Spectral element method for elastic and acoustic

Journal of Computational Physics 327 (2016) 19–38

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Journal of Computational Physics www.elsevier.com/locate/jcp

Spectral element method for elastic and acoustic waves in frequency domain Linlin Shi a , Yuanguo Zhou a , Jia-Min Wang a , Mingwei Zhuang a , Na Liu a,∗ , Qing Huo Liu b,∗ a b

Institute of Electromagnetics and Acoustics, and Department of Electronic Science, Xiamen, 361005, PR China Department of Electrical and Computer Engineering, Duke University, Durham, NC, 27708, USA

a r t i c l e

i n f o

Article history: Received 23 November 2015 Received in revised form 1 September 2016 Accepted 2 September 2016 Available online 21 September 2016 Keywords: Elastic (acoustic) waves Spectral element method (SEM) Perfectly matched layer (PML) Domain decomposition method (DDM) Frequency domain Numerical simulations

a b s t r a c t Numerical techniques in time domain are widespread in seismic and acoustic modeling. In some applications, however, frequency-domain techniques can be advantageous over the time-domain approach when narrow band results are desired, especially if multiple sources can be handled more conveniently in the frequency domain. Moreover, the medium attenuation effects can be more accurately and conveniently modeled in the frequency domain. In this paper, we present a spectral-element method (SEM) in frequency domain to simulate elastic and acoustic waves in anisotropic, heterogeneous, and lossy media. The SEM is based upon the finite-element framework and has exponential convergence because of the use of GLL basis functions. The anisotropic perfectly matched layer is employed to truncate the boundary for unbounded problems. Compared with the conventional finiteelement method, the number of unknowns in the SEM is significantly reduced, and higher order accuracy is obtained due to its spectral accuracy. To account for the acousticsolid interaction, the domain decomposition method (DDM) based upon the discontinuous Galerkin spectral-element method is proposed. Numerical experiments show the proposed method can be an efficient alternative for accurate calculation of elastic and acoustic waves in frequency domain. © 2016 Elsevier Inc. All rights reserved.

1. Introduction Efficient and accurate computation of numerical solutions of elastic (acoustic) wave fields in large-scale heterogeneous media remains an important challenge in computational geophysics. A large collection of numerical methods has been proposed for this purpose. Apart from the integral equation approaches (for example, [6]), there are two main types of partial differential equation based methods: One is based on the strong formulation and the other is based on weak formulation of elastodynamic equations. The finite difference (FD) approach, which belongs to the first type, is one of the most common techniques for the calculation of elastic (acoustic) wave propagation in geophysics [15,42,25,9,14]. The FD approach is simple to implement and very robust, but its main drawback is the requirement of a structured (orthogonal) grid, which makes this technique less flexible, though various techniques have been proposed to overcome this limitation. Besides, significant difficulties arise when anisotropic media and the curved free-surface boundary condition need to be incorporated

*

Corresponding authors. E-mail addresses: [email protected] (N. Liu), [email protected] (Q.H. Liu).

http://dx.doi.org/10.1016/j.jcp.2016.09.036 0021-9991/© 2016 Elsevier Inc. All rights reserved.

20

L. Shi et al. / Journal of Computational Physics 327 (2016) 19–38

[33,12,39,36,45]. The finite element method (FEM), which belongs to the second type, has become another popular numerical technique to model wave propagation problems, due to its superior versatility in geometry and material modeling [10, 32,13,11]. Despite the advantages of FEM, the solutions converge relatively slowly because of its low order, which leads to excessive unknowns and a high computational cost to solve the resulting large linear systems. The spectral element method, which was first proposed for computational fluid dynamics by Patera [35], can be considered as a special high-order FEM. Based upon GLL basis functions, the spectral element method demonstrates a significantly reduced spatial sampling density and the high degree of accuracy with a smaller number of unknowns. Over the past 15 years, the spectral-element method in time domain has been of great interest to many researchers because of its high accuracy and efficiency in a number of fields such as geoacoustics and seismology [38,16,18,19,41,40], and electromagnetism [21,22,50,28,24]. In terms of seismic wave simulations, the time domain SEM has been proved to be considerably effective and of great potential by previous researchers (e.g. Komatitsch & Vilotte [16]; Komatitsch & Tromp [18]; Tromp et al. [41]), and the reader is referred to Tromp et al. [41] for a more detailed review. In this article, the frequency-domain version of SEM is investigated for elastic and acoustic wave simulations. In some applications, frequency-domain techniques can be advantageous over the time-domain approach when narrow band results are desired. For example, in wide-aperture crosshole seismic surveys, useful images can be formed from only a limited number of frequency components, thus modeling in the frequency domain can be of higher efficiency [30]. In addition, another key advantage of the frequency-domain approach is its efficiency in modeling multiple sources. The time-consuming factorization step of LU decomposition in the multiple source modeling only needs to be performed once, thus the computation time for the subsequent sources will be greatly reduced. Other advantages of the frequency-domain approach include accurate inclusion of attenuation effects, no stability limitations, etc. The proposed frequency domain spectral element method is based upon the finite-element framework, and has exponential convergence with the order of basis functions because of the use of GLL basis functions. This spectral accuracy refers to the property that the SEM error decreases exponentially with the order of basis functions. Compared with the conventional finite-element method (FEM), the number of unknowns in the SEM is significantly reduced and higher order accuracy is obtained due to the exponential convergence. Unlike the time domain SEM, however, we need to solve a large scale sparse linear system of equations instead of the ordinary differential equations (ODEs) after the spatial discretization. For this purpose, one can choose a direct (e.g. UMFPACK; MUMPS; PARDISO; GSS) or iterative (e.g. GMRES; Bi-CG; CGS; QMR) sparse matrix linear-equation solver. When tackling frequency-domain full-waveform modeling and inversion problems, direct solvers has obvious advantages because of its efficiency of multiple sources modeling as well as precision of the solutions. The main drawback of the direct approach is the huge memory requirements, which is used to store the LU factors when performing LU decomposition. There has been lot of research devoted to further investigate this problem of memory consumption (especially in 3D wave propagation) over the last decades, as already mentioned in Pratt [37], Operto et al. [34], Wang et al. [44] and so forth. In this work, the governing equations for elastic and acoustic waves are the second-order seismic wave equation with as field variable and the second-order scalar acoustic wave equation with pressure p as field variable, redisplacement u spectively. To deal with the open boundary problems, absorbing boundary conditions (ABCs) are needed to truncate an unbounded domain so that the outgoing waves can be absorbed at the artificial outer computation boundary. The perfectly matched layer (PML), which was first proposed by Berenger [1] for electromagnetic (EM) simulations, has become the most popular ABCs for its remarkable absorption performance. Following Berenger’s work, Chew and Liu extend PMLs to elastic waves application in 1996 using the complex stretched coordinates. Over the last two decades, the PML technique has been greatly enhanced by many researchers for both acoustic and elastic wave propagation simulations [26,27,20,46,47,8,31,48]. However, the conventional field splitting scheme PML is difficult to implement for the FEM technique, and in fact is not necessary in the frequency domain [5,49]. For the frequency-domain SEM technique, we use the anisotropic perfectly matched layer (PML). Its effectiveness is verified based upon several numerical experiments in this study. In order to solve the fluid– solid interaction systems, we develop a domain decomposition technique (DDM) based upon the discontinuous Galerkin spectral-element (DG-SEM) method. In this scheme, one can use spectral elements of different interpolation degrees in the fluid region and solid region to maximize the efficiency. Both conforming and non-conforming meshes can be employed across the interfaces between fluid (acoustic) and solid (elastic) regions. In the implementation, different regions are solved independently, while the adjacent regions are coupled through the fluid–solid interface boundary conditions to guarantee the unique solution. The organization of this article is as follows. In section 2, we will present the detailed formulations of the proposed SEM. Section 3 gives some numerical results to show the capabilities of the method. Finally, the conclusion is given in section 4. 2. Formulation 2.1. Weak form of the frequency-domain SEM In this section, we present a simple, intuitive discretization formulations of momentum equation weak form, which can be straightforward to develop and code for frequency-domain SEM. For a general anisotropic elastic medium, the second and stress tensor τ in frequency domain are order partial differential equations (PDE) for particle displacement vector u given by


21

−ρω2 u − ∇ · τ = f

(1)

τ =C :ε 1

ε = (∇ u + ∇ u T )

(2)

2

ρ is the mass density, and C is the fourth order elasticity tensor whose elements can be written as c i jkl (i , j , k, l = , ε and τ are the particle displacement, strain and stress tensors; the other two variables ω and f denote the 1, 2, 3); u where

angular frequency and force vector (e.g. a point load source or a moment tensor source), respectively. It is easy to incorporate the linear viscoelasticity and consequently attenuation in frequency domain: by the correspondence principle, real frequency-independent moduli are replaced by complex, frequency-dependent quantities. One solves the above equations with appropriate outer boundary conditions. To derive the weak form, within the Galerkin procedure, we test the momentum equation (1) with testing functions in the volume V , which gives

φm u dv −

−ρω2 V

φm (∇ · τ )dv = V

φm f dv

(m = 1, 2, ..., N )

(3)

