Conduction Problem Research Articles

An analysis of the suppression of a pair of odd and even temperature fluctuation in an area occupied by a two-phase one-directional periodically layered composite

The study concerns a heat conduction problem in a two-phase, one-directional periodically layered composite (commonly referred to as a laminate). The model is strictly related to considerations regarding continuous media and was introduced by Professor Czesław Woźniak. The basis for the study is a re-formulation of the heat conduction model for composites, referred to in the literature as the extended tolerance heat conduction model. The model equations consider coefficients responsible for the intensity of exponential and rotational suppression of boundary fluctuations of a temperature field in a limited area or in a half-space occupied by the composite. The study investigated the dependence of the suppression intensity of boundary fluctuations on the geometrical and material properties of the composite. Three special cases of pairs of interacting boundary fluctuations were analysed.

Stable recovery of piecewise constant conductance on spider networks

ABSTRACT We address the discrete inverse conductance problem for well-connected spider networks, that is, to recover the conductance function on a well-connected spider network from the Dirichlet-to-Neumann map. It is well-known that this inverse problem is exponentially ill-posed, requiring the implementation of a regularization strategy for numerical solutions. Our focus lies in exploring whether prior knowledge of the conductance being piecewise constant within a partition of the edge set comprising a few subsets enables stable conductance recovery. To achieve this, we propose formulating the problem as a polynomial optimization problem, incorporating a regularization term that accounts for the piecewise constant hypothesis. We show several experimental examples in which the stable conductance recovery under the aforementioned hypothesis is feasible.

An accelerated sequential function specification method based on LM gradient for transient inverse heat conduction problem

The sequential function specification method (SFSM) and the Levenberg-Marquardt method (LM) are two typical methods for inverse heat conduction problems (IHCP). However, these two methods have poor convergence and low convergence speed. Therefore, the combination of LM with SFSM is depended on to propose a modified Sequential Levenberg-Marquardt gradient method (SLM). The computation results of three shapes of heat fluxes with noise show the SLM is averagely improved over 16.3% more than SFSM in computation accuracy, but equally reduced over 39.9% less than in computation time. On other hand, when the emphasis is on the effect of the Aitken acceleration method, Tikhonov regularization and polynomial order on SLM, a neural network prediction model of the root mean square error is trained by regularization factor, polynomial order and the number of future time steps. Subsequently, the genetic algorithm optimization method is proposed for the minimum root mean square error so that it can improve the computation accuracy and efficiency of IHCP to the maximum extent. Finally, Second-order polynomial order and small order-of-magnitude regularization factor are highly recommended for parameter selection with multiple choices. The ASLM only needs to determine the optimal number of future time steps.

Oxygen-deficient ZnVOH@CC as high-capacity and stable cathode for aqueous zinc-ion batteries

Vanadium oxides have become promising cathodes for aqueous zinc ion batteries (AZIBs) due to their low cost and high theoretical capacity. However, vanadium oxides suffer from the problems of low electrical conductivity and structural instability. To address the challenges, ZnxV2O5·nH2O with oxygen-rich vacancies was synthesized in situ on carbon cloth (ZnVOH@CC) via a one-step hydrothermal method. The introduction of Zn2+ and oxygen vacancies is beneficial to enhance the structural stability and improve the Zn2+ diffusion rate. The pre-intercalated Zn2+/H2O acts as “pillars” to increase the interlayer spacing of vanadium pentoxide (V2O5), weakening the electrostatic interactions between Zn2+ and the ZnVOH host lattice. Benefiting from the above advantages, the ZnVOH@CC composite was used as the cathode for AZIBs, which showed a high capacity of 464.9 mAh/g at 0.1 A/g, a large capacity of 208.1 mAh/g at 1.0 A/g after 1,000 cycles, and a good cycling stability at a high current density of 5.0 A/g after 10,000 cycles. In addition, the assembled ZnVOH@CC//Cu2Se full cell by matching ZnVOH@CC with Cu2Se also exhibits a high reversible capacity of 68 mAh/g at 1.0 A/g and 92.5 % capacity retention after 260 cycles.

Finite element solution of heat conduction in complex 3D geometries

This paper presents the use of the finite element method (FEM) to solve heat conduction problems in complex 3-dimensional geometries not amenable to analytical solutions. Heat conduction is important across engineering domains, but closed-form solutions only exist for basic shapes. For intricate real-world component geometries, numerical techniques like FEM must be applied. The paper outlines the mathematical formulation of FEM, starting from the heat conduction governing equations. The domain is discretized into a mesh of interconnected finite elements. Element equations are derived and assembled into a global matrix system relating nodal temperatures. Boundary conditions are imposed and the matrix equations solved to find the temperature distribution. An example problem analyzes steady state conduction in an L-shaped block with 90-degree corners and surface convection. Results show FEM can capture localized gradients and discontinuities difficult to model otherwise. Detailed temperature contours provide insight. FEM enables robust thermal simulation of complex 3D geometries with localized effects, expanding analysis capabilities beyond basic analytical shapes. Proper application of FEM is critical for accurate results.

