Integral Equation Methods for the Morse-Ingard Equations

2025-05-05 0 0 1.27MB 22 页 10玖币

侵权投诉

Xiaoyu Weia, Andreas Klöcknera,∗, Robert C. Kirbyb

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

Champaign, Illinois, United States

bDepartment of Mathematics, Baylor University, Waco, Texas, United States

Abstract

We present two (a decoupled and a coupled) integral-equation-based methods

for the Morse-Ingard equations subject to Neumann boundary conditions on

the exterior domain. Both methods are based on second-kind integral equation

(SKIE) formulations. The coupled method is well-conditioned and can achieve

high accuracy. The decoupled method has lower computational cost and more

ﬂexibility in dealing with the boundary layer; however, it is prone to the ill-

conditioning of the decoupling transform and cannot achieve as high accuracy

as the coupled method. We show numerical examples using a Nyström method

based on quadrature-by-expansion (QBX) with fast-multipole acceleration.

We demonstrate the accuracy and eﬃciency of the solvers in both two and

three dimensions with complex geometry.

Keywords: The Morse-Ingard Equations, Fast Multipole Method, Integral

Equation Method, Quadrature-by-Expansion

1. Introduction

The Morse-Ingard equations are time-harmonic, steady-state equations

derived from the linearized Navier-Stokes equations [

]. They model the

pressure and the temperature variations of a ﬂuid due to a heat source inside

the ﬂuid, in a regime where both the acoustic wavelength and the thermal

boundary layer thickness are of interest. Speciﬁcally, we consider the following

nondimensionalized Morse-Ingard equations from [

]: in the ﬂuid domain

∗Corresponding author

Email addresses: xywei@illinois.edu (Xiaoyu Wei), andreask@illinois.edu

(Andreas Klöckner), Robert_Kirby@baylor.edu (Robert C. Kirby)

Preprint submitted to Elsevier April 24, 2023

arXiv:2210.12542v2 [math.NA] 21 Apr 2023

with D⊂Rdbounded,











Ω∇2T+iT −iγ−1

γP=S,

(1 −iγΛ)∇2P+γ1−Λ

Ω+Λ

ΩP−γ1−Λ

ΩT=−iγ Λ

ΩS,

(1)

where

is the pressure ﬁeld,

is the temperature ﬁeld,

is the heat source,

and Ω,Λ, γ are dimensionless parameters.

The application that motivated this work is to model trace gas sensors

that utilize optothermal and photoacoustic and eﬀects to aid designing such

sensors. The

uartz-

nhanced

hoto

coustic

pectroscopy (QEPAS) sensor,

for example, employs a quartz tuning fork to detect via the piezoelectric eﬀect

the acoustic pressure waves that are generated when optical radiation from a

laser is periodically absorbed by molecules of a trace gas [

]. The

esonant

pto

hermo

coustic

tection (ROTADE) sensor, on the other hand, uses

the same tuning fork to detect the thermal diﬀusion wave via the indirect

pyroelectric eﬀect [

]. An eﬃcient and accurate solver for the Morse-Ingard

equations on the exterior domain can be a key tool for the modeling and

design of both QEPAS and ROTADE sensors [4, 14, 17].

To suit this application, we consider the thermoacoustic scattering

problem, where the scattered waves obey the homogeneous Morse-Ingard

equations on the exterior domain (

= 0), coupled with sound-hard boundary

conditions in pressure and continuity in heat ﬂux, leading to Neumann

boundary conditions in both Tand P,











∂T

∂n ∂D

=gT,

∂P

∂n ∂D

=gP,

(2)

where the volumetric source

(1)

is accounted for by the incoming waves

it induces in the boundary conditions above.

For solution uniqueness, we also require

T(x)<∞and P(x)<∞,when |x|→∞.(3)

As shown in [

(3)

ensures solution uniqueness by ruling out unphysical

waves from the inﬁnity, similar to what the Sommerfeld radiation condition

does for the Helmholtz equation. In our applications, the wave number

has positive imaginary part. Therefore, boundedness at inﬁnity is suﬃcient.

For general wave numbers, we also provide a direct generalization of the

Sommerfeld condition in Section 2.3.

In this paper, we propose two integral-equation-based methods for (1)

to (3). For both methods, we derive representations of the solution in terms

of layer potentials involving unknown densities, leading to integral equations

that can be reduced to the form of a Fredholm integral equation of the second

kind

(I−A)ρ=f, (4)

where

is a compact integral operator; therefore, our methods are all based

on second-kind integral equation (SKIE) formulations. SKIEs are attractive

because both the condition number and the number of iterations required for

iterative solvers like GMRES [

] are bounded by constant when reﬁning the

mesh. In fact, [

] gives rigorous superlineary GMRES convergence estimates

in this case. The ﬁrst method is based on a direct SKIE formulation to the

original equations using the free-space Green’s functions. While deriving

the analytic formula for the free-space Green’s functions, we also obtain a

decoupling transform that converts the problem into two decoupled Helmholtz

equations, with decoupled Neumann data. Our second method takes advantage

of this transform and uses SKIEs for the Helmholtz equations to obtain the

solution. As such we refer to the ﬁrst method the coupled method and the

second the decoupled method. We solve the SKIEs using a Nyström method

with GMRES, which requires evaluating layer potentials at the boundary

using a suitable singular quadrature scheme. For our numerical experiments,

we use quadrature-by-expansion (QBX) with fast-multipole acceleration

[

]. Consequently, both methods have linear complexity with respect to

the number of degrees of freedoms. It is noteworthy that our methods are

agnostic of the singular quadrature scheme. Our methods are also capable

of using high order discretization and can handle complex geometries. We

demonstrate this through numerical examples in two and three dimensions.

