Revolutionizing Simulation-Based Design: Adaptive Anisotropic Multiscale Computational Methods for Electromagnetics and Engineering

There has been growing demand for high-fidelity and efficient simulations in all areas of computational sciences and engineering and particularly so in computational electromagnetics (CEM), both in industry and academia. While high-performance computing platforms have unlocked applications traditionally beyond feasibility, theoretical challenges inhibit the use of conventional approaches even when computing hardware is readily available.

Among the most severe challenges, resolving non-smooth and multiscale behavior is nearly universal for realistic problems across application domains. Material interfaces, sharp or non-smooth geometric features, etc. drive poor solution regularity, resulting in extremely poor rates of convergence without new techniques. In addition, small variations due to manufacturing tolerances, environment, and all sorts of imperfections, neglections, and failures can have significant impacts on the macroscopic solution behavior and the relevant computables and measurables.

Methods that can resolve fine grain behavior accurately and efficiently will not only revolutionize multiscale simulations, but will also significantly enhance uncertainty quantification, whether for the design of new devices and systems or the analysis of existing ones. The principal objective of this project is the formulation, development, mathematical analysis, testing, and demonstration of a novel synergistic approach to simulation-based design in the presence of multiscale and non-smooth solution behaviors in CEM.

This research provides a means for adaptive modeling, error control, and uncertainty quantification of deterministic and stochastic problems and designs in CEM with exponential solution convergence in smooth, non-smooth, singular, and multiscale environments. It has the potential to significantly enhance accuracy, efficiency, versatility, robustness, and practicality of a broad class of CEM methodologies and techniques, so that they can be made accessible, usable, and beneficial to a broad audience of researchers, practitioners, and students.

The project’s educational activities include advising and training of graduate students, recruiting students from underrepresented groups, developing new educational materials, and participating in various retention/outreach programs.

The research of this project will result in an adaptive, fully anisotropic multiscale hp-element method, aimed to revolutionize simulation-based design. Here, h denotes geometric refinement, p denotes refinement in the expansion order of local shape functions, and hp denotes both enhancements performed simultaneously (a crucial quality), whereas anisotropic or directional refinement is a methodology that allows different h- and/or p-refinements in different directions.

Furthermore, the research will use a refinement-by-superposition approach for adaptive fully anisotropic hp-refinements, which superimposes a set of "child cells" over a "parent cell" without removing the parent cell from the mesh, providing significant gains in accuracy, efficiency, and versatility of the computation, as well as dramatically simpler and easier implementation. The new methodological approach features theoretically guaranteed exponential solution convergence rates in all cases and high efficiency of analysis and design; rigorous multiscale error estimation and error control synergy; ability to accurately and efficiently refine multiscale models and eliminate discretization error fully automatically; and adjoint-based multiscale sensitivity analysis in the presence of uncertain model parameters.

Overall, the project will develop a multistage adaptive anisotropic hp-refinement method based on detecting and predicting global errors, local error contributions, and solution regularity to adaptively instruct the discretization of the micro- and macroscale problems coupled with accelerated high-dimensional uncertainty quantification of multiscale simulations. Extensive theoretical analyses will be conducted to rigorously examine the transaction between computational resources and accuracy, to understand the limits of resource-bounded computation and to feed those findings back into further development and advancement of the novel methodology.

The new adaptive framework can be readily applied to a variety of objectives and as a vital companion tool for emerging and state-of-the-art techniques in, for instance, machine learning and optimization. This extends directly to numerical methods in general, and to multiphysics and other areas of computational sciences and engineering.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.