Nonlinear Heat Source Research Articles

Three-Dimensional Non-Linear Complex Model of Dynamic Memristor Switching

A thermodynamical model of filament growing when a current pulse via memristor flows is introduced. The model is the boundary value problem, which includes nonstationary heat conduction equation with non-linear Joule heat source, Poisson equation, and Shockley-Read-Hall equations taking into account strong electron-phonon interactions in trap ionization and charge transport processes. The charge current, which defines the heating in the model, depends on the rate of the oxygen vacancy generation. The latter depends on the local temperature. The solution of the introduced problem allows one to describe the kinetics of the switch process and the final filament morphology.

Inverse heat transfer analysis for design and control of a micro-heater array

A non-linear inverse heat source identification problem is described and solved. The inverse problem analysis is used in the design of an embedded micro-heater array and to estimate the required control settings, which are the input currents to each heating element, to generate as close as possible to a prescribed temperature profile on the surface of a thin copper film. The purpose of the micro-heater array is to control the local copper microstructure through control of the local temperature field. A finite element model of the micro-heater system is used to define a discrete set of non-linear equations used as a basis for the inverse problem solution. Two methods are explored to solve the inverse problem, a direct minimization method with Tikhonov regularization and a passivity-based feedback control algorithm. A uniform and a linear temperature distribution could be attained in the central region above the micro-heater array, but the temperatures near the edges of the domain could not be controlled due to heat loss at the edges. Thus, to control the temperature field over the full width of the domain, the heater array must extend beyond the domain of interest. Both methods to solve the inverse problem are found to perform well. The regularization method allows for a smoother solution, while the feedback control method is simpler as the coefficient matrix for which the update remains unchanged for each iteration.

Non-linear memristor switching model

We introduce a thermodynamical model of filament growing when a current pulse via memristor flows. The model is the boundary value problem, which includes nonstationary heat conduction equation with non-linear Joule heat source, Poisson equation, and Shockley- Read-Hall equations taking into account strong electron-phonon interactions in trap ionization and charge transport processes. The charge current, which defines the heating in the model, depends on the rate of the oxygen vacancy generation. The latter depends on the local temperature. The solution of the introduced problem allows one to describe the kinetics of the switch process and the final filament morphology.

Three-Dimensional Non-Linear Complex Model of Dynamic Memristor Switching

In recent years significant efforts are being made for implementation of fourth electronic element – resistor with memory called memristor. This element is a passive device whose resistance can be changed by passing through a certain electric current pulse. The memristor has a simple metal-high-k-metal structure. Functionally the memristors can be divided into two-level (one-bit) and multilevel (multibit) ones. If the one-bit device may be used as a cell of non-volatile memory with a large storage time and high resistance to radiation, the multibit memristors on the one hand can increase the density of integration of the non-volatile memory, and on the other hand can be used to form the basis of adaptive neuromorphic (cognitive) computational systems. One of the most interesting and hitherto unsolved problems is the nature of the intermediate (discrete) multi-level states in a memristor. The hypothesis that the effect of memristive switch is directly connected with the formation of a filament, that occurs when certain current pulse pass over a dielectric, is distributed among specialists. However, the kinetics of filament formation during the transition from the high resistance state of the memristor to a low resistive state is still not clear. Another unsolved problem in the way of developing memristor arrays is a description of the forming process of memristor elements – the first switch from the initial high resistance state to a low resistance at high voltages, accompanied by intense Joule heat release. Nature of the forming of memristor matrix is still unclear, and the absence of any plausible physical model constrains the practical implementation of these elements. It is considered that charge transport in high-k is associated with the presence of defects. The most probable and the most widespread defects are oxygen vacancies. Properties of these vacancies are actively studied both experimentally and theoretically. However, many contradictory data can be found in literature, and the role and characteristics of oxygen vacancies in the charge transport through dielectric process are not established yet. It is obvious that rate of change in structure of dielectric (namely the generation rate of oxygen vacancies associated with conductivity) in strong electric field is mainly determined by non-stationary processes of heat and mass transfer. Thus, the kinetics of such processes determines final morphology and properties of memristor. In the presented work we introduce thermodynamical model of filament growing when a current pulse of variable length and value flows. The model is the boundary value problem, which includes nonstationary heat conduction equation with non-linear Joule heat source, Poisson equation, and Shockley-Read-Hall equations taking into account strong electron-phonon interactions in trap ionization and charge transport processes. The charge current, which defines the heating in the model, depends on the rate of the oxygen vacancy generation. The latter depends on the local temperature. Kinetic parameters of our model are obtained from the transport and optical experiments and ab initiosimulations [V.A.Gritsenko, T.V.Perevalov, D.R.Islamov, Physics Reports 613, 1 (2016)]. The solution of the introduced problem allows us to describe the kinetics of the switch process and the final filament morphology. The resulting filamentary structure in turn may explain the memristor switch. End-to-end solution of the task allows us to develop recommendations for optimizing the memristor technology and predict the behavior of memristor arrays, including neuromorphic electronics and non-volatile large memory arrays. The work is supported by Russian Science Foundation (grant #16-19-00002).

