scholarly journals Eulerian formulation of elastic rods

2016 ◽  
Vol 472 (2190) ◽  
pp. 20150547 ◽  
Author(s):  
Alexandre Huynen ◽  
Emmanuel Detournay ◽  
Vincent Denoël
Keyword(s):  
Free Boundary Problems ◽  
Local Equilibrium ◽  
Elastic Rods ◽  
Reference Curve ◽  
Boundary Problems ◽  
Continuous Contact ◽  
Segmentation Strategy ◽  
Eulerian Formulation ◽  
Curvilinear Coordinate ◽  
Isoperimetric Constraints

In numerous biological, medical and engineering applications, elastic rods are constrained to deform inside or around tube-like surfaces. To solve efficiently this class of problems, the equations governing the deflection of elastic rods are reformulated within the Eulerian framework of this generic tubular constraint defined as a perfectly stiff normal ringed surface. This reformulation hinges on describing the rod-deformed configuration by means of its relative position with respect to a reference curve, defined as the axis or spine curve of the constraint, and on restating the rod local equilibrium in terms of the curvilinear coordinate parametrizing this curve. Associated with a segmentation strategy, which partitions the global problem into a sequence of rod segments either in continuous contact with the constraint or free of contact (except for their extremities), this re-parametrization not only trivializes the detection of new contacts but also transforms these free boundary problems into classic two-points boundary-value problems and suppresses the isoperimetric constraints resulting from the imposition of the rod position at the extremities of each rod segment.

scholarly journals Numerical solution of free-boundary problems in fluid mechanics. Part 1. The finite-difference technique

1984 ◽  
Vol 148 ◽  
pp. 1-17 ◽  
Author(s):  
G. Ryskin ◽  
L. G. Leal
Keyword(s):  
Finite Difference ◽  
Free Boundary ◽  
Fluid Mechanics ◽  
Free Boundary Problems ◽  
Numerical Technique ◽  
Equations Of Motion ◽  
Boundary Problems ◽  
Boundary Shape ◽  
Curvilinear Coordinate ◽  
Ultimate Equilibrium

We present here a brief description of a numerical technique suitable for solving axisymmetric (or two-dimensional) free-boundary problems of fluid mechanics. The technique is based on a finite-difference solution of the equations of motion on an orthogonal curvilinear coordinate system, which is also constructed numerically and always adjusted so as to fit the current boundary shape. The overall solution is achieved via a global iterative process, with the condition of balance between total normal stress and the capillary pressure at the free boundary being used to drive the boundary shape to its ultimate equilibrium position.

scholarly journals On optimal regularity of free boundary problems and a conjecture of De Giorgi

2005 ◽  
Vol 58 (8) ◽  
pp. 1051-1076 ◽  
Author(s):  
Herbert Koch ◽  
Giovanni Leoni ◽  
Massimiliano Morini
Keyword(s):  
Free Boundary ◽  
Free Boundary Problems ◽  
Optimal Regularity ◽  
Boundary Problems

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 Bayesian Quickest Detection Problems for Some Diffusion Processes

2013 ◽  
Vol 45 (1) ◽  
pp. 164-185 ◽  
Author(s):  
Pavel V. Gapeev ◽  
Albert N. Shiryaev
Keyword(s):  
Diffusion Processes ◽  
Free Boundary Problems ◽  
Three Dimensional ◽  
Parabolic Type ◽  
Value Functions ◽  
Weighted Likelihood ◽  
Likelihood Ratios ◽  
Boundary Problems ◽  
Detection Delay ◽  
Penalty Costs

We study the Bayesian problems of detecting a change in the drift rate of an observable diffusion process with linear and exponential penalty costs for a detection delay. The optimal times of alarms are found as the first times at which the weighted likelihood ratios hit stochastic boundaries depending on the current observations. The proof is based on the reduction of the initial problems into appropriate three-dimensional optimal stopping problems and the analysis of the associated parabolic-type free-boundary problems. We provide closed-form estimates for the value functions and the boundaries, under certain nontrivial relations between the coefficients of the observable diffusion.

Sign in / Sign up

Export Citation Format

Share Document