Comput Mech (2011) 47:669–680DOI 10.1007/s00466-010-0569-y

ORIGINAL PAPER

A regularized collocation boundary element method for linearporoelasticity

Michael Messner · Martin Schanz

Received: 11 August 2010 / Accepted: 23 December 2010 / Published online: 14 January 2011© Springer-Verlag 2011

Abstract This article presents a collocation boundaryelement method for linear poroelasticity, based on the firstboundary integral equation with only weakly singular ker-nels. This is possible due to a regularization of the stronglysingular double layer operator, based on integration by parts,which has been applied to poroelastodynamics for the firsttime. For the time discretization the convolution quadraturemethod (CQM) is used, which only requires the Laplacetransform of the fundamental solution. Furthermore, sincelinear poroelasticity couples a linear elastic with an acousticmaterial, the spatial regularization procedure applied here isadopted from linear elasticity and is performed in Laplacedomain due to the before mentioned CQM. Finally, the spa-tial discretization is done via a collocation scheme. At theend, some numerical results are shown to validate the pre-sented method with respect to different temporal and spatialdiscretizations.

Keywords Linear poroelasticity · Collocation BEM ·Regularized double layer operator · CQM

1 Introduction

As for boundary element formulations in general, their devel-opment is strongly linked to the availability of fundamentalsolutions. In the field of poroelasticity, Manolis and Beskos[17] first used a set of Laplace domain governing equationsbased on Biot’s theory, for which they derived a fundamen-tal solution. Therewith, these authors developed a Laplacedomain boundary element method, leading to time domain

M. Messner · M. Schanz (B)Institute of Applied Mechanics, Graz University of Technology,Technikerstraße 4, 8010 Graz, Austriae-mail: m.schanz@tugraz.at

results via numerical inversion schemes. Neglecting frictionbetween the two constituents of a poroelastic material,Wiebe and Antes [29] were able to transform the just men-tioned Laplace domain fundamental solution in a closed formback to time domain, allowing them to establish a direct timedomain BEM for their reduced model. The next step was doneby Schanz [24], who moved on by using the convolutionquadrature method (CQM) developed by Lubich [15,16],allowing him to realize a direct time domain formulation forthe complete model just requiring the Laplace transform fun-damental solution. As pointed out in [24], the latter methodhas some advantages over inverse transformations, e.g., norestrictions concerning transient boundary conditions. Moredetails on boundary element methods in linear poroelasticitycan be found in [7,25].

These previous observations were related to the temporalaspect of linear poroelasticity when working with boundaryintegral equations (BIE). However, dealing with the spatialaspect always causes some trouble, too. This is due to the sin-gular behavior of the fundamental solution within the integraloperators. Hence, one has to distinguish whether only the firstor also the second BIE is involved. In the engineering com-munity a common approach is to use collocation schemesfor the discretization of the first boundary integral equation.Such methods are justified by their simplicity and computa-tional efficiency compared to other schemes such as Galer-kin methods. There, to exploit the advantageous symmetryof the method [11], the second boundary integral equationneeds to be included. The latter equation involves a bound-ary integral operator with a hypersingular kernel function(hypersingular operator). Nevertheless, the restriction to thefirst boundary integral equation still requires the evaluation ofthe double layer operator. This is a boundary integral oper-ator with strongly singular kernel, whose evaluation is notstraightforward within a numerical scheme. Due to the strong

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relation between poroelasticity and linear elasticity on oneside and acoustics on the other side, a look at these fields isgiven to get an idea how to cope with this difficulty.

Apart from analytical integration, which is limited to somedistinct element types and, therefore, not desired within ageneral scheme, there exist two main approaches to deal withthis matter. On one hand, there are numerical approaches asintroduced by Martin and Rizzo [19] for two dimensionalwave scattering or by Guiggiani and Gigante [8] which allowsthe evaluation of Cauchy singular boundary integrals aris-ing from three dimensional analysis. The latter techniquehas already been applied to linear poroelasticity by Schanz[24], however, it involves adding and subtracting the singu-lar part of the fundamental solution’s asymptotic behavior. Inthis case, the frequency independent elastostatic fundamentalsolution has to be subtracted from the frequency dependentporoelastodynamic fundamental solution. This may causenumerical instability at high and very low frequencies. Fur-thermore, the implementation differs between collocationand Galerkin type approaches, which is not favorable foran extension to the latter methods. On the other hand, thereare analytical regularization techniques based on integrationby parts, dealing with the kernel as a whole, thus they donot show the above discussed problem of numerical insta-bility. Such an approach was first applied by Maue [20] toscalar wave scattering and by Kupradze [14] to linear elas-ticity in the field of potential theory. Almost at the same timeNedelec [22] proposed a regularization for acoustics as wellas for linear isotropic elastic problems, an extension of whichto the anisotropic case can be found in [2]. Nishimura andKobayashi [23] presented similar results but with respect toa collocation scheme, giving some remarks comparing thetwo approaches. As all previously mentioned regularizationtechniques for the linear elastic system involving integrationby parts, the approach by Han [10], which is based on theoriginal work of Kupradze [14], is also concerned with thehypersingular operator. However, as a side product it yieldsa regularization of the double layer operator. In the herepresented approach, this result is adapted to poroelasticity,resulting in a weakly singular first boundary integro-differ-ential equation in Laplace domain. This is another advantageof this approach—within a numerical scheme the same treat-ment of all operators is possible, since all of them show thesame regularity.

