What can do for scientific computing minntnu edu

What can do for scientific 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

Self-pulsating laser diodes equation where : photon density : electron density in the active region : electron density in the saturable absorption region : injection current 2021/6/3 3

Numerical simulation of the solution 2021/6/3 4

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Bifurcation diagram 2021/6/3 7

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Numerical simulation of the three dimensional photonic crystals 2021/6/3 9

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Maxwell Equation l Maxwell equation E: electric field, H: magnetic field l Time Harmonic modes l Generalized eigenvalue problem 2021/6/3 12

離散化 2021/6/3 13

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Numerical simulation of nanoscale semiconductor 2021/6/3 15

Nanometer l It is a small world, with a plenty rooms. l Why consider quantum effects? Small devices imply significant quantum effect. l 1 nm = m Nano-scale 1 -100 nm l A semiconductor QD 10 nm (QD: hair 1: 10, 000) 2021/6/3 Semiconductor QD 16

Quantum dots l Cross-sections of hetero-structure In. As/Ga. As QDs by Transmission Electron Microscope [Schoenfeld, 00] 150 Å 300 Å 2021/6/3 17

Nano-scale quantum dot fabrication • E-beam, chemical solution, … (bigger QDs) • Molecular beam epitaxy (smaller QDs) 2021/6/3 18

Energy Confinement (band gap engineering) 2021/6/3 19

The Schrödinger equation l ħ: reduced Plank constant l λ: unknown eigenvalue; l m: effective mass; V: confinement potential m and V are discontinuous across the heterojunction l Ben. Daniel-Duke interface cond. l Dirichlet boundary cond. 2021/6/3 u(x, y, z): unknown eigenvector 20

Effective mass models l Constant model l Non-parabolic model the momentum, main energy gap, and spin-orbit splitting in the lth region, respectively. 2021/6/3 21

The eigenvalue problems l Constant effective mass model l Non-parabolic effective mass model: multiply the common denominator 2021/6/3 22

Energy states spectrum and wave functions 2021/6/3 23

Resonances in rail tracks excited by high speed trains l l With new ICE trains crossing Europe at speeds of up to 300 km/h, sound and vibration levels in the trains are an important issue. Hilliges/Mehrmann/Mehl(2 004) first proposed this problem on a project with company SFE Gmb. H in Berlin. 2021/6/3 24

Finite Element Model l A 3 D finite element discretization of the rail with linear isoparametric tetrahedron elements produces an infinitedimensional system of O. D. E. : l where M, D and K are block-tridiagonal matrices 2021/6/3 25

Palindromic Quadratic Eigenprobelm l The system is periodic and leads to a Palindromic QEP: l where 2021/6/3 , and 26

l Finite element discretization l Palindromic Quadratic Eigenvalue Problem where 2021/6/3 , and 28

Partial Differential Equation where unit ball in 2021/6/3 for , and is the . 29

Linear System The discretization forms the following linear system where 2021/6/3 30

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Summary Science and Engineering Applied Mathematics • 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 2021/6/3 -- accelerator • The 3 D 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 Computer certain Science architecture and language) • Robust & efficient implementations • Numerical experiments • Computational and visual results 33