V

During the testing, we choose the basis function as the testing function, as in the regular Galerkin procedure. In addition, the Gauss (divergence) theorem in the second-order tensor form can be written as

· τ ds n

∇ ·τ = V

(4)

S

is the outward normal unit vector and S is the boundary of volume of V . Substituting equation (4) into (3), we can Here, n obtain the weak form in vector form

−ρω

φm u dv +

2 V

∇φm · τ dv − V

φm ( n · τ )ds = S

φm f dv

(5)

V

In the above equation, we can rewrite the integral terms in the following form

φm u dv =

−ρω2 V

∇φm · τ dv =

3

φm ( n · τ )ds =

V

(6)

V

3

V

V

el {

l =1

S

φm ul dv )

el [ ∇φm · (C l1 ∇ u 1 )dv + ∇φm · (C l2 ∇ u 2 )dv + ∇φm · (C l3 ∇ u 3 )dv ]

l =1

V

φm f dv =

el (−ρω2

l =1

3

φm [ n · (C l1 ∇ u 1 )]ds +

S

φm [ n · (C l2 ∇ u 2 )]ds +

S

φm Fp e j φ δ(r − r s )dv =

3

(7)

V

φm [ n · (C l3 ∇ u 3 )]ds}

(8)

S

(l)

el [ F p e j φ φm (r s )]

(9)

l =1

V

where the point load source is used and is located at r s = (xs , y s , z s ); C i j (i , j = 1, 2, 3) are 3 × 3 matrices whose elements come from the fourth order elasticity tensor, which can be expressed as

⎛

⎞

⎛

⎞

⎛

⎞

c 1111 C 11 = ⎝ c 2111 c 3111

c 1112 c 2112 c 3112

c 1113 c 2113 ⎠ , c 3113

c 1121 C 12 = ⎝ c 2121 c 3121

c 1122 c 2122 c 3122

c 1123 c 2123 ⎠ , c 3123

c 1131 C 13 = ⎝ c 2131 c 3131

c 1132 c 2132 c 3132

c 1133 c 2133 ⎠ c 3133

c 1211 C 21 = ⎝ c 2211 c 3211

c 1212 c 2212 c 3212

c 1213 c 2213 ⎠ , c 3213

c 1221 C 22 = ⎝ c 2221 c 3221

c 1222 c 2222 c 3222

c 1223 c 2223 ⎠ , c 3223

c 1231 C 23 = ⎝ c 2231 c 3231

c 1232 c 2232 c 3232

c 1233 c 2233 ⎠ c 3233

c 1311 C 31 = ⎝ c 2311 c 3311

c 1312 c 2312 c 3312

c 1313 c 2313 ⎠ , c 3313

c 1321 C 32 = ⎝ c 2321 c 3321

c 1322 c 2322 c 3322

c 1323 c 2323 ⎠ , c 3323

c 1331 C 33 = ⎝ c 2331 c 3331

c 1332 c 2332 c 3332

c 1333 c 2333 ⎠ c 3333

⎛

⎛

⎞

⎞

⎛

⎛

⎞

⎞

⎛

⎛

⎞

⎞ (10)

Substituting (6), (7), (8), and (9) into (5), we can obtain the new form of weak formulation of the problems described as

22


3

∇φm · (C 1i ∇ u i )dv −

i =1 V

3

φm · [ n · (C 1i ∇ u i )]ds − ρω

2

i =1 S

∇φm · (C 2i ∇ u i )dv −

i =1 V

3

3

3

∇φm · (C 3i ∇ u i )dv −

i =1 V

(11)

φm u 2 dv = F p(2) e j φ φm (r s )

(12)

φm u 3 dv = F p(3) e j φ φm (r s )

(13)

V

φm · [ n · (C 2i ∇ u i )]ds − ρω2

i =1 S

3

(1 )

φm u 1 dv = F p e j φ φm (r s )

V

φm · [ n · (C 3i ∇ u i )]ds − ρω2

i =1 S

V

· τ = 0) or the fixed Note that the surface integral terms will vanish when either the free surface boundary condition (n = 0) is imposed. In order to discretize the weak form above with the SEM approach, constraint boundary condition (u we expand the three components of the displacement vector by using the node-based Gauss–Lobatto–Legendre (GLL) basis functions to arrive at u1 =

N

(i )

u 1 φi (x, y , z)

u2 =

N

i =1

(i )

u3 =

u 2 φi (x, y , z)

i =1

N

(i )

u 3 φi (x, y , z)

(14)

i =1

where {uli } are the expansion coefficients of the components of displacement vector. As we can see, the basis functions we use in (14) are the same as the testing functions in the weak form, which is a requirement for Galerkin procedure. Finally, taking (14) into (11), (12), and (13), we obtain a discretized system of SEM equation in a matrix form given as

M 1 u˜ 1 + K 11 u˜ 1 + K 12 u˜ 2 + K 13 u˜ 3 − P 11 u˜ 1 − P 12 u˜ 2 − P 13 u˜ 3 = F 1

(15)


(16)


(17)

The above equations can be written more compactly to give a 3N × 3N matrix equation

⎛

M 1 + K 11 − P 11 ⎝ K 21 − P 21 K 31 − P 31

K 12 − P 12 M 2 + K 22 − P 22 K 32 − P 32

⎞

K 13 − P 13 ⎠u˜ = F˜ K 23 − P 23 M 3 + K 33 − P 33

(18)

where u˜ = (u˜ 1 , u˜ 2 , u˜ 3 ) T denotes the unknowns of displacement vectors after spatial discretization, M i (i = 1, 2, 3) are the mass matrices, K i j (i , j = 1, 2, 3) are the stiffness matrices, P i j (i , j = 1, 2, 3) are boundary integral matrices and F˜ = ( F 1 , F 2 , F 3 ) T are the forcing vector. Due to the symmetry of the elasticity tensor, the system matrix for the whole SEM computational domain is a symmetric matrix. In the above, the elements of each matrix and vector are as follows:

( M 1 )kl = ( M 2 )kl = ( M 3 )kl = −ρω

φk φl dv

(k, l = 1, 2, ..., N )

V

( K i j )kl =

2

(∇φk ) · C i j ∇φl dv

(i , j = 1, 2, 3; k, l = 1, 2, ..., N )

φk ( n · C i j ∇φl )ds

(i , j = 1, 2, 3; k, l = 1, 2, ..., N )

V

( P i j )kl = S

( F i )k = F p(i ) e j φ φk (r s )

(i = 1, 2, 3; k = 1, 2, ..., N )

(19)

At the element level, for the calculation of these matrices, we need to map each hexahedral element in the physical domain into the standard reference cube element [−1, 1] × [−1, 1] × [−1, 1]. In this standard reference element, we expand the displacement fields by the Gauss–Lobatto–Legendre (GLL) basis functions. It is known that GLL polynomials can be used for interpolating a smooth function with spectral accuracy, thus the numerical error can be decreased exponentially by increasing the interpolation orders. In the reference cube, the pth-order 1D GLL basis functions are defined by ( p)

φi (ξ ) =

(1 − ξ 2 ) L p (ξ ) −1 p ( p + 1) L p (ξi ) (ξ − ξi )

where ξ ∈ [−1, 1], i = 0, 1, ..., p; L p (ξ ) and the p + 1 nodal points, which correspond to (1 − ξi2 ) L p (ξ ) = 0. For 3D problem, the basis basis functions, which reads

(20)

L p (ξ ) are the pth-order Legendre polynomial and its derivative, respectively; p + 1 one-dimensional GLL basis functions, are chosen as the roots of equation function in a reference element can be given as the tensor product of three 1D


( p) ( p) ( p) ( p) ( p) φˆ j = φˆrst (ξ, η, ζ ) = φr (ξ )φs (η)φt (ζ )

23

(21)

where r , s, t = 1, 2, ..., p + 1 and j is the compound index of (r , s, t ). Obviously, there are a total of ( p + 1)3 GLL basis functions in one reference element. Note that the gradient of the basis functions between physical element and reference element are related by

ˆ η, ζ ) ∇φ(x, y , z) = J −1 ∇ˆ φ(ξ,

(22)

with the Jacobian matrix given as

⎛ ⎜

J =⎝

∂y ∂ξ ∂y ∂η ∂y ∂ζ

∂x ∂ξ ∂x ∂η ∂x ∂ζ

∂z ∂ξ ∂z ∂η ∂z ∂ζ

⎞ ⎟ ⎠

(23)

ˆ φˆ denotes the gradient of the GLL basis function φ(ξ, ˆ η, ζ ) in the reference (ξ, η, ζ ) coordinates. Thus the elemental where ∇ matrices for the reference domain are (e)

(e)

(e)

1 1 1

( M 1 )kl = ( M 2 )kl = ( M 3 )kl = −ρω

2

|det ( J )|φˆk φˆl dξ dηdζ

−1 −1 −1 (e) = ( K i j )kl

1 1 1

|det ( J )| · ( J −1 ∇ˆ φˆk ) · (C i j J −1 ∇ˆ φˆl )dξ dηdζ

−1 −1 −1 (e)

1 1

( P i j )kl =

ˆ l (a, η, ζ )) Adηdζ n · C i j ∇φ φˆk (a, η, ζ ) · (

(i , j = 1, 2, 3; k, l = 1, 2, ..., N (e) )