The procedure throughout this article is as follows: InSect. 2, Biot’s theory is recalled in a suitable manner forthe later development of the boundary element formulationin Sect. 3. As pointed out by Lubich [15], advantage is takenof the CQM being especially suitable for the discretization oftime convolutions, where only the Laplace transformed ker-nel is known a priori. Moreover, due to the use of the refor-mulated version by Banjai and Sauter [1], the time domainproblem is decoupled into a set of Laplace domain problems.

Thus, the boundary integral equation is derived in Sect. 3.1and regularized in Sect. 3.2, both in Laplace domain. Then,in Sect. 3.3 the time discretization of the formal time domainBIE is done via CQM, before in Sect. 3.4 for the spatialdiscretization a collocation scheme is used. In Sect. 4, twonumerical examples are presented: First, the proposedmethod is verified by comparing it with a semi-analytic solu-tion, whereas the second example’s aim is to show one of theboundary element methods’ strength, i.e., the capability ofdealing with (semi-)infinite domains such as a half-space.

Some remarks on the notation: Bold lower case letters areused for vectors and bold upper case letters for matrices inthe context of the fundamental solution. Tensors are denotedby bold lower case Greek symbols and the nabla operatoracting with respect to the point x = [x1, x2, x3]� is definedas ∇x := [

∂x1, ∂x1 , ∂x2

]�. Laplace transformed quantities are

identified by L [ f (t)] := f (s) with s ∈ C and time deriva-tives by ∂t f (t) := f (t). Finally, all multiplications are to beunderstood in the sense of matrix multiplications.

2 Biot’s theory

Considering a linearized three-dimensional setting, Biot’stheory [3,4] leads to a set of second order partial differen-tial equations describing the state of a saturated poroelasticcontinuum in some domain � ⊂ R

3.According to Biot’s theory, a poroelastic medium consists

of a solid skeleton with volume V s , intersected by some inter-connected fluid filled pores with volume V f . The fraction ofthe fluid volume over the total volume V = V f + V s isdefined as the porosity

φ = V f

V. (1)

With Biot’s effective stress coefficient α = φ(1 + QR ) the

constitution of the total stress tensor σ is given by

σ = 2με + (λ trε − αp) I, (2)

where λ and μ are the Lamé parameters of the bulk material,and Q and R describe the coupling between the fluid and thesolid phase. The pore pressure is denoted by p and the linearstrain tensor ε is defined as

ε = 1

2

(∇u� + (∇u�)�

)(3)

relating the solid strain state with the solid displacement fieldu. Furthermore, a kind of constitutive equation for the varia-tion of the fluid volume per reference volume ζ is introducedas

ζ = α trε + φ2

Rp. (4)

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This variation of the fluid volume per reference volume hasto obey the continuity equation

ζ + ∇�q = 0, (5)

with the specific flux q = φ(u f − u) where u f is the fluiddisplacement field. The total stress, on the other hand, mustfulfill the balance of momentum for the bulk material

(∇�σ

)� + F = ρu + φρ f (u f − u), (6)

with body forces F, the bulk density ρ, and the fluid den-sity ρ f . A generalized form of Darcy’s law [24] is used todescribe the flux trough the pores

q = −κ(

∇ p + ρ f u + ρa + φρ f

φ(u f − u)

), (7)

where κ denotes the dynamic permeability. The apparentmass densityρa was introduced by Biot himself and is definedas ρa = Cφρ f . For the low frequency range C = 0.66, oth-erwise it is frequency dependent [6].

There are different possibilities to combine the aboveequations in order to obtain the equations of motion. How-ever, as shown by Bonnet [5] it is sufficient to chose u andp to describe the poroelastic continuum in Laplace domain.Moreover, having the later use of the CQM in mind, thisformulation is absolutely sufficient. Thus, assuming homo-geneous initial conditions for the Laplace transformationof the above equations and combining them appropriately,a homogeneous mixed boundary value problem in linearporoelasticity in terms of the generalized displacement fieldug = [

u, p]� is stated by

Bxug(x) = 0 for x ∈ �,ug(x) = gD for x ∈ D, (8)

tg(x) = gN for x ∈ N .

