introduction of computer-aided nano-engineering · taylor series functional ... 01 0 1 0 2 0 2 0 00...

Introduction of Computer-Aided Nano Engineering

Prof. Yan WangWoodruff School of Mechanical Engineering

Georgia Institute of TechnologyAtlanta, GA 30332, U.S.A.yan.wang@me.gatech.edu

Multiscale Systems Engineering Research Group

Topics

Computational Nano Engineering

Modeling & Simulation (M&S)

Approximations in M&S

Computational Nano-EngineeringExtensive 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

Virtual Prototyping at nanoscalesComputer-Aided Nano-Design (CAND)

Computer-Aided Nano-Manufacturing (CANM)

Computer-Aided Nano-Engineering (CANE)

(a) CAD (b) CAE (c) CAPP/CAM (d) CAM

Computational Nano-EngineeringUse Modeling & Simulation tools to systematically resolve the issue of “lack of design” at nano scales.

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

PAST: Discovery-Based Science and Product Development

Design, produce, and scale up nano-based materials with exact properties needed

(based on established understanding and

methods)

Start with existing needs, problems, or

challenges in end-use application

Large numbers of diverse products

based on Nanomaterials By

Design rapidly enter multiple

markets

FUTURE: Application-Based Problem Solving

How to study a systemSystem

Experiment with actual

system

Experiment with a model of actual system

Physical model

Mathematical model

Analytical solution

Simulation

What is Modeling?

Why mathematical modeling?

Advantages

Disadvantages

An example of modeling

Free fall model

Mathematical model

Dependent variable=f(Independent variable)

High dimensional

Parametric systems

“Noisy” systems

( )y f x=

( )1 2, , ...y f x x=

( )1 2( ), ( ), , ( ),

ny f x u x u x u u= …

( ),y f x= γ

Complexity of Mathematical Models

Simple

Complex

Linear

Nonlinear

Algebraic Equation/ Closed-form

Differential Equation

Static

Dynamic

Model Taxonomy

System model

Deterministic Stochastic

Static Dynamic Static Dynamic

Continuous Discrete Continuous Discrete

Simulation-based Designmodel refinement model refinement

mathematical model(first-principles,

empirical, multiscale)

mathematical modelmathematical model(first(first--principles, principles,

empirical, empirical, multiscalemultiscale))

computer simulation

computer computer simulationsimulation

validationvalidationvalidation

Simulation-based designSimulationSimulation--based designbased design

scalable algorithms & scalable algorithms & solverssolvers

data/observationsdata/observationsdata/observations

geometry modeling &geometry modeling &discretizationdiscretization schemesschemes

visualizationvisualizationdata mining/sciencedata mining/science

optimizationoptimizationuncertainty quantificationuncertainty quantification

verificationverificationverificationapproximation approximation error controlerror control

numerical modelnumerical modelnumerical model

parameter inversionparameter inversiondata assimilationdata assimilationmodel/data error controlmodel/data error control

•Molecular Dynamics / Force Field

Modeling & Simulation at Multiple Scales

Various methods used to simulate at different length and time scales

nm μm mm m

pico-snsμs

Length Scales

Time Scales

femto-s

•Tight Binding

•Kinetic Monte Carlo

•Finite Element Analysis

•Dislocation Dynamics

•Quantum Monte Carlo•Self-Consistent Field (Hartree-Fock)

•Density Functional Theory

Approximations in simulation

mathematical models numerical modelsTaylor series

Functional analysis

numerical models computer codesDiscretization (differentiation, integration)

Searching algorithms (solving equations, optimization)

Floating-point representation

Mathematical models Numerical modelsApproximation in Taylor Series

Truncation

Mathematical models Numerical modelsFunctional Analysis

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

Approximate the original by linear combinations of basis functions ’s as

In a vector space (e.g. Hilbert space) with an infinite number of dimensions

( )f x

0( ) ( )

i iif x c x

=≈ ∑ ψ

( )ixψ

0( ) ( )

i iif x c x

== ∑ ψ

An inner product is defined as a “projection”in the vector space, such as

Typically we choose orthogonal basis functions ’s such that

for orthonormal basis functions

, : ( ) ( ) ( )f g f x g x W x dx∞

−∞= ∫

( )ixψ

( , , ), ( ) ( ) ( )

0 ( )i j i j

Constant i j i jx x W x dx

−∞

⎧ ∀ =⎪= = ⎨ ≠⎪⎩∫ψ ψ ψ ψ

1 ( ), ( ) ( ) ( )

i jx x W x dx

∞ ⎧

0 ( )i j i j i j−∞

=⎪= = ⎨ ≠⎪⎩∫ψ ψ ψ ψ

The coefficients ’s are computed by

The computable function is

with truncation!

