Hybrid-Trefftz stress element for bounded and unbounded poroelastic media

Transcription

1 INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2011; 85: Published online 28 October 2010 in Wiley Online Library (wileyonlinelibrary.com) Hybrid-Trefftz stress element for bounded and unbounded poroelastic media J. A. Teixeira de Freitas 1,, and I. D. Moldovan 2 1 Departamento de Engenharia Civil e Arquitectura, Instituto Superior Técnico, Technical University of Lisbon, Lisboa, Portugal 2 Departamento de Engenharia Civil, Faculdade de Engenharia, Universidade Católica Portuguesa, Lisboa, Portugal SUMMARY The equations that govern the dynamic response of saturated porous media are first discretized in time to define the boundary value problem that supports the formulation of the hybrid-trefftz stress element. The (total) stress and pore pressure fields are directly approximated under the condition of locally satisfying the domain conditions of the problem. The solid displacement and the outward normal component of the seepage displacement are approximated independently on the boundary of the element. Unbounded domains are modelled using either unbounded elements that locally satisfy the Sommerfeld condition or absorbing boundary elements that enforce that condition in weak form. As the finite element equations are derived from first-principles, the associated energy statements are recovered and the sufficient conditions for the existence and uniqueness of the solutions are stated. The performance of the element is illustrated with the time domain response of a biphasic unbounded domain to show the quality of the modelling that can be attained for the stress, pressure, displacement and seepage fields using a high-order, wavelet-based time integration procedure. Copyright 2010 John Wiley & Sons, Ltd. Received 12 March 2010; Revised 14 June 2010; Accepted 16 July 2010 KEY WORDS: hybrid-trefftz elements; stress elements; poroelastic media; absorbing boundaries 1. INTRODUCTION The development of the stress and displacement models of the hybrid-trefftz finite element formulation developed for the spectral analysis of homogeneous elastic media [1, 2] was first motivated by the research reported by Degrande [3]. A detailed description of their extension to the modelling of the response of compressible and incompressible biphasic media in both frequency and time domains is given in References [4, 5]. An overall view of this extension in the framework of the mixed and hybrid finite element formulations is reported in Reference [6]. The stress model is applied here to the two-dimensional analysis of unbounded saturated porous media modelled as an elastic solid phase fully permeated by a liquid phase obeying Darcy s law. Biot s theory is used to derive the governing system of differential equations [7]. The free-field solution of the governing system of differential equations is used to set up the finite element approximation basis. This corresponds to the application of the method originally proposed by Trefftz [8] for the solution of boundary-value problems in a finite element context, pioneered by Jirousek [9]. The finite element model is hybrid, in the sense first proposed by Correspondence to: J. A. Teixeira de Freitas, Departamento de Engenharia Civil e Arquitectura, Instituto Superior Técnico, Technical University of Lisbon, Lisboa, Portugal. freitas@civil.ist.utl.pt Copyright 2010 John Wiley & Sons, Ltd.

2 HYBRID-TREFFTZ STRESS ELEMENT FOR POROELASTIC MEDIA 1281 Pian [10], because the solutions that define the Trefftz basis cannot be combined apriorito satisfy in strong form the boundary continuity conditions. The stress model of the hybrid-trefftz finite element formulation used here is derived approximating independently the stress field in the domain of the element and the displacement field on its boundary. The stress field combines the stress components in the solid skeleton and the fluid pressure. Similarly, the boundary displacement field combines the components of the solid displacement and the normal component of the seepage displacement. No particular constraints are placed on the geometry and the topology of the element, which may not be bounded, simply connected or convex. It is assumed that the variables are separated in space and time. The basic equations are first discretized in time to obtain the boundary value problem that is subsequently discretized in space using duality, the vector description of the invariance of the inner-product in the finite element mappings, and enforcing the Trefftz constraint. In the present case, duality consists of defining the independent (generalized) strains and boundary forces associated with the independent approximations of the stress and boundary displacement fields. The procedure is equivalent to the Galerkin enforcement of the compatibility and flux conditions, using the domain and boundary approximation bases as weighting functions. The resulting algebraic finite element solving system is Hermitian, with the exception of the damping term, and its coefficients are defined by boundary integral expressions, a feature typical of (Trefftz-type) solution methods based on approximation functions taken from the solution set of the differential system of equations governing the boundary value problem. Consequently, the Trefftz formulation used here combines the main features of the alternative finite and boundary element methods, while avoiding the use of (singular) fundamental solutions. The assemblage and the solution of the finite element system are characterized and analysed in terms of sparse system manipulation and parallelization. The formulation is specialized next to the analysis of unbounded media. The Trefftz approximation basis is used to recover the Sommerfeld radiation condition for saturated porous media. Two modelling techniques are presented. The first consists of using unbounded elements built on a Trefftz approximation basis that satisfies the radiation condition in strong form. The second is based on the use of absorbing boundary elements designed to enforce this condition in weak form. As the finite element equations are derived from first-principles, the presentation of the formulation is complemented with the definition of the associated energy statements. The sufficient conditions for the existence and uniqueness of the solutions are also stated, for both bounded and unbounded domain modelling. The performance of the stress element in the time domain analysis of the response of a saturated soil is illustrated applying the high-order time integration method presented in Reference [11]. It is shown that the use of a relatively weak finite element approximation in a single time step yields to good levels of accuracy in the modelling of the stress, pressure and displacement fields developing in the mixture at every instant of a relatively wide time interval. 2. DISCRETIZATION IN TIME It is assumed that all variables present in the modelling of the elastodynamic response of saturated porous media are separated in time and space. For instance, the displacement, velocity and acceleration fields are written as follows, u(x,t) = N T n (t)u n (x) (1) n=1 v(x,t) = N T n (t)v n (x) (2) n=1 a(x,t) = N T n (t)a n (x) (3) where T n (t) defines a complete time approximation basis with support 0 t Δt. n=1

