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 PlatesBy Husain Jubran Al-Gahtani CIVIL ENGINEERING KFUPM Husain Jubran Al-Gahtani CIVIL ENGINEERING KFUPM

26-27March 2007 RBF for Plates 1 st Saudi-French Workshop, KFUPM Outline  What is an RBF?  Application to Poisson-Type Problems  Application to Small Deflection of Plates  Application to Large Deflection of Plates  Conclusions

26-27March 2007 RBF for Plates 1 st Saudi-French Workshop, KFUPM What is RBF? Common types: Multi-quadrics (MQ) Reciprocal multi-quadrics (RMQ) 3 rd Order Polynomial Spline (P) Gaussian (GS) where is a shape parameter and

26-27March 2007 RBF for Plates 1 st Saudi-French Workshop, KFUPM What is RBF? Historical background 1971 RBF as an interpolant 1982 Combined w/BEM for comp. mech For potential problems For other PDEs

26-27March 2007 RBF for Plates 1 st Saudi-French Workshop, KFUPM Mesh Versus Meshless

26-27March 2007 RBF for Plates 1 st Saudi-French Workshop, KFUPM Application to Poisson Eq Xb XdXd

26-27March 2007 RBF for Plates 1 st Saudi-French Workshop, KFUPM Application to Poisson Eq The solution can be approximated by Applying the B.C. at Nb boundary points: Xb Xd Nb x (Nb+Nd)

26-27March 2007 RBF for Plates 1 st Saudi-French Workshop, KFUPM Application to Poisson Eq Similarly, applying GDE at Nd domain points: XbXb XdXd Nd x (Nb+Nd)

26-27March 2007 RBF for Plates 1 st Saudi-French Workshop, KFUPM Application to Poisson Eq XbXb XdXd (Nb+Nd) x (Nb+Nd)

26-27March 2007 RBF for Plates 1 st Saudi-French Workshop, KFUPM (36+81) x (36+81+Nd) Example: Torsion of a Beam with Rectangular Section u = 0 on Γ

26-27March 2007 RBF for Plates 1 st Saudi-French Workshop, KFUPM 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

26-27March 2007 RBF for Plates 1 st Saudi-French Workshop, KFUPM 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

26-27March 2007 RBF for Plates 1 st Saudi-French Workshop, KFUPM RBF Solution for

26-27March 2007 RBF for Plates 1 st Saudi-French Workshop, KFUPM RBF Solution for

26-27March 2007 RBF for Plates 1 st Saudi-French Workshop, KFUPM RBF for Small Deflection of Thin Plates

26-27March 2007 RBF for Plates 1 st Saudi-French Workshop, KFUPM RBF for Small Deflection of Thin Plates Applying the 1 st B.C. at Nb boundary points: XbXb XdXd

26-27March 2007 RBF for Plates 1 st Saudi-French Workshop, KFUPM RBF for Small Deflection of Thin Plates Applying the 2 ndt B.C. at Nb boundary points: XbXb XdXd Similarly, applying GDE at Nd points:

26-27March 2007 RBF for Plates 1 st Saudi-French Workshop, KFUPM RBF for Small Deflection of Thin Plates XbXb XdXd (2Nb+Nd) x (2Nb+Nd)

26-27March 2007 RBF for Plates 1 st Saudi-French Workshop, KFUPM SCFree B1: w=0 w=0V =0 B2: M=0 =0 M = 0 RBF for Large Deflection of Plates W-F Formulation For movable edge B1: F =0 B2:

26-27March 2007 RBF for Plates 1 st Saudi-French Workshop, KFUPM RBF for Large Deflection of Plates ( W – F Formulation) Where

26-27March 2007 RBF for Plates 1 st Saudi-French Workshop, KFUPM RBF for Large Deflection of Plates ( W – F Formulation) RBF equations for

26-27March 2007 RBF for Plates 1 st Saudi-French Workshop, KFUPM u-v-w Formulation: RBF for Large Deflection of Plates ( u-v-w Formulation)

26-27March 2007 RBF for Plates 1 st Saudi-French Workshop, KFUPM RBF for Large Deflection of Plates ( u-v-w Formulation) Bending B.C. In-Plane B.C.

26-27March 2007 RBF for Plates 1 st Saudi-French Workshop, KFUPM RBF for Large Deflection of Plates ( u-v-w Formulation)

26-27March 2007 RBF for Plates 1 st Saudi-French Workshop, KFUPM RBF for Large Deflection of Plates ( u-v-w Formulation)

26-27March 2007 RBF for Plates 1 st Saudi-French Workshop, KFUPM 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 a a

26-27March 2007 RBF for Plates 1 st Saudi-French Workshop, KFUPM Simply Supp. Movable Edge N b = 32 N d = 69 Central deflection versus load Example 1 2a2a

26-27March 2007 RBF for Plates 1 st Saudi-French Workshop, KFUPM Example 1 Bending & membrane stresses versus load Bending Membrane Simply Supp. Movable Edge N b = 32 N d = 69 2a2a

26-27March 2007 RBF for Plates 1 st Saudi-French Workshop, KFUPM Central deflection versus load Example 2 Simply Supp. Movable Edge N b = 36 N d = 81 a a

26-27March 2007 RBF for Plates 1 st Saudi-French Workshop, KFUPM Bending Membrane Bending & membrane stresses versus load Example 2 Simply Supp. Movable Edge N b = 36 N d = 81 a a

26-27March 2007 RBF for Plates 1 st Saudi-French Workshop, KFUPM Central deflection versus load Example 3 Clamped Immovable Edge N b = 32 N d = 69

26-27March 2007 RBF for Plates 1 st Saudi-French Workshop, KFUPM Bending Membrane Example 3 Central Bending & membrane stresses Clamped, Immovable Edge N b = 32 N d = 69

26-27March 2007 RBF for Plates 1 st Saudi-French Workshop, KFUPM Bending Membrane Example 3 Edge Bending & membrane stresses Clamped Immovable Edge N b = 32 N d = 69

26-27March 2007 RBF for Plates 1 st Saudi-French Workshop, KFUPM 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