26-27March 2007 RBF for Plates 1 st Saudi-French Workshop, KFUPM RBF-Based Meshless Method for Large...
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Transcript of 26-27March 2007 RBF for Plates 1 st Saudi-French Workshop, KFUPM RBF-Based Meshless Method for Large...
1st Saudi-French Workshop, KFUPM 26-27March 2007 RBF for Plates
RBF-Based Meshless Method forRBF-Based Meshless Method for Large Deflection of Thin PlatesLarge Deflection of Thin Plates
ByBy Husain Jubran Al-GahtaniHusain Jubran Al-Gahtani
CIVIL ENGINEERINGCIVIL ENGINEERINGKFUPMKFUPM
1st Saudi-French Workshop, KFUPM 26-27March 2007 RBF for Plates
Outline
What is an RBF?
Application to Poisson-Type Problems
Application to Small Deflection of Plates
Application to Large Deflection of Plates
Conclusions
1st Saudi-French Workshop, KFUPM 26-27March 2007 RBF for Plates
What is RBF?
Common types:• Multi-quadrics (MQ)
• Reciprocal multi-quadrics (RMQ)• 3rd Order Polynomial Spline (P) • Gaussian (GS)
where is a shape parameter and
2/122 rk
2/122 rk
3r
22rkExp
k
22 )()( kkk yyxxxxr
1st Saudi-French Workshop, KFUPM 26-27March 2007 RBF for Plates
What is RBF?
Historical background
• 1971 RBF as an interpolant
• 1982 Combined w/BEM for comp. mech.
• 1990 For potential problems
• 1990- For other PDEs
1st Saudi-French Workshop, KFUPM 26-27March 2007 RBF for Plates
Application to Poisson Eq
onguB
infuuL
nu
yu
xu
),(
),,,(
Xb
Xd
1st Saudi-French Workshop, KFUPM 26-27March 2007 RBF for Plates
Application to Poisson Eq
j
Nb
jjj
Nd
jj xbxBxdxxu
11
)(
Nbifor
xbgxbxbBBxdxbB iji
Nb
jj
Nd
jjij
,1
)(11
)(),(),( xbgxbxbBBxdxbB
The solution can be approximated by
Applying the B.C. at Nb boundary points:
Xb
Xd
onguB
infuL
)(
)(
Nb x (Nb+Nd)
1st Saudi-French Workshop, KFUPM 26-27March 2007 RBF for Plates
Ndifor
xdfxbxdBLxdxdL iji
Nb
jj
Nd
jjij
,1
)(11
Application to Poisson Eq
)(),(),( dxfxbxdBLxdxdL
Similarly, applying GDE at Nd domain points:
Xb
Xd
Nd x (Nb+Nd)
1st Saudi-French Workshop, KFUPM 26-27March 2007 RBF for Plates
Application to Poisson Eq
)(
)(
),(
),(
),(
),(
d
b
xf
xg
xbxdBL
xbxbBB
xdxdL
xdxbB
Xb
Xd
(Nb+Nd) x (Nb+Nd)
1st Saudi-French Workshop, KFUPM 26-27March 2007 RBF for Plates
2
0
),(
),(
),(
),(
xbxd
xbxb
xdxd
xdxb
(36+81) x (36+81+Nd)
Example: Torsion of a Beam with Rectangular Section
2uu = 0 on Γ
1st Saudi-French Workshop, KFUPM 26-27March 2007 RBF for Plates
a = 1; b = 1;; xf = Flatten[Table[.1 a i , {j, 1, 9}, {i, 1, 9}]];
yf = Flatten[Table[.1 b j , {j, 1, 9}, {i, 1, 9}]]; nf = Length[xf];
xb = Flatten[{Table[.1 a i, {i, 1, 9}], Table[1, {i, 1, 9}],
Table[1 - .1 a i, {i, 1, 9}], Table[0, {i, 1, 9}]}];
yb = Flatten[{Table[0, {i, 1, 9}], Table[.1 b i, {i, 1, 9}],
Table[1, {i, 1, 9}], Table[1 - .1 b i, {i, 1, 9}]}];
nb = Length[xb]; xt = Join[xb, xf]; yt = Join[yb, yf];
nt = nb + nf;
dat = Table[{xt[[i]], yt[[i]]}, {i, 1, nt}];ListPlot[dat, AspectRatio -> Automatic, PlotStyle -> PointSize[0.02]]
Mathematica Code for 2u
1st Saudi-French Workshop, KFUPM 26-27March 2007 RBF for Plates
