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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

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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 electric field equation Electrohydrodynamic ( EHD ) equations Coupled with ion transport equation Electrosmotic flow ( EOF ) equations Coupled with level-set equation 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 ) model 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 Linear wave Absorption contribution 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 peakon-antipeakon problem Preserve Hamiltonians embedded in the above equation Mixed derivative term Three nonlinear terms

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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

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2. Nonlinear system of Partial differential equations (A) Incompressible MHD equations ( 1/3 manpower ) * 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 (2.1) Incompressible Navier-Stokes ( NS ) equations Note : For electrically conducting fluids such as the plasma and Liquid metal

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(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 Note : For electrically charged fluids

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

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* 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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(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 ) (A) Leray-α regularized model where (B) NS-α regularized model (C) NS-ω regularized model 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 ) 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

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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 contours Inlet flow profile Inspiration Expiration

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4. Applications (C) Construction of a cupuncture ( 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

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