Approximate Analytic Solution of the One Phase Stefan Problem for the Sphere

2019 ◽  
pp. 185-201
Author(s):  
R. B. Shorten
Keyword(s):  
Stefan Problem ◽  
Analytic Solution ◽  
Approximate Analytic Solution ◽  
The One

Simple Approximate Solutions for Dynamic Response of Suspension System

2022 ◽  
Vol 21 ◽  
pp. 20-31
Author(s):  
Jacob Nagler
Keyword(s):  
Analytic Solution ◽  
Approximate Solutions ◽  
Quantitative Agreement ◽  
Gravitational Energy ◽  
Suspension System ◽  
Approximate Analytic Solution ◽  
Stiffness Properties ◽  
Coupling Case ◽  
The One ◽  
The Impact

An approximate simplified analytic solution is proposed for the one DOF (degree of freedom) static and dynamic displacements alongside the stiffness (dynamic and static) and damping coefficients (minimum and maximum/critical values) of a parallel spring-damper suspension system connected to a solid mass-body gaining its energy by falling from height h. The analytic solution for the prescribed system is based on energy conservation equilibrium, considering the impact by a special G parameter. The formulation is based on the works performed by Timoshenko (1928), Mindlin (1945), and the U. S. army-engineering handbook (1975, 1982). A comparison between the prescribed studies formulations and current development has led to qualitative agreement. Moreover, quantitative agreement was found between the current prescribed suspension properties approximate value - results and the traditionally time dependent (transient, frequency) parameter properties. Also, coupling models that concerns the linkage between different work and energy terms, e.g., the damping energy, friction work, spring potential energy and gravitational energy model was performed. Moreover, approximate analytic solution was proposed for both cases (friction and coupling case), whereas the uncoupling and the coupling cases were found to agree qualitatively with the literature studies. Both coupling and uncoupling solutions were found to complete each other, explaining different literature attitudes and assumptions. In addition, some design points were clarified about the wire mounting isolators stiffness properties dependent on their physical behavior (compression, shear tension), based on Cavoflex catalog. Finally, the current study aims to continue and contribute the suspension, package cushioning and containers studies by using an initial simple pre – design analytic evaluation of falling mass- body (like cushion, containers, etc.).

scholarly journals Modeling of the COVID-19 pandemic in the limit of no acquired immunity

2021 ◽  
Vol 8 (2) ◽  
pp. 282-303
Author(s):  
J. M. Ilnytskyi ◽  
Keyword(s):  
Analytic Solution ◽  
Effective Vaccine ◽  
Acquired Immunity ◽  
Approximate Analytic Solution ◽  
Contact Rate ◽  
Reproduction Ratio ◽  
Numeric Solution ◽  
The One ◽  
Disease Free ◽  
Rate Threshold

We propose the SEIRS compartmental epidemiology model aimed at modeling the COVID-19 pandemy dynamics. The limit case of no acquired immunity (neither natural nor via vaccination) is considered mimicking the situation (i) when no effective vaccine being developed or available yet, and (ii) the virus strongly mutates causing massive reinfections. Therefore, the only means of suppressing the virus spread are via quarantine measures and effective identification and isolation of infected individuals. We found both the disease-free and the endemic fixed points and examined their stability. The basic reproduction ratio is obtained and its dependence on the parameters of the model is discussed. We found the presence of the contact rate threshold beyond which the disease-free fixed point cannot be reached. Using the numeric solution, the approximate analytic solution of the model, characterized by rescaled contact rate, is obtained. Several possible "quarantine on"/"quarantine off" scenarios are considered and the one combined with flexible adjustment of the identification and isolation rates is found to be the most effective in bringing the second and consequent waves down. The study can be interpreted as a reference point for the case when the natural or acquired immunity, as well as vaccination, are taken into account. It will be a topic of a separate study.

scholarly journals On the Two-phase Fractional Stefan Problem

2020 ◽  
Vol 20 (2) ◽  
pp. 437-458 ◽  
Author(s):  
Félix del Teso ◽  
Jørgen Endal ◽  
Juan Luis Vázquez
Keyword(s):  
Free Boundary ◽  
Stefan Problem ◽  
Free Boundary Problems ◽  
Classical Problem ◽  
Nonlocal Diffusion ◽  
Two Phase ◽  
Boundary Problems ◽  
Numerical Studies ◽  
The One ◽  
Evolution Type

AbstractThe classical Stefan problem is one of the most studied free boundary problems of evolution type. Recently, there has been interest in treating the corresponding free boundary problem with nonlocal diffusion. We start the paper by reviewing the main properties of the classical problem that are of interest to us. Then we introduce the fractional Stefan problem and develop the basic theory. After that we center our attention on selfsimilar solutions, their properties and consequences. We first discuss the results of the one-phase fractional Stefan problem, which have recently been studied by the authors. Finally, we address the theory of the two-phase fractional Stefan problem, which contains the main original contributions of this paper. Rigorous numerical studies support our results and claims.

scholarly journals AN ANALYTIC SOLUTION FOR ONE-DIMENSIONAL DISSIPATIONAL STRAIN-GRADIENT PLASTICITY

2009 ◽  
Vol 50 (3) ◽  
pp. 407-420
Author(s):  
ROGER YOUNG
Keyword(s):  
Boundary Condition ◽  
Strain Gradient ◽  
Analytic Solution ◽  
Strain Gradient Plasticity ◽  
Numerical Results ◽  
Gradient Plasticity ◽  
One Dimensional ◽  
Numerical Result ◽  
Formulation Analysis ◽  
The One

AbstractAn analytic solution is developed for the one-dimensional dissipational slip gradient equation first described by Gurtin [“On the plasticity of single crystals: free energy, microforces, plastic strain-gradients”, J. Mech. Phys. Solids48 (2000) 989–1036] and then investigated numerically by Anand et al. [“A one-dimensional theory of strain-gradient plasticity: formulation, analysis, numerical results”, J. Mech. Phys. Solids53 (2005) 1798–1826]. However we find that the analytic solution is incompatible with the zero-sliprate boundary condition (“clamped boundary condition”) postulated by these authors, and is in fact excluded by the theory. As a consequence the analytic solution agrees with the numerical results except near the boundary. The equation also admits a series of higher mode solutions where the numerical result corresponds to (a particular case of) the fundamental mode. Anand et al. also established that the one-dimensional dissipational gradients strengthen the material, but this proposition only holds if zero-sliprate boundary conditions can be imposed, which we have shown cannot be done. Hence the possibility remains open that dissipational gradient weakening may also occur.

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