One-dimensional Finite Difference Method Research Articles

Estimation of Electromagnetic Radiation Distribution in the Human Heart Rate at Different Frequencies‏‏

The health implications of electromagnetic radiation have been extensively debated due to the sharp increase in the usage of cell phones, Wi-Fi transmitters and other microwave equipment. Electromagnetic radiation affects the autonomic nervous system, heart rate, blood pressure and other cardiovascular functions. This study aims to present a Numerical Simulation of Electromagnetic Radiation (EMR) on the human heart tissue and to explore the effect of different frequencies in the spectral range (900, 1,800 and 2,400 MHz) on Specific Absorption Rate (SAR), power density, the Distribution of Electromagnetic Fields by Matlab program, and Finite-Difference Time-Domain (FDTD) method in One Dimension (1D). A 1-dimensional finite difference method was used to solve Maxwell’s equations in heart tissue. The heart model was subjected to electromagnetic radiation that has dielectric properties according to frequency. Frequency was chosen and the operation of the software program as Matlab and the dielectric attribute has been calculated. Concurring with the Frequency in a private program, are the conductivity, relative permittivity, wavelength and infiltration profundity. The amplitudes of the reflected and transmitted sinusoid waves, relative to the occurrence wave, were portrayed by the reflection coefficient and the transmission coefficient, which relate to the amplitudes of the electric field wave. The electric field, magnetic field and power density were simulated along the line X = Y = 0. The study showed that the impact of electromagnetic radiation depends on the frequency of the wave which hit the heart. It was found that the heart tissue reacts more at 900 MHz compared to 1,800 and 2,400 MHz, and the tissue absorption is higher at the lower frequency. There was a relation between the magnetic field in the y-dimension and the time step the in x-dimension. HIGHLIGHTS The effect of electromagnetic has been theoretically studied where the electromagnetic fields produced from electromagnetic radiation at 900 MHz,1800 MHz, and 2400 MHz on layered biological tissues (heart model) by the finite difference time domain (FDTD) method A one-dimensional finite difference method was used to solve Maxwell's equations in heart tissue The amplitudes of the reflected and transmitted sinusoid waves, relative to the occurrence wave, were portrayed by the reflection coefficient and the transmission coefficient, which relate to the amplitudes of the electric field wave The impact of electromagnetic radiation depends on the frequency of the wave which hit the heart. There was a relation between the magnetic field in the y-dimension and the time step the in x-dimension GRAPHICAL ABSTRACT

A Methodology to Reduce Thermal Gradients Due to the Exothermic Reactions in Resin Transfer Molding Applications

Resin transfer molding (RTM) is among the most used manufacturing processes for composite parts. Initially, the resin cure is initiated by heat supply to the mold. The supplementary heat generated during the reaction can cause thermal gradients in the composite, potentially leading to undesired residual stresses which can cause shrinkage and warpage. In the present numerical study of these processes, a one-dimensional finite difference method is used to predict the temperature evolution and the degree of cure in the course of the resin polymerization; the effect of some parameters on the thermal gradient is then analyzed, namely: the fiber nature, the use of multiple layers of reinforcement with different thermal properties and also the temperature cycle variation. The validity of this numerical model is tested by comparison with experimental and numerical results in the existing literature.

Finite Difference Method Simulation on Effect of Blood Perfusion on Temperature Distribution of Female Human Skin

<p><span lang="EN-US">Bioheat</span><span lang="EN-US"> transfer is a combination of heat transfer and biology process comprising of transfer of heat from human living tissue to the environment. This study aims to estimate the temperature distribution of female human’s skin. In this study, effect of varied blood perfusions of 1.5, 2.0, 2.5 and 3.0 kg/s.m<sup>3 </sup>on bioheat transfer of female human skin, which was modeled into four layers, was simulated. To address this issue, one-dimensional unsteady state finite difference method was applied in this study. Result of this study informed that the finite difference method can be used to solve the bioheat transfer equation and to estimate temperature distribution of different skin layers of female human. In this study, blood perfusion affects temperature distribution in the four layers of unstable women's skin. The highest blood perfusion of 3 kg/s.m<sup>3</sup> resulted in more homogeny temperature distribution of different skin layers. This circumstance occurred since higher blood perfusion increases heat transfer between blood and skin layers.</span></p>

Thermoplastic Composites: Modelling Melting, Decomposition and Combustion of Matrix Polymers

