Multi-time-scale heat transfer modeling of turbid tissues exposed to short-pulsed irradiations

A combined hyperbolic radiation and conduction heat transfer model is developed to simulate multi-time-scale heat transfer in turbid tissues exposed to short-pulsed irradiations. An initial temperature response of a tissue to an ultrashort pulse irradiation is analyzed by the volume-average method in combination with the transient discrete ordinates method for modeling the ultrafast radiation heat transfer. This response is found to reach pseudo steady state within 1 ns for the considered tissues. The single pulse result is then utilized to obtain the temperature response to pulse train irradiation at the microsecond/millisecond time scales. After that, the temperature field is predicted by the hyperbolic heat conduction model which is solved by the MacCormack's scheme with error terms correction. Finally, the hyperbolic conduction is compared with the traditional parabolic heat diffusion model. It is found that the maximum local temperatures are larger in the hyperbolic prediction than the parabolic prediction. In the modeled dermis tissue, a 7% non-dimensional temperature increase is found. After about 10 thermal relaxation times, thermal waves fade away and the predictions between the hyperbolic and parabolic models are consistent.     

A new MHD code with adaptive mesh refinement and parallelization for astrophysics

A new code, named MAP, is written in FORTRAN language for magnetohydrodynamics (MHD) simulations with the adaptive mesh refinement (AMR) and Message Passing Interface (MPI) parallelization. There are several optional numerical schemes for computing the MHD part, namely, modified Mac Cormack Scheme (MMC), Lax–Friedrichs scheme (LF), and weighted essentially non-oscillatory (WENO) scheme. All of them are second-order, two-step, component-wise schemes for hyperbolic conservative equations. The total variation diminishing (TVD) limiters and approximate Riemann solvers are also equipped. A high resolution can be achieved by the hierarchical block-structured AMR mesh. We use the extended generalized Lagrange multiplier (EGLM) MHD equations to reduce the non-divergence free error produced by the scheme in the magnetic induction equation. The numerical algorithms for the non-ideal terms, e.g., the resistivity and the thermal conduction, are also equipped in the code. The details of the AMR and MPI algorithms are described in the paper.     

A spacetime discontinuous Galerkin method for hyperbolic heat conduction

Non-Fourier conduction models remedy the paradox of infinite signal speed in the traditional parabolic heat equation. For applications involving very short length or time scales, hyperbolic conduction models better represent the physical thermal transport processes. This paper reviews the Maxwell-Cattaneo-Vernotte modification of the Fourier conduction law and describes its implementation within a spacetime discontinuous Galerkin (SDG) finite element method that admits jumps in the primary variables across element boundaries with arbitrary orientation in space and time. A causal, advancing-front meshing procedure enables a patch-wise solution procedure with linear complexity in the number of spacetime elements. An h-adaptive scheme and a special SDG shock-capturing operator accurately resolve sharp solution features in both space and time. Numerical results for one spatial dimension demonstrate the convergence properties of the SDG method as well as the effectiveness of the shock-capturing method. Simulations in two spatial dimensions demonstrate the proposed method’s ability to accurately resolve continuous and discontinuous thermal waves in problems where rapid and localized heating of the conducting medium takes place.     

An Explicit Implicit Scheme for Cut Cells in Embedded Boundary Meshes

We present a new mixed explicit implicit time stepping scheme for solving the linear advection equation on a Cartesian cut cell mesh. We use a standard second-order explicit scheme on Cartesian cells away from the embedded boundary. On cut cells, we use an implicit scheme for stability. This approach overcomes the small cell problem—that standard schemes are not stable on the arbitrarily small cut cells—while keeping the cost fairly low. We examine several approaches for coupling the schemes in one dimension. For one of them, which we refer to as flux bounding, we can show a TVD result for using a first-order implicit scheme. We also describe a mixed scheme using a second-order implicit scheme and combine both mixed schemes by an FCT approach to retain monotonicity. In the second part of this paper, extensions of the second-order mixed scheme to two and three dimensions are discussed and the corresponding numerical results are presented. These indicate that this mixed scheme is second-order accurate in \(L^1\) and between first- and second-order accurate along the embedded boundary in two and three dimensions.     

