arXiv subjects

Paulo Akira F. Enabe

Publications and source records attributed to Paulo Akira F. Enabe.

8 recordsLinked to original sources

arXiv2026

PoliVEM: a Python-driven virtual element framework for computational solid mechanics

This work presents PoliVEM, a software framework for the Virtual Element Method (VEM) in computational solid and structural mechanics. A C++17 computational core and a Python interface place one-dimensional beams, two- and three-dimensional elasticity, axisymmetric elasticity, transient diffusion, and finite-strain hyperelasticity in a common implementation. The framework stores vertex, edge, face, and cell degrees of freedom in one hierarchy, constructs the energy, strain, and $L^2$ projections from common polynomial data, and retains the consistency--stabilization split at the element level. The core separates the mesh, material, element, assembler, and solver responsibilities and combines them by composition. A new formulation supplies its projection, discrete form, and stabilization while reusing the mesh representation, degree-of-freedom numbering, boundary-condition treatment, sparse assembly, algebraic solvers, and Python binding pattern. The numerical infrastructure includes polygonal and polyhedral mesh input, higher-order entity numbering, cached projection operators, static condensation, coloured sparse assembly, linear solver selection based on the matrix structure, an incremental Newton method with line search and regularization, and explicit and implicit time integration. Three numerical experiments examine the shared implementation through polynomial reproduction, recovery after static condensation, higher-order degree-of-freedom handling, polygonal and polyhedral meshes, patch tests, stiffness spectra, and comparisons with independent finite element calculations. The results verify the common core for the beam and two- and three-dimensional elasticity formulations. They also identify the scaling of high-order polynomial bases on anisotropic cells and the memory required by sparse direct factorizations as the main limitations of the current implementation.

cs.SE↗

arXiv2026

Anytime PAC-Bayes for Constrained Density-Ratio Networks under Covariate Shift

A unified framework for learning under covariate shift is presented, in which a constrained density-ratio network approximates the Radon-Nikodym derivative $r^\star = dP/dQ$ and feeds an anytime PAC-Bayes generalization certificate. A change-of-measure identity decomposes the gap between target risk and importance-weighted source risk into a ratio-bias term governed by $\|r_θ- r^\star\|_{L^2(Q)}$ and a generalization-gap term governed by the variability of the weighted loss. Normalization and moment-matching identities are enforced as hard integral constraints through an augmented-Lagrangian scheme, with a second-moment penalty controlling the effective sample size. PAC-Bayes is instantiated on the weighted risk in a fixed-time regime that yields Bernoulli-KL bounds, identifies the network-weighted Gibbs posterior as the unique KL-regularized minimizer, and quantifies stability under $L^2(Q)$ perturbations of the learned ratio, and is then strengthened by geometric peeling to an anytime certificate uniform in $t \geq t_{\min}$. A pre-registered two-campaign protocol combining a patch test against analytic ground truth with a real-data deployment validates the framework: the network produces calibrated ratios, reduces target $0/1$ loss against unweighted ERM and classical direct ratio-estimation baselines, and attains the anytime certificate. A single fixed-time coverage failure is recorded, with per-split coverage aligning one-to-one with the magnitude of the label shift, confirming that the covariate-only assumption is operationally tight rather than a defect of the certificate.

cs.LG↗

arXiv2026

An Investigation of Stabilization Scaling in Finite-Strain Virtual Element Methods for Hyperelasticity

Low-order virtual element methods (VEM) compute a consistent finite-strain contribution through polynomial projections and rely on stabilization to control the unresolved modes in the projector kernel. In current hyperelastic VEM practice, stabilization is often defined by integrating a nonlinear surrogate energy over an auxiliary sub-triangulation and scaled through modified Lamé parameters and incompressibility factors; this can introduce sensitivity to the arbitrary internal tessellation, complicate consistent Newton linearization, and, most critically, inject bulk-dependent proxies into shear-type kernel penalties, artificially stiffening isochoric missing modes in the nearly incompressible regime. This work develops a submesh-free, kernel-only stabilization that decouples deviatoric and volumetric channels and is explicitly designed to scale like the current Newton tangent energy on the kernel: the deviatoric term is scaled solely by a shear measure and enhanced by bounded geometry-driven directional weights, while the volumetric term is scaled by an independent bulk measure and can be capped or suppressed as $ν\to 1/2$. A spectral framework is established in which the canonical VEM stability requirement on the kernel is characterized by generalized Rayleigh quotients and eigenvalue bounds, and it is shown under standard polygon regularity assumptions that the deviatoric stabilization is uniformly equivalent to $μ_E|\cdot|_{1,E}^2$ on the kernel with constants independent of mesh size and Poisson ratio. Element-level diagnostics confirm that classical surrogate-based stabilizations assign bulk-driven energy to isochoric kernel modes as $ν\to 1/2$, whereas the proposed decoupled stabilization remains shear-scaled; kernel spectra and Cook's membrane simulations in the nearly incompressible regime further support improved robustness across polygonal mesh families.

math.NA↗

arXiv2026

Profile-Then-Reason: Bounded Semantic Complexity for Tool-Augmented Language Agents

