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 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 What is FEM? 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 lAlA fBAfBA fABfAB n (a) A B fBAfBA n, l (b) A B Particle centroid fABfAB lBlB (a) (b) sr nn st 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. Contact parameters: a) for smooth non-spherical, convex; b) circular particles
15 Principle of DEM (a) F1F1 F2F2 F3F3 F4F4 (b) MM FF (c) a v (d) 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 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 F dn = C n. ń F ds = C t. 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 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 Contact Detection (cont.)
21 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. Contact Detection (cont.) L R2R2 R1R1 (a) L m (b)
22 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 F5F5 1 F2F2 F3F3 F4F4 F6F6 1 apap pp (a)(b)(c)(d)
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