The differential operator is defined as

Bx =[

Bex + s2(ρ − βρ f )I (α − β)∇x

s(α − β)∇�x − β

sρ f �x + sφ2

R

]

(9)

and the generalized tractions as

tg(x) =[

T ex −αnx

sβn�x

β

sρ f n�x ∇x

][u(x)p(x)

], (10)

with Bex being the differential operator of linear elastostatics

and T ex its traction operator reflecting Hooke’s law.

3 Boundary element formulation

3.1 Boundary integral equation

The representation formula related to the homogeneous prob-lem (8) is given by

ug (x) =(V tg

)

�(x)−

(Kug

)

�(x) for x ∈ �, (11)

with the integral operators(V tg

)

�(x) =

∫

U� (y − x) tg (y) dsy, (12)

(Kug

)

�(x) =

∫

(TyU

)�(y − x) ug (y) dsy. (13)

In (12) and (13), U is the complex conjugated fundamentalsolution of the adjoint problem and Ty the related tractionoperator, which naturally appears throughout the derivationof the representation formula. Skipping all arguments forsake of readability, they are defined as

U =⎡

⎣Us U f(

Ps)�

P f

⎤

⎦ Ty =[

T ey sαny

−βn�y

β

sρ f n�y ∇

]

, (14)

where U may be found in full detail in Appendix A.1. Themost important property within the context of this paper is thesingular behavior of the individual parts in U in combinationwith the properties of Ty, which is subject of Sect. 3.2.

Equation (11) describes the unknown field ug inside thedomain for given Cauchy data, which are not known yet. Fortheir computation, a limiting process� � x → x ∈ is per-formed, in which (12) is well defined and commonly referredto as single layer operator

lim��x→x∈

(V tg

)

�(x)

=(V tg

)(x) :=

∫

U� (y − x) tg (y) dsy. (15)

More important for the further understanding of this article,however, is the result of this limiting process performed on(13) due to the strong singularity of the kernel function in itsclassical form. Its derivation is standard technique, thereforeonly the final result is recalled here [28]

lim��x→x∈

(Kug

)

�(x) = [−I (x)+ C (x)] ug (x)

+(Kug

)(x), (16)

where I (x) is the identity operator, the jump term C (x) isdefined as

C (x) := limε→0

∫

y∈�:|y−x|=ε

(TyU

)�(y − x) dsy (17)

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672 Comput Mech (2011) 47:669–680

and the double layer operator as(Kug

)(x) := lim

ε→0

∫

|y−x|≥ε

(TyU

)�(y − x) ug (y) dsy.

(18)

Collecting these previous results, one ends up with theLaplace domain BIE

C (x) ug (x) =(V tg

)(x)−

(Kug

)(x). (19)

Finally, with an inverse Laplace transformation the formaltime domain BIE for a homogeneous poroelastodynamicproblem with homogeneous initial conditions reads as

C (x) ug (x, t) = (V ∗ tg) (x, t)− (

Kug) (x, t), (20)

where the ∗ operator denotes the convolution in time

( f ∗ g) (t) =t∫

0

f (t − τ) g (τ ) dτ for t ∈ (0, T ). (21)

Other than that all operators are defined analogously as inLaplace domain.

3.2 Regularization of the double layer operator

As outlined in the title of this article, the aim is to work withkernels of certain regularity, i.e., showing at most weak sin-gularities. Motivated by the arguments given in the introduc-tion, the same regularization technique as in [11,12], whichitself is based on Han’s work, is used. However, only the nec-essary ingredients for its understanding are recalled, whereasfurther details may be found in [11].

The idea is to integrate (13) by parts first, which leads toa kernel function with higher regularity, before the limitingprocess (16) is performed. As a consequence of this regular-ization, the extension of this operator to the boundary onlyinvolves the jump term of Laplace’ equation. This findingis of special interest, since for the later presented colloca-tion BEM formulation a point-wise evaluation of the BIE isrequired and due to this fact the evaluation of the jump termeven at corner points remains rather simple.

