Introduction of Computer-Aided Nano Engineering

Transcription

1 Introduction of Computer-Aided Nano Engineering Prof. Yan Wang Woodruff School of Mechanical Engineering Georgia Institute of Technology Atlanta, GA 30332, U.S.A.

2 Topics Computational Nano Engineering Modeling & Simulation (M&S) Approximations in M&S

3 Computational Nano-Engineering Extensive applications of CAD/CAM/CAE software tools in traditional manufacturing lead to Scalable processes Cost effective and high-quality products with short time-to-market (a) CAD (b) CAE (c) CAPP/CAM (d) CAM Virtual Prototyping at nanoscales Computer-Aided Nano-Design (CAND) Computer-Aided Nano-Manufacturing (CANM) Computer-Aided Nano-Engineering (CANE)

4 Computational Nano-Engineering Use Modeling & Simulation tools to systematically resolve the issue of lack of design at nano scales. PAST: Discovery-Based Science and Product Development Discover novel Nanostructures, nanoparticles, and nanomaterials through investigator-initiated exploratory research on a broad range of materials Determine nanomaterial properties (chemical, physical, and biological) Identify potential applications of value Assess commercial viability Nanomaterials enter limited markets FUTURE: Application-Based Problem Solving Start with existing needs, problems, or challenges in end-use application Design, produce, and scale up nano-based materials with exact properties needed (based on established understanding and methods) Large numbers of diverse products based on Nanomaterials By Design rapidly enter multiple markets

5 How to study a system System Experiment with actual system Experiment with a model of actual system Physical model Mathematical model Analytical solution Simulation

6 What is Modeling? Multiscale Systems Engineering Research Group

7 Why mathematical modeling? Advantages Disadvantages

8 An example of modeling Free fall model

9 Mathematical model Dependent variable=f(independent variable) High dimensional Parametric systems Noisy systems y y = = f x ( ) f x, x,... ( ) 1 2 ( ( ), ( ),, ( ), ) y = f x u x u x u u 1 2 y = f ( x, γ) n

10 Complexity of Mathematical Models Simple Linear Algebraic Equation / Closed-form Static Complex Nonlinear Differential Equation Dynamic

11 Model Taxonomy System model Deterministic Stochastic Static Dynamic Static Dynamic Continuous Discrete Continuous Discrete

12 Simulation-based Design visualization data mining/science model refinement validation validation optimization uncertainty quantification mathematical mathematical model model (first-principles, (first-principles, empirical, empirical, multiscale) multiscale) parameter inversion data assimilation model/data error control Simulation-based design design geometry modeling & discretization schemes data/observations data/observations computer computer simulation simulation scalable algorithms & solvers numerical numerical model model approximation error control verification verification

13 Modeling & Simulation at Multiple Scales s ms μs ns pico-s femto-s Time Scales Finite Element Analysis Dislocation Dynamics Kinetic Monte Carlo Molecular Dynamics / Force Field Tight Binding Density Functional Theory Self-Consistent Field (Hartree-Fock) Quantum Monte Carlo Length Scales nm μm mm m Various methods used to simulate at different length and time scales

14 Approximations in simulation mathematical models numerical models Taylor series Functional analysis numerical models computer codes Discretization (differentiation, integration) Searching algorithms (solving equations, optimization) Floating-point representation

15 Mathematical models Numerical models Approximation in Taylor Series Truncation

16 Mathematical models Numerical models Functional Analysis Multiscale Systems Engineering Research Group Convert complex functions into simple and computable ones by transformation in vector spaces Fourier analysis Wavelet transform Polynomial chaos expansion Spectral methods Mesh-free methods

17 Mathematical models Numerical models Functional Analysis Approximate the original fx ( ) by linear combinations of basis functions ψ i ( x) s as fx ( ) c ψ ( x) i i i= 0 In a vector space (e.g. Hilbert space) with an infinite number of dimensions N fx ( ) = c ψ ( x) i= 0 i i

18 Mathematical models Numerical models Functional Analysis An inner product f, g is defined as a projection in the vector space, such as f, g : = f( x) g( xw ) ( x) dx Typically we choose orthogonal basis functions ψ i ( x) s such that Constant ( i, j, i = j) ψ, ψ = ψ( x) ψ ( xw ) ( x) dx = i j i j 0 ( i j) for orthonormal basis functions 1 ( i = j) ψ, ψ = ψ( x) ψ ( xw ) ( x) dx = i j i j 0 ( i j) Multiscale Systems Engineering Research Group

19 Mathematical models Numerical models Functional Analysis The coefficients s are computed by The computable function is c i c = with truncation! i f, ψ i ψ, ψ i N fx ( ) c ψ ( x) i i i i= 0

20 Functional Analysis Fourier Series Approximations where ( ) cos ( ω ) sin ( ω ) cos ( 2ω ) sin ( 2ω ) f t = a + a t + b t + a t + b t ( ω ) sin ( ω ) = a + a cos k t + b k t 0 k 0 k 0 k = 1 2π ω = 0 T is called the fundamental frequency. All periodic functions can be approximated by Fourier Series well! Is used in plane-wave density functional theory simulations Because of efficient Fast Fourier Transform (FFT)!