Nonlinear Refractory Plasmonics with Titanium Nitride Nanoantennas.

Titanium nitride (TiN) is a novel refractory plasmonic material which can sustain high temperatures and exhibits large optical nonlinearities, potentially opening the door for high-power nonlinear plasmonic applications. We fabricate TiN nanoantenna arrays with plasmonic resonances tunable in the range of about 950-1050 nm by changing the antenna length. We present second-harmonic (SH) spectroscopy of TiN nanoantenna arrays, which is analyzed using a nonlinear oscillator model with a wavelength-dependent second-order response from the material itself. Furthermore, characterization of the robustness upon strong laser illumination confirms that the TiN antennas are able to endure laser irradiation with high peak intensity up to 15 GW/cm(2) without changing their optical properties and their physical appearance. They outperform gold antennas by one order of magnitude regarding laser power sustainability. Thus, TiN nanoantennas could serve as promising candidates for high-power/high-temperature applications such as coherent nonlinear converters and local heat sources on the nanoscale.

Buoyancy induced model for the flow of 36 nm alumina-water nanofluid along upper horizontal surface of a paraboloid of revolution with variable thermal conductivity and viscosity

The motion of nanofluid (water and 36nm alumina nanoparticles) along upper horizontal surface of a paraboloid of revolution in the presence of nonlinear thermal radiation, Lorentz force and space dependent internal heat source within thin boundary layer is investigated theoretically. It is assumed that buoyancy induces the flow over this kind of surface which is neither a horizontal/vertical nor cone/wedge, hence suitable buoyancy model for this case of fluid flow is presented. The viscosity and thermal conductivity are assumed to vary with volume fraction and suitable models for the case 0%≤ϕ≤0.8% are adopted. The transformed governing equations are solved numerically using Runge-Kutta fourth order along with shooting technique (RK4SM). Good agreement is obtained between the solutions of RK4SM and MATLAB bvp5c for limiting case. The influence of pertinent parameters are illustrated graphically and discussed. It is found that temperature and velocity functions are maximum at higher values of internal space dependent heat source. Local heat transfer rate is maximum at smaller values of internal space dependent heat source.

Critical conditions for thermal explosion in a plate with a nonlinear heat source

The exact analytical solution of the nonstationary heat conduction problem with a nonlinear internal heat source at the nonsymmetric third-type boundary conditions was obtained by the method of variable separation. The relations between the boundary condition parameters and a heat source separating the stationary processes from processes of an unlimited temperature increase (thermal explosion) were found. For known physical properties of medium, the maximum allowable value of the specific power of a heat source has been found. If nonsymmetric boundary conditions are exceeded on wall surfaces, it can lead to thermal destruction.

Influence of nonlinear thermal radiation and Magnetic field on upper-convected Maxwell fluid flow due to a convectively heated stretching sheet in the presence of dust particles

A numerical investigation of two-dimensional MHD boundary layer flow and thermal characteristics of an electrically conducting dusty non-Newtonian fluid over a convectively heated stretching sheet has been considered. The effects of nonlinear thermal radiation, heat source or sink and viscous dissipation are also taken into the account. The Rosseland approximation is used to model the nonlinear thermal radiation. Suitable similarity transformations are used to transform the flow governing equations into a set of nonlinear differential equations of one independent variable. The Shooting method is adopted to solve transformed equations. The effects of various material parameters on the flow and heat transfer in terms of velocity and temperature distributions are drawn in the form of graphs and are briefly discussed. The numerical computations for the Nusselt number and skin friction drag are also carried out for the emerging parameters of interest in the problem. The obtained numerical results show the good agreement with the existing one for limiting case.

