Pseudospectral Method for Seepage Behind Earth Retaining Wall
A higher precision numerical method, spectral method, for seepage behind earth retaining wall is put forward. For most cases, Finite element usually is adopted by its excellent flexibility, but finite element consumes greater times to improve algorithm pr
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PSEUDOSPECTRAL METHOD FOR SEEPAGE BEHIND EARTH RETAINING WALL Nansheng Li, Lihui Xie Department ofHydrauli c Engineering, School ofCivil Engineering, Tongji University, Shanghai 200092, China A higher precision numerical method, spectral method, for seepage behind earth retaining wall is put forward. For most cases, Finite element usually is adopted by its excellent flexibility, but finite element consumes greater times to improve algorithm precision whether h-refinement or p- refinement. Spectral methods generate algebraic equations with full matrices, but in compensation, the high order ofthe basis functions gives high accuracy for a given N. A model of transient two-dimension seepage problem in earth retaining wall is first established. The water-head function is asymptotic expands by Chebyshev series up to N orders. In interested spatial domain, we get discreted equations at different collocation points, viz. pseudo-spectral methods being used in this problem. It is convenient to integrate the ODE's in time through modified Euler finite difference formula. The predicted results are in excellent agreement with the analytical solutions. INTRODUCTION
For seepage problem behind earth retaining wall, the general and widely practicable way to solve seepage equation is finite element method. In most cases, finite element method usually is adopted by its excellent flexibility, but FE method costs greatly to improve algorithm precision . The goal ofthis paper is to set up a more efficient and reliable algorithm to arrive at higher numerical precision. Fortunately, the pseudo-Chebyshev method is the best choice for this goal. As we know, the Chebyshev spectral method has not been used widely in seepage problems now. Though the parameter matrices of pseudo-Chebyshev method are of full matrices, it has much less independent variables for regular interesting domain in compensation than other ordinary algorithm. A typical example is illustrated in which we obtain the useful conclusion, Chebyshev polynomial expands only to 6 terms to reach satisfactory precision. SEEPAGE MODELS OF EARTH RETAINING WALL
The configuration of earth retaining wall is shown in Figure 1. For the most cases of seepage behind earth retaining wall the vertical velocity of seepage flow almost can be neglected, so Dupuit hypothesis is suitable to be used in the seepage problem behind earth retaining wall. Ifthe surface water supply w( x, t) ofsoil dam relates to the time t , then the seepage head h (x,t) certainly is a function of time t . The mathematical description of seepage problem behind retaining wall can be expressed as
Retaining Wall
Earthen embankment
Ground
h(x,t)
51
Figure 1. Configuration of earth retainingwall
w(X,t) K K at
fi h j.J ah -+--=-2
ax
h(O,y) = hi k :; =
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(1)
in 81 in
5,
h=y
H=po
k
ah ay
in
=0
5,
in
84
We assume that the soil mass behind retaining wall is homogeneous and the bottom of free-water layer is horizontal in above description. For the reason of taking Dupuit hypothesis, the boundary condition
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