MPEMathematical Problems in Engineering1563-51471024-123XHindawi Publishing Corporation14238410.1155/2009/142384142384Research ArticleA Wavelet Galerkin Finite-Element Method for the Biot Wave Equation in the Fluid-Saturated Porous MediumZhangXinming1LiuJiaqi2LiuKe'an2VampaVictoria1Harbin Institute of Technology Shenzhen Graduate SchoolShenzhen 518055Chinahit.edu.cn2Department of MathematicsHarbin Institute of TechnologyHarbin, 150001Chinahit.edu.cn200913092009200919012009260720092009Copyright © 2009This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A wavelet Galerkin finite-element method is proposed by combining the wavelet analysis with traditional finite-element method to analyze wave propagation phenomena in fluid-saturated porous medium. The scaling functions of Daubechies wavelets are considered as the interpolation basis functions to replace the polynomial functions, and then the wavelet element is constructed. In order to overcome the integral difficulty for lacking of the explicit expression for the Daubechies wavelets, a kind of characteristic function is introduced. The recursive expression of calculating the function values of Daubechies wavelets on the fraction nodes is deduced, and the rapid wavelet transform between the wavelet coefficient space and the wave field displacement space is constructed. The results of numerical simulation demonstrate that the method is effective.

1. Introduction

The fluid-saturated porous medium is modeled as a two-phase system consisting of a solid and a fluid phase. It is assumed that the solid phase is homogenous, isotropic, elastic frame and the fluid phase is viscous, compressible, and filled with the pore space of solid frame. Compared with the single-phase medium theory, fluid-saturated porous medium theory can describe the formation underground more precisely and the fluid-saturated porous medium elastic wave equation can bring more lithology information than ever. For these reasons, fluid-saturated porous medium theory can be used widely in geophysics exploration and engineering surveying.

In 1956, a theory was developed for the propagation of stress waves in a porous elastic solid containing compressible viscous fluid by Biot [1, 2]. Biot described the Second-Kind P wave in fluid-saturated porous medium firstly. Since then, many researchers paid their attention to the propagation characters of elastic wave in saturated porous medium and obtained many achievements [3, 4]. Complicated equations given in Biot dynamic theory can be solved by analytical methods with some simple boundary conditions. Most dynamic problems in fluid-saturated porous medium are solved using numerical methods, especially using finite-element method. Ghaboussi and Wilson  first proposed a multidimensional finite element numerical scheme to solve the linear coupled governing equations. Prevose  proposed an efficient finite element procedure to analyze wave propagation phenomena in fluid saturated porous medium and presented some numerical results which demonstrate the versatility of the proposed procedure. Simon et al. [7, 8] presented an analytical solution for a transient analysis of a one-dimensional column of a fluid saturated porous elastic solid and presented a comparison of this exact closed-form solution with finite-element method for several transient problems in porous media. Yazdchi et al. [9, 10] combined the finite element method with the boundary element method and the infinite element method, constructed the finite-infinite element method and the finite-boundary element method to deal with the two-phase model in lateral extensive field and obtained better result. Zhao et al.  proposed an explicit finite element method for Biot dynamic formulation in fluid-saturated porous medium. It does not need to assemble a global stiffness matrix and solve a set of linear equations in each time step by using the decoupling-technique. For the problem of local high gradient, finite element method improves the calculation precision by employing the higher-order polynomial or the denser mesh. However, the increment of polynomial order and mesh knots inevitably needs more computational work. Meanwhile, the condition of numerical dissipation will limit the frequency range that can be obtained. To overcome these disadvantages, wavelet analysis is introduced to the finite-element method in this paper. As a new method, the development of wavelet analysis is recent fairly in many fields. Its desirable advantages are the multiresolution analysis property and various basis functions for structure analysis. According to different requirement, the corresponding scaling functions and wavelet functions can be adopted to improve the numerical calculation precision. Especially, those wavelets with compactly supported property and orthogonality, such as Daubechies wavelets, can play an important role in many problems . Because of the compactly supported property, if the Daubechies wavelets are considered as the interpolation functions of the finite element method, the coefficient matrices obtained are sparse matrices and their condition number can be proved independent of the dimension . Moreover, a new method could be provided because of the existence of various basis functions, which can increase the resolution without changing mesh.

