SearcharxivSearch

arXiv subjects

Yann Bernard

Publications and source records attributed to Yann Bernard.

At least 19 recordsLinked to original sources

Virial-based extraction of structures in numerical simulations: The vibes tool

The processes that determine the stellar initial mass function (IMF) and its connection to the core mass function (CMF) are among the major open questions in star formation. The definition of a core remains unclear, yet the way they are extracted from simulations and observations critically shapes the CMF. Nowadays, cores are mostly detected through their density or intensity only. We aim to explore a new way to define cores in 3D numerical simulations based on a direct application of the virial theorem, and break free from some limitations induced by density-based methods. We intend to improve the accuracy and the physical meaning of the extracted cores. We developed vibes, an innovative method that makes full use of the virial theorem to extract overdensities in simulation snapshots. It works by building structures iteratively around density peaks, and applying the virial theorem to the structure at each iteration. Then, the structure boundary is set from the evolution of the its energy as it spatially grows. We used STARFORGE simulations to test the sensitivity of the extraction process to the main working parameters (constraints on the structure shape, iteration step, and peak selection criteria). This sensitivity is observed to be low. We compared our extraction with two density-based extraction algorithms, hop and dendrogram, that are observed to be very sensitive to their input density threshold parameter. Vibes returns structures that are coherent to each other and physically motivated, and it appears much more stable than existing 3D extraction tools. By defining the boundary of the cores on a physical criterion rather than on a user-defined set of density parameters, we expect such extracted cores to be closer to their forsaken definition: gas reservoirs that will form a single star or a close multiple system.

astro-ph.SR

Momentum-conserving self-gravity in the phantom smoothed particle hydrodynamics code. Parallel dual tree traversal for the symmetric fast multipole method

Tree codes that approximate groups of distant particles with multipole expansions are the standard way to accelerate the computation of self-gravity on particles. While momentum-conserving fast multipole methods exist, parallelisation is non-trivial and previous implementations have been limited to self-gravity with fixed softening lengths. We aim for a practical, parallel version of Dehnen's momentum-conserving Cartesian fast multipole method for the computation of the gravitational force in smoothed particle hydrodynamics (SPH) with adaptive gravitational force softening. We parallelise the dual tree walk by replicating the node-node interaction on the parents of each leaf node in the tree. While this duplicates work, it greatly simplifies the parallelisation and can be implemented with relatively minor changes from the previous non-conservative force algorithm in Phantom. We also adapt the tree opening criterion for adaptive softening lengths, such that all interactions within the softening kernel are handled pairwise (as in SPH) rather than with multipole expansions, also allowing the gravity calculation to be performed alongside the SPH force evaluation. We demonstrate that the new code conserves linear momentum to machine precision while giving similar force accuracy and computational performance to the previous (non-symmetric) fast multipole method in Phantom . The new method also gives better conservation of the angular momentum and orbital phase in a binary polytrope evolution. The symmetric fast multipole method is now the default for computing self-gravity in the public code.

astro-ph.IM

The Regularity of Critical Points to Scale-Invariant Curvature Energies in Dimension 4

We consider a class of scale-invariant curvature energies defined on immersed $4$-dimensional manifolds and prove that weak immersions that are critical points of such energies are analytic in any given local harmonic chart. Because of the criticality of this variational problem, the regularity result is obtained through the identification of conservation laws by applying Noether theorem. The resulting identities generate a lower order elliptic system of PDEs to which methods from integrability by compensation and interpolation theory are applied.

math.AP

Structural Equations for Critical Points of Conformally Invariant Curvature Energies in 4d

This paper considers the Euler-Lagrange equations satisfied by the critical points of a large class of conformally invariant extrinsic energies for 4-manifolds immersed into Euclidean space (any codimension). Using invariances and Noether's theorem, we convert the Euler-Lagrange equation in a system of equations with analytically favourable structures. The present paper generalises to the four-dimensional setting ideas originally developed by Tristan Rivi\`ere in his study of the Willmore energy in two dimensions.

math.DG

DAWN. I. Simulating the formation and early evolution of stellar clusters with Phantom N-Body

