A-SLEIPNNIR: A multiscale, anisotropic adaptive, particle level set framework for moving interfaces. Transport equation applications
Introduction
Moving interfaces are at the heart of myriad Scientific and Engineering applications; from crystal growth [1] to tumor growth [2]; from biological systems [3] to shape optimization [4]; from bubble dynamics [5], [6] to simulations of shock-driven, gas-particle flows [7] — the significance of a robust, accurate and computationally inexpensive method capable of representing the interface with precision and flexibility cannot be overstated.
In a first step toward devising such an ideal method, the computational scientist is confronted with a choice between two large classes of schemes concerning spatial discretization: meshfree (or “meshless”) and mesh-based methods. For interface problems, meshless methods [8] stand out due to the natural handling of large deformations, the ease of implementation of higher-order approximations, and the independence between integration points and interpolation nodes; however, this comes at a cost, usually in the form of an increased computational effort (especially in weak-form methods) and/or lack of robustness (particularly in strong-form, collocation methods). Among meshfree methods, Smooth Particle Hydrodynamics (SPH) is arguably the most widely used technique for multiphase/free-surface flows [9]. On the other hand, mesh-based methods [10] still hold a prevalent position in the community, with more than half a century of developments and accomplishments; they are customarily divided into interface tracking and interface capturing methods depending on whether the interface is explicitly described (Lagrangian approach), or defined implicitly by an auxiliary function (Eulerian approach). In the former group we find the front-tracking [11], and Arbitrary Lagrangian Eulerian (ALE) methods [12], [13]. Instances of interface-capturing schemes are the Volume-Of-Fluid (VOF) method [14] and improvements such as the PLIC (‘Piecewise Linear Interface Calculation’) schemes [15] or the hybrid geometric/algebraic VOF approach [16], all of them featuring excellent mass conservation properties; the Phase-Field (PF) method [17], whose diffuse interface technique offers advantages when modeling physical systems such as fracture propagation [18] or biological membranes [19]; the Immersed Boundary Method and the Level Set (LS) method [20], [21], [22], of widespread use in moving-interface problems due to its flexibility when dealing with topologically complex domains. Many other techniques build upon the previous mesh-based methods: thus, the Particle Level Set (PLS) method by Enright and co-workers [23], [24], the Moment-Of-Fluid (MOF) method by Dyadechko & Shashkov [25], the hybrid particle MAC method by Zheng et al. [26], the Cellwise Conservative Unsplit VOF method by Comminal and collaborators [27], or the improved Polygonal Area Mapping (iPAM) approach by Zhang and Fogelson [28], [29]. And yet, in a number of situations (thin filaments, stretching flows, boundary layers, interface merging/breaking, etc.), the spatial resolution required by mesh-based methods takes such a toll on their computational efficiency as to render them unsuitable for everyday engineering practice. In this context, Adaptive Mesh Refinement (AMR) presents itself as an invaluable tool.
Isotropic adaptation has been extensively used to account for the large disparity of scales found in many interface problems. The idea is to compute the solution up to a certain accuracy defined by an error tolerance, according to proper error estimators; in this sense, we highlight the pioneering work on error estimators by Babuška and Rheinboldt [30] within a Finite Element framework. Improvements on mesh-based methods such as the Narrow-Band Level Set method by Gómez, Hernández and López [31], the Adaptive Moment-Of-Fluid (AMF-MOF) method by Ahn & Shashkov [32], the reconnection-based scheme by Bo and Shashkov [33], the 3D AMR-capable level set method by Morgan and Waltz [34], the adaptive, Fixed-Mesh ALE method by Baiges et al. [35], or the Discontinuous-Galerkin AMR algorithm by Papoutsakis and co-workers [36] are all examples of isotropic AMR. When the solution shows directional features, anisotropic mesh refinement [37], [38], [39] has an edge over the isotropic approach: the ability to prescribe not only the size, but also the shape and orientation of the elements comprising the mesh allows for similar levels of accuracy with fewer degrees of freedom. Recent developments of anisotropic AMR for interface problems include the anisotropic Level Set method by Bui and collaborators [40], the domain meshing technique by Dapogny and co-workers [41], the anisotropic Level Set – Immersed Boundary Method by Abgrall et al. [42], the norm-oriented formulation proposed by Brèthes and Dervieux [43], or the adaptive ALE method by Barral, Olivier and Alauzet [44]. Moreover, the development of efficient and scalable, quality anisotropic mesh generators that integrate seamlessly within a Finite Element framework is a topic of active research [45]. Though with a phenomenal potential, anisotropic AMR for moving interfaces is still in its infancy [46].
