SHAPE OPTIMIZATION OF A COUPLED THERMAL LMA FST B eni-Mellal, Universit e Sultan Moulay Slimane, Morocco. Abstract In this work, we consider the… [609754]

SHAPE OPTIMIZATION OF A COUPLED
THERMAL
LMA FST B eni-Mellal, Universit e Sultan Moulay Slimane, Morocco.
Abstract
In this work, we consider the problem of identifying a Robin term in an elliptic
equation with mixed boundary conditions. This desired coecient modelled
the intensity of heat exchange between a hot wire and the cooling water in
which it is placed. We rstly reformulate the inverse problem as a variational
one with a total variation regularization term, then we prove the existence
and uniqueness of a minimizer to the resulting optimization problem in a
suitable functional space. Finally, we provide a Primal-Dual algorithm to
solve the variational problem and show some numerical results that prove
the stability and accuracy of the proposed method in the identication of
the heat exchange.
Keywords: Inverse Robin problem, L1regularization, xed point,
Primal-Dual algorithm.
1. Introduction
Shape optimization problems are always a relevant mathematical research
domain used especially to treat optimal structures arising from technological
processes. These type of problems are modelled by partial dierential equa-
tions, which are extensively studied in []. In many practical circumstances,
the shape under investigation is parameterized by nitely many parameters,
which on the one hand allows the application of standard optimization ap-
proaches, but on the other hand limits the space of reachable shapes unnec-
essarily. Shape calculus, which has been the subject of several monographs
[12, 22, 29] presents a way out of that dilemma. However, so far it is mainly
applied in the form of gradient descent methods, which can be shown to
converge. The major dierence between shape optimization and the stan-
dard PDE constrained optimization framework is the lack of the linear space
Preprint submitted to Elsevier January 25, 2018

structure in shape spaces. If one cannot use a linear space structure, then
the next best structure is the Riemannian manifold structure as discussed
for shape spaces in [5, 6, 19, 20, 21]. The publication [27] makes a link
between shape calculus and shape manifolds and thus enables the usage of
optimization techniques on manifolds in the context of shape optimization.
PDE constrained shape optimization however, is confronted with function
spaces dened on varying domains. The current paper presents a vector
bundle framework based on the Riemannian framework established in [27],
which enables the discus- sion of LagrangeNewton methods within the shape
calculus framework for PDE constrained shape optimization.
Topology optimization methods have long suered from a drawback, in-
herent to its main advantage: the computed optimal structures have so com-
plex topologies that they are, very often, impossible to build by traditional
techniques like casting. This has boosted many research works on the ad-
dition of manufacturing constraints during the optimization, see e.g. [4],
[5], [17], [18], [19], [21], [37], [44]. The picture is completely dierent with
the emergence and maturity of additive manufacturing technologies, which
are able to build structures with a high degree of complexity, thereby allow-
ing to process almost directly the designs predicted by shape and topology
optimization algorithms
Inverse modeling in diusive processes is one of the major themes in the
eld of inverse problems. Often, a distributed diusivity parameter is to be
estimated from observations of the diused state, as in [11, 19, 20]. In many
cases, however, the rough overall structure of the parameter distribution is
known, but the details are missing. In the present paper, we assume that
the distributed diusion parameter to be estimated is piecewise constant in
subdomains with smooth boundaries.
Let us, very brie y comment on the related literature. Most of these en-
deavors focus on control problems related to ordinary dierential equations.
We quote the two surveys papers [12, 27] and [26]. From these publications
already it becomes clear that the notion by which optimality is measured is
an important topic in its own right. The literature on optimal actuator posi-
tioning for distributed parameter systems is less rich but it also dates back for
several decades already. From among the earlier contributions we quote [9]
where the topic is investigated in a semigroup setting for linear systems, [5]
for a class of linear innite dimensional ltering problems, and [11] where the
optimal actuator problem is investigated for hyperbolic problems related to
active noise suppression. In the works [18, 16, 19] the optimal actuator prob-
2

lem is formulated in terms of parameter-dependent linear quadratic regulator
problems where the parameters characterize the position of actuators, with
predetermined shape, for example. By choosing the actuator position in [13]
the authors optimise the decay rate in the one-dimensional wave equation.
Our research may be most closely related to the recent contribution [21],
where the optimal actuator design is driven by exact controllability consider-
ations, leading to actuators which are chosen on the basis of minimal energy
controls steering the system to zero within a specied time uniformly, for a
bounded set of initial conditions. Finally, let us mention that the optimal
actuator problem is in some sense dual to optimal sensor location problems
[14], which is of paramount importance.
2. Conclusion
A typical elliptic partial dierential equation Robin inverse problem usu-
ally involves both Neumann and Dirichlet boundary observations. In this
paper we propose a new method to solve the Robin inverse problem. By
coupling both Neumann and Dirichlet data in one boundary value condition,
and introducing two dierent boundary value problems which depend on
two positive constants, we optimize an objective functional in the problem
domain instead of one on the boundary. Dierentiability of the objective
functional with respect to the Robin coecient is established. A gradient
projection method is investigated for its stability and convergence. Theo-
retical and numerical results show that the proposed method is robust and
eective. Time-dependent inverse Robin problems will be studied in future
works.
Compliance with Ethical Standards
Funding: This research was entirely funded by the respective institu-
tions of the authors.
Con
ict of interest: The authors declare that they have no con
ict of
interest.
Neither human participants nor animals are involved in this research.
Bibliography
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