A time domain coupled boundary element - finite element method for the dynamic response of structures S. Fran¸cois, H.R. Masoumi and G. Degrande K.U.Leuven, Department of Civil Engineering Kasteelpark Arenberg 40, B-3001 Leuven email:
[email protected],
[email protected],
[email protected] Abstract—Dynamic excitations in the built environment as caused by earthquakes, heavy traffic and pile driving may result in structural damage, which is determined by the constitutive behaviour of building materials and foundation soils under cyclic loading. This paper presents a coupled finite element-boundary element approach for the calculation of the dynamic response of structures due to dynamic excitations. Both the non-linear constitutive behaviour and the dynamic interaction between the soil and the structure are accounted for. A time domain finite element formulation is used for the structure, as the non-linear constitutive behaviour of the structural materials requires a direct time integration procedure. The soil is assumed to be linear elastic and a boundary element method is used to fully account for dynamic soil-structure interaction. As both the finite and the boundary element method impose different conditions on the time integration step for reasons of stability and accuracy, an iterative coupling scheme is proposed that allows for a different time step in both subdomains. An interface relaxation technique is employed in order to speed up convergence. Instead of selecting a constant value, an optimal relaxation parameter is computed using Aitken’s method, resulting in a non-stationary Richardson iteration. The numerical behaviour of this scheme is studied in detail. The method is applied to the calculation of the response of a structure due to traffic induced vibrations, where the dynamic interaction between the soil and the structure is fully accounted for. Keywords— Boundary elements
impose different conditions on the time integration step for reasons of accuracy and convergence, an iterative coupling scheme is considered, allowing for a different time step in both subdomains. II. Variational formulation in the time domain The equations of motion of a structure excited by an incident wave field are elaborated (figure 1). Two substructures are defined: the first is the generalised structure Ωb , which can have non-linear characteristics and possibly contains a non-linear or non-homogeneous part of the soil. The second substructure is the linear elastic unbounded soil domain Ωext s . Due to the semi-infinite extent of the soil domain, radiation conditions have to be satisfied. The substructures are coupled through the boundary Σbs between the generalised structure and the soil. The equilibrium of Γbt ∪ Γbu
Ωb Γst ∪ Γsu
Σbs
I. Introduction
R
EPEATED dynamic excitations in the built environment as caused by heavy traffic and pile driving may result in structural damage, which is determined by the constitutive behaviour of building materials and foundation soils under cyclic loading. Recently, a numerical model for the prediction of traffic induced structural vibrations that fully accounts for the dynamic soil-structure interaction has been developed and validated [1]. The model is limited to linear structural behaviour as the analysis is performed in the frequency domain. In the case of structural damage, however, a frequency domain technique is not applicable as superposition is not allowed due to the non-linear constitutive behaviour. Therefore, a direct time integration procedure is required. This paper presents a coupled finite element-boundary element approach in the time domain for the calculation of the structural response due to dynamic excitations as arising from traffic or pile driving. The structure is modelled with finite elements while the soil domain is modelled with boundary elements. Both the non-linear constitutive behaviour and the dynamic interaction between the soil and the structure are accounted for. As both the finite and the boundary element method
Ωext s Γ∞
Fig. 1. Geometry of the subdomains. the structure is expressed in a weak sense by means of the principle of virtual displacements. The sum of the virtual work performed by internal and inertial forces is equal to the total virtual work of external forces [2]: Z Z δub ρb bb dΩ (σ b (ub ) : δb + ρ¨ ub δub ) dΩ = Γbs Ωb Z Z + δub¯tb dΣ + δub tb dΣ (1) Γbt
Σbs
where ub is the structural displacement vector, σ b and b are the internal stress and strain tensors, ρb bb is the volume force acting on Ωb and tb = σ b · nb is the stress vector on a plane with outward unit normal nb . Neumann conditions tb = ¯tb are considered on the boundary Γbt . After the introduction of a finite element discretisation, a Galerkin approach results in the following finite element
equations:
Mb1 b1 Mb2 b1
stresses on the boundary Σbs : Z Qnb2s = − NT b2 (x)Ns (x) dΣ
¨ b1 (t) Mb1 b2 u ¨ b2 (t) u Mb2 b2 t ub1 (t) Kb1 b1 Ktb1 b2 + ub2 (t) Ktb2 b1 Ktb2 b2 ext 0 f b1 (t) + = Qb2 (t) f ext b2 (t)
Σbs
(2)