(24)

−1 −1

Here, i , j denote the indices of the elemental stiffness or surface integral matrices, and N (e) is the number of node degrees of freedom (Node-DOFs) in an element. Note that, for the surface integral above, we have assumed that the element surface is mapped into the reference cube surface ξ = ±1, thus we have a = ±1, and the area element is given as

A = |dη × dζ | = [(

∂ y ∂z ∂ y ∂z 2 ∂z ∂x ∂x ∂ y ∂z ∂x 2 ∂x ∂ y 2 1 ) +( ) +( ) ] 2 d ηd ζ − − − ∂ η ∂ζ ∂ζ ∂ η ∂ η ∂ζ ∂ζ ∂ η ∂ η ∂ζ ∂ζ ∂ η

and similarly for other cases. Note that, in order to evaluate the volume or surface integral in (24), a numerical quadrature can be used. In this paper, the Gauss–Lobatto–Legendre (GLL) points and the corresponding weights are employed for this quadrature. 2.2. Anisotropic perfectly matched layer implementation The goal for this part is to extend an anisotropic perfectly matched layer (PML) to SEM simulation for elastic (acoustic) waves. Note that in the frequency domain, the anisotropic PML follows naturally from the original formulation of Chew and Liu [5] without the need for splitting of fields because the frequency dependence can be easily included in the material properties; in the time domain, however, non-split PML becomes more cumbersome, as in [47] and in [49]. The formulation of anisotropic PML is different from the PML using field splitting. For this type of PML, we need that the waves in the PML are still governed by the momentum equation and Hooke’s law, thus, the PML can be considered as an artificial anisotropic absorptive material. To derive the PML formulation, we first rewrite the momentum equation (1) in the form of

∂ τ11 ∂ τ21 ∂ τ31 + + + f1 ∂ x1 ∂ x2 ∂ x3 ∂ τ12 ∂ τ22 ∂ τ32 −ρω2 u 2 = + + + f2 ∂ x1 ∂ x2 ∂ x3 ∂ τ13 ∂ τ23 ∂ τ33 −ρω2 u 3 = + + + f3 ∂ x1 ∂ x2 ∂ x3 −ρω2 u 1 =

(25)

Following the complex coordinate stretching technique [5], we then introduce complex coordinate-stretching variables

∂ 1 ∂ ⇒ ∂η sη ∂ η sη = aη − j ωη /ω

ωη = K max ωc |η − ηb | p /d pP M L

( p = 0, 1 , 2 , 3 )

(26)

24


where aη ≥ 1 and ωη ≥ 0 (η = x1 , x2 , x3 ) are called the scaling and attenuation factors of the PML, respectively; and ηb is the location of the PML interface to the regular physical domain, ωc is the central frequency and d P M L is the width of PML. The parameter p range from zero to three, especially two is commonly used. For the SEM scheme, we can also use a constant PML profile by setting p = 0 to simplify the implementation since sη for PML will be constant within each element. It is worth noting that, the maximum value of K , i.e., K max can be obtained from a theoretical reflection coefficient relation [3]:

K max = −

( p + 1)c p ln( R )

(27)

2d P M L ωc

Here, c p is the P-wave velocity, R is the theoretical reflection coefficient for a normal-incident plane P-wave with a Dirichlet boundary condition (or called fixed constraint boundary condition), i.e., v = 0 and τ = 0 at the outer boundary of PML layer. Taking the complex coordinate stretching (26) into (25) yields

∂(sx2 sx3 τ11 ) ∂(sx1 sx3 τ21 ) ∂(sx1 sx2 τ31 ) + + + sx1 sx2 sx3 f 1 ∂ x1 ∂ x2 ∂ x3 ∂(sx2 sx3 τ12 ) ∂(sx1 sx3 τ22 ) ∂(sx1 sx2 τ32 ) −ρ sx1 sx2 sx3 ω2 u 2 = + + + sx1 sx2 sx3 f 2 ∂ x1 ∂ x2 ∂ x3 ∂(sx2 sx3 τ13 ) ∂(sx1 sx3 τ23 ) ∂(sx1 sx2 τ33 ) −ρ sx1 sx2 sx3 ω2 u 3 = + + + sx1 sx2 sx3 f 3 ∂ x1 ∂ x2 ∂ x3 −ρ sx1 sx2 sx3 ω2 u 1 =

(28)

By defining the new stress tensor and the mass density in PML

τ˜i j =

sx1 sx2 sx3 s xi

( i , j = 1, 2, 3)

τi j

ρ˜ = ρ sx1 sx2 sx3

(29)

Then, Eq. (28) becomes

∂(τ˜11 ) ∂(τ˜21 ) ∂(τ˜31 ) ˜ + + + f1 ∂ x1 ∂ x2 ∂ x3 ∂(τ˜12 ) ∂(τ˜22 ) ∂(τ˜32 ) ˜ ˜ 2 u2 = −ρω + + + f2 ∂ x1 ∂ x2 ∂ x3 ∂(τ˜13 ) ∂(τ˜23 ) ∂(τ˜33 ) ˜ ˜ 2 u3 = −ρω + + + f3 ∂ x1 ∂ x2 ∂ x3 ˜ 2 u1 = −ρω

(30)

On the other hand, the Hooke’s law (2) can be rewritten as

τi j = c i jkl

∂ ul ∂ xk

(31)

In the above, the Einstein index notation has been used where the repeated indices imply summation. To facilitate the complex coordinate stretching implementation, we expand equation (31) as follows

τi j = c i j11

∂ u1 ∂ u2 ∂ u3 ∂ u1 ∂ u2 ∂ u3 ∂ u1 ∂ u2 ∂ u3 + c i j12 + c i j13 + c i j21 + c i j22 + c i j23 + c i j31 + c i j32 + c i j33 ∂ x1 ∂ x1 ∂ x1 ∂ x2 ∂ x2 ∂ x2 ∂ x3 ∂ x3 ∂ x3

(32)

Applying the complex coordinate stretching to above equation (32) and with the use of Eq. (29), we obtain

τ˜i j = c i j11

sx1 sx2 sx3 ∂ u 1 sxi sx1

+ c i j23

∂ x1

+ c i j12


∂ x2


+ c i j31

∂ x1

+ c i j13


∂ x3


+ c i j32

∂ x1

+ c i j21


∂ x3


+ c i j33

∂ x2

+ c i j22


∂ x2


∂ x3

(33)

Then, a new elasticity tensor in PML can be defined as

c˜ i jkl = c i jkl

sx1 sx2 sx3 sxi sxk

(34)

Inserting equation (34) into equation (33) gives

τ˜i j = c˜ i j11 = c˜ i jkl

∂ ul ∂ xk

∂ u1 ∂ u2 ∂ u3 ∂ u1 ∂ u2 ∂ u3 ∂ u1 ∂ u2 ∂ u3 + c˜ i j12 + c˜ i j13 + c˜ i j21 + c˜ i j22 + c˜ i j23 + c˜ i j31 + c˜ i j32 + c˜ i j33 ∂ x1 ∂ x1 ∂ x1 ∂ x2 ∂ x2 ∂ x2 ∂ x3 ∂ x3 ∂ x3 (35)


25

Now we can rewrite equations (30) and (35) in the tensor form, which gives the modified momentum equation and Hooke’s law in the anisotropic perfectly matched layers (PMLs)

˜ 2 u − ∇ · τ˜ = f −ρω τ˜ = C˜ : ∇ u

(36)

Note that in equation (36), the mass density ρ˜ and the elasticity tensor C˜ are material properties of the PML, which are given in Eq. (29) and (34), respectively. Obviously, in PMLs, the stress tensor is no longer symmetric. We find the governing equations in a PML medium have exactly the same form as the regular solid (elastic) medium, in spite of the modification of the material properties. Furthermore, when setting ax1 = ax2 = ax3 = 1 and ωx1 = ωx2 = ωx3 = 0, equation (36) will be degenerated to the standard partial differential equations (1) and (2) governing a regular elastic medium. In the SEM implementation, for the computational domain containing PML, the elemental matrices of the discretized system will take the form as

˜ 1 )kl = ( M ˜ 2 )kl = ( M ˜ 3 )kl = −ρω ˜ 2 (M ( K˜ i j )kl =

(k, l = 1, 2, ..., N )

φk φl dv V

(∇φk ) · C˜ i j ∇φl dv

(i , j = 1, 2, 3; k, l = 1, 2, ..., N )

φk ( n · C˜ i j ∇φl )ds

(i , j = 1, 2, 3; k, l = 1, 2, ..., N )

V

( P˜ i j )kl =

(37)

S

Similar to the elastic wave treatment, for acoustic waves in a fluid, here we omit the details and only list the anisotropic perfectly matched layer (PML) equation as following: For the regular (non-PML) region

−κ −1 ω2 p = ∇ · (ρ −1 ∇ p )

(38)

and for the PML region

−κ˜ −1 ω2 p = ∇ · (ρ˜ −1 ∇ p )

