r - The Hong Kong Polytechnic University

programming problems (the surface traction is omitted)
*
λ = min D ε&
dV
∫
∫
V
( )
T *
st .. f u dV = 1
V
(5)
* 1 * *
ε&
= ( ∇ u + u ∇)
in V
2
*
u = u
on Γu
From the optimum of limit loading multiplier λ opt , the limit loading can be computed according to the following
equation:
f = λ ⋅f (6)
lim
opt
Nonlinear optimization problems for the plane strain von-Mises yield criterion
In general, the slope stability problems are treated as the plain strain problems in geotechnical engineering. For
the plain strain condition, the von-Mises (or Tresca) yield criterion can be written as (Pastor, 2000)
f ( σ ) =
1
( ) 2 2
σx − σ
y
+ τxy
− c = 0
4
(7)
where c is the cohesion. According to the associated flow rule, the power of dissipation can be formulated as a
function of strain rates as (Capsoni and Corradi, 1997)
D ( ε& ) =
T
ε& Θε& (8)
where
2 2
⎡ c −c
0 ⎤
⎢ 2 2 ⎥
Θ = ⎢ −c
c 0 ⎥
(9)
⎢
2
0 0 c ⎥
⎣
⎦
Therefore, the mathematical programming problem (5) of finding the upper bound solution of limit loading
multiplier can be formulated as the following nonlinear optimization problem:
T
λ = min εΘε & & dV
V
T *
.. ∫ f u dV = 1
V
st
∫
* 1 * *
ε&
= ( ∇ u + u ∇)
2
in V
*
u = u
on Γ
Radial point interpolation method
u
The approximation of the field variables of interest x using radial point interpolation method (RPIM) can be
expressed in the following form (Liu and Gu, 2005):
T
T
−1
⎧Us
⎫
u( x)
= ⎡
⎣Rq ( x) Pm ( x) ⎤
⎦G ⎨ ⎬=
Φ( x)
Us
(7)
⎩ 0 ⎭
where u(x) is the function of field variables, U s ={u 1 , u 2 , …, u n } T is the vector of function values, Ф(x) is the
RPIM shape functions corresponding to the nodal value and given by
Φ ( x T
T
−1
) = ⎣
⎡ R ( x ) P q m ( x ) ⎤
⎦ G = ⎡⎣φ1( x ) φ2( x ) L φn( x ) ⎤⎦
(8)
in which, R q is the moment matrix of the radial basis function (RBF) given by
⎡R1( x1) R2( x1) L Rn
( x1)
⎤
⎢
⎥
⎢
R1( x2) R2( x2) L Rn
( x2)
R
q
= ⎥
(9)
⎢ M M O M ⎥
⎢
⎥
⎢⎣R1( xn) R2( xn) L Rn( xn)
⎥⎦n×
n
and P m the polynomial moment matrix is defined as follows
(10)
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