In thermoplastic composites, the polymeric matrix upon exposure to heat may melt, decompose and deform prior to burning, as opposed to the char-forming matrices of thermoset composites, which retain their shape until reaching a temperature at which decomposition and ignition occur. In this work, a theoretical and numerical heat transfer model to simulate temperature variations during the melting, decomposition and early stages of burning of commonly used thermoplastic matrices is proposed. The scenario includes exposing polymeric slabs to one-sided radiant heat in a cone calorimeter with heat fluxes ranging from 15 to 35 kW/m2. A one-dimensional finite difference method based on the Stefan approach involving phase-changing and moving boundary conditions was developed by considering convective and radiative heat transfer at the exposed side of the polymer samples. The polymers chosen to experimentally validate the simulated results included polypropylene (PP), polyester (PET), and polyamide 6 (PA6). The predicted results match well with the experimental results.

Thermal behavior of external-insulated cold-formed steel non-load-bearing walls exposed to different fire conditions

An external-insulated cold-formed steel (CFS) wall is a potential configuration that overcomes the disadvantage of cavity insulation in the fire performance of such walls and can meet the sound and thermal insulation requirements for building structures by increasing the thickness of insulation material. This paper conducted eight mid-scale non-load-bearing fire experiments on external-insulated wall specimens. Four different fire conditions were tested, including ISO 834 fire, external fire, hydrocarbon fire and a typical realistic design fire. A quasi-steady heat transfer state was identified at the final stage of external fire exposure. The effect of substituting the cavity insulation with external insulation on the fire performance is discussed by comparison with the previous fire experiments on cavity-insulated CFS walls. The results show that external insulation is a more reasonable configuration than cavity insulation for improving the fire resistance of CFS walls. In addition, two types of stud buckling, which include the local buckling of the whole stud section and local buckling of stud hot flange and adjacent web, were identified for non-load-bearing walls in fire, and the corresponding cause and conditions were described. In addition, to achieve high efficiency in the thermal performance modeling of CFS walls in fire, a quick and simplified calculation method was developed to get the temperature profile of steel frame based on a one-dimensional finite difference method. Moreover, an unexposed surface temperature of 900 °C was recommended as the criterion to predict the falling-off of the fire-side face-layer gypsum plasterboards of CFS walls in the heat transfer modeling.

Nonlinear solutions for laterally loaded piles

A nonlinear analysis framework for laterally loaded piles is presented that is as accurate as equivalent three-dimensional nonlinear finite element analysis, but computationally one order of magnitude faster. The nonlinear behavior of sands and clays are account for by using hyperbolic modulus–reduction relationships. These nonlinear–elastic constitutive models are used to calculate the reduced modulus at different points in the soil based on the soil strains induced by lateral pile displacement. The reduced modulus at different points in the soil domain are spatially integrated to calculate the reduced soil resistance parameters associated with the differential equation governing the lateral pile displacement. The differential equations governing the lateral displacements of pile and soil under equilibrium are obtained by applying the principle of virtual work to a continuum-based pile–soil system. These coupled differential equations are solved using the one-dimensional finite difference method following an iterative algorithm. The accuracy of the analysis is verified against equivalent three-dimensional nonlinear finite element analysis, and the validity of the analysis in predicting the field response is checked by comparisons with multiple pile load test results. Parametric studies are performed to gain insights into the lateral pile response.

Hybrid FEM and FDM Approach for Monolithic Catalytic Converters with Pore Diffusion

A transient, three-dimensional model is developed for monolith catalytic converters with washcoat pore diffusion. Due to the unique configuration of monolith converters, the flow within the channel can be considered fully developed laminar flow, and one-dimensional transport equations can be formulated for the flow and species. On the other hand, a three-dimensional equation is necessary for the monolith solid temperature to capture the effects of significant external heat loss or flow maldistribution. The proposed model employs a three-dimensional finite-element method (FEM) for the solid temperature equation and a one-dimensional finite-difference method (FDM) for the flow and species equations. A recently developed asymptotic approach (Bissett, Emiss. Control Sci. Technol. 1(1), 3–16, 2015, Bissett, Emiss. Control Sci. Technol. 5(1), 45–54, 2019) is utilized to solve the washcoat pore diffusion and surface coverages equations. The three-dimensional FEM meshes are chosen so the nodes also serve for the one-dimensional transport equations. All equations are solved fully coupled, without lagging or reliance upon external tools. The developed solver can efficiently simulate general cross-section geometries (circle, oval, rectangle, etc.) for a catalytic monolith under nonuniform inlet distribution and detailed washcoat pore diffusion. The solver is verified by comparing the simulation results to the one-dimensional simulation results for a simple geometry under uniform inlet conditions. The model can be applied to flow through general monolith catalytic reactors with single or dual washcoat layers and a wide range of kinetics.

