A numerical study of heat source reconstruction for the advection-diffusion operator: A conjugate gradient method stabilized with SVD
Résumé
In order to better understand micromechanical phenomena such as viscoelasticity and plasticity, the
thermomechanical viewpoint is of prime importance but requires calorimetric measurements to be
performed during a deformation process. Infrared imaging is commonly used to this aim but does not
provide direct access to the intrinsic volumetric Thermomechanical Heat Sources (THS). An inverse
method is needed to convert temperature fields in the former quantity. The one proposed here relies on a
diffusion-advection heat transfer model. Advection is generally not considered in such problems but due
to plastic instabilities, a heterogeneous and non-negligible velocity field can play a role in the local heat
transfer balance. Discretization of the governing equation is made through appropriate spectral
approach. Spatial regularization is then achieved through regular modal truncation. The objective of the
inversion process lies in a proper identification of the decomposition coefficients (states) which minimize
the residuals. When a Conjugate Gradient Method (CGM) is applied to this nonlinear least square
optimization, the use of Karhunen-Loeve Decomposition (KLD) or Singular Value Decomposition (SVD)
on gradient vectors is shown to produce very good temporal regularization. Two test-cases were
explored for noisy data which show that this algorithm performs very well when compared to the
Tikhonov penalized conjugate gradient method.
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