Dynamic wave propagation in infinite saturated ...

Comput Mech DOI 10.1007/s00466-011-0647-9

ORIGINAL PAPER

Dynamic wave propagation in infinite saturated porous media half spaces Y. Heider · B. Markert · W. Ehlers

Received: 29 June 2011 / Accepted: 4 September 2011 © Springer-Verlag 2011

Abstract From a macroscopic perspective, saturated porous materials like soils represent volumetrically interacting solid–fluid aggregates. They can be properly modelled using continuum porous media theories accounting for both solidmatrix deformation and pore-fluid flow. The dynamic excitation of such multi-phase materials gives rise to different types of travelling waves, where it is of common interest to adequately describe their propagation through unbounded domains. This poses challenges for the numerical treatment and demands special solution strategies that avoid artificial and numerically-induced perturbations or interferences. The present paper is concerned with the accurate and stable numerical solution of dynamic wave propagation problems in infinite half spaces. Proceeding from an isothermal, biphasic, linear poroelasticity model with incompressible constituents, finite elements are used to discretise the near field and infinite elements to approximate the far field. The transient propagation of the poroacoustic body waves to the infinity is thereby modelled by a viscous damping boundary, which, for stability reasons, necessitates an appropriate treatment of the included velocity-dependent damping forces. Keywords Poroacoustics · Infinite elements · Wave propagation · Half-space · Viscous damping boundary

Y. Heider · B. Markert (B) · W. Ehlers Institute of Applied Mechanics (CE), University of Stuttgart, Pfaffenwaldring 7, 70569 Stuttgart, Germany e-mail: [email protected] Y. Heider e-mail: [email protected] W. Ehlers e-mail: [email protected]

1 Introduction Dynamic wave propagation in semi-infinite domains is of great importance, especially in the fields of geomechanics, civil engineering and seismology. As examples, consider the transient excitation of inventory buildings due to on-site construction of a driven pile foundation or the hazardous seismic impacts caused by earthquakes. For the theoretical description of travelling waves through vast areas, the usage of continuum-mechanical approaches is a standard practise. In this regard, as long as fluid-saturated porous domains such as subsurface soil layers are concerned, the theory of porous media (TPM) is proven to provide a comprehensive and elaborated modelling framework. Thereby, saturated porous materials are treated as multiphasic aggregates consisting of interacting solid and fluid constituents, which, independent of their usually unknown microtopology, are assumed to be in a state of ideal disarrangement. By homogenisation over some representative elementary volume (REV), an averaged continuum model is obtained, in which each spatial point is simultaneously occupied by all constituents in the sense of superimposed continua. This way of treating multiphasic porous materials can be traced back to the theory of mixtures (TM), cf. Bowen [14] or Truesdell and Toupin [66], where the TM was extended later by the concept of volume fractions to additionally incorporate information about the local composition of the smeared-out continuum (cf. Goodman and Cowin [33]), which is fundamental to the TPM. The approach has been employed by Drumheller [26] to describe an empty porous solid, and Bowen [15,16] extended this study to fluid-saturated porous media considering compressible as well as incompressible constituents. Subsequent developments were mainly driven by geomechanical issues and have substantially been contributed by the works of de Boer and Ehlers, see [10,27]

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for detailed references. Another popular approach to model porous materials, which is based on a generalisation of the theory of elasticity, is Biot’s Theory (BT) introduced in the early works of Biot [7,8]. In fact, BT, the TM, and the TPM are considered the bases of many later works in the field of porous media dynamics modelling, see Breuer [17], Coussy [22], Lewis and Schrefler [47], Zienkiewicz et al. [78,79], and Diebels and Ehlers [23] among others. The appearance and behaviour of acoustic waves in saturated porous media basically depend on the frequency of the excitation, the hydraulic permeability, and the mechanical properties of the constituent materials, see the review chapter of Corapcioglu and Tuncay [21], the book of Straughan [65] and also [57,56,73,72,64] for further particulars. In general, three apparent modes of bulk waves can be observed in biphasic solid–fluid aggregates, cf. Biot [8,9]: (1) The fast and weakly damped compressional waves (p1) with an in-phase motion of the solid and fluid constituents. The appearance of this type of waves is mainly governed by the compressibility of the constituents, which in case of the materially incompressible twophase model yields a theoretically infinite propagation speed. (2) The slow (p2 or Biot) longitudinal waves with outof-phase motion of solid and fluid. Proceeding from a strong viscous momentum coupling between the constituents associated with a low hydraulic permeability, this highly damped type of waves cannot propagate in the domain under low-frequency excitations, cf. Steeb [64]. In case of a materially incompressible pore fluid, the appearance of the p2-wave is mainly governed by the deformability of the solid skeleton, and they can be observed next to permeable, dynamically-loaded boundaries. (3) The transverse shear waves (s) are transmitted only in the solid skeleton and are mainly governed by its shear stiffness. The present contribution has its main focus on geotechnical problems, where commonly low-frequency excitations and low permeabilities entail very low relative motions between the solid matrix and the viscous pore fluid. Thus, it is accepted that far from the permeable boundaries, the pore fluid is almost trapped in the solid matrix, and therefore, the fluid can be approximately treated as an incompressible material together with the solid phase. Here, only biphasic poroelastic media with intrinsically incompressible solid and fluid constituents in the low-frequency regime are addressed, which gives rise to only two types of bulk waves, viz., the longitudinal p2-waves and the transverse s-waves transmitted through the elastic structure of the solid skeleton. Restricting the presentation to the low-frequency range associated with

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a Poiseuille-like flow on the pore scale, the wave velocities are derived based on the TPM equations in combination with an acceleration wave analysis involving singular surfaces. Against the background of an efficient numerical treatment, one has to take into consideration that in unbounded domains, acoustic body waves are supposed to propagate towards infinity. Thus, it is sensible to divide the semi-infinite domain into a finite near field surrounding the source of vibration and an infinite far field accounting for the energy radiation to infinity. Applying the finite element method (FEM) for the spatial semi-discretisation of the near field and simply truncating the domain by introducing artificial fixed (hard) or free (soft) boundaries causes the incident waves to be reflected back into the near domain. This might be partly overcome by choosing the dimensions of the near field large enough in order to ensure that the reflected waves will not disturb the progressing ones in the region of interest till the end of the computation. However, this commonly entails a huge number of finite elements followed by uneconomic computational costs. In this regard, numerous approaches have been proposed in the literature to efficiently treat unbounded spatial domains, see, e. g., the works by Givoli [32] and Lehmann [46] for an overview. In the following, some of the most popular methods used in geomechanics and earthquake engineering are outlined: (1) The finite element–boundary element coupled scheme (FEM–BEM) as given in the works by, e. g., Yazdchi et al. [75], von Estorff and Firuziaan [69], or Schanz [61]. Herein, the near field is modelled with the FEM, whereas the far-field response is captured using the BEM. Although this method exposes a good accuracy by combining the advantages of both mentioned discretisation techniques, the implementation involves a lot of mathematical complexities as it requires the derivation of fundamental solutions including an analytical treatment of the governing differential balance equations. (2) The combined finite element–infinite element method (FEM–IEM), in which the near field is discretised using the FEM and the far field using the IEM that extends to infinity in one or more directions. The shape functions of the infinite elements towards the infinity, usually called ‘wave propagation functions’ in the dynamic treatment, have exponential form and depend on the frequency of the excitation. Using the FEM–IEM method, the whole problem is solved in the frequency domain. Otherwise, inverse Laplace transformation rules need to be implemented to solve the problem in the time domain, cf. Khalili et al. [42] or Wang et al. [70]. In this connection, a domain decomposition strategy into a near and a far field can be applied for the time stepping in the time domain as given by Nenning and Schanz [58] and

