Material interpolation in finite strain topology optimization: a comprehensive assessment and a generalized energy-strain-combined interpolation scheme

Density-based topology optimization (TO) relies critically on material interpolation schemes to represent the transition between material and void phases. While numerous interpolation strategies have been proposed, particularly for problems involving large deformations, existing studies have predominantly focused on numerical stabilization and implementation aspects. As a result, the physical meaning and consistency of material interpolation under finite strain conditions remain insufficiently understood. This study provides a systematic and physically grounded comparative investigation of representative material interpolation schemes for density-based TO. Five classes of interpolation strategies are considered, including (1) material parameter-based, (2) energy-based, (3) strain-based, (4) parameter-strain-combined, and (5) energy-strain-combined interpolations. These schemes are first examined within the small strain framework, more precisely within geometrically linear elasticity, where their mathematical relationships and potential equivalences can be clearly identified. The formulations are then extended to the finite strain framework, more precisely to geometrically nonlinear formulations with material nonlinearity, enabling an analytical and numerical assessment of how different interpolation strategies influence sensitivity expressions, mesh distortion in void regions, numerical stability, and the generality with respect to constitutive modeling. The results demonstrate that interpolation schemes that appear closely related under small strain assumptions can exhibit fundamentally different characteristics when finite deformation is taken into account. From this comparative and physically motivated analysis, an energy-strain combined interpolation framework can be naturally proposed as a consistent and broadly applicable formulation for finite strain TO, rather than as an ad hoc numerical modification. The present study thereby clarifies the physical rationale underlying material interpolation in finite strain TO and provides a unified perspective for selecting and interpreting interpolation strategies in nonlinear settings.