A direct-forcing immersed boundary method for the thermal lattice Boltzmann method

In this study, a direct-forcing immersed boundary method (IBM) for thermal lattice Boltzmann method (TLBM) is proposed to simulate the non-isothermal flows. The direct-forcing IBM formulas for thermal equations are derived based on two TLBM models: a double-population model with a simplified thermal lattice Boltzmann equation (Model 1) and a hybrid model with an advection–diffusion equation of temperature (Model 2). As an interface scheme, which is required due to a mismatch between boundary and computational grids in the IBM, the sharp interface scheme based on second-order bilinear and linear interpolations (instead of the diffuse interface scheme, which uses discrete delta functions) is adopted to obtain the more accurate results. The proposed methods are validated through convective heat transfer problems with not only stationary but also moving boundaries – the natural convection in a square cavity with an eccentrically located cylinder and a cold particle sedimentation in an infinite channel. In terms of accuracy, the results from the IBM based on both models are comparable and show a good agreement with those from other numerical methods. In contrast, the IBM based on Model 2 is more numerically efficient than the IBM based on Model 1.     

Gibbs Phenomena

In this note we show that when a discontinuous initial value problem for a scalar hyperbolic equation in one space variable is approximated by a difference scheme that is more than first order accurate; it leads to overshoots analogous to the Gibbs phenomenon when discontinuous functions are approximated by sections of Fourier series. A hybrid scheme due to Harten and Zwass removes the overshoots. Similar phenomena occur when solving schemes of hyperbolic equations.To David Gottlieb, master of scientific computation, subtle numerical analyst, Mensch extraordinaire.     

Development of a sixth-order two-dimensional convection–diffusion scheme via Cole–Hopf transformation

In this paper, we develop a two-dimensional finite-difference scheme for solving the time-dependent convection–diffusion equation. The numerical method exploits Cole–Hopf equation to transform the nonlinear scalar transport equation into the linear heat conduction equation. Within the semi-discretization context, the time derivative term in the transformed parabolic equation is approximated by a second-order accurate time-stepping scheme, resulting in an inhomogeneous Helmholtz equation. We apply the alternating direction implicit scheme of Polezhaev to solve the Helmholtz equation. As the key to success in the present simulation, we develop a Helmholtz scheme with sixth-order spatial accuracy. As is standard practice, we validated the code against test problems which were amenable to exact solutions. Results show excellent agreement for the one-dimensional test problems and good agreement with the analytical solution for the two-dimensional problem.     

Size effect on heat transfer in micro gas sensors

Gas gap is usually used as an important thermal insulation in micro gas sensors to reduce the heating power. The heat transport through the gap consists of two parts, heat conduction by air and thermal radiation between surfaces. It is usually regarded that thermal radiation through the gap is negligible compared with conductive heat transfer by air. This work investigates the heat transport by thermal radiation and heat conduction through a broad size range of gas gaps from one nanometer to dozens of micrometers. The result shows that thermal radiation is the major way of heat transfer when the gap is less than 20 nm, which will result in unexpected high energy consumption in the process of minimization. The equivalent thermal conductivity of thermal radiation is computed and a partition map is depicted to demonstrate the relative importance of radiation and conduction on different gap scales under dissimilar surface temperatures. A practical gas sensor heated by a micro hotplate (MHP) is thermally analyzed. The calculation shows that extra energy consumption comes forth as the gap distance reduces to several tens of nanometers.     

Examination of thermo-physical and material property interactions in cereal foams by means of Boltzmann modeling techniques

Cereal foam is a high complex material undergoing several temperature-dependent processes under thermal treatment, such as phase transitions, biochemical reactions and structural changes. Simultaneous heat and mass transfer plays an important role to investigate optimization studies in cereal-based foams. In porous media such as cereal foams, thermal conduction is of minor impact on the overall heat transfer, since the major part of heat is transferred through the gas phase filled with water vapor. This becomes evident comparing the thermal diffusivities of solid and gaseous components of the foam, where the difference is in the order of five magnitudes. The objective of this study is to model the coupled heat and mass diffusion processes in cereal-based foam under thermal treatment by means of Lattice Boltzmann methods. The proposed model is then used to perform parameter variation studies, showing the impact of material property changes offering the possibility on optimizing heat transfer through the foam.     