The poroelastodynamic fundamental solution inherits thesingularities from linear elastostatics and Laplace’ equation[24]. Thus, having a look at the double layer kernel functionwith all arguments skipped

(TyU

)� =⎡

⎣

[T e

y sαny

−βn�y

β

sρ f0

n�y ∇y

] ⎡

⎣Us U f(

Ps)�

P f

⎤

⎦

⎤

⎦

�

=⎡

⎣Ts T f(

Qs)�

Q f

⎤

⎦

�, (22)

reveals that the same holds there as well, i.e., only the solidrelated part Ts becomes strongly singular due to the appli-cation of T e

y onto Us . This observation is easily verified by

means of a decomposition of Us into a singular and a regularpart, respectively

Us(r) = Ussing(r)+ Us

reg(r) with r := |y − x|= 1

μ

[I�y − λ+ μ

λ+ 2μ∇y∇�

y

]�yχ(r)

− 1

μ

[ ((k2

1 + k22

)�y − k2

1k22

)I

−(

k21 + k2

2 − k24 − k2

1k22

k23

)

∇y∇�y

]χ (r), (23)

with the definitions of k1, k2, k3, k4 and χ (r) given in Appen-dix A.1. Us

sing is explicitely given in the second line aboveand contains forth order derivatives of χ (r), which are ofO

(r−1

), whereas Us

reg one line below contains up to at mostsecond order derivatives of χ(r), which are of O

(r1

). Note

that Ussing not only contains the singularity of linear elasto-

statics as already mentioned before, but has the appropriatestructure as required for the regularization procedure as well.

Contrary to Ts , the second term on the main diagonal Q f ,i.e., the application of the normal derivative onto P f , does notlead to a strong singularity for smooth enough surfaces [9].Moreover, the off diagonal terms do not need any treatmenteither, as can be verified by having a look at the definition ofthe fundamental solution given in the Appendix A.1. Due tothis observation, only the regularization of the solid relatedpart is worked out in detail, whereas the remaining terms ofthe double layer kernel can be found in [24].

Together with the central assumption of a closed bound-ary ∂ = {∅}, Stokes theorem, and the definition of theskew symmetric Günter [14] differential operator My, themain ingredients for the regularization of Ts are identifiedand can be found in Appendix A.2. Here, only, the resultingintegration by parts formulas are recalled for the later use

∫

(My v

)udsy = −

∫

v(My u

)dsy (24)

∫

(My v

)� udsy =∫

v� (My u

)dsy. (25)

Next, writing Ts (22) out in more detail together with thebefore introduced decomposition (23), results in the follow-ing representation of the solid related double layer kernel

(Ts

)� =(T e

y

(Us

sing + Usreg

))� + sαPsn�y

=(T e

y Ussing

)� + O(r0). (26)

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Comput Mech (2011) 47:669–680 673

Introducing the same representation of T ey as in [14], this

rewrites as(

Ts)� = (λ+ 2μ) ny∇�

y Ussing

−μ(

ny ×(∇y × Us

sing

))+ 2μMyUs

sing + O(r0),

(27)

which for the structure of Ussing given in (23) can be rear-

ranged to(

Ts)� = My�

2yχ + I

(n�∇y

)�2

yχ

+2μ(MyUs

sing

)� + O(r0). (28)

This is already the desired form of the kernel function, whichis suitable for integration by parts. Plugging (28) into the def-inition of (13)(Ku

)s

�(x) =

∫

[ (My�

2yχ

)u +

(I(

n�∇y

)�2

yχ)

u

+2μ(MyUs

sing

)�u + O(r0)u

]dsy, (29)

the regularization can be performed by application of (24)to the first term on the right hand side and (25) to the thirdterm, respectively. Keeping the symmetry of Us

sing in mind,this yields(Ku

)s

�(x) =

∫

[−�2

yχ(Myu

) +(

I(

n�∇y

)�2

yχ)

u

+2μUssing

(Myu

) + O(r0)u]dsy, (30)

which is an equivalent form of the operator (13) under theassumption of a closed boundary.

What has to be done next, is the limit � � x → x ∈ ,which due to the before performed regularization is ratherstraightforward. For the last integral on the right hand sideof (30) this limit poses no problem. Moreover, since the firstand the third integrals on the right hand side of (30) haveweakly singular kernel functions, they can continuously beextended to the boundary as well. The only part, which needsto be investigated, is the second term containing the normalderivative of �2

yχ . However, with

(n�∇y

)�2

yχ (r) = n�∇yr

4πr2 + O(

r0)

(31)

the solid related jump term Cs (x) reduces to the jump termof Laplace’ equation c (x)

Cs (x) = I (x) c (x) with c (x) = �(x)4π

, (32)

where �(x) is the internal solid angle, which may be com-puted as presented by Mantic [18]. Due to the regularization,the jump term becomes a constant only depending on the

geometry, i.e., does neither depend on the material nor onthe Laplace parameter s. Therefore, the solid related part ofthe limit (16) finally becomes

lim��x→x∈

(Ku

)s

�(x) = −I (x) [−1 + c (x)] u (x)

+(Ku

)s(x) , (33)

with the regularized double layer kernel function (30).

Remark 1 Stressing the assumption of a closed boundaryonce again, for the implementation a more efficient formof (30) is found, which is obtained by augmenting Us

sing to

Us . This is done by adding and subtracting the regular part,making use of (25)(Ku

)s

�(x) =

∫

[−�2

yχ(Myu

) +(

I(

n�∇y

)�2

yχ)

u

+2μUs (Myu

) + O(

r0)

u]dsy . (34)

Since this modification only concerns regular parts of theoperator, the jump term is not affected at all.