where ub2 are the degrees of freedom on the interface Σbs and ub1 are the remaining degrees of freedom. A tangential stiffness matrix is considered. The vector Z Qb2 (t) = NT (3) b2 (x)tb (x, t) dΣ Σbs
represents the vector of soil-structure interaction forces, where Nb2 (x) are the global finite element shape functions related to the degrees of freedom ub2 . A Newmark direct time integration method is employed for the solution of equation (2). The time discretised equilibrium equation nb +1 ˜ b1 b1 K ˜ b1 b2 ub1 K ˜ b2 b1 K ˜ b2 b2 unb b +1 K ( n +1 )2 ˜f b 0 b1 (4) + = Qbn2b +1 ˜f nb +1 b2 is solved iteratively for every time step tnb b = (nb − 1)∆tb . III. Boundary element method The response of the soil domain Ωext is computed by s means of the boundary element method. For convenience, it is assumed that the displacement degrees of freedom of the boundary element mesh correspond to the degrees of freedom ub2 of the structural finite element model, requiring conforming boundary and finite element meshes on the interface. A classic time domain boundary element formulation is considered [3]: T1 uns s
=
U1 tns s
+
ns X
Uns −k+1 tks − Tns −k+1 uks
k=1
(5)
(7)
where Ns (x) is the boundary element traction interpolation function. As a result, a transformation matrix TQ can be defined that relates the boundary tractions ts to the corresponding finite element forces Qb2 as tns s = TQ Qnb2s . In order to facilitate the coupling of the boundary and finite element equations, this transformation is introduced into the boundary element equation (6): 1
˘ Qns + hns T1 uns s = U b2
(8)
˘ 1 = U1 TQ . The term hns is computed as: with U hns =
ns X ˘ ns −k+1 Qk − Tns −k+1 uk U s b2
(9)
k=1
˘ k = Uk TQ . with U IV. FE-BE coupling A. Direct domain coupling In a direct coupling approach, the dynamic equilibrium equations for both subdomains are combined, resulting in a global coupled system of equations. The same time discretisation is considered for the finite and boundary element domains. In a rather straightforward way, the boundary element equation (8) is evaluated at tsns +1 : 1 −1 ˘ Qnb2s +1 = U T1 uns +1 − hns +1
(10)
and introduced in the finite element equation (4): #! " 0 0 ˜ b1 b1 K ˜ b1 b2 ubn1b +1 K −1 + ˜b b K ˜b b ˘1 unb2b +1 K U 0 T1 2 1 2 2 ( n +1 ) ( ) 0 ˜f b b1 −1 = + ˘1 ˜f nb +1 U hnb +1 b 2
where the vectors uns s and tns s collect the boundary displacements and tractions. A time discretisation tns s = (ns − 1)∆ts with a constant time step ∆ts is considered. This discretisation can be different from the time discretisation for the structure, in view of the iterative coupling scheme where the equilibrium of the structure and the soil are evaluated at different time steps. The matrices Tns and Uns are referred to as the influence matrices at time step ns . The second term on the right hand side of equation (5) is a convolutive sum representing the influence of the response at previous time steps on the response at time step ns . For brevity, this term is denoted as hns and equation (5) reduces to: T1 uns s = U1 tns s + hns
tns s
(6)
In order to couple the boundary and finite element domains, the boundary element traction discretisation is introduced into equation (3), enforcing the equilibrium of
(11)
Equation (11) represents the dynamic equilibrium of the coupled soil-structure interaction system in terms of the structural degrees of freedom and can be solved with a classical finite element solver. Direct coupling is the classical approach to couple finite and boundary elements [4], [5], [6], because of its straightforward implementation in a finite element direct time integration scheme. However, the method suffers from some 1 −1 ˘ disadvantages. First, the fully populated matrix U
strongly reduces the sparsity of the system under consideration. As a result, the application of a sparse finite element solver is less efficient, especially in the case where the number of interface degrees of freedom is large with respect to the total number of degrees of freedom. Secondly, the same time discretisation should be employed in both subdomains. This is an important disadvantage as the range of
suitable time steps for the boundary element domain is limited. For a three-dimensional elastodynamic problem, this range is considered to be 0.7Cp /lBE < ∆ts < 1.2Cp /lBE where Cp /lBE represents the time needed for a longitudinal wave with a wave velocity Cp to travel over a boundary element with a size lBE . A smaller time step results in noncausal, unstable boundary element equations while strong numerical damping is observed for a larger time step [7]. For the finite element domain, the time step should result in an accurate solution and satisfy stability conditions based on energy considerations [8]. These considerations often lead to incompatible time steps in both subdomains. B. Iterative coupling As an alternative to a direct coupling approach, staggered and iterative solution procedures can be employed where the equations for both subdomains are solved separately, avoiding the assembly and solution of a global coupled system of equations. This