Referent: Dr. Robert Kotiuga, Boston University, USA
Decades ago, the author obsessed with articulating why topological aspects of 3D finite element (FE) meshes could be deceptively unintuitive, while 2D was so obvious. For EEs, this dichotomy didn’t originate with FE meshes, but is implicit in how nonplanar electronic circuits are communicated. In FE for electromagnetics, the ease with which homology calculations can be reduced to sparse matrix linear algebra, enables certain complexities of 3D geometries to be automated, taking the user out of the loop. Although linking numbers, the problem of making cuts for magnetic scalar potentials, and the “helicity” of a vector field, all helped to articulate how 3D was so much richer than 2D, knotted geometries were largely dismissed in the decades of milling machines and lathes.
The quadratic constraint imposed by the Lorentz force is a game changer which forced the embrace of 3D topological complexities. From magnets made of superconducting tapes for particle accelerators, or compact MRI and fusion devices, to astrophysical object like magnetaurs, this nonlinear constraint unleashed topological considerations far more subtle than the comfortable linear algebra and graph theory associated with homology calculations. Taming this increased complexity, the eigenvalue problem (EVP) for the curl operator has emerged as a very important bridge between topological characterizations of optimal designs involving the Lorentz force, and tools associated with computational linear algebra.
This talk will focus on how the curl EVP differs from the more familiar curlcurl EVP, its unique features for tackling inverse problems, and how a 7D analogy helps formalize unintuitive aspects.