κ˜ = κ

1 sx1 sx2 sx3

⎛ ⎜

s x1 s x2 s x3

ρ˜ = ρ ⎜ ⎝ 0 0

0

0

s x2 s x1 s x3

0

0

s x3 s x1 s x2

⎞ ⎟ ⎟ ⎠

(39)

where we have used the second-order scalar acoustic wave equation with pressure p as the field variable; κ is the adiabatic bulk modulus of the fluid, and ρ is the mass density. The corresponding material parameters κ˜ and ρ˜ in the anisotropic perfectly matched layer (PML) are also given in above equation. In addition, we find that the fluid mass density in the PML is no longer a scalar but a second-order tensor. The SEM discretization for the Eq. (39) will be given in the next section. 2.3. DG-SEM technique for the Acoustic-Solid Interaction In many engineering applications, the calculation of the motion of an elastic solid interacting with a fluid becomes an important problem. In this part, we develop a domain decomposition method (DDM) based on the discontinuous Galerkin spectral-element method (DG-SEM) [2,23,4,29] for the fluid–solid coupling problems. By the DDM technique, the fluid–solid coupling systems are divided into several non-overlapping regions. For each region, one can mesh the model independently; in other words, different meshes and different interpolation degrees can be employed in different regions. In this scheme, each region is solved by the SEM independently, while the adjacent regions are coupled through the fluid–solid interface boundary conditions [18,17] to guarantee the unique solution. In general, the equations describing the motion of the coupled system can be given as following 1 ∇ · (ρ − ∇ χ ) + κ −1 ω 2 χ = 0 f

(40)

−ρs ω2 u − ∇ · τ = f

τ = C : ∇ u

(41) −1

where χ denotes the velocity potential satisfying v = ρ f ∇ χ , p = − j ωχ . For solving the coupled system, some appropriate outer boundary conditions should be considered. For example, if the outer boundary is a free boundary, then for a solid

26


· τ = 0, and we can impose the Dirichlet BC, i.e., p = 0, for region we can impose the free surface boundary condition, i.e., n the outer boundary of fluid regions. Furthermore, to couple the two media at the fluid–solid interface, we need to impose the continuity of traction and of the normal velocity (Komatitsch & Tromp [19]). By multiplying equation (40) by the test functions φm and integrating by parts over the fluid region V f , we obtain

κ −1 ω2 φm χ dv −

Vf

1 ∇φm · (ρ − ∇ χ )dv + f

Vf

1 φm (ρ − ∇ χ ) · nds = 0 f

(42)

fs

Here, we assume that Dirichlet BC is imposed on the fluid region’s outer boundary, thus the surface integral associated with the outer boundary vanishes. On the coupling fluid–solid interface, we replace the normal component of the velocity 1 · v f luid = n · (ρ − n ∇ χ ) in the fluid region with the normal component of the velocity n · v solid = n · ( j ωu ) in the solid region f to correct for the interactions. This gives

κ −1 ω2 φm χ dv −

Vf

1 ∇φm · (ρ − ∇ χ )dv + j ω f

Vf

φm (n1 u 1 + n2 u 2 + n3 u 3 )ds = 0

(43)

fs

where (u 1 , u 2 , u 3 ) and (n1 , n2 , n3 ) are the components of the displacement vector and the outward normal unit vector, respectively. For the solid region, we first rewrite the weak form (5) as

−ρs ω

φm u dv +

2 Vs

∇φm · τ dv − Vs

φm ( n · τ )ds =

φm f dv

(44)

Vs

fs

· τ in the In the above weak form, we will impose the continuity of traction. At this point, we replace the traction term n = j ωχ n in the fluid, where p is the fluid pressure. Thus we have solid with the traction − pn

φm u dv +

−ρs ω2 Vs

Vs

φm f dv

φm (χ n)ds =

∇φm · τ dv − j ω

(45)

Vs

fs

Then, we rewrite the above vector weak form into three scalar weak forms to read

−ρs ω2

φm u 1 dv +

3

Vs

i =1 V

3

−ρs ω2

φm u 2 dv + Vs

3

φm u 3 dv +

i =1 V

Vs

(1 )

(46)

(2 )

(47)

(3 )

(48)

n1 φm χ = F p e j φ φm (r s )

fs

s

i =1 V

−ρs ω2

∇φm · (C 1i ∇ u i )dv − j ω


∇φm · (C 2i ∇ u i )dv − j ω fs

s


∇φm · (C 3i ∇ u i )dv − j ω fs

s

Thus, equations (43), (46), (47), and (48) are the weak form for the whole fluid–solid interaction system. Based upon these weak formulations, the SEM procedure can be employed for the discretization of the fluid–solid coupling problems. Generally, assuming that the whole fluid–solid interaction system have N d regions, the above weak form can be modified as: For the ith fluid region

κ −1 ω2 φm(i) χ (i)dv −

Vi

(i ) 1 ∇φm · (ρ − ∇ χ (i ) )dv + f

Nd

j =1

jω

j =1

Vi

+

Nd

jω

Nd

(i ) ( j )

n2 φm u 2 ds +

jω

j =1

i j

(i ) ( j )

n1 φm u 1 ds

i j (i ) ( j )

n3 φm u 3 ds = 0

(49)

i j

and for the ith solid region

−ρs ω2

(i ) (i ) (i ) (i ) (i ) (i ) φm u 1 dv + [ ∇φm · (C 11 ∇ u 1 )dv + ∇φm · (C 12 ∇ u 2 )dv +

Vi

Vi

Vi

(i ) ∇φm · (C 13 ∇ u (3i ) )dv ] −

Vi Nd j =1

jω i j

(i )

n1 φm

χ ( j) ds = F p(1) e jφ φm(i) (r s )

(50)


−ρs ω2 Vi

Vi

Vi

Nd

(i )

(i )

∇φm · (C 23 ∇ u 3 )dv ] −

27


jω

j =1

Vi

(i )

n2 φm

χ ( j) ds = F p(2) e jφ φm(i) (r s )

(51)

i j


−ρs ω2 Vi

Vi

Vi

Nd

(i )

(i )

∇φm · (C 33 ∇ u 3 )dv ] −

jω

j =1

Vi

(i )

n3 φm

χ ( j) ds = F p(3) e jφ φm(i) (r s )

(52)

i j (i )

(i )

(i )

In the above, we have taken the ith region as the local region and assume it is adjacent to the jth region; u 1 , u 2 , u 3 and ( j)

( j)

( j)

u1 , u2 , u3

denote the components of the displacement vector of the ith and jth region, respectively; similarly, χ (i ) and

χ ( j) are the velocity potential; V i is the volume of ith region and i j denotes the interface of the ith and its adjacent jth (i )

region; φm are the testing functions, where i = 1, 2, ..., N (i ) . Within the SEM procedure, we expand the unknown fields (i ) , u ( j ) , χ (i ) and χ ( j) in terms of the node-based GLL basis functions associated with the ith and jth region, i.e., u Ni

χ (i ) =

χk(i) φk(i) (x, y , z)

k =1

χ ( j) =

Nj

χk( j) φk( j) (x, y , z)

k =1

(i ) = u

Ni

(i )

(i )

e1 u 1,k φk (x, y , z) +

Ni

k =1

k =1

Nj

Nj

( j) = u

( j)

( j)

e1 u 1,k φk (x, y , z) +

k =1

(i )

(i )

e2 u 2,k φk (x, y , z) +

Ni

(i )

(i )

e3 u 3,k φk (x, y , z)

k =1

( j)

( j)

e2 u 2,k φk (x, y , z) +

k =1

Nj

( j)

( j)

e3 u 3,k φk (x, y , z)

(53)

k =1

where N i and N j are the total numbers of nodal degrees of freedom (DOF) of the ith and jth regions, which can be ( j)

(i )

(i )

(i )

( j)

(i )

( j)

( j)

either fluid or solid region; {χk }, {χk }, {u 1,k , u 2,k , u 3,k } and {u 1,k , u 2,k , u 3,k } are the interpolation coefficients for the corresponding basis functions. Substituting Eq. (53) into (49)–(52), we arrive at the linear systems

( M 0(i ) + K 0(i ) )χ˜ (i ) +

Nd

(i , j ) ( j )

(i , j ) ( j )

u˜ 1 + R 2

(R1

(i , j ) ( j )

u˜ 2 + R 3

u˜ 3 ) = 0

(54)

j =1 j =i

and

⎛ ⎜ ⎝

(i )

(i )

(i )

M 1 + K 11 (i )

(i )

(i )

(i )

(i )

K 31 (i )

K 13

(i )

M 2 + K 22

K 21

K 32

(i )

⎛ (i ) ⎞ ⎛ (i , j ) ⎞ Q1 F1 Nd ⎟ ⎜ (i ) ⎟ ⎜ (i , j ) ⎟ ( j ) ⎜ (i ) ⎟ ⎠ ⎝ u˜ 2 ⎠ + ⎝ Q 2 ⎠ χ˜ = ⎝ F 2 ⎠ ⎞⎛

(i )

K 12

K 23 (i )

(i )

M 3 + K 33

(i )

u˜ 1

(i ) u˜ 3

⎞

j =1 j =i

(i , j )

Q3

(55)

(i )

