Data-adaptive spline surfaces for non-separable hyperelastic energy functions

TL;DR

This paper introduces a data-driven, spline-based modeling approach for hyperelastic materials that improves accuracy and calibration efficiency over traditional separable models by directly approximating coupled invariants.

Contribution

It proposes a novel bivariate B-spline surface model on the invariant domain, enabling fast, robust, and accurate calibration without complex optimization or large parameter sets.

Findings

The spline model outperforms separable approaches in accuracy.

Calibration is fast, robust, and requires minimal regularization.

The method is suitable for repeated calibration tasks like uncertainty quantification.

Abstract

Invariant-based models for incompressible isotropic hyperelasticity are typically formulated as functions of the first and second invariants, $W = W(\bar{I}_1, \bar{I}_2)$. A widely used class of models employs separable representations of the form $W(\bar{I}_1, \bar{I}_2) = W_1(\bar{I}_1) + W_2(\bar{I}_2)$, which enable efficient calibration and straightforward enforcement of modeling constraints. However, this decomposition implicitly restricts the coupling between the invariants and may limit the achievable accuracy for complex material responses. Fully coupled data-driven approaches overcome this limitation but often require nonlinear optimization and large parameter sets. In this contribution, we propose a compact alternative: a bivariate B-spline surface defined directly on the physically admissible invariant domain. By aligning the approximation space with physically realizable states, all model parameters contribute meaningfully to the constitutive response. We utilize homogeneous deformation modes to perform a calibration directly from analytical stress relations, eliminating the need for finite element model updating. Owing to the linear dependence of the spline representation on its coefficients, the resulting parameter identification problem reduces to a constrained linear least-squares problem. This enables fast, robust, and initialization-independent calibration, which makes parameter identification practically instantaneous. The results demonstrate that the proposed model improves accuracy compared to separable approaches while requiring only mild regularization in weakly sampled regions. The combination of computational efficiency and the linear structure of a highly expressive spline surface makes the approach particularly attractive for applications requiring repeated calibration, such as uncertainty quantification or interactive material characterization.