Vanja Nikolić

Radboud University | RU · Department of Mathematics

Inspired by medical applications of high-intensity ultrasound we study a coupled elasto-acoustic problem with general acoustic nonlinearities of quadratic type as they arise, for example, in the Westervelt and Kuznetsov equations of nonlinear acoustics. We derive convergence rates in the energy norm of a finite element approximation to the coupled...

In this paper, we investigate a general class of quasilinear wave equations with nonlocal dissipation. We first motivate these in the context of nonlinear acoustics using heat flux laws of Gurtin--Pipkin type within the system of governing equations of sound propagation, as these are known to have finite propagation speeds. Two families of models a...

Accurate simulation of nonlinear acoustic waves is essential for the continued development of a wide range of (high-intensity) focused ultrasound applications. This article explores mixed finite element formulations of classical strongly damped quasilinear models of ultrasonic wave propagation; the Kuznetsov and Westervelt equations. Such formulati...

High-intensity focused ultrasound (HIFU) waves are known to induce localised heat to a targeted area during medical treatments. In turn, the rise in temperature influences their speed of propagation. This coupling affects the position of the focal region as well as the achieved pressure and temperature values. In this work, we investigate a mathema...

Heating generated by high-intensity focused ultrasound waves is central to many emerging medical applications, including non-invasive cancer therapy and targeted drug delivery. In this study, we aim to gain a fundamental understanding of numerical simulations in this context by analyzing conforming finite element approximations of the underlying no...

We propose and analyze a space--time finite element method for Westervelt's quasilinear model of ultrasound waves in second-order formulation. The method combines conforming finite element spatial discretizations with a discontinuous-continuous Galerkin time stepping. Its analysis is challenged by the fact that standard Galerkin testing approaches...

We investigate a nonlinear multiphysics model motivated by ultrasound-enhanced drug delivery. The acoustic pressure field is modeled by Westervelt's quasilinear wave equation to adequately capture the nonlinear effects in ultrasound propagation. The nonlocal attenuation characteristic for soft biological media is modeled by acoustic damping of the...

Motivated by numerical modeling of ultrasound waves, we investigate conforming finite element approximations of quasilinear and possibly nonlocal equations of Westervelt type. These wave equations involve either a strong dissipation or damping of fractional-derivative type and we unify them into one class by introducing a memory kernel that satisfi...

We investigate models for nonlinear ultrasound propagation in soft biological tissue based on the one that serves as the core for the software package k‐Wave. The systems are solved for the acoustic particle velocity, mass density, and acoustic pressure and involve a fractional absorption operator. We first consider a system that incorporates addit...

Ultrasound contrast imaging is a specialized imaging technique that applies microbubble contrast agents to traditional medical sonography, providing real-time visualization of blood flow and vessels. Gas-filled microbubbles are injected into the body, where they undergo compression and rarefaction and interact nonlinearly with the ultrasound waves....

Surfactants that are deposited at aqueous liquid films have the ability to generate surface tension gradients at the air-water interface, and thereby induce Marangoni flow. Combined with the production and depletion of surfactants at different locations of source and drains, out-of-equilibrium surface tension gradients can be sustained, resulting i...

We investigate models for nonlinear ultrasound propagation in soft biological tissue based on the one that serves as the core for the software package k-Wave. The systems are solved for the acoustic particle velocity, mass density, and acoustic pressure and involve a fractional absorption operator. We first consider a system that incorporates addit...

In this work, we investigate a class of quasilinear wave equations of Westervelt type with, in general, nonlocal-in-time dissipation. They arise as models of nonlinear sound propagation through complex media with anomalous diffusion of Gurtin–Pipkin type. Aiming at minimal assumptions on the involved memory kernels – which we allow to be weakly sin...

In ultrasonics, nonlocal quasilinear wave equations arise when taking into account a class of heat flux laws of Gurtin–Pipkin type within the system of governing equations of sound motion. The present study extends previous work by the authors to incorporate nonlocal acoustic wave equations with quadratic gradient nonlinearities which require a new...

When high-frequency sound waves travel through media with anomalous diffusion, such as biological tissues, their motion can be described by nonlinear acoustic equations of fractional higher order. In this work, we relate them to the classical second-order acoustic equations and, in this sense, justify them as their approximations for small relaxati...