Remark 2 The just performed regularization still requires theevaluation of weakly singular integral operators. Within anumerical scheme it does not suffice to just use standardGaussian quadrature, thus in the presented approach, before-hand a Duffy transformation [28] is performed.

3.3 Temporal discretization

Before the CQM is applied to the convolution operator in(20), its application to (21) is presented. Splitting the timeinterval (0, T ) into N +1 equidistant time steps�t , the con-volution at a discrete time tn = n�t is approximated by thequadrature rule

( f ∗ g)(tn) ≈n∑

k=0

ω�tn−k( f ) g(tk). (35)

The derivation of the quadrature weights can be found in [15],whereas here they are assumed to be given by definition

ω�tn ( f ) := 1

2π i

∮f

(γ (z)

�t

)z−(n+1)dz. (36)

For their computation, the closed contour has to lie entirelyin the domain of analyticity of f , but can be chosen of anyshape otherwise. Choosing it to be a circle centered at theorigin with radius R, parameterized by a polar coordinatetransformation z = Re−iϕ , allows to approximate the inte-gral (36) with the trapezoidal rule for L + 1 equal steps

ω�tn ( f ) ≈ R−n

L + 1

L∑

�=0

f (s�) ζ�n

with ζ = e2π iL+1 and s� = γ (Rζ−�)

�t, (37)

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674 Comput Mech (2011) 47:669–680

where γ is the quotient of the characteristic polynomialsresulting from the underlying linear multi-step method [15].The next steps follow the procedure as outlined in [1], thusplugging (37) into (35), extending the outer sum to k = N ,assuming L = N , and setting ω�t

k = 0 for negative indicesdue to causality, results in

( f ∗ g) (tn) ≈N∑

k=0

R−(n−k)

N + 1

N∑

�=0

f (s�) ζ�(n−k) g(tk)

≈ R−n

N + 1

N∑

�=0

f (s�) g (s�) ζ�n

with g (s�) =N∑

k=0

Rk g(tk)ζ−�k . (38)

It is easily verified that exchanging the summation orderof the finite double sum leaves the final result unchanged.Applying this procedure to the convolution operator of theoriginal problem (20)

C(x)ug(x, t) = (V ∗ tg) (x, t)− (

K ∗ ug) (x, t), (39)

yields a set of N +1 Laplace domain problems to be discret-ized in space

C(x)ug(x, s�) = (V tg)(x, s�)−

(Kug

)(x, s�)

� = 0 . . . N . (40)

Time domain results for the original problem, are obtainedvia the formula in the final line of (38). From an engineeringpoint of view, a more detailed presentation of this techniquemay be found in [26].

3.4 Spatial discretization

The computational domain = ∂� is approximated via astandard triangulation by the union of M triangles τm

≈ h =M⋃

m=1

τm . (41)

For the sought after solutions of (40), the following subspaceson h are defined

Sh[k](N ,h) := span{ϕαi [k]}Ii=1 (42)

Sh[k](D,h) := span{ψβj [k]}Jj=1, (43)

where k = 1 . . . 4 are the four poroelastic degrees of free-dom, and i and j the nodes on the Neumann, respectively theDirichlet boundary. The unknown Dirichlet datum is approx-imated by a linear combination of I continuous polynomialshape functions ϕαi [k] of order α ≥ 1, which ensures a mean-ingful regularized double layer operator (30). Contrary, the Jpiecewise discontinuous polynomial shape functions ψβj [k]

of order β ≥ 0 are used to approximate the unknown Neu-mann datum

ug[k] (x) ≈ ugh [k] (x)

=I∑

i=1

ugh,i[k]ϕαi [k] (x) ∈ Sh[k](N ,h) (44)

tg[k] (x) ≈ tgh [k] (x)

=J∑

j=1

tgh,j[k]ψβj [k] (x) ∈ Sh[k](D,h). (45)

Plugging these approximations into (40), yields N + 1 equa-tions with (I + J )×4 unknowns each. Claiming these equa-tions to be satisfied at J collocation points on the Dirichletboundary and I collocation points on the Neumann bound-ary, yields N + 1 algebraic systems of equations, which aresolvable for pairwise different collocation points. On theNeumann boundary, the obvious choice is to make themcoincide with the nodes, whereas on the Dirichlet bound-ary the matter is a little more complicated. Since discon-tinuous elements’ nodes coincide for neighboring elements,this choice does in general not deliver linear independentequations. This matter is solved by indenting the collocationpoints with respect to the nodes, such that their geometricalposition becomes unique, as it is required for all collocationpoints. With the introduced discretization and this choice ofcollocation points, the �th system of equations reads as[

VDD −KDN

VND −(C + KNN)

]

�

[tgD,hug

N,h

]

�

=[

−VDN (C + KDD)

−VNN KND

]

�

[gg

N,hgg

D,h

]

�

� = 0 . . . N .

