A CONTROL ANALYSIS OF DISPLACEMENT IN
FLUID-SOIL-BALLOON INTERACTION PROBLEM
Shoichiro Kato and Mutsuto Kawahara
ABSTRACT
This paper presents a control analysis of displacement for a building with friction piles. The
building constructed on the poor subsoil sometimes comes to be under a differential settlement
condition, i.e., the building tends to be inclined at that time. To control the vertical displacement,
the control device of multi-balloons is introduced on the friction piles, and the water is injected or
removed from the balloons. Coupling through the water, the soil and the balloon, the interaction
problem is numerically solved. The control analysis of vertical displacement in case of a balloon is
performed. One of the optimal control theory, so-called as the Sakawa-Shindo method, is applied in
the control analysis. Using this method, the control flux of the water injected or removed from the
balloon is determined so that the position at the top of the balloon comes to be close to the objective
position.
KEYWORDS
Control Analysis, Interaction Problem, Balloons, Vertical Displacement, Sakawa-Shindo method
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INTRODUCTION
A building with friction piles on poor subsoil often comes to be under the differential settlement,
and inclines at that time. To keep the building horizontal, multi-balloons of control device are
introduced on the friction piles. Then water is injected into or removed from the balloons to control
the vertical displacement of the building. The two-dimensional incompressible Navier-Stokes
equation is introduced for the water flow analysis. The ALE(Arbitrary Lagrangian Eulerian)
method[1,2] and the FS(Fractional Step) method[3,4] are applied for the numerical formulation.
Coupling though the water, soil, balloon and pile, a finite element analysis is performed. A control
analysis of vertical displacement is also carried out for the interaction problem. One of the optimal
control theory, which is the so-called Sakawa-Shindo method[5], is applied for the optimal control
analysis. Control flux of the water fluid, which is injected into or removed from the balloon, is
determined so that position at the top of the balloon comes to be very close to objective position, i.e.,
the performance function converges to a constant value.
MATHEMATICAL MODELS
Water flow analysis
Introducing the Cartesian coordinate system, the two dimensional incompressible Navier-Stokes
equation is used and the ALE method is applied for the water flow analysis. The flux of water fluid
is introduced into the equation of continuity so that the water is injected into or removed from the
balloon. Discretizing by conventional Galerkin method in space, the velocity and pressure are
linearly-interpolated in an arbitrary triangular finite element. The FS method is applied for temporal
descretization, and the one-step explicit scheme is used in time.
Soil deformation analysis
The soil outside a balloon is assumed to be two-dimensional linear elastic body. Assuming the
isotropy of material, the displacement and virtual displacement inside of an arbitrary triangular
finite element are linearly-interpolated.
Balloon and pile deformation analyses
The balloon and the pile attached on the balloon are assumed to be linear elastic truss and
rigid-frame components. The axial, shearing, and bending deformation are considered, although the
bending deformation is only considered in addition to the shearing deformation. The axial
deformation analysis is for the balloon and pile, though the shearing and bending deformation
analysis is for the pile only. Also the finite element formulation is performed for the balloon and
pile deformation analysis. Superposing the finite element equation to that of the soil deformation,
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the balloon, pile and soil deformation analyses are carried out at the same time. The inertia is
neglected in the equations governing soil, balloon, and pile. Because the phenomena of differential
settlement and its control are relatively slow.
INTERACTION PROBLEM
The coupling procedure between the water and soil is introduced in this section. It should be noted
that the soil has already included the balloon and pile, i.e., it comes to be the water-soil interaction
problem. The following continuity condition of the traction is imposed on the interactive boundary.
t  w ti
(1)
s i
where sti and wti represent the traction of the soil in the outer domain, and water in the inner domain.
The continuity condition of displacement should be also imposed on the interactive boundary. To do
this, the displacement must be defined at the boundary of the water flow. Using the displacement at
n+1 and n time steps, the fluid velocity at n+1 time step is approximated in the following form.
u
n 1
i

in1  in
(2)
t
where ηi denotes the displacement of the boundary points. Defining ηi at n+1 time step as the
representative displacement of the water flow boundary, then the following continuity condition of
the displacement is imposed on the interactive boundary.
vi  in1
(3)
where vi denotes the displacement of soil at the boundary. Thus the coupling is performed, and the
interaction problem between the water and soil is calculated in the following procedure.
CONTROL ANALYSIS OF VERTICAL DISPLACEMENT
State equations and performance function
State equation of water flow analysis is expressed in the following form.
Mx1  Ax1  Bx2  Cf1  G1t
(4)
Kx2  Dx1  Eq
(5)
x(t 0 )  x0
(6)
x T  {x1T x2T }, x1T  {u T wT }, x2T  p T .
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State equation of soil deformation analysis, which includes balloon and pile deformation analyses,
is also expressed in the following form.
Hy  If 2  G2t
(7)
The vectors of u, w, p and q represent the velocity components, the pressure and the flux of water
fluid, respectively. The vectors of f1 , f2 and t denote the external forces and the traction on the
interactive boundary. The vector y represents the displacements.
Hence the performance function, which is also called as cost function, is defined as:
J