The Kuznetsov equation is a classical wave model of acoustics that incorporates quadratic gradient nonlinearities. When its strong damping vanishes, it undergoes a singular behavior change, switching from a parabolic-like to a hyperbolic quasilinear evolution. In this work, we establish for the first time the optimal error bounds for its finite ele...

In ultrasonics, nonlocal quasilinear wave equations arise when taking into account a class of heat flux laws of Gurtin--Pipkin type within the system of governing equations of sound motion. The present study extends previous work by the authors to incorporate nonlocal acoustic wave equations with quadratic gradient nonlinearities which require a ne...

High frequencies at which ultrasonic waves travel give rise to nonlinear phenomena. In thermoviscous fluids, these are captured by Blackstock’s acoustic wave equation with strong damping. We revisit in this work its well-posedness analysis. By exploiting the parabolic-like character of this equation due to strong dissipation, we construct a time-we...

Motivated by numerical modeling of ultrasound waves, we investigate robust conforming finite element discretizations of quasilinear and possibly nonlocal equations of Westervelt type. These wave equations involve either a strong dissipation or damping of fractional-derivative type and we unify them into one class by introducing a memory kernel that...

When high-frequency sound waves travel through media with anomalous diffusion, such as biological tissues, their motion can be described by nonlinear wave equations of fractional higher order. These can be understood as nonlocal generalizations of the Jordan-Moore-Gibson-Thompson equations in nonlinear acoustics. In this work, we relate them to the...

High frequencies at which ultrasonic waves travel give rise to nonlinear phenomena. In thermoviscous fluids, these are captured by Blackstock's acoustic wave equation with strong damping. We revisit in this work its well-posedness analysis. By exploiting the parabolic-like character of this equation due to strong dissipation, we construct a time-we...

In various biomedical applications, precise focusing of nonlinear ultrasonic waves is crucial for efficiency and safety of the involved procedures. This work analyzes a class of shape optimization problems constrained by general quasi-linear acoustic wave equations that arise in high-intensity focused ultrasound (HIFU) applications. Within our theo...

In this work, we investigate the global well-posedness and asymptotic behavior of a mathematical model of ultrasound-induced heating based on a coupled system of Westervelt’s nonlinear acoustic wave equation and Pennes bioheat equation. To this end, under Dirichlet–Dirichlet boundary conditions, we prove global existence for sufficiently small and...

In this work, we investigate the inverse problem of determining the kernel functions that best describe the mechanical behavior of a complex medium modeled by a general nonlocal viscoelastic wave equation. To this end, we minimize a tracking-type data misfit function under this PDE constraint. We perform the well-posedness analysis of the state and...

In this paper, we consider several time-fractional generalizations of the Jordan–Moore–Gibson–Thompson (JMGT) equations in nonlinear acoustics as well as their linear Moore–Gibson–Thompson (MGT) versions. Following the procedure described in Jordan (2014), these time-fractional acoustic equations are derived from four fractional versions of the Max...

In this work, we investigate the inverse problem of determining the kernel functions that best describe the mechanical behavior of a complex medium modeled by a general nonlocal viscoelastic wave equation. To this end, we minimize a tracking-type data misfit function under this PDE constraint. We perform the well-posedness analysis of the state and...

Nonlinear sound propagation through media with thermal and molecular relaxation can be modeled by third-order in time wave-like equations with memory. We investigate the asymptotic behavior of a Cauchy problem for such a model, the nonlocal Jordan–Moore–Gibson–Thompson equation, in the so-called critical case, which corresponds to propagation throu...

We investigate the problem of finding the optimal shape and topology of a system of acoustic lenses in a dissipative medium. The sound propagation is governed by a general semilinear strongly damped wave equation. We introduce a phase-field formulation of this problem through diffuse interfaces between the lenses and the surrounding fluid. The resu...

In various biomedical applications, precise focusing of nonlinear ultrasonic waves is crucial for efficiency and safety of the involved procedures. This work analyzes a class of shape optimization problems constrained by general quasi-linear acoustic wave equations that arise in high-intensity focused ultrasound (HIFU) applications. Within our theo...

