A Unified Framework for Interface Conformability Optimisation: Stress, Spectral, and Topological Constraints

Abstract

We develop a mathematical framework for optimising interfaces between discrete rigid elements and continuous irregular substrates. While contact mechanics analyses stress for given geometries and homogenisation theory treats continuous–discrete transitions, the inverse problem—determining optimal element size from substrate topology and material constraints—has lacked systematic treatment. We introduce three independent dimensionless metrics: a stress-based Conformability Index (CI), a spectral Critical Wavelength Ratio (CWR), and a topology-inspired Topological Obstruction Number (TON). Through variational energy minimisation, we show these metrics emerge naturally from a unified principle rather than being ad hoc constructs. An independent derivation via quantisation theory recovers identical scaling exponents, providing internal consistency without empirical data. We derive universal scaling laws—d ∝ t, d ∝ σz^(−1/(H+1)), d ∝ √(σy/E)—with parameter-independent exponents offering falsifiable predictions. Ceramic tile installation serves as the canonical example, while the mathematical structure applies wherever rigid discrete coverings interface continuous irregular fields under structural constraints.