A mechanistic approach to predict built-in temperature in concrete pavements

The built-in temperature difference (BITD), defined as the temperature difference between the top and bottom of a slab at the final setting time, is an important parameter for analysing curling stress and slab deformation in concrete pavements. However, the available methods for estimating this parameter are very limited. A method to predict the BITD in a concrete pavement was therefore developed. To do this, a numerical model was developed to predict the temperature distribution in a concrete slab at early age using a transient one-dimensional finite-difference method. A mathematical equation for predicting concrete final setting time presented in the literature was used and incorporated in the numerical model to predict the BITD. The results of the numerical model showed good agreement with field data. Using the proposed model, the effects of climatic conditions, placement time, pavement thickness and concrete mix proportion on the BITD were also evaluated. The results showed that climatic conditions, placement time and concrete mix proportion have substantial effects on the BITD, whereas pavement thickness has only a slight effect. The proposed model can be used to predict the BITD in concrete slabs for given concrete mix design, placement time, pavement thickness and environmental conditions.

Numerical Modelling of Frost Formation of Flat Surface

Frost formation begins when humid air comes in contact with a cold surface kept below freezing temperature of water. Objective of the present study is to develop a numerical model which can predict frost formation parameters such as rate of frost growth, frost densification and to study effect of ambient conditions on these parameters. The one-dimensional pure implicit finite difference method is adopted for solving differential equations. Numerical code is written in MATLAB 2013. The proposed numerical model is validated against two independent published experimental data with 6.6 and 16.8% deviation. Effect of ambient parameters like wall temperature, ambient temperature, humidity and Reynolds number on frost growth and densification are investigated. Also effect of variable wall temperature and variable ambient temperature on frost growth rate is discussed.

Co-simulation of a HVAC system-integrated phase change material thermal storage unit

A co-simulation environment, consisting of a detailed mathematical model of a thermal energy storage unit which is incorporated with an EnergyPlus simulation model of a full building HVAC system, is described. The two models are integrated using the user-defined plant component feature in EnergyPlus and the Building Controls Virtual Test Bed (BCVTB) environment. The thermal energy storage unit, which consists of encapsulated phase change material in a series of flat plates and a heat transfer working fluid (water), is modelled using a transient one-dimensional forward finite difference method. The thermal storage model is executed within MATLAB and is verified against experimental data, showing a discharging heat transfer accuracy to within 2.5%. The building model, which incorporates a retrofitted ground source heat pump system within a thermally massive building, is simulated in the EnergyPlus environment. The co-simulation arrangement allows for in-depth analysis of the integrated system under dynamic operating conditions, which is currently not possible within the EnergyPlus environment. Moreover, the overall adopted approach, based on generic integration of a detailed mathematical model, using a third party generalised programming environment, into an established building simulation environment, serves as a successful exemplar for other researchers and practitioners working in the field.

Design and economic analysis of a flue gas condenser to recover latent heat from exhaust flue gas

In this study, we examined the event of condensing water vapour in flue gas in order to recoverthe latent heat that is carried by exhaust flue gas in 60 MW natural-gas fired district heating system of a university. With this, we aimed to design theoretically a waste heat recovery unit and conduct an economic analysis of it. For this study, actual operation data of the system were taken. In the design calculations of flue gas condenser, we used one-dimensional Finite-difference Method and made a computer programme. According to calculation results, exhaust flue gas temperature can be decreased to 40 °C with a flue gas condenser that has 80 m2 surface area and is made of 316 quality stainless steel horizontal plain tube bundles. Annual fuel saving was calculated as $ 407,396.16 with Annual Equivalent Analysis Method.

An Efficient Method for Solving Spread Option Pricing Problem: Numerical Analysis and Computing

This paper deals with numerical analysis and computing of spread option pricing problem described by a two-spatial variables partial differential equation. Both European and American cases are treated. Taking advantage of a cross derivative removing technique, an explicit difference scheme is developed retaining the benefits of the one-dimensional finite difference method, preserving positivity, accuracy, and computational time efficiency. Numerical results illustrate the interest of the approach.