High-Intensity Focused Ultrasound (HIFU) waves are known to induce localized heat to a targeted area during medical treatments. In turn, the rise in temperature influences their speed of propagation. This coupling affects the position of the focal region as well as the achieved pressure and temperature values. In this work, we investigate a mathema...

In this paper, we consider several time-fractional generalizations of the Jordan-Moore-Gibson-Thompson (JMGT) equations in nonlinear acoustics as well as their linear Moore-Gibson-Thompson (MGT) versions. Following the procedure described in Jordan (2014), these time-fractional acoustic equations are derived from four fractional versions of the Max...

Inspired by medical applications of high-intensity ultrasound, we study a coupled elasto-acoustic problem with general acoustic nonlinearities of quadratic type as they arise, for example, in the Westervelt and Kuznetsov equations of nonlinear acoustics. We derive convergence rates in the energy norm of a finite element approximation to the coupled...

We prove global solvability of the third-order in time Jordan-More-Gibson-Thompson acoustic wave equation with memory in $\mathbb{R}^n$, where $n \geq 3$. This wave equation models ultrasonic propagation in relaxing hereditary fluids and incorporates both local and cumulative nonlinear effects. The proof of global solvability is based on a sequence...

We analyze the behavior of third-order in time linear and nonlinear sound waves in thermally relaxing fluids and gases as the sound diffusivity vanishes. The nonlinear acoustic propagation is modeled by the Jordan--Moore--Gibson--Thompson equation both in its Westervelt and in its Kuznetsov-type forms, that is, including quadratic nonlinearities of...

Ultrasonic propagation through media with thermal and molecular relaxation can be modeled by third-order in time nonlinear wave-like equations with memory. This paper investigates the asymptotic behavior of a Cauchy problem for such a model, the nonlocal Jordan--Moore--Gibson--Thompson equation, in the so-called critical case, which corresponds to...

This work deals with the convergence analysis of parabolic perturbations to quasilinear wave equations on smooth bounded domains. In particular, we consider wave equations with nonlinearities of quadratic type, which cover the two classical models of nonlinear acoustics, the Westervelt and Kuznetsov equations. By employing a high-order energy analy...

We prove global solvability of the third-order in time Jordan-More-Gibson-Thompson acoustic wave equation with memory in $\mathbb{R}^n$, where $n \geq 3$. This wave equation models ultrasonic propagation in relaxing hereditary fluids and incorporates both local and cumulative nonlinear effects. The proof of global solvability is based on a sequence...

We propose a high-order discontinuous Galerkin scheme for nonlinear acoustic waves on polytopic meshes. To model sound propagation with losses through homogeneous media, we use Westervelt's nonlinear wave equation with strong damping. Challenges in the numerical analysis lie in handling the nonlinearity in the model, which involves the derivatives...

Motivated by the propagation of nonlinear sound waves through relaxing hereditary media, we study a nonlocal third-order Jordan-Moore-Gibson-Thompson acoustic wave equation. Under the assumption that the relaxation kernel decays exponentially, we prove local well-posedness in unbounded two- and three-dimensional domains. In addition, we show that t...

We propose a high-order discontinuous Galerkin scheme for nonlinear acoustic waves on polytopic meshes. To model sound propagation with and without losses, we use Westervelt's nonlinear wave equation with and without strong damping. Challenges in the numerical analysis lie in handling the nonlinearity in the model, which involves the derivatives in...

We present and analyze new multi-species phase-field mathematical models of tumor growth and ECM invasion. The local and nonlocal mathematical models describe the evolution of volume fractions of tumor cells, viable cells (proliferative and hypoxic cells), necrotic cells, and the evolution of matrix-degenerative enzyme (MDE) and extracellular matri...

In this paper, we consider the Jordan–Moore–Gibson–Thompson equation, a third-order in time wave equation describing the nonlinear propagation of sound that avoids the infinite signal speed paradox of classical second-order in time strongly damped models of nonlinear acoustics, such as the Westervelt and the Kuznetsov equation. We show well-posedne...

We present and analyze new multi-species phase-field mathematical models of tumor growth and ECM invasion. The local and nonlocal mathematical models describe the evolution of volume fractions of tumor cells, viable cells (proliferative and hypoxic cells), necrotic cells, and the evolution of MDE and ECM, together with chemotaxis, haptotaxis, apopt...