3 1282 J. A. TEIXEIRA DE FREITAS AND I. D. MOLDOVAN The non-periodic, high-order time integration procedure described in Reference [11] and outlined in Reference [6] yields time integration rules of the form, v n (x)=iω n u n (x) iω 0n u 0 (x) (4) a n (x)=iω n v n (x) iω 0n v 0 (x) (5) where i is the imaginary unit, u 0 and v 0 are the initial conditions of the problem, and ω n and ω 0n are (complex) algorithmic frequencies. The application reported below is obtained applying this time integration method in a single time step using a wavelet time approximation basis defined on the interval [12, 13]. The commonly used periodic (or periodically extended) spectral decomposition method is recovered using a Fourier basis, T n (t)=exp(iω n t)andω n =±2nπ/Δt. Under this assumption, Equations (4) and (5) hold with real forcing frequencies with ω 0n =0, as periodicity renders the discretization independent of the initial conditions of the problem. 3. POROELASTICITY EQUATIONS The discretization of the time dimension of the equations governing the elastodynamic response of a saturated poroelastic body V with boundary Γ generates a set of N fully uncoupled boundary value problems, which can be stated in a matrix form as follows: Dr n +b n +(ω 2 n q iω nc)u n =0 in V (6) ε n =D u n in V (7) e n =fr n in V (8) Nr n = t n on Γ t (9) u n =ū n on Γ u (10) In the pseudo-spectral decomposition of the equilibrium (6), compatibility (7) and elasticity (8) conditions, vectors r n and e n define the independent components of the (total) stress and (solid) strain tensors extended to include the fluid pressure and the fluid content, respectively, r T ={σ xx σ yy σ xy π} e T ={ε xx ε yy 2ε xy ζ} where π and ζ are the pore pressure and the fluid content, respectively, and vector u n combines the displacement components of the solid skeleton and the pore fluid seepage displacement components: u T ={u x u y w x w y } The divergence and gradient operators, D and D, are adjoint in geometrically linear problems, and the mass density, damping and flexibility matrices q, c and f, respectively, are real and symmetric. They are defined in Appendix A for this particular application, as well as the term in the equilibrium condition (6) that combines the effects of the body forces of the mixture with the initial conditions of the problem. This term, b n, is null in the application presented below, as the body forces are neglected and the domain is assumed to be initially at rest. The explicit form of the Neumann and Dirichlet conditions (9) and (10) is also presented in Appendix A. In the Neumann condition, vector t n defines the components of the (total) surface forces and the pressure prescribed on the boundary, and the boundary equilibrium matrix N collects the appropriate components of the unit outward normal vector, n. In the Dirichlet condition, vector ū n defines the prescribed surface displacements in the solid and the outward normal component of the seepage displacement.

4 HYBRID-TREFFTZ STRESS ELEMENT FOR POROELASTIC MEDIA 1283 The Dirichlet conditions can be alternatively expressed in terms of prescribed velocity or acceleration terms, u= v or ü=ā, in which case the prescribed term in Equation (10) is replaced by the following expressions: ū n <> iω 1 n ( v n +iω 0n u 0 ) ū n <> ω 2 n (ā n +iω 0n v 0 )+ω 1 n ω 0nu 0 The notation used to express the boundary conditions (9) and (10) accounts for mixed conditions, as it implies that the Neumann and Dirichlet boundaries, Γ t and Γ u, are complementary and disjoint in each dimension of the space: Γ=Γ t Γ u and =Γ t Γ u. It is possible, also, to account for Robin-type boundaries. This is illustrated below with the absorbing boundary condition used in the modelling of unbounded media. The boundary value problem (6) (10) holds, with appropriate adjustments in the definitions of the generalized body force vector and damping matrix, when the discretization of the time dimension is based on trapezoidal rules, e.g. Tamma et al. [14]. The Newmark-type integration methods lead to a single-frequency spectrum (N =1) and imaginary algorithmic frequencies ω<> iγ(βδt) 1 and ω 2 <> (βδt 2 ) 1 for the velocity and acceleration integration rules, respectively: u=u 0 +Δtv 0 +αδt 2 a 0 +βδt 2 a v=v 0 +(1 γ)δta 0 +γδta 4. DISCRETIZATION IN SPACE System (6) (10) is used to set up the formulation of the hybrid-trefftz stress element. V and Γ identify now the domain and the boundary of a typical element of the mesh, with no particular constraints being placed on its geometry and topology. To simplify the notation, the subscript used to identify the algorithmic frequency, n, is omitted from this point onwards. The interelement continuity conditions are still stated by Equations (9) and (10), with the prescribed terms defining now the forces and displacements of the connecting element, t + +t =0 on Γ i (11) u + u =0 on Γ i (12) with t = t and u =+ū. The Neumann boundary of a stress element is the boundary whereon the displacements are unknown. Therefore, condition (9) applies to the Neumann boundary of the mesh the element may contain and to all its interelement boundaries. Consequently, condition (10) is empty for all elements that do not share the Dirichlet boundary of the mesh Primary approximations The stress model of the hybrid finite element formulation for biphasic media is derived approximating directly the stress and pressure fields in the domain of the element, r=sp in V (13) where vector p defines the amplitudes of the stress and pressure modes collected as columns of matrix S. This approximation is extended to include a particular term to account for the effect of non-trivial initial conditions (typically in multi-step time integration methods), body forces and, eventually, alternative local effects deemed relevant to improve the rate of convergence of the approximation (typically high-stress gradients). As approximation (13) is not constrained to satisfy locally the element Neumann conditions (9) and (11), it becomes necessary to approximate independently the displacements in the solid

