Integral constraints in multiple scales problems with a slowly varying microstructure

Journal article

Asymptotic homogenisation is considered for problems with integral constraints imposed on a slowly-varying microstructure; an insulator with an array of perfectly dielectric inclusions of slowly varying size serves as a paradigm. Although it is well-known how to handle each of these effects (integral constraints, slowly-varying microstructure) independently within multiple scales analysis, additional care is needed when they are combined. Using the flux transport theorem, the multiple scales form of an integral constraint on a slowly varying domain is identified. The proposed form is applied to obtain a homogenised model for the electric potential in a dielectric composite, where the microstructure slowly varies and the integral constraint arises due to a statement of charge conservation. A comparison with multiple scales analysis of the problem with established approaches provides validation that the proposed form results in the correct homogenised model.

Authors: Kent, A; Waters, SL; Oliver, J; Chapman, SJ

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Cambridge University Press
• Journal: European Journal of Applied Mathematics
• Publication date: 2025-04-25
• Acceptance date: 2025-03-17
• DOI: 10.1017/S0956792525000142
• EISSN: 1469-4425
• ISSN: 0956-7925
• Language: English
• Keywords: microstructural variation; integral constraints; asymptotic homogenisation; multiple scales; perfect dielectric