In this paper, the wavelet Galerkin finite element method is applied to the direct simulation of the wave equation in the fluid-saturated porous medium. The scaling functions of Daubechies wavelets are considered as the interpolation basis functions instead of the polynomial functions and the wavelet element is constructed. Because a kind of characteristic function is introduced, the integral difficulty for lacking of the explicit expression for the Daubechies wavelets is solved. Based on the recursive expression of calculating the function values of Daubechies wavelets on the fraction nodes, the rapid wavelet transform between the wavelet coefficient space and the wave field displacement space is constructed and reduces the computational cost. The results of numerical simulation demonstrate the method is effective.

2. Wavelet Galerkin Finite-Element Method2.1. Wavelet Galerkin Finite-Element Method

For purpose of constructing the wavelet Galerkin finite element method, we consider a typical boundary value problem:

L(u,v)=f,B(u,v)|Ω=g, where  L(·,·) is differential operator,  B(·,·) is boundary operator, u,v are the unknown functions in the solving domain Ω, and Ω is the boundary.

Supposing u,v are the exact solutions of (2.1) and (2.2), then one gets

L(u,v)-f0, and if L(u,v)  and f are continuous, (2.3) is equal to

Ω(L(u,v)-f)ϕkdxdy=0.

In fact, because of the derivation of one-dimensional wavelet basis element facilitates a straightforward discussion of multidimensional tensor product wavelet basis element and multiresolution analysis property of wavelet function , the functions u,v can be assumed to consist of a superposition of scaling functions at j  level and wavelet functions at the same and higher levels:

u(x,y)=U1(x)U2(y),v(x,y)=V1(x)V2(y), where

U1(x)=kaj,kϕjk(x)+ij,kai,kψik(x),U2(y)=kbj,kϕjk(y)+ij,kbi,kψik(y),V1(x)=kcj,kϕjk(x)+ij,kci,kψik(x),V2(y)=kdj,kϕjk(y)+ij,kdi,kψik(y).

Upon substituting (2.4) and (2.5) into (2.3), we can obtain an equation system of wavelet coefficients, whose coefficient matrix consists of the following integrals:

Ωϕjqψir,Ωϕjqϕjr,Ωψiqψlr,Ωϕjq(m)ψir(n),Ωϕjq(m)ϕjr(n),Ωψiq(m)ψlr(n).

In conventional finite element method, these integrals would be calculated by Gauss quadrature formulae. However, it is not feasible for most wavelet functions. In many cases, there is no explicit expression for the function, in this paper, we choose the Daubechies wavelet as the basis function, and they cannot be integrated numerically due to their unusual smoothness characteristics. Moreover, the wavelet function is defined in terms of scaling function, so these integrals can be rewritten in terms of scaling function alone.

Define the connection coefficients :

Γp,r0,0=Ωϕ(x-p)ϕ(x-r)dx,Γp,rm,n=Ωd(m)ϕdxm(x-p)d(n)ϕdxn(x-r)dx,=Ωϕ(m)(x-p)ϕ(n)(x-r)dx.

Once these integrals can be calculated, all the integrals in (2.8) can be obtained and eventually construct the stiffness matrix and load matrix of wavelet Galerkin finite element method.

2.2. The Calculation of Wavelet Connection Coefficients

From what has been discussed earlier, the quality matrix, stiffness matrix, and the load matrix are composed of the integral values of Daubechies wavelets. However, it is well known that Daubechies wavelets have no explicit expression. In order to solve this problem, a kind of characteristic function is introduced:

χ[0,1](x)={10x1,0otherwise. Set ξ=2x  then   χ[0,1](ξ2)={10ξ2,0otherwise.

So the trivial two-scale equation of characteristic function is obtained:

χ[0,1](ξ2)=χ[0,1](ξ)+χ[1,2](ξ)=χ[0,1](ξ)+χ[0,1](ξ-1).

Set τk,s0,0=Rχ[0,1](x)ϕ(x-k)ϕ(x-s)dx.

