1,587 research outputs found
Detecting interfaces in a parabolic-elliptic problem from surface measurements
Assuming that the heat capacity of a body is negligible outside certain inclusions the heat equation degenerates to a parabolic-elliptic interface problem. In this work we aim to detect these interfaces from thermal measurements on the surface of the body. We deduce an equivalent variational formulation for the parabolic-elliptic problem and give a new proof of the unique solvability based on Lions’s projection lemma. For the case that the heat conductivity is higher inside the inclusions, we develop an adaptation of the factorization method to this time-
dependent problem. In particular this shows that the locations of the interfaces are uniquely determined by boundary measurements. The method also yields to a numerical algorithm to recover the inclusions and thus the interfaces. We demonstrate how measurement data can be simulated numerically by a coupling of a finite element method with a boundary element method, and finally we present some numerical results for the inverse problem
Sensitivity analysis of a parabolic-elliptic problem
We consider the heat flux through a domain with subregions in which the thermal capacity approaches zero. In these subregions the parabolic heat equation degenerates to an elliptic one. We show the well-posedness of such parabolic-elliptic differential equations for general non-negative L-infinity-capacities and study the continuity of the solutions with respect to the capacity, thus giving a rigorous justification for modeling a small thermal capacity by setting it to zero. We also characterize weak directional derivatives of the temperature with respect to capacity as solutions of related parabolic-elliptic problems
Sampling methods for low-frequency electromagnetic imaging
For the detection of hidden objects by low-frequency electromagnetic imaging the Linear Sampling Method works remarkably well despite the fact that the rigorous mathematical justification is still incomplete. In this work, we give an explanation for this good performance by showing that in the low-frequency limit the measurement operator fulfills the assumptions for the fully justified variant of the Linear Sampling Method, the so-called Factorization Method. We also show how the method has to be modified in the physically relevant case of electromagnetic imaging with divergence-free currents. We present numerical results to illustrate our
findings, and to show that similar performance can be expected for the case of conducting objects and layered backgrounds
Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization
We introduce a new variational method for the numerical homogenization of
divergence form elliptic, parabolic and hyperbolic equations with arbitrary
rough () coefficients. Our method does not rely on concepts of
ergodicity or scale-separation but on compactness properties of the solution
space and a new variational approach to homogenization. The approximation space
is generated by an interpolation basis (over scattered points forming a mesh of
resolution ) minimizing the norm of the source terms; its
(pre-)computation involves minimizing quadratic (cell)
problems on (super-)localized sub-domains of size .
The resulting localized linear systems remain sparse and banded. The resulting
interpolation basis functions are biharmonic for , and polyharmonic
for , for the operator -\diiv(a\nabla \cdot) and can be seen as a
generalization of polyharmonic splines to differential operators with arbitrary
rough coefficients. The accuracy of the method ( in energy norm
and independent from aspect ratios of the mesh formed by the scattered points)
is established via the introduction of a new class of higher-order Poincar\'{e}
inequalities. The method bypasses (pre-)computations on the full domain and
naturally generalizes to time dependent problems, it also provides a natural
solution to the inverse problem of recovering the solution of a divergence form
elliptic equation from a finite number of point measurements.Comment: ESAIM: Mathematical Modelling and Numerical Analysis. Special issue
(2013
Capillary Condensation in Confined Media
We review here the physics of capillary condensation of liquids in confined
media, with a special regard to the application in nanotechnologies. The
thermodynamics of capillary condensation and thin film adsorption are first
exposed along with all the relevant notions. The focus is then shifted to the
modelling of capillary forces, to their measurements techniques (including SFA,
AFM and crack tips) and to their influence on AFM imaging techniques as well as
on the static and dynamic friction properties of solids (including granular
heaps and sliding nanocontacts). A great attention is spent in investigating
the delicate role of the surface roughness and all the difficulties involved in
the reduction of the probe size to nanometric dimensions. Another major
consequence of capillary condensation in nanosystems is the activation of
several chemical and corrosive processes that can significantly alter the
surface properties, such as dissolution/redeposition of solid materials and
stress-corrosion crack propagation.Comment: 28 pages - To appear in 2010 in the Handbook of Nanophysics - Vol 1 -
Edited by Klaus Sattler - CRC Pres
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