generalized Bernoulli exterior free boundary problem
Towards a Lagrange-Newton approach for PDE constrained shape optimization
The novel Riemannian view on shape optimization developed in [Schulz, FoCM,
2014] is extended to a Lagrange-Newton approach for PDE constrained shape
optimization problems. The extension is based on optimization on Riemannian
vector space bundles and exemplified for a simple numerical example.Comment: 16 pages, 4 figures, 1 tabl
Branching Structures in Elastic Shape Optimization
Fine scale elastic structures are widespread in nature, for instances in
plants or bones, whenever stiffness and low weight are required. These patterns
frequently refine towards a Dirichlet boundary to ensure an effective load
transfer. The paper discusses the optimization of such supporting structures in
a specific class of domain patterns in 2D, which composes of periodic and
branching period transitions on subdomain facets. These investigations can be
considered as a case study to display examples of optimal branching domain
patterns.
In explicit, a rectangular domain is decomposed into rectangular subdomains,
which share facets with neighbouring subdomains or with facets which split on
one side into equally sized facets of two different subdomains. On each
subdomain one considers an elastic material phase with stiff elasticity
coefficients and an approximate void phase with orders of magnitude softer
material. For given load on the outer domain boundary, which is distributed on
a prescribed fine scale pattern representing the contact area of the shape, the
interior elastic phase is optimized with respect to the compliance cost. The
elastic stress is supposed to be continuous on the domain and a stress based
finite volume discretization is used for the optimization. If in one direction
equally sized subdomains with equal adjacent subdomain topology line up, these
subdomains are consider as equal copies including the enforced boundary
conditions for the stress and form a locally periodic substructure.
An alternating descent algorithm is employed for a discrete characteristic
function describing the stiff elastic subset on the subdomains and the solution
of the elastic state equation. Numerical experiments are shown for compression
and shear load on the boundary of a quadratic domain.Comment: 13 pages, 6 figure
Shape optimization of pressurized air bearings
Use of externally pressurized air bearings allows for the design of mechanical systems requiring extreme precision in positioning. One application is the fine control for the positioning of mirrors in large-scale optical telescopes. Other examples come from applications in robotics and computer hard-drive manufacturing. Pressurized bearings maintain a finite separation between mechanical components by virtue of the presence of a pressurized flow of air through the gap between the components. An everyday example is an air hockey table, where a puck is levitated above the table by an array of vertical jets of air. Using pressurized bearings there is no contact between “moving parts” and hence there is no friction and no wear of sensitive components.
This workshop project is focused on the problem of designing optimal static air bearings subject to given engineering constraints. Recent numerical computations of this problem, done at IBM by Robert and Hendriks, suggest that near-optimal designs can have unexpected complicated and intricate structures. We will use analytical approaches to shed some light on this situation and to offer some guides for the design process.
In Section 2 the design problem is stated and formulated as an optimization problem for an elliptic boundary value problem.
In Section 3 the general problem is specialized to bearings with rectangular bases.
Section 4 addresses the solutions of this problem that can be obtained using variational formulations of the problem.
Analysis showing the sensitive dependence to perturbations (in numerical computations or manufacturing constraints) of near-optimal designs is given in Section 5.
In Section 6, a restricted class of “groove network” designs motivated by the original results of Robert and Hendriks is examined.
Finally, in Section 7, we consider the design problem for circular axisymmetric air bearings
Shape Optimization Problems for Metric Graphs
We consider the shape optimization problem where is the one-dimensional Hausdorff measure and is an
admissible class of one-dimensional sets connecting some prescribed set of
points . The cost
functional is the Dirichlet energy of defined
through the Sobolev functions on vanishing on the points . We
analyze the existence of a solution in both the families of connected sets and
of metric graphs. At the end, several explicit examples are discussed.Comment: 23 pages, 11 figures, ESAIM Control Optim. Calc. Var., (to appear
Shape Optimization Problems with Internal Constraint
We consider shape optimization problems with internal inclusion constraints,
of the form \min\big\{J(\Omega)\ :\ \Dr\subset\Omega\subset\R^d,\
|\Omega|=m\big\}, where the set \Dr is fixed, possibly unbounded, and
depends on via the spectrum of the Dirichlet Laplacian. We analyze the
existence of a solution and its qualitative properties, and rise some open
questions.Comment: 18 pages, 0 figure
Shape optimization of disc-type flywheels
Techniques were developed for presenting an analytical and graphical means for selecting an optimum flywheel system design, based on system requirements, geometric constraints, and weight limitations. The techniques for creating an analytical solution are formulated from energy and structural principals. The resulting flywheel design relates stress and strain pattern distribution, operating speeds, geometry, and specific energy levels. The design techniques incorporate the lowest stressed flywheel for any particular application and achieve the highest specific energy per unit flywheel weight possible. Stress and strain contour mapping and sectional profile plotting reflect the results of the structural behavior manifested under rotating conditions. This approach toward flywheel design is applicable to any metal flywheel, and permits the selection of the flywheel design to be based solely on the criteria of the system requirements that must be met, those that must be optimized, and those system parameters that may be permitted to vary
- …