Control of geometry error in hp finite element (FE) simulations of electromagnetic (EM) waves
The success of high accuracy Finite Element (FE) simulations for complex, curvilinear geometries depends greatly on a precise representation of the geometry and a proper mesh generation scheme. Sizable errors are introduced into the numerical predictions when the order of the geometric approximation is too low with respect to the polynomial order of the discretization. In hp finite element methods, preserving exponential convergence rates for problems over curved domains requires the use of either exact geometry elements or higher order (iso- or superparametric) geometry representations. Radiation of electromagnetic (EM) waves from various sources, including cell phones, and their absorption into the human body, has become a raising public concern. This has motivated us to select the problem of scattering and absorption of EM waves on the human head, as a driving application for the research on the geometry induced errors in FE simulations. Maxwell equations are discretized using H(curl)- conforming elements that turn out to be more sensitive to geometry induced errors than standard H1 - conforming (continuous) finite elements. In this dissertation, we review the theoretical framework for a general class of parametric H1 , H(curl) and H(div)- conforming elements, with both exact and isoparametric geometry description. A systematic way of computing H1− and H(curl)− discretization errors, accounting for the error in geometry approximation, is proposed. The technique is illustrated with numerical examples and compared with the customary error evaluation neglecting the geometry approximation error. Two general geometry representation schemes have been addressed: CADlike geometric modeling and geometry reconstruction from discrete data. A number of novel geometrical modeling techniques are explored and implemented in the presented Geometric Modeling Package (GMP). The package is used to generate an exact representation of complex objects, and provides a foundation for a multi-block hp mesh generator. The package allows for maintaining a continuous interface with adaptive codes to update the geometry information during mesh refinements. In addition, an approaches have been developed to accelerate preparation of geometry data by extracting topology information of a meshed model from existing mesh generation toolkits. The geometric model needs to be sufficiently smooth enough to produce a finite element mesh free of local geometric discontinuities which create numerical artifacts in the EM solutions. An efficient biquartic G1 surface reconstruction scheme is developed in this dissertation for general unstructured meshes. The polyvii nomial parameterizations are inexpensive to compute and guarantee high regularity of parametrization necessary in FE computations. The new geometric representation techniques have been incorporated into a 3D hp 1 coupled Finite Element/Infinite Element (FE/IE) codedeveloped in Dr. Demkowicz group at the Institute for Computational Engineering and Sciences (ICES). The new GMP and the coupled FE/IE hp code have been verified using the Mie series solution for the problem of scattering a plane EM wave on a dielectric sphere. The accuracy of FE/IE approximation has then been assessed using the precise definition of the solution error incorporating the effects of geometry approximation. Finally, an explicit a posteriori error estimator for time-harmonic Maxwell equations and arbitrary hp meshes on curvilinear geometries is implemented in the hp FE code. The estimator is used to drive an h-adaptive strategy to solve the head problem. The computed Spatial-peak and average Specific Absorption Rate (SAR) values have been compared with results obtained by other numerical methods.