Bayesian Hybrid Surrogate Modeling of Algorithmic Tangent Stiffness Matrices in Elastoplastic Materials

The algorithmic tangent stiffness is the local derivative of the discrete constitutive stress update with respect to the strain increment. It is therefore central tothe convergence, stability, and computational cost of implicit finite-element analyses of plastic materials. This paper develops a physics-informed Bayesian hybridsurrogate framework for predicting stress and algorithmic tangent responses fromexpensive constitutive integrations. The reference constitutive structure is smallstrain associative von Mises plasticity with isotropic hardening, integrated by animplicit return-mapping algorithm. Its consistent tangent is derived by linearizingthe discrete update, rather than by differentiating a continuous-time rate equation in isolation. A Gaussian-process surrogate is then defined for the constitutive correction,with its tangent obtained as the derivative of the posterior mean. The proposedhybrid construction combines an analytical elastic or mechanistic baseline with aprobabilistic correction and imposes symmetry and thermodynamic admissibilitythrough parameterization, constraints, or explicit diagnostics. Bayesian calibration,heteroscedastic observation noise, active learning, and finite-element equilibriumcoupling are formulated in a unified notation. The work is a mathematical andcomputational framework; it does not claim a new experimental data set or reportnumerical validation results. A verification protocol is specified to assess stress error,tangent error, posterior calibration, dissipation, and global Newton convergencebefore engineering deployment.