(46)

Upper case subscripts indicate the boundaries, which the vec-tors and matrices are related to and, therewith, also theirdimension. Note that due to the block structure of the systemmatrix, a Schur complement can be defined

SNN := VNDV−1DDKDN − (C + KNN), (47)

which is used within the solution procedure for the linearsystems of equations, either with a direct or indirect solver.

4 Numerical examples

In both examples presented in this section, for the underlyinglinear multi-step method within the CQM, the BDF2 is cho-sen. The Dirichlet datum is continuously approximated bypiecewise linear and the Neumann datum discontinuouslyby piecewise constant shape functions. The relevant mate-rial data are given in Table 1. All computations have been

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Comput Mech (2011) 47:669–680 675

Table 1 Material data for rock (Berea sandstone) and a sandy soil [13]

Rock Soil

λ (N/m2) 0 1.44 × 108

μ (N/m2) 7.2 × 109 9.78 × 107

ρ (kg/m3) 2458 1884φ (−) 0.19 0.48α (N/m2) 0.778 0.981ρ f (kg/m3) 1000 1000R (N/m2) 4.885 × 108 1.206 × 109

κ (m4/N s) 1.9 × 10−10 3.5 × 10−9

performed by using the HyENA C++ library for solving par-tial differential equations using boundary element methods[21].

4.1 Wave propagation in a 1D-column

For sake of validation, wave propagation in a one-dimensional column is studied and the obtained results arecompared with the semi-analytic solution by Schanz andCheng [27]. A closed form of the time domain referencesolution is in general not possible, but can only be obtainednumerically—in this case via CQM. A column with length� = 3 m times a cross sectional area of A = 1 m × 1 m isconsidered (Fig. 1). To come as close as possible to the one-dimensional case, the surrounding sidewalls and the bottomface of the column are supported by impermeable roller bear-ings, i.e., zero normal displacement and no flux are allowedto happen, whereas in-plane motion is not restricted at all.The top face of the column is subjected to a step stress loadin normal direction of t1 = −1 N/m2 for t > 0 and zero pres-sure at all times. Furthermore, material data of rock (Table 1),with Poisson’s ratio artificially set to zero are used, resultingin λ = 0.

Fig. 1 Poroelastic column

(a) (b)

(c)

Fig. 2 Different spatial discretizations of the considered column

Introducing the dimensionless Courant–Friedrichs–Levynumber βCFL

βCFL := c1�t

h, (48)

allows to compare different spatial and temporal discretiza-tions. In the definition above, c1 is the fast compression wavespeed for t → 0,�t the chosen time step, and h marks themesh size, which is defined to be equal to the triangles’ hypot-enuses. The different meshes used are depicted in Fig. 2 andcharacterized as follows: COL1 has 224 elements and 114nodes, COL2 consists of 896 elements and 450 nodes, andCOL3 is made up by 2016 elements on 1010 nodes.

Figures 3 and 4 show displacement and stress results forthe solid, respectively pressure and flux results for the fluidphase with βCFL = 0.3 (its influence is studied afterwards,where one can see that values between 0.125 and 0.5 yieldgood results). The displacement u1 at the center of the topface and the flux q through it are plotted in Figs. 3a and4b, respectively. The traction results t1 and the pressure p inFigs. 3b and 4a are observed at the midpoint of the bottomface. These results show that COL1 is to coarse and COL3obviously delivers the best results. However, COL2 gives sat-isfactory results as well, which is why this mesh is chosennext to study the influence of the βCFL number.

For lower βCFL values (more elements per wave length)the CQM tends to become less stable, whereas for higher val-ues (less elements per wave length) the numerical damping

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676 Comput Mech (2011) 47:669–680

(a)

(b)

Fig. 3 Solid displacement and traction results for βCFL = 0.3 com-pared with the semi-analytic solution

increases more and more. These observations are confirmedby Fig. 5. There, again a one-dimensional wave propagationis investigated, however, now keeping the spatial discreti-zation constant (COL2) and varying βCFL by changing thetime step�t . While in Fig. 5a instability of the displacementresults is not visible for the lowest βCFL value, the tractionresults in Fig. 5b start to become unstable even at the nexthigherβCFL value (for sake of visibility not the whole range isplotted). On the other side, the displacement results clearlyshow the damping at higher βCFL values. Commenting onthe skipped results, the fluid pressure essentially behaves thesame as the solid traction, whereas the flux becomes evenmore unstable for lowerβCFL values, than the traction results.