Context. Simulating stellar dynamics in a molecular cloud environment is numerically challenging due to the strong coupling between young stars and their surrounding gas, and the large range of length and time scales. Aims. This paper is the first of a suite aimed at investigating the complex early stellar dynamics in star-forming regions. We present a new simulation framework which is the key to generating a larger set of simulations, enabling statistical analysis. Methods. Methods originating from the stellar dynamics community, including regularisation and slowdown methods (SDAR), have been added to the hydrodynamical code Phantom to produce simulations of embedded cluster early dynamics. This is completed by a novel prescription of star formation to initialise stars with a low numerical cost, but in a way that is consistent with the gas distribution. Finally, a prescription for H ii region expansion has been added to model the gas removal. Results. We have run testcase simulations following the dynamical evolution of stellar clusters from the cloud collapse to a few Myr. Our new numerical methods fulfil their function by speeding up the calculation. The N-body dynamics with our novel implementation never appear as a bottleneck. Our first simulations show that massive stars largely impact the star formation process and shape the dynamics of the resulting cluster. Depending on the position of these massive stars and the strength of their feedback, they can prematurely dismantle part of the cloud or trigger a second event of cloud collapse, preferentially forming low-mass stars. This stochastic behaviour confirms the need for statistical studies. Conclusions. Our new Phantom N-Body framework enables efficient simulation of the formation and evolution of star clusters. It enables the statistical analysis needed to build models of the dynamical evolution of embedded star clusters.

astro-ph.GA

Analysis of Critical Points of Conformally Invariant Curvature Energies in 4d

We discuss a large class of conformally invariant curvature energies for immersed hypersurfaces of dimension 4. The class under study includes various examples that have appeared in the recent literature and which arise from different contexts. We show that under natural small-energy hypotheses, critical points satisfy improved energy estimates. Nearly all the PDEs which we consider are quasilinear and fourth-order in the mean curvature. We approach the problem \`a la T. Rivi\`ere by generating first from Noether's theorem divergence-free "potentials", and then by exhibiting an underlying analytically favourable algebraic structure relating them. We also consider local Palais-Smale sequences and show they converge to a solution of a constrained Euler-Lagrange equation with Lagrange multiplier appearing in the form of a TT-tensor.

math.DG

A New Conformal Invariant for four-Dimensional Hypersurfaces

A new conformally invariant energy for four-dimensional hypersurfaces is devised. It renders possible the study of a large class of curvature energies, and we show that their critical points are smooth. As corollaries, we obtain the regularity of the critical points of the four-dimensional analogues of the Willmore energy, of the $Q$-curvature energy, but also that Bach-flat hypersurfaces are smooth, along with relevant estimates.

math.DG

Regularity of four dimensional Willmore-type hypersurfaces

A four dimensional conformally invariant energy is studied. This energy generalises the well known two-dimensional Willmore energy. Although not positive definite, it includes minimal hypersurfaces as critical points. We compute its first variation and by applying the Noether theorem to the invariances, we derive some conservation laws which are satisfied by its critical points and with good analytical dispositions. In particular, we show that its critical points are smooth. We also investigate other possible four dimensional generalisations of the Willmore energy, and give strong evidence that critical points of such energies do not include minimal hypersurfaces.

math.DG

Energy Estimates for the Tracefree Curvature of Willmore Surfaces and Applications

We prove an $ε$-regularity result for the tracefree curvature of a Willmore surface with bounded second fundamental form. For such a surface, we obtain a pointwise control of the tracefree second fundamental form from a small control of its $L^2$-norm.Several applications are investigated. Notably, we derive a gap statement for surfaces of the aforementioned type. We further apply our results to deduce regularity results for conformal minimal spacelike immersions into the de Sitter space $S^{4,1}$.

math.DG

Concentration-compactness and finite-time singularities for Chen's flow

Chen's flow is a fourth-order curvature flow motivated by the spectral decomposition of immersions, a program classically pushed by B.-Y. Chen since the 1970s. In curvature flow terms the flow sits at the critical level of scaling together with the most popular extrinsic fourth-order curvature flow, the Willmore and surface diffusion flows. Unlike them however the famous Chen conjecture indicates that there should be no stationary nonminimal data, and so in particular the flow should drive all closed submanifolds to singularities. We investigate this idea, proving that (1) closed data becomes extinct in finite time in all dimensions and for any codimension; (2) singularities are characterised by concentration of curvature in $L^n$ for intrinsic dimension $n \in \{2,4\}$ and any codimension (a Lifespan Theorem); and (3) for $n = 2$ and in one codimension only, there exists an explicit small constant $\varepsilon_2$ such that if the $L^2$ norm of the tracefree curvature is initially smaller than $\varepsilon_2$, the flow remains smooth until it shrinks to a point, and that the blowup of that point is an embedded smooth round sphere.