5 1284 J. A. TEIXEIRA DE FREITAS AND I. D. MOLDOVAN skeleton and the normal component of the seepage displacement on the Neumann boundary of the element, in the form: u=bq on Γ t (14) Vector q represents generalized displacements as non-nodal hierarchical functions are used to set up the boundary basis, B. The same approximation is assigned to the connecting element, to fulfil condition (12). The constraints that are assumed for domain and boundary approximation bases S and B are stated before they are defined for the current application to saturated porous media Constraints Linear independence and completeness are the only constraints placed on the functions used to define the boundary displacement approximation basis, B. The same constraints apply to the stress and pressure approximation modes that form the domain approximation basis, S, with the additional requirement that this approximation is selected from the solution set of the free-field wave equation defined by the homogeneous form of the domain equilibrium, compatibility and elasticity conditions (6) to (8), DkD U+(ω 2 q iωc)u=o (15) where k=f 1 is the local stiffness matrix. It is this Trefftz condition that leads to boundary integral expressions for all matrices and vectors present in the system governing the response of the element Dependent approximations In consequence of the Trefftz constraint, the stress approximation (13) is associated with uniquely defined approximations for the strain and displacement fields, that satisfy the domain conditions of the problem in a strong form: e=ep in V (16) u=up in V (17) DS+(ω 2 q iωc)u=o (18) E=fS (19) E=D U (20) Although the domain equilibrium condition is locally satisfied, the surface forces that equilibrate the primary approximation (13) on the stress and pressure fields, t=tp on Γ (21) T=NS (22) will not equilibrate locally, in general, either the prescribed boundary values or the force induced by a connecting element, as required by conditions (9) and (11), respectively. Similarly, the dependent displacement approximation (17) is not subject to satisfy locally the conformity conditions (10) and (12) on the Dirichlet boundary of the element and on its interelement boundary, respectively. In addition, it may be inconsistent with the primary, independent boundary approximation (14). As for the flux continuity conditions, these conformity conditions are only met under full convergence of the finite element solutions.

6 HYBRID-TREFFTZ STRESS ELEMENT FOR POROELASTIC MEDIA Element geometry and topology As Trefftz bases lead to boundary integral definitions for all finite element coefficients, the only requirement in terms of numerical implementation is to define, in parametric form, the geometry of the element boundaries. Consequently, the elements may not be convex, simply connected or bounded. Although the standard technique of mapping master elements can still be used, the option that is followed to enhance flexibility in the definition of shape consists of assigning the (parametrically defined) boundaries to master nodes, that is, the nodes that are necessary and sufficient to define the domain. The geometry and the topology of the element are implicitly defined by assigning the boundaries to each element of the mesh. This technique, described in detail in Reference [15], is useful mainly in the description of complex or varying shapes. Triangular and trapezoidal elements are used in standard, twodimensional applications, with alternative shapes being implemented only to model adequately the geometry of layers or the boundary of the domain. In the present context, the only constraint that is assumed in terms of geometry is in the design of unbounded elements. They are assumed to be either rectangular or sectorial, as illustrated below, in order to obtain analytical expressions for the finite element coefficients they are associated with, and thus avoid the implementation of computationally heavy numerical integration procedures, necessary to attain similar levels of accuracy. 5. DOMAIN APPROXIMATION BASIS This section addresses the definition of the stress and pressure modes used to set up the primary approximation (13). It is also used to recover the Sommerfeld condition and justify the two main options to model unbounded domains, namely formulating elements that satisfy this condition at infinity or using this condition to formulate absorbing boundary elements Helmholtz equation It is assumed that the displacements in the solid and fluid phases are proportional and that they derive from potential fields [16]. Therefore, and according to the organization defined above for the displacement vector, the general term in the dependent approximation (17) is [ Us U w ] [ ] Φ j Φ 3 = γ j Φ j γ 3 Φ 3 involving two P-waves, for j =1 and j =2, and one S-wave, j =3, as and represent the gradient and rotation vectors, respectively: T ={ x y } T ={ y x } It is shown in Appendix B that assumption (23) reduces the Trefftz constraint (15) to the Helmholtz equation, (23) 2 Φ j +β 2 j Φ j =0 (24) where the wave number β j (complex, in general) depends on the (algorithmic) forcing frequency, ω, and the mechanical properties of the element. It is this direct dependency of the approximation basis on the characteristics of the elastodynamic problem that ensures the relatively high rates of convergence that can be attained with Trefftz elements.

