شبيه سازی

1. شبيه سازی Simulation

2. مقايسه! • Experimental • Provide useful quantitative information • Are common as they use real system • Considerable Time and cost usage (repetition!) • Numerical simulation (Computer Simulation) • Virtual systems • To predict the behaviour of a real system • More flexible in application • Micro and Macro scale results at any time

3. Experimental Simulation • In-lab experiment that is as much like some real situation as possible. • Small scale equipment • Example: • ground-based flight (Pesticide application), Dam, rainfall and Silo simulators • behaves as closely as possible to a real one • still under researcher control

4. Experimental Simulation • Still fairly precise. • More realistic than in-lab experiment. • Not a natural setting – interaction may not be normal. • Extrapolation of Results may lead to uncertainty and ERRORS

5. Computer (Numerical) Simulation • Creating a complete & closed system that models the operation of the real system without users. • Example: • Plant growth simulations (Agronomy researchers) • Engineering Models • Continuum Models • Discrete Element Models

6. Continuum approach • The behaviour of a mechanical system can be expressed by differential equations • Mechanical system is divided into Finite elements • Derived constitutive equations for elements are linked together to solve the problem • Application for Stress and heat analysis • Finite Element Method (FEM) • Boundary Element Method (BEM)

7. What is FEM? • Full name: Finite Element Method General Concepts • FEM cuts a structure into several elements of the structure • The nodes at the each end of an element are reconnected as if nodes were pins that hold elements together • This results in a set of simultaneous algebraic equations Applications of FEM • Desire to understand how various elements behave with arbitrary shape, loads, and support conditions • Can be contained within a single computer program for users to input data such as geometry, boundary conditions, and element selection • Handle complex restraints, which allow indeterminate structures to be solved Disadvantages of FEM • FEM obtains only approximate solutions • Many input data are required

8. Discrete Element Method • Continuum models based on Continuity • Increases in computer speed • Calculation of the position of individual particles • DEM Useful for particulate materials • Grains, Soil , Powder, Fruits • Solid systems also can be modelled • Need good Programming skills

9. Main steps of DEM • Particle and environment generation • Search for contact • Contact detection between pairs of discrete particles • Calculation of contact force • Update particle motion due to unbalanced force • Circulation

10. DEM Development • Appeared in 1979 by Cundall and Strack • Shape representation • Circle (2D) • Sphere (3D) • Ellipse (2D) • Ellipsoid (3D) • Polygon (2 & 3D) • Combined Primitives

11. DEM Application • Geomechanics (Soil & rock) • Granular storage & flow (Silo) • Powder Technology • Fruits and Vegetable (handling) • Processing Operation ( ball mills) • Continuous System BUT composed of individual ( say particles) in Microscopic level ( Asphalt, Biomaterials, solid structures) • Combined DEM & FEM

12. DEM Limitations • Matching Real and Model particle shape • Not for very large spatial domain where millions of particles involved • Need for physical properties • Running time concern

13. Theoretical aspects • Contact models between contacting bodies • Contact area • Contact point • Contact vector • Contact Displacement and deformation • Normal • Tangetial

14. Particle centroid A st A fAB lA B fBA fBA n, l n fAB lB sr B n (b) (b) (a) (a) Contact parameters: a) for smooth non-spherical, convex; b) circular particles Normal (n) and tangential (st and sr) displacements for two particles in contact due to: (a) relative translation and (b) relative rotation of the particles.

15. (a) F1 F3 (b) F2 F4  M  a F v (d) (c) Principle of DEM The DEM procedure for contact force calculations and updating the dynamic situation of particles; a) Recognising the formation of contact points due to relative velocities and position of particles. b) Application of force-displacement law for each contact point to calculate the contact force. c) The moment of contact forces about the particle centroid is calculated and the resultant force and moment on particle centroid is determined. d) Application of Newton’s second law of motion to calculate the particle acceleration and velocity.

16. Kn k c Remove when sliding occur n n c n k t k t c t m c t m ( a) (b) Contact Models DEM contact models for cohesionless materials: a) the maximum frictional force based on the sum of spring and dashpot; b) the maximum frictional force calculated only from spring (elastic) force

17. Damping • Contact Damping Fdn= Cn . ń Fds = Ct . ś • Global Damping (act on Absolute Velocity & rotation)

18. Time step • In DEM the time step is the time during which force is transmitted from one contact point to another along the particle boundary. • The time step should be as large as possible to increase the efficiency of simulation and still be smaller than the critical time step to justify the assumption of constant acceleration within each time step and to ensure stability of the calculations • The idea is based on the assumption that the selected time step is small enough so that no new contacts take place in the current time step except those that have already been recognised at the beginning of the time step.

19. Contact Detection • the most important step prior to any mechanical calculation is determination of which surfaces are in contact and the type of contact. • It is estimated that more than 80% of the computational time can be spent on this task. • In a very simple approach each particle is checked against every other particle to determine any probable contact. • The computational time for this simple procedure with n particles will be proportional to , which is too long if there are hundreds of particles in the simulation

20. Contact Detection (cont.) • For densely pack a “link list“ algorithm • the simulation space is divided into relatively large cells • A separate list of particles for each cell is provided, including the particles in the home cell and surrounding cells. • The particles within a cell and its neighbouring cells are considered as potential contacting bodies. • Therefore, contact detection for such list would be an efficient process regarding time consumption

21. m R1 R2 L L (b) (a) Contact Detection (cont.) • For loosely pack a grid search • Small cells, so that in each cell one particle can be occupied Contact detection between particles; a) Circular shape b) Polygonal shape.

22. 2 6 1 3 4 5 F2 p F6 1 1 F3 F5 ap F4 (a) (b) (c) (d) Calculation cycle • Simulation steps in DEM; a) Particle and environment generation, b) Contact search and detection, c) Calculation of contact force, d) update the particle accelerations.

24. visualisation