Geometry-independent spline finite element method to analyze two-dimensional heat conduction and elasticity problems

Traditional isogeometric analysis involves redundant geometric calculations and is more complex than the finite element method when imposing boundary conditions. To address this, a geometry-independent spline finite element method is proposed, which integrates the geometric construction techniques of isogeometric analysis and the element construction techniques of finite element method. Firstly, the geometrical precision element is constructed by using non-uniform rational B-spline and B-splines to describe the geometry and the physical field, where a transformation matrix is introduced to construct shape functions with interpolation properties. Secondly, calculation formats for two-dimensional heat conduction and elasticity problems are derived by parameter mapping. Finally, the feasibility and accuracy of the method are verified through several numerical examples. Unlike in isogeometric analysis, in the proposed method, geometry and physics are separated, and the shape functions have interpolation properties, which reduce redundant geometry calculations and allow the boundary conditions to be applied directly to the nodes. The numerical results indicate that the method has a higher convergence rate compared to the original isogeometric analysis.

A hybrid shape functions based temporal finite element method to solve transient heat conduction problems

A novel temporal finite element method is presented to solve Fourier and non-Fourier heat conduction problems. Two kinds of temporal finite element models are developed using Gurtin variational principle and weighted residual technique, respectively. A kind of hybrid shape function with polynomial and trigonometric basis is stressed to give more flexible and appropriate description of temporal behavior of temperature. A recursive algorithm is developed by which the temperature solution at a specific time can be obtained only via a matrix power product with the initial condition, and a criterion of stability analysis is derived, which is can numerically be conducted when the constitution of shape function and time step size are prescribed. The proposed approach is available for Fourier and non-Fourier heat transfer problems, and can conveniently be combined with well-developed numerical algorithms for the boundary value problem, such as FE, SBFEM, etc. Various numerical examples, including those with the nonlinear thermal conductivity, radiative boundary condition, etc. are provided to illustrate the efficiency of proposed approaches, and impacts of the temporal FE model, constitution of shape functions, and step size, etc. are taken into account. Satisfactory results are achieved in comparison with the analytical and the ABAQUS based numerical solutions.

Regularized kernel function methods for the backward heat conduction problem

In this letter, we are interested in the regularized numerical techniques for backward heat conduction problems, which aim at the detection of the previous status of physical field from its present information. It is well known that backward heat conduction problems are ill-posed. The small perturbation in the final status may result in large change in the previous status of physical field. It is difficult to get the stable numerical solutions of backward heat conduction problems. Based on the reproducing kernel theory, the letter aims at finding a new regularized kernel function methods for backward heat conduction problems.

Physics-Informed Fully Convolutional Networks for Forward Prediction of Temperature Field and Inverse Estimation of Thermal Diffusivity

Abstract Physics-informed neural networks (PINNs) are a novel approach to solving partial differential equations (PDEs) through deep learning. They offer a unified manner for solving forward and inverse problems, which is beneficial for various engineering problems, including heat transfer analysis. However, traditional PINNs suffer from low accuracy and efficiency due to the fully-connected neural network framework and the method to incorporate physical laws. In this paper, a novel physics-informed learning architecture, named physics-informed fully convolutional networks (PIFCNs), is developed to simultaneously solve forward and inverse problems in thermal conduction. The use of fully convolutional networks (FCNs) significantly reduces the density of connections. Thus, the computational cost is reduced. With the advantage of the nodal-level match between inputs and outputs in FCNs, the output solution can be used directly to formulate discretized PDEs via a finite difference method, which is more accurate and efficient than the traditional approach in PINNs. The results demonstrate that PIFCNs can flexibly implement Dirichlet and Neumann boundary conditions to predict temperature distribution. Remarkably, PIFCNs can also estimate unknown thermal diffusivity with an accuracy exceeding 99%, even with incomplete boundaries and limited sampling data. The results obtained from PIFCNs outperform those obtained from PINNs.

