Heat Equation Research Articles

Discrete heat equation for a periodic layered system with allowance for the interfacial thermal resistance: General formulation and dispersion analysis

Discrete heat equation for a periodic layered system with allowance for the interfacial thermal resistance: General formulation and dispersion analysis

A lemma on C 0-semigroups on time scales and approximate controllability of the heat dynamic equation

In this paper, we present a lemma that allows us to characterize a broad class of C 0-semigroups on time scales, which can be applied to prove existence and uniqueness of solutions of systems of partial differential equations where the time domain is a time scale. The result obtained is applied to study the controllability of the heat equation on time scales. The lemma presented in this paper can be seen as a unification of the one proved by H. Leiva in [28], for semigroups in [0,∞).

Comparison principle for stochastic heat equations driven by α-stable white noises

Comparison principle for stochastic heat equations driven by α-stable white noises

Ultrafast bimetallic interface reinforcement in additively manufactured multi-material via electropulsing

The bimetallic interface in additively manufactured multi-material was subjected to intense thermal cycling, and thus forming sizable heat affected zone and drastic mutation of chemical component, which led to weak metallurgical bonding. In this investigation, the pulsed current (1255 Hz; 64 μs; 35.4 V; 191 A; 1000 ℃; 20 min) was used to rapidly broaden the compositional mutation zone from 16 μm to 45 μm, and the heat affected zone was completely eliminated. However, the roles of equivalent heat treatment (1000 ℃; 20 min) were opposite with the pulsed one. The width of compositional mutation zone became narrower and the morphology heat affected zone did not change after equivalent heat treatment. Furthermore, the hot compression experiment proved the better bimetallic interface quality of pulsed sample. The rapid regulation of microstructure and composition gradient is caused by increasing interfacial diffusion flux and the promotion of electron wind force to the solute atom.

Internal null controllability for the one-dimensional heat equation with dynamic boundary conditions

Abstract The primary focus of this paper is to establish the internal null controllability for the one-dimensional heat equation featuring dynamic boundary conditions. This achievement is realized by introducing a new Carleman estimate and an observability inequality for the corresponding backward system. In conclusion, the paper includes a set of numerical experiments that serve to confirm the validity of the theoretical findings and underscore the effectiveness of the designed control with a minimal $L^{2}$-norm.

Multi-Objective Topology Optimization of Conjugate Heat Transfer Using Level Sets and Anisotropic Mesh Adaptation

This study proposes a new computational framework for the multi-objective topology optimization of conjugate heat transfer systems using a continuous adjoint approach. It relies on a monolithic solver for the coupled steady-state Navier–Stokes and heat equations, which combines finite elements stabilized by the variational multi-scale method, level set representations of the fluid–solid interfaces and immersed modeling of heterogeneous materials (fluid–solid) to ensure that the proper amount of heat is exchanged to the ambient fluid by solid objects in arbitrary geometry. At each optimization iteration, anisotropic mesh adaptation is applied in near-wall regions automatically captured by the level set. This considerably cuts the computational effort associated with calling the finite element solver, in comparison to traditional topology optimization algorithms operating on isotropic grids with a comparable refinement level. Given that we operate within the constraint of a specified number of nodes in the mesh, this allows not only to improve the accuracy of interface representation and motion but also to retain the high fidelity of the numerical solutions at the grid points just adjacent to the interface. Finally, the remeshing and resolution steps both run within a highly parallel environment, which makes it possible for the proposed algorithm to tackle large-scale problems in three dimensions with several tens of millions of state degrees of freedom. The developed solver is validated first by minimizing dissipation in a flow splitter device, for which the method delivers relevant optimal designs over a wide range of volume constraints and flow rate distributions over the multiple outlet orifices but yields better accuracy compared to reference data from literature obtained using uniform meshes (in the sense that the layouts are more smooth, and the solutions are better resolved). The scheme is then applied to a two-dimensional heat transfer problem, using bi-objective cost functionals combining flow resistance and thermal recoverable power. A comprehensive parametric study reveals a complex arrangement of optimal solutions on the Pareto front, with multiple branches of symmetric and asymmetric designs, some of them previously unreported. Finally, the algorithmic developments are substantiated with several three-dimensional numerical examples tackled under fixed weights for heat transfer and flow resistance, for which we show that the optimal layouts computed at low Reynolds number, that are intrinsically relevant to a broad range of microfluidic application, can also serve as smooth solutions to high-Reynolds-number engineering problems of practical interest.