21 Functional Analysis Fourier Series Approximations Inner products 1 T ik t ω0 ck = f () t e dt T iω0 ( ) () 0 ik 0 f () t ce ω Computational complexity:o(n 2 ) FFT Complexity: O(Nlog 2 N) T 1 a = f () t dt 0 T 0 T 2 ak = cos k 0t f t dt k 1,2,... T = 0 T 2 b = sin ( kω t) f () t dt k 0 T t F iω0 = f t e dt 0 ( ω ) () ( ) Fourier Series Fourier Transform Discrete Fourier Transform ik 0n F ω = fe ( k = 0,, N 1) t iω0t = f () t F k ( iω0) e dω0 k = 2π 1 k N 1 n = 0 1 N = ikω0n f = Fe ( n = 0,, N 1) n n N 1 k = 0 k

22 Functional Analysis Wavelet Transform Wavelet basis functions ψ ab, ( x ) 1 x b = ψ a a where ψ( x) is a continuous function in both the real and reciprocal spaces called mother wavelet, a is the scale factor, and b is the translation factor. ψ( x) satisfies Admissibility Regularity condition

23 1 Example Continuous Wavelet Functions -Mexican hat Mexican hat wavelet 2 ψ( x) = π ( 1 x ) e 3 1/4 2 x / FFT of Mexican hat wavelet Time (t) Frequency (Hz)

24 1 Example Continuous Wavelet Functions -Morlet Mexican hat wavelet ψ ( ) x 2 /2 ( ) = cos 5 x e x Time (t) FFT of Mexican hat wavelet Frequency (Hz)

25 k Functional Analysis Wavelet Transform Continuous Transform f a, b = + f( x) ψ ( x) dx where ψ * () x is the complex conjugate of ψ () x. ab, ab, Inverse Transform Discrete Transform ( ) *, ( ) da f x = f (,) a b () x db ψ 0 ab, 2 c a ψ j ab ψ j 2, k ( x ) 1 x k = ψ j j 2 2

26 Admissibility ˆ Functional Analysis Wavelet Transform ( ) ( ) 0 + iωx where ψ ω ψ x e dx is the Fourier transform of. = This implies or equivalently ψ No information loss in reconstruction must be a wave with zero mean -- wave- 2 ˆ( ψω) d ω < ω 2 ˆ( 0) 0 ψ( x) ψω= = ( x) dx = 0

27 Functional Analysis Wavelet Transform Regularity condition N 1 1 ( k) k 0 k 1 f x + ( a b = ) f x ψ ( ) dx + O x, 0 (0) ( ) k 0 k! a = a = = 1 N 1 ( k) k+ 1 k+ 2 f (0) M a + O( a ) k a k = 0 k! (1) 2 1 ( N) N+ 1 k+ 2 f(0) M a + f (0) M a f (0) M a + Oa ( ) 0 1 N a 0! 1! N! wavelet functions should have some smoothness and concentration in both time and frequency domains vanishing moments M 1,, M N as the scale factor a increases (admissibility: M 0 =0) Fast Decay -- -let Multiscale Systems Engineering Research Group

28 Approximations in M&S mathematical models numerical models Taylor series Functional analysis numerical models computer codes Discretization (differentiation, integration) Searching algorithms (solving equations, optimization) Floating-point representation

29 Numerical Model Computer Code Compute integrals Multiscale Systems Engineering Research Group Quadrature Approximate the integrand function by a polynomial of certain degree f(x) f(x) f(x) f(x i ) f(x i ) f(x i ) a x 1 x 2 x 3 n=0 b x a n=1 b x a n=2 b x

30 Numerical Model Computer Code Compute integrals Quadrature Approximate the integral by the weighted sum of regularly sampled functional values e.g. Simpson s 3/8 rule f(x) f(x 3 ) f(x 0 ) f(x 1 ) f(x 2 ) x 3 (3) 3h I f x dx = f x + f x + f x + f x 8 x 0 ( ) ( ) 3 ( ) 3 ( ) ( ) x 0 x 1 x 2 x 3 x

31 Numerical Model Computer Code Compute integrals Monte Carlo simulation Let p(u) denote uniform density function over [a, b] Let U i denote i th uniform random variable generated by Monte Carlo according to the density p(u) Then, for large N b a b a fxdx ( ) fu ( ) N N i = 1 i Variance reduction (importance sampling) to improve efficiency

32 Numerical Model Computer Code Compute derivatives Finite-divided-difference methods Approximated derivatives come from Taylor series e.g. forward-finite-difference 2 2 ( ) = ( ) + '( ) ( ) + ( ) = ( ) + '( ) + ( ) f x f x f x x x O h f x f x h O h i+ 1 i i i+ 1 i i i ( ) f ( x ) f x i+ 1 i f ' ( xi ) = + O h h ( )