 


1 tf
* T
y

y
S y  y *  q T Rq dt

t
2 0
(8)
where y* denotes objective vector of displacement, and S, R are weighting matrices. Introducing the
constraint conditions of eq.(4), eq.(5), and eq.(7) into eq.(8), the extended performance function is
defined as:

 


1 tf
* T
y

y
S y  y *  q T Rq dt

t
2 0
tf
  p1T Mx1  Ax1  Bx2  Cf1  G1t dt
Jc 
t0
  p2T Kx2  Dx1  Eq dt
tf
t0
  p3T Hy  If 2  G2t dt
tf
(9)
t0
where p1, p2, p3 represent the Lagrange multipliers.
In the optimal control problem, the necessary condition for stationary condition is that first variation
of the extended performance function Jc is equal to zero as follow.
J c  0.
(10)
Minimization by the Sakawa-Shindo method
The Sakawa-Shindo method is applied for the minimization of the performance function. Hence the
modified performance function K(i) is defined as:
K
(i )
J
(i )
c




T
1 t f (i 1)
(i )
  q
 q W (i ) q (i 1)  q (i ) dt
2 t0
(11)
where (i) is iteration counts. The matrix W(i) represents the weighting matrix, and it is renewed
during the iteration. To minimize performance function with respect to the control flux q, the
following equation is obtained, and the flux is renewed by eq.(13).
K (i )
 0.
q (i )
(12)
4
q ( i 1)


tf
t0
{(W ( i )  R )q ( i )  E T p2( i ) }dt

tf
t0
(13)
W ( i ) dt
NUMERICAL EXAMPLE
In figure 1, the domain in the center circle is for the water flow analysis, and the other part is for the
soil deformation analysis. The lateral displacement is set to be zero on the boundary A-B and C-D,
and the vertical displacement is set to be zero on B-D. The boundary A-C is assumed as ground
surface, i.e., the vertical displacement is free. The pressure at the top of the balloon is set to be a
constant value. The velocity components and pressure are set to be zero at initial time. The
objective value of displacement at the top of balloon is set to be 0.2[m] up to the z direction. The
control flux, which is a constant value in time, is imposed around the top of balloon. The numerical
results are shown in figure 2 and figure 3.
12.0[m]
C
B
D
12.0[m]
A
Z
X
Figure 1: Finite element mesh
5
O bjective
C alculated
500.0
1.5
300.0
1.4
[m]
400.0
200.0
100.0
1.3
1.2
1.1
0.0
1.0
1
2
3
4
5
6
7
8
1
2 3
4 5
6 7
8 9 10 11 12 13 14 15 16 17 18 19 20
Tim e step
Iteration count
Figure 2: Performance function
Figure 3: Z coordinate at the top of balloon
CONCLUSIONS
A control analysis of vertical displacement is performed in case of only a balloon with water
injected into. The position at the top of balloon is raised up to the objective vertical level(1.4m) by
the fluid force, and the convergence of performance function and control flux is confirmed for the
control analysis. The performance function converges to a constant value which must mainly be the
square sum of residual of water fluid flux injected into. In this case the control flux assumed to be
constant in time, but it should be changed for actual problem. The near future subjects will be the
analyses in cases of a balloon with water removed from, multi-balloons system, and the control flux
changed in time.
References
[1]
Hirt,C.W., Amsden,A.A. and Cook,J.L.(1972).An arbitrary Lagrangian Eulerian computing
method for all flow speeds. J. Comp. Phys. 14.227-253
[2]
A.Maruoka, A.Anju and M.Kawahara. (1994).An Arbitrary Lagrangian-Eulerian Finite
Element Method for Fluid-Structure Interaction Problem. Proc. of Computational methods in
water resources X.1231-1238
[3]
Chorin,A.J.(1968).Numerical simulation of the Navier-Stokes equations. Math. Comp.
22.745-762
[4]
A.Maruoka, A.Anju and M.Kawahara.(1993).A Fractional Step Finite Element Analysis of
Incompressible Navier-Stokes Equation. Proc. of the 5th Int. Symp. on Computational Fluid
Dynamics. 1.19-26
[5]
Yoshiyuki Sakawa and Yuji Shindo.(1980).On Global Convergence of an Algorithm for
Optimal Control. IEEE Transactions on Automatic Control.25:6.1149-1153
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