High-order ALE schemes for incompressible capillary flows

Montefuscolo, Felipe; Sousa, Fabrício S.; Buscaglia, Gustavo C.

doi:10.1016/j.jcp.2014.08.030

Cited by 17 publications

(23 citation statements)

References 28 publications

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“…Before proceeding to the proof, we note that we kept the second term of the right-hand side so that when n is orthogonal to the face defined by c and b and " D 0, (22) reduces to the trivial inequality c b n > jc bj. This follows from the fact that in this case c b D jc bjn.…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

“…The bound on jn x;K j follows as an immediate consequence of such lemma by identifying jKj D jc bj=2, jT j D c b a=6 in (22), and noticing that from (26) in Lemma 6.2 a n x C "jaj > :…”

Section: Proofmentioning

confidence: 99%

“…Strategies to move the vertices of the mesh are often included in r-adaption methods (e.g., [16,17]) or arbitrary Lagrangian-Eulerian algorithms (e.g., [18][19][20]). These include methods that compute the motion of vertices by solving a linear [9,21,22] or non-linear elasticity systems [10,23], or mass-spring systems (e.g., [18]). There are also methods that prescribe the velocity for the vertices [24], including methods based on radial basis function interpolation [25], the solution of Winslow equations [26], the solution of the biharmonic operator [27], Delaunay mesh adaptation [28], and edge-vertex-based connectivity changes [29].…”

Section: Introductionmentioning

confidence: 99%

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Universal meshes for smooth surfaces with no boundary in three dimensions

Kabaria

Lew

2018

Numerical Meth Engineering

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We introduce a method to mesh the boundary of a smooth, open domain in R 3 immersed in a mesh of tetrahedra. The mesh follows by mapping a specific collection of triangular faces in the mesh to . Two types of surface meshes follow: (a) a mesh that exactly meshes , and (b) meshes that approximate to any order, by interpolating the map over the selected faces; that is, an isoparametric approximation to . The map we use to deform the faces is the closest point projection to . We formulate conditions for the closest point projection to define a homeomorphism between each face and its image. These are conditions on some of the tetrahedra intersected by the boundary, and they essentially state that each such tetrahedron should (a) have a small enough diameter, and (b) have two of its dihedral angles be acute. We provide explicit upper bounds on the mesh size, and these can be computed on the fly.We showcase the quality of the resulting meshes with several numerical examples. More importantly, all surfaces in these examples were meshed with a single background mesh. This is an important feature for problems in which the geometry evolves or changes, because it could be possible for the background mesh to never change as the geometry does. In this case, the background mesh would be a universal mesh [1] for all these geometries. We expect the method introduced here to be the basis for the construction of universal meshes for domains in three dimensions. it defines then a triangulation with curved triangles that exactly lie on the surface. Additionally, and as we explain here, by constructing a finite element space over the set of positive faces, and then interpolating the map that defines the homeomorphism, we obtain an isoparametric approximation of the domain. In this way, the method could be used with standard finite element tools available in many commercial codes, and approximations of the geometry of any order can be constructed. For finite element spaces of the Lagrange type, this is tantamount to moving nodes on the positive faces to the surface.The motivation behind this method is that, as the geometry of the domain evolves, the same background mesh would be deformed to conform to the geometry of the domain for the entire (or for periods that do not depend on the mesh size or time step) of the simulation. Hence, no new vertices or faces are introduced in the background mesh as the geometry evolves. For finite element methods, this has the positive consequence of keeping intact the sparsity pattern of the matrix associated with the mesh as the domains evolves. For similar reasons, the method provides algorithmic advantages if iterating over the geometry of the domain is necessary, a feature that could be advantageous in explored in applications such as topology optimization or some fluid-structure interaction problems. It is important to note that we see the surface generation method presented here a step towards the method to generate tetrahedral meshes for deforming and moving domains from a fixed background mesh...

