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
雷射光上多重訊號的保密通訊
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
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
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
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
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
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 :
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
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.
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
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
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
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
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