Solving Heat Equations Research Articles

A High-Order Numerical Scheme for Efficiently Solving Nonlinear Vectorial Problems in Engineering Applications

In scientific and engineering disciplines, vectorial problems involving systems of equations or functions with multiple variables frequently arise, often defying analytical solutions and necessitating numerical techniques. This research introduces an efficient numerical scheme capable of simultaneously approximating all roots of nonlinear equations with a convergence order of ten, specifically designed for vectorial problems. Random initial vectors are employed to assess the global convergence behavior of the proposed scheme. The newly developed method surpasses methods in the existing literature in terms of accuracy, consistency, computational CPU time, residual error, and stability. This superiority is demonstrated through numerical experiments tackling engineering problems and solving heat equations under various diffusibility parameters and boundary conditions. The findings underscore the efficacy of the proposed approach in addressing complex nonlinear systems encountered in diverse applied scenarios.

Numerical Solution of Heat Equation using Modified Cubic B-spline Collocation Method

In this paper, a collocation method is presented based on the Modified Cubic B-spline Method (MCBSM) for the numerical solution of the heat equation. The PDE is fully discretized by using the Modified Cubic B-spline basis collocation for spatial discretization and the finite difference method is used for the time discretization. A numerical example from PDE is used to evaluate the accuracy of the proposed method. The numerical results are evaluated in comparison to the exact solutions. The findings consistently indicate that the suggested technique provides good error estimates. We also discovered that our proposed method was unconditionally stable. Hence, based on the results and the efficiency of the method, the method is suitable for solving heat equation.

Anisotropic damage behavior in fiber-based materials: Modeling and experimental validation

This study presents a thermodynamically consistent continuum damage model for fiber-based materials that combines elastoplasticity and damage mechanisms to simulate the nonlinear mechanical behavior under in-plane loading. The anisotropic plastic response is characterized by a non-quadratic yield surface composed of six sub-surfaces, providing flexibility in defining plastic properties and accuracy in reproducing material response. The damage response is modeled based on detailed uniaxial monotonic and cyclic tension-loaded experiments conducted on specimens extracted from a paper sheet in various directions. To account for anisotropic damage, we propose a criterion consisting of three sub-surfaces representing tension damage in the in-plane material principal directions and shear direction, where the damage onset is determined through cyclic loading tests. The damage evolution employs a normalized fracture energy concept based on experimental observation, which accommodates an arbitrary uniaxial loading direction. To obtain a mesh-independent numerical solution, the model is regularized using the implicit gradient enhancement by utilizing the linear heat equation solver available in commercial finite-element software. The study provides insights into the damage behavior of fiber-based materials, which can exhibit a range of failure modes from brittle-like to ductile, and establishes relationships between different length measurements.

Double Formable Integral Transform for Solving Heat Equations

Chemistry, physics, and many other applied fields depend heavily on partial differential equations. As a result, the literature contains a variety of techniques that all have a symmetry goal for solving partial differential equations. This study introduces a new double transform known as the double formable transform. New results on partial derivatives and the double convolution theorem are also presented, together with the definition and fundamental characteristics of the proposed double transform. Moreover, we use a new approach to solve a number of symmetric applications with different characteristics on the heat equation to demonstrate the usefulness of the provided transform in solving partial differential equations.

Aerothermal Computations for Laminar–Turbulent Transition Onset Measurement Using Infrared Imaging Technique

This paper deals with the computation of heat exchange in transitional boundary layers for the prediction of the laminar–turbulent transition detection using infrared thermography. An aerothermal coupling between a heat equation solver and a boundary-layer solver is presented here. This coupling is first used to provide guidelines in the academic framework of a flat plate, to achieve an enhanced temperature gradient at the transition onset in order to improve infrared imaging, and thus the transition position detection. The method is then successfully applied on a real case, by comparing the numerical predictions to measurements obtained during transonic wind-tunnel tests on a two-dimensional wing model.

