MORTIS: Quadrature-compatible cofactor gauges and optimal material margins in finite elasticity

Abstract

We develop a discrete design theory for cofactor reference forms that preserve the complete finite-element elasticity tangent. A fixed continuous potential generates the coefficient, while the variation space and quadrature determine compatibility. We classify the exact local potential spaces for all-order tensor elements, anisotropic spaces, cubic serendipity and quadratic tetrahedra. At a stress-free Mooney-Rivlin state with a common positive material-coefficient sum, a convex loss expresses the attainable quadrature-point material margin. The optimum is attained; under stated space and sampling assumptions, its ideal value is attained precisely when the physical identity potential is compatible. A conforming fixed-candidate example proves a strict margin loss, linear in curvature amplitude and uniform in cell size, under displacement enrichment. Explicit positive candidates accompany this restriction. A Gram identity gives the sharp constant-coefficient defect at a prescribed margin in the stated norm. Compatible and reference-defect-corrected forms retain the original tangent through a deformation-independent external-boundary Hessian; equal potential traces give equal exact assembled gauges. Quadratic tensor geometry admits a stress-free physical coercivity bound uniform in mesh size and displacement order under explicit material, shape, quadrature and boundary assumptions. Certified curved tetrahedral shear families with zero recovered cell pressure establish finite-deformation regimes, while a loaded pressure-curvature counterexample identifies their limitation. The theory and observations distinguish local exactness, material certificates, complete-reference coercivity and unchanged equations.