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What can do for scient ific computing ?

黃聰明

臺灣師範大學 數學系

min@ntnu.edu.tw

虎尾科技大學 2012 年 12 月 14 日

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雷射光上多重訊號的保密通訊

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Self-pulsating laser diodes eq uation

where

: photon density

: electron density in the active region

: electron density in the saturable absorption region : injection current

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1 1 1 1 1 2

1 1

1 12

2 2 2 2 2 1

2 2

2 21

1 1 1 1 2 2 2 2 1 1

( )

( )

( ) (

sin(2 )

)

g

s

g

s

g g th

s

dN k N N N

N N S

dt V T

dN k N N N

N N

a b

dt V S T

N V

dS k N N k N N G S C

f

dt

t

 

 

S

N1

N2

sin(2 ) a b ft

(4)

Numerical simulation of the solu

tion

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Bifurcation diagram

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Numerical simulation of the thre e dimensional photonic crystal

s

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光子晶體

光子晶體是週期性的介電質材料

在介電係數呈週期性排列的三維介電材料中,電磁波經介 電函數散射後,某些波段的電磁波強度會因破壞性干涉而 呈指數衰減,無法在系統內傳遞,相當於在頻譜上形成能 隙,導致色散關係具有帶狀結構,此即所謂的光子能帶結 構 (photonic band structures)

具有光子能帶結構的介電物質,就稱為光能隙系統 (photo nic band-gap system) ,或簡稱光子晶體 (photonic crysta ls) 。

科學家利用光子晶體的這個性質來做光導,也就是利用光 子晶體把特定波長的光限制在某個區域或路徑中。這種光 導可以用在光通訊上傳遞光訊號或作為雷射的共振腔。

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Maxwell Equation

Maxwell equation

E: electric field, H: magnetic field

Time Harmonic modes

Generalized eigenvalue problem



0 )

( E H

H E

E H

t t

E E

Bx Ax

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離散化

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Numerical simulation of nan

o-scale semiconductor

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Nanometer

It is a small world, with a plenty rooms.

Why consider quantum effect s?

Small devices imply significa nt quantum effect.

1 nm = m

Nano-scale  1-100 nm

A semiconductor QD  10 nm

(QD:hair  1:10,000) Semiconduct or QD

109

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Quantum dots

Cross-sections of hetero-structure InAs/GaAs QDs by Transmission Electron Microscope [Schoenfeld, 00]

150 Å

300 Å

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Nano-scale quantum dot fabri cation

E-beam, chemical solution,… (bigger QDs)

Molecular beam epitaxy (smaller QDs)

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Confinement

(band gap engineering)

Energy

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ħ: reduced Plank constant

λ: unknown eigenvalue; u(x,y,z): unknown eigenvector

m: effective mass; V: confinement potential

m and V are discontinuous across the heterojunction

BenDaniel-Duke interface cond.

Dirichlet boundary cond.

The Schrödinger equation

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Effective mass models

Constant model

Non-parabolic model

the momentum, main energy gap, and spin-orbit splitti ng in the lth region, respectively.Pl,gl,l :

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The eigenvalue problems

Constant effective mass model

Non-parabolic effective mass model:

multiply the common denominator

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Energy states spectrum and w ave functions

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Resonances in rail tracks excit ed by high speed trains

With new ICE trains crossi ng Europe at speeds of up to 300 km/h, sound and vib ration levels in the trains ar e an important issue.

Hilliges/Mehrmann/Mehl(2 004) first proposed this pro blem on a project with com pany SFE GmbH in Berlin.

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Finite Element Model

A 3D finite element discretization of the rail with linear is oparametric tetrahedron elements produces an infinite-di mensional system of O.D.E.:

where M, D and K are block-tridiagonal matrices

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F x

K x

D x

M 

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Palindromic Quadratic Eigenpr obelm

The system is periodic and leads to a Palindr omic QEP:

where , and

1 0 1

0,

2 AT A A x

n

Cn

A

A0, 1 A0T A0

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Surface Acoustic Wave Filter ( 表面聲波濾波器 )

為手機的關鍵零組件。表面聲波濾波器 (SAW Filter) 可被 廣泛應用在各種無線通訊系統、電視機、錄放影機及全球 衛星定位系統接收器上。主要功用在於把雜訊濾掉,比傳 統的 LC 濾波器安裝更簡單、體積更小。

SAW 元件主要作用原理是利用壓電材料的壓電特性,利 用輸入與輸出換能器( Transducer )將電波的輸入訊號 轉換成機械能,經過處理後,再把機械能轉換成電的訊 號,以達到過濾不必要的訊號及雜訊,提昇收訊品質的目 標。

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Finite element discretization

Palindromic Quadratic Eigenvalue Problem

where , and

1 0 1

0,

2 AT A A x

n

Cn

A

A0, 1 A0T A0

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Partial Differential Equation

where for , and is the unit ball in .

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2 1 2

,

0 

 

n

p n

d n 2 B1

R2

1 1

1

on 0

in 0

in 0

n B u

B u

B u

u u

d p

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Linear System

The discretization forms the following linear system where

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Discretization scheme -- non-uniform mesh, uniform mesh

-- finite-diff. on cylindrical cord. and

curvilinear cord.

-- finite-volume on Cartisian and

cylindrical coordinate

Large-scale matrix computation

-- matrix reduction

-- polynomial eigenvalue solver

-- deflation scheme -- accelerator

Summary

Science and Engineering

Applied Mathematics

Computer Science

The 3D model: Schrödinger eqs. with

constant or non-parabolic effective mass

approx.

Concerning issues: eng.

level & wave ft.

Computed results verifications,

explanations, applications Practical algorithms (for a certain

architecture and language)

Robust & efficient implementations

Numerical experiments

Computational and visual results

參考文獻

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