A new approach to grid generation

Liao, Guojun; Anderson, Dale

doi:10.1080/00036819208840084

Cited by 63 publications

(84 citation statements)

References 6 publications

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“…The significance of this work stems from the fact that the existence proofs in [49] are constructive in nature, thus motivating associated numerical methods such as [86,88,89], discussed below. Specifically, Dacorogna and Moser linearize (3.11) by expanding φ as a perturbation from the identity: φ(x) = x+ν(x).…”

Section: The Deformation Methodsmentioning

confidence: 99%

“…As indicated above, these concepts from [49] have been used to develop an algorithm for mesh transformation by Liao and co-workers [86,88,89], who noted that f (x) could be chosen to provide a transformation which contracts elements where f is small and expands them where it is large, forming the basis of an adaptive meshing algorithm. Liao and Su [90] then extended the ideas to allow for the tracking of time-varying volume elements.…”

Section: The Deformation Methodsmentioning

confidence: 99%

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Velocity-Based Moving Mesh Methods for Nonlinear Partial Differential Equations

Baines

Hubbard

Jimack

2011

Commun. comput. phys.

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Abstract. This article describes a number of velocity-based moving mesh numerical methods for multidimensional nonlinear time-dependent partial differential equations (PDEs). It consists of a short historical review followed by a detailed description of a recently developed multidimensional moving mesh finite element method based on conservation. Finite element algorithms are derived for both mass-conserving and non mass-conserving problems, and results shown for a number of multidimensional nonlinear test problems, including the second order porous medium equation and the fourth order thin film equation as well as a two-phase problem. Further applications and extensions are referenced.

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Section: The Deformation Methodsmentioning

confidence: 99%

Section: The Deformation Methodsmentioning

confidence: 99%

Velocity-Based Moving Mesh Methods for Nonlinear Partial Differential Equations

Baines

Hubbard

Jimack

2011

Commun. comput. phys.

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“…A simple and straightforward approach for determining the mesh transformation x(ξ, t) is to specify its Jacobian J (the determinant of (∂x)/(∂ξ)). This idea for mesh adaption has been used by many researchers, e.g., see [3,6,16,17,18]. While specifying the Jacobian is appealing in a number of aspects, the differential equations obtained directly are not necessarily easy to solve, nor is mesh nonsingularity guaranteed [6].…”

Section: Introductionmentioning

confidence: 99%

“…While it is not always clear how a mesh which drifts away from the desired one will be corrected at a later time for such methods, numerical results clearly demonstrate that they can be quite successful. Examples of methods in this class include the moving finite element (MFE) method of Miller and Miller [21,22], the deformation map method advocated by Liao and Anderson [18] and Semper and Liao [23], and the arbitrary Lagrangian-Eulerian (ALE) method first proposed by Hirt, Amsden, and Cook for the solution of fluid dynamic problems [11].…”

Section: Introductionmentioning

confidence: 99%

A Moving Mesh Method Based on the Geometric Conservation Law

Cao¹,

Huang²,

Russell³

2002

SIAM J. Sci. Comput.

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Abstract.A new adaptive mesh movement strategy is presented, which, unlike many existing moving mesh methods, targets the mesh velocities rather than the mesh coordinates. The mesh velocities are determined in a least squares framework by using the geometric conservation law, specifying a form for the Jacobian determinant of the coordinate transformation defining the mesh, and employing a curl condition. By relating the Jacobian to a monitor function, one is able to directly control the mesh concentration. The geometric conservation law, an identity satisfied by any nonsingular coordinate transformation, is an important tool which has been used for many years in the engineering community to develop cell-volume-preserving finite-volume schemes. It is used here to transform the algebraic expression specifying the Jacobian into an equivalent differential relation which is the key formula for the new mesh movement strategy. It is shown that the resulting method bears a close relation with the Lagrangian method. Advantages of the new approach include the ease of controlling the cell volumes (and therefore mesh adaption) and a theoretical guarantee for existence and nonsingularity of the coordinate transformation. It is shown that the method may suffer from the mesh skewness, a consequence resulting from its close relation with the Lagrangian method. Numerical results are presented to demonstrate various features of the new method.

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“…The main ways to follow in order to obtain adaptive meshes are: 1) mesh regenerating, 2) moving the inner nodes, 3) adding or extracting a number of nodes, 4) a composed method and taking into account the previous 3 methods already presented. The author suggest the reader to consider references [1], [2] for the case 1), [5], [8], [9] for the case 2), [3], [4], [7] for the case 3), [6] for the case 4).…”

Section: Introductionmentioning

confidence: 99%

Adaptive Method Using Controlled Grid Deformation

Florin¹

2011

INCAS BULLETIN

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A new approach to grid generation

Cited by 63 publications

References 6 publications

Velocity-Based Moving Mesh Methods for Nonlinear Partial Differential Equations

Velocity-Based Moving Mesh Methods for Nonlinear Partial Differential Equations

A Moving Mesh Method Based on the Geometric Conservation Law

Adaptive Method Using Controlled Grid Deformation

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