Exact domain truncation for the Morse-Ingard equations

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Exact domain truncation for the Morse-Ingard

equations?

Robert C. Kirbya,∗, Xiaoyu Weib, Andreas Kl¨ocknerb

aDepartment of Mathematics, Baylor University; Sid Richardson Science Building; 1410

S. 4th St.; Waco, TX 76706; United States

bDepartment of Computer Science, University of Illinois at Urbana-Champaign,

Champaign, Illinois, United States

Abstract

Morse and Ingard [1] give a coupled system of time-harmonic equations for

the temperature and pressure of an excited gas. These equations form a

critical aspect of modeling trace gas sensors. Like other wave propagation

problems, the computational problem must be closed with suitable far-ﬁeld

boundary conditions. Working in a scattered-ﬁeld formulation, we adapt a

nonlocal boundary condition proposed in [2] for the Helmholtz equation to

this coupled system. This boundary condition uses a Green’s formula for the

true solution on the boundary, giving rise to a nonlocal perturbation of stan-

dard transmission boundary conditions. However, the boundary condition

is exact and so Galerkin discretization of the resulting problem converges to

the restriction of the exact solution to the computational domain. Numer-

ical results demonstrate that accuracy can be obtained on relatively coarse

meshes on small computational domains, and the resulting algebraic systems

may be solved by GMRES using the local part of the operator as an eﬀective

preconditioner.

Keywords: Far ﬁeld boundary conditions, Finite element, Multiphysics,

Thermoacoustics,

MSC[2010] 65N30, 65F08

?This work was supported by NSF SHF-1909176 and SHF-1911019.

∗Corresponding author

Email addresses: robert_kirby@baylor.edu (Robert C. Kirby),

xywei@illinois.edu (Xiaoyu Wei), andreask@illinois.edu (Andreas Kl¨ockner)

Preprint submitted to Elsevier October 26, 2022

arXiv:2210.13765v1 [math.NA] 25 Oct 2022

1. Introduction

Laser absorption spectroscopy is used for detecting trace amounts of gases

in diverse application areas such as air quality monitoring, disease diagnosis,

and manufacturing [3, 4, 5]. In photoacoustic spectroscopy, a laser is ﬁred

between the tines of a small quartz tuning fork, and in the presence of a

particular gas, acoustic and thermal waves are generated. These waves, in

turn, interact with the tuning fork to generate an electric signal via pyroelec-

tric and piezoelectric eﬀects. Two variants of these sensors are the so-called

QEPAS (quartz-enhanced photoacoustic spectroscopy) and ROTADE (reso-

nant optothermoacoustic detection) models [6, 7]. In QEPAS, the acoustic

wave dominates the signal, while the thermal wave is more important in RO-

TADE. In many experimental conﬁgurations, both eﬀects appear. While a

full model of the sensor is an eventual goal, obtaining eﬃcient and accurate

models of the gas itself is a critical step.

Earlier work on modeling this problem [8, 9, 10] simpliﬁed the model to

a single PDE that included an empirically-determined damping term to ac-

count for otherwise-neglected processes. This approach is only accurate in

particular regimes, and the empirical corrections depend strongly on geom-

etry as well as physical parameters, which limits the model’s utility in an

optimal design context.

Hence, work began on coupled models including both thermal and acous-

tic eﬀects. A ﬁnite element discretization of the coupled pressure-temperature

system of Morse and Ingard [1] was ﬁrst addressed in [11], where the diﬃ-

culty of solving the linear system was noted. Kirby and Brennan gave a

more rigorous treatment in [12], with analysis of the ﬁnite element error and

preconditioner performance. Kaderli et al derived an analytical solution for

the coupled system in idealized geometry in [13]. Their technique involves

reformulating the system studied in [12] by an algebraic simpliﬁcation that

eliminates the temperature Laplacian from the pressure equation. In [14],

this reformulation was seen to lose coercivity but still retain a G˚arding-type

inequality, leading to optimal-order ﬁnite element convergence theory and

preconditioners. Work by Saﬁn et al [15] began a more robust multi-physics

study, coupling the Morse-Ingard equations for atmospheric pressure and

temperature to heat conduction of the quartz tuning fork, although vibra-

tional eﬀects were still not considered. They also applied a perfectly-matched

layer (PML) [16] to truncate the computational domain, and a Schwarz-type

preconditioner that separates out the PML region was used to eﬀectively re-

duce the cost of solving the linear system. They also include some favorable

comparisons between the computational model and experimental data.

Previous numerical analysis of this problem in the cited literature has

focused on volumetric discretizations based on ﬁnite elements. In [17], we

derived a boundary integral formulation for a scattered-ﬁeld form of the

Morse-Ingard equations. As with other wave problems, this problem writes

the solution as the sum of a Morse-Ingard solution that satisﬁes the forcing

(evaluated by means of a fast volumetric convolution with a Green’s func-

tion) plus a ﬁeld that satisﬁes Morse-Ingard with no volumetric forcing but

Neumann data on the tuning fork such that the sum satisﬁes homogeneous

boundary conditions. We then formulated a second-kind integral equation

for the scattered ﬁeld and approximated it with a boundary integral method.

In this work we return to ﬁnite element discretization, but we make use of the

results we obtained considering the integral form of the equations to make

signiﬁcant advances in imposing a far-ﬁeld condition.

In [2], we developed a novel nonlocal boundary condition for truncating

the domain of Helmholtz scattering problems. This condition, which uses

Green’s representation of the solution on the artiﬁcial boundary to give a

nonlocal Robin-type condition involving layer potentials, is exact – the solu-

tion of resulting BVP agrees exactly with the restriction of the solution of the

original problem to the computational domain. The variational form of the

problem inherits a G˚arding-type inequality from the Helmholtz operator so

that a Galerkin ﬁnite element method yields optimal asymptotic convergence

rates. Moreover, empirical results suggest that the standard local operator

with transmission boundary conditions serves as an excellent preconditioner.

