A Developement of Numerical Schemes of Asymmetric Fields Based on Discrete Differential Forms and Homogenization Methods, and its Applications into Industrial Problems

Reference No.	2023a021
Type/Category	Grant for General Research-Short-term Joint Research
Title of Research Project	A Developement of Numerical Schemes of Asymmetric Fields Based on Discrete Differential Forms and Homogenization Methods, and its Applications into Industrial Problems
Principal Investigator	Jun Masamune（Tohoku University・Professor）
Research Period	May 12,2023. ～ May 14,2023.
Keyword(s) of Research Fields	Discrete Differential Form; Homogenization Method; Asymmetric Field; Semiconductor Device; Optimal Design
Abstract for Research Report	In recent years, differential forms have been used as mathematical models of physical phenomena, and their numerical computations. The differential form distincts flow (1-form) and flux (2-forms), which clarifies the relation between the domain and the physical quantity, and which becomes also dvantages when treating measurable flux in mathematical models. Mathematical models and numerical computations based on the differential form are called by various names depending on the research fields, such as ”discrete exterior differential calculus”, ”discrete differential geometry”, ”difference forms”, etc. In the joint research project last year, we focused ourselves on finite element exterior calculus, and tried to understand the mathematical basis of the method. Moreover, we discussed the advantage of finite element exterior calculus, where discretized equations can keep the mathematical structure of differential equations, for example, in case of numerical computations of electromagnetic field problems with finite element exterior calculus. During the discussion, one of the private company participants proposed that the optimal design of semiconductor devices may become a practical example of the applications of numerical computations based on discrete differential forms. Some participants from the university suggested that mathematical models based both on discrete differential forms and on homogenization methods may be applicable into the optimal design of semiconducor devices because the devices have the periodic structures, which are familiar with the homogenization methods. By following the results mentioned above, we organize next joint research project to further understand numerical analysis based on the discrete differential form and to further discuss their applications into industrial problems. We have the following main participants of the project, Dr. Fukagawa from the industrial side, and Prof. Uemura, Prof. Tagami, and I from the academic side: Dr. Fukagawa has achieved results in the implementation of large-scale numerical computations using discrete differential forms, while Prof. Uemura, Prof. Tagami, and I have organized a research group on mathematical analysis of homogenization methods.
Organizing Committee Members (Workshop) Participants (Short-term Joint Usage)	Jun Masamune（Tohoku University・Professor） Hiroki Fukagawa（DeepFlow, Inc・Representative Director ） Toshihiro Uemura（Kansai University・Professor） Daisuke Tagami（Kyushu University・Associate professor）