Discussion Forum

Hiii, I am going to introduced my model physics and setup first and error in solving this is discussed towards the end. I have made a bubble-blood-vessel 3-D model. This contains blood vessel (10 um diameter & 2um in thickness) filled with blood and surrounded by infinite blood. In the center of the blood is a bubble (4 um diameter). The bubble is in the form of spherical cavity inside the blood. I am applying the ultrasound wave to this model, which causes the bubble to oscillate. The oscillation of bubble induces displacement and stresses in the vessel. The bubble wall is given a normal velocity, which is calculated using the ideal gas equation in global ODE in COMSOL. The ideal gas equation used is: (p_0+2σ/(ρR_0 )) (R_0/R)^3k-2σ/R-(4ηRt ̇)/R-(4k_S Rt ̇)/R^2-p_0-p_a sin(2pift)=p_b p_0 is ambient pressure, ρ is density of blood, R_0 is bubble initial radius, R is bubble radius at any time, Rt bubble velocity at any time, p_a is ultrasound pressure, f is ultrasound frequency, p_b is pressure on bubble wall and σ, k, η, k_S are other constants.

The bubble wall normal velocity inside moving mesh settings is set equal to Rt. p_b is the change in pressure exerted by blood on bubble wall. p_b is calculated using the boundary probe for pressure and using it ensures double coupling between bubble and blood. At each time step, p_b from previous time step is used for solving equation.

The model is set up according to the paper " A three-dimensional model of an ultrasound contrast agent gas bubble and its mechanical effects on microvessels" by Nazanin Hosseinkhah (paper attached). The model is producing the desired result at 1MHz frequency. I am using "Automatic Newton" in non-linear settings for this problem. But, when the frequency is increased to 3.25 MHz, the model is not converging and giving error. 3.25 MHz is the resonance frequency of bubble for this setup, which results in high amplitude oscillation at this frequency. The model fails during simulation when pressure on bubble surface (p_b) reaches a high value (plot 2) in " 3.25 MHz & 2.5 Po(204um length) - FAILED ". The time step start decreasing at high pressure(p_b) and it finally gives following error. "Nonlinear solver did not converge. Maximum number of Newton iterations reached. Time : 7.2295012969103006e-07 Last time step is not converged." The model converged once when i tried fine meshing in " 3.25 MHz & 2.5 Po(204um length)", but the solver took 2 hours to solve. The model which converged has around 1,00,000 elements but in paper they are building model using much less elements(around 20,000). I think the problem is with meshing. I have tried changing meshing but it is not working. Can you suggest some better meshing technique or other changes for this model to converge easily in much less time??

Also, I am using boundary probe to feed pressure changes(p_b) on bubble surface at each time step back to the equation. Is there any other better method to feed pressure changes(p_b) back to the equation??

Thanks & Regards Rohit Singh rohitsingh@ku.edu