Quasilinear Heat Research Articles

Direct microstability optimization of stellarator devices.

Turbulent transport is regarded as one of the key issues in magnetic confinement nuclear fusion, both for tokamaks and stellarators. In this work, we show that a significant decrease in a microstability-based proxy, as opposed to a geometric one, for the turbulent heat flux, namely the quasilinear heat flux, can be obtained in an efficient manner by coupling stellarator optimization with linear gyrokinetic simulations. This is accomplished by computing the quasilinear heat flux at each step of the optimization process, as well as the deviation from quasisymmetry, and minimizing their sum, leading to a balance between neoclassical and the turbulent transport proxy.

An optimal control problem for single-spot pulsed laser welding

We consider an optimal control problem for a single-spot pulsed laser welding problem. The distribution of thermal energy is described by a quasilinear heat equation. Our emphasis is on materials which tend to suffer from hot cracking when welded, such as aluminum alloys. A simple precursor for the occurrence of hot cracks is the velocity of the solidification front. We therefore formulate an optimal control problem whose objective contains a term which penalizes excessive solidification velocities. The control function to be optimized is the laser power over time, subject to pointwise lower and upper bounds. We describe the finite element discretization of the problem and a projected gradient scheme for its solution. Numerical experiments for material data representing the EN AW 6082-T6 aluminum alloy exhibit interesting laser pulse patterns which perform significantly better than standard ramp-down patterns.

Isotope effects under the influence of global radial electric fields in a helical configuration

Isotope effects under the influence of a radial electric field are examined in a helical magnetic field configuration. We perform global gyrokinetic simulations with additional poloidal rotations to estimate quasi-linear heat flux due to ion temperature gradient mode under the mixing length model. In single-ion-species plasmas, the mass number dependency of heat flux agrees with gyro-Bohm scaling in the absence of a radial electric field. Favorable mass number dependencies violating gyro-Bohm scaling are observed in the presence of a global radial electric field or a heavy hydrogen component in multi-ion-species plasmas. The radial electric field and the heavy hydrogen component affect the heat flux through an increase of wavelength as well as mode stabilization. Poloidal Mach number characterizes the transition from unfavorable to favorable mass number dependency under radial electric fields. While the heat flux is independent of mass number for a given poloidal Mach number, the heat flux decreases for higher mass numbers in a given radial electric field. The heat flux is also independent of average mass number in multi-ion-species plasmas because the heavy hydrogen component effectively enhances the light hydrogen heat flux. The present results are potentially relevant to the violation of gyro-Bohm scaling observed in the recent deuterium experiments in the Large Helical Device.

Corrigendum: Quasilinear turbulent particle and heat transport modeling with a neural-network-based approach founded on gyrokinetic calculations and experimental data (2021 Nucl. Fusion 61 116041)

Corrigendum: Quasilinear turbulent particle and heat transport modeling with a neural-network-based approach founded on gyrokinetic calculations and experimental data (2021 Nucl. Fusion 61 116041)

Application of agent-based approach for heat conduction processes simulation

Heat transfer processes are the basis of the most technological processes of the energy sector. Thus, the development of modern approaches for computer simulation and visualization of the phenomena of thermal energy transfer in various objects are of great importance and is relevant. Classical models in the form of differential equations of various types describe processes in continuous space and time. So, it is difficult to apply classical models to study nonlinear phenomena, and processes in inhomogeneous media in the presence of discontinuous solutions at the boundaries. In these cases, simplifying assumptions are used, thus, the adequacy of the models is reduced. It is of great interest to apply fundamentally different approaches to describe transfer processes, which include discrete dynamic models. The purpose of this project is to study the possibilities to apply discrete approaches to simulate nonlinear heat transfer processes under conditions of material inhomogeneity and the presence of volume sources of variable power. The paper studies the possibilities to apply the agent-based approach to simulate models of complex systems. This approach allows us to consider a continuum as a set of interacting elements (agents). The behavior of the elements is completely described by local dependencies. At the same time, the laws of functioning of individual elements are accepted as deterministic and they correspond to the fundamental principles of the theory of heat transfer. The possibility to apply a discrete approach for simulating the process of heat transfer by the molecular mechanism has been studied. The general methodology to develop an agent-based deterministic model is described. Its applicability to describe quasi-linear and nonlinear heat conduction processes is considered. The examples of simulation of combustion processes complicated by exothermic and endothermic effects are considered. The advantages and disadvantages of the proposed method are indicated. The results of the study have shown that discrete agent models are a good alternative to classical continuum approaches to study heat transfer processes in inhomogeneous media. The results obtained do not contradict modern approaches to the description of thermal processes. It has also been found that the simulation algorithms used in the agent-based approach are quite universal and easily adapt to changes under the conditions of problem setting. The analysis of the results makes it possible to recommend a discrete agent-based approach to develop simulation models of complex technological processes and systems.

