Manipulating Boundaries and Viscous Regions of Unstructured Meshes Using Winslow's Equations

Masters, James S.; Karman, Steve L.

doi:10.2514/1.j051276

Cited by 11 publications

(3 citation statements)

References 8 publications

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“…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%

“…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]. It is a common practice to employ mesh quality improving optimization techniques, such as in [30,31].…”

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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“…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%

Universal meshes for smooth surfaces with no boundary in three dimensions

Kabaria

Lew

2018

Numerical Meth Engineering

View full text Add to dashboard Cite

show abstract

“…These include methods that compute the motion of vertices by solving a linear or non‐linear elasticity systems , or mass‐spring systems (e.g., ). There are also methods that prescribe the velocity for the vertices , including methods based on radial basis function interpolation , the solution of Winslow equations , the solution of the biharmonic operator , Delaunay mesh adaptation, and edge‐vertex‐based connectivity changes . It is a common practice to employ mesh quality improving optimization techniques, such as in .…”

Section: Introductionmentioning

confidence: 99%

Universal meshes for smooth surfaces with no boundary in three dimensions

Kabaria

Lew

2016

Numerical Meth Engineering

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Summary We introduce a method to mesh the boundary Γ of a smooth, open domain in double-struckR3 immersed in a mesh of tetrahedra. The mesh follows by mapping a specific collection of triangular faces in the mesh to Γ. Two types of surface meshes follow: (a) a mesh that exactly meshes Γ, and (b) meshes that approximate Γ to any order, by interpolating the map over the selected faces; that is, an isoparametric approximation to Γ. The map we use to deform the faces is the closest point projection to Γ. 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 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. Copyright © 2016 John Wiley & Sons, Ltd.

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Unstructured Mesh Manipulation in Response to Surface Deformation

Masters¹,

Tatum²

2016

AIAA Journal

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Manipulating Boundaries and Viscous Regions of Unstructured Meshes Using Winslow's Equations

Cited by 11 publications

References 8 publications

Universal meshes for smooth surfaces with no boundary in three dimensions

Universal meshes for smooth surfaces with no boundary in three dimensions

Universal meshes for smooth surfaces with no boundary in three dimensions

Unstructured Mesh Manipulation in Response to Surface Deformation

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