A key component to all methods dealing with moving interfaces is the discretization of the transport operator (“total” or “material derivative”) representing the bulk motion of a certain scalar or vector quantity. This is especially relevant in Fluid Mechanics, where a great number of applications (viscoelastic, multiphase, combustion or turbulent flows, to name a few) can be formulated as a system of convection–reaction–diffusion equations in which the convective terms, dominant for high Peclet numbers, arise as a well-known source of numerical difficulties [47], [48], [49]. In contrast to Eulerian techniques, semi-Lagrangian schemes [50], [51] offer excellent stability properties by circumventing the Courant–Friedrichs–Lewy (CFL) condition, facilitating spatial mesh refinement along with large time steps.
Our purpose in this paper is to introduce ‘A-SLEIPNNIR’ (“Adaptive Semi-Lagrangian Ensemble Implementation of Particle level set for Newtonian and non-Newtonian Interfacial Rheology”), a robust, accurate and efficient technique suitable for moving interface problems. This method features a unified approach leveraging the techniques outlined above: a particle level set method for interface capturing [52], (an)isotropic refinement for spatial resolution [53], and semi-Lagrangian schemes for transport equations [54]. Especially relevant to the method is the role played by the “massless” particles that are passively advected by the velocity field (in a Lagrangian manner), helping us to correct the global level set function at the sub-mesh scale (thus acting as “markers”) by defining local level sets; attending to that dual character, they are denoted interchangeably “Lagrangian particles” and “marker particles” throughout the paper. As such particle correction relies on the level set function keeping a signed-distance character, a re-distancing step, accomplished here by an eikonal-based reinitialization procedure, is mandatory, at least within the region around the interface where particles are used. To the best of our knowledge, there is no previous work on particle level set methods with anisotropic mesh refinement. The merits of the proposed framework are showcased by a series of benchmark problems for interface transport in 2D configurations with analytically supplied, divergence-free velocity fields, while a future work will explore the flexibility of the technique in multiphase flows of incompressible, Newtonian and non-Newtonian fluids. The structure of the paper is straightforward: after this Introduction, we present in Section 2 the method proper, describing the numerical tools involved in A-SLEIPNNIR and showing the full time-stepping algorithm; then, we test in Section 3 our method in benchmark problems, analyzing its numerical behavior and comparing the results with the literature; finally, we summarize the main findings of this study in Section 4, highlighting the advantages and drawbacks of the technique.
Section snippets
A-SLEIPNNIR for interface transport problems
This Section presents the mathematical background for the numerical solution of moving interfaces in pure advection problems for which we solve the interface transport equation according to the A-SLEIPNNIR technique. We first provide a description of the interface capturing method employed to tackle the problem at hand; then, we discuss the semi-Lagrangian approach used to advect the level set function, together with the time and space discretizations in a finite element framework; after that,
Numerical results
In this section, we carry out a series of 2D numerical experiments to assess the capabilities of the A-SLEIPNNIR method in passive interface transport using analytically supplied, divergence-free velocity fields.
Remark 4 In the simulations performed in this paper, we choose a fixed, small time step size Δt that, along with a sufficiently high-order explicit Runge–Kutta method, produce a time discretization error low enough to be negligible when compared to the spatial error. The advantage of such
Conclusions
In this work we have introduced A-SLEIPNNIR, a technique suitable for moving interface problems. As shown by an in-depth comparison with the existing literature, the unique combination of a particle level set method for interface capturing, a second-order accurate reinitialization procedure, anisotropic mesh refinement for spatial adaptation, along with a quadratic, semi-Lagrangian advection scheme translates into a computationally cheap method offering state-of-the-art shape preservation,
Acknowledgements
Financial support from project MTM2015-67030-P from Ministerio de Educación, Cultura y Deporte, is gratefully acknowledged. The authors declare no conflict of interest.
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