A COMPARISON OF SERIAL AND PARALLEL SOLUTIONS OF TWO DIMENSIONAL HEAT CONDUCTION EQUATION

We study a comparison of serial and parallel solution of 2D-parabolic heat conduction equation using a Crank-Nicolson method with an Alternating Direction Implicit (ADI) scheme. The two-dimensional Heat equation is applied on a thin rectangular aluminum sheet. The forward difference formula is used for time and an averaged second order central difference formula for the derivatives in space to develop the Crank-Nicolson method. FORTRAN serial codes and parallel algorithms using OpenMP are used. Thomas tridigonal algorithm and parallel cyclic reduction methods are employed to solve the tridigonal matrix generated while solving heat equation. This paper emphasize on the run time of both algorithms and their difference. The results are compared and evaluated by creating GNU-plots (Command-line driven graphing utility).

Solving Fundamental Solution of Non-Homogeneous Heat Equation with Dirichlet Boundary Conditions

In this study, we developed a solution of nonhomogeneous heat equation with Dirichlet boundary conditions. moreover, the non-homogeneous heat equation with constant coefficient. since heat equation has a simple form, we would like to start from the heat equation to find the exact solution of the partial differential equation with constant coefficient. to emphasize our main results, we also consider some important way of solving of partial differential equation specially solving heat equation with Dirichlet boundary conditions. the main results of our paper are quite general in nature and yield some interesting solution of non-homogeneous heat equation with Dirichlet boundary conditions and it is used for problems of mathematical modeling and mathematical physics.

Coupled thermo-mechanical swelling of a thermo-responsive hydrogel hollow cylinder under extension-torsion: Analytical Solution and FEM

In this article, the mechanical swelling behavior of poly-(N-isopropylacrylamide) hydrogel is scrutinized considering a hollow circular cylinder subjected to temperature variation-extension-torsion. Accordingly, an analytical solution is presented to consider the general combined loading on temperature-sensitive hydrogel cylinder for two approaches, considering same temperature for whole structure and solving heat equation to compute internal temperature of structure. Additionally, to evaluate the proposed solution, finite element analysis has been conducted for same problem which revealed excellent conformity for various case studies. Therefore, a user-material subroutine, UHYPER, is implemented to be employed in finite element analysis in order to define swelling of PNIPAM hydrogels. This subroutine was validated using free swelling and constraint swelling with previous and their analytical solutions. Regarding the complexity of material and loading namely, nonlinear finite behavior of hydrogel as well as combined cooling-extension-torsional loading, various factors, cross-linked density, axial stretch, torsional twist and temperature variation, were investigated to clarify the swelling behavior of hydrogel. Solving heat equation, in the second approach, enables us to apply various thermal conditions on this structure, while in previous studies, the temperature of whole structure is considered to vary simultaneously with the same function. This approach can help researchers to examine diverse thermo-mechanicals problem for temperature-sensitive hydrogels.

Solving heat equations under convection boundary conditions by a high-performance space-time boundary shape functions method

To be a numerical method, the time-dependent convection boundary conditions are hard to be fulfilled exactly, which will deteriorate the accuracy of numerical solution. With this in mind, we develop novel algorithms to find the solutions for 1-D and 2-D heat equations, which can exactly satisfy the initial condition and convection boundary conditions. A new idea of space-time boundary shape functions (STBSFs) with two free parameters is introduced, whose existence are proven and they can automatically and exactly satisfy all the specified conditions. We let the STBSFs be the bases of the solution, which not only satisfy all the prescribed conditions automatically, but also can find solution simply by a collocation technique. Numerical examples confirm the high-performance of the STBSF methods (STBSFMs), which provide very accurate solutions and the CPU time is very saving.

Advantages, effectiveness and efficiency of using neodymium nanoparticles by 3d finite element method (FEM) as an optothermal human cancer cells, tissues and tumors treatment under synchrotron radiation

In the current study, thermoplasmonic characteristics of Neodymium nanoparticles with spherical, core–shell and rod shapes are investigated. In order to investigate these characteristics, interaction of synchrotron radiation emission as a function of the beam energy and Neodymium nanoparticles were simulated using 3D finite element method. Firstly, absorption and extinction cross sections were calculated. Then, increases in temperature due to synchrotron radiation emission as a function of the beam energy absorption were calculated in Neodymium nanoparticles by solving heat equation. The obtained results show that Neodymium nanorods are more appropriate option for using in optothermal human cancer cells, tissues and tumors treatment method.   Scanning Electron Microscope (SEM) image of Neodymium nanoparticles with 50000x zoom. Â