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Schanz [61]. Here, the Newmark integration method can directly be applied to the near field to evaluate the temporal behaviour, as it is discretised with the FEM. However, as the IEM is applied to the far field together with the inverse Laplace transformation, the temporal response of the far field can simply be calculated in the time domain using the convolution quadrature method (CQM). (3) In the framework of absorbing boundary condition (ABC) schemes, the method of perfectly matched layers (PML) appears as an important strategy to simulate the response of unbounded half-space dynamics, see, e. g., Basu [5], Basu and Chopra [6], and Oskooi et al. [59], for details. This method was originally developed for electromagnetic wave problems and later extended to elastic wave propagation in infinite domains. The PML is an unphysical wave absorbing layer, which is placed adjacent to the truncated near-field boundaries that are supposed to extend to infinity. The purpose of the PML is basically to prevent wave reflections back into the near domain. This scheme is applied mostly when solving wave equations in the frequency domain as the solution is based on frequency-dependent stretching functions. In the current contribution, the simulation of wave propagation into infinity is realised in the time domain. Here, the near field is discretised with the FEM, whereas the spatial discretisation of the far field is accomplished using the mapped IEM in the quasi-static form as given in the work by Marques and Owen [54]. This insures the representation of the far-field stiffness and its quasi-static response instead of implementing rigid boundaries surrounding the near field, cf. Wunderlich et al. [74]. This mapped IEM has already been successfully applied by Schrefler and Simoni to simulate the isothermal and non-isothermal consolidation of unbounded biphasic porous media, see [62,63]. In particular, they have performed a coupled analysis under quasi-static conditions, where infinite elements with different decay functions are applied to the solid displacement, the pore pressure and the temperature fields. Moreover, they have calibrated the numerical results by comparison with respective analytical reference solutions. However, in dynamical applications, some additional effects must be taken into account. In fact, when body waves approach the interface between the FE and the IE domains, they partially reflect back to the near field as the quasi-static IE cannot capture the dynamic wave pattern in the far field. To overcome this, the waves are absorbed at the FE–IE interface using the viscous damping boundary (VDB) scheme, which basically belongs to the ABC class. The idea of the VDB is based on the work by Lysmer and Kuhlemeyer [48], in which velocity- and parameter-dependent damping forces are

introduced to get rid of artificial wave reflections. Lysmer and Kuhlemeyer verified the VDB scheme by studying the reflection and refraction of elastic waves at the interface between two domains, where the arriving elastic energy should be absorbed. For more information and different applications, see, e. g., the works by Haeggblad and Nordgren [36], Wunderlich et al. [74] and Akiyoshi et al. [1]. For the numerical treatment, saturated, materially incompressible porous media are represented by a set of differential-algebraic equations (DAE), which belongs to the class of strongly coupled problems discussed by, e. g., Felippa et al. [31] and Matthies et al. [55] or more generally by Markert [51]. In our implementation, the velocity-dependent and with that solution-dependent damping terms of the VDB method enter the weak formulation of the initial-boundaryvalue problem in form of a boundary integral. Hence, an unconditionally stable numerical solution requires the damping forces to be treated implicitly as weakly imposed Neumann boundary conditions (cf. Ehlers and Acartürk [28]). In the literature, a number of solution strategies are available to solve this type of DAE. For instance, implicit monolithic approaches and semi-explicit–implicit splitting schemes, cf. Markert et al. [53], or time and coupled space-time discontinuous Galerkin methods, cf. Chen et al. [19,20]. In this work, an implicit monolithic solution strategy is implemented, where the spatial semi-discretisation is carried out using the FEM and IEM resulting in a time-continuous DAE system. For the time discretisation, a diagonal implicit Runge–Kutta (DIRK) time-stepping algorithm is applied. Here, the method of choice is the second-order accurate and L-stable TR-BDF2 method (Bank et al. [4]) as a composite integration scheme combining the advantages of both the trapezoidal rule (TR) and the second-order backward difference formula (BDF2). In summary, this work is structured as follows: In Sect. 2, the basics of the TPM model, the governing balance equations, and the derivation of bulk wave velocities are introduced. Section 3 is concerned with the numerical treatment, such as the derivation of the weak formulation, treatment of the far field and the weakly imposed VDB, the spatial semi-discretisation using the FEM and IEM, and the time discretisation applying the TR-BDF2 scheme together with the numerical generation of the stiffness and mass matrices. The discussed formulations and schemes are applied in Sect. 4 to a one- and a two-dimensional wave propagation example, followed by the conclusions in the last section.

2 Theoretical basics For the sake of mathematical modelling of the physical phenomena of wave propagation in saturated porous media, fundamentals of multiphasic continuum theories are briefly

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2.2 Kinematics of multi-phase continua

