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Meshfree Methods and Simulations of Material Failures

Meshfree Methods and Simulations of Material Failures Shaofan Li Department of Civil and Environmental Engineering, University of California at Berkeley Collaborators Dr. Bo C Simonsen, Technical University of Denmark; Dr. Daniel C. Simkins,

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Meshfree Methods and Simulations of Material Failures

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  1. Meshfree Methods and Simulations of Material Failures ShaofanLi Department of Civil and Environmental Engineering, University of California at Berkeley

  2. Collaborators Dr. Bo C Simonsen, Technical University of Denmark; Dr. Daniel C. Simkins, University of South Florida; Dr. Sergey N. Medyanik, Northwestern University

  3. Table of Contents • Introduction: What is Meshfree Method • Simulations of Large Deformations • Simulations of Strain Localizations • Simulations of Dynamics Shear Band Propagations • Simulations of Ductile Fracture 6. Simulations of Impact and penetrations 7. Conclusions

  4. I. Introduction of meshfree methods Mesh vs. Meshfree

  5. Meshfree Methods • Smooth Particle Hydrodynamics (SPH) • Element Free Galerkin (EFG) • Reproducing Kernel Particle Method (RKPM) 1. Unknown is represented by convolving a smooth kernel with dependent variable 2. Discretize by evaluating integral via nodal integration

  6. Meshfree Interpolation

  7. RKPM Kernel • f(x-y) is a smooth compactly-supported function, e.g. cubic spline • PT(x) = [1 x y z xy xz …] is vector of monomial terms • b(x) (called the normalizer) is used to regain discrete partition of unity

  8. Moment Equation Note and P(0) = (1, 0, 0, …, 0) then

  9. By Taylor theorem, therefore

  10. Compare FEM shape function with Meshfree shape function (a) FEM ; (b) Meshfree

  11. Compare FEM shape function with Meshfree shape function (a) FEM ; (b) Meshfree

  12. Meshfree discretization and Meshfree shape functions

  13. (a) Shape function (b) Derivative (c) Derivative (d) Derivative “The Cloud”: 3-D Meshfree shape function and its first derivatives generated by the tri-linear polynomial basis, P(X) = (1, x1 ,x2 , x3 , x1x2 ,x2x3 , x3x1 , x1x2x3 )

  14. 1.2 A Few Virtues of Meshfree Methods Convergence Property as as Reproducing Property For That is Non-local Interpolation (Discrete Convolution)

  15. * Large Support Size It enables meshfree discretization/interpolation to endure large mesh distortion and sustain computation without remeshing; is deformation map

  16. II. Simulation of large deformations

  17. Example 2.1 : Compression of A Rubber Cylinder FEMMeshfree 50% Compression 65% Compression 85% Compression 90% Compression

  18. Simulation of Compression of a rubber block

  19. Example 2.2 A pinched cylindrical shell Material Properties Computational Parameters Number of Particles: 30300 Time Step:

  20. Deformation Sequence of A Pinched Cylinder (a) (b) (c) (c) (d) (e)

  21. Example 2.3 Hemispheric Shell with Pinched Load Summary: Material: Elastic-plastic material; Geometry: Hemisphere shell with radius of 1 inch, thickness of 0.04 inch. Particles: 12,300

  22. (c) t = 3.010-3s (b) t = 1.510-3s (a) t = 0.510-3s (f) t = 7.510-3s (d) t = 4.510-3s (e) t = 6.010-3s

  23. Example 2.4 The snap-through of a conic shell Material Properties Computational Parameters Number of Particles: 12300 Time Step:

  24. The snap-through of a 3D conic shell (a) (b) (c) (c) (d) (e)

  25. Excessive mesh distortion (hourglassing) ABAQUS/Explicit RKPM/Explicit

  26. III. Simulations of Strain Localizations

  27. Meshfree Simulation of Strain Localization Hardening Softening From Reid, Gilbert, and Hahn [1966]

  28. The Thirty-one Hole Problem

  29. Meshfree Methods: Element-free Galerkin (EFG) Fleming and Belytechko [1997]

  30. Shearband Path for a Plate with 31 Holes (FEM vs. Meshfree Methods) (b) 60  90 mesh (c) 90  60 mesh (a) 60  60 mesh (d) 60  90 particles (e) 90  60 particles (c) 60  60 particles

  31. Load-deflection Curves

  32. Problem Statement: Compression Test

  33. FEM Meshfree (a) 20  20 (b) 30  20 (c) 40  20 (d) 50  20

  34. 4. Meshfree Multiscale Computations = Scale 0 + Scale 1 + Scale 2 + …... 1-D Meshfree Multiscale Basis Functions for Polynomial Basis P = ( 1, x, x2, x3 )

  35. A Meshfree Hierarchical basis

  36. [0,0] [1,0] [0,1] [2,0] [1,1] [0,2] Hierarchical Partition of Unity of Quadratic Basis

  37. Multi-resolution Analysis/Approximation (MRA)

  38. Multiscale Analysis Spectral Meshfree Adaptive Simulations

  39. Multiscale Analysis (a) Total Scale(b) Low Scale(c ) High Scale(d) Adaptive Pattern Total Particles: 3321 Particles used for adaptive calculation: 361

  40. Example 4.2 Multiresolution Analysis Total Scale Low Scale High Scale

  41. Tension of A Bar with A Hole Meshfree (particles) Zoom in Kinematically Admissible Mode I 4272 particles/layer with 3 layers in thickness direction Mesh-based (element distortion)

  42. Symmetry slip line solution Shear Plane Development for a bar with A Hole in Tension (I)

  43. Simulation of anti-symmetric slip line solution

  44. Anti-symmetry slip line solution

  45. How to simulate curve shearband Curved Shear Band Formation in Double-Notched Bar in Tension 13364 particles Reference 1: Ewing, D.J.F. and Hill, R. J. Mech. Phys. Solids, 15, 115 (1967)

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