Impact Mechanics of Elastic Structures With Point Contact

2014 ◽  
Vol 136 (4) ◽  
Author(s):  
Róbert Szalai
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Point Contact ◽  
Simple Criterion ◽  
Modeling Framework ◽  
Regular Behavior ◽  
Impact Phenomena ◽  
New Form ◽  
Delay Differential ◽  
The Continuum

This paper introduces a modeling framework that is suitable to resolve singularities of impact phenomena encountered in applications. The method involves an exact transformation that turns the continuum, often partial differential equation description of the contact problem into a delay differential equation. The new form of the physical model highlights the source of singularities and suggests a simple criterion for regularity. To contrast singular and regular behavior the impacting Euler–Bernoulli and Timoshenko beam models are compared.

scholarly journals A free boundary model of epithelial dynamics

2018 ◽  
Author(s):  
Ruth E Baker ◽  
Andrew Parker ◽  
Matthew J Simpson
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Partial Differential Equation ◽  
Free Boundary ◽  
Continuum Limit ◽  
Equation Model ◽  
Free Boundary Condition ◽  
Partial Differential ◽  
Cell Dynamics ◽  
The Continuum

AbstractIn this work we analyse a one-dimensional, cell-based model of an epithelial sheet. In this model, cells interact with their nearest neighbouring cells and move deterministically. Cells also proliferate stochastically, with the rate of proliferation specified as a function of the cell length. This mechanical model of cell dynamics gives rise to a free boundary problem. We construct a corresponding continuum-limit description where the variables in the continuum limit description are expanded in powers of the small parameter 1/N, where N is the number of cells in the population. By carefully constructing the continuum limit description we obtain a free boundary partial differential equation description governing the density of the cells within the evolving domain, as well as a free boundary condition that governs the evolution of the domain. We show that care must be taken to arrive at a free boundary condition that conserves mass. By comparing averaged realisations of the cell-based model with the numerical solution of the free boundary partial differential equation, we show that the new mass-conserving boundary condition enables the coarsegrained partial differential equation model to provide very accurate predictions of the behaviour of the cell-based model, including both evolution of the cell density, and the position of the free boundary, across a range of interaction potentials and proliferation functions in the cell based model.

scholarly journals Delay differential equations for the spatially-resolved simulation of epidemics with specific application to COVID-19

2021 ◽  
Author(s):  
Nicola Guglielmi ◽  
Elisa Iacomini ◽  
Alexander Viguerie
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Computational Time ◽  
Partial Differential ◽  
Predictive Capacity ◽  
Spatially Resolved ◽  
The Stability ◽  
Delay Differential

In the wake of the 2020 COVID-19 epidemic, much work has been performed on the development of mathematical models for the simulation of the epidemic, and of disease models generally. Most works follow the susceptible-infected-removed (SIR) compartmental framework, modeling the epidemic with a system of ordinary differential equations. Alternative formulations using a partial differential equation (PDE) to incorporate both spatial and temporal resolution have also been introduced, with their numerical results showing potentially powerful descriptive and predictive capacity. In the present work, we introduce a new variation to such models by using delay differential equations (DDEs). The dynamics of many infectious diseases, including COVID-19, exhibit delays due to incubation periods and related phenomena. Accordingly, DDE models allow for a natural representation of the problem dynamics, in addition to offering advantages in terms of computational time and modeling, as they eliminate the need for additional, difficult-to-estimate, compartments (such as exposed individuals) to incorporate time delays. In the present work, we introduce a DDE epidemic model in both an ordinary- and partial differential equation framework. We present a series of mathematical results assessing the stability of the formulation. We then perform several numerical experiments, validating both the mathematical results and establishing model’s ability to reproduce measured data on realistic problems.

Dynamics of Gyroelastic Continua

1984 ◽  
Vol 51 (2) ◽  
pp. 415-422 ◽  
Author(s):  
G. M. T. D’Eleuterio ◽  
P. C. Hughes
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Angular Momentum ◽  
Volume Element ◽  
Basis Functions ◽  
Vibration Modes ◽  
Linear Elastic ◽  
Operator Notation ◽  
General Motion ◽  
The Continuum

This paper introduces the idea of distributed gyricity, in which each volume element of a continuum possesses an infinitesimal quantity of stored angular momentum. The continuum is also assumed to be linear-elastic. Using operator notation, a partial differential equation is derived that governs the small displacements of this gyroelastic continuum. Gyroelastic vibration modes are derived and used as basis functions in terms of which the general motion can be expressed. A discretized approximation is also developed using the method of Rayleigh-Ritz. The paper concludes with a numerical example of gyroelastic modes.

Qxygen transfer in gravity flow sewers

2000 ◽  
Vol 42 (3-4) ◽  
pp. 417-422 ◽  
Author(s):  
T.Y. Pai ◽  
C.F. Ouyang ◽  
Y.C. Liao ◽  
H.G. Leu
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Dissolved Oxygen ◽  
Flow Velocity ◽  
Molecular Diffusion ◽  
Eddy Diffusion ◽  
Partial Differential ◽  
Diffusion Term ◽  
Sewer Pipes ◽  
Gravity Sewer

Oxygen diffused to water in gravity sewer pipes was studied in a 21 m long, 0.15 m diameter model sewer. At first, the sodium sulfide was added into the clean water to deoxygenate, then the pump was started to recirculate the water and the deoxygenated water was reaerated. The dissolved oxygen microelectrode was installed to measure the dissolved oxygen concentrations varied with flow velocity, time and depth. The dissolved oxygen concentration profiles were constructed and observed. The partial differential equation diffusion model that considered Fick's law including the molecular diffusion term and eddy diffusion term were derived. The analytic solution of the partial differential equation was used to determine the diffusivities by the method of nonlinear regression. The diffusivity values for the oxygen transfer was found to be a function of molecular diffusion, eddy diffusion and flow velocity.

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