Substituting ϕ(x)=kakϕ(2x-k) into (2.13), one obtains

τk,s0,0=Rχ[0,1](x)lalϕ(2(x-k)-l)mamϕ(2(x-s)-m)dx=Rχ[0,1](x)lmalamϕ(2x-2k-l)ϕ(2x-2s-m)dx=12Rχ[0,1](ξ2)lmalamϕ(ξ-2k-l)ϕ(ξ-2s-m)dξ=12lmalamR(χ[0,1](ξ)+χ[0,1](ξ-1))ϕ(ξ-2k-l)ϕ(ξ-2s-m)dξ=12lmalamRχ[0,1](ξ)ϕ(ξ-2k-l)ϕ(ξ-2s-m)dξ+12lmalamRχ[0,1](ξ-1)ϕ(ξ-2k-l)ϕ(ξ-2s-m)dξ=12lmalamτ2k+l,2s+m0,0+12lmalamτ2k+l-1,2s+m-10,0=12pr(ap-2kar-2s+ap-2k+1ar-2s+1)τp,r0,0.

It is not difficult to show that we will require the solution of an eigenvalue problem having the form

τk,s0,0=12Aτp,r0,0,2N-1k,s0,2N-1p,r0, where A is a (2N-1)×(2N-1) partitioned matrix, each submatrix is also a (2N-1)×(2N-1) matrix, in which  asr=ap-2kar-2s+ap-2k+1ar-2s+1.

Considering the requirement of numerical simulation set

1i=s+88,1j=r+88,1m=k+88,1n=p+88, then Akp is changed to  Amn, in which ãij=an-2m+8aj-2i+8+an-2m+9aj-2i+9.

However, the eigenvalue problem does not uniquely define the solution, it is essential to introduce an additional condition to define the solution uniquely.

It is well known that the Daubechies wavelets satisfy

1=kϕ(x-k).

By multiplying (2.17) by itself, and subsequently multiplying the product by the characteristic function  χ[0,1](x), one obtains

1=klϕ(x-k)ϕ(x-l),χ[0,1](x)=klχ[0,1](x)ϕ(x-k)ϕ(x-l).

Now, a single integration yields a first normalization condition:

1=klτk,l0,0.

So, the unique solution of the eigenvalue problem is defined.

The same step can be followed to calculate

τk,sm,n=Rχ[0,1](x)ϕ(m)(x-k)ϕ(n)(x-s)dx.

Substituting ϕ(m)(x)=2mlalϕ(m)(2x-k) and ϕ(n)(x)=2nqaqϕ(n)(2x-k) into (2.20), one gets

τk,sm,n=Rχ[0,1](x)2m+nlalϕ(m)(2(x-k)-l)qaqϕ(n)(2(x-s)-q)dx=2m+nRχ[0,1](x)lqalaqϕ(m)(2x-2k-l)ϕ(n)(2x-2s-q)dx=2m+n-1Rχ[0,1](ξ2)lqalaqϕ(m)(ξ-2k-l)ϕ(n)(ξ-2s-q)dξ=2m+n-1lqalaqR(χ[0,1](ξ)+χ[0,1](ξ-1))ϕ(m)(ξ-2k-l)ϕ(n)(ξ-2s-q)dξ=2m+n-1lqalaqRχ[0,1](ξ)ϕ(m)(ξ-2k-l)ϕ(n)(ξ-2s-q)dξ+2m+n-1lqalaqRχ[0,1](ξ-1)ϕ(m)(ξ-2k-l)ϕ(n)(ξ-2s-q)dξ=2m+n-1lqalaqτ2k+l,2s+qm,n+2m+n-1lqalaqτ2k+l-1,2s+q-1m,n=2m+n-1pr(ap-2kar-2s+ap-2k+1ar-2s+1)τp,rm,n, namely,

τk,sm,n=2m+n-1Aτp,rm,n.

The polynomial reproducing property is employed to construct the additional condition:

xm=kpkϕ(x-k),xn=lplϕ(x-l),

Explicit form for calculating the coefficients pkpl can be found in .

By differentiating (2.23) m times, one obtains

m!=kpkϕ(m)(x-k).

By differentiating (2.24) n times, one gets

n!=lplϕ(n)(x-l).

