One-dimensional Heat Equation Research Articles

Numerical study of different methods applied to the one-dimensional transient heat equation

This article aims to compare the results obtained by applying three numerical methods: Explicit Euler, Crank-Nicolson,and Multi-stage (R11), in the one-dimensional heat diffusion transient equation with different initial and boundary conditions. The discretization process was performed using the finite difference method. In order to guarantee the convergence of the methods used, consistency and stability were verified by Lax theorem. The results are presented in graphs and tables that contain the data of the analytical solution and the numerical solutions. It was observed that the results obtained by R11 method generated solutions with minor errors.

Separation of singularities for the Bergman space and application to control theory

In this paper, we solve a separation of singularities problem in the Bergman space. More precisely, we show that if P⊂C is a convex polygon which is the intersection of n half planes, then the Bergman space on P decomposes into the sum of the Bergman spaces on these half planes. The result applies to the characterization of the reachable space of the one-dimensional heat equation on a finite interval with boundary controls. We prove that this space is a Bergman space of the square which has the given interval as a diagonal. This gives an affirmative answer to a conjecture raised in [17].

A Unified Analytical Approach to Fixed and Moving Boundary Problems for the Heat Equation

Fixed and moving boundary problems for the one-dimensional heat equation are considered. A unified approach to solving such problems is proposed by embedding a given initial-boundary value problem into an appropriate initial value problem on the real line with arbitrary but given functions, whose solution is known. These arbitrary functions are determined by imposing that the solution of the initial value problem satisfies the given boundary conditions. Exact analytical solutions of some moving boundary problems that have not been previously obtained are provided. Moreover, examples of fixed boundary problems over semi-infinite and bounded intervals are given, thus providing an alternative approach to the usual methods of solution.

Analysis of Different Boundary Conditions on Homogeneous One-Dimensional Heat Equation

Partial differential equations involve results of unknown functions when there are multiple independent variables. There is a need for analytical solutions to ensure partial differential equations could be solved accurately. Thus, these partial differential equations could be solved using the right initial and boundaries conditions. In this light, boundary conditions depend on the general solution; the partial differential equations should present particular solutions when paired with varied boundary conditions. This study analysed the use of variable separation to provide an analytical solution of the homogeneous, one-dimensional heat equation. This study is applied to varied boundary conditions to examine the flow attributes of the heat equation. The solution is verified through different boundary conditions: Dirichlet, Neumann, and mixed-insulated boundary conditions. the initial value was kept constant despite the varied boundary conditions. There are two significant findings in this study. First, the temperature profile changes are influenced by the boundary conditions, and that the boundary conditions are dependent on the heat equation’s flow attributes.

Computer Simulation System of Nonlinear Thermal Conductivity

The article discusses the computer simulation system of nonlinear processes that described by the one-dimensional nonstationary heat equation with power-law nonlinearity. The solution of the nonlinear heat equation is realized by meshless scheme, which is based on combination of the collocation method and radial basis functions. Radial basis functions are used to approximate the solution of a partial differential equation, while the collocation method is used to determine unknown coefficients. The computer simulation system allows is used the following radial basis functions: Gaussian, multiquadric, inverse quadratic, and inverse multiquadric. The computer simulation system allows setting the initial and boundary conditions of the boundary-value problem. In the computer simulation system, it is possible to set such parameters of the solution as the exponent in the nonlinear heat equation, the coefficient of thermal conductivity, the density, the specific heat at constant pressure, the size of the domain of the boundary-value problem, the distance between interpolation nodes, the time interval of the nonstationary boundary-value problem, the time step, and the shape parameter of the radial basis function. The visualization of an approximate solution of the one-dimensional nonstationary heat equation with power-law nonlinearity is realized in the form of the three-dimensional surface in the computer simulation system. The computer simulation system allows visualization of the solution at chosen time steps as three-dimensional plots. The results of numerical calculations are demonstrated by the benchmark problem that simulates the stopping of a heat wave front in a heat equation with power-law nonlinearity.

An accuracy-preserving numerical scheme for parabolic partial differential equations subject to discontinuities in boundary conditions

In this paper, we develop a method to alleviate the loss of accuracy that occurs when parabolic partial differential equations (PDEs), subject to discontinuous boundary conditions, are solved numerically. Employing the Keller box finite-difference method, we consider a benchmark case involving the linear one-dimensional transient heat equation, subject to discontinuous heat flux at one of the boundaries. The method we develop constitutes an improvement on earlier work which involved dropping a spatial grid point at each time step; moreover, we demonstrate that our new approach is able to maintain second-order accuracy for the solution and the numerical scheme, whereas it is in general not even possible to calculate the accuracy for either using the earlier approach. Furthermore, our method can also be used in principle for discontinuities in non-linear steady-state boundary-layer problems, such as those that occur in fluid mechanics and electrochemistry; some examples of these are given.

