yuying@uiuc.edu 10 th US National Congress on Computational Mechanics Large deflection analysis of planar solids based on the Finite Particle Method 1, 2 Presenter: Ying Yu Advisors: Prof. Glaucio H. Paulino Prof. Yaozhi Luo 2 Date: July 17 th 1 1 Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, U.S. & NSF 2 Space Structures Research Center, Zhejiang University, China
Outline Motivation of the Finite Particle Method (FPM) Fundamentals of the FPM Numerical Examples Concluding Remarks Future Work 2
Motivation of the FPM Structural nonlinear problems: large deformation, large rotation Discontinuous problems: Methods: fracture, collapse, fragmentation Finite Element Method (FEM): general method Mesh-free Methods: (DEM, SPH, and others) suited for particulate material, such as sand and concrete 3
Motivation of the FPM Finite Particle Method (FPM) Based on Vector Mechanics The procedure of this method is simple and unified. Without special considerations, strong nonlinear and discontinuous problems can be solved. Ting, E.C., Shi, C., Wang, Y.K.: Fundamentals of a vector form intrinsic finite element: Part I. Basic procedure and a planar frame element. J. Mech. 20, 113 122 (2004) 4
Fundamentals of the FPM 1) Discretization of structure. particle cell Assumptions: Particle Cell ( element like) The relationship between particle and element The particle motion undergoes a time history. 5
Fundamentals of the FPM 2) Discrete Path Path unit (interval) Assumptions: The reference effect due configuration to geometrical for stress changes analysis within of a the continuous time body interval is its t1-t2 configuration can be neglected at time t1. The deformation use of infinitesimal of the continuous strain and body engineering is infinitesimal stress (small for deformation) evaluating stresses. and computing virtual work is feasible. 6
7 Fundamentals of the FPM 3) Particle Motion Equation n n & ext ext int F f f i1 i1 M d ext F y ext F x
Fundamentals of the FPM 4)Evaluation of deformations and internal forces a) Evaluate the rigid body motion b) Use reverse motion to remove rigid translation and rotation c) Use deformed coordinate to reduce element degree of freedom f) Use forward motion to get the element back to the original position g) Transform the internal force back to global coordinate d) Calculate the internal force at the deformed coordinate system 8
Fundamentals of the FPM 5) Time Integration To avoid iteration, explicit time integration is used.. If a second order, explicit, central difference time integrator is adopted: Velocity: 1 dn ( dn 1 dn 1 ), 2t Acceleration: Displacement: 1 dn 2 ( dn 1 2 dn dn 1 ). t d n1 2 n n 2 t ext ext int ( F f f ) 2 t m i1 i1 4 2t dn d 2 t 2 t 9 n1.
Fundamentals of the FPM 6) Flow Chart 10
Numerical examples 1. A square plane subjected to an initial angular velocity Goal: a) verify the accuracy of internal force evaluation b) effect of the Young s modulus 11
Numerical examples 1. A square plane subjected to an initial angular velocity Animation The Young s modulus and initial angular velocity are the same in these two cases. 12
Numerical examples 1. A square plane subjected to an initial angular velocity Displacement of point 1 in x direction 13
Numerical examples 2. A square frame under tension and compression Goal: test the capability of FPM in simulating large deformation of planar solids 14
Numerical examples 2. A square frame under tension Modeled with 228 particles 15
Numerical examples 2. A square frame under compression Modeled with 228 particles 16
Numerical examples 2. A square frame under tension and compression Compare with analytical solution of Euler beam under tension under compression 17
Remarks on the Finite Particle Method 1. FPM is based on the combination of the vector mechanics and numerical calculations. It enforces equilibrium on each particle. 2. No iterations are used to follow nonlinear laws, and no matrices are formed. The procedures are quite simple and robust. 3. The examples demonstrate performance and applicability of the proposed method on large deflection analysis of planar solids. 18
Future work 1. Expand the present work into 3D; 2. Use FPM in failure and collapse simulation. 19
References Ying Yu, Yaozhi Luo. Motion analysis of deployable structures based on the rod hinge element by the finite particle method. Proc. IMechE Part G: J. Aerospace Engineering. 2009, 223: 1-10. Ying Yu, Yaozhi Luo. Finite particle method for kinematically indeterminate bar assemblies. J Zhejiang Univ Sci A. 2009, 10 (5): 667-676. Ying Yu, Glaucio Paulino, Yaozhi Luo. Finite particle method for progressive failure simulation of framed structures. (finished and will be submitted for publication ) 20
Thank you!! 21
a) Evaluate the rigid body motion translation rotation 1 3 3 i1 i 22
b) Using reverse motion to remove rigid translation and rotation R cos( ) sin( ) sin( ) cos( ) d u u u r i i i d ( )( ), ( 1,2,3) i d1 R I xi x1 i 23
c) Using deformation coordinate to reduce element degree of freedom eˆ 1 d l1 1 u 2x d d m1 u u 2 2y eˆ 2 m l1 1 xˆ Qˆ ( x x ) 1 Qˆ eˆ l m T 1 1 1 T ˆ m 2 1 l e 1 24
d) Calculate the internal force at the deformation coordinate uˆ N uˆ N uˆ N uˆ 1 1 2 2 3 3 vˆ N vˆ N vˆ N vˆ 1 1 2 2 3 3 xˆ ˆ 1 y1 0 uˆ vˆ vˆ 0 1 1 2 uˆ N uˆ N uˆ vˆ N vˆ 2 2 3 3 3 3 Principle of virtual work: f ˆ 2x f 2xa f 2x f 2xa u 2x T f ˆ 3x f3xa f 3x f3xa [ t Β EBdA] u A 3x f f f f v 3y 3ya 3y 3ya 3y f f f f f f 1x 1y 2x 2 y 3x 3y T 25
g) Transform the internal force back to global coordinate Qˆ eˆ l m T 1 1 1 T ˆ m 2 1 l e 1 ˆ f ix T fix Q, i 1, 2,3 f ˆ iy f iy 26
f) Using forward motion to get the element back to the original position Transform the internal force back to the original direction fix f ix R, i 1, 2,3 fiy fiy 27