Scientific computing topics under Scientific computing topics under current studycurrent study

Tony W. H. SheuScientific Computing and Cardiovascular Simulation Lab.Computational Mechanics and Scientific Visualization Lab.

National Taiwan UniversityTEL: 886-2-33665746 FAX: 886-2-23929885E-mail: twhsheu@ntu.edu.twhttp://ccms.ntu.edu.tw/~twhsheu

Dec. 8. 2010 in NTU for the interaction of CQSE and TIMS colleagues

Contents(1) Nonlinear partial differential equations 1.1 Westervelt equation

1.2 Camassa-Holm equation 1.3 Schrodinger equation

(2) Nonlinear system of partial differential equations for incompressible fluid flow

2.1 Incompressible Navier-Stokes ( NS ) equations

Coupled with magnetic induction equation Magnetohydrodynamic ( MHD ) equations

Coupled with electricfield equation

Electrohydrodynamic ( EHD ) equations

Coupled with iontransport equation

Electrosmotic flow ( EOF ) equations

Coupled with level-setequation

Free ( interface ) flow equations

2.2 Subgrid models for the simulation of flow turbulence - Leray-α differential model - NS-α differential model - NS-ω differential model

(3) Maxwell’s equations(4) Applications

High-intensity focussed ultrasound ( HIFU ) for liver tumor ablation Inspiration / expiration in human lung airway Construction of acupuncture ( 針 ) & moxibustion ( 灸 ) mod

el Free surface flow over an irregular obstacle

1. Nonlinear partial differential equations

(A) Westervelt equation ( one-manpower )

- Challenge : Computationally efficient linearization of the last term in case of a focused high-frequency and sound field - Application : Coupled with the hydrodynamics and energy equations in HIFU study

2 22 4 40 0 0 0

1( ) 0tt ttt ttp p p p

c c c

Linear wave Absorptioncontribution

Nonlinear contribution

(B) Camassa-Holm equation ( 1/3 manpower )

- Academic topics under investigation Resolve oscillations due to the highly dispersive term so as to capture the cusp ( peakon or soliton ) profile Clarify the debate if the dissipative behavior is present in the p

eakon-antipeakon problem Preserve Hamiltonians embedded in the above equation

Mixedderivative term

Three nonlinear terms

3 2 0t txx x x xx xxxu u uu u u uu

xxxuu

(C) Schrodinger equation ( one manpower )

- Academic topics under investigation

Preserve the Hamiltonian

and the particle number properties imbedded in

the above equation

Explore the time-evolving behavior of the momentum given by

2 2| | 0ti a

24

0 0

1 1( ) | | | |

2 2

L L

xH dx a dx 2

0

1| |

2

Ldx

0

1Im

2

L

xdx

2. Nonlinear system of Partial differential equations

21 1( ) ( )

0

tu u u p u g B B

u

(A) Incompressible MHD equations ( 1/3 manpower )

* Hydrodynamic field equations

* Magnetic field equations2( )

0tB u B B

B

Academic topics under current investigation - enforce divergence-free condition for the momentum equations - enforce divergence-free condition for the magnetic induction equations

0u 0B

(2.1) Incompressible Navier-Stokes ( NS ) equations

Note : For electrically conducting fluids such as the plasma andLiquid metal

21 1( )

0

tu u u p u g qE

u

(B) Incompressible EHD equations ( one manpower )

* Hydrodynamic field equations

* Electric field equations

2

( ( )) 0tq q E u

E

q

Academic topics under current investigation - Reveal the bifurcation types and the route to chaos in the unipolar injection problem - Resolve sharp solution profile in the EHD flow field

Note : For electrically charged fluids

21( ) ( ( ) )

0

etu u u p u T u E

u

(C) Incompressible EOF ( 電泳 ) equations ( 1/3 manpower )

* Hydrodynamic equations

* Energy equation

21( ) ( ( ) )p t ec T u T k T T u E

Note : For the fluid with ion

0

0

( ( ) ) 0

2( ( ) ) sinh( )

( )

T

en eT

kT

E

* Electrosmotic equations

Academic topic under current investigation - Simulation of the 3D large-scale EOF microchannel flow problem in parallel CPU and GPU processors

2

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

( ) Re ( ) ( )

0

0

t

t

D Ku u u p

We Fr

u

u

(D) Incompressible Interface / free surface flow equations ( one man power )

Academic topics under current investigation - Preserve either the area or volume of the liquid and gas phases - Resolve contact discontinuity oscillations near the interface/free surface

(2.2) Subgrid turbulence models ( one manpower )

21

Re0

tu u u p u

u

(A) Leray-α regularized model

where 22

1 uu u

(B) NS-α regularized model (C) NS-ω regularized model

2

2

1( )

Re0

tu u u p u f

u

u u u

2

2

1( )

Re

0

tu u u p u f

u

u u u

Academic topic under current investigation - Examine how well these regularized NS equations can be applied to model flow turbulence

3. Maxwell equations ( two manpower )

1

0

1

0

t

t

H E

H

E E

E

Academic topics under current investigation - Preserve Hamiltonians and conserved quantities - Enforce Gauss law ( divergence-free conditions for and , or ) - Preserve symplecticity and energy in the above equations - Optimize the numerical dispersion relation

E H

0H E

4. Applications

(A) High-intensity focused ultrasound ( HIFU ) of liver tumor ablatian

The time-evolving volume with the temperature higher than 45 °C in the liver tumor (ultrasound is imposed in the first 5 seconds)

Animation

4. Applications

(B) Inspiration / expiration in human lung airways

Velocity profile

Pressure contoursInlet flow profile

InspirationExpiration

4. Applications

(C) Construction of acupuncture ( 針 ) & moxibustion ( 灸 ) model

Acupuncture needle is combined with one-column needle and one curl handle, which covers on the columned needle.

Temperature distribution on the calf due to moxibustion practice

4. Applications

(D) Free surface flow over an irregular obstacle