7 1286 J. A. TEIXEIRA DE FREITAS AND I. D. MOLDOVAN 5.2. Stress and displacement approximation modes Depending on the geometry of the finite element mesh, the solutions of the Helmholtz equation used to set up the Trefftz basis are defined either in Cartesian or in cylindrical co-ordinate systems, Φ j =exp(ik jx x)exp(ik jy y) with k 2 jx +k2 jy =β2 j (25) Φ j = W n (β j r)exp(inθ) with n =0,±1,...,±d V (26) where the radial term, W n, is the solution of the Bessel equation of order n: z 2 j W n +z j W n +(z2 j n2 )W n =0 with z j =β j r Testing on single-phase problems [2, 17] has shown that the alternative bases perform well in terms of accuracy and rate of convergence when implemented on adequate regular elements, that is, rectangular (sectorial) elements for Cartesian (cylindrical) bases. However, the same tests also show that the cylindrical basis is more robust, as it is virtually insensitive to gross shape distortion [18]. The corresponding stress and displacement solutions used to set up approximations (13) and (17) are presented in Appendix C. They recover the propagation expected for poroelastic media, namely two families of dilational waves and a single shear wave family for each displacement potential, implying the following dimension for the domain approximation bases, where d V represents the highest order in the approximation: N V =3 (2d V +1) (27) Interior (exterior) problems are modelled by identifying W n (β j r) with the Bessel functions of the first (second) kind of order n, usually denoted by J n (Y n ). These regular (singular) functions are applied in the implementation of bounded (multiply connected) elements. To improve the condition number of the solving systems, the local co-ordinate systems are taken as baricentric and parallel to the principal directions of bounded elements, as shown in Figure 1 for a general, multiply connected element. In addition, the approximation functions are scaled to ensure finite element coefficients with norms close to unity Sommerfeld condition Unbounded domains subject to incoming (outgoing) waves are modelled identifying W n (β j r)in definition with the Hankel functions of the first (second) kind of order n, Hn 1 (H n 2 ), under condition Im(β j )<0. Complex arguments are involved in all instances for dissipative media. It is shown in Appendix D that, under these conditions and for outgoing waves, the stress, strain and displacement fields (13), (16) and (17) tend to zero as distance r tends to infinity, thus opening the option of modelling unbounded media with semi-infinite elements. In addition, it is shown in Appendix E that the first-order approximation of the asymptotic expansion of the Trefftz basis leads to the following relation on surface forces and displacements, t= iω c a u (28) t n y e r θ x e y y e r θ x e x r a r b Figure 1. Local co-ordinate systems in bounded and unbounded elements.

8 HYBRID-TREFFTZ STRESS ELEMENT FOR POROELASTIC MEDIA 1287 which recovers the Sommerfeld radiation condition written for saturated porous media [16]. This relation, which also holds for Cartesian coordinate systems [3, 19], is used below to model the response of unbounded domains using absorbing boundary elements. In the applications reported in Reference [4], the unbounded elements are sectorial and placed symmetrically to the local system of reference, with origin at the centre of the sector, as shown in Figure 1. Similarly, the absorbing boundary elements are circular and placed at a distance, r b, consistent with the condition that supports the derivation of condition (28). 6. BOUNDARY APPROXIMATION BASIS The selection of the approximation functions used to set up the boundary displacement approximation (14) depends on the objectives set in terms of the implementation of the hybrid-trefftz element. The usual option, rooted in the work of T.H.H. Pian on hybrid stress elements and followed by J. Jirousek in the extension to the Trefftz variant, consists of deriving the element to emulate the conventional (conform displacement) element, governed by equation Kq=Q (29) where K is the element stiffness matrix and q and Q the nodal displacement and force vectors, respectively. When this option is followed, matrix B in the displacement approximation (14) is defined by the boundary mapping of the shape functions (now termed as frame functions) that characterize the alternative isoparametric elements. The option followed here is designed to exploit the flexibility offered by hybrid-trefftz elements, both in terms of the description of the geometry and topology of the element and the use of naturally hierarchical approximation bases. The ultimate goal is to implement the formulation on coarse meshes of high-order elements while preserving a structure that enhances adaptive refinement and parallelization. To this end, naturally hierarchical bases are also used to approximate the displacements on the (extended) Neumann boundary of the element, as stated by Equation (14): B=[IB m (η)] on Γ t (30) In two-dimensional poroelastic applications, matrix I is the third-order identity matrix to account for the independent approximation of the two components of the solid displacement and the outward normal component of the fluid seepage. Under mixed boundary conditions, definition (30) is modified to contain only the unknown components in the boundary displacement approximation (14). In Equation (30), η identifies the parametric description of the geometry of the boundary and B m (η) is the approximation function. They are defined as Chebyshev polynomials to match the structure of the domain Bessel basis and enhance numerical stability [17]: B m =cos(m cos 1 η) with m =0,1,2,...,d Γ (31) However, to obtain analytical (or semi-analytical) solutions for the definitions of the finite element coefficients, boundary mappings of the domain approximation functions (26) are used in the implementation of unbounded sectorial elements. Trigonometric bases are applied on the circular sides of the element and Bessel functions are implemented on its radial sides: B m =exp(imθ) with m =0,±1,±2,...,±d Γ (32) B m = W m (β j r) with m =0,1,2,...,d Γ These alternative definitions for the boundary displacement approximation functions lead to frame functions that are discontinuous at the element vertices, the gap tending to zero with convergence of finite element solution. As it is shown next, this apparent drawback is compensated by the fact that the interelement continuity conditions are enforced in a stronger form.