Fig. 1 REV of saturated sand showing the granular micro-structure and the biphasic TPM macro model with ϕ = ϕ S ∪ ϕ F

introduced. This includes the basic concepts of the TPM, the kinematics of biphasic media, the governing balance and constitutive relations, and the derivation of body wave velocities. For a more detailed discussion, the interested reader is referred to, e. g., [11,27,53]. 2.1 Theory of porous media (TPM) The TPM is used for the continuum-mechanical modelling of a biphasic porous body consisting of an immiscible solid skeleton saturated by a single interstitial fluid. In this regard, the heterogeneous solid aggregate is assumed to be statistically distributed over a representative elementary volume (REV). Following this, homogenisation is applied to the REV, cf. Fig. 1, yielding a macroscopic multiphasic continuum ϕ with overlaid and interacting solid and fluid phases ϕ α (α = S : solid phase; α = F : pore-fluid phase), so that we have ϕ = ϕ S ∪ ϕ F at any macroscopic spatial point. Following this, the concept of volume fractions is introduced in order to integrate constituent microscopic information. Thus, a volumetric averaging process of all constituents is prescribed over the REV, and the incorporated physical fields of the micro-structure are represented by their local volume proportions. In particular, the volume fraction n α := dv α /dv of ϕ α is defined as the ratio of the partial volume element dv α to the total volume element dv of ϕ. The saturation constraint for the case of a fully saturated medium is given by  S  n : solidity, α S F n = n + n = 1 with (1) n F : porosity, α and assumed to be satisfied during the whole deformation process. Additionally, the current treatment proceeds from 0 < n α < 1, where the transition of the multiphasic material into a dense single-phase medium (pure solid or fluid) is not a case of study. The introduction of the n α is furthermore associated with two density functions, i. e., a material (effective or intrinsic) density ρ αR and a partial density ρ α relating the local mass of ϕ α to the partial or the bulk volume element, respectively. It is easily concluded that ρ α = n α ρ αR , which underlines the general compressibility of porous solids under drained conditions through possible changes of the volume fractions.

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Proceeding with the kinematic formulations of multiphasic continuum mechanics, which have been adopted from mixture theories, cf. [14,38], one makes use of the concept of superimposed continua with internal interactions and individual states of motion. Thus, starting from different reference positions Xα at time t0 , each constituent is supposed to follow its unique individual Lagrangian (material) motion function χ α and has its own velocity and acceleration fields, viz.: x = χ α (Xα , t) ⇔ Xα = χ −1 α (x, t), dα x d2 x  , (vα )α = xα = α 2 . vα := xα = dt dt 

(2)

Therein, χ −1 α represents the unique inverse (Eulerian or spatial) motion function and ∂( q) dα ( q ) = + grad ( q ) · vα ( q )α := dt ∂t (3) ∂( q) q with grad ( ) = ∂x indicates the material time derivative following the motion of ϕ α . By virtue of (2)1 , each spatial point x of the current configuration at time t is simultaneously occupied by a material point of both constituents, reflecting the overlay of the partial continua. In porous media theories, it is convenient to proceed from a Lagrangian description of the solid matrix via the solid displacement u S and velocity v S as the kinematical variables. However, the pore-fluid flow is expressed either in a modified Eulerian setting via the seepage velocity vector w FS describing the fluid motion relative to the deforming skeleton, or by an Eulerian description using the fluid velocity v F itself. In particular, we have 

u S = x − X S , v S = (u S )S = x S , 

(4)

v F = x F , w FS = v F − v S . Please note that in the governing balance relations used for the numerical treatment, cf. Sect. 2.3, v F is adopted as the primary unknown for describing the pore-fluid motion rather than w FS , since this choice has positive impacts on the numerical stability of the monolithic implicit treatment as discussed in [53]. 2.3 Governing balance and constitutive relations The considered biphasic model excludes thermal effects as well as any mass exchanges (inert ϕ α ) and proceeds from intrinsically incompressible constituents (ρ α R = const. ). In particular, the arising purely mechanical, binary model with

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α = {S, F} is governed by the following constituent balance equations: • Partial mass balance −→ partial volume balance: (ρ α )α + ρ α div vα = 0 −→ (n α )α + n α div vα = 0 .

pˆ EF = −

(6)

where k F > 0 denotes the conventional hydraulic conductivity (Darcy permeability) in m/s and γ FR = ρ FR g is the effective fluid weight with g = |b| = const. as the scalar gravitational acceleration. Moreover, using (3), the material time derivative with respect to the fluid motion can be written as

Herein, div ( q ) is the divergence operator related to grad ( q ), Tα = (Tα )T is the symmetric partial Cauchy stress assuming non-polar constituents, b is the mass-specific body force acting on the overall aggregate, and pˆ α denotes the direct momentum production, which can be interpreted as the volume-specific local interaction force between the percolating pore fluid and the solid skeleton. Due to the overall conservation of momentum, pˆ S + pˆ F = 0 must hold for any closed multiphasic system. From (5) with α = S and (n S ρ S R )S = (n S )S ρ S R , one directly obtains the solidity as a secondary variable by analytical integration: (n S )S = −n S div v S −→ n S =

S n 0S

det F−1 S

(7)

S being the initial volume fraction of ϕ S at time t and with n 0S 0 F S = ∂ x/∂ X S as the solid deformation gradient. Moreover, proceeding from a small strain approach, n S can be written in geometrically linear form as S (1 − div u S ) . n S ≈ n 0S

(8)

To continue, according to the principle of effective stresses, see [12] for references, Tα and pˆ F can be split into effective field quantities, the so-called extra terms indicated by the subscript ( q ) E , and parts that are governed by the pore-fluid pressure p : Tα = TαE − n α p I , pˆ F = pˆ EF + p grad n F

with

(10)

as the geometrically linear solid strain tensor and μ S , λ S being the macroscopic Lamé constants of the porous solid matrix. Moreover, it follows from a dimensional analysis that in a macroscopic porous media approach div T EF  pˆ EF (cf. [25,50]), such that in good approximation T F ≈ −n F p I.

(11)

( q )F = ( q )S + grad ( q ) · w FS ,

(12)

where in the geometrically linear case, the nonlinear convective term can be omitted by magnitude arguments as grad ( q ) · w FS  ( q )S yielding ( q )F ≈ ( q )S . In summary, inserting the aforementioned constitutive and kinematic relations into (6), the ‘convectiveless’ governing set of coupled partial differential equations (PDE) reads: • Momentum balance of the overall aggregate: ρ S (v S )S +ρ F (v F )S = div (T SE − p I)+(ρ S + ρ F ) b. (13) • Momentum balance of the pore fluid: ρ F (v F )S = −n F grad p+ρ F b−

(n F )2 γ FR w FS . kF

(14)

• Volume balance of the overall aggregate: div (v S +

(9)

with I being the 2nd-order identity tensor. With regard to a thermodynamically consistent model, admissible constitutive equations for the response functions TαE and pˆ EF must be provided. Restricting the presentation to the small strain regime, the solid extra stress is determined by the Hookean elasticity law T SE = 2μ S ε S + λ S (ε S · I) I ε S = 21 (grad uS + gradTuS )

(n F )2 γ FR w FS , kF

(5)

• Partial momentum balance: ρ α (vα )α = div Tα + ρ α b + pˆ α .

