Adaptive hp-finite element for transient analysis

摘要

Even though a tremendous effort has been devoted to the area of transient analysis, many of these models fall short of achieving the overall desired objective due to oversights and rigid simplifying assumptions. Some of those shortfalls are summarized as follows: 1.(i) The grid used is too fine or too coarse and is usually fixed in size;2.(ii) No error estimates are being used to trigger a grid size change;3.(iii) A fixed polynomial degree is being used to interpolate the approximate solution of the differential equation at hand (finite element approach);4.(iv) An approximate differential equation is used to model the physical phenomena rather than the exact one (finite difference approach).In our study, a self-adaptive grid refinement technique coupled with the p-version of the finite element method is investigated. This type of approach is called the hp-version of the finite element. In the modeling process, the approximate solution to the exact differential equation achieves convergence by applying two distinct solution enrichment strategies. One strategy is to solve the problem using a very coarse grid and to enrich the quality of the approximation by increasing the degree p of the interpolating polynomial shape functions for those elements where it is needed. The other strategy involves local grid h refinements for those elements where it is needed. Both strategies would interact synergistically to produce an optimal computational model that produces an accurate simulation of the transient phenomena addressed with minimal computational effort. The triggering for the increase or decrease in polynomial degree p and placement or removal of local mesh h refinements is based on the calculation of a reliable a posteriori error estimate over each element at the end of each time-step. Acase study of particular interest is the simulation of wave propagation in a semi-infinite media.