approach shows some advantages over the direct coupling approach. First, the finite element equations for the structure remain symmetric and sparse while the boundary element equations are not. Therefore, the use of a dedicated solver for each subdomain results in a reduced computational effort. In a staggered solution approach [9], [10], the equations for each subdomain are solved once for each time step. For the first subdomain (e.g. the structure), the boundary conditions are estimated from previous time steps. The solution is applied as a boundary condition to the second subdomain (e.g. the soil) in the same time step. A suitable predictor operator and a sufficiently small time step are prerequisites for convergence. Rizos et al. [11], [12] applied a staggered FE-BE method to compute the seismic response of foundations. The unbounded soil domain is modelled using time domain boundary elements with higher order B-spline fundamental solutions. O’Brien and Rizos [13] applied the methodology to the prediction of railway induced vibrations. However, staggered solution methods require a sufficiently small time step in order to achieve convergence. The maximum time step depends on the mechanical behaviour of both subdomains and might result in an unstable boundary element equation. Therefore, staggered solution procedures are not considered in the present work. In an iterative procedure, the equations for each subdomain are solved iteratively in each time step, updating the boundary conditions at the interface between the subdomains until convergence is achieved. In order to improve the convergence speed, a relaxation operator is often applied within the iterative procedure to the predicted boundary conditions on the interface. Interface relaxation techniques for static FE-BE and BE-BE coupling are discussed in detail by El-Gebeily et al. [14] and Elleithy et al. [15], [16], [17]. An iterative solution approach for two-dimensional transient FE-BE coupling has been introduced by Soares et al. [18]. An advantage over the direct and staggered approach is that different time steps can be employed in each subdomain, which is beneficial for both stability and ac-
curacy. The resulting method is denoted as a sequential Neumann-Dirichlet algorithm, since the finite element domain is solved with Neumann boundary conditions on the interface, while the boundary element domain is solved with Dirichlet boundary conditions on the interface. In order to speed up convergence, a boundary relaxation is applied to the boundary displacements. Hagen [19], [20] and Von Estorff and Hagen [21] extended this iterative method in the framework of fluid-soilstructure interaction to allow for the coupling of multiple BE domains to a FE domain. Furthermore, a relaxation of both boundary displacements and interaction forces is introduced. A parametric study has been performed to study the optimal choice of the relaxation parameter [19]. In the present work, a similar iterative scheme is proposed. The original sequential Neumann-Dirichlet algorithm of Soares et al. [18] is considered, as in the case of a dynamic soil-structure interaction problem only one BE domain is coupled to a FE domain. A modified version of the algorithm is applied where the interaction forces instead of the interface displacements are relaxed. This allows for a detailed study of the convergence properties of the numerical scheme and results in an optimal choice for the relaxation parameter. C. Sequential Neumann-Dirichlet algorithm The sequential Neumann-Dirichlet algorithm is outlined in table I and is discussed in the following. As different time steps are selected for the finite and boundary element domains, the equilibrium of both domains is considered at times tnb b +1 and tns s +1 . Hence, the algorithm requires a time interpolation of the interface forces and displacements, where it is assumed that tnb b +1 < tns s +1 . The interpolation scheme is similar to the boundary element time interpolation, where a linear interpolation is used for the displacements and a constant interpolation for the interaction forces (figure 2). The equations for both subdomains are solved in a sequential, iterative way. First, the finite element equations with Neumann boundary conditions on the soil-structure interface are considered for iteration step k + 1: ( nb +1 ) ˜ b1 b2 ˜ b1 b1 K ub1 (k+1) K +1 ˜ ˜ unb2b(k+1) Kb2 b1 Kb2 b2 ( n +1 ) ˜f b 0 b1 + (12) = +1 Qbn2b(k) ˜f nb +1 b2 +1 where Qnb2b(k) represents the soil-structure interaction force from the previous iteration k. For the first iteration, +1 +1 Qnb2b(k=0) = Qnb2b . The displacements unb1b(k+1) on the interface are found through solution of equation (12) (figure 2, step À): −1 +1 ˜ ˜ −1 K ˜ b2 b1 K ˜ b2 b2 − K unb2b(k+1) = K b1 b1 b1 b2 n +1 b ˜ b2 b1 K ˜ −1 ˜f nb +1 + Qnb +1 × ˜f b2 − K (13) b1 b1 b1 b2 (k)
˜ ˜ −1 K ˜b b − K ˜b b K The matrix (K b1 b1 b1 b2 ) is denoted as the 2 2 2 1 Schur complement or discretised Stekolov-Poincarr´e oper-
TABLE I Outline of the sequential Neumann-Dirichlet algorithm.