r2 = (x - xi)^2 + (y - yi)^2; r = Sqrt[r2]; phi = Sqrt[r2 + .2];u = Sum[c[i] phi /. {xi -> xt[[i]], yi -> yt[[i]]}, {i, 1, nt}];gde = D[u, {x, 2}] + D[u, {y, 2}];Do[eq[i] = u == 0. /. {x -> xb[[i]], y -> yb[[i]]}, {i,1,nb}];Do[eq[i + nb] = gde == -2. /. {x -> xf[[i]], y -> yf[[i]]},{i, 1, nf}];sol = Solve[Table[eq[i], {i, 1, nt}]];un = u /. sol[[1]]
Mathematica Code for 2u
1st Saudi-French Workshop, KFUPM 26-27March 2007 RBF for Plates
RBF for Small Deflection of Thin Plates
D
qw )(
00:)(1 nVorwwB
0:)(2 nMornwwB
yx
wnn
y
wn
x
wnwvDM yxyxn
2
2
22
2
222 21
xy
wvvnvnn
y
wvnnDV yxxxyn 2
322
3
32 11211
3
32
2
322 11121
x
wvnn
yx
wvvnvnn yxyxy
1st Saudi-French Workshop, KFUPM 26-27March 2007 RBF for Plates
RBF for Small Deflection of Thin Plates
j
Nb
jjj
Nb
jjj
Nd
jj xbxBxbxBxdxxw
2
11
11
)(
NbixbxbBB
xbxbBBxdxbB
ji
Nb
jj
ji
Nb
jj
Nd
jjij
,1,0211
1111
1
Applying the 1st B.C. at Nb boundary points:
Xb
Xd
1st Saudi-French Workshop, KFUPM 26-27March 2007 RBF for Plates
RBF for Small Deflection of Thin Plates
NdiD
qxbxdB
xbxdBxdxd
iji
Nb
jj
ji
Nb
jj
Nd
jjij
,1)(21
111
Applying the 2ndt B.C. at Nb boundary points:
Xb
Xd
NbixbxbBB
xbxbBBxdxbB
ji
Nb
jj
ji
Nb
jj
Nd
jjij
,1,0221
1211
2
Similarly, applying GDE at Nd points:
1st Saudi-French Workshop, KFUPM 26-27March 2007 RBF for Plates
RBF for Small Deflection of Thin Plates
DqxbxdBxbxdBxdxd
xbxbBBxbxbBBxdxbB
xbxbBBxbxbBBxdxbB
/
0
0
),(),(),(
),(),(),(
),(),(),(
21
22122
21111
Xb
Xd
(2Nb+Nd) x (2Nb+Nd)
1st Saudi-French Workshop, KFUPM 26-27March 2007 RBF for Plates
S C Free
B1: w=0 w=0 V =0
B2: M=0 =0 M = 0
RBF for Large Deflection of Plates
2
2
2
2224
y
w
x
w
yx
wEF
yx
w
yx
F
y
w
x
F
x
w
y
F
h
q
D
hw
22
2
2
2
2
2
2
2
24 2
n
w
W-F Formulation
For movable edge
B1: F =0
B2:
0
n
F
1st Saudi-French Workshop, KFUPM 26-27March 2007 RBF for Plates
),(4 wwNLF ),(4 FwNLD
qw
2
2
2
222
),(y
w
x
w
yx
wwwNL
RBF for Large Deflection of Plates ( W – F Formulation)
yx
w
yx
F
y
w
x
F
x
w
y
FFwNL
22
2
2
2
2
2
2
2
2
2),(
Where
1st Saudi-French Workshop, KFUPM 26-27March 2007 RBF for Plates
RBF for Large Deflection of Plates ( W – F Formulation)
),(/
0
0
),(),(),(
),(),(),(
),(),(),(
21
22122
21111
FwNLDqxbxdBxbxdBxdxd
xbxbBBxbxbBBxdxbB
xbxbBBxbxbBBxdxbB
w
w
w
),(
0
0
),(),(),(
),(),(),(
),(),(),(
2
2
wwNLxbxdn
xbxdxdxd
xbxbn
xbxbn
xdxbn
xbxbn
xbxbxdxb
w
w
w
),(/4 FwNLDqw
RBF equations for ),(4 wwNLF
RBF equations for
1st Saudi-French Workshop, KFUPM 26-27March 2007 RBF for Plates
u-v-w Formulation:
02
222
2
2
1
22
2
2
2
2
21
y
w
x
w
y
w
yx
w
yxy
u
v
Eh
y
w
yx
w
yxv
x
w
x
w
x
u
v
Eh
02
222
2
2
212
22
2
22
1
y
w
y
w
yx
w
x
w
yx
uv
yv
Eh
x
w
y
w
x
w
yx
w
xyx
u
v
Eh
2
222
222
2
212
2
)1 y
w
yv
x
w
x
u
x
w
vD
Eh
y
w
x
w
xy
u
yx
w
vD
Eh
D
qw
2
222
222
2
212 x
w
x
uv
y
w
yy
w
vD
Eh
RBF for Large Deflection of Plates ( u-v-w Formulation)
1st Saudi-French Workshop, KFUPM 26-27March 2007 RBF for Plates
RBF for Large Deflection of Plates ( u-v-w Formulation)
),,(/
),(
),(
3
22
11
wvuNLDqw
wNLvuL
wNLvuL
00 nVorw
0 nMornw
Bending B.C.In-Plane B.C.
