26-27March 2007 RBF for Plates 1 st Saudi-French Workshop, KFUPM RBF-Based Meshless Method for Large Deflection of Thin Plates Large Deflection of Thin.

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


Outline

What is an RBF?

Application to Poisson-Type Problems

Application to Small Deflection of Plates

Application to Large Deflection of Plates

Conclusions


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


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


Mesh Versus Meshless


Application to Poisson Eq

onguB

infuuL

nu

yu

xu

),(

),,,(

Xb

Xd



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)


Ndifor

xdfxbxdBLxdxdL iji

Nb

jj

Nd

jjij

,1

)(11


)(),(),( dxfxbxdBLxdxdL

Similarly, applying GDE at Nd domain points:

Xb

Xd

Nd x (Nb+Nd)



)(

)(

),(

),(

),(

),(

d

b

xf

xg

xbxdBL

xbxbBB

xdxdL

xdxbB

Xb

Xd

(Nb+Nd) x (Nb+Nd)


2

0

),(

),(

),(

),(

xbxd

xbxb

xdxd

xdxb

(36+81) x (36+81+Nd)

Example: Torsion of a Beam with Rectangular Section

2uu = 0 on Γ


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


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


RBF Solution for 2u


xz xz

RBF Solution for 2u


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



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



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:



DqxbxdBxbxdBxdxd

xbxbBBxbxbBBxdxbB

xbxbBBxbxbBBxdxbB

/

0

0

),(),(),(

),(),(),(

),(),(),(

21

22122

21111

Xb

Xd

(2Nb+Nd) x (2Nb+Nd)


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


),(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


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


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)



),,(/

),(

),(

3

22

11

wvuNLDqw

wNLvuL

wNLvuL

00 nVorw

0 nMornw

Bending B.C.In-Plane B.C.

0v

0u



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

)(



),,(

0

0

)(

)(

0

0

/

0

0

0

0

0

0

3

2

1

)(

wvuNL

wNL

wNL

Dqw

w

w

v

v

u

u

L


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


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


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


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


w

Example 2

Simply Supp.

Movable Edge

Nb = 36

Nd = 81

a

a


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


w


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


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


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


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

26-27March 2007 RBF for Plates 1 st Saudi-French Workshop, KFUPM RBF-Based Meshless Method for Large Deflection of Thin Plates Large Deflection of Thin.

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