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This 7 page Class Notes was uploaded by Shanel Lubowitz on Monday October 26, 2015. The Class Notes belongs to ENGR0011 at University of Pittsburgh taught by Staff in Fall. Since its upload, it has received 29 views. For similar materials see /class/229414/engr0011-university-of-pittsburgh in Engineering and Tech at University of Pittsburgh.
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Date Created: 10/26/15
MONTE CARLO Cainuiations Computer can be used like a simple graphing calculator X 1 x2x3 x Using variables and functions can automate complex calculations e x 2 3 n Must be aware of limitations of precision and accuracy Optimization Developing models of real physical data is a critical aspect in Materials Science calculations 2 leY fxacexpbx c Reducing system to a finite number of parameters 39 Quantitative methods to relate model to data Signai Processing Operations on data to convert into something useful g i Interpolation extrapolation m Frequency domain analysis PSD convolution i L Other statistical methods EVERYTHING UNTIL THIS POINT HAS BEEN GENERAL OR ABSTRACTED NOT SPECIFIC TO MATERIALS SCIENCE New approach to modeling random stochastic processes Uses random numbers to simulate a complex system Software packages exist but are not standard offtheshelf SOFI39WARE 0 Design AI AbInitio First principles Monte Carlo Stochastic Finite elements Diff Eq C 3rd PARTY TODAY Symbolic math Curve fitting optimization 1995 Integration Root finding MATHEMATICAMAPLE 1985 Linear algebra MATLAB Statistics Graphing CALC EXCEL 1975 Arithmetic Atomicmolecular fluctuations and motion determine many materials processes Advances in computation continue to allow more realistic complex simulations Timr39 Lime large scale FEMFD largcescnle plasticity elssneny ncsiimnspm Materials science spans many orders of year E m e magnitude in spatial and temporal scale quot 0 ll l 10 O 10 meiaucn models m mdwmmu We must consider the behaVIor of incliViclual Wmlpiassenyncm atoms as well as their collective macro homogrnization methods behaVIor csow 239 a m m s n s a a a 17 mm consistent models in conjunction 0 3 with advanced constitutive laws w 10 in kirictic mullislate Putts models CC bylk Ginzburg Landau ly pe m diffusion and kinetic phase eld a Warm 7 models microscopic g dimmitln j iicld kinetic models a phase eld 5 o dynamics m dislocation dynamics 2 dislocql39nn 10 7 6 3 dynamics 10 topological network and E molccillax vertex models bounderv g dynamics 5 Debye nequency spring models s 15 quot3 H ms 5 b iimc scale molecular dynamics Debye um MelropolisMomeCarlo 0 14 G s local electron density runcricnol lheorv Figure 12 39 39 A iliv L x A minis scienuel MODELING SIMULATION k 0 0 3 g gt39 6f H a 2 lt lt H X x PLASTICITY GRAIN BOUNDARIES Stress strain dislocation generation 39 Many bOdy problem on atomic scale Can measure properties of system macro Cannot make measurements via experiment scale No analytical expressions are available Don t consider positions of individual atoms or disocations Individual atom positions must be considered ATOMISTIC approach MEANFIELD CONTINUUM approach GLOBAL MODEL GLOBAL CONFIGURATION i A LOCAL CONFIGURATION LOCAL BEHAVIOR GAS LIQUID SOLID U numbu of toms In cube pm length nl39oubc System composed of identical atoms large number 1023 in real systems Each atom has 6 degrees of freedom rx ry rz px py pz Each behaves independently and they simultaneously obey classical ENSEMBLE andlor quantum mechanics laws vibrations F ma wrt Local interactions with neighbors collision mechanics energy exchange bonds Covalenrt ionic metallic hydrogen VDW AN ENSEMBLE 1 E exp Obeys certain statistical mechanics laws R Eexp k for example a IeXpE dr 39 ka Can be described in terms of average PARTITION FUNCTION properties N V T E IL Cp P ARTIFICIAL ENSEMBLE SIMULATION If we set up an ensemble of independently acting atoms in the computer Their collective behavior will mimic or simulate the behavior of a real system 1 rs l 1 Set up ensemble of independent trials or s simple laws pecies gt each obeys 2 Run simulation that applies these laws via RANDOM FLUCTUATIONS many times 3 Statistically sample results to get meaningful global information DETERMINI STIC PROCESS STiOiCHASTIC PROCESS eg Classical mechanics heat flow eg Stock market Brownian motion Analytic form of governing laws No analytic form each step is random Single answer determined completely by Startan Statistical characterization of larger structure conditions This is the essence of Monte Carlo modeling Von Neumann 1944 Diffusion in fissile materials 100 gt X 100 Free motion Gas 100 0 x Lattice sites Solid 1 Single atom placed at center 5050 2 Take n random steps record positions 3 Multiple trials will give statistical behavior Mainm initial conditions loop move atom to new spot end plot results calculate statistics Get new position VK FunctionxNew yNew getNewxym pick random direction xNew m yNew m correct for collision with wall end
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