9 1288 J. A. TEIXEIRA DE FREITAS AND I. D. MOLDOVAN 7. FINITE ELEMENT EQUATIONS The finite element equations are derived using duality, which ensures the invariance of the inner product in the finite element mapping. The generalized strain defined by the dual transformation of the primary stress and pressure approximation (13), e= S edv where S =Ŝ T denotes the transpose conjugate of matrix S, is used to enforce on average, in the sense of Galerkin, the local compatibility and elasticity conditions (7) and (8): e= S (D u)dv (33) e= S (fr)dv (34) Similarly, the generalized surface force defined by the dual transformation of the primary boundary displacement approximation (14) is used to enforce on average the Neumann condition (9), extended to include the interelement flux continuity condition (11): f= B tdγ t (35) The remaining conditions in system (6) (10) are either locally satisfied or explicitly enforced Weak kinematic admissibility condition In order to combine in a single statement the weak enforcement of the local kinematic admissibility conditions (7) and (10), definition (33) is first integrated by parts, to mobilize the boundary term and yield the following result, where definition (22) is used: e= (DS) udv + T udγ (36) The dependent displacement approximation (17) is enforced in the domain term, together with constraint (18). Next, the boundary term is uncoupled in its Neumann and Dirichlet parts to enforce the independent boundary displacement approximation (14) and the Dirichlet condition (10), respectively. In the resulting expression of the finite element kinematic admissibility condition, e=( ˆω 2 M+i ˆωC)p+Aq+ē (37) M and C are the (Hermitian) mass and damping matrices, respectively, A is the boundary compatibility matrix and vector ē defines the contribution of prescribed displacements: M = U qudv (38) C = A = ē = U cudv (39) T BdΓ t (40) T ūdγ u (41)

10 HYBRID-TREFFTZ STRESS ELEMENT FOR POROELASTIC MEDIA 1289 As vectors e and p are unknown, both with dimension N V, condition (37) shows that the element kinematic indeterminacy number identifies with the (necessarily non-negative) dimension of the boundary approximation basis: β= N Γ 0 This number is null only for a single-element mesh with an empty Neumann boundary Weak static admissibility condition As the domain equilibrium condition (6) is locally satisfied by the primary and dependent stress and displacement approximation (13) and (17) under constraint (18), the finite element static admissibility condition reduces to the weak enforcement (35) of the Neumann conditions for the surface force estimate (21) that equilibrates the assumed stress and pressure fields, to yield, using definition (40): f=a p (42) When continuous frame functions are used, Equation (35) defines equivalent nodal forces and the equation above leads to a equilibrium condition that lumps at each node the contribution of the forces on all boundaries of the mesh that contain that node. When discontinuous frame functions are used, Equation (35) defines boundary force resultants and Equation (42) enforces the equilibrium of those forces strictly on the particular boundary where they are defined. As vector f is prescribed and has dimension N Γ, result (42) shows that the static indeterminacy number of the element is non-negative if the following relation holds, α= N V (d V ) N Γ (d Γ ) 0 (43) under the linear independence conditions assumed above. This relation is used to select the order of the domain approximation, d V, for a given order d Γ of the boundary approximation Duality and virtual work The finite element compatibility and equilibrium conditions (37) and (42) recover the duality of the local conditions (6) and (7). It can be verified that, in consequence, their inner-product, q f (e ē) p+p (ω 2 M iωc)p=0 (44) recovers the virtual work equation written in terms of the element total mechanical energy, M=2(T E) C+W=0 (45) where T and E are the (spectral decomposition of the) element kinetic and strain energies, and C and W are the (corresponding) dissipation and the external work, respectively: T= 1 2 ω2 u qu dv = 1 v qv dv 2 E= 1 2 e r dv W= C=iω u b dv + u cu dv = u t dγ t + u cv dv ū t dγ u

11 1290 J. A. TEIXEIRA DE FREITAS AND I. D. MOLDOVAN 7.4. Weak elasticity condition The finite element description of elasticity is obtained enforcing the local condition (8) in definition (34) for the assumed stress field (13), to yield, where the (Hermitian) flexibility matrix is defined as follows: F= S fs dv e=fp (46) To obtain a boundary integral finite element model, a feature shared by all Trefftz-type methods, conditions (19) and (20) are enforced in the definition above and the resulting expression is integrated by parts to enforce the Trefftz constraint (15). The following equivalent definition for the flexibility matrix is found: F=D+ ˆω 2 M+i ˆωC (47) where results (38) and (39) hold and the elastodynamic matrix is defined as follows: D= T U dγ (48) 7.5. Finite element system The compatibility and elasticity equations (37) and (46) are equated to eliminate the generalized strains e as independent variables and enforce result (47) to eliminate the terms defined by domain integral expressions. The finite element system is obtained combining the resulting equation with the boundary equilibrium condition (42): [ ]{ } { } D A p ē A = (49) O q f It can be readily verified that all terms in this system have boundary integral expressions. They are computed numerically, with the Gauss Legendre quadrature rule and the provision of subdividing each boundary, as the stress and displacement approximation functions can be highly oscillatory for high frequencies. Closed-form solutions can be obtained [2] for rectangular and sectorial elements (see Appendix F). Condensation of system (49) on the boundary displacement degrees-of-freedom leads to the emulation (29) of the conventional displacement element, with K=A D 1 A Q= f A D 1 ē and assemblage for a given finite element mesh is implemented in the usual manner, enforcing nodal connectivity and summing the corresponding nodal forces. The option followed here consists in preserving at mesh level the sparse structure of the elementary system (49). To assemble the mesh, the same boundary approximation (14) is assigned to edges shared by connecting elements, as stated by the continuity condition (12). This condition plays the role of nodal connectivity in the conventional approach and is used to define the first set of equations (kinematic admissibility conditions) in system (49). The second (and dual) set of equations in the assembled system defines the force continuity condition (11). The procedure is similar to the assemblage of meshes for elastostatic problems, e.g. in Reference [15], as the structure of the elementary systems is identical. It can be summarized as follows, assuming that only the non-null coefficients are stored: Compute the elementary elastodynamic matrices (48) and collect them in block-diagonal form according to the domain element numbering; To enforce the displacement continuity condition (12), compute the elementary