F3

(i )

where (u˜ 1 , u˜ 2 , u˜ 3 ) T and χ˜ (i ) denote the unknowns of displacement vectors and the velocity potential for the ith region in the linear systems; M 0 and K 0 are the mass matrices and stiffness matrices for the fluid, while M i (i = 1, 2, 3) and (i , j ) K i j (i , j = 1, 2, 3) are the mass matrices and stiffness matrices for the solid region, respectively. Note that, matrices R 1 , (i , j )

(i , j )

(i , j )

(i , j )

(i , j )

R 2 , R 3 , Q 1 , Q 2 , and Q 3 are obtained from the interface integrations, thus they can be viewed as the couplings between fields of the ith region and fields of the jth region. Formulations of the above matrices and vectors are given below: (i )

( M 0 )kl = Vi

(i )

κ −1 ω2 φk(i) φl(i)dv

( K 0 )kl = − Vi

ρ −f 1 ∇φk(i) · ∇φl(i)dv

(k, l = 1, 2, ..., N i )

(56)

(k, l = 1, 2, ..., N i )

(57)

28


(i , j )

(R1

)kl = j ω

(i ) ( j )

(k = 1, 2, ..., N i ; l = 1, 2, ..., N j )

(58)

(i ) ( j )

(k = 1, 2, ..., N i ; l = 1, 2, ..., N j )

(59)

(i ) ( j )

(k = 1, 2, ..., N i ; l = 1, 2, ..., N j )

(60)

n1 φk φl ds

i j (i , j )

(R2

)kl = j ω

n2 φk φl ds

i j (i , j )

(R3

)kl = j ω

n3 φk φl ds

i j (i )

(i )

(i )

M1 = M2 = M3 (i )

(61)

(i ) (i )

( M 1 )kl = −ρs ω2 (i ) ( K pq )kl =

(Q 1

(62)

( p , q = 1, 2, 3; k, l = 1, 2, ..., N i )

(63)

(i ) ( j )

(k = 1, 2, ..., N i ; l = 1, 2, ..., N j )

(64)

(i ) ( j )

(k = 1, 2, ..., N i ; l = 1, 2, ..., N j )

(65)

(i ) ( j )

(k = 1, 2, ..., N i ; l = 1, 2, ..., N j )

(66)

∇φk(i ) · (C pq ∇φl(i ) )dv

Vi

(i , j )

(k, l = 1, 2, ..., N i )

φk φl dv Vi

)kl = − j ω

n1 φk φl ds

i j (i , j )

(Q 2

)kl = − j ω

n2 φk φl ds

i j (i , j )

(Q 3

)kl = − j ω

n3 φk φl ds

i j (i )

(1 )

(i )

(k = 1, 2, ..., N i )

(67)

(i )

(2 ) j φ ( i )

(k = 1, 2, ..., N i )

(68)

(i )

(3 ) j φ ( i )

(k = 1, 2, ..., N i )

(69)

( F 1 )k = F p e j φ φk (r s ) ( F 2 )k = F p e φk (r s ) ( F 3 )k = F p e φk (r s )

Based upon the above formulations, to solve the whole fluid–solid coupling system, we can choose either an iterative or direct linear solver. Anisotropic perfectly matched layer (PML) can also be employed for the coupling system, which has been presented in subsection 2.2. For the DG-SEM technique, the calculation of the surface integral on the fluid–solid interface is a very critical issue. Because the meshes can be conforming or non-conforming across the interfaces between the fluid and solid regions, we evaluate the surface integral in two cases: If the mesh is conforming, to evaluate the surface integral at an elemental level, we used the Gauss–Lobatto–Legendre (GLL) points of an interface surface and the corresponding weights for the numerical quadrature. On the other hand, for the non-conforming mesh, one cannot use this numerical quadrature any more. In this case, we can evaluate the surface integral over the shared area between two hexahedral elements from the two adjacent regions. Obviously, the shared area between two hexahedral elements is always a flat or curved polygon (depending on whether the interface is flat or curved). To approximate the surface integral over a polygon, we can decompose a polygon with M vertices into M − 2 triangles. Thus, we only need to carry out the surface integral on those triangles. In our implementation, a higher degree quadrature rules on triangles [7] has been used. 3. Numerical results In this section, we present five numerical tests to validate the proposed SEM technique. In the first test, we study the convergence of SEM solution for a homogeneous cube model with isotropic and anisotropic materials, respectively. In the second test, we consider an open boundary problem, i.e., a full-space model, to verify our PML implementation and the high accuracy of SEM. Then, in the third test, we study a layered half-space model to validate the SEM simulation for an inhomogeneous model. The last two simulations, the Karst cave model and the layered solid–fluid model with curved surface, are used for validating the DG-SEM implementation for the solid–fluid interaction problem. We use a sparse direct solver UMFPACK for solving the linear systems after the discretization. Results based upon the FEM commercial software COMSOL Multiphysics are used for comparison for all our SEM simulations; they show significant advantages of our method. Note that COMSOL Multiphysics is a highly optimized commercial software, while the proposed SEM technique is in a developing code package without thorough optimization, so we would expect even more obvious advantages from the proposed SEM if the code is further optimized. In addition, analytical solutions for a homogeneous medium [43] are used as references for the second test. All of the following examples run on an IBM x3850X5 machine with eight Xeon E7-8870 processors. Each of the processors has ten cores operating at 2.4 GHz.


29

Fig. 1. SEM results with interpolation orders 3, 4, and 6, as well as the reference results obtained by the second-order FEM for U z component of displacement field along a cut line with the end-points at (−153, 75, 55) and (153, 75, 55) for the homogeneous cube model.

Fig. 2. Error convergence of the SEM for the homogeneous isotropic cube model (G=4) (a), the homogeneous anisotropic cube model (G=4) (b), and the fullspace homogeneous model (G=4.5) (c). The relative error was calculated along the cut lines with the end-points at (−153, 75, 55) m and (153, 75, 55) m for (a) and (b), and with the end-points (−192, 120, 0) m and (192, 120, 0) m for (c), respectively.

3.1. SEM convergence validation model In order to verify the accuracy and convergence of the proposed SEM, we first consider a simple homogeneous isotropic solid cube centered at the origin with a size of 306 m × 306 m × 306 m. The medium has a P-wave velocity of c p = 2650 m/s and an S-wave velocity c s = 1530 m/s. The mass density is 2200 kg/m3 . Free surface boundary condition is imposed on all sides of the model. The source used is a point load oriented along the direction (1, 1, 1) and is located at the center of the model, i.e. (0, 0, 0) m, with an operating frequency 12.5 Hz. This model is divided into 27 hexahedron elements. We first calculate the displacement field with the basis function order being 3, 4, and 6, respectively, and show the results along a cut line with the end-points at (−153, 75, 55) m and (153, 75, 55) m. These results are compared to the solution obtained by the second-order FEM with 398, 933 tetrahedron elements, leading to 1, 633, 476 degrees of freedom (DoFs). As illustrated in Fig. 1, we find the SEM solution converges rapidly to the reference FEM results with the increasing order in the basis functions. In order to further study the convergence behavior of SEM, we have calculated the average relative error along the cut line. The relative error versus the order of SEM basis functions is plotted in Fig. 2 (a) (G=4). The reference used is obtained by a very high SEM order ( N = 12). We did not use the FEM solution as the reference because its

30


Fig. 3. Convergence of U x component of displacement obtained using the second-order FEM with increasing degrees of freedom (DoFs) for the anisotropic cube model. Reference computed using third-order FEM, as well as the SEM results (N = 9), are plotted for comparison.