Furthermore, under the assumption of isotropic lingering flow conditions at low Reynolds numbers, the percolation process is appropriately described by a linear Darcy-type filter law, which can be traced back to the simple but thermodynamically consistent ansatz

 kF   grad p − (b − (v ) ) ) = 0. F S γ FR   n F w FS

(15)

Note that the chosen primary unknowns for this set of PDE are u S , v F , and p. Hence, v S (u S ) as well as T SE (u S ), n S (u S ), n F (u S ), and w FS represent the secondary variables of the problem. Moreover, the fluid momentum balance (14) is solved with respect to the filter velocity n F w FS and replaced into the volume balance of the overall aggregate (15). This modification guarantees that Eqs. 13–15 after spatial semidiscretisation will yield a system of DAE with differential index 1, which is desirable for a smooth numerical solution. For particulars on index reduction methods, see, e. g., [3,37,53] among others. Additionally, a reduction in the order of the PDE to order-one in time is achieved using (u S )S = v S ,

(16)

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order m of jump-free spatial derivatives of  via n = m + 1, where the order of time derivatives is coupled with that of spatial derivatives. In this work, velocity but not acceleration fields are continuous across I. Hence, the singular surface is of order two. Following this, the terms in the balance relations are tested in regard of their continuity over I. Starting with the jump-free fields, we have [[vα ]] = 0, [[ p]] = 0, [[ρ α R ]] = 0, [[n α ]] = 0.

Fig. 2 Biphasic porous body B = B+ ∪ B− in the actual configuration divided by an immaterial singular surface I

which eliminates the second time derivative of the solid displacement from (13), and allows the applicability of a wide range of time-stepping algorithms such as diagonal-implicit DIRK methods. 2.4 Derivation of bulk wave velocities According to Hadamard [35], acoustic waves in poroelastic continua are assumed to be non-destructive and defined as isolated, non-material surfaces (wave fronts) that move relative to the material constituents. As accelerations but not velocities are considered to be discontinuous across the wave fronts, acoustic waves are also called acceleration waves. For a more detailed discussion, we refer to [40,49,71]. Figure 2 illustrates a wave-front surface I, which divides a domain B into two sub-domains B + and B − . Therein, nI is the normal unit vector to I in the actual configuration pointing from B − to B + . Assuming that vI is the velocity of I in the actual configuration, the velocity in which the surface I moves relative to the constituent ϕ α and perpendicular to I is referred to as wI α and expressed by wI α = wI α nI with wI α = (vI − vα ) · nI .

(17)

In this connection, for a scalar-valued field function (x, t), which is continuous and sufficiently differentiable over B + and B − , the following limits on I with x ∈ I can be defined:  + = lim (x + ε nI , t), 

−

ε→0

= lim (x − ε nI , t). ε→0

(18)

The jump of  across I is then given as [[]] =  + −  − .

(19)

If [[]] = 0, a jump exists and the surface I is called singular with respect to . Otherwise, it is a jump-free case. The order n of the singular surface is determined by the maximum

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(20)

Therein, additional to the velocity vα and the pore pressure p, the constant material densities ρ α R and the volume fractions n α are continuous over I. Unlike the case of a barotropic fluid with ρ FR = ρ FR ( p), the pore pressure p is independent of ρ FR in the case of materially incompressible fluid yielding that [[grad p]] = 0, cf. [44] and also [52]. Next, according to Hadamard’s lemma [35], the compatibility condition, and Maxwell’s theory for second-order singular surfaces, one finds the following jump expressions: [[(vα )S ]] = (wI α )2 ξ α , [[div vα ]] = [[grad vα ]] · I = −wI α ξ α · nI , [[div T SE ]] = [[grad T SE ]] I

= 2μ S [[grad ε S ]] I+λ S [[grad (ε S · I)]]

(21)

with [[grad ε S ]] =

1 2

(ξ S ⊗ nI + nI ⊗ ξ S ) ⊗ nI ,

[[grad (ε S · I)]] = (nI ⊗ nI ) ξ S .

(22)

Herein, ⊗ is the dyadic tensor product operator and ξ α is an amplitude vector of the discontinuity resulting from the compatibility condition. Following this, applying the continuity condition to the momentum balance of the overall aggregate (13) and inserting the resulting jump relations from Eqs. 20– 22 reads ρ S [[(v S )S ]] + ρ F [[(v F )S ]] = [[div T SE ]], yielding

 ρ S (wI S )2 I − μS (I + nI ⊗ nI ) − λS (nI ⊗ nI ) ξ S +ρ F (wI F )2 ξ F = 0.

(23)

In order to distinguish between the transversal and the longitudinal modes of motion, the rotational and divergence concepts are employed. Therefore, (23) is multiplied by the vector mI perpendicular to nI , i. e. mI · nI = 0, in order to obtain the transversal component of the motion. This yields   S ρ (wI S )2 − μS ξ S · mI +ρ F (wI F )2 ξ F · mI = 0. (24) As discussed in the introduction, in geomechanical dynamic problems, such as under seismic loading, it is convenient to proceed from moderately small values of k F associated with negligible relative accelerations of the solid

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and the fluid, i. e. an insignificant time derivative of the seepage flow in the low frequency range, cf. [53,79].

3.1 Governing weak formulation

In this case, we may write x S ≈ x F , wI S ≈ wI F and ξ S ≈ ξ F . Consequently, relation (24) simplifies to   S (25) (ρ + ρ F ) (wI S )2 − μS ξ S · mI = 0,

Following the FEM procedure for deriving the weak formulation,e. g. [76], Eqs. 13–15 are weighted by independent test functions and integrated over the spatial domain occupied by the overall aggregate body B. Therefore, the surface  = ∂ is split into Dirichlet (essential) and Neumann (natural) boundaries, respectively, yielding  = u ∪ t for the momentum balance of the overall aggregate,  = v F ∪ t F for the fluid momentum balance, and  =  p ∪ v for the volume balance of the overall aggregate. Next, applying the product rule and the Gaußian integral theorem, one obtains the weak forms of the uv p -formulation. In particular, the weak form of the overall aggregate momentum balance G u S , which will be used in later sections for the treatment of the far-field response, reads

S grad δu S · (T E − p I) dv − δu S · t da +





yielding cs := wI S =

μS ρS + ρF

(26)

as the shear-wave propagation speed. The resulting constant shear-wave velocity is suitable for the implementation of the viscous damping boundary scheme. In fact, the latter simplification makes the behaviour of the biphasic aggregate similar to a single-phase elastic material, which is also commonly used in geophysical applications under low-frequency assumptions, cf. [73]. In analogy, the velocity of the longitudinal waves is obtained by a scalar multiplication of Eq. 23 by a vector nI , where nI · nI = 1, yielding   S ρ (wI S )2 − 2μS − λS ξ S · nI + ρ F (wI F )2 ξ F · nI = 0.

(27)

In the low frequency range with negligible seepage flow acceleration, the pressure-wave speed c p := wI S ≈ wI F can be expressed similar to a single-phase elastic material as 2μ S + λ S cp = . (28) ρS + ρF In this contribution, (26) and (28) are used in the numerical examples for the treatment of the two modes of wave damping and the application of the viscous damping boundary method.