Boundary Control and Observation to Inverse Coefficient Problem for Heat Equation With Unknown Source and Initial Value

This article investigates an inverse problem of determining the spatially variable diffusion coefficient of a one-dimensional heat equation with an unknown spatial varying source term and initial value. The big challenge of the problem comes from the multiple unknowns and very limited available data that are only boundary control and boundary observation at one end, in addition to the ill-posed nature of the inverse problem. We first design a switch <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">on/off</small> boundary control and show that the diffusion coefficient can be uniquely determined by the boundary observation. Next, a stable numerical algorithm for reconstruction of the diffusion coefficient is proposed by means of the matrix pencil method and optimal perturbation regularization technique. Finally, some numerical examples are presented to illustrate the effectiveness of the proposed identification algorithm.

A comparison of predictive control strategies for a highly segmented injection mold tempering

High precision, warpage free injection molded parts are highly in demand despite posing a great challenge. According to the pvT-behavior of the polymer, local variations of temperatures and pressures lead to inhomogeneous shrinkage which results in warpage. Since the pressure is dependent on the flow path, a homogenization of the specific volume is aimed by controlling the local mold temperatures. Therefore, a mold with 18 individually controllable tempering zones was developed, each zone equipped with ceramic heating elements and liquid CO2 expansion chambers. Due to the high mass of the mold, thermal processes underlie significant delays. The precise actuation of the tempering zones therefore requires a suitable control strategy. Two control strategies for the tempering zones were investigated on a basis of experimental trials. The tempering zones were initially controlled by a PID controller since it is easy to use and has low hardware requirements. To yield superior control precision, a model predictive control approach (MPC) has been recently developed and implemented, based on a discretized one-dimensional Fourier heat equation. The control accuracy of both control strategies was tested with an asymmetrical mold tempering experiment, which showed that the MPC is far more precise than the PID control approach.

Radial Basis Functions Based Differential Quadrature Method for One Dimensional Heat Equation

In this paper, Radial basis functions based differential quadrature method has been presented for solving one-dimensional heat equation. First, the given solution domain is discretized using uniform discretization grid point in both direction and the derivative involving the spatial variable, x is replaced by the sum of the weighting coefficients times functional values at each grid points. Next by using properties of linear independence of vector space and Multiquadratic radial basis function we can find all waiting coefficient at each grid points of solution domain and we obtain first order linear system of ordinary differential equation with N by N square coefficient Matrices. Then, the resulting first order linear ordinary differential equation is solved by fifth-order Runge-Kutta method. To validate the applicability of the proposed method, one model example is considered and solved for different values of the shape parameter ‘c’ and mesh sizes in the direction of the temporal variable; t and fixed value of mesh size in the direction of spatial variable, x. Numerical results are presented in tables in terms of root mean square (E²), maximum absolute error (E^∞) and condition number K (A) of the system matrix. The numerical results presented in tables and graphs confirm that the approximate solution is in a good agreement with the exact solution.

Usage of the finite difference method for solving one-dimensional heat equations*

In this paper the one-dimensional heat equations with the heat generation arising in the associated fractal transient conduction is investigated. Analytical solutions are obtained by using the finite difference method (FDM). The method, in general, is easy to implement and it yields good results. In addition, the uniqueness and stability results are also discussed. Some illustrative examples are included to demonstrate the validity and applicability of the technique.

Explicit and Implicit Crandall’s Scheme for the Heat Equation with Nonlocal Nonlinear Conditions

In this paper, explicit and implicit Crandall's formulas are applied for finding the solution of the one-dimensional heat equation with nonlinear nonlocal boundary conditions. The integrals in the boundary equations are approximated by the composite Simpson quadrature rule. Here nonlinear terms are approximated by Richtmyer's linearization method. Finally, some numerical examples are given to show the effectiveness of the proposed method.

Coupled method for the solution of a one-dimensional heat equation with axial symmetry

In this article, we implement the Projected Differential Transform Method (PDTM) coupled with Laplace Transform Method (LTM), hereby referred to as LPDTM, to solve a one-dimensional heat model (equation) with axial symmetry. Using the proposed framework (LPDTM), the exact solution (results) are obtained with convenience. It is noted that the suggested methodology performs very well and gives rapidly converging series solutions with less computational activities.

