arXiv subjects

Vanja Nikolić

Publications and source records attributed to Vanja Nikolić.

At least 19 recordsLinked to original sources

arXiv2026

Discontinuous Galerkin approximations of the Jordan-Moore-Gibson-Thompson equation in the vanishing relaxation limit

The Jordan-Moore-Gibson-Thompson (JMGT) equation models nonlinear acoustic wave propagation in thermally relaxing media and in the vanishing relaxation limit approaches the damped Westervelt equation. We investigate discontinuous Galerkin spatial discretizations of the JMGT equation on simplicial meshes and analyze their behavior uniformly with respect to the relaxation parameter. Under practically relevant mixed Neumann and absorbing boundary conditions, we derive a priori error estimates independent of the relaxation parameter. These estimates enable a rigorous singular limit analysis, yielding convergence of the semi-discrete JMGT approximations to the corresponding Westervelt pressure profile at a linear rate. This also sheds light on the expected behavior of exact solutions in the vanishing relaxation limit. For the fully discrete problem, we propose a Newmark-type method based on a reformulation as a coupled second-/first-order system. Numerical experiments support the theoretical findings and demonstrate the robustness of the approach in the small-parameter regime.

math.NA↗

arXiv2026

Discontinuous Galerkin approximation of a nonlinear multiphysics problem arising in ultrasound-enhanced drug delivery

Motivated by simulations of ultrasound-enhanced drug delivery, this work presents the numerical analysis of a mathematical model that captures the influence of ultrasound waves on the diffusivity of the drug. The system under study consists of the Westervelt wave equation, accounting for the nonlinear propagation of ultrasound, coupled to a convection-diffusion equation modeling the drug concentration. In particular, drug delivery is affected by ultrasound through a pressure-dependent diffusion coefficient. The Westervelt equation is supplemented by linear absorbing boundary conditions as a means of reducing spurious reflections off the boundaries of computational domains. For spatial discretization of this multiphysics system, we employ a discontinuous Galerkin approach on simplicial meshes. Under suitable assumptions on the exact pressure and the mesh size, we first establish well-posedness, non-degeneracy, and optimal convergence rates in the energy norm for the semi-discrete pressure subproblem. The smallness of the semi-discrete pressure is then used to establish the well-posedness and convergence of the wave--convection-diffusion system under suitable regularity of the exact concentration. Finally, theoretical findings are illustrated through numerical experiments.

math.NA↗

arXiv2026

Multiharmonic algorithms for contrast-enhanced ultrasound

Harmonic generation plays a crucial role in contrast-enhanced ultrasound, both for imaging and therapeutic applications. However, accurately capturing these nonlinear effects is computationally very demanding when using traditional time-domain approaches. To address this issue, in this work, we develop algorithms based on a time discretization that uses a multiharmonic Ansatz applied to a model that couples the Westervelt equation for acoustic pressure with a volume-based approximation of the Rayleigh--Plesset equation for the dynamics of microbubble contrast agents. We first rigorously establish the existence of time-periodic solutions for this Westervelt-ODE system. We then derive a multiharmonic representation of the system under time-periodic excitation and develop iterative algorithms that rely on the successive computation of higher harmonics under the assumption of real-valued or complex solution fields. In the real-valued setting, we characterize the approximation error in terms of the number of harmonics and a contribution owing to the fixed-point iteration. Finally, we investigate these algorithms numerically and illustrate how the number of harmonics and presence of microbubbles influence the propagation of acoustic waves.

math.NA↗

arXiv2025

Optimal Neumann boundary and distributed control of the Westervelt equation with time-fractional attenuation

Optimal control of nonlinear acoustic waves is relevant in many medical ultrasound technologies, ranging from cancer therapy to targeted drug delivery, where it can help guide the precise deposition of acoustic energy. In this work, we study Neumann boundary and distributed control problems for tracking a prescribed pressure field governed by the Westervelt equation with time-fractional dissipation. This model captures nonlinear ultrasonic wave propagation in biological media and accounts for the experimentally observed power-law attenuation. We begin by extending the existing well-posedness theory for time-fractional equations to include inhomogeneous Neumann boundary data used as control inputs, which requires constructing an appropriate data extension and regularization. Using these analytical results for the forward problem, we prove the existence of globally optimal controls and analyze the stability of the optimization problem with respect to perturbations in the target pressure field and to vanishing regularization parameters. Finally, we investigate the associated adjoint equation, which has state-dependent coefficients, and use it to derive first-order necessary optimality conditions.

math.OC↗

arXiv2025

Finite element discretization of nonlinear models of ultrasound heating

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 nonlinear models that describe ultrasound-heat interactions. These models are based on a coupling of a nonlinear Westervelt--Kuznetsov acoustic wave equation to the heat equation with a pressure-dependent source term. A particular challenging feature of the system is that the acoustic medium parameters may depend on the temperature. The core of our new arguments in the \emph{a prior} error analysis lies in devising energy estimates for the coupled semi-discrete system that can accommodate the nonlinearities present in the model. To derive them, we exploit the parabolic nature of the system thanks to the strong damping present in the acoustic component. Theoretically obtained optimal convergence rates in the energy norm are confirmed by the numerical experiments. In addition, we conduct a further numerical study of the problem, where we simulate the propagation of acoustic waves in liver tissue for an initially excited profile and under high-frequency sources.