Characterization of Pseudo-Differential Operators Associated with the Coupled Fractional Fourier Transform

The main aim of this article is to derive certain continuity and boundedness properties of the coupled fractional Fourier transform on Schwartz-like spaces. We extend the domain of the coupled fractional Fourier transform to the space of tempered distributions and then study the mapping properties of pseudo-differential operators associated with the coupled fractional Fourier transform on a Schwartz-like space. We conclude the article by applying some of the results to obtain an analytical solution of a generalized heat equation.

On a family of nonzero solutions to the heat equation ut = △u on ℝn+1 which vanish on an arbitrary n-dimensional hyperplane P ⊂ ℝn+1

Motivated by the paper Rosenbloom–Widder [A temperature function which vanishes initially, Amer. Math. Mon. 65(8) (1958) 607–609], we construct a more general family of nonzero solutions to the [Formula: see text]-dimensional heat equation [Formula: see text] on [Formula: see text] with zero initial data. These solutions are analytically more manageable than the well-known Tychonoff power series solution. Nonzero solutions to the [Formula: see text]-dimensional heat equation [Formula: see text] on [Formula: see text] with zero initial data (i.e. vanishing on the hyperplane [Formula: see text]) can be easily obtained as a consequence of the examples on [Formula: see text] Finally, given an arbitrary hyperplane [Formula: see text], we can construct a family of nonzero solutions which vanish on [Formula: see text]

Multigoal-oriented a posteriori error control for heated material processing using a generalized Boussinesq model

In this work, we develop a posteriori error control for a generalized Boussinesq model in which thermal conductivity and viscosity are temperature-dependent. Therein, the stationary Navier–Stokes equations are coupled with a stationary heat equation. The coupled problem is modeled and solved in a monolithic fashion. The focus is on multigoal-oriented error estimation with the dual-weighted residual method in which an adjoint problem is utilized to obtain sensitivity measures with respect to several goal functionals. The error localization is achieved with the help of a partition-of-unity in a weak formulation, which is specifically convenient for coupled problems as we have at hand. The error indicators are used to employ adaptive algorithms, which are substantiated with several numerical tests such as one benchmark and two further experiments that are motivated from laser material processing. Therein, error reductions and effectivity indices are consulted to establish the robustness and efficiency of our framework.

Numerical investigation on thermal-hydraulic performance of leeward cut annular finned tubes in crossflow heat exchangers: An insight on fin material saving

The objective of the current study is to analyse the hydro-thermal characteristics of leeward cut fins with various trimming values and finalise a specific fin that benefits by maximum material saving and negligible compromise in heat transfer. The leeward section of a fin is cut in two ways: angular and chordwise. For angular cutting, four different angles (15°, 30°, 60°, and 90°) are considered, while for chordwise cutting, a fin is trimmed along a chord present at D/3, 2D/3, and D mm from the vertical diameter. The spacing between fins is varied (2–4 mm). For turbulence modelling, the transition SST model is implemented. Streamlines reveal that the intensity of a vortex formed downstream of the tube depends on the fin-trimming value. For a higher fin cutoff, a lower pressure drop is obtained. The heat transfer rate analysis reveals that angular cutoffs of 15° (8.3 % material saving) and 30° (16.6 % material saving) have 2.5 %–3.1 % less heat transfer than the standard case. Chordwise cut at D gives 1.1 % less heat transfer but saves only 2.7 % fin material than the standard case of no cutoff. Z/E factor, a ratio of heat transfer for a unit temperature drop to power provided, is used to determine hydro-thermal performance. The value of the Z/E ratio is the highest for the benchmark case. However, for angular cutoffs of 15° and 30°, the Z/E factor is less by 4 and 6.7 %, respectively, than the standard case. Examining the outcomes of this study, we recommend using an angular cutoff of 30° or lower to get significant material savings for an equivalent heat transfer.

