Boundary Element Simulation of Thermal Waves

Abstract

In this paper we survey computational techniques based on boundary integral formulations for the simulation of thermal waves. Time-harmonic solutions to diffusion problems appear in many physical situations of interest and give rise to many interesting problems related to material characterization, parameter assessment or detection of defects. We review the main direct, indirect and mixed integral numerical methods for a model of scattering of thermal waves by many obstacles and discuss how multiple scattering techniques and other physical tools can be understood as iterative methods or used as preconditioners. We also deal with some transient problems that can be solved with boundary element methods using the Laplace transform and with coupled finite and boundary element schemes for nonhomogeneous obstacles. Photothermal Techniques and Thermal WavesMathematically speaking a thermal wave is just a timeharmonic solution to the heat equation, i.e., a function of the form a(x) cos(ω t) + b(x) sin(ωt) that solves the linear equations of heat diffusion. The problem of scattering of thermal waves appears in the study of structural properties of composite materials by non-destructive testing.

Domain reconstruction using photothermal techniques

“…For our particular type of complex wave-numbers and a i > 0, problem (9) and (10) is always uniquely solvable (see [10] or [28]). Efficient numerical strategies to solve (9) and (10) by means of boundary element techniques are discussed in the survey [29].…”

“…We describe below how this is accomplished. A rigorous justification for our transmission problem can be found in [21] (see also [29]). For related problems arising in the study of coated materials see [19].…”

“…When X i 6 ¼ ;, none of the wave-numbers can be a Dirichlet eigenvalue of the Laplace operator in X i , because they are complex. Therefore, problem (14)- (16) can be solved numerically at low cost by using elementary boundary integral formulations based on single layer potentials, as suggested in [28] (see also [29] for other formulations).…”

“…Applications to the recovery of dielectric coating factors or detection of corrosion are studied in [6,18]. The properties of thermal waves and their multiple usages are discussed in [1,23,24,29]. We refer to [12,20] and references therein for theoretical results on inverse problems involving heat equations.…”

Thermal wave scattering by two overlapping and parallel cylinders

“…In the last years several studies on the propagation of thermal waves in materials with plane [2,3], cylindrical [4,5] and spherical [6] inclusions have been published. The results of these works show that photothermal techniques are suitable to study the thermal properties of multilayer samples, fiber reinforced composites, and particulate reinforced composites, respectively.…”