Large language model agents that use external tools are often implemented through reactive execution, in which reasoning is repeatedly recomputed after each observation, increasing latency and sensitivity to error propagation. This work introduces Profile--Then--Reason (PTR), a bounded execution framework for structured tool-augmented reasoning, in which a language model first synthesizes an explicit workflow, deterministic or guarded operators execute that workflow, a verifier evaluates the resulting trace, and repair is invoked only when the original workflow is no longer reliable. A mathematical formulation is developed in which the full pipeline is expressed as a composition of profile, routing, execution, verification, repair, and reasoning operators; under bounded repair, the number of language-model calls is restricted to two in the nominal case and three in the worst case. Experiments against a ReAct baseline on six benchmarks and four language models show that PTR achieves the pairwise exact-match advantage in 16 of 24 configurations. The results indicate that PTR is particularly effective on retrieval-centered and decomposition-heavy tasks, whereas reactive execution remains preferable when success depends on substantial online adaptation.

cs.AI↗

arXiv2026

Mass-Lumped Virtual Element Method with Strong Stability-Preserving Runge-Kutta Time Stepping for Two-Dimensional Parabolic Problems

This paper presents a mass-lumped Virtual Element Method (VEM) with explicit Strong Stability-Preserving Runge--Kutta (SSP-RK) time integration for two-dimensional parabolic problems on general polygonal meshes. A diagonal mass matrix is constructed via row-sum operations combined with flooring to ensure uniform positivity. Stabilization terms vanish identically under row summation, so the lumped weights derive solely from the $L^2$ projector and are computable through a small polynomial system at cost $\mathcal{O}(N_k^3)$ per element. The resulting lumped bilinear form satisfies $L^2$-equivalence with constants independent of the number of element edges, yielding a symmetric positive definite discrete inner product. A mesh-robust spectral estimate is established, showing that the largest eigenvalue of the discrete diffusion operator scales like $h^{-2}$, with constants depending only on the space dimension, polynomial degree, and mesh regularity. This yields the classical diffusion-type CFL condition $Δt=\mathcal{O}(h^2)$ for forward Euler stability and extends to higher-order SSP-RK schemes, ensuring the preservation of stability properties inherited from the forward Euler step. Numerical experiments on distorted quadrilateral, serendipity, and Voronoi meshes validate the theoretical predictions: for $k=1$, the lumped VEM attains optimal convergence rates, namely $\mathcal{O}(h)$ in the $H^1$-seminorm and $\mathcal{O}(h^2)$ in the $L^2$-norm, without degradation due to mesh distortion or diagonal mass approximation, while the SSP-RK methods remain stable under the predicted $Δt\propto h^2$ scaling. Additional tests on accuracy versus efficiency and on heterogeneous anisotropic diffusion further illustrate the practical competitiveness and robustness of the proposed formulation.

math.NA↗

arXiv2025

Virtual Element Method Applied to Two Dimensional Axisymmetric Elastic Problems

This work presents a Virtual Element Method (VEM) formulation tailored for two-dimensional axisymmetric problems in linear elasticity. By exploiting the rotational symmetry of the geometry and loading conditions, the problem is reduced to a meridional cross-section, where all fields depend only on the radial and axial coordinates. The method incorporates the radial weight $r$ in both the weak formulation and the interpolation estimates to remain consistent with the physical volume measure of cylindrical coordinates. A projection operator onto constant strain fields is constructed via boundary integrals, and a volumetric correction term is introduced to account for the divergence of the stress field arising from axisymmetry. The stabilization term is designed to act only on the kernel of the projection and is implemented using a boundary-based formulation that guarantees stability without affecting polynomial consistency. Furthermore, an a priori interpolation error estimate is established in a weighted Sobolev space, showing optimal convergence rates. The implementation is validated through patch tests that demonstrate the accuracy, consistency, and robustness of the proposed approach.

math.NA↗

arXiv2025

A Hybrid Virtual Element Method and Deep Learning Approach for Solving One-Dimensional Euler-Bernoulli Beams

A hybrid framework integrating the Virtual Element Method (VEM) with deep learning is presented as an initial step toward developing efficient and flexible numerical models for one-dimensional Euler-Bernoulli beams. The primary aim is to explore a data-driven surrogate model capable of predicting displacement fields across varying material and geometric parameters while maintaining computational efficiency. Building upon VEM's ability to handle higher-order polynomials and non-conforming discretizations, the method offers a robust numerical foundation for structural mechanics. A neural network architecture is introduced to separately process nodal and material-specific data, effectively capturing complex interactions with minimal reliance on large datasets. To address challenges in training, the model incorporates Sobolev training and GradNorm techniques, ensuring balanced loss contributions and enhanced generalization. While this framework is in its early stages, it demonstrates the potential for further refinement and development into a scalable alternative to traditional methods. The proposed approach lays the groundwork for advancing numerical and data-driven techniques in beam modeling, offering a foundation for future research in structural mechanics.

cs.LG↗

arXiv2024

Equivalence of D and D' properties in Banach spaces

This work explores the equivalence of two sequential properties, $D$ and $D'$, for dual Banach spaces under the weak* topology. Property $D$ ensures that any totally scalarly measurable function is also scalarly measurable, while property $D'$ states that every weakly* sequentially closed subspace of $X^*$ is weakly* closed. These properties, which are central to the study of the interplay between topology and measurability in Banach spaces, was left as an open question. By examining the topological and measurable structures induced by the Baire $σ$-algebra, we prove that properties $D$ and $D'$ are indeed equivalent. The proof utilizes the relationship between total sets, weak* closures, and scalar measurability, extending previous results on sequential properties of dual Banach spaces. Additionally, we revisit the failure of property $D$ in nonseparable Banach spaces with $M$-basic $\ell_1^+$-systems, providing a topological reinterpretation of this phenomenon. These findings contribute to a deeper understanding of the weak* topology and measurable mappings in Banach space theory.

math.FA↗