However (2.25) can be multiplied by (2.26), and subsequently multiplying the product by the characteristic function  χ[0,1](x),

m!n!χ[0,1](x)=klχ[0,1](x)pkplϕ(m)(x-k)ϕ(n)(x-l).

By integrating (2.27), one obtains the additional condition.

m!n!=klpkplτk,lm,n.

Then, the unique solution of the eigenvalue problem is defined.

3. Wavelet Galerkin Finite-Element Solution of 1D Elastic Wave Equation in Fluid-Saturated Porous Medium

From the Biot theory, the 1D differential equation governing wave propagation in the fluid-saturated porous medium, without fluid viscosity, can be expressed as

x((λ+2μ+αM)ux)+x(αMωx)=ρü+ρfω̈-f1,x(αMux)+x(Mωx)=ρfü+mω̈-f2, where u is the solid displacement and  ω  is the relative fluid to solid displacement. β is the porosity, ρ=(1-β)ρs+βρf is the bulk density of solid-fluid mixture, and   ρs  and ρf are the densities of solid and fluid, respectively. Also t is time and λb,μ are the Lame coefficients,  λ=λb+α2M, where α is the effective stress parameter and M is the compressibility of pore fluid. α=1-Kb/Ks, M=Ks/[α+β(Ks/Kf-1)] where Ks,Kf,Kb are the bulk change modulus of the solid, fluid, and skeleton, respectively. Moreover  Kb=λb+2μ/3,  m=ρf/β, Finally f  is seismic focus, and f1=(1-β)f, f2=β(2β-1)f.

Multiplying both sides of the fluid-saturated porous medium wave equation by the Daubechies wavelets basis function  ϕjk(x)=2j/2ϕ(2jx-k), and integrating them at  [0,L], we can get

0L(x((λ+2μ+αM)ux)+x(αMωx))ϕjk(x)dx=0L(ρü+ρfω̈-f1)ϕjk(x)dx0L(x(αMux)+x(Mωx))ϕjk(x)dx=0L(ρfü+mω̈-f2)ϕjk(x)dx.

By using integration by part

(λ+2μ+αM)uxϕjk(x)|0L-0L(λ+2μ+αM)uxϕjk(x)xdx+αMωxϕjk(x)|0L-0LαMωxϕjk(x)xdx=0L(ρü+ρfω̈-f1)ϕjk(x)dx,αMuxϕjk(x)|0L-0LαMuxϕjk(x)xdx+Mωxϕjk(x)|0L-0LMωxϕjk(x)xdx=0L(ρfü+mω̈-f2)ϕjk(x)dx.

Set

u(x,t)=l=2-2N-2jL0al(t)ϕjl(x),ω(x,t)=l=2-2N-2jL0bl(t)ϕjl(x).

Upon substituting (3.5) into (3.3) and (3.4), one gets

(λ+2μ+αM)uxϕjk(x)|0L+αMωxϕjk(x)|0L-0L((λ+2μ+αM)l=2-2N-2jL0al(t)ϕjl(x)xϕjk(x)x+αMl=2-2N-2jL0bl(t)ϕjl(x)xϕjk(x)x)dx=0L(ρl=2-2N-2jL0al′′(t)ϕjl(x)+ρfl=2-2N-2jL0bl′′(t)ϕjl(x)-f1)ϕjk(x)dx,αMuxϕjk(x)|0L+Mωxϕjk(x)|0L-0L(αMl=2-2N-2jL0al(t)ϕjl(x)xϕjk(x)x+Ml=2-2N-2jL0bl(t)ϕjl(x)xϕjk(x)x)dx=0L(ρfl=2-2N-2jL0al′′(t)ϕjl(x)+ml=2-2N-2jL0bl′′(t)ϕjl(x)-f2)ϕjk(x)dx.

By rearranging, (3.6) and become

(λ+2μ+αM)uxϕjk(x)|0L+αMωxϕjk(x)|0L-((λ+2μ+αM)l=2-2N-2jL0al(t)+αMl=2-2N-2jL0bl(t))0Lϕjl(x)xϕjk(x)xdx=(ρl=2-2N-2jL0al′′(t)+ρfl=2-2N-2jL0bl′′(t))0Lϕjl(x)ϕjk(x)dx-f10Lϕjk(x)dx,αMuxϕjk(x)|0L+Mωxϕjk(x)|0L-(αMl=2-2N-2jL0al(t)+Ml=2-2N-2jL0bl(t))0Lϕjl(x)xϕjk(x)xdx=(ρfl=2-2N-2jL0al′′(t)+ml=2-2N-2jL0bl′′(t))0Lϕjl(x)ϕjk(x)dx-f20Lϕjk(x)dx.

