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
k22 )()( 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
Mesh Versus Meshless
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:
XbXd
onguBinfuL
)()(
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
xbxdBLxbxbBB
xdxdLxdxbB
Xb
Xd
(Nb+Nd) x (Nb+Nd)
1st Saudi-French Workshop, KFUPM 26-27March 2007 RBF for Plates
20
),(),(
),(),(
xbxdxbxb
xdxdxdxb
(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 Solution for 2u
1st Saudi-French Workshop, KFUPM 26-27March 2007 RBF for Plates
xz xz
RBF Solution for 2u
1st Saudi-French Workshop, KFUPM 26-27March 2007 RBF for Plates
RBF for Small Deflection of Thin Plates
Dqw )(
00:)(1 nVorwwB
0:)(2 nMornwwB
yxwnn
ywn
xwnwvDM yxyxn
2
2
22
2
222 21
xywvvnvnn
ywvnnDV yxxxyn 2
322
3
32 11211
3
32
2
322 11121
xwvnn
yxwvvnvnn 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
NdiDqxbxdB
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
DqxbxdBxbxdBxdxdxbxbBBxbxbBBxdxbBxbxbBBxbxbBBxdxbB
/00
),(),(),(),(),(),(),(),(),(
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
yw
xw
yxwEF
yxw
yxF
yw
xF
xw
yF
hq
Dhw
22
2
2
2
2
2
2
2
24 2
nw
W-F Formulation
For movable edge
B1: F =0
B2:
0nF
1st Saudi-French Workshop, KFUPM 26-27March 2007 RBF for Plates
),(4 wwNLF ),(4 FwNLDqw
2
2
2
222
),(yw
xw
yxwwwNL
RBF for Large Deflection of Plates ( W – F Formulation)
yxw
yxF
yw
xF
xw
yFFwNL
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
FwNLDqxbxdBxbxdBxdxdxbxbBBxbxbBBxdxbBxbxbBBxbxbBBxdxbB
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
wvuNLDqwwNLvuLwNLvuL
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)
),,(
00)()(
00
/000000
3
2
1)(wvuNL
wNLwNL
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