Taming infinities in the modelling of radiofrequency miniaturized devices and designing physics-inspired regularization techniques

Modern telecommunication and signal processing applications have motivated extreme miniaturization of acoustic, electronic, photonic, plasmonic, polariton, and molecular devices. The complexity of the design, optimization, modelling and simulation of contemporary devices mirrors challenges encountered in their sophisticated fabrication processes. The intricate topologies of individual devices and mazelike interwoven circuitry of “smart” and “cognitively-enabled” systems and system-of-systems, alike, have been blurring device boundaries, making precise interpretation of their operation and interconnectivity an arduous undertaking. The prospect of impending realization of ultra-small devices, harvesting quantum-phenomena, has been adding further pressure to modelling- and design rules which are yet to be devised, fine-tuned, tested, and established on firm bases. There is an urgent need for a new type of “mathematics” to respond to the immense structural- and functional complexity of the devices, thus calling for a genuine paradigm change in device modelling and design. Miniaturization of micro-acoustic devices in telecommunication engineering applications involves a plethora of physical and mathematical foundational challenges which need to be addressed masterfully. This presentation uses the physical phenomena involved in today's micro-acoustic devices to communicate the variety of enabling methodological advances which have been achieved. The developed techniques exhibit a promising generality with a wide-range of applications including terahertz- and quantum level devices. Considering major commercially available analysis tools; i.e., the Finite Element Method (FEM), Finite Difference Method FDM (spectral-or time domain formulations), and the Boundary Element Method (BEM), this presentation focuses on the BEM. However, BEM involves problem-specific Dyadic Green's Functions (DGFs) which are plagued with strong- and hyper-strong singularities severely obscuring numerical calculations, in particular in the vicinity of sharp edges and corners of bounding surfaces. Field problems involving (i) elasto-electric, (ii) elasto-electro-magnetic, (iii) thermo-elasto-electric, and (iv) thermo-elasto-electro-magnetic phenomena are considered following a unified approach. Uni-, bi-, and for the first time, tri-, and quad-anisotropic and inhomogeneous media, supporting the simultaneous interaction of thermal, acoustic, and electro-magnetic waves at radiofrequencies are considered. The proposed analysis techniques are based on the conjecture that “linearized governing- and constitutive equations in mathematical physics are diagonalizable.” The validity of this statement is established in the case of thermo-elasto-electro-magnetics. A further recent rigorous result demonstrates the existence of a system of equations supplementing diagonalized forms. The derivation of the diagonalized- and supplementary partial differential equations utilizes classical operators in mathematical physics employing conveniently defined “scaffolding” matrices allowing finitary algebras. The proposed diagonalized- and associated supplementary equations have enabled systematizing the description of the physical phenomena involved and led to two novel algebraic- and exponential regularization techniques. The exponential renormalization method generalizes and puts on a firm basis earlier ad hoc regularization techniques. The results suggest further generalizations to attack renormalization challenges encountered in quantum electrodynamics. Additionally, systematic advancements of the standard formulations in FDM, FEM and BEM are presented. Thereby, the key unifying tool has been the concept of the “resolution of identity” in the functional analysis and a generalization thereof. The construction of Green's functions-inspired wavelets and frames, and the development of (property preserving) conservative numerical techniques with applications in fields, waves and signal processing shall enrich the presentation. In conclusion, a thorough discussion of challenges, still persisting in photonic-, plasmonic-, polariton- and quantum physics-based device modelling, shall pave the way for the construction of future accelerated algorithms.