math.NA↗

arXiv2025

Robust fully discrete error bounds for the Kuznetsov equation in the inviscid limit

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 element approximation as well as a semi-implicit fully discrete approximation that are robust with respect to the vanishing damping parameter. The core of the new arguments lies in devising energy estimates directly for the error equation where one can more easily exploit the polynomial structure of the nonlinearities and compensate inverse estimates with smallness conditions on the error. Numerical experiments are included to illustrate the theoretical results.

math.NA↗

arXiv2025

Combined DG-CG finite element method for the Westervelt equation

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 for wave problems do not allow for bounding the discrete energy at all times. By means of redesigned energy arguments for a linearized problem combined with Banach's fixed-point argument, we show the well-posedness of the scheme, \emph{a priori} error estimates, and robustness with respect to the strong damping parameter $δ$. Moreover, the scheme preserves the asymptotic preserving property of the continuous problem; more precisely, we prove that the discrete solutions corresponding to $δ>0$ converge, in the singular vanishing dissipation limit, to the solution of the discrete inviscid problem. We use several numerical experiments in $(2 + 1)$ dimensions to validate our theoretical results.

math.NA↗

arXiv2025

Homotopy trust-region method for phase-field approximations in perimeter-regularized binary optimal control

We consider optimal control problems that have binary-valued control input functions and a perimeter regularization. We develop and analyze a trust-region algorithm that solves a sequence of subproblems in which the regularization term and the binarity constraint are relaxed by a non-convex energy functional. We show how the parameter that controls the distinctiveness of the resulting phase field can be coupled to the trust-region radius updates and be driven to zero over the course of the iterations in order to obtain convergence to stationary points of the limit problem under suitable regularity assumptions. Finally, we highlight and discuss the assumptions and restrictions of our approach and provide the first computational results for a motivating application in the field of control of acoustic waves in dissipative media.

math.OC↗

arXiv2024

Mathematical models for nonlinear ultrasound contrast imaging with microbubbles

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. Therefore, the propagation of sound through a bubbly liquid is a strongly nonlinear problem that can be modeled by a nonlinear acoustic wave equation for the propagation of the pressure waves coupled via the source terms to a nonlinear ordinary differential equation of Rayleigh-Plesset type for the bubble dynamics. In this work, we first derive a hierarchy of such coupled models based on constitutive laws. We then focus on the coupling of Westervelt's acoustic equation to Rayleigh-Plesset type equations, where we rigorously show the existence of solutions locally in time under suitable conditions on the initial pressure-microbubble data and final time. Thirdly, we devise and discuss numerical experiments on both single-bubble dynamics and the interaction of microbubbles with ultrasound waves.

math.AP↗

arXiv2024

Westervelt-based modeling of ultrasound-enhanced drug delivery

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 time-fractional type. Additionally, acoustic medium parameters are allowed to depend on the temperature of the medium. The wave equation is coupled to the nonlinear Pennes heat equation with a pressure-dependent source to account for ultrasound waves heating up the tissue. Finally, the drug concentration is obtained as the solution to an advection-diffusion equation with a pressure-dependent velocity. Toward gaining a rigorous understanding of this system, we set up a fixed-point argument in the analysis combined with devising energy estimates that can accommodate the time-fractional damping. The energy arguments are, in part, carried out by employing time-weighted test functions to reduce the regularity assumptions on the initial temperature. The analysis reveals that different smoothness of the initial pressure, temperature, and concentration fields is needed as well as smallness of the pressure-temperature data in order to ensure non-degeneracy of the system and establish well-posedness. Our theoretical considerations are complemented by a numerical investigation of the system under more realistic boundary conditions. The numerical experiments, performed in different computational scenarios, underline the importance of considering nonlinear effects when modeling ultrasound-targeted drug delivery.

math.AP↗

arXiv2024

Asymptotic-preserving finite element analysis of Westervelt-type wave equations

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 satisfies non-restrictive regularity and positivity assumptions. As the involved damping parameter is relatively small and can become negligible in certain (inviscid) media, it is important to develop methods that remain stable as the said parameter vanishes. To this end, the contributions of this work are twofold. First, we determine sufficient conditions under which conforming finite element discretizations of (non)local Westervelt equations can be made robust with respect to the dissipation parameter. Secondly, we establish the rate of convergence of the semi-discrete solutions in the singular vanishing dissipation limit. The analysis hinges upon devising appropriate energy functionals for the semi-discrete solutions that remain uniformly bounded with respect to the damping parameter.

math.NA↗

arXiv2024

Application of the thin-film equations in modelling of Marangoni flow patterns amongst surfactant source and drain locations

