Scientific computing topics under
current study
Tony W. H. Sheu
Scientific Computing and Cardiovascular Simulation Lab.
Computational Mechanics and Scientific Visualization Lab.
National Taiwan University
TEL: 886-2-33665746
FAX: 886-2-23929885
E-mail: twhsheu@ntu.edu.tw
http://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
Coupled with electric
field equation
Coupled with ion
transport equation
Coupled with level-set
equation
Magnetohydrodynamic ( MHD ) equations
Electrohydrodynamic ( EHD ) equations
Electrosmotic flow ( EOF ) equations
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 ) model

Free surface flow over an irregular obstacle
1. Nonlinear partial differential equations
(A) Westervelt equation ( one-manpower )
1
 p  2 ptt
c0
2
Linear wave


4
0
c
pttt
Absorption
contribution


2
(
p
)tt  0
4
0c0
Nonlinear
contribution
- 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
(B) Camassa-Holm equation ( 1/3 manpower )
ut  utxx  3uux  2uxuxx  uuxxx  0
Mixed
derivative
term
Three nonlinear terms
- Academic topics under investigation
 Resolve oscillations due to the highly dispersive term uuxxx
so as to capture the cusp ( peakon or soliton ) profile
 Clarify the debate if the dissipative behavior is present in the
peakon-antipeakon problem
 Preserve Hamiltonians embedded in the above equation
(C) Schrodinger equation ( one manpower )
i t  2  a | |2   0
- Academic topics under investigation
2
1 L
1 L
 Preserve the Hamiltonian H ( )  0 |  x | dx  0 a |  |4 dx
and the particle number
2
2
1
|  | dx
2 0
L
2
properties imbedded in
the above equation

Explore the time-evolving behavior of the momentum given by
1 L

Im    x dx 
2 0

2. Nonlinear system of Partial differential equations
(2.1) Incompressible Navier-Stokes ( NS ) equations
(A) Incompressible MHD equations ( 1/3 manpower )
Note : For electrically conducting fluids such as the plasma and
Liquid metal
* Hydrodynamic field equations
1
1

2
u

(
u


)
u



p



u

g

(  B)  B
 t



 u  0

* Magnetic field equations
 Bt    (u  B)    2 B

  B  0
Academic topics under current investigation
- enforce divergence-free condition   u  0 for the momentum equations
- enforce divergence-free condition   B  0 for the magnetic induction
equations
(B) Incompressible EHD equations ( one manpower )
Note : For electrically charged fluids
* Hydrodynamic field equations
1
1

2
u

(
u


)
u



p



u

g

qE
 t



  u  0

* Electric field equations
 qt    ( q ( E  u ))  0

E  

2


  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
(C) Incompressible EOF (電泳) equations ( 1/3 manpower )
Note : For the fluid with ion
* Hydrodynamic equations
e
1

2
u

(
u

)
u



p



u



(

(
T
)

u
)

E
 t



 u  0

* Energy equation
 c p (Tt  u  T )    (k (T )T ) 
1

u  e   E 
2
* Electrosmotic equations

  ( (T ) )  0

2 en0
 e

sinh(
)
  ( (T ) )  
0
kT


E  (   )

Academic topic under current investigation
- Simulation of the 3D large-scale EOF microchannel flow problem in
parallel CPU and GPU processors
(D) Incompressible Interface / free surface flow equations
( one man power )

1
1    2 ( ) D  1 K ( ) ( ) 1
p 

 2
 u t  (u )u  
 ( )
Re
 ( )
We
 ( )
Fr


  u  0
 t  u   0
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 )
(A)
Leray-α regularized model
1 2

u t  u u  p   u
Re


u  0
where
 u
2
1

u
u
2
(B) NS-α regularized model
(C) NS-ω regularized model
1 2

u t  u  (  u )  p  Re  u  f

 u  0

 2 u  u  u
1 2

u t  u  (  u )  p  Re  u  f

  u  0

 2 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
 H t     E

   H  0

  E  1  E
  t 
 
    E  0
Academic topics under current investigation
- Preserve Hamiltonians and conserved quantities
- Enforce Gauss law ( divergence-free conditions for E and H
, or  H   E  0 )
- Preserve symplecticity and energy in the above equations
- Optimize the numerical dispersion relation
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
Inspiration
Expiration
Inlet flow profile
Pressure contours
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