  + δu S · ρ S (v S )S + ρ F (v F )S − ρ b dv = 0

(30)



with δu S being the test function corresponding to the primary variable u S or v S . Moreover, t = (T SE − p I) n is the external load vector acting on the Neumann boundary t of the overall aggregate with outward-oriented unit surface normal n. For the weak formulation of the other balance relations, the interested reader is referred to [53]. 3.2 Spatial semi-discretisation For the spatial semi-discretisation, the continuous control space occupied by the overall aggregate B is subdivided into Ne non-overlapping finite elements yielding an approximate discrete domain h . The FEM is applied to the near field of the domain yielding a FE mesh with Nx nodes for the geometry approximation, on which the following discrete trial and test functions are defined: Nu 

Nu(i) (x) u(i) (t) ∈ Suh (t) ,

i=1

In the following, the governing set of PDE (13–15) with primary unknowns u S , v F , p is treated in a fully coupled manner and referred to as uv p -formulation. Additionally, the secondary variable v S (u S ) from (16) is considered together with the primary variables to obtain a coupled equation system of order one in time. Consequently, the vector of unknowns u of the three-field problem is given by

see [39,53] for more details.

t

uh (x, t) = uh (x, t) +

3 Numerical treatment

T  u = u(x, t) with u = u S , v S , v F , p ,



(29)

δuh (x) =

Mu 

Mu(i) (x) δu(i) ∈ Tuh .

(31)

i=1

Therein, uh are the approximated Dirichlet boundary conditions of the considered problem, Nu denotes the number of FE nodes used for the approximation of the respective fields in u, and Nu(i) represents the global basis functions at node i which depend only on the spatial position x, whereas the degrees of freedom u(i) are the time-dependent nodal coefficients. Moreover, Mu is the number of FE nodes used for the test functions δu S , δv F , and δp, respectively, Mu(i)

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Fig. 3 Geometrical mapping of a quadrilateral infinite element

Fig. 4 Viscous damping boundary method (VDB): FEM–IEM coupling with VDB at the interface (left), and rheological model with applied damping forces (right)

denotes the global basis functions, and δu(i) represents the corresponding nodal values of the test functions. Furthermore, Suh (t) and Tuh are the discrete, finite-dimensional trial and test spaces. In the current contribution, the BubnovGalerkin procedure is applied using the same basis functions Nu(i) ≡ Mu(i) for the approximation of u and δu. For more details, the interested reader is referred to, e. g., [2,18,29,34,53] among others. The spatial discretisation of the infinite, far domain is carried out using the mapped IE as introduced in [54]. Here, one distinguishes between the ansatz functions for the geometry modelling Minf(i) (mapping functions), which are multiplied by decay functions in order to extend to infinity and cover the unbounded domain (interpolation from the parameter space (ξ, η) to the real space (x1 , x2 )), and the shape functions for the primary variables N inf(i) , which are treated similarly to the FE shape functions (Ninf(i) = Nu(i) ). The significant advantage of using this type of IE treatment is that it leads to integrals over unity elements in the parameter space, where the Gauß quadrature rule can be directly implemented. This entails only minor modifications when using common FE programs for the numerical treatment, cf. [30] for exemplary applications. In the current work, two types of singly mapped infinite elements, which extend to infinity in one direction are implemented [54]. They correspond to linear and quadratic ansatz functions: 5-node isoparametric infinite elements that

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utilise five of the shape functions of the 8-node mixed FE and for which five mapping functions Minf(i) are derived. 4-node isoparametric infinite elements that originate from the 6-node Lagrangian FE (linear in ξ and quadratic in η) with two nodes corresponding to η = +1 positioned in the infinity (cf. Fig. 3). The infinite elements are assigned to the variables which are defined zero in the infinity, i. e. u S , v S , and v F . 3.3 Treatment of the far field In the course of dynamic problems in infinite half spaces, it is essential to represent the stiffness of the unbounded far region instead of implementing rigid boundaries surrounding the near field. This can be accomplished by exploiting the mapped IE in the static form, which are usually used when steady-state static loading conditions are applied to problems with boundless domains, cf. [36,54]. In the presence of dynamic waves, additional effort is necessary at the boundaries to prevent wave reflections and to imitate the dynamic response of the unbounded domain. Following this approach, the near field is discretised using the FEM, the far field is discretised with the IEM, and the outgoing dynamic waves towards the infinity are absorbed using a local VDB layer placed at the FE–IE interface, cf. Fig. 4. This treatment with the VDB is usually referred to as a doubly asymptotic approximation (DAA) and frequently

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applied in soil-structure interaction problems to study the interplay between a bounded and an unbounded region or between a structure and a surrounding acoustic medium, cf. [36,67,74]. Accordingly, the weak form (30) is split for the treatment of the far field into two parts: a quasi-static part discretised in space with the aforementioned IE method and a dynamic part replaced by damping forces integrated over the FE–IE interface  F I . In particular, one obtains

grad δu S · (T SE − p I) dv − δu S · ρ b dv +



  quasi-static(→ IE)

+ δu S · (ai ρ ci ) v S da = 0. (32) F I   dynamic(→ VDB)

In saturated, materially incompressible solid–fluid aggregates, there exist two major types of bulk waves that have to be damped out, namely compressional waves (p2- or p-waves) known as Biot’s slow waves, and shear waves (s-waves) that propagate only through the solid matrix. Hence, in order to develop boundary conditions that ensure the absorption of the elastic energy arriving at a certain boundary, Lysmer & Kuhlemeyer [48] developed damping expressions for the boundary conditions. For two-dimensional (2-d) problems, the damping relations read σ = (a ρ c p ) v S1

: p-waves,

τ = (b ρ cs ) v S2

: s-waves.

