Numerical Methods in Soil Mechanics BEM FEM soil structure

Applications of Staggered BEM-FEM Solutions to Soil-Structure
Interaction
D.C. Rizos, M. ASCE and Z.Y. Wang
University of Nebraska Lincoln, Lincoln NE 68588

Abstract:
The present work introduces a direct time domain BEM-FEM formulation for 3-D SoilStructure Interaction analysis. The proposed BEM that is based on the B-Spline family
of fundamental solutions computes the dynamic response of the soil domain through a
superposition of the characteristic B-Spline impulse responses. Standard direct
integration FEM procedures are used to compute the dynamic response of the structure.
A staggered solution process is proposed for the coupling of the two methods. The
proposed methodology is applied on the problem of the dynamic through-the-soil
interaction of massive foundations, and the examples presented in this work demonstrate
the accuracy and efficiency of the method.
Introduction
The idea of coupling the FEM and the BEM finds its origins in the work of McDonal and
Wexley in the beginning of 1970’s on the microwave theory. The first organized
formulation was presented by Zienkiewicz and his coworkers (1977) for analysis of
solids. Coupled BEM-FEM procedures are of three general types. The first one is a
Boundary Element (BE) approach, which considers the Finite Element (FE) subdomains
as equivalent BE subregions by transforming the force-displacement relations of the FEM
to “BEM-like” traction-displacement relations. The second approach is the FE, in which,
the BE equations are considered as a special case of the FE procedures. Staggered
BEM-FEM solutions have been implemented in fluid-structure interaction analysis. The
coupling of the FEM with the BEM for wave propagation and soil-structure interaction
problems follows similar procedures. The solutions are obtained in either a direct time
domain, or a frequency (transform) domain approach. Most of the coupled FEM-BEM
solutions reported in the literature are in the frequency domain and adopt the FE or BE
approach. One can mention the work of Bielak et al. (1984), Gaitanaros and Karabalis
(1986), Aubry and Clouteau (1992), and Chuhan at al. (1993), among others. Only a few
publications have dealt with the time domain BEM-FEM techniques for SSI and wave

propagation problems. Karabalis and Beskos (1985) and Spyrakos and Beskos (1986)
have reported on 2-D and 3-D flexible foundations following the BE approach. Fukui
(1987) and Estorff and Kausel (1989,1990) reported on more generally applicable
coupling formulations for 2-D scattering of anti-plane waves and 2-D plane-strain.
This work employs the B-Spline BEM formulation for 3-D wave propagation and SSI in
elastic media reported by Rizos (1993) and Rizos and Karabalis (1994,1998). A staggered
solution algorithm for the coupling with standard FEM processes in the direct time
domain is introduced.


BEM Formulation
The direct time domain BEM employed in this work is developed based on a special case
of the Stokes fundamental solutions for the infinite elastodynamic space that assumes that
the time variation of the body forces in the domain is defined by the B-Spline
polynomials. The derived B-Spline fundamental solutions accommodate virtually any
order of parametric representation of the time dependent variables without excessive
computational effort and implicitly satisfy the continuity conditions of the Stokes
fundamental solutions. A detailed formulation and integration of the B-Spline
fundamental solutions and of the associated BEM can be found in the work of Rizos
(1993), and Rizos and Karabalis (1994, 1998).
Under the assumption of a small displacement field in a linear isotropic and
homogeneous space, the Navier-Cauchy equations of motion can be expressed in the
Love integral identity form as,

{ [

]

}

cij (î )u i (î , t ) = ∫ U ijB x, t ; î t (n )i (x, t ) − TijB [x, t ; î ui (x, t )] dS

(1)

S

where S is the bounding surface of the elastodynamic domain, î, and x represent the
receiver and source points, respectively, and the tensor cij is known as the “jump” term
that depends on the geometric characteristics of the domain in the neighborhood of the
receiver point. The tensors ui (x, t ) , and t (n )i (x, t ) are the displacement and traction fields
of the actual elastodynamic state, and the tensors U ijB , and TijB are the B-Spline
fundamental solutions of the infinite elastodynamic space (Rizos and Karabalis 1998).
Appropriate spatial and temporal discretization schemes along with a transformation of
tractions to forces are applied on equation (1) and the system of algebraic equations is
derived as
~ N
N
T*u N = G *L−1f soil
+ RN = Gf soil
+ RN

(2)

where T* and U* are coefficient matrices derived on the basis of the B-Spline
fundamental solutions and depend only on the first and/or second time steps, L is the
traction-force transformation matrix obtained on a virtual displacement approach and
vector RN represents the influence of the past time steps on the current step N and is
always known. Equation (2) can be solved in a time marching scheme to obtain the
B-Spline impulse response of the elastodynamic system. The B-Spline impulse response
of all degrees of freedom due to unit force excitations can be collected in a matrix form

that represents the time dependent flexibility matrix, BR(t), of the elastodynamic system.
The response u(t) to an arbitrary excitation f(t) can be calculated by an appropriate
superposition of the B-Spline impulse responses as
j

u(t ) = ∑ BR(t − t i )f (τ i ),
i =1

[ ]

t ∈ t1 , t j

and τ i =

ti + ti +1 + L + t i +k −1
,
k −1

k >1

(3)


where k is the order of the B-Spline fundamental solutions. Matrices BR are independent
of the actual external excitations and need to be computed only once for each soil region.
FEM Formulation
The FEM system of equations for the structural model is also solved in a time marching
scheme using Newmark’s algorithm. The FEM equations relate incremental forces,
˜f iFE , to displacements ˜u iFE on discrete nodes in the FE model at time interval i, and
are presented symbolically as,

