scholarly journals A Phase-Field Approximation of the Willmore Flow with Volume and Area Constraints

2012 ◽  
Vol 44 (6) ◽  
pp. 3734-3754 ◽  
Author(s):  
Pierluigi Colli ◽  
Philippe Laurençot
Keyword(s):  
Phase Field ◽  
Field Approximation ◽  
Willmore Flow

A Phase-field Approximation of the Perimeter under a Connectedness Constraint

2019 ◽  
Vol 51 (5) ◽  
pp. 3902-3920
Author(s):  
Patrick W. Dondl ◽  
Matteo Novaga ◽  
Benedikt Wirth ◽  
Stephan Wojtowytsch
Keyword(s):  
Phase Field ◽  
Field Approximation

scholarly journals A phase-field approximation of the Steiner problem in dimension two

2019 ◽  
Vol 12 (2) ◽  
pp. 157-179 ◽  
Author(s):  
Antonin Chambolle ◽  
Luca Alberto Davide Ferrari ◽  
Benoit Merlet
Keyword(s):  
Phase Field ◽  
Transportation Problem ◽  
Numerical Approximation ◽  
Unit Length ◽  
Field Approximation ◽  
Two Dimensions ◽  
Steiner Problem ◽  
Limit Case ◽  
Fixed Parameter ◽  
Γ Convergence

AbstractIn this paper we consider the branched transportation problem in two dimensions associated with a cost per unit length of the form {1+\beta\,\theta}, where θ denotes the amount of transported mass and {\beta>0} is a fixed parameter (notice that the limit case {\beta=0} corresponds to the classical Steiner problem). Motivated by the numerical approximation of this problem, we introduce a family of functionals ({\{\mathcal{F}_{\varepsilon}\}_{\varepsilon>0}}) which approximate the above branched transport energy. We justify rigorously the approximation by establishing the equicoercivity and the Γ-convergence of {\{\mathcal{F}_{\varepsilon}\}} as {\varepsilon\downarrow 0}. Our functionals are modeled on the Ambrosio–Tortorelli functional and are easy to optimize in practice. We present numerical evidences of the efficiency of the method.

Phase field approximation of dynamic brittle fracture

2014 ◽  
Vol 54 (5) ◽  
pp. 1141-1161 ◽  
Author(s):  
Alexander Schlüter ◽  
Adrian Willenbücher ◽  
Charlotte Kuhn ◽  
Ralf Müller
Keyword(s):  
Brittle Fracture ◽  
Phase Field ◽  
Field Approximation ◽  
Dynamic Brittle Fracture

scholarly journals Phase-field approach to implicit solvation of biomolecules with Coulomb-field approximation

2013 ◽  
Vol 139 (2) ◽  
pp. 024111 ◽  
Author(s):  
Yanxiang Zhao ◽  
Yuen-Yick Kwan ◽  
Jianwei Che ◽  
Bo Li ◽  
J. Andrew McCammon
Keyword(s):  
Phase Field ◽  
Coulomb Field ◽  
Field Approximation ◽  
Implicit Solvation ◽  
Field Approach

scholarly journals On a phase field approximation of the planar Steiner problem: Existence, regularity, and asymptotic of minimizers

2018 ◽  
Vol 20 (1) ◽  
pp. 69-106 ◽  
Author(s):  
Matthieu Bonnivard ◽  
Antoine Lemenant ◽  
Vincent Millot
Keyword(s):  
Phase Field ◽  
Field Approximation ◽  
Steiner Problem
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