We propose a self-adaptive absorbing technique for quasilinear ultrasound waves in two- and three-dimensional computational domains. As a model for the nonlinear ultrasound propagation in thermoviscous fluids, we employ Westervelt's wave equation solved for the acoustic velocity potential. The angle of incidence of the wave is computed based on the...

We study the Jordan--Moore--Gibson--Thompson (JMGT) equation, a third order in time wave equation that models nonlinear sound propagation, in the practically relevant setting of Neumann and absorbing boundary conditions. In the analysis, we pay special attention to dependencies on the coefficient $\tau$ of the third order time derivative that plays...

We study the spatial discretization of Westervelt's quasilinear strongly damped wave equation by piecewise linear finite elements. Our approach employs the Banach fixed-point theorem combined with a priori analysis of a linear wave model with variable coefficients. Degeneracy of the semi-discrete Westervelt equation is avoided by relying on the inv...

In this paper, we consider the Jordan-Moore-Gibson-Thompson equation, a third order in time wave equation describing the nonlinear propagation of sound that avoids the infinite signal speed paradox of classical second order in time strongly damped models of nonlinear acoustics, such as the Westervelt and the Kuznetsov equation. We show well-posedne...

We propose a self-adaptive absorbing technique for quasilinear ultrasound waves in two- and three-dimensional computational domains. As a model for the nonlinear ultrasound propagation in thermoviscous fluids, we employ Westervelt's wave equation solved for the acoustic velocity potential. The angle of incidence of the wave is computed based on the...

We study the Blackstock equation which models the propagation of nonlinear sound waves through dissipative fluids. Global well-posedness of the model with homogeneous Dirichlet boundary conditions is shown for small initial data. To this end, we employ a fixed-point technique coupled with well-posedness results for a linearized model and appropriat...

We study the Blackstock equation which models the propagation of nonlinear sound waves through dissipative fluids. Global well-posedness of the model with homogeneous Dirichlet boundary conditions is shown for small initial data. To this end, we employ a fixed-point technique coupled with well-posedness results for a linearized model and appropriat...

The goal of this work is to improve focusing of high-intensity ultrasound by modifying the geometry of acoustic lenses through shape optimization. The shape optimization problem is formulated by introducing a tracking-type cost functional to match a desired pressure distribution in the focal region. Westervelt's equation, a nonlinear acoustic wave...

The goal of this work is to improve focusing of high-intensity ultrasound by modifying the geometry of acoustic lenses through shape optimization. The shape optimization problem is formulated by introducing a tracking-type cost functional to match a desired pressure distribution in the focal region. Westervelt's equation, a nonlinear acoustic wave...

We are interested in shape sensitivity analysis for an optimization problem arising in medical applications of high intensity focused ultrasound. The goal is to find the optimal shape of a focusing acoustic lens so that the desired acoustic pressure at a kidney stone is achieved. Coupling of the silicone acoustic lens and nonlinearly acoustic fluid...

We show higher interior regularity for the Westervelt equation with strong nonlinear damping term of the $q$-Laplace type. Secondly, we investigate an interface coupling problem for these models, which arise, e.g., in the context of medical applications of high intensity focused ultrasound in the treatment of kidney stones. We show that the solutio...

We show higher interior regularity for the Westervelt equation with strong nonlinear damping term of the $q$-Laplace type. Secondly, we investigate an interface coupling problem for these models, which arise, e.g., in the context of medical applications of high intensity focused ultrasound in the treatment of kidney stones. We show that the solutio...

We investigate the Westervelt equation with several versions of nonlinear damping and lower order damping terms and Neumann as well as absorbing boundary conditions. We prove local in time existence of weak solutions under the assumption that the initial and boundary data are sufficiently small. Additionally, we prove local well-posedness in the ca...

Efficient time integration methods based on operator splitting are introduced for the Westervelt equation, a nonlinear damped wave equation that arises in nonlinear acoustics as mathematical model for the propagation of sound waves in high intensity ultrasound applications. For the first-order Lie-Trotter splitting method a global error estimate is...