In this article, we introduce the finite element method to analyze a membrane with a Koch snowflake-shaped boundary. The fractal nature of this domain presents unique challenges due to its intricate boundary structure. Our approach involves discretizing the domain, estimating the error, and proving convergence. With these aspects addressed, we solve a shape optimization problem to determine the optimal thickness of the membrane. These findings provide valuable insights into how fractal boundaries affect structural performance and optimization.
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Riane et al. (2026) studied this question.
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