The pressure-based Westervelt equation is given by (Dirkse [444]):

(8–277)

where:

= pressure

= diffusivity of sound, which is related to the dynamic viscosity , the bulk viscosity , the thermal conductivity , the specific heat capacity at constant volume , and the specific heat capacity at constant pressure as follows:

= dimensionless nonlinearity coefficient, which is related to the nonlinearity parameter as follows:

= mass density of fluid

= speed of sound

= mass source rate

The velocity potential-based Kuznetsov equation is given by (Hoffelner and Kaltenbacher [445]):

(8–278)

where:

= velocity potential

= mass source

The acoustic pressure and particle velocity are related to the scalar velocity potential by

and

The finite element formulations are obtained by testing Equation 8–277 and Equation 8–278 using the Galerkin procedure.

Multiplying the Westervelt equation (Equation 8–277) by a testing function and integrating over the volume of the domain with some manipulation yield the following:

(8–279)

The nonlinear matrix equation is given by:

(8–280)

where:

Since is on the order of 10^-10, the second surface integral in Equation 8–279 is neglected if the surface is not rigid.

Multiplying the Kuznetsov equation Equation 8–278 by a testing function and integrating over the volume of the domain with some manipulation yield the following:

(8–281)

The nonlinear matrix equation is given by:

(8–282)

where:

The nonlinear acoustics equations are iteratively solved using the Newmark method (see Transient Analysis). The right hand side of Equation 8–280 or Equation 8–282 is iteratively updated with the solved degree of freedom and its first and second order time derivatives until convergence is achieved.

For harmonic analysis to resolve the nonlinear acoustic equation, the Westervelt equation needs to be converted into a different format. The multiharmonic expansion is utilized to illustrate contributions coming from each harmonic, as shown in the following equation (Du and Jensen [446]).

(8–283)

By substituting this equation into the Westervelt equation and performing a few mathematical operations, the following cascaded equation is formed.

(8–284)

where is the maximum requested number of harmonics and represents the Kronecker delta function, which equals 1 when and 0 otherwise. Multiplying the Westervelt equation (Equation 8–284) by a testing function and integrating over the volume of the domain with some manipulation yield the following:

(8–285)

The nonlinear matrix equation is given by:

(8–286)

where:

The fixed-point iteration method is used to solve the nonlinear acoustic equations iteratively. In the first iteration, the initial degree of freedom is solved for each harmonic (from fundamental frequency to desired maximum frequency) by neglecting the contribution coming from higher harmonics, which is the second part of the nonlinear source vector, . In subsequent iterations, by including that contribution, the solution is sought through an iterative process.

The coupling conditions between the fluid and structure domains are given by Equation 8–150 and Equation 8–151.

For the Westervelt equation, the coupled matrix equations are:

(8–287)

For the definitions of parameters and matrices, see Equation 8–152 - Equation 8–157 and Equation 8–280.

For the Kuznetsov equation, the coupled matrix equations are:

(8–288)

For the definitions of parameters and matrices, see Equation 8–282.

In equations Equation 8–287 and Equation 8–288:

is the coupling matrix is given by:

The surface integral with the factor is neglected on the FSI interface.

The Westervelt equation, with the pressure as a degree of freedom, can be employed to simulate the scattered signal produced by the nonlinear interaction of the primary ultrasound waves at the difference frequency in harmonic analysis.

The difference frequency, denoted as , is related to the interaction of two primary frequencies, and , through a nonlinear process. can be expressed as:

(8–289)

In the context of the Westervelt equation, the difference frequency arises due to the nonlinear term in the equation. This term accounts for the generation of harmonics and difference frequencies as the sound wave propagates through the medium. Similarly, in the harmonic implementation of the Westervelt equation, a harmonic expansion is employed to derive the equation for wave propagation at the difference frequency. The following linearized equation governs the computation of pressure .

(8–290)

where and represent the pressure values obtained from linear equations at frequencies and respectively. Following this cascaded approach, the first step involves computing and . Subsequently, the obtained values are utilized to form the source vector on the right-hand side of the equation mentioned above.