Impurity effects on quasi-linear heat transport induced by interaction of TEM and ITG turbulence

The impurity effects on quasi-linear turbulent heat transport induced by ion temperature gradient and trapped electron modes in tokamak plasmas are numerically studied, using a full gyrokinetic description for main and impurity ions. The unstable modes in independent and hybrid states are all taken into account. The fluxes integrated over the complete spectrum rather than taking an individual characteristic k θ ρ s of poloidal wave-vector are systematically investigated. The results indicate that the injected impurity enhances the heat transport of main ions and electrons, and the maximum variations of the former are shown to be approximately twice as high as those of the latter, i.e. ΔQ i−sum ∼ 2ΔQ e−sum for the core plasmas of the parameters in this work. In contrast, the impurity ions can induce reduction of the edge ion heat fluxes, while the electron heat fluxes are almost unchanged. Such distinct behaviors are found to be associated with the transition of instabilities types and dominated regimes of heat transport. Significantly, reversed magnetic shear, large temperature ratios of electrons to main ions as well as high mass numbers of impurity ions contribute to the improvement of confinement and formation of energy pinch. The newly updated code identifies individual contributions to heat transport from conductive and convective fluxes, both of which are composed of diagonal, off-diagonal diffusive and non-diffusive components and thus is expected to be a powerful tool for experimental data analysis as well. The phase shift (larger than π/2) is demonstrated to be a reasonable mechanism for the induction of energy pinch.

Existence of solution for a class of heat equation with double criticality

In this paper, we study the following class of quasilinear heat equations{ut−ΔΦu=f(x,u)inΩ,t>0,u=0on∂Ω,t>0,u(x,0)=u0(x)inΩ, where ΔΦu=div(φ(x,|∇φ|)∇φ) and Φ(x,s)=∫0|s|φ(x,σ)σdσ is a generalized N-function. We suppose that Ω⊂RN(N≥2) is a smooth bounded domain that contains two open regions ΩN and Ωp with Ω‾N∩Ω‾p=∅. Under some appropriate conditions, the global existence will be done by combining the Galerkin approximations with the potential well theory. Moreover, the large-time behavior of the global weak solution is analyzed. The main feature of this paper consists that −ΔΦu behaves like −ΔNu on ΩN and −Δpu on Ωp, while the continuous function f:Ω×R→R behaves like eα|s|NN−1 on ΩN and |s|p⁎−2s on Ωp as |s|→∞.

Solving an inverse heat convection problem with an implicit forward operator by using a projected quasi-Newton method

We consider the quasilinear 1D inverse heat convection problem of determining the enthalpy-dependent heat fluxes from noisy internal enthalpy measurements. This problem arises in the accelerated cooling process of producing thermomechanically controlled processed heavy plates made of steel. In order to adjust the complex microstructure of the underlying material, the Leidenfrost behavior of the hot surfaces with respect to the application of the cooling fluid has to be studied. Since the heat fluxes depend on the enthalpy and hence on the solution of the underlying initial boundary value problem (IBVP), the parameter-to-solution operator, and thus the forward operator of the inverse problem, can only be defined implicitly. To guarantee well-defined operators, we study two approaches for showing existence and uniqueness of solutions of the IBVP. One approach deals with the theory of pseudomonotone operators and so-called strong solutions in Sobolev–Bochner spaces. The other theory uses classical solutions in Hölder spaces. Whereas the first approach yields a solution under milder assumptions, it fails to show the uniqueness result in contrast to the second approach. Furthermore, we propose a convenient parametrization approach for the nonlinear heat fluxes in order to decouple the parameter-to-solution relation and use an iterative solver based on a projected quasi-Newton (PQN) method together with box-constraints to solve the inverse problem. For numerical experiments, we derive the necessary gradient information of the objective functional and use the discrepancy principle as a stopping rule. Numerical tests show that the PQN method outperforms the Landweber method with respect to computing time and approximation accuracy.

Inverse extremum problem for a model of endovenous laser ablation

Abstract An inverse extremum problem (optimal control problem) for a quasi-linear radiative-conductive heat transfer model of endovenous laser ablation is considered. The problem is to find the powers of the source spending on heating the fiber tip and on radiation. As a result, it provides a given temperature distribution in some part of the model domain. The unique solvability of the initial-boundary value problem is proved, on the basis of which the solvability of the optimal control problem is shown. An iterative algorithm for solving the optimal control problem is proposed. Its efficiency is illustrated by a numerical example.

