Adaptive meshes

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. 

Note that the codes usually do not support moving objects with either adaptive meshes or a stationary mesh. 

Solutions to models with different length scales may contain regions such as shocks, steep fronts and other near discontinuities. Adaptive meshing strategies, in which a spatial mesh network is adjusted dynamically so as to capture the local behavior accurately, will be described. The algorithm will be tested on an example of filtration combustion. 

Several different methods may be used to generate and adaptive mesh. Saltzman (1) categorized them as follows i) fixed meshes, ii) selectively refined meshes, and iii) moving meshes. 

It is obvious that a strong coupling between the solution of the governing equations and the generation of the adaptive mesh is of prime importance for a case like this one, since a priori knowledge on how the solution will evolve is not available. 

We did not attempt to perform two-dimensional calculations on an equispaced grid. According to our estimates, it would have taken approximately 3 hours of CPU time to integrate the governing equations from time t 0.0 till t = 1.0 on a Cray-1 computer as compared to 25 minutes of CPU time using the adaptive mesh. 

Ab initio calculations addition of hydroxide ion to formaldehyde, 212 energy surfaces, 201-211 Adaptive continuation method, hydrogen-air flame, 417 Adaptive mesh... 

Kinoshita, T, Inoue, O., A parallel adaptive mesh approach for flowfields with shock waves. Shock Waves 12 (2002) 167-175. 

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... 

Russell, R. D., and Christiansen, J., Adaptive mesh selection strategies for solving boundary value problems," SIAM J. Numer. Anal. 15(1), 59-80(1978). 

To track it correctly, the Lagrange PI finite elements, with adaptive mesh,... 

If the OCFE solution procedure is to be used regularly for system analysis, a careful study of the problem of locating elements is necessary. Finlayson (1980) gives a basis for this study in a cursory examination of variable grid spacing and elements sizes, or adaptive meshes as he calls them. The simplest procedure is based on physical information about the solution profile such as in the present case the point where CO is depleted within the reactor bed. 

With this implicit boundary-condition specification, the eigenvalue computation can be easily incorporated into a Newton iteration to solve the entire problem. Time marching and adaptive meshing can also be incorporated [158,159]. 

In this section we consider problems in which there is convective and diffusive transport in one spatial dimension, as well as elementary chemical reaction. The computational solution of such problems requires attention to discretization on a mesh network and solution algorithms. For steady-state situations the computational problem is one of solving a boundary-value problem. In chemically reacting flow problems it is not uncommon to have steep reaction fronts, such as in a flame. In such a case it is important to provide adequate mesh resolution within the front. Adaptive mesh schemes are used to accomplish this objective. 

A great scale disparity is apparent from the species profiles, whose peak values range over several orders of magnitude. The spatial-scale disparity is also apparent. The very high gradients and curvatures are especially clear in the low-concentration species shown in the lower right-hand panel. Without good adaptive mesh placement, the numerical solution would be unable to capture these critical features of the flame structure. 

In a similar spirit, Alkire, Braatz and co-workers developed coupled hybrid continuum-KMC simulations to study the electrodeposition of Cu on flat surfaces and in trenches (Drews et al., 2003b, 2004 Pricer et al., 2002a, b). A 3D KMC simulation accounted for the surface processes as well as diffusion in the boundary layer next to the surface, whereas a ID or 2D continuum model (with adaptive mesh) was applied to simulate the boundary layer farther away. In... 

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... 

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