Numerical integrator for highly oscillatory differential equations based on the Neumann series

AbstractWe propose a third-order numerical integrator based on the Neumann series and the Filon quadrature, designed mainly for highly oscillatory partial differential equations. The method can be applied to equations that exhibit small or moderate oscillations; however, counter-intuitively, large oscillations increase the accuracy of the scheme. With the proposed approach, the convergence order of the method can be easily improved. Error analysis of the method is also performed. We consider linear evolution equations involving first- and second-time derivatives that feature elliptic differential operators, such as the heat equation or the wave equation. Numerical experiments consider the case in which the space dimension is greater than one and confirm the theoretical study.

Comprehensive investigation of the explicit, positivity preserving methods for the heat equation

In this paper-series, we investigate the performance of 12 explicit non-conventional algorithms. All of them have the convex combination property, thus they are unconditionally stable and preserve the positivity of the solution when they applied to the heat equation. In this part of the series, we construct several 2D systems to find how the errors depend on the time step size. Sweeps for other key parameters will be presented in the next part of the series.

Comprehensive investigation of the explicit, positivity preserving methods for the heat equation

In this paper-series, we investigate the performance of 12 explicit non-conventional algorithms in several 2D systems. All of them have the convex combination property, thus they are unconditionally stable and preserve the positivity of the solution when they are applied to the heat equation. In the first part of the series, we examined how the errors depend on the time step size and running times. Now we present additional numerical test results, where sweeps for parameters such as the stiffness and the wavelength of the initial function will be performed.

Fractional nonlinear heat equations and characterizations of some function spaces in terms of fractional Gauss–Weierstrass semi–groups

AbstractWe present a new proof of the caloric smoothing related to the fractional Gauss–Weierstrass semi–group in Triebel-Lizorkin spaces. This property will be used to prove existence and uniqueness of mild and strong solutions of the Cauchy problem for a fractional nonlinear heat equation.

Travelling wave solution of fourth order reaction diffusion equation using hybrid quintic hermite splines collocation technique

AbstractFourth order extended Fisher Kolmogorov reaction diffusion equation has been solved numerically using a hybrid technique. The temporal direction has been discretized using Crank Nicolson technique. The space direction has been split into second order equation using twice continuously differentiable function. The space splitting results into a system of equations with linear heat equation and non linear reaction diffusion equation. Quintic Hermite interpolating polynomials have been implemented to discretize the space direction which gives a system of collocation equations to be solved numerically. The hybrid technique ensures the fourth order convergence in space and second order in time direction. Unconditional stability has been obtained by plotting the eigen values of the matrix of iterations. Travelling wave behaviour of dependent variable has been obtained and the computed numerical values are shown by surfaces and curves for analyzing the behaviour of the numerical solution in both space and time directions.

BEM Modeling for Stress Sensitivity of Nonlocal Thermo-Elasto-Plastic Damage Problems

The main objective of this paper is to propose a new boundary element method (BEM) modeling for stress sensitivity of nonlocal thermo-elasto-plastic damage problems. The numerical solution of the heat conduction equation subjected to a non-local condition is described using a boundary element model. The total amount of heat energy contained inside the solid under consideration is specified by the non-local condition. The procedure of solving the heat equation will reveal an unknown control function that governs the temperature on a specific region of the solid’s boundary. The initial stress BEM for structures with strain-softening damage is employed in a boundary element program with iterations in each load increment to develop a plasticity model with yield limit deterioration. To avoid the difficulties associated with the numerical calculation of singular integrals, the regularization technique is applicable to integral operators. To validate the physical correctness and efficiency of the suggested formulation, a numerical case is solved.