Quasi-linear heat transport induced by ITG turbulence in the presence of impurities

The impurity effects on turbulent transport induced by ion temperature gradient (ITG) turbulence are numerically studied in tokamak plasmas, using a gyrokinetic quasi-linear model. The characteristics of the instability and heat fluxes in the presence of impurity ions are investigated for a broad parameter regime, including temperature and density gradients of main and impurity ions, concentration, charge and mass numbers of impurity ions, magnetic shear as well as wave vector spectrum. The heat fluxes are demonstrated to depend not only on the saturation amplitude of the instability but also on the phase shift between and . The peaking factor of temperature/density profile, defined as the ratio of major radius to gradient scale length when the total turbulent heat flux equals zero, is fitted with linear/quadratic functions. In addition, the contributions from diagonal and off-diagonal terms to heat fluxes are identified in detail, i.e. the main ion heat diffusion are proved to be dominated by off-diagonal (diagonal) terms for regions of weak (strong) ITG. In general, steep temperature gradients of main ions as well as hollow density profiles of impurity ions significantly enhance instability and heat fluxes. However, it is interesting to find that the effect of impurity ions with positive density gradient may transit from the enhancement to reduction of the quasi-linear heat flux of main ions in regions of steep ITG, corresponding to transport barriers (e.g. pedestal of H-mode and I-mode plasmas). Both strong and weak positive magnetic shear decrease heat transport.

A non-conservation stochastic partial differential equation driven by anisotropic fractional Lévy random field

We study a non-conservation second-order stochastic partial differential equation (SPDE) driven by multi-parameter anisotropic fractional Lévy noise (AFLN) and under different initial and/or boundary conditions. It includes the time-dependent linear heat equation and quasi-linear heat equation under the fractional noise as special cases. Unique existence and expressions of solution to the equation are proved and constructed. An AFLN is defined as the derivative of an anisotropic fractional Lévy random field (AFLRF) in certain sense. Comparing with existing noise systems, our non-Gaussian fractional noises are essentially observed from random disturbances on system accelerations rather than from those on system moving velocities. In the process of proving our claims, there are three folds. First, we consider the AFLRF as the generalized functional of sample paths of a pure jump Lévy process. Second, we build Skorohod integration with respect to the AFLN by white noise approach. Third, by combining this noise analysis method with the conventional PDE solution techniques, we provide solid proofs for our claims.

SOLVABILITY OF THE INITIAL-BOUNDARY VALUE PROBLEM FOR THE QUASILINEAR EQUATION OF HEAT CONDUCTIVITY IN DOMAINS THAT CAN BE TRANSFORMED INTO RECTANGLES

In this paper, we study the initial-boundary-value problem for the quasilinear heat equation in regions that are reduced to rectangular. Mathematical modeling of many processes taking place in the real world leads to the study of the problems of equations of mathematical physics, when the areas are not rectangular. The theory of nonlinear problems is an actively developing section of the theory of modern differential equations. In the theory of nonlinear equations, a special place is occupied by the study of unbounded solutions or, in other words, modes with exacerbation. Nonlinear evolutionary problems that allow unlimited solutions are globally unsolvable: solutions grow unlimitedly over a finite period of time. In this paper, the initial-boundary-value problem for the quasilinear heat equation in regions that can be reduced to rectangular ones, the existence of a solution is proved by the Galerkin method. The uniqueness of the solution was proved by the obtained a priori estimates. Sufficient conditions for the destruction of the solution in a finite time in a bounded domain are obtained. The exponential decay of the solution with an infinite increase in time is proved. In the final time, it was proved that the solution is localized, i.e. disappears (nullifies).

Determination of thermophysical properties and density volume fractions of Al2O3/Y-ZrO2 layered composite materials using transient thermography and two-stage inverse nonlinear heat conduction analysis

In the present study, transient thermography, a nondestructive imaging technique, is applied to evaluate the transient temperature response in a graded medium without the use of embedded thermocouples. A layered composite sample was fabricated from Al2O3 and Y-ZrO2 powders using powder metallurgy (PM). This sample was irradiated on one side with a direct current laser while the transient temperature was measured along its depth by a midinfrared camera. Also, a MATLAB code based on the truly meshless radial point interpolation method (t-RPIM) was developed and implemented to solve the problem of quasilinear transient heat transfer in PM solids. In the t-RPIM formulation, the Cartesian transformation method and the Crank-Nicolson scheme were used for the evaluation of domain integrals and time discretization, respectively, thereby yielding a truly mesh-free technique. In the conducted experiment, the thermophysical properties were assumed to be independent of temperature because of the small amount of temperature increase. These properties and the volume fractions of the constituent powders were determined using a combination of the t-RPIM and the damped Gauss-Newton method in an inverse analysis. Good agreement was found between the measured temperature and the reconstructed temperature profile using the identified thermal parameters and volume fractions, thus validating the accuracy and ability of the applied t-RPIM as a tool in an inverse scheme to solve the inverse transient heat conduction problem in nonhomogeneous media.