МОДЕЛЮВАННЯ ПРОЦЕСУ СФЕРОЇДИЗАЦІЇ ЧАСТИНОК ПОРОШКУ ПЛАЗМОВО-ДУГОВИМ МЕТОДОМ

Багато сучасних технологічних процесів вимагають використання металевих, керамічних та метало-керамічних порошків, частинки яких мають ідеальну сферичну форму. Отже, виникає задача ефективної сфероїдизації порошків, що використовуються. Найбільш ефективним способом сфероїдизації є обробка порошку в потоці низькотемпературної плазми. Для отримання частинок ідеальної сферичної форми необхідно, щоб до кінця дистанції частинка мала мінімальну швидкість і температуру, щоб уникнути деформації під час удару об поверхню для збору порошку. Крім того, в процесі польоту частинка повинна повністю розплавитись, але не досягати температури випаровування, а потім затвердіти. Задача моделювання процесу сфероїдизації частинок порошку плазмово-дуговим методом зводиться до визначення швидкості і температури частинки в потоці низькотемпературної плазми з урахуванням зміни її агрегатного стану. Визначення швидкості частинки проведено з урахуванням того, що єдиною силою, що діє на неї, є сила аеродинамічного опору. Швидкість плазмового потоку апроксимуємо експоненціальною функцією. В результаті, отримуємо неявний розв’язок диференціального рівняння для визначення швидкості частинки вздовж дистанції. Для дослідження температурного режиму розглядаємо п'ять ділянок дистанції. На першій з них (розігрів частинки до температури плавлення) використовуємо для визначення температури рівняння теплового балансу. Друга ділянка – процес плавлення частинки. Тут ми визначаємо час плавлення частинки, виходячи з рівняння теплового балансу, за умови відсутності теплового потоку на межі фазового переходу. Третя ділянка – політ частинки в розплавленому стані. Для моделювання процесу розв’язуємо крайову задачу для одновимірного рівняння теплопровідності для сферичного тіла, апроксимуючи температуру плазми кубічними сплайнами. Для четвертої ділянки (затвердіння частки) використана та ж модель, що і для другого. А на п'ятому (охолодження частки) - такий самий підхід, що й на третьому. На завершення наведені результати для частинок титану діаметром 10 мкм.

Numerical Solution of Parabolic Partial Differential Equation by Using Finite Difference Method

In the real world, many physical problems like heat equation, wave equation, Laplace equation and Poisson equation are modeled by partial differential equations (PDEs). A PDE of the form ut = α uxx, (α &gt; 0) where x and t are independent variables and u is a dependent variable; is a one-dimensional heat equation. This is an example of a prototypical parabolic equation. The heat equation has analytic solution in regular shape domain. If the domain has irregular shape, computing analytic solution of such equations is difficult. In this case, we can use numerical methods to compute the solution of such PDEs. Finite difference method is one of the numerical methods that is used to compute the solutions of PDEs by discretizing the given domain into finite number of regions. Here, we derived the Forward Time Central Space Scheme (FTCSS) for this heat equation. We also computed its numerical solution by using FTCSS. We compared the analytic solution and numerical solution for different homogeneous materials (for different values of diffusivity α). There is instantaneous heat transfer and heat loss for the materials with higher diffusivity (α) as compared to the materials of lower diffusivity. Finally, we compared simulation results of different non-homogeneous materials.

VARIATIVE SOLUTION OF THE COEFFICIENT INVERSE PROBLEM FOR THE HEAT EQUATIONS

One of the main types of inverse problems for partial differential equations are problems in which the coefficients of the equations or the quantities included in them must be determined using some additional information. Such problems are called coefficient inverse problems for partial differential equations. Coefficient inverse problems (identification problems) have become the subject of close study, especially in recent years. Interest in them is caused primarily by their important applied values. They find applications in solving problems of planning the development of oil fields (determining the filtration parameters of fields), in creating new types of measuring equipment, in solving problems of environmental monitoring, etc. The standard formulation of the coefficient inverse problem contains a functional (discrepancy), physics. When formulating the statements of inverse problems, the statements of direct problems are assumed to be known. The solution to the problem is sought from the condition of its minimum. Inverse problems for partial differential equations can be posed in variational form, i.e., as optimal control problems for the corresponding systems. A variational statement of one coefficient inverse problem for a one-dimensional heat equation is considered. By the solution of the boundary value problem for each fixed control coefficient we mean a generalized solution from the Sobolev space. The questions of correctness of the considered coefficient inverse problem in the variational setting are investigated.

Fast Detection of Heat Accumulation in Powder Bed Fusion Using Computationally Efficient Thermal Models.