convergence is inadequate. It is observed that the error decreases exponentially with the increase of the interpolation order. Note that this exponential convergence of error (thus the spectral accuracy) is an important feature of SEM, thus leading to its high degree of accuracy. For the same size model, we now consider the effects of elastic material anisotropy. We use a homogenous vertical transverse isotropy (VTI) medium with elastic coefficients c 1111 = 9.07 × 1010 N/m2 , c 3333 = 9.38 × 1010 N/m2 , c 1122 = 5.81 × 1010 N/m2 , c 1133 = 5.10 × 1010 N/m2 , c 2323 = 1.504 × 1010 N/m2 and density 3640 kg/m3 . The frequency is 25 Hz and all the other parameters remain the same as the previous simulation. In our numerical tests, we found that simulations for elastic anisotropy are sometimes very difficult for the lower order FEM (e.g. first-order or second-order FEM). We calculated the displacement field along the cut line with the end-points at (−153, 75, 55) m and (153, 75, 55) m using the second-order FEM with different degrees of freedom. Fig. 3 shows the convergence of the U x component of displacement obtained using the second-order FEM with increasing degrees of freedom (DoFs) for the anisotropic cube model. The reference result is obtained by a higher-order (third order) FEM. For comparison, SEM solution with interpolation order N = 9 corresponding to 65, 856 degrees of freedom is also plotted in the figure. It is observed that the SEM solution matches excellently with the reference results. On the other hand, we find that the FEM solutions converge very slowly to the reference results. For the simple anisotropy model, to achieve similar accuracy, the number of unknowns (i.e. degrees of freedom) used by FEM is 156 times more than SEM, thus hardly acceptable. Thus, lower-order FEM is not a good choice for modeling elastic material anisotropy due to its poor convergence. In contrast, SEM is capable of simulating anisotropic media accurately thanks to its excellent convergence behaviors. As shown in Fig. 2 (b), with increasing of the SEM interpolation order, the relative error along the cut line decreases exponentially (G=4). In other words, SEM has spectral accuracy with the order of basis functions for anisotropic media. 3.2. Full-space homogeneous model Next, to validate the anisotropic PML as well as efficiency of the proposed SEM technique, we study a full-space homogeneous model with open boundaries (i.e., an unbounded medium). The cube considered is centered at the origin with a size of 640 m × 640 m × 640 m. The isotropic elastic medium has a P-wave velocity 2000 m/s, an S-wave velocity 1154.7 m/s and a mass density 2200 kg/m3 , respectively. The model is discretized using 5 × 5 × 5 = 125 hexahedron elements as shown in Fig. 4 (a). To mimic the full unbounded space wave propagation, we use the anisotropic PML on all sides of the model. In our simulation, the PML region consists of only one layer of spectral elements with thickness of 128 m. We place a point load source with direction (1, 0, 0) at the center of the model, i.e. (0, 0, 0) m. The frequency used is 18 Hz. In Fig. 5, we show the components of displacement along a cut line with the end-points at (−192, 120, 0) m and (192, 120, 0) m obtained based upon three methods, i.e., SEM with interpolation order N = 10, FEM, and analytical solution. The agreement among the three results is excellent. In order to further investigate the high-efficiency and accuracy of SEM, we compared the relative errors and computational costs between SEM and 2nd-order FEM as illustrated in Table 1. In the simulation we have used 40 cores and assume that parallel efficiency of FEM for this case is 85%. As we can see, to achieve about 1.6% accuracy, the number of unknowns and CPU time used by FEM is 13.97 and 61 times more than SEM, respectively. Thus, the proposed SEM is much more efficient than FEM method, although our SEM codes are not optimized. The sampling density in terms of the number of PPWs (i.e. points per wavelength) corresponding to the seven to ten SEM interpolation orders are 3.5, 4.0, 4.51, and 5.01, respectively. As shown in Table 1, to reach about 0.74% and 0.49% error for displacement component U x and U y , SEM interpolation order N = 10 is needed, which implies the sampling density of 5.01 PPW is required.


31

Fig. 4. Meshes for the full-space homogeneous model with 5 × 5 × 5 hexahedron elements (a), and the half-space layered model with 10 × 10 × 10 hexahedron elements (b).

Fig. 5. Comparison of real U x component (a), imaginary U x component (b), real U y component (c) and imaginary U y component (d) of displacement field along a cut line with the end-points at (−192, 120, 0) m and (192, 120, 0) m for the full-space homogeneous model. The field in PML part and the U z component of displacement which is zero by symmetry are not shown in the figure.

Table 1 Relative errors and computational costs of FEM and SEM for the full-space homogeneous model.

FEM SEM (N=9) SEM (N=10)

PPWs

Unknowns

Error (U x )

Error (U y )

Memory (GB)

CPU (s)

Computing type

7.82 4.51 5.01

3, 570, 096 255, 552 352, 947

0.0192 0.0161 0.0074

0.0249 0.0181 0.0049

344 18 33

21, 339 (40 cores) 11, 891 37, 952

Parallelization Serialization Serialization

Fig. 2 (c) shows the relative error along the cut line versus the order of SEM basis functions where the analytical solution is chosen as a reference (G=4.5). We again observe that the relative error decreases exponentially, which implies the spectral accuracy of SEM.

32


Fig. 6. Displacement field distribution along the cut plane X = 105 m for the half-space layered model: (a) amplitude of U x ; (b) amplitude of U y ; (c) amplitude of U z .

3.3. Half-space layered model In the third test, we study a layered half-space model to examine SEM simulation for an inhomogeneous model. The computational domain consists of three solid medium layers. Specifically, the upper, middle, and lower layer are defined as the blocks [−400, 400] m × [−400, 400] m × [90, 400] m with material parameters {c p , c s , ρ } = {2540 m/s, 1466.4 m/s, 2200 kg/m3 }; [−400, 400] m × [−400, 400] m × [−90, 90] m with material parameters {c p , c s , ρ } = {3000 m/s, 1847.5 m/s, 2400 kg/m3 } and [−400, 400] m × [−400, 400] m × [−400, −90] m with material parameters {c p , c s , ρ } = {2540 m/s, 1466.4 m/s, 2200 kg/m3 }, respectively. Thus, the whole computation domain is [−400, 400] m × [−400, 400] m × [−400, 400] m. In this simulation, the PML absorbing boundary condition is employed on all sides of the model except for the top free surface. The thickness of PML is 185 m which contains two layers of spectral elements. As illustrated in Fig. 4 (b), the mesh is composed of 10 × 10 × 10 = 1000 hexahedron elements, thus resulting in 397, 953 unknowns if an interpolation order N = 5 is used in each element. A point load source with direction (1, 0, 0) is placed at (0, 0, 0) m. The frequency used is 15 Hz. For comparison, we also simulated the same model by FEM with 1, 052, 659 tetrahedron elements, leading to 4, 364, 461 degrees of freedom (DoFs). In Fig. 6, we demonstrate the displacement field distribution along the cut plane X = 105 m. From the plotted amplitude of U x , U y and U z , the effects of material discontinuity can be observed. Fig. 7 shows the comparison of the components of displacement along a cut line with the end-points (105, 0, −400) m and (105, 0, 400) m between the SEM and FEM simulations. The agreement is very good, thus demonstrating that the SEM is capable of accurately simulating the inhomogeneous medium. We list the computational costs between FEM and SEM for the simulations in Table 2. It is observed that FEM uses more memory cost than it does in the former simulation due to its unsymmetrical system matrix. 3.4. Karst cave model We then consider two simulations of Acoustic-Solid Interaction problems for validating our DG-SEM implementation. The first fluid–solid model is a Karst cave model with the configuration shown in Fig. 8 (a). The Karst cave model is composed of two solid regions and one fluid region. The fluid region is a block of [−172.5, 172.5] m × [−172.5, 172.5] m × [0, 345] m with material parameters {c p , ρ } = {1500 m/s, 1000 kg/m−3 }. Solid region 1 and solid region 2 are defined as the blocks [−635, 635] m × [−635, 635] m × [0, 460] m with material parameters {c p , c s , ρ } = {2000 m/s, 1154.7 m/s, 2200 kg/m3 } and [−635, 635] m × [−635, 635] m × [−460, 0] m with material parameters {c p , c s , ρ } = {2540 m/s, 1466.4 m/s, 3350 kg/m3 }, respectively. We use the anisotropic PML with thickness of 230 m on all sides of the model except for the top free surface. The computational domain is divided into 968 hexahedron elements as shown in Fig. 8 (b). We use different interpolation orders for solid and fluid regions, i.e., N = 6 for solid region and N = 5 for fluid region, respectively. The total number of degrees of freedom (DoFs) for the simulation is 649, 240, which is composed of 645, 144 unknowns from solid regions as well as 4096 unknowns from the fluid region. The sampling density in solid regions and fluid region are 6.0 and 6.52 PPWs, respectively. We use a power point source placed at (0, 0, 170) m in the fluid region and the frequency used for this simulation is 10 Hz. Fig. 9 shows displacement field distribution along the cut plane X = 90 m, which passes through the fluid and two solid regions. We plotted the displacement instead of pressure field in the fluid region. It can be clearly observed that, at interfaces of fluid and solid regions, only the normal component continuity of displacement is preserved. For comparison purpose, we then calculate the same model with second-order FEM and third-order FEM respectively. In the second-order FEM simulation, we use 1, 749, 202 tetrahedron elements with 6, 995, 418 unknowns. For the third-order FEM simulation, we use 513, 021 tetrahedron elements and 6, 936, 173 unknowns. Fig. 10 (a–b) shows the comparison of different displacement components calculated by the second-order FEM, third-order FEM, and SEM along a cut line defined by two points (−635, 0, 370) m and (635, 0, 370) m. Fig. 10 (c–d) and Fig. 10 (e–f) show the comparison of pressure and displacement


33

Fig. 7. Comparison of real U x component (a), imaginary U x component (b), real U z component (c) and imaginary U z component (d) of displacement field along a cut line with the end-points at (105, 0, −400) m and (105, 0, 400) m for the half-space layered model. Table 2 Computational costs of FEM and SEM for the half-space model.

FEM SEM (N=5)

PPWs

Unknowns

Memory (GB)

CPU (s)

Computing type

10.26 6.1

4, 364, 461 397, 953

717 32

30, 446 (40 cores) 3729

Parallelization Serialization

Fig. 8. Geometric structure (a) and the mesh used in SEM (b) for the Karst cave model.

components along a cut line defined by points (−172.5, 0, 260) m and (172.5, 0, 260) m, and another cut line defined by points (−635, 0, −50) m and (635, 0, −50) m, respectively. Note that the results obtained by the third-order FEM are used as a reference in the figure. It is observed that the agreement between SEM and the reference is excellent, while results from the second-order FEM have relatively poor agreement due to its slow convergence. We summarized the computational costs between FEM and SEM for this simulation in Table 3. With higher accuracy, as we can see, the number of unknowns, memory, and CPU time in SEM is 9.28%, 7.76% and 0.8% (assumed that parallel efficiency of FEM is 85%) of those in FEM. Thus, the proposed SEM is also much more efficient than the conventional FEM in calculating fluid–solid interaction problems.