If select L=1,  j=0, (3.5) become

u(x,t)=l=1-2N0al(t)ϕ(x-l)ω(x,t)=l=1-2N0bl(t)ϕ(x-l)

Set

A=(a1-2N,a2-2Na0)B=(b1-2N,b2-2Nb0),R=(A,B)T=(a1-2N,a2-2Na0,b1-2N,b2-2Nb0)T.

Then, (3.7) can be changed into an equation system of coefficient  R:

M̅R̈+PR=F+Q, where

M̅=(ρEρfEρfEmE),P=(-(λ+2μ+αM)G-αMG-αMG-MG),F=(F1F2),E=(01𝔄𝔄dx01𝔅𝔄dx01ϕ(x)𝔄dx01𝔄𝔅dx01𝔅𝔅dx01ϕ(x)𝔅dx01𝔄ϕ(x)dx01𝔅ϕ(x)dx01ϕ(x)ϕ(x)dx),G=(01dx01𝔇dx01ϕ(x)dx01𝔇dx01𝔇𝔇dx01ϕ(x)𝔇dx01ϕ(x)dx01𝔇ϕ(x)dx01ϕ(x)ϕ(x)dx),F1=f1(01𝔄dx,01𝔅dx,,01ϕ(x)dx)T,F2=f2(01𝔄dx,01𝔅dx,,01ϕ(x)dx)T,Q=(Q1,Q2)T,Q1=((λ+2μ+αM)ux+αMωx)(𝔄,𝔅ϕ(x))T|01,Q2=(αMux+Mωx)(𝔄,𝔅ϕ(x))T|01, where 𝔄 denote ϕ(x-1+2N), 𝔅 denote  ϕ(x-2+2N), denote ϕ(x-1+2N) and 𝔇 denote ϕ(x-2+2N).

Using the second-order center difference to approximate the two derivatives in (3.10), we can obtain

M̅Rn+1-2Rn+Rn-1(Δt)2+PRn=F+Q.

Arranging (3.12), we have

M̅Rn+1=(2M̅-(Δt)2P)Rn-M̅Rn-1+(Δt)2F+(Δt)2Q, given the initial conditions:

ak(0)=bk(0)=0,ak(1)=bk(1)=0.

So, we can obtain the wavelet coefficients at each time level by solving (3.13) and (3.14) with some boundary conditions, and then substitute the wavelet coefficients into (3.8), the wave field displacements can be obtained.

4. Rapid Wavelet Transform

In order to obtain the wave field displacements conveniently and quickly, the fast wavelet transform between the wavelet coefficients space and the wave field displacements space is constructed as follows:

U=ΦP,U is the wave field displacement vector,  P is the wavelet coefficient vector, Φ is the wavelet transform matrix.

For the sake of simplicity, take the DB2 wavelet as the example. There are 7 nodes in solution field:

U=(u(18),u(14),u(38),u(12),u(58),u(34),u(78))T,P=(p-2,p-1,p0)T,Φ=(ϕ(18+2)ϕ(18+1)ϕ(18)ϕ(14+2)ϕ(14+1)ϕ(14)ϕ(38+2)ϕ(38+1)ϕ(38)ϕ(12+2)ϕ(12+1)ϕ(12)ϕ(58+2)ϕ(58+1)ϕ(58)ϕ(34+2)ϕ(34+1)ϕ(34)ϕ(78+2)ϕ(78+1)ϕ(78)).

It is important for constructing the fast wavelet transform to solve the function values of the Daubechies wavelets on the fraction nodes. So, the recursive expression of calculating the function values of Daubechies wavelets on the fraction nodes is deduced to save the computational cost.

Φ(2ni+p2n)={AΦ(2n-1i+q2n-1)if2·p2n1BΦ(2n-1i+q2n-1)if2·p2n>1 in which i=0,1,2N-2, p=1:2:2n-1, q=pmod2n-1, n controls the mesh partition.

