Dear Jie, > it seems that the L2-projection method also evaluates the stress tensor at > the quadrature points and then finds the cell-averaged stress. The only > difference compared with the heuristic method is that it solves AX = Y > where A is mass matrix, Y is integration of cell-averaged stress component > and X is the nodal stress component that we want. Is that right? >
No, this is not correct. I can think of only two simplified situations where the least-squares projection is reduced to either of the two calculation approaches you've mentioned in this thread: 1. If you use a FE_DGQ of degree 0, then the projection is reduced to the cell-average value (a constant, and thus obviously piece-wise discontinuous). 2. If your grid is uniform and Cartesian, and you're using linear FE_Q elements then you can compute the equivalent average (continuous) nodal value using the geometric arguments you put forward earlier. However, if you add any complexity to your problem then immediately you run into issues: - If your grid is non-uniform or unstructured, then your have more geometric considerations to make: If you have a large cell adjacent to a small cell, then is the weighting of the values that come from the large cell equal to the small cell? - What if you have higher order continuous elements, and the support points no longer all coincide with the cell vertices? - What if your finite element has no support points? In any of these situations your geometric averaging idea will either fail or it will not produce a smoothed field that is representative of the underlying quadrature data. Like a simple linear regression <https://en.wikipedia.org/wiki/Linear_least_squares_(mathematics)>, the least-squares method will minimise the error of your function (as given by the quadrature data) projected onto the given finite element space (be it continuous or discontinuous, low or high order, etc.). Does that make sense? Regards, Jean-Paul -- The deal.II project is located at http://www.dealii.org/ For mailing list/forum options, see https://groups.google.com/d/forum/dealii?hl=en --- You received this message because you are subscribed to the Google Groups "deal.II User Group" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. For more options, visit https://groups.google.com/d/optout.
