1 / 34

RBF-Based Meshless Method for Large Deflection of Thin Plates By

RBF-Based Meshless Method for Large Deflection of Thin Plates By Husain Jubran Al-Gahtani CIVIL ENGINEERING KFUPM. Outline. What is an RBF? Application to Poisson-Type Problems Application to Small Deflection of Plates Application to Large Deflection of Plates Conclusions.

ahava
Download Presentation

RBF-Based Meshless Method for Large Deflection of Thin Plates By

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. RBF-Based Meshless Method for Large Deflection of Thin Plates By Husain Jubran Al-GahtaniCIVIL ENGINEERINGKFUPM 1st Saudi-French Workshop, KFUPM

  2. 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

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

  4. 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

  5. Mesh Versus Meshless 1st Saudi-French Workshop, KFUPM

  6. Xb Xd Application to Poisson Eq 1st Saudi-French Workshop, KFUPM

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

  8. Application to Poisson Eq Xb Xd Similarly, applying GDE at Nd domain points: Nd x (Nb+Nd) 1st Saudi-French Workshop, KFUPM

  9. Application to Poisson Eq Xb Xd (Nb+Nd) x (Nb+Nd) 1st Saudi-French Workshop, KFUPM

  10. Example: Torsion of a Beam with Rectangular Section u = 0 on Γ (36+81) x (36+81+Nd) 1st Saudi-French Workshop, KFUPM

  11. 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, {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]] 1st Saudi-French Workshop, KFUPM

  12. Mathematica Code for 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]] 1st Saudi-French Workshop, KFUPM

  13. RBF Solution for 1st Saudi-French Workshop, KFUPM

  14. RBF Solution for 1st Saudi-French Workshop, KFUPM

  15. RBF for Small Deflection of Thin Plates 1st Saudi-French Workshop, KFUPM

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

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

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

  19. RBF for Large Deflection of Plates W-F Formulation S C Free B1: w=0 w=0 V =0 B2: M=0 =0 M = 0 For movable edge B1: F =0 B2: 1st Saudi-French Workshop, KFUPM

  20. RBF for Large Deflection of Plates ( W – F Formulation) Where 1st Saudi-French Workshop, KFUPM

  21. RBF for Large Deflection of Plates( W – F Formulation) RBF equations for RBF equations for 1st Saudi-French Workshop, KFUPM

  22. RBF for Large Deflection of Plates ( u-v-w Formulation) u-v-w Formulation: 1st Saudi-French Workshop, KFUPM

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

  24. RBF for Large Deflection of Plates ( u-v-w Formulation) 1st Saudi-French Workshop, KFUPM

  25. RBF for Large Deflection of Plates ( u-v-w Formulation) 1st Saudi-French Workshop, KFUPM

  26. 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 1st Saudi-French Workshop, KFUPM

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

  28. 2a Simply Supp. Movable Edge Nb = 32 Nd = 69 Example 1 Bending Membrane Bending & membrane stresses versus load 1st Saudi-French Workshop, KFUPM

  29. a a Simply Supp. Movable Edge Nb = 36 Nd = 81 Example 2 Central deflectionversus load 1st Saudi-French Workshop, KFUPM

  30. a a Simply Supp. Movable Edge Nb = 36 Nd = 81 Example 2 Bending Membrane Bending & membrane stresses versus load 1st Saudi-French Workshop, KFUPM

  31. Clamped Immovable Edge Nb = 32 Nd = 69 Example 3 Central deflectionversus load 1st Saudi-French Workshop, KFUPM

  32. Clamped, Immovable Edge Nb = 32 Nd = 69 Example 3 Bending Membrane Central Bending & membrane stresses 1st Saudi-French Workshop, KFUPM

  33. Clamped Immovable Edge Nb = 32 Nd = 69 Example 3 Bending Membrane Edge Bending & membrane stresses 1st Saudi-French Workshop, KFUPM

  34. 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 1st Saudi-French Workshop, KFUPM

More Related