Anisotropic hyperelastic homogenization stops guessing directions: six learnable paired features cut stress error sevenfold and keep constrained energies solvable beyond the data

Surrogates for anisotropic hyperelastic homogenization no longer rely on a prescribed directional description; the directions themselves are learned from the response of the representative volume element (RVE), the microscopic cell that would otherwise be solved repeatedly. Earlier practice fixed the directions, either through structural tensors or by adding more fixed features to recover expressiveness. On a multicavity periodic cell, learning the features lowers the median stress error from about three parts in a thousand to about four parts in ten thousand, roughly sevenfold, while thirty-two fixed features remain about three times less accurate than six learned ones.

Learning the direction parameters works because each feature pairs two perpendicular directional contributions, each carrying an exponent that couples it to the area ratio. A convex scalar core alone cannot separate deformation states its inputs do not distinguish, because the core is monotone in every feature and cannot reverse the ordering the features impose. The same paired features feed both an input-convex neural network (ICNN) and an input-convex Kolmogorov–Arnold network (ICKAN), and both cores improve by a similar factor, so the gain comes from adapting the directional representation.

The constrained energies stay solvable beyond the data, whereas an unconstrained energy that satisfies only potential structure and reference normalization loses rank-one convexity. Inside the fitting range the unconstrained energy is more accurate instead, reaching about seven parts in a thousand in stress error, roughly six times below the constrained models. In a confined compression case both constrained energies reach 500 MPa, while the unconstrained energy stops when Newton fails to converge in the increments to 250 MPa and 225 MPa, where its minimum rank-one curvature turns negative at individual points among 384 Gauss points.

The work covers two square periodic cells, plane-strain kinematics, and two structural loadings: a tensile coupon and a confined compression case. Agreement with the nested full-order model FE2 measures surrogate error within the adopted homogenization model, not error against a structure in which every microscopic cell is resolved explicitly, and the reported timings cover online workflows only. The authors propose passing the determinant to the core as an input that requires convexity but not monotonicity, and extending the construction to three-dimensional cells.

The authors state that the accuracy cost of the constraints is about a factor of 3.2 to 5.5 on the multicavity cell but 76 to 126 on the single-cavity cell, and that neither wider cores nor more features remove it. Six features suffice on both cells, but the authors note this count need not generalize to other microstructures. Global nonnegative energy is not part of the construction guarantee; it is certified separately for the trained parameters through a sufficient bound, and the thirty-two-feature models fail that certificate.