0( ) ( )

i iif x c x

=≈ ∑ ψ

Functional Analysis – Fourier Series Approximations

where is called the fundamental frequency.

All periodic functions can be approximated by Fourier Series well!Is used in plane-wave density functional theory simulationsBecause of efficient Fast Fourier Transform (FFT)!

( ) ( ) ( ) ( ) ( )

( ) ( )

0 1 0 1 0 2 0 2 0

0 0 01

cos sin cos 2 sin 2

cos sink kk

f t a a t b t a t b t

a a k t b k t

ω ω ω ω

ω ω∞

= + + + + +

= + +⎡ ⎤⎣ ⎦∑

Functional Analysis – Fourier Series Approximations

Inner products

Computational complexity:O(N2)

FFT Complexity: O(Nlog2N)

( ) ( ) ( )

( ) ( )

2cos 1,2,...

a f t dtT

a k t f t dt kT

b k t f t dtT

1 Tik t

kc f t e dtT

ω−= ∫ ( ) ( ) 0

0i tF i f t e dtωω

∞−

−∞

= ∫ ( )0

0, , 1N

ik nk n

F f e k Nω−

= = −∑ …

( ) 0ik tk

f t c e ω∞

=−∞

= ∑ ( ) ( ) 0

i tf t F i e dωω ωπ

−∞

= ∫ ( )0

10, , 1

f Fe n NN

= = −∑ …

Fourier Series Fourier Transform Discrete Fourier Transform

Functional Analysis – Wavelet Transform

Wavelet basis functions

where 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.

satisfiesAdmissibility

Regularity condition

⎛ ⎞−= ⎜ ⎟

⎝ ⎠ψ ψ

( )xψ

Example Continuous Wavelet Functions- Mexican hat

( ) 21/4 2 /22( ) 1

xx x e− −= −ψ π

0 10 20 30 40 50 60 70 80 90 100-0.2

0.2FFT of Mexican hat wavelet

Frequency (Hz)

-10 -8 -6 -4 -2 0 2 4 6 8 10-0.4

1Mexican hat wavelet

Time (t)

Example Continuous Wavelet Functions- Morlet

( )2 /2( ) cos 5xx e x−=ψ

-10 -8 -6 -4 -2 0 2 4 6 8 10-1

1Mexican hat wavelet

Time (t)

0 10 20 30 40 50 60 70 80 90 100-0.05

0.05FFT of Mexican hat wavelet

Frequency (Hz)

Continuous Transform

where is the complex conjugate of .

Inverse Transform

Discrete Transform

,, ( ) ( )

a bf a b f x x dx

−∞= ∫ ψ

a bxψ ,

( )a b

( ) , 20

1( , ) ( )

daf x f a b x db

+∞ +∞

−∞= ∫ ∫

( )2 ,

j jk j

⎛ ⎞−= ⎜ ⎟

⎝ ⎠ψ ψ

Admissibility

where is the Fourier transform of .

This implies or equivalently“No information loss in reconstruction”

“must be a wave with zero mean” -- wave-

ˆ( )d

ψ ωω

+∞< ∞∫

( ) ( )ˆ i xx e dx∞

−∞

= ∫ ωψ ω ψ ( )xψ

2ˆ( 0) 0ψ ω = = ( ) 0x dx

−∞

=∫ ψ

Regularity condition

wavelet functions should have some smoothness and concentration in both time and frequency domains vanishing moments M1, …, MN as the scale factor a increases (admissibility: M0=0)“Fast Decay” -- -let

( ) ( ) 1

( ) 1 2

(1) 2 ( )

1 1, 0 (0) ( )

(0) ( )!

