A preconditioned finite element solution of the coupled pressure-temperature equations used to model trace gas sensors. (English) Zbl 1401.35040
Summary: Quartz enhanced photoacoustic spectroscopy (QEPAS) is a technique for detecting trace gases which relies on a quartz tuning fork resonator to amplify and measure the weak acoustic pressure waves that are generated when a laser heat source periodically interacts with a gas sample. At low ambient pressures, the same tuning fork can instead detect thermal diffusion waves generated by this laser-gas interaction in a process called resonant optothermoacoustic detection (ROTADE). In this paper, we present a unified computational model for QEPAS and ROTADE sensors that is based on a coupled system of Helmholtz equations for pressure and temperature in a fluid domain surrounding the tuning fork. In the tuning fork itself, the standard heat equation is used to solve for temperature. We employ the perfectly matched layer (PML) approach to absorb outgoing waves and prevent reflections off of the boundary of the computational domain. The resulting linear system is highly ill conditioned, but Krylov subspace solvers can be used to solve the system effectively if one employs an appropriate parallel block preconditioner. This method reduces the problem to that of solving a scalar Helmholtz problem with PML, which we precondition by coupling an algebraic multigrid solver in the interior of the computational domain to a direct solver in the PML region. Numerical results indicate that the preconditioner for the scalar Helmholtz problem with PML is both scalable and mesh-independent. Simulations show that the coupled pressure-temperature waves can strongly differ from the solution to the acoustic wave equation at low ambient pressures. In particular, interactions between the pressure and temperature solutions of the coupled system contribute to the reduced sensitivity of ROTADE sensors which has been experimentally observed in certain parameter regimes.
MSC:
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 35J57 | Boundary value problems for second-order elliptic systems |
| 35K05 | Heat equation |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
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