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Scientific computing topics under current study

Explore the study of nonlinear partial differential equations, including topics such as the Westervelt equation, Camassa-Holm equation, and Schrodinger equation. Also examine applications in areas such as liver tumor ablation, lung airway dynamics, and acupuncture modeling.

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Scientific computing topics under current study

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  1. 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

  2. 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 electric field equation Electrohydrodynamic ( EHD ) equations Coupled with ion transport equation Electrosmotic flow ( EOF ) equations Free ( interface ) flow equations Coupled with level-set equation

  3. 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

  4. 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 Linear wave Absorption contribution Nonlinear contribution

  5. (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 peakon-antipeakon problem • Preserve Hamiltonians embedded in the above equation Mixed derivative term Three nonlinear terms

  6. (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

  7. 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 * Magnetic field equations Academic topics under current investigation - enforce divergence-free condition for the momentum equations - enforce divergence-free condition for the magnetic induction equations

  8. Note : For electrically charged fluids (B) Incompressible EHD equations ( one manpower ) * Hydrodynamic field equations * Electric field equations 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

  9. (C) Incompressible EOF(電泳) equations ( 1/3 manpower ) Note : For the fluid with ion * Hydrodynamic equations * Energy equation

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

  11. (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

  12. (2.2) Subgrid turbulence models ( one manpower ) (A) Leray-α regularized model where (C) NS-ω regularized model (B) NS-α regularized model Academic topic under current investigation - Examine how well these regularized NS equations can be applied to model flow turbulence

  13. 3. Maxwell equations ( two manpower ) 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

  14. 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

  15. 4. Applications (B) Inspiration / expiration in human lung airways Velocity profile Expiration Inspiration Inlet flow profile Pressure contours

  16. 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.

  17. Temperature distribution on the calf due to moxibustion practice

  18. 4. Applications (D) Free surface flow over an irregular obstacle

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