440 research outputs found
Optimal Reliability for Components under Thermomechanical Cyclic Loading
We consider the existence of optimal shapes in the context of the
thermomechanical system of partial differential equations (PDE) using the
recent approach based on elliptic regularity theory. We give an extended and
improved definition of the set of admissible shapes based on a class of
sufficiently differentiable deformation maps applied to a baseline shape. The
obtained set of admissible shapes again allows one to prove a uniform Schauder
estimate for the elasticity PDE. In order to deal with thermal stress, a
related uniform Schauder estimate is also given for the heat equation. Special
emphasis is put on Robin boundary conditions, which are motivated from
convective heat transfer. It is shown that these thermal Schauder estimates can
serve as an input to the Schauder estimates for the elasticity equation. This
is needed to prove the compactness of the (suitably extended) solutions of the
entire PDE system in some state space that carries a c2-H\"older topology for
the temperature field and a C3-H\"older topology for the displacement. From
this one obtains he property of graph compactness, which is the essential tool
in an proof of the existence of optimal shapes. Due to the topologies employed,
the method works for objective functionals that depend on the displacement and
its derivatives up to third order and on the temperature field and its
derivatives up to second order. This general result in shape optimization is
then applied to the problem of optimal reliability, i.e. the problem of finding
shapes that have minimal failure probability under cyclic thermomechanical
loading.Comment: 32 pages 1 figur
A triviality result in the AdS/CFT correspondence for Euclidean quantum fields with exponential interaction
We consider scalar quantum fields with exponential interaction on Euclidean
hyperbolic space in two dimensions. Using decoupling
inequalities for Neumann boundary conditions on a tessellation of
, we are able to show that the infra-red limit for the generating
functional of the conformal boundary field becomes trivial.Comment: 13 pages, 1 figur
The Feynman graph representation of convolution semigroups and its applications to L\'{e}vy statistics
We consider the Cauchy problem for a pseudo-differential operator which has a
translation-invariant and analytic symbol. For a certain set of initial
conditions, a formal solution is obtained by a perturbative expansion. The
series so obtained can be re-expressed in terms of generalized Feynman graphs
and Feynman rules. The logarithm of the solution can then be represented by a
series containing only the connected Feynman graphs. Under some conditions, it
is shown that the formal solution uniquely determines the real solution by
means of Borel transforms. The formalism is then applied to probabilistic
L\'{e}vy distributions. Here, the Gaussian part of such a distribution is
re-interpreted as a initial condition and a large diffusion expansion for
L\'{e}vy densities is obtained. It is outlined how this expansion can be used
in statistical problems that involve L\'{e}vy distributions.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ106 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Minimal failure probability for ceramic design via shape control
We consider the probability of failure for components made of brittle mate-
rials under one time application of a load as introduced by Weibull and Batdorf
- Crosse and more recently studied by NASA and the STAU cooperation as an
objective functional in shape optimization and prove the existence of optimal
shapes in the class of shapes with a uniform cone property. The corresponding
integrand of the objective functional has convexity properties that allow to
derive lower-semicontinuity according to Fujii (Opt. Th. Appl. 1988). These
properties require less restrictive regularity assumptions for the boundaries
and state functions compared to [arXiv:1210.4954]. Thereby, the weak
formulation of linear elasticity can be kept for the abstract setting for shape
optimization as presented in the book by Haslinger and Maekinen
- …