5. Numerical Simulation

To verify the correctness and accuracy of the wavelet Galerkin finite element method, two examples are given to compare the results obtained by this method with an analytical solution. An one-dimensional column of length l  as sketched in Figure 1 is considered. It is assumed that the side walls and the bottom are rigid, frictionless, and impermeable. At top, the stress σy  and the pressure p are prescribed. The boundary conditions are

Model of fluid saturated porous medium.

u|y=0=ω|y=0=0,σ|y=l=-P0f(t),p|y=l=0.

For this model, if the permeability tends to infinity, that is,  κ, the analytical solutions in time domain are 

uy=P0E(d1λ2-d2λ1)n=0(-1)-n{d2[(t-λ1(l(2n+1)-y))H(t-λ1(l(2n+1)-y))-(t-λ1(l(2n+1)+y))H(t-λ1(l(2n+1)+y))]-d1[(t-λ2(l(2n+1)-y))H(t-λ2(l(2n+1)-y))-(t-λ2(l(2n+1)+y))H(t-λ2(l(2n+1)+y))]},p=P0d1d2E(d1λ2-d2λ1)n=0(-1)-n[H(t-λ1(l(2n+1)-y))+H(t-λ1(l(2n+1)+y))-H(t-λ2(l(2n+1)-y))+H(t-λ2(l(2n+1)+y))], where  E  is Young modulus, assuming a Heaviside step function as temporal behavior, that is,  f(t)=H(t), and together with vanishing initial conditions:

di=Eλi2-(ρ-ρf)(α-Q)λi(i=1,2),Q=β2ρfρα+βρf(κ),ρα=0.66βρf.

However  λi  are the characteristic roots of following characteristic equation

EQρfλ4-(E-β2M+(ρ-Qρf)Qρf+(α-Q)2)λ2+β2(ρ-Qρf)M=0.

Supposing A=EQρf,B=E-β2M+(ρ-Qρf)Qρf+(α-Q)2,C=β2(ρ-Qρf)M, one gets

λ1=-λ3=B+B2-4AC2A,λ2=-λ4=B-B2-4AC2A.

In the first example, the length of column is chosen as  l=1000m, and three very different materials, a rock (Berea sandstone), a soil (coarse sand), and a sediment (mud) are chosen. The material data are given in Table 1. In Figures 2, 3, 4, we record the pressure  p(t,y=995m), five meters behind the excitation (y=l=1000m). The numerical results (plotted with dot) are compared with the analytical solution (5.3), shown as solid lines in Figures 2, 3, 4. In the second example, the length of column is chosen as  l=10m. We choose a material-soil, Figures 5, 6 demonstrate the numerical results—the displacements uy(t,y=5m)  and the pressure  p(t,y=5m). All the figures show that the numerical solutions are perfectly close to the analytical solutions, so the method developed in this paper has a very high degree of calculating accuracy.

The parameters of fluid saturated porous medium.

Kb(Pa)G(Pa)ρ(kg/m3)βKs(Pa)ρf(kg/m3)Kf(Pa)
rock8.0×1096.0×10925480.193.6×101010003.3×109
soil2.1×1089.8×10718840.481.1×101010003.3×109
sediment3.7×1072.2×10713960.763.6×101010002.3×109

The pressure of rock (l=1000m,  y=995m).

The pressure of soil (l=1000m,  y=995m).

The pressure of sediment (l=1000m,  y=995m).

The displacement of soil (l=1000m,  y=5m).

The pressure of soil (l=1000m,  y=5m).

6. Conclusion

In this article, the wavelet Galerkin finite element method is constructed by combining the finite element method with wavelet analysis, and is applied to the numerical simulation of the fluid-saturated porous medium elastic wave equation. For the beautiful and deep mathematic properties of Daubechies wavelets, such as the compactly supported property and vanishing moment property, the wavelet Galerkin finite element method has the feature of quick iterative rate and high numerical precision. Moreover, contrasts to h- or p-based FEM, a new refine algorithm can be presented because of the multi-resolution property of the wavelet analysis. The algorithm can increase the numerical precision by adopting various wavelet basis functions or various wavelet spaces, without refining the mesh.

