Periodic Reporting for period 3 - 321 (from Cubic To Linear complexity in computational electromagnetics)
Período documentado: 2020-09-01 hasta 2022-02-28
Computational Electromagnetics (CEM) is the scientific field at the origin of all new modeling and simulation tools required by the constantly arising design challenges of emerging and future technologies in applied electromagnetics. CEM is at the interface between electrical engineering, advanced computing, and applied mathematics and it focuses on developing fast and efficient solvers to characterize the electromagnetic interactions, radiation, and scattering of large, multiscale, and complex objects. Computational Electromagnetics is the underpinning of a plethora of electrical, electronic, optical, wireless, geophysical sensing, and biomedical applications. Because electromagnetic theory has strong predictive power, EM simulators play a dominant role in the advancement of today's physics and engineering science.
As in many other technological fields, however, the trend in all emerging technologies in electromagnetic engineering is going towards miniaturized, higher density and multi-scale scenarios. Computationally speaking this translates into a steep increase of the number of degrees of freedom. Given that the design cost (the cost of a multi-right-hand side problem dominated by matrix inversion) can scale as badly as cubically with these degrees of freedom, this can sensibly compromise the practical impact of CEM on future and emerging technologies.
For this reason, the CEM scientific community has been looking for years for an FFT-like paradigm shift: a dynamic fast direct solver providing a design cost that would scale only linearly with the degrees of freedom. Such a fast solver is considered today a Holy Grail of the discipline.
The Grand Challenge of 321 is to tackle this Holy Grail in Computational Electromagnetics by investigating a dynamic Fast Direct Solver for Maxwell Problems that would run in a linear-instead-of-cubic complexity for an arbitrary number and configuration of degrees of freedom.
To attain this, the project’s objectives include the investigation of new modelling and solution strategies for a large plethora of scenarios and frequencies, new fast solution strategies, and impacting applications in dosimetry, bioelectromagnetism, and neuroimaging.
The impact of the FFT’s quadratic-to-linear paradigm shift shows how computational complexity reductions can be groundbreaking on applications. The cubic-to-linear paradigm shift, which the 321 project aims for, will have such a rupturing impact on electromagnetic science and technology.
As in many other technological fields, however, the trend in all emerging technologies in electromagnetic engineering is going towards miniaturized, higher density and multi-scale scenarios. Computationally speaking this translates into a steep increase of the number of degrees of freedom. Given that the design cost (the cost of a multi-right-hand side problem dominated by matrix inversion) can scale as badly as cubically with these degrees of freedom, this can sensibly compromise the practical impact of CEM on future and emerging technologies.
For this reason, the CEM scientific community has been looking for years for an FFT-like paradigm shift: a dynamic fast direct solver providing a design cost that would scale only linearly with the degrees of freedom. Such a fast solver is considered today a Holy Grail of the discipline.
The Grand Challenge of 321 is to tackle this Holy Grail in Computational Electromagnetics by investigating a dynamic Fast Direct Solver for Maxwell Problems that would run in a linear-instead-of-cubic complexity for an arbitrary number and configuration of degrees of freedom.
To attain this, the project’s objectives include the investigation of new modelling and solution strategies for a large plethora of scenarios and frequencies, new fast solution strategies, and impacting applications in dosimetry, bioelectromagnetism, and neuroimaging.
The impact of the FFT’s quadratic-to-linear paradigm shift shows how computational complexity reductions can be groundbreaking on applications. The cubic-to-linear paradigm shift, which the 321 project aims for, will have such a rupturing impact on electromagnetic science and technology.
The work performed in the first half of the project has been focusing on both low and high frequency stabilizations. Among our activities, we have investigated and obtained the first integral formulation for electromagnetics which is Hermitian positive definite (HPD) independently of the sign of the original problem. Standard strategies can achieve such a desirable outcome only at the cost of increasing the condition number and thus making the modelling slow and unstable. Instead we have obtained here an HPD formulation for which not only the condition number does not worsen but is even improved by several orders of magnitude.
A significant challenge arising when solving electric problems containing loops and handles (most notably in electronic circuits) with full wave high frequency solvers is to handle the breakdown present in the lower part of the frequency spectrum. One common option is the use of magnetic formulations that do not suffer from this breakdown. A stable discretization of these equations, however, is extremely expensive in the low frequency limit. This problem has been solved in this project, with the first formulation that decouples frequency and integration precision for magnetic type equations. This resulted in the magnetic complement of the Calderón equation quite popular for electric formulations. Moreover, the linear combination of the two formulations leads to a high frequency stable equation that can model electromagnetic problems from zero to high frequency.