Hence, one only needs to apply the action of the resulting nonlocal operator,

say, by a fast multipole method, in order to obtain fast GMRES convergence.

In this paper, extend this approach from the Helmholtz operator to

the Morse-Ingard system, making using of several results developed in our

boundary-integral formulation in [17]. In Section 2, we recall the Morse-

Ingard equations. Then, Section 3 addresses far-ﬁeld boundary conditions for

the system and appropriate boundary conditions for domain truncation. By

means of the transformation to a decoupled Helmholtz system, we are able to

state an analogous far-ﬁeld condition and associated transmission-type con-

dition for the Morse-Ingard system. This allows a comparison to the ad hoc

transmission boundary conditions used in [12, 14]. Moreover, we can derive

an exact analog of the nonlocal Helmholtz boundary condition for Morse-

Ingard. Although one may directly solve the decoupled Helmholtz equations

rather than the coupled form of Morse-Ingard, formulating boundary con-

ditions and directly simulating the coupled system serves several purposes.

First, the pointwise transformation, thought it only involves a 2 ×2 matrix,

is quite ill-conditioned and seems to limit the accuracy we obtain on ﬁne

meshes for the decoupled form compared to the coupled system. Second, a

more complete model of trace gas sensors [18] involves coupling Morse-Ingard

to the tuning fork vibration, which in turn requires modeling the ﬂuid ﬂow.

Domain truncation will still be required, but coupling of pressure and tem-

perature to the ﬂuid and tuning fork may limit the utility of the decoupled

system. Additionally, as noted in [17], solving for the acoustic mode while

neglecting the thermal mode turns out to be an eﬀective approximation.

After developing the boundary conditions in Section 3, we derive a ﬁnite el-

ement formulation for the Morse-Ingard system in Section 4. We discuss the

structure of the linear system and approaches to preconditioning in Section 5

and we then provide some numerical in Section 6 before oﬀering some ﬁnal

conclusions in Section 7.

2. The Morse-Ingard equations

The Morse-Ingard equations of thermoacoustics are a system of partial

diﬀerential equations for the temperature and pressure of an excited gas.

The model begins from a time-domain formulation. After assuming time-

periodic forcing and performing nondimensionalization and some algebraic

manipulations, we arrive at the form given in [13] and further analyzed in [14]:

−M∆T−iT +iγ−1

γP=−S,

γ1−Λ

MT−(1 −iγΛ) ∆P−γ1−Λ

M+Λ

MP=iγ Λ

MS. (1)

Here, Tand Pare the non-dimensional temperature and pressure, re-

spectively, within the gas. Sis a volumetric forcing function, modeling for

example a laser pulse. γis the ratio of speciﬁc heat of the gas at constant

pressure to that at constant volume. The dimensionless number Mmeasures

the ratio of the product of the characteristic thermal conduction scale and

forcing frequency to sound speed, and Λ does similarly for the viscous length

scale. Typical values of parameters

γ= 7/5

M= 3.664152973215096 ·10−5

Λ=5.370572762330994 ·10−5

(2)

are taken as in [13, 17].

We let Ωc⊂Rd(with d= 2,3) be a bounded domain representing the

tuning fork, and let its boundary be called Γ. The complement of Ωcwill

be the domain Ω on which we pose (1). On Γ, we impose homogeneous

Neumann boundary conditions,

∂T

∂n = 0,∂P

∂n = 0.(3)

which posits that the tuning fork is thermally insulated from the gas, and

that the tuning fork is sound-hard. More advanced models, in which the gas

heats the tuning fork or the acoustic waves couple to tuning fork deformation,

generalize this condition [15].

A suitable far-ﬁeld condition is required to close the model, which re-

quires some appropriate decay at inﬁnity akin to the Sommerfeld radiation

condition for the Helmholtz operator. Numerical methods based on volu-

metric discretization on a truncated domain have posed either some kind of

transmission-type condition [12, 14] or perfectly-matched layers [15].

In [17], we gave a boundary integral method for Morse-Ingard based on a

scattered-ﬁeld formulation, which turns the volumetric inhomogeneity into an

inhomogeneous Neumann condition on Γ. We discuss this in greater detail in

Section 3.2. To arrive at the scattered-ﬁeld formulation, we split the solution

into incoming and scattered waves via

T=Ti+Ts, P =Pi+Ps,(4)

where Tiand Pisatisfy (1) with the given forcing function Sbut have some

inhomogeneous boundary conditions on Γ. Then, Tsand Psare chosen to

satisfy (1) with homogeneous forcing S= 0 and such that the combined

waves satisfy (3). The incoming waves Tiand Pican be constructed by

volumetric convolution of Swith a free-space Green’s function. With these

in hand, their normal derivatives on Γ can be computed, and the negative of

these used as boundary conditions for Tsand Ps. Consequently, we drop the

superscripts ‘s’ for the scattered ﬁeld and, for the rest of the paper, consider

the system of PDE

−M∆T−iT +iγ−1

γP= 0,

γ1−Λ

MT−(1 −iγΛ) ∆P−γ1−Λ

M+Λ

MP= 0,(5)

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ExactdomaintruncationfortheMorse-Ingardequations?RobertC.Kirbya,,XiaoyuWeib,AndreasKlocknerbaDepartmentofMathematics,BaylorUniversity;SidRichardsonScienceBuilding;1410S.4thSt.;Waco,TX76706;UnitedStatesbDepartmentofComputerScience,UniversityofIllinoisatUrbana-Champaign,Champaign,Illinois,UnitedStat...

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