Exact solutions for the insulated and perfect conductivity problems with concentric balls

Zhiwen Zhao; Zhiwen Zhao

doi:10.3934/mine.2023060

Mathematics in Engineering

2023, Volume 5, Issue 3: 1-11. doi: 10.3934/mine.2023060

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Research article

Exact solutions for the insulated and perfect conductivity problems with concentric balls

Zhiwen Zhao ^,

Beijing Computational Science Research Center, Beijing 100193, China

Received: 28 July 2022 Revised: 05 November 2022 Accepted: 05 November 2022 Published: 16 November 2022

The insulated and perfect conductivity problems arising from high-contrast composite materials are considered in all dimensions. The solution and its gradient, respectively, represent the electric potential and field. The novelty of this paper lies in finding exact solutions for the insulated and perfect conductivity problems with concentric balls. Our results show that there appears no electric field concentration for the insulated conductivity problem, while the electric field for the perfect conductivity problem exhibits sharp singularity with respect to the small distance between interfacial boundaries of the interior and exterior balls. This discrepancy reveals that concentric balls is the optimal structure of insulated composites, but not for superconducting composites.
- exact solutions,
- the insulated and perfect conductivity problems,
- concentric balls,
- the electric field,
- high-contrast composites,
- blow-up
Citation: Zhiwen Zhao. Exact solutions for the insulated and perfect conductivity problems with concentric balls[J]. Mathematics in Engineering, 2023, 5(3): 1-11. doi: 10.3934/mine.2023060

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Abstract

The insulated and perfect conductivity problems arising from high-contrast composite materials are considered in all dimensions. The solution and its gradient, respectively, represent the electric potential and field. The novelty of this paper lies in finding exact solutions for the insulated and perfect conductivity problems with concentric balls. Our results show that there appears no electric field concentration for the insulated conductivity problem, while the electric field for the perfect conductivity problem exhibits sharp singularity with respect to the small distance between interfacial boundaries of the interior and exterior balls. This discrepancy reveals that concentric balls is the optimal structure of insulated composites, but not for superconducting composites.

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