rbf-based meshless method for large deflection of thin plates by

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