The powder bed fusion (PBF) process is a type of Additive Manufacturing (AM) technique which enables fabrication of highly complex geometries with unprecedented design freedom. However, PBF still suffers from manufacturing constraints which, if overlooked, can cause various types of defects in the final part. One such constraint is the local accumulation of heat which leads to surface defects such as melt ball and dross formation. Moreover, slow cooling rates due to local heat accumulation can adversely affect resulting microstructures. In this paper, first a layer-by-layer PBF thermal process model, well established in the literature, is used to predict zones of local heat accumulation in a given part geometry. However, due to the transient nature of the analysis and the continuously growing domain size, the associated computational cost is high which prohibits part-scale applications. Therefore, to reduce the overall computational burden, various simplifications and their associated effects on the accuracy of detecting overheating are analyzed. In this context, three novel physics-based simplifications are introduced motivated by the analytical solution of the one-dimensional heat equation. It is shown that these novel simplifications provide unprecedented computational benefits while still allowing correct prediction of the zones of heat accumulation. The most far-reaching simplification uses the steady-state thermal response of the part for predicting its heat accumulation behavior with a speedup of 600 times as compared to a conventional analysis. The proposed simplified thermal models are capable of fast detection of problematic part features. This allows for quick design evaluations and opens up the possibility of integrating simplified models with design optimization algorithms.

Измерение существенно нестационарных тепловых потоков градиентным датчиком на основе висмута

In this work we performed calibration of the gradient heat flux sensor made of bismuth single crystal. The value of the volt-watt sensitivity of the sensor is found and a data processing method based on the one-dimensional heat equation for a thin plate is presented. An experimental study of the heat flux on the frontal surface of the cylinder after a laser discharge was carried out. The data obtained as a result of the experiment were processed by the proposed method.

A Dufort–Frankel scheme for one-dimensional uncertain heat equation

Uncertain heat equation (for short, UHE) is a type of second-order uncertain PDEs driven by Liu processes. In most cases, since it is tough to get the analytic solution for a UHE, we must find a way to obtain its numerical solution. A forward difference scheme has been designed to solve UHE, but our paper will show that this method may exhibit instability in some situations. We explore another approach, unconditionally stable and namely Dufort–Frankel method. Moreover, this paper will use Dufort–Frankel method to calculate the expected value and extreme value of the solution for a UHE.

Numerical simulation of the period of surface micro-protrusions generated on titanium and stainless steel targets by femtosecond laser irradiation

This article reports our simulation results on a period of surface micro-protrusions, which are generated on titanium and stainless steel 304 target surfaces by femtosecond pulsed laser irradiation. The period of the generated micro-protrusions for varying laser fluence level has been estimated using an approach derived from the linear hydrodynamic Kuramoto Sivashinsky model. Some of the parameters, needed for calculating the period of surface micro-protrusions, have been estimated by numerically solving one-dimensional heat equations. Temperature evolution inside the target upon irradiation with a femtosecond laser pulse has been simulated using the two temperature model until the time electron and lattice subsystems attain thermal equilibrium. Thereafter, temperature evolution in the target has been simulated by defining a single temperature of the target at every position and time. We have validated our theoretical model by comparing simulated variation of period of surface micro-protrusions with incident laser fluence, ablation depth per pulse, and time required for thermalization between electrons and lattice subsystems with the reported experimental data for titanium target. Subsequently, the validated model has been used to simulate the period of surface micro-protrusions which are generated on the stainless steel 304 target via femtosecond laser irradiation.

Thermoporoelasticity analysis of variable thickness and elastically restrained functionally graded auxetic metamaterial circular plate resting on an auxetic material circular plate

The main purpose of this research is to investigate the thermoporoelastic behavior of non-homogeneous circular plate made up of auxetic metamaterial rested on an auxetic material plate. The plate is non-uniform, restrained with flexible support at outer edge and subjected to structure-structure interaction and thermo-mechanical load. The plate material is graded in transverse direction and the Poisson's ratio is constant. Based on the linear thermoporoelasticity theory, the governing state equations in terms of displacements and temperature are acquired. At first, the one-dimensional heat equation including quadratic type heat source is solved to compute the nonlinear distribution of temperature across the thickness. Then, the state equations are solved semi-analytically using state-space (SS) method and differential quadrature (DQ) rule to predict the elastic field quantities. A comprehensive parametric discussion is accomplished to elucidate the effects of key parameters on the steady-state response of the plate to mechanical and thermal loads. The computed results indicate that the auxeticity of the structure and supporting medium, as well as the thermal environment, play a significant role in the thermoporoelastic behavior of the plate.