GPU-accelerated study of the inertial cavitation threshold in viscoelastic soft tissue using a dual-frequency driving signal

Numerical investigation of the translational motion of bubbles: The comparison of capabilities of the time-resolved and the time-averaged methods

“…From the available initial value problem solvers, the RKCK45 method was chosen, which is based on the fourth-order Runge–Kutta–Cash–Karp method with fifth-order error estimation. The high processing power of GPUs enables to carry out a considerable amount of numerical simulations within a reasonable time [100] , [101] , [102] .…”

Weakly nonlinear focused ultrasound in viscoelastic media containing multiple bubbles

“…(10 ), and into Eq. (4) to obtain the following: The velocity and displacement can be related as follows: where the Lagrange derivative is the differential operator The deviatoric stress tensor is based on the Zener viscoelastic model: [41] , [42] , [45] , [46] , [43] , [44] , [47] where is the relaxation time of the liquid phase, is the velocity gradient tensor, and is the deformation gradient tensor. When , Eq.…”

“…The subscript 0 represents initial values. Note that the Keller–Miksis equation can be used for cases in which the Mach number is less than one [46] , [47] , and the Gilmore–Akulichev model [46] , [47] can be used with larger Mach numbers. The variable is given by where is the distance from the center of each bubble.…”

“…Newtonian fluid is the most famous model in the field of fluid dynamics, but only viscosity is considered. The Kelvin–Voigt [32] , [33] , [34] , [35] , [36] , [38] , [19] , [37] , [39] , [30] , [31] and Zener models [41] , [42] , [45] , [46] , [43] , [44] , [47] have been used as mathematical models of liquid with viscoelasticity. The Kelvin–Voigt model corresponds to the generalized Newtonian fluid with the elasticity introduced while the relaxation effect is omitted.…”

“…Zilonova et al [46] investigated the interaction of two bubbles with the effects of drag force and translation based on the Zener model and Gilmore–Akulichev equation [53] . Gilmore–Akulichev equation has broader applications than the Keller–Miksis equation [46] , [47] . Filonets and Solovchuk [47] investigated the single bubble behavior excited by a dual-frequency ultrasound based on the Zener model and Gilmore–Akulichev equation, and they showed that the elasticity of a liquid increases the threshold pressure of bubble collapse.…”