Adaptive mesh refinement

M.J. Berger and J. Olinger, Adaptive Mesh Refinement for Hyperbolic Partial Differential Equations, J. Comput. Phys. S3 (1984). 

M.J. Berger and P. Colella, Local Adaptive Mesh Refinement for Shock Hydrodynamics, UCRL-97196, Lawrence Livermore National Laboratory, Livermore, CA, 1987. 

Computational fluid dynamics (CFD) programs are more specialized, and most have been designed to solve sets of equations that are appropriate to specific industries. They can then include approximations and correlations for some features that would be difficult to solve for directly. Four major packages widely used are Fluent (http //www.fluent.com/), CFX (now part of ANSYS), Comsol Multiphysics (formerly FEMLAB) (http //www.comsol.com/), and ANSYS (http //www.ansys.com/). Of these, Comsol Multiphysics is particularly useful because it has a convenient graphical-user interface, permits easy mesh generation and refinement (including adaptive mesh refinement), allows the user to add phenomena and equations easily, permits solution by continuation methods (thus enhancing... 

Solution adaptive mesh refinement is another powerful feature that enables faster and more-accurate results. This method allows the user to modify the grid interactively based on intermediate solutions so that grid points can be put where they are required and can make the most contribution to the solution accuracy. Although not new, solution adaptive mesh refinement has become increasingly easy to use due to more-sophisticated user interfaces and the ability to handle hybrid meshes in the new generation of CFD packages. 

Berger M, Oliger J (1984) Adaptive mesh refinement for hyperbolic partial differential equations. J ComputPhys 53 484-512... 

Provatas N., Goldenfeld N. and Dantzig J., Efficient Computation of Dendritic Microstructures Using Adaptive Mesh Refinement, Phys. Rev. Lett. 80, 3308 (1998). 

In the considered test case, a gel (/gei = 5 mm) is immersed in a solution bath (total length I = 15 mm). The applied discretisation of the gel-solution domain is depicted in Fig. 11. In order to resolve the large gradients/jumps, an adaptive mesh refinement is applied. 

James and Williams (1985) employed a Galerkin finite elanent schane to provide flexible numerical solutions to the PB equation in one and two dimensions, using a Newton sequence for the solution of the set of nonlinear equations arising from the finite-element discretization procedure the procedure was shown to be applicable for different geometries. Dyshlovenko (2002) applied a finite-element solution of the PB equation with an adaptive mesh refinement procedure. The procedure allowed the gradual improvement of the solution and adjustment of the geometry of the problem. The approach was successfully applied to several problems... 

Chapter II, Blast Wave Reflections and Interactions, presents a number of articles on the interaction of blast waves with real surfaces. For example, Rayevsky et al. have studied the normal reflection of a blast wave from a rigid wall coated with polyurethane foam. They found that, contrary to intuition, the foam layer significantly increased the peak reflected pressure on the wall. Lyakhov and coworkers report on shock reflections from a body with a hot or cold gas layer. Kuhl et al. present a detailed simulation of a double-Mach reflection from a dusty wall. By using a nondiffusive numerical scheme and adaptive mesh refinement, they were able to directly calculate the mixing in the unstable wall jet and dusty boundary layer flow. Similarity coordinates were used to average the fluctuating flow and thereby determine the dusty boundary layer profiles. Shepherd et al. report on the repeated reflections of detonation-driven blast waves in containers. 

J. A. Bennett and M. E. Botkin Shape Optimization Of Two-Dimensional Structures With Geometric Problem Description and Adaptive Mesh Refinement, General Motors Research Laboratories, May 2,1983. 

Figure 6.1 NOBEL adaptive mesh refinement allows isentropic refinement limited to a 2 1 ratio. Adjacent cells may not differ by more than one level.

Forward and adjoint methods—PDE methods with adaptive mesh refinement... 

Primary droplet size and velocity can be extracted from simulations, as illustrated in Fig. 18.10. The primary liquid fragment can be well resolved through adaptive mesh refinement. In simulations the droplet size is determined by... 

P. MacNeice, K. M. Olson, C. Mobarry, R. de Fainchtein, and C. Packer, Comput. Phys. Commun., 126, 330 (2000). PARAMESH A Parallel Adaptive Mesh Refinement Community Toolkit. 

Due to the scale-invariance of the barycentric coordinates Ai, A2, A3, the condition number of the matrix, arising from the expansion (6), is independent of the local mesh width h, see [1]. We remark that this observation is very crucial when working with adaptive mesh refinement, where very small cells may appear. Indeed, the representation (6) is, due to its robustness, particularly suited for adaptive mesh refinement, even for strongly distorted meshes. 

In general, our results in Figure 12 show, that for problems with smooth solutions higher order schemes are more efficient than low order schemes with globally refined meshes, especially when highly accurate results are desired. In order to enhance the accuracy of the proposed ADER schemes even more, especially for solutions with discontinuities, we combine the ADER schemes with the ideas of adaptive mesh refinement in order to reduce numerical smearing. Details of our adaptive mesh strategy are discussed below. 

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