1. Initialization 1.1 ns ← 1 and nb ← 1 2. Time stepping procedure 2.1 Initial time step for the BE domain: ns ← 1 and t1s = ∆ts 2.2 Loop over all time steps: nb ← nb + 1 2.3 While tnb b > tns s : update the BE time step 2.3.1 ns ← ns + 1 and tns s = tsns −1 + ∆ts 2.3.2 Compute the BE interaction forces in the previous time step 2.4 Iterative solution 2.4.1 Solve the FE problem with Neumann boundary conditions on the interface Σbs 2.4.2 Time extrapolation of the boundary displacements to time tns s 2.4.3 Solution of the boundary element problem at time tns s with Dirichlet conditions on Σbs 2.4.4 Time interpolation to obtain interaction forces at time tnb b 2.4.5 Relaxation of interaction forces 2.4.6 Convergence check.
The boundary displacements are found as the solution of equation (12) for which finite element solvers are employed that account for the sparsity and symmetry of the system matrix. Nevertheless, equation (13) is used in the analysis of the numerical behaviour of the algorithm under consideration. The interface displacements are subsequently imposed as a Dirichlet boundary condition on the BE domain. This requires an extrapolation of the boundary displacements to time tns s +1 (figure 2, step Á): +1 unb2s(k+1) =
1 nb +1 ub2 (k+1) − βunb2s 1−β
where β = (tns s +1 − tnb b +1 )/∆ts . The solution of the Dirichlet boundary problem is written as (figure 2, step Â): 1 −1 +1 +1 ˘ Qnb2s(k+1) = U Tunb2s(k+1) − hns +1
(15)
1 −1 ˘ Again, the inverse U of the soil flexibility is never ac-
tually computed, but equation (15) is usefull in the study of the numerical behaviour of the iterative coupling algorithm. Next, a constant time interpolation of the soil-structure interaction force from tns s +1 to tnb b +1 is performed (figure 2, step Ã): +1 +1 ¯ nb b(k+1) Q = Qnb2s(k+1) 2
ub2
(14)
(16)
+1 unb2s(k+2)
where the bar indicates an unrelaxed interaction force. The interaction forces are finally relaxed by means of a relaxation parameter ω (figure 2, step Ä):
+1 unb2b(k+2) +1 unb2s(k+1)
Á +1 unb2b(k+1)
n +1
nb +1 +1 ¯ b Qnb2b(k+1) = ωQ b2 (k+1) + (1 − ω) Qb2 (k)
t À
Â
Qb2 +1 ¯ nb b(k+1) Q 2
Ã
In order to study the numerical behaviour of the iterb +1 ative coupling scheme, the interface displacements un(k) are eliminated from equations (12) to (16). The resulting evolutive equation is written as:
+1 Qnb2b(k)
t tnb
tns
tnb +1
The iterative procedure is performed until the boundary displacements and interaction forces converge. In the case of a non-linear structure, these iterations are combined with the Newton-Raphson iteration for the finite element equations. C.1 Convergence properties
+1 Qnb2s(k+1)
Ä +1 Qnb2b(k+1)
(17)
tns +1
Fig. 2. Force and displacement interpolation in the iterative sequential Neumann-Dirichlet algorithm. ator Sb2 b2 and represents the condensation of the dynamic stiffness matrix of the finite element model on the interface. As the evaluation of the Schur complement is computationally expensive, equation (13) is never evaluated explicitly.
+1 +1 Qnb2b(k+1) = Qnb2b(k) 1 −1 nb +1 1 −1 ˜nb +1 1 ˘ Sb2 b2 f b2 − K21 K11 ˜f b1 +ω U T 1−β −1 ns +1 h − T1 nb +1 1 −1 ˘ 1 U + T Q + S−1 (18) b2 b2 b2 (k)
This evolutive equation is identified as a Richardson iteration. The convergence properties of the iterative scheme
mainly depend on the relaxation parameter ω and on the spectral properties of the operator matrix: 1 −1 1 −1 ˘ 1 ˘ A= U U T1 S−1 b2 b2 + T
(19)
1 −1 ˘ T1 , which is the Schur complement of The factor U the boundary element domain, can be seen as a preconditioner [22]. In the case where the relaxation parameter ω is constant during the iteration, the Richardson iteration is stationary. In the linear case, the scheme is stable if [23]: 0