Analytic Modelling of 2-D Transient Heat Conduction with Heat Source Under Mixed Boundary Constraints by Symplectic Superposition

Abstract Many studies have been conducted on 2-D transient heat conduction, but analytic modelling is still uncommon for the cases with complex boundary constraints due to the mathematical challenge. With an unusual symplectic superposition method, this paper reports new analytic solutions to 2-D isotropic transient heat conduction problems with heat source over a rectangular region under mixed boundary constraints at an edge. With the Laplace transform, the Hamiltonian governing equation is derived. The applicable mathematical treatments, e.g., the variable separation and the symplectic eigenvector expansion in the symplectic space, are implemented for the fundamental solutions whose superposition yields the ultimate solutions. Benchmark results obtained by the present method are tabulated, with verification by the finite element solutions. Instead of the conventional Euclidean space, the present symplectic-space solution framework has the superiority on rigorous derivations without pre-determining solution forms, which may be extended to more issues with the complexity caused by mixed boundary constraints.

Numerical Computation of 2D Domain Integrals in Boundary Element Method by (α, β) Distance Transformation for Transient Heat Conduction Problems

When the time-dependent boundary element method, also termed the pseudo-initial condition method, is employed for solving transient heat conduction problems, the numerical evaluation of domain integrals is necessitated. Consequently, the accurate calculation of the domain integrals is of crucial importance for analyzing transient heat conduction. However, as the time step decreases progressively and approaches zero, the integrand of the domain integrals is close to singular, resulting in large errors when employing standard Gaussian quadrature directly. To solve the problem and further improve the calculation accuracy of the domain integrals, an (α, β) distance transformation is presented. Distance transformation is a simple and efficient method for eliminating near-singularity, typically applied to nearly singular integrals. Firstly, the (α, β) coordinate transformation is introduced. Then, a new distance transformation for the domain integrals is constructed by replacing the shortest distance with the time step. With the new method, the integrand of the domain integrals is substantially smoothed, and the singularity arising from small time steps in the domain integrals is effectively eliminated. Thus, more accurate results can be obtained by the (α, β) distance transformation. Different sizes of time steps, positions of source point, and shapes of integration elements are considered in numerical examples. Comparative studies of the numerical results for the domain integrals using various methods demonstrate that higher accuracy and efficiency are achieved by the proposed method.

SINGULAR INTEGRAL EQUATIONS FOR A CRACK SUBJECTED NORMAL STRESS IN A HEATED PLATE

In this paper, a crack in a heated plate is investigated, subjected to normal stress. Employing the relationship between the uniform and perturbation fields, as well as complex potential functions and stresses, the problems of heat conduction and heat stress are modeled as singular integral equations. The derivatives of the crack opening displacement function and the temperature jump function serve as the unknown functions. Gauss integration rules are applied to solve the obtained equations numerically. Analysis of the stress intensity factors(SIFs) for some particular crack configurations is presented.

Noninvasive reconstruction of internal heat source in biological tissue using adaptive simulated annealing algorithm

The heat distribution information of human lesions is of great value for disease analysis, diagnosis, and treatment. It is a typical inverse problem of heat conduction that deriving the distribution of internal heat sources from the temperature distribution on the body surface. This paper transforms such an inverse problem of bio-heat transfer into a direct one, thereby avoiding complex boundary conditions and regularization processes. To noninvasively reconstruct the internal heat source and its corresponding 3D temperature field in biological tissue, the adaptive simulated annealing (ASA) algorithm is used in the simulation module, where the position P(x, y, z) of point heat source in biological tissue and its corresponding temperature T are set as the optimization variables. Under a certain optimized sample, one can obtain the simulated temperature distributing on the surface of the module, then subtract the simulated temperature from the measured temperature of the same surface which was measured using a thermal infrared imager. If the sum of absolute values of the difference is smaller, it indicates that the current sample is closer to the true location and temperature of the heat source. When the values of optimization variables are determined, the corresponding 3D temperature field is also confirmed. The simulation results show the simulated position and temperature of the heat source are very approximate with those of the real experimental module. The method presented in this paper has enormous potential and promising prospects in clinical research and application, such as tumor hyperthermia, disease thermal diagnosis technology, etc.

Analytical Solution for Transient Thermal Behavior of Two Semisolids With Contact Resistance and Interfacial Heat Generation

Abstract The problem of two dissimilar semi-infinite solids at different initial temperatures brought into contact has a well-known simple analytical solution. In this work, this problem is reexamined with the additional simultaneous complications of both contact resistance and surface heat generation. While contact resistance is always present to some degree due to surface asperities or oxidation layers, heat generation at the contact interface can also occur in certain situations. These situations can occur in applications such as ultrasonic welding or arise in situations involving electromagnetic radiation passing through an optically transparent medium, but dissipating as heat at an interface with an opaque material that is in contact with the transparent material. In this paper, an analytical solution to the unsteady conduction problem is developed that accounts for both contact resistance and interfacial heat generation. The solution confirms that the initially warmer object rapidly decreases in temperature in the vicinity of the interface as heat flows into the cooler object and the heat generated at the interface preferentially flows to the cooler material. After a short time, however, the temperatures of both materials at the interface increase in temperature above even the initial temperature of the initially hotter material. An experiment was performed that verified the analytical solution.