Measurement and modeling of strain waves in germanium induced by ultrafast laser pulses

Transient reflectivity measurements are used to probe the strain waves induced by ultrashort laser pulses in bulk [100] germanium. The measurement signals are compared to purely analytical model functions based on the known material parameters for germanium. The modeling includes (i) a derivation of analytical solutions to the wave equation for strain waves coupled to the diffusion equation for heat and charge carriers and (ii) an expression for the impact on reflection coefficients that are caused by perturbations to the dielectric function but extended to cover a non-isotropic, uniaxial dielectric tensorial form. The model is held up against transient reflectivity measurements with an s- and a p-polarized probe and with a probe wavelength in the range of 502–710 nm. Excellent agreement is found when comparing the oscillatory shape of the measurement signals to the models. As for the magnitude of the oscillations, the models reproduce the overall trends of the experiment when using the previously published values for the elasto-optical tensor measured under static strain.

An Implicitly Extended Crank–Nicolson Scheme for the Heat Equation on a Time-Dependent Domain

We consider a time-stepping scheme of Crank–Nicolson type for the heat equation on a moving domain in Eulerian coordinates. As the spatial domain varies between subsequent time steps, an extension of the solution from the previous time step is required. Following Lehrenfeld and Olskanskii (ESAIM: M2AN 53(2):585–614, 2019), we apply an implicit extension based on so-called ghost-penalty terms. For spatial discretisation, a cut finite element method is used. We derive a complete a priori error analysis in space and time, which shows in particular second-order convergence in time under a parabolic CFL condition. Finally, we present numerical results in two and three space dimensions that confirm the analytical estimates, even for much larger time steps.

Coupled responses of thermomechanical waves in functionally graded viscoelastic nanobeams via thermoelastic heat conduction model including Atangana–Baleanu fractional derivative

Accurately characterizing the thermomechanical parameters of nanoscale systems is essential for understanding their performance and building innovative nanoscale technologies due to their distinct behaviours. Fractional thermal transport models are commonly utilized to correctly depict the heat transfer that occurs in these nanoscale systems. The current study presents a novel mathematical thermoelastic model that incorporates a new fractional differential constitutive equation for heat conduction. This heat equation is useful for understanding the effects of thermal memory. An application of a fractional-time Atangana–Baleanu (AB) derivative with a local and non-singular kernel was utilized in the process of developing the mathematical model that was suggested. To deal with effects that depend on size, nonlocal constitutive relations are introduced. Furthermore, in order to take into consideration, the viscoelastic behaviour of the material at the nanoscale, the fractional Kelvin–Voigt model is utilized. The proposed model is highly effective in properly depicting the unusual thermal conductivity phenomena often found in nanoscale devices. The study also considered the mechanical deformation, temperature variations, and viscoelastic characteristics of the functionally graded (FG) nanostructured beams. The consideration was made that the material characteristics exhibit heterogeneity and continuous variation across the thickness of the beam as the nanobeam transitions from a ceramic composition in the lower region to a metallic composition in the upper region. The complicated thermomechanical features of simply supported viscoelastic nanobeams that were exposed to harmonic heat flow were determined by the application of the model that was constructed. Heterogeneity, nonlocality, and fractional operators are some of the important variables that contribute to its success, and this article provides a full study and illustration of the significance of these characteristics. The results that were obtained have the potential to play a significant role in pushing forward the design and development of tools, materials, and nanostructures that have viscoelastic mechanical characteristics and graded functions.

Generalized thermoelastic damping model for small-scale beams with circular cross section in the framework of nonlocal dual-phase-lag heat equation

Generalized thermoelastic damping model for small-scale beams with circular cross section in the framework of nonlocal dual-phase-lag heat equation