math.DG

Analysis of the Inhomogeneous Willmore Equation

We study a class of fourth-order geometric problems modelling Willmore surfaces, conformally constrained Willmore surfaces, isoperimetrically constrained Willmore surfaces, bi-harmonic surfaces in the sense of Chen, among others. We prove several local energy estimates and derive a global gap lemma.

math.DG

Rigidity and stability of spheres in the Helfrich model

The Helfrich functional, denoted by H^{c_0}, is a mathematical expression proposed by Helfrich (1973) for the natural free energy carried by an elastic phospholipid bilayer. Helfrich theorises that idealised elastic phospholipid bilayers minimise H^{c_0} among all possible configurations. The functional integrates a spontaneous curvature parameter c_0 together with the mean curvature of the bilayer and constraints on area and volume, either through an inclusion of osmotic pressure difference and tensile stress or otherwise. Using the mathematical concept of embedded orientable surface to represent the configuration of the bilayer, one might expect to be able to adapt methods from differential geometry and the calculus of variations to perform a fine analysis of bilayer configurations in terms of the parameters that it depends upon. In this article we focus upon the case of spherical red blood cells with a view to better understanding spherocytes and spherocytosis. We provide a complete classification of spherical solutions in terms of the parameters in the Helfrich model. We additionally present some further analysis on the rigidity and stability of spherocytes.

math-ph

Ends of Immersed Minimal and Willmore Surfaces in Asymptotically Flat Spaces

We study ends of an oriented, immersed, non-compact, complete Willmore surfaces, which are critical points of the integral of the square of the mean curvature, in asymptotically flat spaces of any dimension; assuming the surface has $L^2$-bounded second fundamental form and satisfies a weak power growth on the area. We give the precise asymptotic behavior of an end of such a surface. This asymptotic information is very much dependent on the way the ambient metric decays to the Euclidean one. Our results apply in particular to minimal surfaces.

math.DG

Noether's Theorem and the Willmore Functional

Noether's theorem and the invariances of the Willmore functional are used to derive conservation laws that are satisfied by the critical points of the Willmore energy subject to generic constraints. We recover in particular previous results independently obtained by R. Capovilla and J. Guven, and by T. Riviere. Several examples are considered in details.

math.DG

On the Structure of Minimizers of Causal Variational Principles in the Non-Compact and Equivariant Settings

We derive the Euler-Lagrange equations for minimizers of causal variational principles in the non-compact setting with constraints, possibly prescribing symmetries. Considering first variations, we show that the minimizing measure is supported on the intersection of a hyperplane with a level set of a function which is homogeneous of degree two. Moreover, we perform second variations to obtain that the compact operator representing the quadratic part of the action is positive semi-definite. The key ingredient for the proof is a subtle adaptation of the Lagrange multiplier method to variational principles on convex sets.

math-ph

Analysis of Constrained Willmore Surfaces

This paper studies the regularity of constrained Willmore immersions into $\R^{m\ge3}$ locally around both "regular" points and around branch points, where the immersive nature of the map degenerates. We develop local asymptotic expansions for the immersion, its first, and its second derivatives, given in terms of residues which are computed as circulation integrals. We deduce explicit "point removability" conditions ensuring that the immersion is smooth. Our results apply in particular to Willmore immersions and to parallel mean curvature immersions in any codimension.

math.DG

Asymptotic Analysis of Branched Willmore Surfaces

We consider a closed Willmore surface properly immersed in ${\R}^m$ (m>2) with square-integrable second fundamental form, and with one point-singularity of finite arbitrary integer order. Using the "conservative" reformulation of the Willmore equation introduced in a previous paper by the second author, we show that, in an appropriate conformal parametrization, the gradient of the Gauss map of the immersion has bounded mean oscillations if the singularity has order one, and is bounded if the order is at least two. We develop around the singular point local asymptotic expansions for the immersion, its first and second derivatives, and for the mean curvature vector. Finally, we exhibit an explicit condition ensuring the removability of the point-singularity

math.AP

Energy Quantization for Willmore Surfaces and Applications

We prove a bubble-neck decomposition together with an energy quantization result for sequences of Willmore surfaces into an arbitrary euclidian space with uniformly bounded energy and non-degenerating conformal type. We deduce the strong compactness of Willmore closed surfaces of a given genus modulo the Möbius group action, below some energy threshold

math.AP