12 HYBRID-TREFFTZ STRESS ELEMENT FOR POROELASTIC MEDIA 1291 compatibility matrices (40) and assign the columns (rows) according to the boundary (domain) element numbering; Compute the elementary vector ē using equation (41), defined on the Dirichlet boundary of the assembled mesh, and store the coefficients according to the domain element numbering; Compute the coefficients of vector f using Equation (35), defined on the Neumann boundary of the assembled mesh, and store according to the boundary element numbering, and set to zero the coefficients of vector f on all interelement boundaries to implement the weak form of the force continuity condition (11). Besides preserving the sparsity of the elementary systems (49), this option simplifies adaptive refinement and parallelization. Adaptive refinement is implemented by adding the rows and columns associated with the extension of the approximation bases (13) and (14). Suitability to parallelization follows from the fact that the generalized stresses p are strictly element dependent and the generalized boundary displacements q are shared by at most two connecting elements. Neither of these techniques is used to obtain the results presented below because the number of degrees-of-freedom involved is relatively low. 8. MODELLING OF UNBOUNDED DOMAINS This section addresses the basic aspects of the implementation of the two techniques used to model unbounded domains, namely with semi-infinite elements and with absorbing boundary conditions, e.g. Tsynkov [20]. Their relative merits in the context of the finite element formulation presented here are discussed in detail in Reference [4] Unbounded elements System (49) holds for unbounded elements, provided that they are implemented using the Hankel solution of the Trefftz basis to represent outgoing waves. Therefore, the numerical techniques presented in References [2, 17] to model unbounded homogeneous domains can be directly extended to saturated porous media. One technique consists of partitioning the unbounded domain in sectorial elements (assuming that a cylindrical basis is used), as shown in Figure 2 with r b =, and using semi-analytical integration procedures in the calculation of the coefficients of system (49). All integrals are bounded, in consequence of using a domain approximation basis that satisfies the Sommerfeld condition. The second technique consists of using a single unbounded element, as shown in the same figure, with a basis so defined as to satisfy locally the boundary conditions. This leads to derivation of analytical expressions for all coefficients in system (49) associated with the unbounded element, at the cost of using a high-order approximation basis in the domain to ensure condition (43) on the element static indeterminacy number. The computation of the system coefficients for unbounded sectorial elements is summarized in Appendix G. Figure 2. Symmetrically loaded half-space.

13 1292 J. A. TEIXEIRA DE FREITAS AND I. D. MOLDOVAN 8.2. Absorbing boundary elements The alternative approach consists of using Equation (28) to model an absorbing boundary condition. The displacement approximation (14) is extended to the absorbing boundary, Γ a, placed at a finite distance, r a, in the illustration presented in Figure 2 (in the absence of the outer layer of unbounded elements): u=b a q a on Γ a (50) As before, matrix B a collects hierarchical displacement approximation modes, constrained only by linear independence and completeness, and the dual transformation is used to enforce on average the (Robin-type) absorbing boundary condition (28): B a (t+iωc au)dγ a =0 (51) The compatibility condition (7) is still enforced in the weak form (33) or (36), which is rewritten to state explicitly the absorbing boundary of the element and the assumed approximation (50), extending thus result (37) into form: e=( ˆω 2 M+i ˆωC)p+A a q a +Aq+ē (52) A a = T B a dγ a (53) Equations (42) and (46) still hold as the finite element static admissibility and elasticity conditions. They are combined with the absorbing boundary condition (51), written for the assumed surface force and displacement fields (21) and (50), respectively, and with the extended kinematic admissibility condition (52) to establish the finite element solving system: D A a A p ē Aa D a O q a = 0 (54) A O O q f D a =iω Ba c ab a dγ a (55) In terms of numerical implementation, system (54) is treated as summarized above for bounded problems. The assembled solving system remains sparse and strongly localized as the additional boundary displacement vector, q a, is now strictly element dependent. The emulation of the conventional displacement element can be implemented using the condensed form (29) Absorbing boundary elements versus unbounded elements The study reported in Reference [4] shows that the formulation is not strongly sensitive to the positioning of the absorbing boundary, meaning that spurious reflections are not observed for relatively short distances to the origin of the perturbation, as shown in the test reported below. This distance is, in formal terms, determined by the range that renders admissible the first-order asymptotic approximation of the Hankel function used in the implementation of unbounded elements, as assumed in Appendix D to recover the Sommerfeld condition (28). The difficulty in establishing a formal definition for this distance may suggest the use of unbounded elements, as their definition does not depend on the asymptotic approximation of the basis functions. However, this advantage is only apparent because it is necessary to choose the distance where the unbounded elements are placed in the mesh, distance r a in Figure 2. This distance influences the finite element solution because the Hankel basis used to set up the unbounded element constrains the flow to outgoing waves. This fact, the added difficulty of computing unbounded integrals when closed forms are not available and, mainly, the relative insensitivity of the absorbing boundary position justify the option