A compact LOD method and its extrapolation for two-dimensional modified anomalous fractional sub-diffusion equations

A Crank–Nicolson-type compact locally one-dimensional (LOD) finite difference method is proposed for a class of two-dimensional modified anomalous fractional sub-diffusion equations with two time Riemann–Liouville fractional derivatives of orders (1−α) and (1−β)(0<α,β<1). The resulting scheme consists of simple tridiagonal systems and all computations are carried out completely in one spatial direction as for one-dimensional problems. This property evidently enhances the simplicity of programming and makes the computations more easy. The unconditional stability and convergence of the scheme are rigorously proved. The error estimates in the standard H1- and L2-norms and the weighted L∞-norm are obtained and show that the proposed compact LOD method has the accuracy of the order 2min{α,β} in time and 4 in space. A Richardson extrapolation algorithm is presented to increase the temporal accuracy to the order min{α+β,4min{α,β}} if α≠β and min{1+α,4α} if α=β. A comparison study of the compact LOD method with the other existing methods is given to show its superiority. Numerical results confirm our theoretical analysis, and demonstrate the accuracy and the effectiveness of the compact LOD method and the extrapolation algorithm.

Thermal diffusivity imaging

The tip of a rod is heated with a torch and brought into contact with the center of a metal sheet. A thermal camera is then used to image the temperature profile of the surface as a function of time. The infrared camera is capable of recording radiometric data with 1 mK resolution in nearly 105 pixels, so thermal diffusion can be monitored with unprecedented precision. With a frame rate of approximately 10 Hz, the pace of the data acquisition minimizes the loss of accuracy due to inevitable cooling mechanisms. We report diffusivity constants equal to 1.23 ± 0.06 cm2/s in copper and 0.70 ± 0.05 cm2/s in aluminum. The behavior is modeled with a straightforward but oddly under-utilized one-dimensional finite difference method.

Examining rhyolite lava flow dynamics through photo-based 3D reconstructions of the 2011–2012 lava flowfield at Cordón-Caulle, Chile

During the 2011–2012 eruption at Cordón-Caulle, Chile, an extensive rhyolitic flowfield was created (in excess of 0.5km3 in volume), affording a unique opportunity to characterise rhyolitic lava advance. In 2012 and 2013, we acquired approximately 2500 digital photographs of active flowfronts on the north and east of the flowfield. These images were processed into three-dimensional point clouds using structure-from-motion and multi-view stereo (SfM–MVS) freeware, from which digital elevation models were derived. Sequential elevation models—separated by intervals of three hours, six days, and one year—were used to reconstruct spatial distributions of lava velocity and depth, and estimate rheological parameters. Three-dimensional reconstructions of flowfronts indicate that lateral extension of the rubbly, 'a'ā-like flowfield was accompanied by vertical inflation, which differed both spatially and temporally as a function of the underlying topography and localised supply of lava beneath the cooled upper carapace. Compressive processes also drove the formation of extensive surface ridges across the flowfield. Continued evolution of the flowfield resulted in the development of a compound flowfield morphology fed by iterative emplacement of breakout lobes. The thermal evolution of flow units was modelled using a one-dimensional finite difference method, which indicated prolonged residence of magma above its glass transition across the flowfield. We compare the estimated apparent viscosity (1.21–4.03×1010Pas) of a breakout lobe, based on its advance rate over a known slope, with plausible lava viscosities from published non-Arrhenian temperature–viscosity models and accounting for crystallinity (~50vol.%). There is an excellent correspondence between viscosity estimates when the lava temperature is taken to be magmatic, despite the breakout being located >3km from the vent, and advancing approximately nine months after vent effusion ceased. This indicates the remarkably effective insulation of the lava flow interior, providing scope for significant evolution of rhyolitic flow fields long after effusive activity has ceased.

