Civil Engineering Reference
In-Depth Information
8.9.3.3 The Herrmann theorem and optimal sampling points
The LSF has additional justification in self-adjoint problems, for which the minimisation of
the total potential energy (TPE) is equivalent to the LSF of the approximation stresses to the
exact solution.
1
2
∫
() ()
T
T
TPE
()
u
=
LuALu
−
fu
d
1
2
(
)
(
)
∫
T
(
)
ALu
and( )
LSF
u
=
L uu
−
(
−
)
d
where
L
= linear operator, matrix
A
is symmetric,
f
T
u
= external work and
u
= exact
solution.
Herrmann Theorem
LSF
=
TPE
+ constant
Minimisation of the TPE and the LSF projection onto exact solutions are equivalent problems.
Noting that
δTPE( )
u
= 0
, a proof can be established by variation, i.e. δTPE = δLSF.
In 3D, the number and the positions of the sampling points in terms of barycentre (volume)
co-ordinates for tetrahedral elements T4, T10 and T20 for stress recovery are as follows.
1
4
1
4
1
4
1
Tsampling point at the centre
41
:
,
,
,
4
Tsampling pointsat(b, a, a, a),
10 4
:
(a,b,a,a), (a,a,b,a),
5
−
5
535
20
+
(a,a,a,b), a =
,
b
=
20
1
4
1
4
1
4
1
4
1
2
1
6
1
6
1
6
Tsampling pointsat
20 5
:
,
,
,
,
,
,
,
,
1
6
1
2
1
6
1
6
1
6
1
6
1
2
1
6
1
6
1
6
1
6
1
2
,
,
,
,
,
,
,
,
,
,
,
The basis polynomials adopted in the construction of the smoothed stress field are
T
4
:
P
P
=
=
[ ,
1
1
x,y,z
]
222
T10:
[,
x, y, z, xy,yz, xz
,,x ,y ,z
]
2222333 2
2
2
2
2
2
T20:
P
=
[,
1
x, y, z, xy,yz, xz,x
,y ,z ,x ,y ,z ,x y, yz,z xzx,xy
,
,,yz,xyz]
The rate of convergence of the recovery stress is affected by the smoothness of the solu-
tion as well as the FE mesh. If the exact solution is smooth without any singularity, the