A CCD-ADI method for two-dimensional linear and nonlinear hyperbolic telegraph equations with variable coefficients

In this paper, a combined compact finite difference method (CCD) together with alternating direction implicit (ADI) scheme is developed for two-dimensional linear and nonlinear hyperbolic telegraph equations with variable coefficients. The proposed CCD-ADI method is second-order accurate in time variable and sixth-order accurate in space variable. For the linear hyperbolic equation, the CCD-ADI method is shown to be unconditionally stable by using the Von Neumann stability analysis. Numerical results for both linear and nonlinear hyperbolic equations are presented to illustrate the high accuracy of the proposed method.     

Simulations of the wind field in Athens with the nonhydrostatic mesoscale model MEMO

MEMO is a fully vectorized nonhydrostatic mesoscale model using terrain-following coordinates. The numerical solution is based on second-order discretization applied on a staggered grid which is allowed to be non-equidistant in all directions. Special care is taken that conservative propeprties are preserved within the discrete model equations. The discrete pressure equation is solved with a direct elliptic solver in conjuction with a generalized conjugate gradient method. Advective terms are treated with an explicit, monotonicity-preserving discretization scheme with only small implicit diffusion. Turbulent diffusion is described using an one-equation turbulence model, while at roughness height similarity theory is applied. An efficient scheme is applied to calculate radiative transfer. The algebraic surface heat budget equation and an one dimensional heat conduction equation are solved to obtain the surface temperature over land and the soil temperature. In the frame of the APSIS exercise A the model MEMO was used to simulate the mesoscale flow in the Athens basin on May 25, 1990. Being in satisfactory agreement with observations, the results indicate that weak pressure gradients accompanied by warm advection aloft may lead to stagnant conditions and thus to severe air pollution episodes in Athens.     

HPS整流管芯片散热效率计算仿真研究

针对传统散热效率计算方法存在计算时间过长、计算效率过低以及计算误差偏高等问题,提出了一种新的散热效率计算方法———基于傅里叶导热方程的散热效率计算方法。通过对高功率半导体整流管芯片进行分析,引用傅里叶导热方程计算出整流管芯片的传热热阻,根据传热热阻随着温度的变化,获取高功率半导体整流管芯片散热系数。根据高功率半导体整流管芯片散热系数,构建高功率半导体整流管芯片散热模型,利用建立的模型分析转速、冷气流入口的压力和速度以及冷却孔的分布等对转子温度场的以及散热效率的影响,优化散热路径,完成高功率半导体整流管芯片散热计算。实验结果表明,所提方法有效减少了计算时间,提高了计算效率,与此同时,降低了计算误差,使计算结果更为准确。     

Simulation of radiation expansion of laser plasma in an external magnetic field

Radiation expansion of laser plasma in an external magnetic field is investigated in the paper. A two-dimensional system of ideal magneto-hydrodynamics with radiation transfer in a cylindrical system of coordinates was solved numerically using second-order conservative TVD difference scheme by space and time. A multigroup flux-limited diffusion scheme was applied for the solution of the radiative transfer equation. At the initial moment, the heating of a target, consisting of vapors of aluminum was implemented by a short-action laser pulse with a duration time of 30 nanoseconds and Gaussian profile by space with a half-thickness of 0.03 centimeters. Cases that take into account, as well as those that do not take into account, the radiation transfer and the magnetic field effects are considered. The numerical simulations show that inclusion of radiation transfer changes the dynamics of laser expansion quantitatively and qualitatively.     

Efficient transfer and concentration of energy between explosive dual bubbles via time-delayed interactions

The dynamics of a high heat flux thermal bubble is constrained by the thermal energy carried on the bubble surface right after the bubble formation because of thermal isolation of vapor. This article proposes a way by assigning time delays between dual bubbles to transfer effectively energy from one bubble into the other, thus, breaks energy limitation that one single bubble can usually carry. Experiment result has demonstrated that the useful work as large as 40% can be transferred from one bubble into the other for the ignition time delay set between 2 and 3 μs in a dual bubble system. At the same time, the total extractable useful work in a dual bubble system is 20% higher than twice that of a single-bubble system with the same input heat energy. This phenomenon opens up a new way to transfer or concentrate energies from distributed energy sources with limit energy density into a much higher one for higher power application.     

On the Linear Stability of the Fifth-Order WENO Discretization

We study the linear stability of the fifth-order Weighted Essentially Non-Oscillatory spatial discretization (WENO5) combined with explicit time stepping applied to the one-dimensional advection equation. We show that it is not necessary for the stability domain of the time integrator to include a part of the imaginary axis. In particular, we show that the combination of WENO5 with either the forward Euler method or a two-stage, second-order Runge–Kutta method is linearly stable provided very small time step-sizes are taken. We also consider fifth-order multistep time discretizations whose stability domains do not include the imaginary axis. These are found to be linearly stable with moderate time steps when combined with WENO5. In particular, the fifth-order extrapolated BDF scheme gave superior results in practice to high-order Runge–Kutta methods whose stability domain includes the imaginary axis. Numerical tests are presented which confirm the analysis.     

Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method II: Two dimensional case

A class of fifth-order weighted essentially non-oscillatory (WENO) schemes based on Hermite polynomials, termed HWENO (Hermite WENO) schemes, for solving one dimensional non-linear hyperbolic conservation law systems, was developed and applied as limiters for the Runge-Kutta discontinuous Galerkin (RKDG) methods in [J. Comput. Phys. 193 (2003) 115]. In this paper, we extend the method to solve two dimensional non-linear hyperbolic conservation law systems. The emphasis is again on the application of such HWENO finite volume methodology as limiters for RKDG methods to maintain compactness of RKDG methods. Numerical experiments for two dimensional Burgers’ equation and Euler equations of compressible gas dynamics are presented to show the effectiveness of these methods.     

Simulation of three-dimensional nonideal MHD flow at high magnetic Reynolds number

A conservative TVD scheme is adopted to solve the equations governing the three-dimensional flow of a nonideal compressible conducting fluid in a magnetic field.The eight-wave equations for magnetohydrodynamics(MHD) are proved to be a non-strict hyperbolic system,therefore it is difficult to develop its eigenstructure.Powell developed a new set of equations which cannot be numerically simulated by conservative TVD scheme directly due to its non-conservative form.A conservative TVD scheme augmented with a ne...     

热泡式喷墨的建模与分析

喷墨过程的数值建模是建立在一维热传导公式,温度—压强关系式和能量、物质转换关系的平衡的基础上的。一维热传导公式用来考虑气泡和它周围液体之间的能量交换以及气—液交界层的温度分布。温度—压强关系式和能量转换关系用来考虑气泡的增长和破裂过程。气泡产生的初始温度、初始压强以及控制电压等参数都在本模型中进行分析,分析的结果为设计喷墨打印机提供了最基础的资料。     

热泡式喷墨的建模与分析

喷墨过程的数值建模是建立在一维热传导公式，温度-压强关系式和能量、物质转换关系的平衡的基础上的。一维热传导公式用来考虑气泡和它周围液体之间的能量交换以及气-液交界层的温度分布。温度-压强关系式和能量转换关系用来考虑气泡的增长和破裂过程。气泡产生的初始温度、初始压强以及控制电压等参数都在本模型中进行分析，分析的结果为设计喷墨打印机提供了最基础的资料。     

Simulation of a simple reacting gas flow with an upwind scheme

Good results have been obtained using the Random Choice Method (RCM) in the computation of reacting gas flow problems. The RCM is an unfamiliar method and difficult to program. The question arises as to whether a simpler difference approximation can obtain as effective results with less computational difficulty. Among all difference schemes upwind methods have been proven to have excellent properties. Thus, such methods serve as models for the effectiveness of all difference schemes.A standard upwind scheme modified to include a fractional heat conduction step is used to compute solutions of one dimensional compressible fluid flow equations with a finite heat conduction coefficient. The gas is assumed to be chemically reacting and thus to deposit energy in the field. Comparison is made to the known qualitative behavior of the solutions for different ratios of the reaction rate and the heat conduction coefficient. This difference scheme is seen to compare unfavorably with the RCM.