Double multiple-relaxation-time lattice Boltzmann model for solid–liquid phase change with natural convection in porous media

In this paper, a double multiple-relaxation-time lattice Boltzmann model is developed for simulating transient solid–liquid phase change problems in porous media at the representative elementary volume scale. The model uses two different multiple-relaxation-time lattice Boltzmann equations, one for the flow field and the other for the temperature field with nonlinear latent heat source term. The model is based on the generalized non-Darcy formulation, and the solid–liquid interface is traced through the liquid fraction which is determined by the enthalpy-based method. The present model is validated by numerical simulations of conduction melting in a semi-infinite space, solidification in a semi-infinite corner, and convection melting in a square cavity filled with porous media. The numerical results demonstrate the efficiency and accuracy of the present model for simulating transient solid–liquid phase change problems in porous media.

Existence of Solutions for Impulsive Parabolic Partial Differential Equations

We discuss the propagation of heat along a homogeneous rod of length A under the influence of a nonlinear heat source and impulsive effects at fixed times. This problem is described by an initial-boundary value problem for a nonlinear parabolic partial differential equation subjected to impulsive effects at fixed times. Using Green's function, we convert the problem into a nonlinear integral equation. Sufficient conditions are provided that enable the application of fixed point theorems to prove existence and uniqueness of solutions.

Backward heat equations with locally lipschitz source

Let be in and let , be a locally Lipschitz function with respect to the variable and . In this article, we consider the following backward heat problem with a nonlinear heat source The problem is ill-posed in the sense of Hadamard. Hence a regularization is in order. We use the -dimensional Fourier transform to construct an approximate solution for this problem. We also prove error estimates for our problem and give comments on similar problems.

Finding unknown heat source in a nonlinear Cauchy problem by the Lie-group differential algebraic equations method

We consider an inverse heat source problem of a nonlinear heat conduction equation, for recovering an unknown space-dependent heat source under the Cauchy type boundary conditions. With the aid of measured initial temperature and initial heat flux, which are disturbanced by random noise causing measurement error, we develop a Lie-group differential algebraic equations (LGDAE) method to solve the resultant differential algebraic equations. The Lie-group numerical method has a stabilizing effect to retain the solution on the associated manifold, which thus naturally has a regularization effect to overcome the ill-posed property of the nonlinear inverse heat source problem. As a consequence, we can quickly recover the unknown heat source under noisy input data only through a few iterations. The initial data used in the recovery of heat source are assumed to be the analytic continuation ones which are not given arbitrarily. Certainly, the measured initial data belong to this type data.

Mathematical modeling of biological tissue cryodestruction

New statements of one- and two-dimensional Stefan-type boundary problems with nonlinear heat sources arising in cryosurgery are considered in the paper. An effective method for their full research for cases of sufficiently extended cryoinstruments with a cylindrical shape of a cooling surface is offered. Some results of the computer simulation are present.

Two-dimensional boundary problems of Stefan’s type in cryomedicine

In this paper we propose a new formulation of two-dimensional boundary problems of Stefan’s type with non-linear heat source of a special kind, arising in cryomedicine. These models correspond to the processes of biological tissue cooling by cryoinstruments with cylindrical and conical forms of the cooling surface. An effective method for the study of these models is offered.

Identification of the thermal growth characteristics of coagulated tumor tissue in laser‐induced thermotherapy

We consider an inverse problem arising in laser‐induced thermotherapy, a minimally invasive method for cancer treatment, in which cancer tissues are destroyed by coagulation. For the dosage planning, numerical simulations play an important role. To this end, a crucial problem is to identify the thermal growth kinetics of the coagulated zone. Mathematically, this problem is a nonlinear and nonlocal parabolic heat source inverse problem. The solution to this inverse problem is defined as the minimizer of a nonconvex cost functional in this paper. The existence of the minimizer is proven. We derive the Gateaux derivative of the cost functional, which is based on the adjoint system, and use it for a numerical approximation of the optimal coefficient. Numerical implementations are presented to show the validity of the optimization schemes. Copyright © 2012 John Wiley & Sons, Ltd.