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Fig. 9. Displacement field distribution along the cut plane X = 90 m for the Karst cave model: (a) amplitude of U x ; (b) amplitude of U y ; (c) amplitude of U z .

Fig. 10. Comparison of displacement and pressure along the given cut lines for the Karst cave model. (a–b) Re (U z ) and Im(U z ) along the cut line with the end-points (−635, 0, 370) m and (635, 0, 370) m; (c–d) Re ( P ) and Im( P ) along the cut line with the end-points at (−172.5, 0, 260) m and (172.5, 0, 260) m; (e–f) Re(U z ) and Im(U z ) along the cut line with the end-points at (−635, 0, −50) m and (635, 0, −50) m. The reference is obtained by 3rd-order FEM. Table 3 Computational costs of FEM and SEM for the Karst cave model.

FEM SEM (N=[6 5])

PPWs (solid)

PPWs (fluid)

Unknowns

Memory (GB)

CPU (s)

Computing type

10 6

12.99 6.52

6, 995, 418 649, 240

722 56

22, 794 (80 cores) 12, 332

Parallelization Serialization

3.5. Layered solid–fluid model with a curved surface To investigate the proposed method for incorporating topographic variations as well as the anisotropic PML implementation in DG-SEM for fluid–solid interaction, we consider a layered solid–fluid model with a curved surface. The curved surface model is composed of two solid regions and one fluid region as shown in Fig. 11 (a). The solid region 1 with a curved top surface is defined as a block [−632.5, 632.5] m × [−632.5, 632.5] m × [287.5, 632.5] m with material parameters {c p , c s , ρ } = {2000 m/s, 1154.7 m/s, 2200 kg/m3 }. The other two regions, fluid region 2 and solid region 3 are defined as the blocks [−632.5, 632.5] m × [−632.5, 632.5] m × [−57.5, 287.5] m with material parameters


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Fig. 11. Conforming mesh (a) and non-conforming mesh (b) used in SEM for the layered solid–fluid curved surface model. Table 4 Computational costs of FEM and SEM for the layered solid–fluid interaction model.

FEM SEM (conforming mesh) SEM (non-conforming mesh)

PPWs (solid)

PPWs (fluid)

Unknowns

Memory (GB)

CPU (s)

Computing type

10.04 6.0 5.47

13.04 7.8 6.4

8, 483, 871 753, 625 505, 363

829 86 45

25, 518 (80 cores) 14, 953 9422

Parallelization Serialization Serialization

{c p , ρ } = {1500 m/s, 1000 kg/m3 } and [−632.5, 632.5] m × [−632.5, 632.5] m × [−632.5, −57.5] m with material parameters {c p , c s , ρ } = {2540 m/s, 1466.4 m/s, 3350 kg/m3 }, respectively. Except for the top free surface, we use a second order PML (i.e., p = 2) with thickness of 230 m on the other sides of the model. The outer boundary conditions behind the PML layer are Dirichlet BC and free surface BC for fluid and solid region, respectively. We used both conforming mesh (mesh 1) and non-conforming mesh (mesh 2) in our SEM simulations, as shown in Fig. 11. The conforming mesh is composed of 1331 hexahedron elements, with an interpolation order 6 for both solid and fluid regions; the number of unknowns is 753, 625. In contrast, the non-conforming mesh is composed of 1314 hexahedron elements, with an interpolation order 5 and 6 for solid and fluid regions, respectively; the number of unknowns is 505, 363. A explosion point source is placed at (0, 0, 280) m in the fluid region and the simulation frequency is 10 Hz. For comparison, we have also computed the same model based upon FEM. In FEM simulation, we use the second-order discretization with 2, 548, 326 tetrahedron elements and 8, 483, 871 unknowns. The computational costs between FEM and SEM with conforming as well as non-conforming mesh are listed in Table 4. It is shown that, in the simulations, 45 GB memory, 9422 s CPU time; 86 GB memory, 14, 953 s CPU time and 829 GB memory, 25, 518 s × 80 × 85% = 1, 735, 224 s CPU time are required for SEM with the non-conforming mesh, SEM with conforming mesh, and FEM, respectively. Note that, we have used 80 cores for FEM simulation and assumed that the parallel efficiency of FEM is 85%. Compared with the conforming mesh, we use less computational costs as well as a lower sampling density (PPWs) in the case of non-conforming mesh due to the relatively flexible adjustment of the spectral element size and interpolation order in DG-SEM. In Fig. 12, we demonstrate the displacement field distribution along the given cut plane X = 200 m, which passes through both fluid and solid regions. For this case, planes z = −57.5 and z = 287.5 m are the fluid–solid interface. We can observe that U x and U y (tangential) components of displacement are discontinuous while U z (normal) component is continuous through the two interfaces. We then compare the SEM results to those based upon FEM. In Fig. 13 (a–b), we show the comparison of different displacement components calculated by SEM using a conforming mesh (denoted by SEM-mesh 1) with interpolation order N = [6, 6, 6], as well as a non-conforming mesh (denoted by SEM-mesh 2) with interpolation order N = [5, 6, 5], and FEM along a cut line defined by points (−632.5, 0, 400) m and (632.5, 0, 400) m. The agreement among the three results is excellent. Similarly, Fig. 13 (c–d) and Fig. 13 (e–f) show the comparison of pressure and displacement components along the other two cut lines defined by points (−632.5, 0, 150) m and (632.5, 0, 150) m and by points (−632.5, 0, −100) m and (632.5, 0, −100) m, respectively. The overall agreement is very good. Notice that, there are some small discrepancies between SEM and FEM along the line defined by (−632.5, 0, −100) m and (632.5, 0, −100) m as shown in Fig. 13 (e–f), which may be caused by the spurious reflections from PML layers either in SEM or FEM simulations. 4. Conclusion The frequency domain spectral-element method (SEM) is proposed for elastic and acoustic wave simulations under the framework of the higher-order finite element method. The proposed method has spectral accuracy due to the use of Gauss–Lobatto–Legendre (GLL) basis functions. Compared with the conventional finite-element method (FEM), the number of unknowns in SEM is significantly reduced and higher order accuracy is obtained due to its exponential convergence.

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Fig. 12. Displacement field distribution along the cut plane X = 200 m for the layered solid–fluid curved surface model: (a) amplitude of U x ; (b) amplitude of U y ; (c) amplitude of U z .

Fig. 13. Comparison of displacement and pressure along the given cut lines for the layered solid–fluid curved surface model. (a–b) Re (U z ) and Im(U z ) along the cut line with the end-points at (−632.5, 0, 400) and (632.5, 0, 400); (c–d) Re ( P ) and Im( P ) along the cut line with the end-points at (−632.5, 0, 150) and (632.5, 0, 150); (e–f) Re (U z ) and Im(U z ) along the cut line with the end-points at (−632.5, 0, −100) and (632.5, 0, −100).

Especially for some anisotropic media, we find the convergence of conventional FEM (i.e., second-order FEM) is very slow, leading to inaccurate and even erroneous results. In contrast, the SEM is capable of simulating any anisotropic media accurately thanks to its excellent convergence behaviors. To simulate wave propagation in an unbounded medium, an anisotropic perfectly matched layer (PML) is employed to truncate the computational domain in our SEM technique. For a small fullspace homogeneous model, to achieve the same accuracy, the proposed SEM is about 61 times faster than the conventional FEM. The effects of inhomogeneity and anisotropy as well as free-surface topography can be easily and accurately accommodated in SEM implementation. To address the solid–fluid interaction problems, we proposed a domain decomposition method (DDM) based upon the discontinuous Galerkin spectral-element (DG-SEM) method. In this scheme, one can use different interpolation degrees in spectral elements in fluid region and solid region to maximize the efficiency. Both conforming and non-conforming meshes can be employed across the interfaces between fluid (acoustic) and solid (elastic) regions. We presented two numerical examples to validate the efficiency and accuracy of the approach. It is confirmed that the proposed DDM technique is more efficient compared with the conventional FEM for solid–fluid interaction simulations. Note that, with an appropriate transmission condition (e.g. central flux or Riemann solver), we can also solve the solid– solid interaction model with the proposed DDM technique. Future research will focus on extensions of the DDM approach