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Section: Proof Of the Main Resultsmentioning

confidence: 99%

“…The bound on jn x;K j follows as an immediate consequence of such lemma by identifying jKj D jc bj=2, jT j D c b a=6 in (22), and noticing that from (26) in Lemma 6.2 a n x C "jaj > :…”

Section: Proofmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Universal meshes for smooth surfaces with no boundary in three dimensions

Kabaria

Lew

2018

Numerical Meth Engineering

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“…To our best knowledge, we are not aware of any ALE algorithm with the exact mass conservation for incompressible multiphase fluid flows (see, for example, previous works), which is critical for the accuracy of long‐term simulation of multiphase problems. Although Montefuscolo et al presented high‐order ALE formulations based on div‐stable ( P 2 / P 1 ,

P_{1}^{+} false/ P_{1}

) finite elements with small mass losses, the total mass error is a function of time step, and it requires a relatively small time step. However, the compatibility relation between the vertex velocity and the rate of change of a volume is noted within the Lagrangian cell‐centered scheme for two‐dimensional compressible flows, and the reconnection‐based arbitrary Lagrangian‐Eulerian (ReALE) formulation in which the Lagrangian solution of two‐dimensional compressible flow is conservatively interpolated onto a new grid …”

Section: Introductionmentioning

confidence: 99%

An arbitrary Lagrangian‐Eulerian framework with exact mass conservation for the numerical simulation of 2D rising bubble problem

Guventurk

Şahin

2017

Numerical Meth Engineering

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An arbitrary Lagrangian-Eulerian framework, which combines the advantages of both Lagrangian and Eulerian methods, is presented to solve incompressible multiphase flow problems. The incompressible Navier-Stokes equations are discretized using the side-centered unstructured finite volume method, where the velocity vector components are defined at the midpoint of each cell face, while the pressure term is defined at element centroids. The pressure field is treated to be discontinuous across the interface with the discontinuous treatment of density and viscosity. The surface tension term at the interface is treated as a force tangent to the interface and computed with several different approaches including the use of Legendre polynomials. In addition, the several different discretizations of interface kinematic boundary conditions are investigated. For the application of the interface kinematic boundary condition, a special attention is given to satisfy both local and global discrete geometric conservation law to conserve the total mass of both species at machine precision.The mesh vertices are deformed by solving the linear elasticity equations due to the normal displacement of interface. The resulting algebraic equations are solved in a fully coupled manner, and a one-level restricted additive Schwarz preconditioner with a block-incomplete factorization within each partitioned subdomain is used for the resulting fully coupled system. The method is validated by simulating the classical benchmark problem of a single rising bubble in a viscous fluid due to buoyancy.The results of numerical simulations are found out to be in an excellent agreement with the earlier results in the literature. The mass of the bubble is conserved, and discontinuous pressure field is obtained to avoid errors due to the incompressibility condition in the vicinity of the interface, where the density and viscosity jumps occur.

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“…High-order Arbitrary Lagrangian-Eulerian (ALE) schemes for capillary flows proposed by Montefuscolo, Sousa and Buscaglia (2014) are used here for solving the fluid-solid interaction problem with slip velocity (DONEA; GIULIANI; HALLEUX, 1982; DONEA;…”

Section: Introductionmentioning

confidence: 99%

Finite element methods for multuphase flow in microscales

Sanchez¹

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I am deeply and extremely grateful to my family, the perfect treasure I have. Each of them well deserves a vast letter of acknowledgements for their contributions, in many senses, to my life. However, for completeness, I would like to mention my mother Rosa, who has always been there for me, my pillar, my brother Jhonatan, for his patience during my journeys, and my father Carlos for his unconditional help. My aunts, Luz, Carmenza, Martha and Amparo, because they have taught me what family and real love are. My cousins and uncles, for all the cheering up and good mood. I would also like to specially thank my advisor Gustavo, because his patience with my occurrences has been way way way admirable during this period of doctoral studies that ends, as well as special thanks to my coadvisor Fabricio. Finally, many thanks to my colleagues from the LMACC, so nice people I had the fortune to meet, specially Sabrina, Alfredo, Miguel, Hugo and Larissa. Finally, I gratefully acknowledge the financial support received from São Paulo Research Foundation (FAPESP) under grants #2014/14720-8, #2013/07375-0 (CEPID) and #2016/20115-5 (BEPE). The finishing of this work would not have been possible without this important support. "Si alguna vez me cruzas por la calle Regálame tu beso y no te aflijas Si ves que estoy pensando en otra cosa No es nada malo, es que pasó una brisa La brisa de la muerte enamorada Que ronda como un ángel asesino Mas no te asustes siempre se me pasa Es solo la intuición de mi destino." (Fito Páez, Al lado del camino)

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High-order ALE schemes for incompressible capillary flows

Cited by 17 publications

References 28 publications

Universal meshes for smooth surfaces with no boundary in three dimensions

Universal meshes for smooth surfaces with no boundary in three dimensions

An arbitrary Lagrangian‐Eulerian framework with exact mass conservation for the numerical simulation of 2D rising bubble problem

Finite element methods for multuphase flow in microscales

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