0v
0u
1st Saudi-French Workshop, KFUPM 26-27March 2007 RBF for Plates
RBF for Large Deflection of Plates ( u-v-w Formulation)
j
Nb
j
juj
Nd
j
ju xbxBxdxxu
11
)(
j
Nb
j
jwj
Nb
j
jwj
Nd
j
jw xbxBxbxBxdxxw
2
11
11
)(
j
Nb
j
jvj
Nd
j
jv xbxBxdxxv
11
)(
1st Saudi-French Workshop, KFUPM 26-27March 2007 RBF for Plates
RBF for Large Deflection of Plates ( u-v-w Formulation)
),,(
0
0
)(
)(
0
0
/
0
0
0
0
0
0
3
2
1
)(
wvuNL
wNL
wNL
Dqw
w
w
v
v
u
u
L
1st Saudi-French Workshop, KFUPM 26-27March 2007 RBF for Plates
Numerical Examples
1- All quantities are made dimensionless
2- Plate is until the central deflection
exceeds 100% of the plate thickness.
3- RBF solution for Maximum values of deflection & stress are compared
to those obtained by Analytical & FEM
axx /
2244 /,/,/,/,/ EhahwwEhqaqayyaxx
a
a
1st Saudi-French Workshop, KFUPM 26-27March 2007 RBF for Plates
0
0.2
0.4
0.6
0.8
1
1.2
0 0.5 1 1.5 2 2.5
FEM
RBF
Analytical
Simply Supp.
Movable Edge
Nb = 32
Nd = 69
q
w
Central deflection versus load
Example 12a
1st Saudi-French Workshop, KFUPM 26-27March 2007 RBF for Plates
0
0.5
1
1.5
2
2.5
0 0.5 1 1.5 2 2.5
FEM
RBF
Analytical
Example 1
Bending & membrane stresses versus load
Bending
Membrane
q
Simply Supp.
Movable Edge
Nb = 32
Nd = 69
2a
1st Saudi-French Workshop, KFUPM 26-27March 2007 RBF for Plates
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 4 8 12 16 20 24 28 32
FEM
RBF
Central deflection versus load
w
Example 2
Simply Supp.
Movable Edge
Nb = 36
Nd = 81
a
a
1st Saudi-French Workshop, KFUPM 26-27March 2007 RBF for Plates
0
1
2
3
4
5
6
7
0 4 8 12 16 20 24 28 32
FEM
RBFBending
Membrane
Bending & membrane stresses versus load
Example 2
Simply Supp.
Movable Edge
Nb = 36
Nd = 81
a
a
1st Saudi-French Workshop, KFUPM 26-27March 2007 RBF for Plates
w
Central deflection versus load
Example 3
Clamped
Immovable EdgeNb = 32
Nd = 69
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 2 4 6 8 10 12
FEM
RBF
Analytical
q
1st Saudi-French Workshop, KFUPM 26-27March 2007 RBF for Plates
0
0.5
1
1.5
2
2.5
3
3.5
0 2 4 6 8 10 12
FEM
RBF
Analytical
Bending
Membrane
Example 3
Central Bending & membrane stresses
Clamped, Immovable Edge
Nb = 32
Nd = 69q
1st Saudi-French Workshop, KFUPM 26-27March 2007 RBF for Plates
0
1
2
3
4
5
6
7
0 2 4 6 8 10 12
FEM
RBF
Analytical
Bending
Membrane
Example 3
Edge Bending & membrane stresses
Clamped
Immovable EdgeNb = 32
Nd = 69q
1st Saudi-French Workshop, KFUPM 26-27March 2007 RBF for Plates
Conclusions
RBF-Based collocation method offers a simple yet efficient method for solving non-linear problems in computational
mechanics
The proposed method is easy to program
The solution is obtained in a functional form which enables determining secondary solutions by direct differentiation
RBF offers an attractive solution to three-dimensional problems