14 HYBRID-TREFFTZ STRESS ELEMENT FOR POROELASTIC MEDIA 1293 to use the latter approach in the test reported here. The only information that is thus missed, and that is directly provided by appropriately placed unbounded elements, is the far-field characterization of the solid and fluid phases of the mixture. 9. ASSOCIATED ENERGY FUNCTIONALS The solutions produced by the stress model of the hybrid-trefftz finite element formulation are analysed next. In order to gain in generality, system (54) is extended to include the effect of enriching the domain approximation (13) with a particular solution term, r=sp+r p in V used to model the effect of non-trivial body forces or initial conditions (a necessity in time-stepping methods). The dependent approximations (16) and (17) are extended accordingly, e=ep+e p u=up+u p in V in V under the assumption that the particular stress, strain and displacement fields satisfy the domain conditions of the problem. This extension of the Trefftz constraint is not strictly necessary, being sufficient to ensure that the domain equilibrium condition is locally satisfied. Under this extension, the procedure described in Sections 7 and 8 leads to the following form for the stipulation vector of system (54), D A a A p ē e p Aa D a O q a = f a A O O q f p f e p = T u p dγ (56) f a = Ba t p dγ a f p = B t p dγ t The analysis of this system, which can be adapted to system (49) written for bounded or unbounded meshes, distinguishes non-hermitian and Hermitian forms. This property depends both on the material properties and on the time integration procedure being used, as the forcing frequency, ω, may be real (periodic spectral decomposition), imaginary (trapezoidal rules) or complex (in the non-periodic spectral decomposition used here). The alternative definition (47) for the elastodynamic matrix is useful to qualify this property, as the flexibility, mass and damping matrices are Hermitian. The local damping matrix, c a, present in definition (55) for the absorbing boundary matrix is symmetric and complex, in general, as it depends on the dissipation coefficient and on the forcing frequency Non-Hermitian systems It can be shown that system (56) is equivalent to the (computationally irrelevant) mathematical programming problem, Min z subject to system (56) (57)

15 1294 J. A. TEIXEIRA DE FREITAS AND I. D. MOLDOVAN where the objective function is real and quadratic: z =Re[ p Dp q a D aq a +p (ē e p )+q ( f f p ) q a f a] The definitions given above for the elastodynamic matrix in terms of the mass, damping and flexibility matrices and for the generalized boundary forces and displacements can be used to show that the objective function defines the real (conservative) part of the total mechanical energy (45) of the assembled finite element mesh, with the damping term now extended to account for the contribution of the absorbing boundary: C=iω u cu dv +iω u c a u dγ a Noting that the first equation of system (56) defines the finite element kinematic admissibility conditions and the remaining two equations define the static admissibility conditions, it may be stated that: Optimal solutions: If program (56) has optimal solutions, the optimal solutions are simultaneously (weak) statically and kinematically admissible and the objective function is null at optimality. A theorem on the multiplicity of optimal solutions to quadratic programming problems [21] yields the following statement when extended [22] to program (57): Multiple solutions: If(p,q a,q) is an optimal solution to the program (57), then the feasible solution (p,q a,q)+α(δp,δq a,δq) is also an optimal solution if the following conditions hold: Re(Δp DΔp+Δqa D aδq a )=0 p (D D )Δp+qa (D a Da )Δq a +(ē e p ) Δp ( f f p ) Δq+fa Δq a =0 D A a A Δp 0 Aa D a O Δq a = 0 A O O Δq 0 (58) It can be verified that Equation (58) leads to a symmetric quadratic form on the real and imaginary parts of vectors Δp and Δq a. Thus, if the (real) eigenvalues of the Hessian matrix of the quadratic form (58) are non-null for the forcing frequency, ω, the domain solution and the absorbing boundary solution are unique: Δp=0 and Δq a =0. Under this condition, the boundary solution is also unique, Δq=0, if the augmented boundary compatibility matrix is full-rank: [ ] A f f p It is noted that matrices D and D a in system (56) are block-diagonal and that they derive from linearly independent bases. The same constraint applies to the boundary approximation basis, which is constrained to satisfy condition (43) at the element level Hermitian systems The sufficient conditions stated above can be specialized for Hermitian systems. It is useful, however, to state explicitly [22] the possible alternatives for program (57). It can be verified that the (Hermitian) system (56) is also associated with the following pair of dual quadratic programming problems, MinP= 1 2 p Dp 1 2 q a D aq a q a f a +Re[q ( f f p )] subject to: Dp A a q a Aq=ē e p

16 HYBRID-TREFFTZ STRESS ELEMENT FOR POROELASTIC MEDIA 1295 MinP = 1 2 p Dp 1 2 q a D aq a +Re[p (ē e p )] subject to: A a p+d aq a = f a A p = f f p which recover the theorems on the stationary conditions of the potential and complementary potential energies, P and P, under the finite element kinematic and static admissibility conditions, respectively. It can be also verified that the objective function of the unconstrained quadratic program, MinH= 1 2 p Dp+ 1 2 q a D aq a +Re[p (A a q a +Aq+ē e p )+q a f a q ( f f p )] represents the Hellinger Reissner functional, and recovers the corresponding stationary condition. 10. NUMERICAL APPLICATION A comprehensive set of tests on the performance of the stress model of the hybrid-trefftz finite element formulation for the elastodynamic analysis of bounded and unbounded porous media is presented in [4, 18]. They characterize the patterns and rates of convergence in both frequency and time domains. They also show that the element is basically insensitive to distortion and that the solutions are marginally affected by the ratio of the wavelength to the characteristic length of the element. The testing problem defined in Figure 3 is selected to illustrate the levels of accuracy that can be attained using relatively coarse finite element and time approximation bases. The same application is reported in Freitas and Moldovan (2010; submitted) to provide a direct comparison of the results obtained with the alternative stress and displacement models of the hybrid-trefftz formulation. The soil specifications are taken from the tests used in Reference [3] (water-saturated Molsand with an incompressible solid matrix): water density, 1000kgm 3 ; mixture density, 2650kgm 3 ; Biot s first and second coefficients, 1.0 and Nm 2 ; Young s modulus, Nm 2 ; bulk modulus, Nm 2 ; the Poisson ratio, 0.333; porosity, 0.388; tortuosity, 1.0; hydraulic conductivity, 0.01ms 1. The load is transmitted to the solid phase and free drainage is assumed for the fluid. The 18-element mesh used in the analysis is defined in Figure 3. The semi-length of the loaded strip, L =2m, defines the radial length of the elements, meaning that the test is solved with four absorbing boundary elements placed at distance 5L from the origin of the load. Bessel functions of the first kind are used to define the domain approximation (13). The order of the approximation is d V =13 in each element, leading to a total of 1458 stress and pressure Figure 3. Testing problem and finite element mesh.