A Nonlinear Frequency Domain Model for Limit Cycles in Thermoacoustic Systems with Modal Coupling

For prediction of limit cycle oscillations of linearly unstable thermo-acoustic systems, a frequency-domain, low-order system with explicit modal coupling is developed. To this purpose, a model for the nonlinear heat source dynamics is obtained from unsteady computational fluid dynamics in combination with feed-forward neural network identification. From the neural network, an equivalent representation for the input-output relation in Volterra series form is derived, where Volterra kernels are computed in terms of the weights of the neural network. Then the kernels are transformed into the frequency domain to obtain the higher order transfer functions, through which the modes are coupled. In this way nonlinear energy exchange among the modes can be described explicitly. Comparison with a Galerkin time domain simulation shows that deviations from purely sinusoidal behaviour in the limit cycle are captured correctly, while the computational cost is drastically reduced.

Nonlinear identification of unsteady heat transfer of a cylinder in pulsating cross flow

Unsteady heat transfer of a cylinder in pulsating cross flow is investigated. The heat transfer process is considered as a nonlinear single-input, single-output system, with large amplitude velocity perturbations as “input” and total heat transfer rate from the cylinder surface to the fluid as “output”. Quantitatively accurate, low-order models of the heat source dynamics are obtained with a variety of nonlinear system identification methods from time series data generated with unsteady CFD computations. A polynomial type, equation error identification scheme is found to yield very accurate results. In order to obtain the heat source response function in the frequency domain, the equation error model is converted into the frequency domain using harmonic balance as well as harmonic probing approaches. In the harmonic balance approach, a set of nonlinear algebraic equations is solved for the coefficients of the harmonic ansatz. In the harmonic probing method, a recursive relation is obtained to deliver the higher order transfer functions of the nonlinear heat source. Alternatively, a nonlinear black box identification technique is used to identify the heat source dynamics. It is found to capture the amplitudes of higher harmonics with increased accuracy.

Boundary element analysis of nonlinear transient heat conduction problems involving non-homogenous and nonlinear heat sources using time-dependent fundamental solutions

A new method for the boundary element analysis of unsteady heat conduction problems involving non-homogenous and/or temperature dependent heat sources by the time-dependent fundamental solution is presented. Nonlinear terms are converted to a fictitious heat source and implemented in the present formulation. The domain integrals are efficiently treated by the recently introduced Cartesian transformation method. Similar to the dual reciprocity method, some internal grid points are considered for the treatment of the domain integrals. In the present method, unlike the dual reciprocity method, there is no need to find particular solution for the shape functions in the interpolation computations and the form of the shape functions can be arbitrary and sufficiently complicated. In the present method, at each time step the temperature at boundary nodes and some internal grid points is computed and used as pseudo-initial values for the next time step. Most of the generated matrices are constant at all time steps and computations can be carried out fast. An example with different forms of heat sources is presented to show the efficiency and accuracy of the proposed method.

Single-Shot, High-Speed, Thermal-Interface Characterization of Semiconductor Laser Arrays

Through a detailed characterization of thermally induced output power degradation it is possible to use junction heating as a tool to resolve thermal interfaces on mus timescales using a single-shot characterization technique. In this work, the deleterious effect junction heating has on the optical output power of a laser array is characterized and then used to infer the time-dependent junction temperature in response to current pulses of varying widths. The extracted parameters are also used numerically to model the laser as a temperature-dependent heat source for thermal simulations. This treatment allows realistic packaging and emitter-placement studies to be parametrically performed by incorporating the relationship between temperature and output power/efficiency for each emitter. In this respect, once the temperature behavior of a single emitter is quantified, the operating temperature and output power performance can be accurately predicted for any realistic physical arrangement of laser array and packaging. The experimental method presented in this work is also compared to other techniques and numerical simulations using the nonlinear heat source; this demonstrates the utility of this approach and the convenience of using easily measured parameters in thermal simulations.

Resonant vibration and heating of ring plates with piezoactuators under electromechanical loading and shear deformation

The bending vibration and dissipative heating of a viscoelastic isotropic ring plate with piezoceramic actuators under electromechanical loading and shear deformation are studied by solving a coupled problem. The temperature dependence of the complex characteristics of the passive and piezoactive materials is taken into account. The nonlinear problem of thermoviscoelasticity is solved by time stepping with discrete orthogonalization used at each iteration to integrate the equations of elasticity and using an explicit finite-difference scheme to solve the heat-conduction equation with a nonlinear heat source. The effect of shear deformation, fixation conditions for the plate, the geometry of the piezoactuators, and the dissipative-heating temperature on the active damping of the forced vibration of a circular plate subject to uniform transverse monoharmonic compression is studied