Solving Heat Equations Research Articles

Dual Series Method for Solving Heat Equation with Mixed Boundary Conditions

Paper is devoted to determine the solution of a non-stationary heat equation in an axial symmetry cylindrical coordinates subject to a nonhomogeneous mixed discontinuous boundary conditions of the first and of the second kind inside the disk of an finite surface cylinder. The solution of the given mixed boundary value problem is obtained with the aid of a classical methods and based on the application of a new type of a dual series equations (DSE) with a Bessel function of the first kind of order zero as a kernel. The solution of obtained DSE which is discussed in this paper is introduced to a Fredholm integral equation of the first kind.

Axially symmetric temperature problem for the body system cylinder-layer

The solution of the axially symmetric temperature problem for the body system cylinder-sphere, which lies on a rigid base with a circular notch in the case of isotropic materials was built. The thermal contact between the bodies is assumed as non-ideal. The developed method for contact problems solution is based on using Hankel integral transforms and Fourier method of separation of variables for solving heat equations. The solution of boundary value problem for finding temperature fields is reduced to the determination of some constants from the system of linear algebraic equations. As a result, formulas for determining the temperature fields at different temperature conditions on the lateral surfaces of the cylinder and the sphere were obtained. The influence of the contact conductance on the temperature fields distribution in the contact area between two bodies was investigated. Numerical calculations and solution analysis indicate that the contact conductance significantly affects the temperature fields distribution in the contact area of two bodies.

A finite difference alternating segment scheme of parallel computations for solving heat equation

In this paper, a numerical scheme named alternating segment Crank-Nikolson is used for solving heat equation. This scheme can be used directly on parallel computations. Truncation error and stability of the presented method is analyzed. Comparison in accuracy with the fully implicit Crank-Nikolson scheme is presented in numerical experiment.

One-dimensional heat transfer study of a fiber-reinforced wind epoxy composite system in vacuum assisted resin transfer molding process

To investigate the influence of curing behavior of a wind-epoxy resin in vacuum assisted resin transfer molding (VARTM) process, numerical analysis of the heat transfer study of VARTM process was established to characterize temperature distribution in one dimension by directly solving heat equation and was compared with the result of experiment. Differential scanning calorimeter (DSC) was applied to test curing kinetic parameters of the epoxy system, which was required to evaluate internal thermal source and analyze heat transfer equations. Two models, such as nth order curing model and autocatalytic model, were established to solve the heat transfer equation. Combining the theoretical results with nth order curing model and experiment, it can be known that in early stage, temperature distribution correlates well with the experiment results due to the dominant chemical-controlled reaction, while great discrepancy appears in the latter stage due to diffusion-controlled reaction taking over. The result of the heat equation solved by autocatalytic model correlates well with the experiment results. POLYM. COMPOS., 35:1031–1037, 2014. © 2013 Society of Plastics Engineers

Schwarz Waveform Relaxation for Heat Equations with Nonlinear Dynamical Boundary Conditions

We are interested in solving heat equations with nonlinear dynamical boundary conditions by using domain decomposition methods. In the classical framework, one first discretizes the time direction and then solves a sequence of state steady problems by the domain decomposition method. In this paper, we consider the heat equations at spacetime continuous level and study a Schwarz waveform relaxation algorithm for parallel computation purpose. We prove the linear convergence of the algorithm on long time intervals and show how the convergence rate depends on the size of overlap and the nonlinearity of the nonlinear boundary functions. Numerical experiments are presented to verify our theoretical conclusions.

Development of an efficient numerical method for solving heat equations applying He's variational iteration method

Development of an efficient numerical method for solving heat equations applying He's variational iteration method

FPGA computation of the 3D heat equation

Efficient solution of the heat equation is one of the recursive topics in computational physics. Over the years, different software solutions have been proposed, taking advantage of today’s impressive computing power of parallel machines. In this work, we consider a hybrid software–hardware approach making use of a field-programmable gate array platform as a heat equation solver that can be easily attached to a PC using a PCI bus with the goal of obtaining a portable system to be used during field experiments. The system has been successfully used for the non-destructive inspection of soils in mine detection applications based on infrared thermography techniques.

Variable late Neogene exhumation of the central European Alps: Low‐temperature thermochronology from the Aar Massif, Switzerland, and the Lepontine Dome, Italy

Several recent studies proposed an important increase in exhumation rate in the western European Alps since circa 5–4 Ma. In order to assess potential spatial differences in exhumation histories, we present new apatite fission track (AFT) and apatite (U‐Th)/He (AHe) ages from the central Aar Massif (Guttannen area, Switzerland) and the western Lepontine Dome (Formazza area, Italy). Internal U/Th zoning in apatites explains alpha‐ejection‐corrected AHe ages that are older than the corresponding AFT ages in this study. A qualitative interpretation of AFT and AHe age‐elevation relationships suggests a two‐phase (9–7 and 5–3 Ma) exhumation scenario affecting the central Alps, with a stronger expression of the Pliocene signal in the Formazza area. However, a quantitative evaluation of exhumation scenarios using the 3‐D heat equation solver Pecube highlights the existence of several other likely scenarios, casting doubt on the validity of a qualitative interpretation of the age‐elevation relationships. In Formazza, scenarios suggested by quantitative modeling include continuous denudation at a rate of ∼750 m/Ma and a one‐step exhumation rate change from 300 to 1000 m/Ma at 5 Ma. In Guttannen, they include continuous denudation at a rate of ∼400 m/Ma with valley deepening and two periods of higher exhumation rate (increasing from 300 to 700 m/Ma repeatedly at 9–7 and at 5–3 Ma). Contingent upon further flexural isostatic modeling, the magnitude of exhumation recorded in the axial region of the Alps since circa 5 Ma does not appear sufficient to solely explain the denudation recorded in the North Alpine Foreland Basin.