Nondivergence form quasilinear heat equations driven by space-time white noise

We give a Wong-Zakai type characterisation of the solutions of quasilinear heat equations driven by space-time white noise in 1+1 dimensions. In order to show that the renormalisation counterterms are local in the solution, a careful arrangement of a few hundred terms is required. The main tool in this computation is a general ‘integration by parts’ formula that provides a number of linear identities for the renormalisation constants.

Parallel algorithm for numerical solution of heat equation in complex cylindrical domain

In this article we present a parallel algorithm for simulation of the heat conduction process inside the so-called pulse cryogenic cell. This simulation is important for designing the device for portion injection of working gases into ionization chamber of ion source. The simulation is based on the numerical solving of the quasilinear heat equation with periodic source in a multilayered cylindrical domain. For numerical solution the Alternating Direction Implicit (ADI) method is used. Due to the non-linearity of the heat equation the simple-iteration method has been applied. In order to ensure convergence of the iteration process, the adaptive time-step has been implemented. The parallelization of the calculation has been realized with shared memory application programming interface OpenMP and the performance of the parallel algorithm is in agreement with the case studies in literature.

SPDEs with fractional noise in space: Continuity in law with respect to the Hurst index

In this article, we consider the quasi-linear stochastic wave and heat equations on the real line and with an additive Gaussian noise which is white in time and behaves in space like a fractional Brownian motion with Hurst index $H\in (0,1)$. The drift term is assumed to be globally Lipschitz. We prove that the solution of each of the above equations is continuous in terms of the index $H$, with respect to the convergence in law in the space of continuous functions.

Grow-up for a quasilinear heat equation with a localized reaction

We study the behaviour of global solutions to the quasilinear heat equation with a reaction localizedut=(um)xx+a(x)up,m,p>0 and a(x) being the characteristic function of an interval. We prove that there exists an exponent p0=max⁡{1,m+12} such that all global solutions are bounded if p>p0, while for p≤p0 all the solutions are global and unbounded. In the last case, we prove that if p<m the grow-up rate is different to the one obtained when a(x)≡1, while if p>m the grow-up rate coincides with that rate, but only inside the support of a; outside the interval the rate is smaller.

A Distributed-Parameter Approach for the Surface Temperature Estimation of an LED Heated Silicon Wafer

A variety of processes in the semiconductor industry require heating of the silicon wafer to the desired temperature. This process is widely referred to as rapid thermal processing. However, the contactless measurement of the surface temperature of the wafer is still a major challenge, especially in the case of rotating wafers. Measurements using infrared cameras are not suitable due to the fact that silicon is transparent in this wavelength range. Special sensors based on the principle of pyrometry are available, but such sensors can only measure the surface temperature at one single point. This paper presents an observer approach that estimates the wafer’s surface temperature by using the temperature measurement of only one pyrometer. The approach is based on a mathematical model capturing the dynamical behavior of the wafer’s temperature. It relies mainly on the quasi-linear heat equation. Real world experiments demonstrate the achieved accuracy of the proposed approach.

Solution of the Problem of Initiating the Heat Wave for a Nonlinear Heat Conduction Equation Using the Boundary Element Method

The paper is devoted to constructing approximate heat wave solutions propagating along the cold front at a finite speed for a nonlinear (quasi-linear) heat conduction equation with a power nonlinearity. The coefficient of the higher derivatives vanishes on the front of the heat wave, i.e., the equation degenerates. One- and two-dimensional problems about the initiation of a heat wave by the boundary mode specified on a given fixed manifold are studied. Algorithms for solving this problem based on the boundary element method and a special change of variables as a result of which the unknown function and the independent spatial variable exchange their roles are proposed. The solution of the transformed problem in the form of a converging power series is constructed. These algorithms are implemented in computer programs, and test computations are performed. Their results are compared with truncated power series mentioned above and with the known exact solutions; the results are in good agreement.

A General Decay and Optimal Decay Result in a Heat System with a Viscoelastic Term

We consider a quasilinear heat system in the presence of an integral term and establish a general and optimal decay result from which improves and generalizes several stability results in the literature.