Principal-stretch-based hyperelastic potentials require a derivative transformation when implemented through the invariant-based Abaqus UHYPER interface. For compressible models with coupled distortional and volumetric effects, an additive energy split is unavailable. The built-in UHYPER_STRETCH interface of Abaqus is restricted to potentials of the Valanis–Landel form and therefore cannot be used for such coupled models. This work develops a model-independent transformation of principal-stretch-based isotropic potentials into the invariant derivatives required by UHYPER, including distortional–volumetric coupling and regular limits at repeated modified eigenvalues. The transformation separates model-dependent scalar derivatives from reusable kinematic relations. Distinct, double and triple eigenvalues are treated by implicit differentiation, a constraint-preserving local parameterization and an invariant-based series expansion, respectively. A complete Fortran implementation is provided for the Hyperfoam model and verified against its built-in Abaqus counterpart. Three homogeneous loading cases target the different eigenvalue branches, while a non-homogeneous three-dimensional example tests their operation in spatially varying deformation fields. The homogeneous stresses agree to the displayed nine decimal places. Additional fixed-increment comparisons show practically coincident final-increment force-residual histories and identical final-increment iteration counts for five increment sizes, although the full-step totals differ slightly in two cases. The framework enables energy-based implementation beyond the Valanis–Landel class without user programming of the stress tensor or the fourth-order consistent tangent.
The background methodology is a two-level mixed finite element approach for modeling flow in heterogeneous porous media. In the context of domain decomposition of the flow region Ω , composite divergence-compatible finite element pairs in H(div,Ω)×L2(Ω) are utilized for discretizing flux and pressure fields. The two-level characteristic of the method is expressed in terms of refined discretizations inside the subdomains versus coarser flux components interacting with interfaces of the subdomains, with respect to mesh size, polynomial degree, or both. The principle is to accurately resolve the small-scale features of local variables inside the subregions (pressure and fluxes with vanishing normal traces), whilst keeping the dimension of trace flux components as low as possible. In the current research, we demonstrate that approximations to local Steklov problems associated with the subdomains can be used to a better a priori adjustment of the boundary fluxes to the phenomena under analysis. These are eigenvalue problems enforcing the spectral constraint over the boundary of the subdomains. The approach follows the logic of spectral analysis. For each subdomain, a Steklov problem is solved by a mixed finite element method, and the first significant modes, which are accurately approximated, are used to select refined components to enrich a given initial coarse guess of the trace space (and the associated flux fields of trace type as well). The technique is designed for piecewise regular permeability tensors, and its effectiveness is demonstrated for cases of intricate maze-type porous structures.
This paper introduces a parallel framework enabling the simultaneous solution of multiple local domains using the non-intrusive Generalized Finite Element Method with global–local enrichment (IGL-GFEMgl). A global problem is first solved with a coarse mesh using the standard Finite Element Method. Localized phenomena, such as cracks, are detailed represented at the local level, as a subdomain of the global model. A mesoscale intermediate subdomain works as bridge between the global and a local scale problem. When several local problems are necessary, multiple mesoscale are associated with each one of the local subproblems through GFEMgl strategy. The method couples global and meso models by transferring displacements and forces, using the Iterative Global–Local (IGL) strategy. Simulations are run on an in-house academic computing system that handles the multiple meso and local subdomains, and on Abaqus, which solves the global problem. Several GFEMgl problems are distributed across multiple processors, taking advantage of the independence between domains during the local solution stage. This allows the integration of advanced academic algorithms with commercial software to solve complex structural problems. Numerical examples are presented to demonstrate the simulation’s performance and evaluate the influence of the main parameters related to the proposed strategy. The investigations indicate that the overall convergence behavior is governed by the local subdomain requiring the largest number of coupling iterations to satisfy the prescribed convergence criterion, while the introduction of additional local subdomains and the respective mesoscale problems does not significantly affect the number of coupling iterations required. Furthermore, the parallel processing of mesoscales significantly reduces computational time compared to sequential implementations, demonstrating the effectiveness of the proposed approach.
The accuracy of isoparametric finite elements can deteriorate substantially when the geometric map is distorted, even when standard mesh-quality indicators remain within commonly accepted limits. Conventional scalar indicators such as aspect ratio, skewness, taper, or Jacobian ratio diagnose element shape but do not identify which physical polynomial modes are lost or amplified by the mapping. This paper develops an a priori, local Gram-matrix framework for geometry-induced interpolation errors. For each relevant physical monomial, the residual between its pullback through the element map and its isoparametric interpolant is formed, and the interaction of these residuals is measured by a local Gram operator in the appropriate seminorm. The resulting approximation indicator — the Gram-Channel Error Estimator (GCEE) — is a quadratic form in the local Taylor coefficients of the target field. All Gram entries are derived in closed form by computer algebra and translated automatically into numerical code, so that the indicator carries no tuned constants. The framework recovers the expected behavior of bilinear quadrilaterals, explains the order degradation of serendipity Q8 elements through the missing ξ2η2 mode, and clarifies the selective robustness of Lagrangian Q9 elements under bilinear angular distortion. It also distinguishes parasitic curvature from geometrically consistent curved sides, and quantifies the quarter-point trade-off of tangential mid-side node offsets. Numerical tests for interpolation and a manufactured Poisson problem show that the Gram-based indicator predicts both global effectivity and local ranking of the error: effectivity indices remain within a few percent of unity with respect to the local best-approximation error, and element-wise rank correlations reach 0.93–0.98 in the best-resolved cases, and remain informative in regimes where purely geometric quality indicators become uncorrelated — or even negatively correlated — with the actual error distribution. For plane-stress elasticity the framework extends to a constitutive-weighted block Gram form that couples the displacement components through the elasticity matrix. An open-source implementation with all symbolic derivations and numerical experiments accompanies the paper.
This paper proposes a Peridynamic Differential Operator-informed neural network (PDDO-INN) framework for the bending analysis of Reissner–Mindlin plates. In this approach, the transverse displacement and two rotational fields are approximated by a neural network, while the Peridynamic Differential Operator is used to construct the particle-based plate operators and the corresponding discrete equilibrium residual. Instead of relying only on automatic differentiation as in conventional physics-informed neural networks, the assembled PDDO equilibrium system is embedded directly into the loss function, providing a discrete operator-informed physical constraint suitable for particle discretizations, curved boundaries, and complex plate geometries. Several learning strategies are investigated, including purely data-driven learning, data-boundary learning, one-stage data-physics learning, and the proposed two-stage PDDO-INN. Numerical examples involving a fully clamped square plate, an elliptical plate with different support conditions, and a perforated circular plate demonstrate that the proposed framework can reasonably reproduce displacement and rotational responses, reduce boundary-condition violations, and produce smooth solution fields with fewer visible zero-energy-mode oscillations than direct particle-based PDDO solutions. The results indicate that PDDO-INN provides an effective data-physics learning framework for Reissner–Mindlin plate analysis and offers potential for future inverse and data-assisted computational mechanics problems.