(33)

These equations are formulated for incident primary (p) and secondary (s) waves that act at an angle θ from the x1 -axis, cf. Fig. 4, right. In (32) and (33), ρ = ρ S + ρ F is the density of the overall aggregate, v S1 and v S2 represent the nodal solid velocities in x1 - and x2 -direction, c p and cs are the velocities of the p- and s-waves given in Eqs. 26 and 28, and a, b are dimensionless parameters (cf. [36]) given as 8 8 (5 + 5 c + 2 c2 ), b = (3 + 2 c), a = 15π 15π μS with c = . λ S + 2μ S

(34)

The implementation of the method is fairly simple since one adds nothing more than dashpots with damping constants (a ρ c p ) and (b ρ cs ) to the degrees of freedom (DOF) of the FE–IE interface elements (Fig. 4, right). The effectiveness of the VDB depends strongly on the wave incident angle θ . Indeed, it is shown in [48] that a nearly perfect absorption of the incident waves can only be achieved for θ < 60, whereas some reflections occur for bigger angles. In the weak formulation, the damping terms are written in an integral form over

the boundary  F I , which for 2-d problems reads ⎡ ⎤  

a ρ c 0 p r1 ⎦ v S da. = δu S · ⎣ r = r2 0 bρc F I

(35)

s

Due to the dependency on the primary unknown v S , the arising damping terms in (35) enter the weak formulation of the problem in a form of nonlinear boundary integrals. Thus, an unconditional stability of the numerical solution requires that these terms are treated implicitly by integrating over the interface at the current time step in the sense of a weakly-imposed Neumann boundary condition, cf. Sect. 3.4 for details. 3.4 Time discretisation In this paper, the resulting space-discrete coupled DAE system is integrated in time using monolithic implicit time-integration schemes. Therefore, let y = y(t) = [u S , v S , v F , p]T ∈ Rm with m = dim(y) represent all nodal degrees of freedom u(i) (t) of the FE and IE mesh. Then, the nonlinear, semi-discrete DAE system of the uv p -formulation can be expressed in an abstract form as !

˙ = 0. F (t, y, y)

(36)

This system of equations includes nonlinear terms, and thus, is solved using iterative solution techniques such as the Newton-Raphson method. In particular, starting from a known equilibrium state, i. e. z = {tn , y n , y˙ n }, yields    dF  dF  ! k  y + y˙ k = 0 (37) F lin = F zk +   dy zk dy˙ zk     K M with k being a Newton iteration index. Using the obtained approximation, the increments y k and y˙ k followed by solution updates y k+1 = y k + y k and y˙ k+1 = y˙ k + y˙ k are calculated at each iteration step. The procedure is repeated until the desired accuracy has been reached, i. e., e. g., the condition y k ≤ TOL is satisfied. In the course of the Newton-Raphson procedure for elastic problems, the partial derivative of the residual vector F with respect to y˙ yields the generalised mass matrix M, and the partial derivative with respect to y yields the generalised stiffness matrix K. It is worth mentioning that in this contribution, the derivatives in (37) are generated numerically,1 which enables the implicit treatment of the VDB terms, cf. [45,60] for additional details. For the considered case of a materially incompressible solid and fluid aggregates, the global mass matrix is singular 1

The numerically generated stiffness and mass matrices are expressed via differential quotients as

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as it has no entry corresponding to the pore-pressure time derivative p˙ (algebraic coupling), which implies that explicit monolithic schemes are not applicable. Hence, proceeding from a strongly coupled problem governed by realistic permeabilities 0 < k F  1, only monolithic implicit or splitting time-integration schemes can be applied, cf. [53]. In case of the monolithic solution, Taylor-Hood-like elements are chosen for the FE treatment with quadratic approximations of u S and v S and linear approximations of v F and p, which fulfil the inf-sup or Ladyszenskaya–Babuška–Brezzi (LBB) condition, see [34] for references. An analogue treatment for the IE is carried out, where the 5-node isoparametric elements are assigned to the solid displacement and velocity (u S and v S ), and the 4-node isoparametric elements are used for the approximation of the pore-fluid velocity v F . For the pore pressure p in the far field, quadrilateral finite linear interpolations are assigned inside the infinite elements, which is compatible with the discretisation in the usual finite elements. In this paper, the resulting DAE are discretised in time using a 1-step, s-stage, diagonally-implicit Runge-Kutta method (DIRK), which is appropriate for first-order DAE systems and provides suitable means at moderate storage and computational costs [24,53]. In the DIRK context, the method of choice is the 3-stage TR-BDF2 method [4,41] as a composite integration scheme combining the advantages of both the trapezoidal rule (TR) and the 2nd-order backward difference formula (BDF2). 4 Numerical examples As applications of the discussed formulations and algorithms, two numerical examples are introduced and implemented in the FE package PANDAS.2 For a simple and abstract representation, a number of abbreviations related to the different numerical schemes is introduced in Table 1. 4.1 Dynamic wave propagation in saturated poroelastic column The aim of this example is to test and verify the suggested infinite domain modelling procedure by comparing the numerical results with analytical solutions. To this end, an elasFootnote 1 continued  F (tn , y n +  en , y˙ n ) − F (tn , y n , y˙ n ) ∂ F  ≈ , ∂ y z  en   F (tn , y n , y˙ n +  en ) − F (tn , y n , y˙ n ) ∂F  ≈ ∂ y˙ z  en with en and en as explicit perturbation vectors of y n and y˙ n , respectively, and  > 0 as a small parameter, which can be determined based on the CPU precision, cf. [45]. 2

Porous media Adaptive Nonlinear finite element solver based on Differential Algebraic Systems, see http://www.get-pandas.com.

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Table 1 Abbreviations of the numerical schemes Abbreviations

Description

Ref.

Reference solution

FE–IE

Finite element discretisation of the near field and infinite elements for the far field together with VDB scheme, cf. Fig. 5, right Finite element discretisation with finite (fixed) boundaries, cf. Fig. 5, middle

FE-fix

tic, infinitely long column taken from an unbounded soil half space is analysed under plane strain conditions. The top boundary is perfectly drained and the lateral boundaries are impermeable, frictionless, but rigid. The applied load and the boundary conditions are illustrated in Fig. 5. The material parameters given in Table 2 are taken from the related literature on poroelastic soil dynamics [13]. Moreover, an analytical solution based on a Laplace transformation of a one-dimensional, infinite, half-space problem is used for the comparison of the different numerical solutions, see [13] for particulars. The spatial discretisation with the FE–IE method is carried out using mixed-order (quadratic- and linear) interpolations for both the finite and the infinite elements as discussed in Sect. 3.4. In this regard, the numeration and location of the connecting nodes must coincide in order to insure the continuity across the FE–IE interface. The time discretisation is performed using the implicit monolithic TR-BDF2 scheme with constant time-step size t = 10−3 s for all cases of study. In order to examine the efficiency of the suggested FE–IE treatment, the domain is truncated at a certain distance l1 from the top before the propagating pressure waves are damped out. Thus, based on the analytical treatment, it is shown in Fig. 6 how far the pressure waves can propagate inside the domain for different values of the permeability k F . Due to the assumed incompressibility of the pore fluid, only one type of p-waves appears, which damps out after a certain depth. The performance of the proposed infinite domain treatment depends on a number of factors. In addition to the chosen damping relations and parameters (cf. Sect. 3.3), the implementation of the IE for the quasi-static behaviour of the far field and the pore-pressure approximation with the FEM inside the IE directly influence the accuracy of the results. In this context, the IE extension distance (or the pole of the mapped IE) represented by l2 in Fig. 5 (right) affects the efficiency of the quasi-static IE treatment in an attempt to improve the fit of the far-field response to the considered decay patterns, cf. [54] or [77] for more details. In this connection, a large increase of the IE extension leads to a coarse mesh for the pore-pressure approximation inside the IE, which is undesired especially under low permeability

