Moving meshes by the deformation method

Liao, Guojun; Xue, Jiaxing

doi:10.1016/j.cam.2005.07.022

Cited by 13 publications

(18 citation statements)

References 9 publications

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“…It is shown in [8] that under reasonable smoothness assumptions, we do indeed have J(ϕ(x)) = f (x). In particular, J(ϕ(x)) > 0, which in turn implies that T is invertible.…”

Section: Construction Of a Transformation With Prescribed Jacobian Dementioning

confidence: 92%

Construction of differentiable transformations

Liao

Cai²,

Liu

et al. 2009

Applied Mathematics Letters

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a b s t r a c tConstruction of invertible transformations using differential equations is an interesting and challenging mathematical problem with important applications. We briefly review the existing method by means of harmonic maps in 2D and propose a method of constructing differentiable, invertible transformations between domains in two and three dimensions. Preliminary numerical results demonstrate the effectiveness of the method.

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“…It is shown in [8] that under reasonable smoothness assumptions, we do indeed have J(ϕ(x)) = f (x). In particular, J(ϕ(x)) > 0, which in turn implies that T is invertible.…”

Section: Construction Of a Transformation With Prescribed Jacobian Dementioning

confidence: 92%

Construction of differentiable transformations

Liao

Cai²,

Liu

et al. 2009

Applied Mathematics Letters

Self Cite

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“…Further, [32] combines the algorithm with a least-squares finite element approach to solving the divcurl system (3.18) directly for ν(x,t), and apply the resulting algorithm to problems where the boundary deforms. The algorithm has also been applied in three space dimensions [91], to track moving shock waves [87], and in image registration [41]. Delzanno et al [51] however note some situations where the quality of the grids obtained compares unfavourably with a Monge-Kantorovich transformation approach.…”

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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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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“…In this section, we summarize the adaptive moving grid by deformation method of Liao et al [24–28] and discuss its practical implementation. The advantage of this particular moving grid method as compared with many others is that it allows a much simpler construction of the grid mapping and can be easily extended to higher dimensions.…”

Section: Deformation Moving Grid Methodsmentioning

confidence: 99%

“…The advantage of this particular moving grid method as compared with many others is that it allows a much simpler construction of the grid mapping and can be easily extended to higher dimensions. The description in this subsection mainly follows the presentation of the cited references [24–28], but is rewritten for clarity and consistency with later adaptation to the GDM framework.…”

Section: Deformation Moving Grid Methodsmentioning

confidence: 99%

“…In contrast, the uniform reference grid maintained by a moving grid method allows the same topology preserving principle to be applied as in the uniform grid case. In this work, we adapt a particular moving grid method, the deformation moving grid method , which was developed by Liao et al [24–28], to solve the level set PDEs required in GDMs. In particular, we develop the methods necessary to compute signed distance functions, to handle topology preservation, and to adapt the grid in an image-driven fashion, which improves the results in many applications.…”

Section: Introductionmentioning

confidence: 99%

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A Moving Grid Framework for Geometric Deformable Models

Han

Prince

2009

Int J Comput Vis

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Geometric deformable models based on the level set method have become very popular in the last decade. To overcome an inherent limitation in accuracy while maintaining computational efficiency, adaptive grid techniques using local grid refinement have been developed for use with these models. This strategy, however, requires a very complex data structure, yields large numbers of contour points, and is inconsistent with the implementation of topology-preserving geometric deformable models (TGDMs). In this paper, we investigate the use of an alternative adaptive grid technique called the moving grid method with geometric deformable models. In addition to the development of a consistent moving grid geometric deformable model framework, our main contributions include the introduction of a new grid nondegeneracy constraint, the design of a new grid adaptation criterion, and the development of novel numerical methods and an efficient implementation scheme. The overall method is simpler to implement than using grid refinement, requiring no large, complex, hierarchical data structures. It also offers an extra benefit of automatically reducing the number of contour vertices in the final results. After presenting the algorithm, we demonstrate its performance using both simulated and real images.

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Moving meshes by the deformation method

Cited by 13 publications

References 9 publications

Construction of differentiable transformations

Construction of differentiable transformations

Velocity-Based Moving Mesh Methods for Nonlinear Partial Differential Equations

A Moving Grid Framework for Geometric Deformable Models

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