An Efficient Dynamically Adaptive Mesh for Potentially Singular Solutions

Ceniceros, Héctor D.; Hou, Thomas Y.

doi:10.1006/jcph.2001.6844

Cited by 147 publications

(157 citation statements)

References 48 publications

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“…However, local refinement methods require complicated data structures and fairly technical methods to communicate information among different levels of refinements. In the mapping approach [14,15], the mesh points are moved continuously in the whole domain to concentrate in regions where the solution has the largest variations or where the moving interfaces are located. These solution-adaptive or geometry-adaptive meshes maps can be used to compute accurately the sharp variation or the moving interface problems.…”

Section: Introductionmentioning

confidence: 99%

Fictitious boundary and moving mesh methods for the numerical simulation of rigid particulate flows

Wan

Turek

2007

Journal of Computational Physics

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In this paper, we investigate the numerical simulation of rigid particulate flows using a new moving mesh method combined with the multigrid fictitious boundary method (FBM) [9,10]. With this approach, the mesh is dynamically relocated through a special partial differential equation to capture the region near the surface of the moving particles with high accuracy. The complete system is realized by solving the mesh movement and physical partial differential equations alternately. The flow is computed by an ALE formulation with a multigrid finite element solver, and the solid particles are allowed to move freely through the computational mesh which is adaptively aligned by the moving mesh method based on an arbitrary grid. The important aspect is that the data structures of the underlying undeformed mesh, in many cases a tensorproduct mesh, are preserved while only the spacing between the grid points is adapted in each step. Numerical results demonstrate that the interaction between the fluid and the particles can be accurately and efficiently handled by the presented method. It is also shown that the presented method directly improves upon the previous pure multigrid FBM to solve particulate flows with many moving rigid particles.

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Section: Introductionmentioning

confidence: 99%

Fictitious boundary and moving mesh methods for the numerical simulation of rigid particulate flows

Wan

Turek

2007

Journal of Computational Physics

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“…One approach is to combine grid adaptivity with multiscale modeling. We use a dynamic adaptive coarse grid [24] to capture the isolated small scale features, such as the flow channels and use the multiscale finite element method to capture the small scale feature within each adaptive coarse grid block. By doing this, we take into account the local flow orientation and anisotropy in upscaling the saturation equation.…”

Section: Applicationsmentioning

confidence: 99%

Multiscale Computations for Flow and Transport in Porous Media

Hou

2009

Series in Contemporary Applied Mathematics

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Abstract. Many problems of fundamental and practical importance have multiple scale solutions. The direct numerical solution of multiple scale problems is difficult to obtain even with modern supercomputers. The major difficulty of direct solutions is the scale of computation. The ratio between the largest scale and the smallest scale could be as large as 10 5 in each space dimension. From an engineering perspective, it is often sufficient to predict the macroscopic properties of the multiple-scale systems, such as the effective conductivity, elastic moduli, permeability, and eddy diffusivity. Therefore, it is desirable to develop a method that captures the small scale effect on the large scales, but does not require resolving all the small scale features. This paper reviews some of the recent advances in developing systematic multiscale methods with particular emphasis on multiscale finite element methods with applications to flow and transport in heterogeneous porous media.

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“…However, the obtained equations are complicated and massive computations are required. An alternative approach, as suggested by Ceniceros and Hou [36], is to consider a functional defined in the computational domain directly:Ẽ…”

Section: Mesh Generationmentioning

confidence: 99%

An alternating Crank–Nicolson method for the numerical solution of the phase‐field equations using adaptive moving meshes

Tan

Huang

2007

Numerical Methods in Fluids

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SUMMARYAn alternating Crank-Nicolson method is proposed for the numerical solution of the phase-field equations on a dynamically adaptive grid, which automatically leads to two decoupled algebraic subsystems, one is linear and the other is semilinear. The moving mesh strategy is based on the approach proposed by Li et al. (J. Comput. Phys. 2001; 170:562-588) to separate the mesh-moving and partial differential equation evolution. The phase-field equations are discretized by a finite volume method in space, and the mesh-moving part is realized by solving the conventional Euler-Lagrange equations with the standard gradient-based monitors. The algorithm is computationally efficient and has been successfully used in numerical simulations.

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An Efficient Dynamically Adaptive Mesh for Potentially Singular Solutions

Cited by 147 publications

References 48 publications

Fictitious boundary and moving mesh methods for the numerical simulation of rigid particulate flows

Fictitious boundary and moving mesh methods for the numerical simulation of rigid particulate flows

Multiscale Computations for Flow and Transport in Porous Media

An alternating Crank–Nicolson method for the numerical solution of the phase‐field equations using adaptive moving meshes

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