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Fig. 5 Geometry of the infinite half space (left), FE discretisation with fixed boundary (middle), FE discretisation with IE at the bottom boundary together with VDB (right) Table 2 Material parameters of saturated, elastic soil used in the numerical examples [13]

Table 3 Two considered cases with varying permeability and geometry of the infinite domain model

Parameter

Symbol

Value

SI unit

Case

k F (m/s)

l1 (m)

1st Lamé constant of ϕ S

μS

5.583 × 106

N/m2

(1)

10−2

3

λS

8.375 × 106

N/m2

Effective solid density

ρ SR

2000

kg/m3

Effective fluid density

ρ FR

1000

kg/m3

Initial solidity

S n 0S

0.67

–

Darcy permeability

kF

10−1. . . 10−3

m/s

2nd Lamé constant of

ϕS

Fig. 6 Analytical solution of the solid displacement u S2 for different values of the permeability k F at t = 0.15 s (left), and the corresponding truncated domains (right)

conditions. A comparable situation is found if for the approximation of a steep pore-pressure gradient, a fine FE mesh is required. However, for the shear-wave propagation through

(2)

10−1

8

l2 (m)

Abbreviations

0.15

FE–IE (1)1

3

FE–IE (2)1

1

FE–IE (1)2

8

FE–IE (2)2

the solid skeleton, the influence of the pore-pressure discretisation is of less importance. In the following discussion, two cases are compared using different values of the permeability k F and the IE extension l2 . The specific values are given in Table 3. In case (1) with lower permeability, it is clear from Fig. 7 that the FE–IE treatment results in a more accurate solid displacement solution than FE-fix. This situation becomes even more prevalent with progressing calculation time. In addition, the FE–IE with shorter l2 yields more accurate approximations than that with longer extension length. The error can be quantified by calculating, for instance, the relative displacement error3 ERRu at a point with coordinate x2 = 1 m and time t = 0.15 s . In this regard, the FE-fix treatment leads to ERRu ≈ 92 %, FE–IE(2)1 yields ERRu ≈ 4.0 %, and FE–IE(1)1 gives the best solution with ERRu ≈ 0.03 %. Moreover, it is obvious from Figs. 7 and 8 that ERRu changes according to the point position and time of observation. Although increasing the IE extension has a positive impact on the solid displacement solution in the far field (cf. [54]), the pore-pressure approximation with the FEM inside the IE ERRu := |(u S2 − u a )/u a | with u a being the analytical displacement solution and u S2 the numerical displacement solution.

3

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Fig. 7 Displacement time history u S2 at x2 = 1 m for case (1) and FE = 20 elem/m

Fig. 8 Displacement solution u S2 for case (1) with FE = 20 elem/m at t = 0.15 s

plays an important role in the overall accuracy of the coupled problem. This coupling between the solid displacement and the pore-pressure solution becomes more evident for lower values of the permeability. In this connection, Fig. 9 shows that the FE–IE(1)1 case with shorter IE extension l2 , and thus, a denser FE discretisation of the pore pressure is better than the FE–IE(2)1 case with longer IE extension. Consequently, Fig. 8 shows that the coarse pore-pressure mesh leads to a poor FE–IE(2)1 approximation of the solid displacement in the coupled problem at the bottom of the truncated domain (x2 = 0 m ). A similar study is carried out for case (2) with higher permeability as given in Table 3. In this regard, Figs. 10, 11, and 12 show that the FE–IE treatment again leads to better results than the FE-fix treatment for both the solid displacement and the pore-pressure fields. For a quantitative comparison between the different schemes, the relative displacement error ERRu is evaluated, for instance, at a point with x2 = 1m

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Fig. 9 Pore-pressure solution p for case (1) with FE = 20 elem/m at t = 0.15 s

Fig. 10 Displacement time history u S2 at x2 = 1 m for case (2) and FE = 10 elem/m

Fig. 11 Displacement solution u S2 for case (2) with FE = 10 elem/m at t = 0.15 s

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Fig. 12 Pore-pressure solution p for case (2) with FE = 10 elem/m at t = 0.15 s

and t = 0.15 s . The FE-fix treatment leads to ERRu ≈ 33 %, FE–IE(1)2 results in ERRu ≈ 10.4 %, and FE–IE(2)2 yields the best solution with ERRu ≈ 9.0 %. Moreover, Figs. 10 and 12 show that ERRu changes as for case (1) according to the point location and time of observation. In this regard, the FE–IE(2)2 treatment with relatively long IE extension l2 yields at different times a better displacement solution than FE–IE(1)2 with short l2 , cf. Fig. 10. This finding is in agreement with the quasi-static IE scheme behaviour, where increasing the IE extension distance improves the displacement solution in the far field, cf. [54]. However, as shown in Fig. 12, the pore-pressure solution with finer FE discretisation (shorter IE extension) is slightly better than that with longer l2 . Here, as the permeability parameter k F is higher in case (2) than in case (1), the influence of the pore

pressure on the solid displacement is less important (weaker u S − p coupling). Thus, increasing the IE extension distance improves the accuracy of the numerical displacement solution. Based on the latter results, we conclude that for case (2), the role of the IE extension on the accuracy of the infinite domain treatment is dominant over the influence of the pore-pressure FE approximation within the IE. For the sake of completeness, we mention an additional source of error, which stems from the fact that for high permeabilities, such as in case (2), the assumption of negligible relative accelerations of the solid and the pore-fluid constituents is somehow violated, cf. Sect. 2.3. For further analysis of the suggested VDB treatment with different wave types, a two-dimensional wave propagation problem giving rise to shear wave damping in a soil-structure interaction problem is introduced in the next section.