17 1296 J. A. TEIXEIRA DE FREITAS AND I. D. MOLDOVAN modes according to definition (27). Approximation (14) on the Dirichlet boundary (extended to include the interelement boundaries) is implemented with the Chebyshev polynomials, with degree d Γ =5, and the absorbing boundary approximation (50) with trigonometric functions with order d Γ =3, for a total of 726 independent displacement modes according to results (31) and (32). The time-domain analysis is implemented in a single time increment, Δt =0.5s. The time basis is defined by the fourth Daubechies family of wavelet and scale functions, which generates 32 pseudo-spectral problems (56). In crude terms, this is equivalent to solve the problem in the same number of time increments. The frequency norms of these pseudo-spectral problems range from to rads 1. The ratio of the characteristic element length to the wavelength, L FE /λ= Re(β j L FE )/2π ranges from 6 to 230. As the domain conditions of the problem are locally satisfied in a strong form by the Trefftz basis, assessment of the quality of the solution, in time and in space, should focus on the implementation of the boundary and interelement continuity conditions. It is recalled that the Neumann and interelement surface force and pressure continuity conditions, and the absorbing boundary conditions, are explicitly enforced in the finite element solving system (56). The Dirichlet and interelement conditions on the two solid displacement components and on the normal component of the solid fluid relative displacement are not explicitly enforced. Continuity is attained upon convergence of the finite element solution. The stress and pore pressure fields obtained with this approximation in time and space are shown in Figure 4. They are defined on the deformed shape of the medium at selected instants of the duration of the test. The results show that the interelement equilibrium conditions and the Neumann boundary conditions are correctly enforced. The enforcement of the absorbing boundary condition is weak in the early stage of the test due to the low (third-) order of the approximation. However, it is sufficient to inhibit spurious reflexions throughout the test, as shown in the representations of the displacement fields in the solid and fluid phases of the mixture presented in Figures 5 and 6. These results show that the interelement and Dirichlet conditions are also enforced adequately, although these conditions are not explicitly enforced in the solving system (56). The results presented in Figures 4 6 show that the formulation models correctly the main characteristics of the dynamic response of the water-saturated Molsand under the loading conditions defined in Figure 3: only dilatational waves are visible along the axis of symmetry, which dampen strongly in depth; the effect of shear waves increases with the distance from this axis, causing a counterclockwise, elliptic motion; the displacements are initially limited to the zone directly affected by the loading and later gradually propagate through the medium, and a similar pattern is recovered by the modelling of the corresponding stress waves. The variation in time of the stress and displacement components shown in Figure 7 is presented to better illustrate the travelling wave front and the effects of consolidation and swelling. The sampling points are placed on the axis of symmetry at depths 0.5 (solid lines), 2.5 (broken) and 5.0m (dashed). The stresses are released immediately after full unloading, at instant t =0.25s, whereas the release of pressure is slower and dependent on the velocity of the liquid exuding the compressed solid matrix. The solutions of all spectral problems are obtained with the same domain and boundary approximation bases and the same finite element mesh. No smoothing is used in the representation of the stress, pressure and displacement fields. The animation of the dynamic response of the unbounded biphasic medium can be accessed using address id=satsoilfilms. Accuracy of the finite element solutions at every instant of the analysis can be improved refining the mesh and/or increasing the order of the approximation in space and time. 11. CLOSURE The stress model of the hybrid-trefftz finite element formulation is applied to the time domain of saturated poroelastic media. The formulation is written for both bounded and unbounded media and,

18 HYBRID-TREFFTZ STRESS ELEMENT FOR POROELASTIC MEDIA 1297 σ xx σ yy σ xy π t (s) t = 0.25 s Figure 4. Stress and pore pressure fields (Nm 2 ). in the latter case, particularized for modelling with unbounded elements and absorbing boundary elements. The time dimension of the problem is discretized first to establish the boundary value problem to be solved coupling the basic concepts of the Trefftz and finite element methods. The formulation can be implemented with alternative time integration methods. However, the structure of the resulting boundary value problem and, consequently, the Trefftz approximation basis are dictated by the method selected for the time domain analysis. The option followed here is to implement a pseudo-spectral decomposition that can be implemented with alternative time approximation basis, namely with trigonometric, polynomial or digital functions, appropriate for periodic and non-periodic solutions and in the latter cases, for small and large time-stepping, respectively.