Acknowledgment

This work was supported by the China Postdoctoral Science Foundation, under Grant no. 20080430930 and by the Natural Science Foundation of Guangdong Province, China, under Grant no. 07300059.

BiotM. A.Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency rangeThe Journal of the Acoustical Society of America195628168178MR013405610.1121/1.1908239BiotM. A.Theory of propagation of elastic waves in a fluid-saturated porous solid. II. Higher frequency rangeThe Journal of the Acoustical Society of America195628179191MR013405710.1121/1.1908241PlonaT. J.Observation of a second bulk compressional wave in a porous medium at ultrasonic frequenciesApplied Physics Letters198036425926110.1063/1.914452-s2.0-36749117593KumarR.HundalB. S.Symmetric wave propagation in a fluid-saturated incompressible porous mediumJournal of Sound and Vibration20052881-236137310.1016/j.jsv.2004.08.0462-s2.0-24344439245GhaboussiJ.WilsonE. L.Variational formulation of dynamics of fluid saturated porous elastic solidsJournal of the Engineering Mechanics Division197298EM4947963PrevostJ. H.Wave propagation in fluid-saturated porous media: an efficient finite element procedureInternational Journal of Soil Dynamics and Earthquake Engineering1985441832022-s2.0-002214500810.1016/0261-7277(85)90038-5SimonB. R.ZienkiewiczO. C.PaulD. K.An analytical solution for the transient response of saturated porous elastic solidsInternational Journal for Numerical and Analytical Methods in Geomechanics1984838139810.1002/nag.1610080406ZBL0539.73128SimonB. R.WuJ. S. S.ZienkiewiczO. C.PaulD. K.Evaluation of u-w and u-π finite element methods for the dynamic response of saturated porous media using one-dimensional modelsInternational Journal for Numerical & Analytical Methods in Geomechanics19861054614822-s2.0-0022860938ZBL0597.73108YazdchiM.KhaliliN.VallippanS.Non-linear seismic behavior of concrete gravity dams using coupled finite element-boundary element methodInternational Journal for Numerical Methods in Engineering19994410113010.1002/(SICI)1097-0207(19990110)44:1<101::AID-NME495>3.0.CO;2-4KhaliliN.n.khalili@unsw.edu.auYazdchiM.ValliappanS.Wave propagation analysis of two-phase saturated porous media using coupled finite-infinite element methodSoil Dynamics and Earthquake Engineering19991885335532-s2.0-001744013710.1016/S0267-7261(99)00029-9ZhaoC.LiW.WangJ.An explicit finite element method for dynamic analysis in fluid saturated porous medium-elastic single-phase medium-ideal fluid medium coupled systems and its applicationJournal of Sound and Vibration20052823–51155116810.1016/j.jsv.2004.03.0722-s2.0-14744304186DaubechiesI.Orthonormal bases of compactly supported waveletsCommunications on Pure and Applied Mathematics1988417909996MR95174510.1002/cpa.3160410705ZBL0644.42026JaffardS.LaurencopP.ChuiC.Orthonormal wavelets, analysis of operators and applications to numerical analysisWavelets: A Tutorial in Theory and Applications1992New York, NY, USAAcademic Press543601ZBL0764.65066KoJ.KurdilaA. J.PilantM. S.A class of finite element methods based on orthonormal, compactly supported waveletsComputational Mechanics1995164235244MR1345949ZBL0830.65084DahmenW.MicchelliC. A.Using the refinement equation for evaluating integrals of waveletsSIAM Journal on Numerical Analysis1993302507537MR121140210.1137/0730024ZBL0773.65006LattoA.ResnikoffH. L.TenenbaumE.The evaluation of connection coefficients of compactly supported wavelets1991AD910708Aware Inc.YouheZ.JizengW.XiaojingZ.Applications of wavelet Galerkin FEM to bending of beam and plate structuresApplied Mathematics and Mechanics19981986977062-s2.0-002848539510.1007/BF02457749ZBL0919.73305SchanzM.ChengA. H.-D.Transient wave propagation in a one-dimensional poroelastic columnActa Mechanica20001451–411810.1007/BF014536412-s2.0-0034478398ZBL0987.74039