1 1 1 1(0) (0) ... (0) ( )

Nk k k

f a b f O xka

f a f a f a

O aM Ma

⎡ ⎤= ≈ +⎢ ⎥

⎣ ⎦⎡ ⎤

= +⎢ ⎥⎣ ⎦⎡ ⎤

= + + + +⎢ ⎥

⎣ ⎦

∫ ψ

Approximations in M&Smathematical models numerical models

Taylor series

Functional analysis

numerical models computer codesDiscretization (differentiation, integration)

Searching algorithms (solving equations, optimization)

Floating-point representation

Numerical Model Computer CodeCompute integrals

QuadratureApproximate the integrand function by a polynomial of certain degree

bax2 x3

QuadratureApproximate the integral by the weighted sum of regularly sampled functional values

• e.g. Simpson’s 3/8 rule

( ) ( ) ( ) ( ) ( )3

(3)0 1 2 3

hI f x dx f x f x f x f x⎡ ⎤≈ = + + +⎣ ⎦∫

f(x1)f(x2)

x1 x2 x3

Monte Carlo simulationLet p(u) denote uniform density function over [a, b]

Let Ui denote i th uniform random variable generated by Monte Carlo according to the density p(u)

Then, for “large” N

Variance reduction (importance sampling) to improve efficiency

( ) ( )b N

b af x dx f U

−≈ ∑∫

Numerical Model Computer CodeCompute derivatives

Finite-divided-difference methodsApproximated derivatives come from Taylor series

• e.g. forward-finite-difference

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2 21 1' 'i i i i i i if x f x f x x x O h f x f x h O h+ += + − + = + +

( ) ( ) ( ) ( )1' i ii

f x f xf x O h

h+ −

Floating-Point Representation

How does computer represent numbers?

Imperfect worldPerfect world

Ariane 5

Exploded 37 seconds after liftoff

Cargo worth $500 million

WhyComputed horizontal velocity as floating point number

Converted to 16-bit integer

Worked OK for Ariane 4

Overflowed for Ariane 5• Used same software

Do you trust your computer?

Single precision: f = 1.172603…

Double precision: f = 1.1726039400531…

Extended precision: f = 1.172603940053178…

Correct one is: f = -0.8273960599468213

yxyyyyxxyyxf

25.5)212111(75.333),( 8462226 ++−−−+=

f(x = 77617, y = 33096) = ?

Rump’s function:

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

1101.0 ≠×

IEEE 754 Standard

If E = 2w–1 and T ≠0, then v is NaN regardless of S.If E = 2w–1 and T =0 , then v = (–1)S×∞.If 1≤E ≤2w–2, then v = (–1)S×2E–bias×(1+21–p×T);

normalized numbers have an implicit leading significand bit of 1.If E =0 and T ≠0, v = (–1)S×2emin×(0+21–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=2w–1–1 and emin = 2–2w–1=1–bias

w bits t = p – 1 bits

Distribution of Values

6-bit IEEE-like formatw = 3 exponent bits

t = 2 fraction/mantissa bits

bias = 3

-15 -10 -5 0 5 10 15Denormalized Normalized Infinity

Distribution of Values(zoom-in view)

6-bit IEEE-like formatw = 3 exponent bits

t = 2 fraction/mantissa bits

bias = 3

-1 -0.5 0 0.5 1Denormalized Normalized Infinity

Round-Off Errors

Overflow error – “not large enough”Underflow error – “not small enough”Rounding error – “chopping”

http://www.cs.utah.edu/~zachary/isp/applets/FP/FP.html

Special Numbers

Expression Result

0.0 / 0.0 NaN

1.0 / 0.0 Infinity

NaN < 1.0 false

NaN == NaN false

0.0 == -0.0 true

NaN + 1.0 NaN

Infinity + Infinity Infinity

NaN == 1.0 false

-1.0 / 0.0 -Infinity

Infinity + 1.0 Infinity

NaN > 1. 0 false

standard range of values permitted by the encoding (from 1.4e-45 to 3.4028235e+38 for float)

Floating point Hazards

The result is 2.600000000000001

The result is 0.29 28

This expression does NOT equal to this expression when

0.0 - f -f

! (f >= g)

f is 0

f < g f or g is NaN

f == f f is NaN

f + g – g 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));

Comparing Floating Point Numbers

Try to avoid floating point comparison directlyTesting 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

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

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).

Random Error

Determines the precision of any measurement

Always present in every physical measurement

Better apparatus

Better procedure

Repeat

Estimate

Systematic ErrorDetermines the accuracyof 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

0 20 40 60 80 100

Uncertainties in Modeling & SimulationAleatory 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

Input Uncertainties

Conflicting Information

Conflicting Beliefs

Lack of Introspection

Measurement Errors

Lack of information

about dependency

Summary

Modeling is abstraction

M&S always has approximationsinvolved, which are important sources of epistemic uncertainty.

Computer tricks us

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

introduction of computer-aided nano-engineering · taylor series functional ... 01 0 1 0 2 0 2 0 00...

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