Efficient software–hardware 3D heat equation solver with applications on the non-destructive evaluation of minefields

This paper targets the efficient computational solution of the heat transfer processes that take place in the soil and at the soil–air interface and its use in non-destructive evaluation (NDE) techniques. In particular, the problem of the detection of plastic antipersonnel mines is considered. To this aim we projected a 3D finite-difference (FD) thermal model of the soil on a FPGA platform using Handel-C and VHDL. A speedup factor of 34 over a purely software solution is achieved, obtaining processing times that permit the use of the system on the field.

Non-destructive soil inspection using an efficient 3D software–hardware heat equation solver

Soil inspection by means of non-destructive evaluation (NDE) methods constitutes a subject of great research interest with practical implications beyond a particular domain of application. NDE methods often require the modelling of physical phenomena involving complex sets of equations whose computational solution has demanding requirements in terms of memory and computing power. One case of particular importance is that of the heat transfer equation, present in multiple situations. In this article, we will focus on NDE for soil inspection, with a particular application on the detection of shallowly buried antipersonnel mines in sandy soils. To this aim, a quasi-inverse procedure for the estimation of the depth of burial of the mine targets, and a full-inverse procedure for the characterization of the physical properties of other targets will be proposed. The efficient computational solution of these procedures is ensured via a software–hardware heat equation solver presented here.

Parallel finite difference schemes for heat equation based upon overlapping domain decomposition

Efficient parallel finite difference schemes (PFDS) are proposed for solving heat equation numerically. They are based upon the overlapping domain decomposition method. We give the optimal convergence order in error estimate analysis, which shows that we just need to iterate once or twice at each time level to reach the optimal convergence order. Numerical experiments also confirm the theoretical analysis.

Spin injection in thermally assisted magnetic random access memory

An integrated thermal, micromagnetic, spin-momentum-transfer (SMT) model was developed to study the effect of SMT on the programming current required for thermally assisted magnetic random access memory (MRAM). The thermal portion of the model is used to compute Joule heating by the spin-polarized current, and it is based on a Crank–Nicolson inhomogeneous heat equation solver. The magnetic portion of the model is based on a micromagnetic Langevin dynamic Landau–Lifshitz–Gilbert solver including SMT torque. Simulations of thermally assisted magnetization reversal of 0.09-μm MRAM elements, heated by passing current through the barrier separating the pinned and free layers, were performed. The free layer of the MRAM elements was switched using a magnetic field at fixed heating-SMT current bias. Results suggest that a spin-polarized heating current can be used to lower the programming current required to write thermally assisted MRAM if the direction of the heating current is properly synchronized with the reversal field.

Evolution of passive margin escarpments: What can we learn from low‐temperature thermochronology?

Recent studies integrating geomorphology, thermochronology, cosmogenic erosion rate estimates, and numerical modeling suggest that escarpment evolution may take place following two dramatically different modes: (1) parallel retreat from the escarpment's original position at the continent‐ocean boundary to its present‐day inland position and (2) formation‐in‐place by progressive downwearing of a plateau initially located between the coast and a preexisting inland drainage divide. Using a three‐dimensional finite element model to solve the heat transfer equation, we show that the mode of migration of a passive margin escarpment can be constrained by low‐temperature (apatite (U‐Th)/He) thermochronology. We first couple the heat equation solver to a surface processes model that predicts the two different escarpment evolution modes from only slightly different initial conditions. We predict (U‐Th)/He age distributions that are markedly different for the two scenarios. We perform a thorough investigation of the model behavior to determine under which circumstances thermochronological data can be used to constrain passive margin escarpment dynamics. These conditions include (a) a tall escarpment, (b) a high geothermal gradient, and/or (c) a low flexural rigidity of the lithosphere. We demonstrate that to determine the rate and mode of escarpment migration from low‐temperature thermochronology, one needs to collect samples along transects perpendicular as well as parallel to the escarpment. Tightest constraints on escarpment development are provided by (in ascending order) the minimum (U‐Th)/He age encountered seaward of the escarpment, the location of where the minimum age is found, the slope of the age‐distance relationship (in a direction perpendicular to the coast), and the slope of the age‐elevation relationship (from a transect parallel to the escarpment). We finally demonstrate that there are situations where thermochronological data sets do not provide constraints on the mode of escarpment migration, such as along the escarpment of southeastern Australia, where migration has possibly been very rapid. Using the Neighborhood Algorithm method, we are, however, able to extract from an existing apatite (U‐Th)/He data set very useful constraints on the evolution of the southeastern Australian escarpment, including the duration of the migration event (<15 Myr), the local geothermal gradient (32°–40°C km−1), and the effective elastic thickness of the underlying lithosphere (6–8 km).

On the performance of pure and impure parallel functional programs

This paper reports on the memory performance of parallel scientific algorithms, written in both pure and impure functional styles. The Id programming language is used, since it allows both pure and impure parallel functional programs to be expressed. The non-strict storage model of Id is introduced. The study focuses on two algorithms: the Dongarra Sorensen Eignensolver and the NAS FT three dimensional heat equation solver, based on FFTs. This study verifies the claim that functional languages allow a composition of programs from modules, exploiting the inter- and intra-module parallelism without the need for rewrinting these modules. But it also shows that memory use of pure functional programs can be excessive, and theat impure functional programs can be as memory-efficient as imperative implementations.