Obviously, this is a huge topic. What I have concentrated on are a few aspects that serve as components to the overall framework.
Fundamental Analysis
Solvers
I've worked on 3D h-version Finite Element solvers and a basic 2D p-version solver for linear mechanics problems. Element types covered include: linear and quadratic triangles, Quads and Tetrahedra elements. For p-version, I used Quad with hierarchical polynomials of orders 1 through 8.
As you may be aware of, there are quite some numerical treatments involved in a Finite Element solver for it to run fast and efficient. Some of the specific areas that I've particularly focused include: Node/element reordering algorithms for bandwidth/profile reduction of the overall stiffness matrix; out-of-core skyline solver to handle HUGE models for which the resulting matrices are too big to be solved in-memory.
Error Estimators
An error estimator serves to provide important feekbacks in an adaptive system and to direct the analysis path by offering quantitive measures of current analysis errors and refinement indications as necessary. I've spent quite some time on projection based error estimators. In fact, half of my thesis is about the projection technique and its various uses including error estimation, solution transfer (rezoning) as well as FEM postprocessing. There are various forms around the projection philosophy - among which Local and Global projection are the most popular.
Solution Transfer
For analyses involving large deformation where mesh updates are necessary in the process, the history dependent solutions need to be transfered after each mesh update. In this area, my work has resulted in a modular rezoning system that employes an effective tree structure and efficient parametric inversion algorithms for accurate solution mapping. The system was tested with different FE applications for various solution processing purposes.
Superconvergent Techniques
Research in Pointwise superconvergent techniques forms the second half of my Ph.D thesis - in which a class of recovery techniques were developed to extract primary and secondary solutions in 2D and 3D domains with rigorous pointwise accuracies that are on the order of the accuracy of the total strain energy -- the best accuracy attainable from FE solutions. Treatments for 2D cases are extensive and are backed by various test examples. Exploration into the practical application of the formulations have also been successful in that some guidlines of their reliable use have been found and verified in this research.
Projection
and Superconvergent Techniques for Adaptive Finite Element Analysis
(Thesis abstract)
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