Papers by Keyword: Diffusion Equation

Abstract: As chloride ion diffusion in concrete theoretical model without considering the shortcomings of the concrete maturity,the improved maturity model is used to establish the relationship between the diffusion coefficient of concrete maturity. With the mathematical derivation, getting new chloride ion diffusion theoretically, which considers the maturity of the concrete, the chloride ion binding capacity, the degradation effects of chlorine ion diffusion equation,and consequently broadens the scope of the fick's second law of diffusion, besides, its form and solution completely consistent with the one in the second law of fick equation. By the calculation example,it can be found that the model is more accurate than the traditional ones, and it also applies to the high content of high-performance concrete.

Abstract: Based on the group explicit strategies, a new alternating group explicit algorithm for solving 2D diffusion equation was presented. The four new difference schemes which were constructed to implement the algorithm in parallel, can be converted to explicit computation. The stability and truncation error analyses were provided. Finally, numerical experiment was performed to examine that the the presented algorithm is efficient and accurate.

Abstract: Given the time-domain measurement of picoseconds pulses and frequency-domain measurement of high-frequency modulation have limitations to realize, this paper introduces the experiment research which use square-wave modulation laser source as incident light source of the diffusion optical tomography in homogeneous medium. Based on the finite-element method, the forward model can be established. The analytical solution of the homogeneous medium can obtained from the derivation of the diffusion equation, after that, we build the optical experiment platform, derive the experimental solution from the platform. Finally, the consistent result is obtained by the comparison of the simulation, the analytical solution and the experimental solution. The result clearly demonstrated the accuracy and effectiveness of the proposed method which use square-wave modulation laser source as incident light source to measure the light intensity.

Abstract: The absorption and transport scattering coefficients of biological material determine the radial dependence of the diffuse reflectance light that is due to a point source. In order to noninvasive determinate the optical scattering and absorption coefficients of biological material, we must know the radial dependence of the diffuse reflectance. The diffusion approximation of the radiative transfer equation is a model used widely to describe photon migration in biological material. An analysis of the steady state diffusion equation together with its solution of the diffuse reflectance light for the slab geometry and for a semi-infinite diffusing biological material is reported. The result has been compared with that obtained from Monte Carlo simulations. The comparison has shown that the solution about the diffuse reflectance light on surface of biological material is the same as that obtained from Monte Carlo simulations.

Abstract: The so-called continuity equation derived by Fick is one of the most fundamental and extremely important equations in physics and/or in materials science. As is well known, this partial differential equation is also called the diffusion equation or the heat conduction equation and is applicable to physical phenomena of the conservation system. Incorporating the parabolic law relevant to a random movement into it, Boltzmann obtained the ordinary differential equation (B-equation). Matano then applied the B-equation to the analysis of the nonlinear problem for the interdiffusion experiment. The empirical Boltzmann-Matano (B-M) method has been successful in the metallurgical field. However, the nonlinear B-equation was not mathematically solved for a long time. Recently, the analytical solutions of the B-equation were obtained in accordance with the results of the B-M method. In the present study, an applicable limitation of the B-equation to the interdiffusion problems is investigated from a mathematical point of view.

Abstract: A new failure model was developed to describe the failure wave formation and propagation in shocked glass. The progressive percolation of microcracks into the stressed body gives rise to the failure wave phenomenon which could be regarded as a diffusion process. The propagation of failure front is governed by a nonlinear diffusion equation. The dynamic damage constitutive relation is built up to compute the stress state in the material through the failure process. Numerical results are presented and compared to lateral stress gauge measurements in shocked glasses. It is shown that the proposed model can capture the essence of the failure wave phenomenon.

Abstract: With the development of damage mechanics, many researchers have used it to analyze the constitutive equation of concrete. Since the special environment in the cold marine regions, the offshore structures are common to subject to the comprehensive effects of freeze-thaw action and chloride erosion. This might cause concrete materials degradation and reduce the mechanical performance of concrete seriously. In this paper, based on the analysis and mechanical experiments of concrete materials under the comprehensive effects of freeze-thaw action and chloride ion erosion, the damage evolution equation of concrete elastic modulus along with the freeze-thaw cycles and chloride ion contents was established. The effects of chloride ion were investigated during the process of concrete degradation. According to the damage evolution equation, a new constitutive equation of concrete under freeze-thaw action and chloride erosion was established. And then, by means of the element simulation analysis of concrete beams when subjected to the comprehensive actions, the feasibility and applicability of the equation was examined and discussed. In this equation, both the freeze-thaw action and chloride ion erosion were considered together. It will be more suitable for analyzing the durability of concrete structures in the real cold marine regions. It will also provide some references for concrete constitutive theory.

Abstract: Based on the diffusion approximation theory, when the laser pulse transmitting through the turbid media, its energy will be attenuated and the pulse shape will be changed by the scattering and absorption. In this paper, Mathematics equations of the ultra short Gauss laser pulse in different pulse width tp are given, the reflective pulses with the boundary condition of semi-infinite homogeneous media are discussed. We get the simulation results of reflective intensity and the reflective pulse shape of different tp based on the diffusion equation. From the results, we know that the ultra short Gauss laser pulse will be widened by the diffusive scattering. Besides, we find that the various medium parameters will influence the reflection of the Gauss laser pulse very differently. With the boundary condition of semi-infinite homogeneous media, the influence of the absorption, the scattering, and the anisotropy coefficient will be also changed by the different tp of ultra short Gauss laser pulse. All the conditions mentioned above have been investigated in the present paper. This study will be very useful for the measurement of optical properties of tissue.

Abstract: A possibility of a modification of the Jackson-Hunt theory of an oriented structure formation is analysed. A new model for the formation of a concentration field ahead of growing regular lamellae with respect to the solid / liquid interface shape is presented. A coordinate system applied in the model is attached to the solid / liquid interface to be advancing in the z - direction, identically with interface moving at a constant velocity, v . The solution to a diffusion equation is given for the improved formulation of the boundary conditions. The boundary conditions are related to the interplay between the diffusion required for phase separation and the formation of the interphase between both lamellae. The boundary conditions are formulated to establish the stability of lamellar structure formation under steady-state conditions. It is assumed that stable growth of the lamellae is ensured by the separation of concentration fields within a boundary layer ahead of the solid / liquid interfaces of both the α and β " phases. Coupled lamellar growth with the presence of a leading phase protrusion is defined. The general mass balance is analysed for a solute concentration in the liquid, taking into account a planar solid / liquid interface. A local mass balance is also ensured but it requires envisaging a protrusion of the minor eutectic phase. The existence of a lead distance is confirmed experimentally for the (Pb)-(Cd) eutectic system. The difference in undercooling is also considered as a phenomenon associated with the separation of concentration fields and the existence of a protrusion to relax the assumption of an isothermal interface (ideally coupled growth) given by the Hunt and Jackson theory.

Abstract: The diffusion of molecules in liquids and dense gases is demonstrated to be nonclassical for long time intervals. This means that the time dependence of the mean-square displacement of molecules is nonlinear. This result was obtained by molecular dynamics simulations over a wide range of density of the medium. The problem of plateau values of the diffusion coefficient is discussed. Nonclassical diffusion equations are derived and discussed.