ˆ ˜u FE = ˜fˆ FE
K
i
i

(4)

ˆ is the dynamic stiffness matrix and ^ indicates quantities related to the
where K
Newmark’s process.
BEM-FEM Coupling
At the interface of the FE and BE models, the compatibility conditions of the
displacement vector, u, and force vector, f, need to be satisfied at all time instances tj,
BE
(t j ) = u intFE (t j )
u int

f intBE (t j ) = −f intFE (t j )

(5)

The coupling of the two models is achieved through a staggered solution approach
according to which the solution of one method serves as initial conditions to the other at
every time step, as depicted in Figure 1. In view of equation (3) the BEM solver
evaluates displacements by a superposition of the B-Spline impulse responses without
solving any system of equations. The FEM solver, however, needs to solve for the
unknown forces at the BE-FE interface. It is apparent that this coupling scheme increases
the efficiency of the solution by reducing the computing time. The proposed scheme has
shown superior accuracy and stability for the examples examined so far.
fFE

f intFE (t j ) = −f intBE (t j )

fBE

FEM

BEM

Solver
Structure

Solver
Soil
uBE

uFE

FE
(t j ) = u intBE (t j )
u int

uBE

Figure 1 Staggered Solution Scheme


Numerical Examples
Analysis of Rigid Surface Massive Foundations
This example examines the through-the-soil interaction of two adjacent massive rigid

foundations. The “structural component” of this soil-structure system pertains only to
inertia forces, which are not known a priori, and demonstrates the compatibility of the BE
solver with the proposed solution scheme. The footings are square of side 2b=5 and their
weights correspond to mass ratio M=10. The constants of the surrounding soil medium
are Poisson’s ratio ν = 1/3, mass density ρ=10.368 lb.sec2/ft4 and modulus of elasticity
E=2.5898x109 lb/ft2. The surface of the half space is modeled by 8 node quadrilateral
elements and each footing covers an area of 4x4 elements. The rigid conditions are
implemented according to the rigid surface element introduced by Rizos (1999). The
frequency domain solutions are due to harmonic forcing functions of unit magnitude
applied on one foundation. The B-Spline impulse response matrices of the foundation
system are obtained only once. Subsequently, for each excitation, the solution is obtained
in the time domain through the procedure outline above and the maximum amplitude of
the steady state is defined. This approach is very efficient since the BE solver reduces to
a mere superposition of pre-computed quantities, as implied by Equation (3). Figure 2
shows the maximum amplitude of response of the excited and the unloaded foundations
as a function of the dimensionless frequency. In this example the footings are spaced at a
distance d/b=0.25 apart. The results are compared to the ones reported by Huang (1993)
and the accuracy is evident. Other modes of vibration as well as distance ratios have
shown the same accuracy.
1.2E-10

Excited - Proposed Work
Unloaded-Proposed Work

1.0E-10

Excited - Huang
Unloaded - Huang
Posin(ωt)

Amplitude

8.0E-11

2b
d

Half Space

6.0E-11

4.0E-11

2.0E-11

0.0E+00
0

0.5

1

1.5

2

2.5

3

3.5

4

Dimensionless Frequency ao

Figure 2 Through-the-Soil interaction of Massive Rigid Foundations


Hollow Rigid Surface Foundation
This example examines a square footing of size 2a=5 ft with a centered square hole of
size 2d for which d/a=0.75. The mass of the foundation varies so that the mass ratio M
takes the values of M=1,3,5 and 10. The footing rests on the elastic half-space described
in the previous example. A series of harmonic excitations of various frequencies are
applied at its center in order to define the amplitude of the steady state response. The
amplitudes of the vertical mode of vibration are shown in Figure 3 as function of the
dimensionless frequency for all mass ratios considered. A comparison with a solution
reported by Huang (1993) for mass ratio M=3 is also shown. All modes of vibration have
been examined, as well as a number of d/a ratios. The proposed method compared
favorably and always converged for the considered frequency range.
1.2E-10
M=10

M=5

M=3

M=1

Huang M=3

M=10
Posin(ωt)

1.0E-10

Amplitude (ft)

M=5

2d

2a

8.0E-11

6.0E-11

Half Space

M=3

4.0E-11

M=1

2.0E-11

0.0E+00

0

1

2

3

4

5

Dimensionless Frequency a0

Figure 3 Vertical Response of Hollow Foundation
Conclusions
A methodology is developed for the efficient coupling of the Finite Element with the
Boundary Element Method for 3-D wave propagation and Soil-Structure Interaction
Analysis in the direct time domain. The method uses the newly developed B-Spline
BEM along with standard FEM processes. The coupling is obtained through a staggered
scheme, which satisfies the compatibility and equilibrium conditions at the interface
boundary between the BEM and FEM domains. This article presented the first attempt to
implement the method and the problem of analysis of massive foundations was selected.
In such problems, kinematic and inertial interaction effects are present. Although the
FEM domain does not contain elastic or damping forces of a real structure, this class of
problems verifies the suitability of the B-Spline BEM method to such staggered solution
schemes. It has been shown that the proposed methodology is accurate and stable for the


class of problems considered, and is very efficient. The present formulations will be

further developed to account for elastic and damping forces of a structure.
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