Simulation of Interaction of Synchrotron Radiation Emission as A Function of the Beam Energy and Plutonium Nanoparticles Using 3d Finite Element Method (Fem) as an Optothermal Human Cancer Cells, Tiss

In the current study, thermoplasmonic characteristics of Plutonium nanoparticles with spherical, core–shell and rod shapes are investigated. In order to investigate these characteristics, interaction of synchrotron radiation emission as a function of the beam energy and Plutonium nanoparticles were simulated using 3D finite element method. Firstly, absorption and extinction cross sections were calculated. Then, increases in temperature due to synchrotron radiation emission as a function of the beam energy absorption were calculated in Plutonium nanoparticles by solving heat equation. The obtained results show that Plutonium nanorods are more appropriate option for using in optothermal human cancer cells, tissues and tumors treatment method.

VENUS: A Vertical-Cavity Surface-Emitting Laser Electro-Opto-Thermal NUmerical Simulator

The properties of vertical-cavity surface-emitting lasers (VCSELs) are investigated by means of a multiphysical Vcsel Electro-opto-thermal NUmerical Simulator (VENUS). VENUS includes a three-dimensional vectorial electromagnetic code, a description of the quantum well optical response, a heat equation solver, and a quantum-corrected drift-diffusion simulator. The proposed suite includes coupling mechanisms often overlooked in VCSEL simulation and allows to reassess the impact of parameters, which can be critically dependent on implementation details inaccessible in commercial codes. The agreement with experimental results holds the promise of the application of this framework to the computer-aided design of innovative VCSEL concepts.

Modify Variational Iteration Method for Solving Heat Equations

The Variational Iteration method (VIM) is one of the semi approximation procedure to solve numerous differential problems. Many literatures shows that this method need to be rectify to get preferable results. This research would like to modify VIM on the replacement the initial value in each steps. Although this method enact the algorithm more complex, but modify VIM gives good outcomes peculiarly in long time intervals. We compare the solutions with the analytical solution using separation variable. The results denotes that both of the VIM and modify VIM have small error in solving heat equation in Dirichlet boundary. It could be seen that the modify VIM give better solutions instead of VIM.

An explicit/implicit Runge–Kutta-based PFEM model for the simulation of thermally coupled incompressible flows

A semi-explicit Lagrangian scheme for the simulation of thermally coupled incompressible flow problems is presented. The model relies on combining an explicit multi-step solver for the momentum equation with an implicit heat equation solver. Computational cost of the model is reduced via application of an efficient strategy adopted for the solution of momentum/continuity system by the authors in their previous work. The applicability of the method to solving thermo-mechanical problems is studied via various numerical examples.

Human Cancer Cells, Tissues and Tumors Treatment Using Dysprosium Nanoparticles

In present study, thermoplasmonic characteristics of dysprosium nanoparticles with spherical, coreshell and rod shapes are investigated. In order to investigate these characteristics, interaction of synchrotron radiation emission as a function of the beam energy and dysprosium nanoparticles were simulated using 3D finite element method. Firstly, absorption and extinction cross-sections were calculated. Then, increases in temperature due to synchrotron radiation emission as a function of the beam energy absorption were calculated in dysprosium nanoparticles by solving heat equation. The results show that the dysprosium nanorods are more appropriate option for using in optothermal human cancer cells, tissues and tumors treatment method.

Effectiveness of Einsteinium Nanoparticles in Optothermal Human Cancer Cells, Tissues and Tumors Treatment under Synchrotron Radiation

In the current study, thermoplasmonic characteristics of Einsteinium nanoparticles with spherical, core-shell and rod shapes are investigated. In order to investigate these characteristics, interaction of synchrotron radiation emission as a function of the beam energy and Einsteinium nanoparticles were simulated using 3D finite element method. Firstly, absorption and extinction cross sections were calculated. Then, increases in temperature due to synchrotron radiation emission as a function of the beam energy absorption were calculated in Einsteinium nanoparticles by solving heat equation. The obtained results show that Einsteinium nanorods are more appropriate option for using in optothermal human cancer cells, tissues and tumors treatment method. Scanning Electron Microscope (SEM) image of Einsteinium nanoparticles with 50000x zoom.