The rest of this paper is organized as follows. We ﬁrst present the deriva-

tion for the free-space Green’s functions and the decoupling transform in

Section 2. Then we give the SKIE formulations for both the decoupled and

coupled methods in Section 3, and present some details of our numerical

implementation in Section 4. After that, we present the numerical results in

Section 5, and give some concluding discussion in Section 6.

2. Analysis of the problem

2.1. Thermal and acoustic modes

In order to obtain an integral equation formulation, we require the free-

space Green’s function for the Morse-Ingard equations (1). Following the

derivation for the analytic solution to the Morse-Ingard equations in a cylin-

drically symmetric geometry in [

], we ﬁrst identify the eigenmodes by looking

for particular solutions where

is an eigenfunction of

∇2

, s.t.

∇2T

−k2T

Substituting into the ﬁrst equation of (1) yields

P=γ

γ−1[(1 + iΩk2)T+iS].(5)

Consider the homogeneous case by letting

= 0, then

is also an

eigenfunction of

∇2

, where the constant

γ−1

(1 +

Ω

). Substituting

(5) and P=mT into (1) yields

(1 −iγΛ)(−k2mT ) + γ1−Λ

Ω+Λ

Ω(mT )−γ1−Λ

ΩT= 0.(6)

For (6) to have nontrivial solution, the coeﬃcient of Tmust vanish, so that

(iΩ + γΩΛ)k4+ (1 −iγΩ−iΛ)k2−1=0.(7)

Let

to be a complex constant such that

= 4(

Ω+

ΩΛ)+(1

−iγ

Ω

−i

Λ)

Based on physical interpretation, we classify the roots of

(7)

into two groups:

1. ktcorresponding to the thermal modes that attenuate rapidly:

t=i

2Ω 1−iγΩ−iΛ + Q

1−iγΛ, mt:= γ

γ−1(1 + iΩk2

t).(8)

2. kpcorresponding to the acoustic modes that attenuate slowly:

p=i

2Ω 1−iγΩ−iΛ−Q

1−iγΛ, mp:= γ

γ−1(1 + iΩk2

p).(9)

Since the eigenfunctions of

∇2

form a basis of

, we obtain the funda-

mental set of solutions to the homogeneous problem under radial symmetry,

denoted Ud,(d= 2,3),

T(r)

P(r)∈Ud.

In two dimensions (d= 2),

U2= span (J0(kpr)

mpJ0(kpr),"H(1)

0(kpr)

mpH(1)

0(kpr)#,J0(ktr)

mtJ0(ktr),"H(1)

0(ktr)

mtH(1)

0(ktr)#),

(10)

where

is the Bessel function of the ﬁrst kind of order zero,

H(1)

is the

Hankel function of the kind of order zero. Similarly, in three dimensions

(d= 3),

U3= span (j0(kpr)

mpj0(kpr),"h(1)

0(kpr)

mph(1)

0(kpr)#,j0(ktr)

mtj0(ktr),"h(1)

0(ktr)

mth(1)

0(ktr)#),

(11)

where

is the spherical Bessel function of the ﬁrst kind of order zero,

h(1)

the spherical Bessel function of the third kind of order zero. We also note

that h(1)

0has simple closed form

h(1)

0(r) = j0(r) + iy0(r) = sin r

r−icos r

r=−i

reir.(12)

Note that the choice of basis is not unique. We deliberately chose the above

basis functions so that the condition in (3) can be easily enforced.

2.2. The decoupled equations

From the above radial symmetry solutions, it is obvious that the Morse-

Ingard equations are a linear superposition of two Helmholtz-type equations.

To get the actual change of variables that decouples the PDE system, we

solve for

t∈C

such that the sum of the ﬁrst equation and

times the second

equation in (1) reduces to a scalar Helmholtz-type PDE

a1(t)∇2T+a2(t)∇2P+a3(t)T+a4(t)P=a5(t)S, (13)

where

= Ω,

= (1

−iγ

Λ)

i−γ1−Λ

Ωt

−iγ−1

γ1−Λ

Ω+Λ

Ωt

= 1

−iγ Λ

Ωt

are all linear functions of

. The condi-

tion under which

(13)

becomes a scalar Helmholtz-type PDE is

a1a4

a2a3

which is a quadratic equation of tand admits two roots

t±=(2Λγ−Λ−Ωγ+i)Ω ∓iΩQ

2γ(Λ −Ω)(iΛγ−1) .(14)

Letting

= Ω

−iγ

Λ)

= Ω

t−

−iγ

Λ)

, we ﬁnd the

decoupled scalar PDEs,

∇2Vt+k2

tVt=a5(t+)S,

∇2Vp+k2

pVp=a5(t−)S, (15)

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

IntegralEquationMethodsfortheMorse-IngardEquationsXiaoyuWeia,AndreasKlöcknera,,RobertC.KirbybaDepartmentofComputerScience,UniversityofIllinoisatUrbana-Champaign,Champaign,Illinois,UnitedStatesbDepartmentofMathematics,BaylorUniversity,Waco,Texas,UnitedStatesAbstractWepresenttwo(adecoupledandacoupled...

展开>> 收起<<

Integral Equation Methods for the Morse-Ingard Equations.pdf

共22页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Integral Equation Methods for the Morse-Ingard Equations

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: