Analytic approach to stochastic cellular automata: exponential and inverse power distributions out of Random Domino Automaton

Inspired by extremely simplified view of the earthquakes we propose the stochastic domino cellular automaton model exhibiting avalanches. From elementary combinatorial arguments we derive a set of nonlinear equations describing the automaton. Exact relations between the average parameters of the model are presented. Depending on imposed triggering, the model reproduces both exponential and inverse power statistics of clusters.Comment: improved, new material added; 9 pages, 3 figures, 2 table

Radiation heat transfer model using Monte Carlo ray tracing method on hierarchical ortho-Cartesian meshes and non-uniform rational basis spline surfaces for description of boundaries

The paper deals with a solution of radiation heat transfer problems in enclosures filled with nonparticipating medium using ray tracing on hierarchical ortho-Cartesian meshes. The idea behind the approach is that radiative heat transfer problems can be solved on much coarser grids than their counterparts from computational fluid dynamics (CFD). The resulting code is designed as an add-on to OpenFOAM, an open-source CFD program. Ortho-Cartesian mesh involving boundary elements is created based upon CFD mesh. Parametric non-uniform rational basis spline (NURBS) surfaces are used to define boundaries of the enclosure, allowing for dealing with domains of complex shapes. Algorithm for determining random, uniformly distributed locations of rays leaving NURBS surfaces is described. The paper presents results of test cases assuming gray diffusive walls. In the current version of the model the radiation is not absorbed within gases. However, the ultimate aim of the work is to upgrade the functionality of the model, to problems in absorbing, emitting and scattering medium projecting iteratively the results of radiative analysis on CFD mesh and CFD solution on radiative mesh

Interaction Of Radiation With Other Heat Transfer Modes

Heat radiation solution of multimode heat transfer problems involving radiation is discussed. Two cases are analyzed: (i)radiation contributes to the boundary condition of steady state or transient heat conduction in opaque solids forming self irradiating cavities, (ii) a volumetric energy generation term due to heat radiation appears in the energy transfer equations in semi transparent solids or fluids. Heat radiation problems addressed encompass transparent and emitting-absorbing media. The solution of the radiation problems is by the BEM. Heat conduction is solved using BEM while the heat convection problems are discretized by the finite volume technique. The coupling conditions linking radiation with other heat transfer modes are discussed and iterative procedures of solving conjugate heat transfer problems are addressed. Numerical examples will be shown during the presentation

Bem-Fvm Solution Of Conjugate Heat Transfer Problems In Czochralski Crystal Growth Process

A technique of solving multi-mode heat transfer problems in semi transparent, phase changing medium is addressed. Simultaneous heat conduction, free and forced convection, radiation and change of phase during Czochralski crystal growth process are modeled. Heat conduction/convection problem is solved using a commercial code based on the Finite Volume Method with the enthalpy technique applied to capture the moving boundary problem. The model of energy, mass and momentum transfer in the melt encompasses both the forced and natural convection. The problem of radiation is solved using the an in-home code based on the boundary element method. Spectral dependent properties of the crystal and melt are accounted for. The condensed phase is treated as an emitting absorbing one the gas filling the apparatus is treated as a transparent medium. The coupling between the CFD model and the radiation is twofold. Heat radiation generates volumetric heat sources that arise in the energy transport equations in the condensed phase. The temperature field coming from the solution of the convection/conduction problem enters a forcing term in the equations of radiative heat exchange. The coupling is taken into account within an iterative loop by solving the convection and radiation equations in a sequence. Numerical example is included

Proper Orthogonal Decomposition And Modal Analysis For Acceleration Of Transient Fem Thermal Analysis

A method of reducing the number of degrees of freedom and the overall computing times in finite element method (FEM) has been devised. The technique is valid for linear problems and arbitrary temporal variation of boundary conditions. At the first stage of the method standard FEM time stepping procedure is invoked. The temperature fields obtained for the first few time steps undergo statistical analysis yielding an optimal set of globally defined trial and weighting functions for the Galerkin solution of the problem at hand. Simple matrix manipulations applied to the original FEM system produce a set of ordinary differential equations of a dimensionality greatly reduced when compared with the original FEM formulation. Using the concept of modal analysis the set is then solved analytically. Treatment of non-homogeneous initial conditions, time-dependent boundary conditions and controlling the error introduced by the reduction of the degrees of freedom are discussed. Several numerical examples are included for validation of the approach. Copyright © 2004 John Wiley & Sons, Ltd

Reduction Of The Dimensionality Of Transient Fem Solutions Using Proper Orthogonal Decomposition

A method of reducing the number of degrees of freedom in FEM analysis has been devised. As in the case of FEM, Galerkin weighted residuals is based on weak formulation. The distinct feature of the method is the usage of a set of globally defined trial (and weighting) functions defined as a linear combination of the shape functions. The coefficients of this combination are evaluated employing the Proper Orthogonal Decomposition method ensuring optimal approximation properties of the trial functions. The resulting set of ODEs of much lower dimensionality than the standard FEM is then solved analytically using the modal analysis technique. In the included numerical examples, the number of unknowns is reduced by several orders of magnitude while the maximum error is of the order of 1%. The technique leads also to a significant reduction of the execution times. A method of controlling the error introduced by the POD is proposed. © 2003 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved

Solving Inverse Heat Conduction Problems Using Trained Pod-Rbf Network Inverse Method

The article presents advances in the approach aiming to solve inverse problems of steady state and transient heat conduction. The regularization of ill-posed problem comes from the proper orthogonal decomposition (POD). The idea is to expand the direct problem solution into a sequence of orthonormal basis vectors, describing the most significant features of spatial and time variation of the temperature field. Due to the optimality of proposed expansion, the majority of the basis vectors can be discarded practically without accuracy loss. The amplitudes of this low-order expansion are expressed as a linear combination of radial basis functions (RBF) depending on both retrieved parameters and time. This approximation, further referred as trained POD-RBF network is then used to retrieve the sought-for parameters. This is done by resorting to least square fit of the network and measurements. Numerical examples show the robustness and numerical stability of the scheme

Explicit Calculation Of Smoothed Sensitivity Coefficients For Linear Problems

A technique of explicit calculation of sensitivity coefficients based on the approximation of the retrieved function by a linear combination of trial functions of compact support is presented. The method is applicable to steady state and transient linear inverse problems where unknown distributions of boundary fluxes, temperatures, initial conditions or source terms are retrieved. The sensitivity coefficients are obtained by solving a sequence of boundary value problems with boundary conditions and source term being homogeneous except for one term. This inhomogeneous term is taken as subsequent trial functions. Depending on the type of the retrieved function, it may appear on boundary conditions (Dirichlet or Neumann), initial conditions or the source term. Commercial software and analytic techniques can be used to solve this sequence of boundary value problems producing the required sensitivity coefficients. The choice of the approximating functions guarantees a filtration of the high frequency errors. Several numerical examples are included where the sensitivity coefficients are used to retrieve the unknown values of boundary fluxes in transient state and volumetric sources. Analytic, boundary-element and finite-element techniques are employed in the study. © 2003 John Wiley and Sons, Ltd

Application of the ASTM D5470 standard test method for thermal conductivity measurements of high thermal conductive materials

Purpose: The purpose of the present study was to demonstrate the procedure for determining the thermal conductivity of a solid material with relatively high thermal conductivity, using an original self-designed apparatus. Design/methodology/approach: The thermal conductivity measurements have been performed according to the ASTM D5470 standard. The thermal conductivity was calculated from the recorded temperature values in steady-state heat transfer conditions and determined heat flux. Findings: It has been found from the obtained experimental results that the applied standard test method, which was initially introduced for thermal conductivity measurements of thermal interface materials (TIMs), is also suitable for materials with high thermal conductivity, giving reliable results. Research limitations/implications: The ASTM D5470 standard test method for measurement of thermal conductivity usually gives poor results for high conductive materials having thermal conductivity above 100 W/mK, due to problems with measuring heat flux and temperature drop across the investigated sample with reasonably high accuracy. Practical implications: The results obtained for the tested material show that the presented standard test method can also be used for materials with high thermal conductivity, which is of importance either for the industrial or laboratory applications. Originality/value: The thermal conductivity measurements have been carried out using an original self-designed apparatus, which was developed for testing broad range of engineering materials with high accuracy