842 research outputs found
Recovery of a space-dependent vector source in thermoelastic systems
In this contribution, an inverse problem of determining a space-dependent vector source in a thermoelastic system of type-I, type-II and type-III is studied using information from a supplementary measurement at a fixed time. These thermoelastic systems consist of two equations that are coupled: a parabolic equation for the temperature
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and a vectorial hyperbolic equation for the displacement
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. In this latter one, the source is unknown, but solely space dependent. A spacewise-dependent additional measurement at the final time ensures that the inverse problem corresponding with each type of thermoelasticity has a unique solution when a damping term
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(with
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componentwise strictly monotone increasing) is present in the hyperbolic equation. Despite the ill-posed nature of these inverse problems, a stable iterative algorithm is proposed to recover the unknown source in the case that
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is also linear. This method is based on a sequence of well-posed direct problems, which are numerically solved at each iteration, step by step, using the finite element method. The instability of the inverse source problem is overcome by stopping the iterations at the first iteration for which the discrepancy principle is satisfied. Numerical results support the theoretically obtained results
Inverse Design Based on Nonlinear Thermoelastic Material Models Applied to Injection Molding
This paper describes an inverse shape design method for thermoelastic bodies.
With a known equilibrium shape as input, the focus of this paper is the
determination of the corresponding initial shape of a body undergoing thermal
expansion or contraction, as well as nonlinear elastic deformations. A
distinguishing feature of the described method lies in its capability to
approximately prescribe an initial heterogeneous temperature distribution as
well as an initial stress field even though the initial shape is unknown. At
the core of the method, there is a system of nonlinear partial differential
equations. They are discretized and solved with the finite element method or
isogeometric analysis. In order to better integrate the method with
application-oriented simulations, an iterative procedure is described that
allows fine-tuning of the results. The method was motivated by an inverse
cavity design problem in injection molding applications. Its use in this field
is specifically highlighted, but the general description is kept independent of
the application to simplify its adaptation to a wider range of use cases.Comment: 22 pages, 13 figure
On Stability of Hyperbolic Thermoelastic Reissner-Mindlin-Timoshenko Plates
In the present article, we consider a thermoelastic plate of
Reissner-Mindlin-Timoshenko type with the hyperbolic heat conduction arising
from Cattaneo's law. In the absense of any additional mechanical dissipations,
the system is often not even strongly stable unless restricted to the
rotationally symmetric case, etc. We present a well-posedness result for the
linear problem under general mixed boundary conditions for the elastic and
thermal parts. For the case of a clamped, thermally isolated plate, we show an
exponential energy decay rate under a full damping for all elastic variables.
Restricting the problem to the rotationally symmetric case, we further prove
that a single frictional damping merely for the bending compoment is sufficient
for exponential stability. To this end, we construct a Lyapunov functional
incorporating the Bogovski\u{i} operator for irrotational vector fields which
we discuss in the appendix.Comment: 27 page
Numerical modelling of thermoelastic processes using nonlinear theories with thermal relaxation time
A nonlinear model of thermoelasticity is solved for a ring-shaped body with special attention given to the effect of relaxation time on the body dynamics. A spectral Galerkin method is used to accurately resolve spatial structures of temperature and displacement. A sequence of solutions is obtained for gradually increasing initial amplitude of the thermoelastic wave, showing a transition from a steady harmonic wave to singular solutions. Larger values of the relaxation time are shown to result in smoother temperature profiles. The dynamics generated by a localised heat release from an external source is studied. It is observed that the source excites a standing-like wave of the displacement while temperature monotonically decays
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