33 Floating-Point Representation How does computer represent numbers? Perfect world Imperfect world

34 Ariane 5 Exploded 37 seconds after liftoff Cargo worth $500 million Why Computed horizontal velocity as floating point number Converted to 16-bit integer Worked OK for Ariane 4 Overflowed for Ariane 5 Used same software

35 Do you trust your computer? Rump s function: f ( x, y) = y + x (11x f(x = 77617, y = 33096) =? 6 y Single precision: f = y 5.5y Double precision: f = Extended precision: f = Correct one is: f = y ) x 2 y

36 Another Story On February 25, 1991 A Patriot missile battery assigned to protect a military installation at Dhahran, Saudi Arabia But failed to intercept a Scud missile 28 soldiers died an error in computer arithmetic

37 IEEE 754 Standard w bits E T t = p 1 bits If E = 2 w 1 and T 0, then v is NaN regardless of S. If E = 2 w 1 and T =0, then v = ( 1) S. If 1 E 2 w 2, then v = ( 1) S 2 E bias (1+2 1 p T); normalized numbers have an implicit leading significand bit of 1. If E =0 and T 0, v = ( 1) S 2 emin (0+2 1 p T); denormalized numbers have an implicit leading significand bit of 0. If E =0 and T =0, then v = ( 1) S 0 (signed zero) where bias=2 w 1 1 and emin = 2 2 w 1 =1 bias

38 Distribution of Values 6-bit IEEE-like format w = 3 exponent bits t = 2 fraction/mantissa bits bias = Denormalized Normalized Infinity

39 Distribution of Values (zoom-in view) 6-bit IEEE-like format w = 3 exponent bits t = 2 fraction/mantissa bits bias = Denormalized Normalized Infinity

40 Round-Off Errors Overflow error not large enough Underflow error not small enough Rounding error chopping

41 Special Numbers Expression Result 0.0 / 0.0 NaN 1.0 / 0.0 Infinity -1.0 / 0.0 -Infinity NaN NaN Infinity Infinity Infinity + Infinity Infinity NaN > 1. 0 false NaN == 1.0 false NaN < 1.0 false NaN == NaN false 0.0 == -0.0 true standard range of values permitted by the encoding (from 1.4e-45 to e+38 for float)

42 Floating point Hazards This expression does NOT equal to this expression when f -f f is 0 f < g! (f >= g) f or g is NaN f == f true f is NaN f + g g f g is infinity or NaN double s=0; for (int i=0; i<26; i++) s += 0.1; System.out.println(s); double d = 29.0 * 0.01; System.out.println(d); System.out.println((int) (d * 100)); The result is The result is

43 Comparing Floating Point Numbers Try to avoid floating point comparison directly Testing if a floating number is greater than or less than zero is even risky. Instead, you should compare the absolute value of the difference of two floating numbers with some pre-chosen epsilon value, and test if they are "close enough If the scale of the underlying measurements is unknown, the test abs(a/b - 1) < epsilon is more robust. Don t use floating point numbers for exact values

44 Uncertainties in M&S Model errors due to approximations in truncation or sampling Taylor approximation Functional analysis Numerical errors due to floating-point representation Round-off errors

45 Two types of uncertainties Aleatory uncertainty (variability, irreducible uncertainty, random error) inherently associated with the randomness/fluctuation (e.g. environmental stochasticity, inhomogeneity of materials, fluctuation of measuring instruments) can only be reduced by taking average of multiple measurements. Epistemic uncertainty (incertitude, reducible uncertainty, systematic error) imprecision comes from scientific ignorance, inobservability, lack of knowledge, etc. can be reduced by additional empirical effort (such as calibration).

46 Random Error Determines the precision of any measurement Always present in every physical measurement Better apparatus Better procedure Repeat Estimate

47 Systematic Error Determines the accuracy of any measurement May be present in every physical measurement calibration uniform or controlled conditions (e.g., avoid systematic changes in temperature, light intensity, air currents, etc.) Identify & eliminate or reduce chronological data trial

48 Uncertainties in Modeling & Simulation Aleatory Uncertainty: inherent randomness in the system. Also known as: stochastic uncertainty variability irreducible uncertainty Epistemic Uncertainty: due to lack of perfect knowledge about the system. Also known as: Incertitude system error reducible uncertainty Lack of data Conflicting Beliefs Measurement Errors Input Uncertainties Lack of information about dependency Conflicting Information Lack of Introspection

49 Summary Modeling is abstraction M&S always has approximations involved, which are important sources of epistemic uncertainty. Computer tricks us

50 Further Readings Goldberg, D. (1991) What every computer scientist should know about floating-point arithmetic, ACM Computing Surveys, 23(1), 5-48