Estimation of Multiple Parameters in Semitransparent Mediums Based on an Improved Grey Wolf Optimization Algorithm

This work investigates the inverse coupled radiation–conduction problem for estimating thermophysical parameters and source terms by an improved grey wolf optimization (GWO). The transient coupled radiation–conduction heat transfer problem in participating slab media is solved by the finite volume method. The radiative intensities on both boundaries are adopted as known measurement information in the inverse model. To overcome the disadvantages of the original GWO algorithm, an improved grey wolf algorithm (IGWO) is developed by introducing the weight strategy and nonlinear factors. Three benchmark functions are adopted to prove that the IGWO has a faster convergence speed and higher estimation accuracy than the original one. The IGWO is applied to inverse the thermophysical parameters and source terms based on the coupled radiation–conduction model; the results indicate that the IGWO is accurate and effective for estimating refractive index, absorption coefficient, and source terms.

Inverse conductivity problem with one measurement: uniqueness of multi-layer structures

In this paper, we study the recovery of multi-layer structures in inverse conductivity problem by using one measurement. First, we define the concept of Generalized Polarization Tensors (GPTs) for multi-layered medium and show some important properties of the proposed GPTs. With the help of GPTs, we present the perturbation formula for general multi-layered medium. Then we derive the perturbed electric potential for multi-layer concentric disks structure in terms of the so-called generalized polarization matrix, whose dimension is the same as the number of the layers. By delicate analysis, we derive an algebraic identity involving the geometric and material configurations of multi-layer concentric disks. This enables us to reconstruct the multi-layer structures by using only one partial-order measurement.

Development and Generalization of the Method of Reflections in Problems of Electrostatics and Thermal Conductivity of Plane-Layered Media

Development and Generalization of the Method of Reflections in Problems of Electrostatics and Thermal Conductivity of Plane-Layered Media

Preparation and Properties of Stearic Acid/Nanocellulose Composite Phase Change Energy Storage Materials

To solve the problems of liquid phase leakage and poor thermal conductivity of organic phase change energy storage materials, a novel composite phase change energy storage material (SA/CNF) based on stearic acid (SA) and nanocellulose (CNF) was prepared in this study and added graphene to enhance its thermal conductivity. For the prepared composite phase change materials (SFs) with different ratios, shape stability, and testing experiments showed that the maximum adsorption rate of CNF on SA reached 80 %. The properties of SA/CNF were then characterized by various instruments. The results show that SA and CNF are compounded together by physical interaction. More than 93 % of SA in the composite phase change energy storage material SA/CNF is able to store and release heat through the phase change process, and the latent heat of phase change of pure SA is 206.3 J/g, while the latent heat of phase change of the composite phase change material SA/CNF with 20 % of CNF is 156.2 J/g, and they both have good cycling stability. The addition of graphene enhances the thermal conductivity of SA/CNF to a large extent, and the thermal conductivity at a graphene ratio of 5 % is 261 % higher than that of SA/CNF without graphene. SA/CNF is an environmentally friendly energy storage material with very high application prospects.

Gradient estimates for the insulated conductivity problem: The non-umbilical case

We study the insulated conductivity problem with inclusions embedded in a bounded domain in Rn, for n≥3. The gradient of solutions may blow up as ε, the distance between the inclusions, approaches to 0. We established in a recent paper optimal gradient estimates for a class of inclusions including balls. In this paper, we prove such gradient estimates for general strictly convex inclusions. Unlike the perfect conductivity problem, the estimates depend on the principal curvatures of the inclusions, and we show that these estimates are characterized by the first non-zero eigenvalue of a divergence form elliptic operator on Sn−2.

Numerical Reconstruction of Heat Source on a Semi-Infinite Rod

Inverse heat conduction on an unbounded domain has always been the focus and difficulty in engineering and mechanical fields. In this paper, we consider the inverse problem of identifying the heat source by using the terminal measured temperature data. Being different from other ordinary heat conduction problems, our mathematical model is defined in a semi-infinite region, which may lead to many numerical methods that cannot be used directly. Based on the artificial boundary method, we first introduce the truncated boundary and derive the exact boundary conditions on it. Then we use the finite difference method to establish an approximate difference scheme for the forward problem. For the inverse problem, we design the Landweber and conjugate gradient iterative algorithm and do numerical experiments. Numerical results verify the correctness and effectiveness of our algorithm.