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 in Marangoni flow patterns that drive e.g., self-organization of amphiphile myelin assemblies. Here, a mathematical model based on the thin-film equations is proposed to simulate these flow patterns. The model equations are based on the surfactant source and drain concentrations, film-height and surfactant bulk concentration. We present a numerical scheme for approximating the model equations and discuss the numerically observed properties of the model.

physics.flu-dyn↗

arXiv2024

$L^\infty$ blow-up in the Jordan-Moore-Gibson-Thompson equation

The Jordan-Moore-Gibson-Thompson equation \[ τu_{ttt} + αu_{tt} = βΔu_t + γΔu + (f(u))_{tt} \] is considered in a smoothly bounded domain $Ω\subset\mathbb{R}^n$ with $n\leq 3$, where $τ>0,β>0,γ>0$, and $α\in\mathbb{R}$. Firstly, it is seen that under the assumption that $f\in C^3(\mathbb{R})$ is such that $f(0)=0$, gradient blow-up phenomena cannot occur in the sense that for any appropriately regular initial data, within a suitable framework of strong solvability, an associated Dirichlet type initial-boundary value problem admits a unique solution $u$ on a maximal time interval $(0,T_{max})$ which is such that \[ \mbox{if $T_{max}<\infty$, then } \limsup_{t\nearrow T_{max}} \|u(\cdot,t)\|_{L^\infty(Ω)}=\infty. \] This is used to, secondly, make sure that if additionally $f$ is convex and grows superlinearly in the sense that \[ f''\ge 0 \mbox{ on $\mathbb{R}$,} \qquad \frac{f(ξ)}ξ \to +\infty \mbox{ as $ξ\to +\infty$} \qquad \mbox{and} \qquad \int_{ξ_0}^\infty \frac{dξ}{f(ξ)} < \infty \mbox{ for some $ξ_0>0$,} \] then for some initial data the above solution must undergo some finite-time $L^\infty$ blow-up in the style described above.

math.AP↗

arXiv2024

Existence of solutions to k-Wave models of nonlinear ultrasound propagation in biological tissue

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 additional viscosity in the equation for momentum conservation. By constructing a Galerkin approximation procedure, we prove the local existence of its solutions. In view of inverse problems arising from imaging tasks, the theory allows for the variable background mass density, speed of sound, and the nonlinearity parameter in the systems. Secondly, under stronger conditions on the data, we take the vanishing viscosity limit of the problem, thereby rigorously establishing the existence of solutions for the limiting system as well.

math.AP↗

arXiv2024

Limiting behavior of quasilinear wave equations with fractional-type dissipation

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 singular -- we prove the well-posedness of such wave equations in a general theoretical framework. In particular, the Abel fractional kernels, as well as Mittag-Leffler-type kernels, are covered by our results. The analysis is carried out uniformly with respect to the small involved parameter on which the kernels depend and which can be physically interpreted as the sound diffusivity or the thermal relaxation time. We then analyze the behavior of solutions as this parameter vanishes, and in this way relate the equations to their limiting counterparts. To establish the limiting problems, we distinguish among different classes of kernels and analyze and discuss all ensuing cases.

math.AP↗

arXiv2023

Nonlinear acoustic equations of fractional higher order at the singular limit

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 classical second-order acoustic equations and, in this sense, justify them as their approximations for small relaxation times. To this end, we perform the singular limit analysis for a class of corresponding nonlocal wave models and determine their behavior as the relaxation time tends to zero. We show that, depending on the nonlinearities and assumptions on the data, these models can be seen as approximations of the Westervelt, Blackstock, or Kuznetsov wave equations. The analysis rests upon the uniform bounds for the solutions of the equations with fractional higher-order derivatives, obtained through a testing procedure tailored to the coercivity property of the involved (weakly) singular memory kernels.

math.AP↗

arXiv2023

The vanishing relaxation time behavior of multi-term nonlocal Jordan-Moore-Gibson-Thompson equations

The family of Jordan-Moore-Gibson-Thompson (JMGT) equations arises in nonlinear acoustics when a relaxed version of the heat flux law is employed within the system of governing equations of sound motion. Motivated by the propagation of sound waves in complex media with anomalous diffusion, we consider here a generalized class of such equations involving two (weakly) singular memory kernels in the principal and non-leading terms. To relate them to the second-order wave equations, we investigate their vanishing relaxation time behavior. The key component of this singular limit analysis are the uniform bounds for the solutions of these nonlinear equations of fractional type with respect to the relaxation time. Their availability turns out to depend not only on the regularity and coercivity properties of the two kernels, but also on their behavior relative to each other and the type of nonlinearity present in the equations.

math.AP↗

arXiv2022

Time-weighted estimates for the Blackstock equation in nonlinear ultrasonics

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-weighted energy framework for investigating its local solvability. In this manner, we obtain the small-data well-posedness on bounded domains under less restrictive regularity assumptions on the initial conditions compared to the known results. Furthermore, we prove that such initial boundary-value problems for the Blackstock equation are globally solvable and that their solution decays exponentially fast to the steady state.

math.AP↗