As per the project’s plan we are also extending these schemes to penetrable cases. We have obtained a full wave high frequency solver that encompasses the Eddy Current regime without instabilities, this will be quite impacting in electromagnetic compatibility (EMC)/electromagnetic interference (EMI) applications and several strategies to extend our results to scenarios containing inhomogeneities and anisotropies have been achieved. These modelling challenges are all very relevant for numerous applications including, but not limited to, bioelectromagnetic assessments. Among the new schemes we have obtained, we have notably obtained the first full wave integral equation solver attaining stable forward results in electroencephalography-based (EEG) neuroimaging. This will have a substantial impact on several applications, including diagnostics and treatment in epilepsy as well as in brain stimulation and neural implants.
A significant challenge arising when solving electric problems containing loops and handles (most notably in electronic circuits) with full wave high frequency solvers is to handle the breakdown present in the lower part of the frequency spectrum. One common option is the use of magnetic formulations that do not suffer from this breakdown. A stable discretization of these equations, however, is extremely expensive in the low frequency limit. This problem has been solved in this project, with the first formulation that decouples frequency and integration precision for magnetic type equations. This resulted in the magnetic complement of the Calderón equation quite popular for electric formulations. Moreover, the linear combination of the two formulations leads to a high frequency stable equation that can model electromagnetic problems from zero to high frequency.
As per the project’s plan we are also extending these schemes to penetrable cases. We have obtained a full wave high frequency solver that encompasses the Eddy Current regime without instabilities, this will be quite impacting in electromagnetic compatibility (EMC)/electromagnetic interference (EMI) applications and several strategies to extend our results to scenarios containing inhomogeneities and anisotropies have been achieved. These modelling challenges are all very relevant for numerous applications including, but not limited to, bioelectromagnetic assessments. Among the new schemes we have obtained, we have notably obtained the first full wave integral equation solver attaining stable forward results in electroencephalography-based (EEG) neuroimaging. This will have a substantial impact on several applications, including diagnostics and treatment in epilepsy as well as in brain stimulation and neural implants.
In this first half of the project we have already obtained:
1) The first existing modelling tool (for now in the metallic case) that renders any initial electromagnetic problem well-conditioned, fast solvable and Hermitian Positive Definite. HPD systems have several favorable properties including sharp convergence bounds, simplified error analyses, and spectral bounds. The effects of these properties in other scenarios, such as high order methods, will be investigated in the following. This technique can also be applied to filtering via manipulation of integral transforms (including wavelets, frames, and related transforms).
2) The first magnetic modelling framework that requires computationally feasible and low order integration precisions for arbitrary frequencies. The approach we have found can be applied in several other scenarios. For example, second kind models are very common for EMC/EMI modelling in circuit like structures that show non simple connectivity and we expect that circuit solvers will be able to take advantage from the new methodology and we will explore this impact in the following months.
3) A new family of integral models that handle anisotropies, inhomogeneities, and losses in an extremely stable and computationally efficient way. These technologies will play a game changing role in several application scenarios and we are currently exploiting their impact in neuroimaging and bioelectromagnetic assessments.
1) The first existing modelling tool (for now in the metallic case) that renders any initial electromagnetic problem well-conditioned, fast solvable and Hermitian Positive Definite. HPD systems have several favorable properties including sharp convergence bounds, simplified error analyses, and spectral bounds. The effects of these properties in other scenarios, such as high order methods, will be investigated in the following. This technique can also be applied to filtering via manipulation of integral transforms (including wavelets, frames, and related transforms).
2) The first magnetic modelling framework that requires computationally feasible and low order integration precisions for arbitrary frequencies. The approach we have found can be applied in several other scenarios. For example, second kind models are very common for EMC/EMI modelling in circuit like structures that show non simple connectivity and we expect that circuit solvers will be able to take advantage from the new methodology and we will explore this impact in the following months.
3) A new family of integral models that handle anisotropies, inhomogeneities, and losses in an extremely stable and computationally efficient way. These technologies will play a game changing role in several application scenarios and we are currently exploiting their impact in neuroimaging and bioelectromagnetic assessments.