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to solid–solid interaction problems, which may greatly reduce the memory costs for large-scale simulations and can be a robust and efficient tool for multi-scale problems in elastodynamics. Acknowledgements We would like to thank YingShi Chen and QiWei Zhan for numerous helpful discussions. This research was supported by the National Natural Science Foundation of China (grant number 41390453 and 11301438), and the Natural Science Foundation of Fujian Province, China (grant number 2013J05060). References [1] J.P. Berenger, A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys. 114 (1994) 185–200. [2] B. Cockburn, F. Li, C.W. Shu, Locally divergence-free discontinuous Galerkin methods for the Maxwell equations, J. Comput. Phys. 194 (2004) 588–610. [3] F. Collino, C. Tsogka, Application of the PML absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media, Geophysics 66 (2001) 294–301. [4] J. Chen, Q.H. Liu, M. Chai, J.A. Mix, A nonspurious 3-D vector discontinuous Galerkin finite-element time-domain method, IEEE Microw. Wirel. Compon. Lett. 20 (2010) 1–3. [5] W.C. Chew, Q.H. Liu, Perfectly matched layers for elastodynamics: a new absorbing boundary condition, J. Comput. Acoust. 4 (1996) 341–359. [6] W.C. Chew, M.S. Tong, B. Hu, Integral Equation Methods for Electromagnetic and Elastic Waves, Morgan & Claypool, London, UK, 2009. [7] D.A. Dunavant, High degree efficient symmetrical Gaussian quadrature rules for the triangle, Int. J. Numer. Methods Eng. 21 (1985) 1129–1148. [8] G. Festa, J.P. Vilotte, The Newmark scheme as velocity-stress time-staggering: an efficient PML implementation for spectral-element simulations of elastodynamics, Geophys. J. Int. 161 (2005) 789–812. [9] R.W. Graves, Simulating seismic wave propagation in 3D elastic media using staggered-grid finite differences, Bull. Seismol. Soc. Am. 86 (1996) 1091–1106. [10] B. Guo, I. Babuska, The h-p version of the finite element method, Comput. Mech. 1 (1986) 21–41. [11] S. Ham, K.J. Bathe, A finite element method enriched for wave propagation problems, Comput. Struct. 94 (2012) 1–12. [12] S. Hestholm, Elastic wave modeling with free surfaces: stability of long simulations, Geophysics 68 (2003) 314–321. [13] A.V. Idesman, M. Schmidt, J.R. Foley, Accurate finite element modeling of linear elastodynamics problems with the reduced dispersion error, Comput. Mech. 47 (2011) 555–572. [14] A. JafarGandomi, H. Takenaka, Efficient FDTD algorithm for plane-wave simulation for vertically heterogeneous attenuative media, Geophysics 72 (2007) H43–H53. [15] K.R. Kelly, R.W. Ward, S. Treitel, R.M. Alford, Synthetic seismograms: a finite difference approach, Geophysics 41 (1976) 2–27. [16] D. Komatitsch, J.P. Vilotte, The spectral-element method: an efficient tool to simulate the seismic response of 2D and 3D geological structures, Bull. Seismol. Soc. Am. 88 (1998) 368–392. [17] D. Komatitsch, C. Barnes, J. Tromp, Wave propagation near a fluid–solid interface: a spectral element approach, Geophysics 65 (2000) 623–631. [18] D. Komatitsch, J. Tromp, Spectral-element simulations of global seismic wave propagation-I. Validation, Geophys. J. Int. 149 (2002) 390–412. [19] D. Komatitsch, J. Tromp, Spectral-element simulations of global seismic wave propagation-II. 3-D models, oceans, rotation, and self-gravitation, Geophys. J. Int. 150 (2002) 303–318. [20] D. Komatitsch, J. Tromp, A perfectly matched layer absorbing boundary condition for the second-order seismic wave equation, Geophys. J. Int. 154 (2003) 146–153. [21] J.-H. Lee, Q.H. Liu, An efficient 3-D spectral element method for Schrodinger equation in nanodevice simulation, IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 24 (2005) 1848–1858. [22] J.-H. Lee, T. Xiao, Q.H. Liu, A 3-D spectral-element method using mixed-order curl conforming vector basis functions for electromagnetic fields, IEEE Trans. Microw. Theory Tech. 54 (2006) 437–444. [23] J.-H. Lee, J. Chen, Q.H. Liu, A 3-D discontinuous spectral element time-domain method for Maxwell’s equations, IEEE Trans. Antennas Propag. 57 (2009) 2666–2674. [24] N. Liu, L. Tobon, Y.F. Tang, Q.H. Liu, Mixed spectral element method for 2D Maxwell’s eigenvalue problem, Commun. Comput. Phys. 17 (2015) 458–486. [25] Q.H. Liu, E. Scheon, F. Daube, C. Randall, H.L. Liu, P. Lee, A three-dimensional finite difference simulation of sonic logging, J. Acoust. Soc. Am. 100 (1996) 72–79. [26] Q.H. Liu, J. Tao, The perfectly matched layer (PML) for acoustic waves in absorptive media, J. Acoust. Soc. Am. 102 (4) (1997) 2072–2082. [27] Q.H. Liu, The PSTD algorithm for acoustic waves in inhomogeneous, absorptive media, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 45 (1999) 1044–1055. [28] M. Luo, Q.H. Liu, Spectral element method for band structures of three-dimensional anisotropic photonic crystals, Phys. Rev. E 80 (2009) 056702. [29] M. Luo, Y. Lin, Q.H. Liu, Spectral methods and domain decomposition for nanophotonic applications, Proc. IEEE 101 (2013) 473–483. [30] H. Maurer, S. Greenhalgh, S. Latzel, Frequency and spatial sampling strategies for crosshole seismic waveform spectral inversion experiments, Geophysics 74 (2009) WCC79–WCC89. [31] K.C. Meza-Fajardo, A.S. Papageorgiou, A nonconvolutional, split-field, perfectly matched layer for wave propagation in isotropic and anisotropic elastic media: stability analysis, Bull. Seismol. Soc. Am. 98 (2008) 1811–1836. [32] F. Moser, L.J. Jacobs, J. Qu, Modeling elastic wave propagation in waveguides with the finite element method, NDT E Int. 32 (1999) 225–234. [33] T. Ohminato, B.A. Chouet, A free-surface boundary condition for including 3D topography in the finite-difference method, Bull. Seismol. Soc. Am. 87 (1997) 494–515. [34] S. Operto, J. Virieux, P. Amestoy, J. L’Excellent, L. Giraud, H.B.H. Ali, 3D finite-difference frequency-domain modeling of visco-acoustic wave propagation using a massively parallel direct solver: a feasibility study, Geophysics 72 (5) (2007) SM195–SM211. [35] A.T. Patera, A spectral element method for fluid dynamics: laminar flow in a channel expansion, J. Comput. Phys. 54 (1984) 468–488. [36] Z. Pei, L. Fu, W. Sun, T. Jiang, B. Zhou, Anisotropic finite-difference algorithm for modeling elastic wave propagation in fractured coalbeds, Geophysics 77 (2012) C13–C26. [37] G.R. Pratt, Frequency-domain elastic wave modeling by finite differences: a tool for crosshole seismic imaging, Geophysics 55 (5) (1990) 626–632. [38] G. Seriani, E. Priolo, Spectral element method for acoustic wave simulation in heterogeneous media, Finite Elem. Anal. Des. 16 (1994) 337–348. [39] B.K. Sinha, E. Simsek, Q.-H. Liu, Elastic-wave propagation in deviated wells in anisotropic formations, Geophysics 71 (2006) D191–D202. [40] P. Tong, D. Komatitsch, T.-L. Tseng, S.-H. Hung, C.-W. Chen, P. Basini, Q. Liu, A 3-D spectral-element and frequency-wave number hybrid method for high-resolution seismic array imaging, Geophys. Res. Lett. 41 (2014) 7025–7034. [41] J. Tromp, D. Komatitsch, Q. Liu, Spectral-element and adjoint methods in seismology, Commun. Comput. Phys. 3 (2008) 1–32.

38


[42] J. Virieux, P-SV wave propagation in heterogeneous media: velocity-stress finite-difference method, Geophysics 51 (1986) 889–901. [43] C.-Y. Wang, J.D. Achenbach, Three-dimensional time-harmonic elastodynamic Green’s functions for anisotropic solids, Proc. R. Soc. Lond. A 449 (1995) 441–458. [44] S. Wang, M.V. de Hoop, J. Xia, X.S. Li, Massively parallel structured multifrontal solver for time-harmonic elastic waves in 3-D anisotropic media, Geophys. J. Int. 191 (2012) 346–366. [45] S.L. Wei, M.W. Zhuang, J.Y. Zhou, Q.H. Liu, A 2-D enlarged cell technique (ECT) for elastic wave modeling on a curved free surface, Geophys. J. Int. 201 (2015) 485–495. [46] Y.Q. Zeng, Q.H. Liu, G. Zhao, Multidomain pseudospectral time-domain (PSTD) method for acoustic waves in lossy media, J. Comput. Acoust. 12 (3) (2004) 277–299. [47] Y.Q. Zeng, Q.H. Liu, A multidomain PSTD method for 3D elastic wave equations, Bull. Seismol. Soc. Am. 94 (3) (2004) 1002–1015. [48] W. Zhang, Y. Shen, Unsplit complex frequency-shifted PML implementation using auxiliary differential equations for seismic wave modeling, Geophysics 75 (2010) T141–T154. [49] Y. Zheng, X. Huang, Anisotropic perfectly matched layers for elastic waves in Cartesian and curvilinear coordinates, Massachusetts Institute of Technology. Earth Resources Laboratory, 2002. [50] J.-H. Lee, Q.H. Liu, A 3-D spectral-element time-domain method for electromagnetic simulation, IEEE Trans. Microwave Theory 55 (5) (2007) 983–991.