43:) is a method of adapting the accuracy of a solution within certain sensitive or turbulent regions of simulation, dynamically and during the time the solution is being calculated. When solutions are calculated numerically, they are often limited to predetermined quantified grids as in the Cartesian plane which constitute the computational grid, or 'mesh'. Many problems in numerical analysis, however, do not require a uniform precision in the numerical grids used for graph plotting or computational simulation, and would be better suited if specific areas of graphs which needed precision could be refined in quantification only in the regions requiring the added precision. Adaptive mesh refinement provides such a dynamic programming environment for adapting the precision of the numerical computation based on the requirements of a computation problem in specific areas of multi-dimensional graphs which need precision while leaving the other regions of the multi-dimensional graphs at lower levels of precision and resolution.
170:, the solution (water height) might only be calculated for points every few feet apart—and one would assume that in between those points the height varies smoothly. The limiting factor to the resolution of the solution is thus the grid spacing: there will be no features of the numerical solution on scales smaller than the grid-spacing. Adaptive mesh refinement (AMR) changes the spacing of grid points, to change how accurately the solution is known in that region. In the shallow water example, the grid might in general be spaced every few feet—but it could be adaptively refined to have grid points every few inches in places where there are large waves.
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The image above shows the grid structure of an AMR calculation of a shock impacting an inclined slope. Each of the boxes is a grid; the more boxes it is nested within, the higher the level of refinements. As the image shows, the algorithm uses high resolution grids only at the physical locations and
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which advances those cells in time. Finally, a correction procedure is implemented to correct the transfer along coarse-fine grid interfaces, to ensure that the amount of any conserved quantity leaving one cell exactly balances the amount entering the bordering cell. If at some point the level of
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Advanced mesh refinement has been introduced via functionals. Functionals allow the ability to generate grids and provide mesh adaptation. Some advanced functionals include the
Winslow and modified Liao functionals.
495:"A subcycling/non-subcycling time advancement scheme-based DLM immersed boundary method framework for solving single and multiphase fluid--structure interaction problems on dynamically adaptive grids"
66:. The use of AMR has since then proved of broad use and has been used in studying turbulence problems in hydrodynamics as well as in the study of large scale structures in astrophysics as in the
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All tagged cells are then refined, meaning that a finer grid is overlaid on the coarse one. After refinement, individual grid patches on a single fixed level of refinement are passed off to an
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In addition, the AMR methods have been developed and applied to a wide range of fluid mechanics problems, including two-phase flows, fluid-structure interactions, and wave energy converters.
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Complete control of grid resolution, compared to the fixed resolution of a static grid approach, or the
Lagrangian-based adaptivity of
446:"A parallel cell-centered adaptive level set framework for efficient simulation of two-phase flows with subcycling and non-subcycling"
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Compared to pre-tuned static meshes, the adaptive approach requires less detailed a priori knowledge on the evolution of the solution.
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536:"Reynolds-Averaged Navier--Stokes simulation of the heave performance of a two-body floating-point absorber wave energy system"
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refinement in a cell is greater than required, the high resolution grid may be removed and replaced with a coarser grid.
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can be used - in which the grid is more finely spaced in some regions than others, but maintains its shape over time.
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If the region in which higher resolution is desired remains localized over the course of the computation, then
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This dynamic technique of adapting computation precision to specific requirements has been accredited to
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313:"Star formation with 3-D adaptive mesh refinement: the collapse and fragmentation of molecular clouds"
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This allows the user to solve problems that are completely intractable on a
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core down to an effective resolution of 131,072 cells per initial cloud
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Zeng, Yadong; Xuan, Anqing; Blaschke, Johannes; Shen, Lian (2022).
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271:"Local adaptive mesh refinement for shock hydrodynamics"
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This article is about the use of adaptive meshing in
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493:Zeng, Yadong; Bhala, Amneet; Shen, Lian (2022).
317:Journal of Computational and Applied Mathematics
371:"Grid generation and adaptation by functionals"
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344:Huang, Weizhang; Russell, Robert D. (2010).
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269:Berger, Marsha J.; Colella, Philipp (1989).
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221:Berger, Marsha J.; Oliger, Joseph (1984).
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346:Adaptive Moving Mesh Method
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399:Popinet, Stéphane (2015).
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471:10.1016/j.jcp.2021.110740
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125:Richardson extrapolation
37:adaptive mesh refinement
311:Klein, Richard (1999).
168:shallow water equations
540:Computers & Fluids
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