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

Dondl, Patrick; Novaga, Matteo; Wirth, Benedikt; Wojtowytsch, Stephan

doi:10.1137/18m1225197

Cited by 6 publications

(6 citation statements)

References 10 publications

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“…To the best of our knowledge, this is the first contribution proposing a theoretical characterization of connected perimeter. However, motivated by the numerical applications, there have been however several contributions on the approximation of such perimeter, or on the approximation of other (sometimes higher-order) related energies, see for instance [5,6,7,3,4]. text We will constantly use in this work the notion of set of finite perimeter and its main properties, for which we refer to [1].…”

Section: Introductionmentioning

confidence: 99%

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Connected perimeter of planar sets

Dayrens¹,

Masnou²,

Novaga³

et al. 2019

Preprint

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We introduce a notion of connected perimeter for planar sets defined as the lower semicontinuous envelope of perimeters of approximating sets which are measure-theoretically connected. A companion notion of simply connected perimeter is also studied. We prove a representation formula which links the connected perimeter, the classical perimeter, and the length of suitable Steiner trees. We also discuss the application of this notion to the existence of solutions to a nonlocal minimization problem.

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Section: Introductionmentioning

confidence: 99%

“…Then i) if E is simple, there exists a sequence E n of smooth simply connected sets such that E n → E and P (E n ) → P (E), ii) if E is indecomposable, there exists a sequence E n of smooth connected sets such that E n → E and P (E n ) → P (E). In particular it holds that (7) P r C (E) = P C (E), P r S (E) = P S (E).…”

Section: Introductionmentioning

confidence: 99%

Connected perimeter of planar sets

Dayrens¹,

Masnou²,

Novaga³

et al. 2019

Preprint

Self Cite

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“…If a is taken too large, the time-step size must become small, even if the curvature energy is treated in an implicit fashion. For more numerical experiments illuminating see also Dondl et al 1,2…”

Section: Topological Constraintmentioning

confidence: 99%

“…It is also possible to use our algorithm in the case of functionals controlling only the perimeter of sets contained in a bounded domain normalΩ ⊂ R 2. 1,2 To simplify the presentation, we focus here on the 3D case with a control of Willmore’s energy.…”

Section: Introductionmentioning

confidence: 99%

Keeping it together: A phase-field version of path-connectedness and its implementation

Dondl

Wojtowytsch

2021

Journal of Algorithms & Computational Technology

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We describe the implementation of a topological constraint in finite-element simulations of phase-field models, which ensures path-connectedness of preimages of intervals in the phase-field variable. The constraint takes the form of an energetic penalty for a suitable geodesic distance between all pairs of points in the domain. The main application of our method presented here is a discrete steepest descent of a phase-field version of a bending energy with spontaneous curvature and additional surface area penalty. This leads to disconnected surfaces without our topological constraint but connected surfaces with the constraint. Numerically, our constraint is treated by first transforming the double integral over all pairs of points in the domain to a weighted graph structure and then using Dijkstra’s algorithm to calculate the distance between discrete connected components.

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“…This connected perimeter used here was introduced in [DMNP19]. A phase-field variant of the connected perimeter was developed in [DNWW19].…”

Section: Introductionmentioning

confidence: 99%

Connected Coulomb columns: analysis and numerics

et al. 2021

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We consider a version of Gamow’s liquid drop model with a short range attractive perimeter-penalizing potential and a long-range Coulomb interaction of a uniformly charged mass in R 3 . Here we constrain ourselves to minimizing among the class of shapes that are columnar, i.e., constant in one spatial direction. Using the standard perimeter in the energy would lead to non-existence for any prescribed cross-sectional area due to the infinite mass in the constant spatial direction. In order to overcome this issue we use a connected perimeter instead. We prove existence of minimizers for this connected isoperimetric problem with long-range interaction and study the shapes of minimizers in the small and large cross section regimes. For an intermediate regime we use an Ohta–Kawasaki phase field model with connectedness constraint to study the shapes of minimizers numerically.

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

Cited by 6 publications

References 10 publications

Connected perimeter of planar sets

Connected perimeter of planar sets

Keeping it together: A phase-field version of path-connectedness and its implementation

Connected Coulomb columns: analysis and numerics

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