A Moving Mesh Method Based on the Geometric Conservation Law

@article{Cao2002AMM,
  title={A Moving Mesh Method Based on the Geometric Conservation Law},
  author={Weiming Cao and Weizhang Huang and Robert D. Russell},
  journal={SIAM J. Sci. Comput.},
  year={2002},
  volume={24},
  pages={118-142}
}
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… 

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References

SHOWING 1-10 OF 31 REFERENCES
A high dimensional moving mesh strategy
Moving Mesh Strategy Based on a Gradient Flow Equation for Two-Dimensional Problems
TLDR
The results demonstrate the potential of the mesh movement strategy to concentrate the mesh points so as to adapt to special problem features and to also preserve a suitable level of mesh orthogonality.
Analysis and computation of adaptive moving grids by deformation
TLDR
It is proved that the resulting grid has the prescribed cell sizes and that no “mesh tangling” occurs, and that the method accurately redistributes the nodes and does not tangle the mesh.
FINITE VOLUME METHOD FOR PREDICTION OF FLUID FLOW IN ARBITRARILY SHAPED DOMAINS WITH MOVING BOUNDARIES
In this paper a method is presented that can be used for both the Lagrangian and the Eulerian solution of the Navier–Stokes equations in a domain of arbitrary shape, bounded by boundaries which move
Geometric Conservation Law and Its Application to Flow Computations on Moving Grids
Boundary-conforming coordinate transformations are used widely to map a flow region onto a computational space in which a finite-difference solution to the differential flow conservation laws is
An Arbitrary Lagrangian-Eulerian Computing Method for All Flow Speeds
A new numerical technique is presented that has many advantages for obtaining solutions to a wide variety of time-dependent multidimensional fluid dynamics problems. The method uses a finite
Numerical Solution of the Quasilinear Poisson Equation in a Nonuniform Triangle Mesh
A finite-difference method using a nonuniform triangle mesh is described for the numerical solution of the nonlinear two-dimensional Poisson equation�·(���) +S= 0, where � is a function of �
An Adaptive Grid Method and Its Application to Steady Euler Flow Calculations
TLDR
An adaptive remeshing procedure based on a cell volume deformation method that offers direct cell volume control through the specification of the transformation Jacobian in order to solve the compressible Euler equations.
Moving finite elements
TLDR
This book brings together most of the work done over the last decade or so which has been stimulated by Miller's original idea, and discusses the interrelationships between the techniques and the established ideas of the method of characteristics, Hamilton's equations, the Legendre transformation, and grid equidistribution.
Moving mesh partial differential equations (MMPDES) based on the equidistribution principle
This paper considers several moving mesh partial differential equations that are related to the equidistribution principle. Several of these are new, and some correspond to discrete moving mesh
...
1
2
3
4
...