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![Page 1: Scientific computing topics under current study Tony W. H. Sheu Scientific Computing and Cardiovascular Simulation Lab. Computational Mechanics and Scientific.](https://reader036.fdocuments.us/reader036/viewer/2022062712/56649cb75503460f9497c83c/html5/thumbnails/1.jpg)
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: [email protected]://ccms.ntu.edu.tw/~twhsheu
Dec. 8. 2010 in NTU for the interaction of CQSE and TIMS colleagues
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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
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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
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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
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(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
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(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
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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
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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
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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
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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
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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
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(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
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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
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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
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4. Applications
(B) Inspiration / expiration in human lung airways
Velocity profile
Pressure contoursInlet flow profile
InspirationExpiration
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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.
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Temperature distribution on the calf due to moxibustion practice
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4. Applications
(D) Free surface flow over an irregular obstacle