A problem, frequently attributed to collocation methods,is the lack of long time stability compared to other methods,such as Galerkin type approaches. However, some long timecomputations have shown that with the collocation formula-tion chosen in this work, this problem does not seem to beas significant, as reported in [11] for the the same problem

flux

(a)

(b)

Fig. 4 Fluid pressure and flux results for βCFL = 0.3 compared withthe semi-analytic solution

in linear elasticity. The long time result for the fluid pressurebehaves almost identical to the traction solution depicted inFig. 6, whereas the result for the solid displacement seem tobehave a little better and the flux result is marginally worsein quality.

Remark 3 A novelty of this boundary element formulationis that poroelastodynamic flux results can be computed, e.g.,Fig. 4b. Such results have not yet been reported in literature.

4.2 Wave propagation in a half-space

The case of a poroelastic half-space is considered next. Inthe engineering community, this is a classical problem to bestudied with boundary elements. When dealing with such adomain, two obvious error sources are introduced: First, thediscretization error due to the necessary truncation of theinfinite boundary at some finite extent as depicted in Fig. 7and, second, the implication this truncation has on the double

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Comput Mech (2011) 47:669–680 677

(a)

(b)

Fig. 5 Comparison of different time steps (βCFL values) for COL2

Fig. 6 Long time traction solution for βCFL = 0.3

layer operator. This results in violation of its definition as aCauchy principal value in the classical formulation, or ina violation of the assumption of a closed boundary withinthe here presented approach. Note that this error is strictlydue to the discretization, on the continuous level, where theregularization was performed, ∂ = {∅} still holds for thehalf-space, too.

(a)

(b)

Fig. 7 Half-space with a finite discretization

Comparing the results obtained by the here presentedapproach with results by Schanz [24], shows that they areof at least the same quality. Schanz also developed a CQMbased collocation BEM, however there the double layer oper-ator is evaluated numerically [8] and, furthermore, all dataare continuously approximated by piece-wise linear shapefunctions with double nodes taking care of discontinuities.

The discretized half-space in Fig. 7 is a flat patch of10 m × 10 m using a triangulation with 484 nodes and 882regular triangles, which outer normal points in the positivex3-direction. For the material data, properties of soil as listedin Table 1 are assumed. The marked area of 0.476 m×0.476 mat the center in Fig. 7b is loaded with a traction step loadof t3 = −1 N/m2 for t > 0, whereas all remaining ele-ments are traction free. Furthermore, zero pressure is appliedon the entire surface. The observation point for the verticaldisplacement u3 is located on the diagonal at 5.39 m fromthe loaded area. The results are depicted in Fig. 8 versustime.

The arrival time of the fast compression wave at the obser-vation point computes to t = 0.0032 s, which is in rathergood agreement with the BEM results, whereas the slow com-pression wave is not visible for the given material parameters.Due to the similar wave speed of the shear- and the Rayleighwave, a distinction at such a short distance is not possible,which makes the identification of their arrival times impos-sible. Nevertheless, their computed arrival times are markedin Fig. 8.

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678 Comput Mech (2011) 47:669–680

Fig. 8 Vertical displacement u3 at the observation point versus time(βCFL = 0.5)

The oscillation after all waves have passed results fromthe CQM, which is as nearly all time stepping techniques notcapable to resolve jumps. Computing the arrival time of thereflected Rayleigh wave from the most distant point, resultsin t = 0.096 s. In Fig. 8, one can observe that at this timethe solution reaches already a static value. The explanationfor this phenomenon are slight reflections on the truncation,which however, are highly damped. In fact, multiple oscilla-tions cannot be observed at all.

Remark 4 The first example was computed using the regu-larized version of the double layer operator (34) mentioned inRemark 1. However, the additional modification mentionedtherein, seems to affect the half-space solution too much (theassumption of a closed boundary is violated), which is whythe computation of the second example was performed using(30).

5 Conclusions

Based on Biot’s theory a regularized double layer operatorfor the u, p formulation in Laplace domain has been derived.As a result of integration by parts a weakly singular represen-tation of the first boundary integral equation was obtained,starting from where a CQM based collocation BEM wasestablished.

At this point it should be pointed out that neither the col-location scheme, nor the CQM are restrictions to the usedregularization technique, since it is performed on a continu-ous level. In fact, the resulting weakly singular first bound-ary integral equation can also be used within a Galerkinscheme, too. Furthermore, since the regularization techniquewas used to remove the strong singularity inherited from lin-ear elastostatics, the proposed method must also work for

other formulations than the one presented in this article, e.g.,all poroelastic BEM formulations listed in the introduction.Additionally, since this regularization technique is just anintermediate result of the hypersingular operator’s regular-ization, an extension to a symmetric Galerkin formulationcomes along quite naturally.

