From: Wolfgang Bangerth
-There is not much to be said about the results of this program, apart
-from that they look nice. All images were made using GMV from the
-output files that the program wrote to disk. The first picture shows
-the displacement as a vector field, where one vector is shown at each
-vertex of the grid:
-
-
-
-You can clearly see the sources of x-displacement around x=0.5 and
-x=-0.5, and of y-displacement at the origin. The next image shows the
-final grid after eight steps of refinement:
-
-
-
-Finally, the x-displacement and y-displacement are displayed separately:
-
-Introduction
-
-
-The main emphasis in this example is the handling of locally refined
-grids. The approach to adaptivity chosen in deal.II us to use grids in which
-neighboring cells may be refined a different number of times. This then
-results in nodes on the interfaces of cells which belong to one
-side, but are unbalanced on the other. The common term for these is
-“hanging nodes”.
-
-
-
-To guarantee that the global solution is continuous at these nodes as
-well, we have to state some additional constraints on the values of
-the solution at these nodes. In the program below, we will show how we
-can get these constraints from the library, and how to use them in the
-solution of the linear system of equations.
-
-
-
-The locally refined grids are produced using an error estimator class
-which estimates the energy error with respect to the Laplace
-operator. This error estimator, although developed for Laplace's
-equation has proven to be a suitable tool to generate locally refined
-meshes for a wide range of equations, not restricted to elliptic
-problems. Although it will create non-optimal meshes for other
-equations, it is often a good way to quickly produce meshes that are
-well adapted to the features of solutions, such as regions of great
-variation or discontinuities. Since it was developed by Kelly and
-co-workers, we often refer to it as the “Kelly refinement
-indicator” in the library, documentation, and mailing list. The
-class that implements it is called
-KellyErrorEstimator
. Although the error estimator (and
-its
-implementation in the deal.II library) is capable of handling variable
-coefficients in the equation, we will not use this feature since we
-are only interested in a quick and simple way to generate locally
-refined grids.
-
-
-
-Since the concepts used for locally refined grids are so important,
-we do not show much additional new stuff in this example. The most
-important exception is that we show how to use biquadratic elements
-instead of the bilinear ones which we have used in all previous
-examples. In fact, The use of higher order elements is accomplished by
-only replacing three lines of the program, namely the declaration of
-the fe
variable, and the use of an appropriate quadrature formula
-in two places. The rest of the program is unchanged.
-
-
-
-The only other new thing is a method to catch exceptions in the
-main
function in order to output some information in case the
-program crashes for some reason.
-
diff --git a/deal.II/examples/step-8/doc/results.dox~ b/deal.II/examples/step-8/doc/results.dox~
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-
-Results
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-It should be noted that intuitively one would have expected the -solution to be symmetric about the x- and y-axes since the x- and -y-forces are symmetric with respect to these axes. However, the force -considered as a vector is not symmetric and consequently neither is -the solution. -
-