Simulation-based Study of Layered Aluminum Crystal Microstructures Subjected to Shock Loading

A one-dimensional finite difference method allowing for anisotropic deformation is used in conjunction with a nonlinear thermoelastic-viscoplastic material model to compute the shock response of various microstructural instantiations of pure aluminum at peak stresses exceeding the Hugoniot Elastic Limit (HEL). Single crystals and layered bi-materials consisting of grains with alternating orientations relative to the direction of shock propagation -- specifically [100], [111], or low-symmetry orientations -- are impacted to peak shock stresses on the order of 5GPa. The [111] orientation [111] is observed to be stiffest both plastically and elastically, while the [100] orientation is found to be most compliant. Layered bi-materials that only demonstrate pure longitudinal waves exhibit average shock stresses, entropy production, and internal energy in between values computed for their single crystal constituents. Layered bi-materials that generate both quasi-longitudinal and quasi-transverse waves results in lower peak stresses and higher internal energy than their single crystal constituents. In bi-material systems, stress fluctuations decrease in frequency with increasing layer thickness, and peak stress amplitudes increase with layer thickness. Average dissipation depends on orientation but is relatively insensitive to layer thickness. Results of the computational method may ultimately be used to guide design of metallic systems with microstructures tailored for optimal impact resistance.

A compact locally one-dimensional method for fractional diffusion-wave equations

This paper is concerned with numerical methods for a class of multi-dimensional fractional diffusion-wave equations with a time fractional derivative of order $$\alpha $$ ( $$1<\alpha <2$$ ). A compact locally one-dimensional (LOD) finite difference method is proposed for the equations. The resulting scheme consists of one-dimensional tridiagonal systems, and all computations are carried out completely in one spatial direction as for one-dimensional problems. The unconditional stability and $$H^{1}$$ norm convergence of the scheme are rigorously proved for the three-dimensional case. The error estimates show that the proposed compact LOD method converges with the order $$(3-\alpha )$$ in time and $$4$$ in space. Numerical results confirm our theoretical analysis and illustrate the effectiveness of this new method.

A higher-order compact LOD method and its extrapolations for nonhomogeneous parabolic differential equations

A higher-order compact locally one-dimensional (LOD) finite difference method for two-dimensional nonhomogeneous parabolic differential equations is proposed. The resulting scheme consists of two one-dimensional tridiagonal systems, and all computations are implemented completely in one spatial direction as for one-dimensional problems. The solvability and the stability of the scheme are proved almost unconditionally. The error estimates are obtained in the discrete H1,L2 and L∞ norms, and show that the proposed compact LOD method has the accuracy of the second-order in time and the fourth-order in space. Two Richardson extrapolation algorithms are presented to increase the accuracy to the fourth-order and the sixth-order in both time and space when the time step is proportional to the spatial mesh size. Numerical results demonstrate the accuracy of the compact LOD method and the high efficiency of its extrapolation algorithms.

MBE Growth of 2.3μm InGaAsSb/AlGaAsSb Strained Multiple Quantum Well Diode Lasers

2.3 μm InGaAsSb/AlGaAsSb lasers with multiple quantum wells(MQWs) have been demonstrated.The growth temperature of quantum wells is 440°C,and the growth quality of InGaAsSb/AlGaAsSb MQWs is examined by X-ray diffraction and Photoluminescence(PL) at room temperature.The energy band structure of MQWs was calculated by one-dimensional finite-difference method(1D-FDM)

Calculation of Microstructure in Vacuum Carburizing Incorporating Kinetics Modeling of Grain-boundary Cementite

In the vacuum carburization of steels, short-time carburization is usually followed by a diffusion period to eliminate the filmy cementite (θ) grown on austenite (γ) grain boundaries. The surface C content in γ was found to increase past the solubility of θ due to radical reaction with decomposed hydro-carbon gas. The thermodynamic condition was recognized as the metastable equilibrium of supersaturated γ with graphite. Based on these results, a calculation model for not only C concentration profiles but also quantitative evaluation of grain-boundary θ was developed.The existing model estimates the amount of θ with the equilibrium fractions for local C contents in a framework of the finite difference method (FDM). However, with the corresponding surface condition of equilibrium γ plus θ with graphite, it overrates the amount of θ observed after several minutes of carburization. In the developed model, a parabolic law was assumed for the thickening of θ and a rate controlling process was considered for partitioning of Si. The local Si concentrations at the γ/θ interface were determined from the C isoactivity condition with supersaturated γ. The change in the rate coefficient (αSi) was also validated using multicomponent diffusion simulations. Coupled with the CALPHAD software, a one-dimensional FDM program calculates the increment of θ using the updated local equilibrium. Predictions of the C profile and θ volume fraction corresponded much better with the experimental results than existing models.