Finite Volume Method with Explicit Scheme Technique for Solving Heat Equation

The Finite Volume Method is one of the numerical procedures that uses integrals on each volume control. It can be used to solve complex problems. This research aims to implement the Finite Volume method with explicit scheme for solving the heat equations. The research methods are modelling of heat equations, integration on each control volume, discretization using explicit scheme, solving linier algebra system, simulation, and comparing with analytical solutions. The separation variable method is used to compare the results. The results showed that the finite volume method provides a good solution with an average error of 0.0016. This showed that both solutions are similar.

Assessment of Thermal Stress on Well Integrity as a Function of Size and Material Properties

Wellbore integrity is critical to long term carbon storage. During CO2 injection, changes in temperature may result in large stress variations that can damage the well, threatening its integrity. The different materials comprising the wellbore and near-wellbore environment (namely the casing, cement and surrounding rock) possess different thermal properties. Consequently, there can be considerable variations in both material properties and thermal gradients across these layers, resulting in different amounts of contraction and expansion of these materials. This may generate sufficient thermal stresses to lead to fracture within the cement and/or host rock, or delamination of the cement/casing or cement/rock interfaces. Downsized wellbore samples have been used in laboratory studies to investigate the failure modes and failure criteria during thermal cycling operations. However, it is not clear to what extent such results can be reliably used to predict well failure at field scale conditions. In this work, we conduct a parameter study involving the size and material properties of the wellbore samples subjected to different rates of thermal loading, with the objective of predicting how these parameters can affect the thermal stress in field scale well conditions.A state-of-the-art parallel multiscale, multiphysics code named GEOS, developed at Lawrence Livermore National Laboratory, was used to study the thermal response of wellbore materials. A finite element solver considering linear elastic materials was coupled with a finite volume heat equation solver to simulate the wellbore deformation and fracture during thermal cycling. To understand the effect of wellbore size on thermal fracturing, simulations were conducted at different height to diameter ratios. The wellbore sample size was systematically varied from the lab scale to field scale with several intermediate scales. The change in thermal stresses were compared for different scales. Additionally, the effect of elastic modulus and cooling rates were studied.

Analysis of the Modal Content Into Large-Mode-Area Photonic Crystal Fibers Under Heat Load

Yb-doped double-cladding photonic crystal fibers are one of the key enabling factors for the development of high-power fiber lasers, due to their capability to provide a very large-mode-area together with the effective suppression of high-order modes, while allowing strong pump absorption and efficient conversion. Thermal effects are currently considered as the main bottleneck for future power scaling; since beyond a certain average power, they allow guidance of high order modes and energy transfer to them, causing a sudden degradation of the beam quality. In this paper, the effects of the heat load on the modes of double cladding fibers are thoroughly analyzed with a full-vector modal solver based on the finite-element method with integrated steady-state heat equation solver. Fibers with different inner cladding designs are compared to provide a deeper understanding of the mechanisms beyond the mode reconfinement and coupling. The influence of the fiber design on the robustness of the single-mode regime with respect to fiber heating has been demonstrated, providing a clear picture of the complex interaction between modes. On the basis of simulation results it has been possible to group fiber modes into three families characterized by peculiar reaction to heating.

Method of Lines Transpose: High Order L-Stable ${\mathcal O}(N)$ Schemes for Parabolic Equations Using Successive Convolution

We present a new solver for nonlinear parabolic problems that is L-stable and achieves high order accuracy in space and time. The solver is built by first constructing a one-dimensional heat equation solver that uses fast $\mathcal O(N)$ convolution. This fundamental solver has arbitrary order of accuracy in space and is based on the use of the Green's function to invert a modified Helmholtz equation. Higher orders of accuracy in time are then constructed through a novel technique known as successive convolution (or resolvent expansions). These resolvent expansions facilitate our proofs of stability and convergence, and permit us to construct schemes that have provable stiff decay. The multidimensional solver is built by repeated application of dimensionally split independent fundamental solvers. Finally, we solve nonlinear parabolic problems by using the integrating factor method, where we apply the basic scheme to invert linear terms (that look like a heat equation), and make use of Hermite--Birkhoff in...