RBFBased Meshless Method for Large Deflection of Thin

RBF-Based Meshless Method for Large Deflection of Thin Plates By Husain Jubran Al-Gahtani CIVIL ENGINEERING KFUPM 26 -27 March 2007 1 st Saudi-French Workshop, KFUPM RBF for Plates

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

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

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

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

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

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

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

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

Mathematica Code for 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 -. 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 -. 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}]; List. Plot[dat, Aspect. Ratio -> Automatic, Plot. Style -> Point. Size[0. 02]] 26 -27 March 2007 1 st Saudi-French Workshop, KFUPM RBF for Plates

Mathematica Code for r 2) = x - xi)^2 + (y - yi)^2; r = Sqrt[r 2 ; [ phi = Sqrt[r 2 +. 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[[ 26 -27 March 2007 1 st Saudi-French Workshop, KFUPM RBF for Plates

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

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

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

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

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

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

RBF for Large Deflection of Plates W-F Formulation S B 1: w=0 B 2: M=0 26 -27 March 2007 C Free For movable edge w=0 V =0 B 1: M=0 B 2: =0 F =0 1 st Saudi-French Workshop, KFUPM RBF for Plates

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

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

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

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

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

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

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

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

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

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

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

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

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

Conclusions q RBF-Based collocation method offers a simple yet efficient method for solving non-linear problems in computational mechanics q The proposed method is easy to program q The solution is obtained in a functional form which enables determining secondary solutions by direct differentiation q RBF offers an attractive solution to three-dimensional problems 26 -27 March 2007 1 st Saudi-French Workshop, KFUPM RBF for Plates