4.2 Wave propagation in an elastic structure-soil system In this example, the dynamic response of an elastic block founded on an elastic soil half space and subjected to a horizontal shear loading is discussed (Fig. 13). Such soil-structure interaction problems have been intensively studied in the literature, cf. [68,69,43] among others. In this regard, the overall behaviour is affected by a number of factors such as the properties of the subsystems and the structural embedment into the foundation soil. In the current problem, the block (4 × 2 m2 ) is considered to be in a welded contact with the soil beneath and discretised with the same type of finite elements as the soil. The applied shear impulse force is given by

Fig. 13 Geometry, boundary conditions and loading path of the 2-d block-soil problem with FE discretisation and fixed boundary (left), and with FE discretisation and IE boundary together with VDB (right)

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Comput Mech Table 4 Material parameters of elastic concrete

Parameter

Symbol

Value

SI unit

1st Lamé constant

μS

1.25 × 1010

N/m2

2nd Lamé constant

λS

8.30

Effective density

ρ SR

2800

Initial volume fraction

S n 0S F k

0.99

–

10−6

m/s

Darcy permeability

× 109

N/m2 kg/m3

Fig. 14 Time sequence  of solid displacement u S = u 2S1 + u 2S2 as 3-d surface plots for case (1) with E B /E S = 1

f (t) = 104 [1 − cos(20 π t)] [1 − H (t − τ )] [ N/m2 ] (38) with H (t − τ ) being the Heaviside step function and τ = 0.1 s . The dynamic response of the soil and the efficiency of the considered infinite boundary treatment are investigated for two cases: Case (1): the material parameters of the block and the soil are the same (cf. Table 2) with k F = 10−2 m/s . Case (2): the block is considered to be made of concrete4 with parameters given in Table 4, whereas the soil parameters are taken from Table 2 with k F = 10−2 m/s. A benchmark solution, referred to as Ref. FE, is generated by taking l1 = l2 = 40 m as in Fig. 13 (left), where the choice of large dimensions guarantees that no reflected waves 4

The numerical simulation of concrete using multi-phase material S = 0.99, i. e., the concrete is treated model is considered by choosing n 0S as an almost singlephasic, linear elastic, solid skeleton.

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propagate back to points A, B and C during the analysis. The efficiency of the proposed boundary treatment is evaluated by comparing the displacements at different points in the domain (A, B and C) for two types of boundaries, i. e., for FE–IE with VDB as in Fig. 13, right, and for FE-fix as in Fig. 13, left, with l1 = l2 = 20 m. In case (1) with unified material parameters, the stiffness ratio between the concrete block and soil is E B /E S = 1, with E B and E S being the elasticity moduli5 of the concrete and the soil solid skeleton, respectively.6 In this case, the vibration of the block damps out in a weak manner and results in a successive wave transition into the supporting soil. In this connection, Fig. 14 shows exemplary contour plots of the computed solid displacement field, which makes the wave propagation and the weak damping behaviour apparent. In Fig. 15, the time history of the horizontal displacement u S1 of point C at the top of the block with weakly damped 5

They are computed from E S = μ S (2 μ S + 3 λ S )/(μ S + λ S ).

6 E B /E S  1 would represent the case of a soft block founded on a rigid base [68], which is not a case of study in this contribution.

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Fig. 15 Horizontal displacement time history at point C for case (1) with E B /E S = 1

Fig. 16 Horizontal displacement time history at point A for case (1) with E B /E S = 1

motion is depicted. Therein, a good agreement among the different solution strategies is obtained as far as the reflected waves do not propagate back to point C. However, the solution with FE-fix deteriorates after a certain time due to the interfere of the reflected waves. For the two points A and B in the soil domain, the time history of the horizontal displacement u S1 is plotted in Figs. 16 and 17. The efficiency of the FE–IE scheme in impeding the reflecting waves is obvious by comparison with the reference solution Ref. FE. However, the FE-fix solution violates this agreement due to the overlapping of progressing and reflected waves. In case (2), the stiffness of the concrete block is higher than that of soil (E B /E S = 2.1 × 103 ), which leads to a strong damping of the block motion. For this case, Fig. 18 shows exemplary contour plots of the computed horizontal solid displacement for the discussed three different boundary cases, and clearly reveals the influence of the boundary conditions on the reflected waves. The maximum response of the block decreases in comparison with case (1) and the motion damps out very strongly. Accordingly, only one wave corresponding to the impulse loading appears and radiates towards the infinity. Moreover, Fig. 19 shows that due to the high stiffness difference between the block and the soil beneath, the reflected waves do not propagate into the block as the displacement at point C(2, −2) is not disturbed. For a point B(−5, 10) in the domain, Fig. 20 shows the role of the damping boundary in reducing the effect of the reflecting waves. Apparently, the proposed VDB method can significantly but not perfectly prevent wave reflections back to the near field. This is due to the fact that the absorption of the approaching waves cannot be completely accomplished over the whole range of the FE–IE boundary, which is an inherent disadvantage of the VDB treatment originating from the varying wave incident angles along the boundary. A possible remedy is to model curved boundaries, which mimic the shape of the incident wave fronts, and thus, guarantee small wave incident angles. For a detailed discussion of this issue, we refer to the pioneering work of Lysmer and Kuhlemeyer [48].

5 Conclusions

Fig. 17 Horizontal displacement time history at point B for case (1) with E B /E S = 1

In this paper, general absorbing boundary conditions for the numerical simulation of dynamic wave propagation in porous media infinite half spaces have been discussed. The investigations proceed from a saturated biphasic solid– fluid aggregate with intrinsically incompressible constituents. Moreover, proceeding from a geometrically linear treatment, the TPM has been used to formulate the governing balance laws and the thermodynamically consistent constitutive equations. As a result, one has to deal with a

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Fig. 18 Time sequence of displacement u S1 contour plots for case (2): Ref. FE (left), FE–IE with VDB (middle) and FE-fix (right)

Fig. 19 Horizontal displacement time history at point C for case (2) with E B /E S  1

Fig. 20 Horizontal displacement time history at point B for case (2) with E B /E S  1

strongly coupled three-field variational problem, which gives rise to two types of bulk waves, namely, pressure and shear waves. For the treatment of soil-structure interaction problems, the surrounding semi-unbounded domain is split into a near field around the structure (source of vibration), which is discretised using the mixed FEM, and a far field that extends to infinity, which is discretised in space with mapped IE in

the quasi-static form. The wave reflection at the near-field/ far-field interface is eliminated by introducing a viscous damping boundary (VDB) layer, which entails nonlinear terms in the problem residual. These terms have to be integrated numerically over the FE–IE interface in a weakly imposed manner. Based on that, an unconditional numerical stability required the use of an implicit monolithic time-stepping scheme.

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Unlike the application to singlephasic elastic materials, the efficiency of the VDB implementation in case of multiphase media is affected by additional factors. For instance, the accuracy of the FE discretisation of the pore-pressure variable inside the IE, the proper implementation of the mapped IE scheme, and the permeability parameter that governs the solid–fluid interaction, and thus, the strength of the coupled multi-field problem. The results of the one- and two-dimensional examples show that the proposed VDB treatment can significantly suppress the spuriously reflected waves, and therefore, provides a suitable representation of the reference solution at moderate implementation complexity. Acknowledgements The author Y. Heider would like to thank the Ministry of Higher Education of Syria for the financial support of his research stay at the Institute of Applied Mechanics (CE), University of Stuttgart, Germany.

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