An adaptive grid is a grid network that automatically clusters grid points in regions of high flow field gradients; it uses the solution of the flow field properties to locate the grid points in the physical plane. The adaptive grid evolves in steps of time in conjunction with a time-dependent solution of the governing flow field equations, which computes the flow field variables in steps of time. During the course of the solution, the grid points in the physical plane move in such a fashion to ‘adapt’ for regions of large flow field gradients. Hence, the actual grid points in the physical plane are constantly in motion during the solution of the flow field, and become stationary only when the flow solution approaches a steady state.
Grid adaptation is used for both steady and unsteady type of problems. In case of steady flow problems grid is adapted after predetermined number of iterations and grid adaptation will stop at the point when solution is converged. In case of time accurate solutions the grid point motion and refinement are performed in conjunction with time accurate solution of the physical problem. This requires time accurate coupling of PDEs of the physical problem and those describing the grid movement or grid adaptation.
For the computations of newer configurations dependence on best practice guidelines for mesh generation and previous experience leaves the door open to large amounts of numerical error. Grid adaptation methods can produce substantial improvements in solution quality and promises better results because no limitations exist that define the limit on grid resolution that can be attained.
There are three basic types of grid adaptation techniques, that are $h$-method, $r$-method and $p$-method. There are some some mixed types of approaches one can find, such as $rp$-adaptation or $hp$-adaptation. Out of this $r$ and $h$ type of grid adaptation techniques are more popular in finite volume and finite difference schemes.
$h$ type:-
The $h$ method involves automatic refinement or coarsening of the spatial mesh based on a posteriori error estimates or error indicators.
$r$ type:-
Instead of making local topological changes to the mesh and its connectivity, r-adaptive methods make local changes to the resolution by moving the locations of a fixed total number of mesh points.
$p$ type:-
Very much popular method of grid adaptation in finite element approach rather than finite volume or finite element method. It reduces the error in the solution by enrichment of the polynomial of interpolating functions with the same geometric element order.Here no new mesh, geometry to be calculated and another advantage of this method is that it can better approximate irregular or curved boundaries with less sensitivity to aspect ratio and skew. Because of the this its very famous in structural application.
$Driving-sources-of-grid-adaptation$
$1. Feature-based-adaptation$ Feature based approximately largely used approach of grid adaptation employs feature of solution as driving force for grid adaptation. 469
These often uses features of the solution such as solution gradients and solution curvature. Flow regions that have large solution gradients are resolved with more points and regions of minimal significance are coarsened. This leads to refinement of the region which is physically specific such as boundary layer, shocks,separation lines, stagnation points, etc. In some cases, gradient-based refinement can actually increase the solution error so there are some issues regarding feature based adaptation like robustness and others.
$2.Truncation-error-based-adaption$ Truncation error is the difference between partial differential equation and its discretized equation. Truncation error is more suitable approach to find where adaptation should occur. General concept behind truncation error based adaptation is to equidistribute the error over the domain of simulation to reduce total discretization error. For simple equations evaluation of truncation error is easiest job but for complex schemes its difficult so different approach is needed for that purpose. For simple discretization schemes, the truncation error can be computed directly. For more complex schemes where direct evaluation of the truncation is difficult, an approach for estimating the truncation error is needed.
$3.Adjoint-based-adaptation$ Next promising approach is the adjoint approach. Its very good in estimating the local contribution of each cell or element to the discretization error in any solution functionals of interest such as lift, drag, and moments. So it is useful in targeted grid adaptation as per requirement of the solution so thats its also called as goal oriented adaptation.
All the best!
$References:-$
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