The major assumption within the applied regularizationtechnique was the requirement of domains with closed bound-ary. This restriction is a consequence of Stokes theorem onwhich the whole technique relies. However, the numericalexample considering the poroelastic half-space shows thatdisregarding the violation of this assumption the results areof at least the same quality as obtained with other formula-tions found in literature.

Together with the point-wise evaluation of the jump termwithin the framework of the presented weakly singular firstboundary integral equation, the proposed collocation schemeallows a flexible approximation of the Cauchy data with basi-cally any meaningful combination of trial spaces and choiceof collocation points. In the way it was presented here, a sys-tem matrix with block structure results, which maintains thephysical properties of the operators.

As a final remark one may note that the proposed methodalso works for linear elasticity and viscoelasticity, since thereone has to deal with the same singularity.

Acknowledgments The authors gratefully acknowledge the financialsupport of the Austrian Science Fund (FWF) within the DoktoratskollegW1208—Numerical Simulations in Technical Science.

A Appendix

A.1 Poroelastodynamic fundamental solution

The Laplace domain fundamental solution

U(r) =⎡

⎣Us(r) U f (r)

(Ps

)�(r) P f (r)

⎤

⎦ with r := |y − x| (A.1)

is recalled from [24]. There, the solid displacement relatedpart of the fundamental solution is given as

Us = 1

4πr(ρ − βρ f )

[R1

k24 − k2

2

k21 − k2

2

e−k1r − R2k2

4 − k21

k21 − k2

2

e−k2r

+(

Ik23 − R3

)e−k3r

]

with R j = 3∇yr ∇�y r −I

r2 +k j3∇yr ∇�

y r −I

r+k2

j ∇yr ∇�y r

Us = 1

8πμr (λ+2μ)

[(λ+ μ)∇yr∇�

y r +I (λ+ 3μ)]+O

(r0

),

(A.2)

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Comput Mech (2011) 47:669–680 679

where k1, k2, and k3 are the wave numbers related to thefast/slow compressional, and the shear wave [24], whereasthe abbreviation k4 = k3

√μ/λ+2μ is just introduced for nota-

tional convenience. For the regularization, however, a moresuitable form of (A.2) is needed, which is obtained by thefollowing decomposition

Us(r) = Ussing(r)+ Us

reg(r) with r := |y − x|= 1

μ

[I�y − λ+ μ

λ+ 2μ∇y∇�

y

]�yχ (r)

− 1

μ

[ ((k2

1 + k22

)�y − k2

1k22

)I

−(

k21 + k2

2 − k24 − k2

1k22

k23

)

∇y∇�y

]χ(r), (A.3)

with

χ (r) = 1

4πr

[e−k1r

(k22 − k2

1)(k23 − k2

1)+ e−k2r

(k22 − k2

1)(k22 − k2

3)

+ e−k3r

(k21 − k2

3)(k22 − k2

3)

]

= − 1

4π(k2

1 − k22

) (k2

1 − k23

) (k2

3 − k22

) + O(

r2).

(A.4)

The remaining parts of (A.1) are defined as

U f (r) = ρ f (α − β)∇yr

4πrβ(λ+ 2μ)(k21 − k2

2)

[(k1 + 1

r

)e−k1r

−(

k2 + 1

r

)e−k2r

]= O

(r0

)(A.5)

Ps(r) = U f (r)

s= O

(r0

)(A.6)

P f (r) = sρ f

4πrβ(k2

1 − k22

)[(k2

1 −k24)e

−k1r −(k22 − k2

4)e−k2r

]

= sρ f

4πβr+ O

(r0

). (A.7)

Even though (A.7) is weakly singular, the application of thetraction operator (14) does not lead to a strong singularity[9] in the double layer kernel, thus no special representationof it is required.

A.2 Stokes theorem and integration by part formulas

In R3, Stokes theorem for a differentiable vector field a(y)with

y ∈ = � \� is given by∫

(∇y × a,ny)

dsy =∫

∂

(a, ν) dγy, (A.8)

where ν is the unit tangent vector along the contour ∂.Obviously, for a closed contour ∂ = {∅} the right hand sidevanishes, which allows to rewrite (A.8) for this special caseas∫

(ny × ∇y, a

)dsy = 0. (A.9)

With the definition of the Günter [14] derivative My :=(

ny∇�y

)� − ny∇�y from (A.9) the following formula can be

derived∫

(My a

)dsy = 0. (A.10)

This formula is helpful for the derivation of the integration bypart formulas [11] for the regularization of the double layerkernel function. Applying (A.10) to a vector a = vu yields∫

(My v

)udsy = −

∫

v(My u

)dsy, (A.11)

and for the double contraction of My with uv�∫

(My v

)� udsy =∫

v� (My u

)dsy (A.12)

is found.

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