A mesh-free discretization in which a single neural network represents the displacement and phase fields and is trained by minimizing the incremental energy directly is proposed.
Abstract
Phase-field modeling of brittle fracture removes the need to track cracks explicitly by recasting their evolution as the minimization of an energy functional. In return it requires a discretization dense enough to resolve a localization band whose width is set by a regularization length and whose path is not known in advance. We propose a mesh-free discretization in which a single neural network represents the displacement and phase fields and is trained by minimizing the incremental energy directly. The coordinates enter the network through a multiresolution feature encoding built from $C^1$ quadratic B-spline grids, so the finest scale the representation can express is set by choice rather than reached through slow training, and the energy is estimated by stratified Monte Carlo integration on points redrawn at every optimizer iteration. This pairing proves critical, since the crack fails to advance both when the integration points are held fixed and when the encoding is too coarse to represent the band, while each ingredient tolerates a wide range of settings once the other is in place. Because the representation is globally $C^1$, the second- and the fourth-order fracture energy densities run on the identical discretization. Across six problems, from single-edge-notched tension and shear to a thick-walled ring on a single spline patch, the computed load-displacement curves follow staggered finite element references at matched regularization length, with peak loads within about 1% on the single-edge-notched tests and within 8% where the crack pattern changes topology. On a public benchmark dataset of random multi-crack configurations the method classifies the active or dormant state of 90% of the seeded cracks in twenty zero-shot runs, where the deep Ritz baseline of the dataset authors fails.
Large-deformation crystal plasticity finite element method (CPFEM) simulations are often limited by accumulated mesh distortion, which degrades accuracy and numerical stability, while adaptive remeshing introduces discrete topology changes that impede gradient-based inverse analysis. We present OmniRemesh, a unified framework that addresses these forward and inverse challenges through two developments. First, a structure-driven remeshing method dynamically redistributes local mesh resolution according to both microstructural geometry and the evolving mechanical state. By refining grain boundaries and localized deformation regions while retaining a coarser mesh elsewhere, the method maintains mesh quality and physical consistency, improves the accuracy and robustness of large-deformation calculations, and resolves grain-scale heterogeneity without uniformly dense discretization. Second, a frozen-remeshing-branch strategy locally fixes the mesh sequence within a parameter trust region and periodically updates it as the parameters evolve. This treatment provides approximate automatic-differentiation sensitivities despite topology changes, enabling efficient inverse calibration of constitutive parameters against both macroscopic and local observables. Numerical examples demonstrate accurate and stable CPFEM simulations up to 80\% tensile deformation. The inverse calibration successfully recovers both macroscopic and local responses. OmniRemesh thus provides a practical framework for large-deformation CPFEM and remeshing-aware constitutive calibration.
This paper presents a robust, fully Lagrangian framework based on the Particle Finite Element Method (PFEM) capable of simulating multiphase flows with an arbitrary number of immiscible phases. Interface-tracking methods can sometimes suffer from numerical diffusion or allow the underlying mesh resolution to prematurely dictate topological changes. To address these limitations, we introduce a dynamic mesh adaptation strategy that naturally preserves sharp geometric interfaces without relying on classical constrained triangulation. A node-empty disk is assigned to each segment of the discretized interface, ensuring that the edge is part of the Delaunay triangulation. Our approach decouples the interface physics from the grid size, allowing the integration of sub-grid physical models to properly govern topological changes independently of the user-defined mesh size. The capabilities and accuracy of the framework are validated against standard multiphase benchmarks, closely matching references while maintaining a remarkably low overall node count. We demonstrate the scalability and geometric versatility of the method, in particular with a challenging 16-phase Rayleigh-Taylor simulation.
F'elix Ruyffelaere, Michel Henry, Jonathan Lambrechts et al.· 0 citations
Fracture mechanics is essential for predicting the failure of materials and structures. The Phase-Field Fracture (PFF) method has emerged as a powerful computational tool, overcoming the limitations of classical discrete approaches by modeling cracks as diffuse damage bands. This formulation eliminates the need for explicit crack tracking and naturally handles complex topologies like branching and merging. However, despite its success, the method faces significant numerical and physical challenges, particularly regarding the robust enforcement of inequality constraints in some regularization models, the accurate measurement of geometric quantities distorted by strain localization, and the high computational cost of cycle-by-cycle fatigue simulations.
The computational challenge of fatiguewhere phase-field fracture approaches typically rely on cycle-by-cycle simulations to accumulate damage over thousands of cycles, leading to prohibitive computational costsis addressed in this thesis through a framework that enables fatigue analysis using only monotonic loading. This is made possible by a specialized energy-controlled solver designed to robustly track the complete crack equilibrium path, capturing instabilities such as snap-back. By performing a single quasi-static simulation, the method numerically extracts the compliance rate of the structure with respect to the crack area, which is then used to bridge the results with Linear Elastic Fracture Mechanics (LEFM) and integrate Paris' law. This allows for the prediction of fatigue life with drastically reduced computational effort while maintaining the topological flexibility of the phase-field method.
To ensure the accuracy of the geometric quantities extracted from these simulations, the work addresses the issue of crack area overestimation caused by numerical strain localization, which also affects the force and displacement values returned by the simulations. The Double Gradient Correction Method (DGCM) is proposed, founded on the principle of energy equipartition, where the energy contributions from the phase-field and its gradient are equal as the length-scale parameter approaches zero. Since numerical artifacts primarily affect the phase-field term while leaving the gradient term largely unperturbed, the crack area can be accurately approximated as twice the gradient-dependent energy. This method is inherently mesh-independent and readily applicable to the entire domain, including 3D simulations.
Finally, to enhance the robustness of constrained phase-field models beyond standard formulations, the Latent Variable Proximal Point (LVPP) algorithm is presented. In phase-field fracture simulations, the phase-field variable must remain within an admissible range typically between 0 (undamaged) and 1 (fully damaged) and satisfy an irreversibility condition to ensure damage does not decrease. While some standard models naturally respect the phase-field interval constraint, other regularizations require these physical constraints to be explicitly and robustly imposed. To address this, the LVPP framework reformulates these inequality-constrained problems as a sequence of unconstrained saddle-point problems via an auxiliary latent variable. This allows for the use of efficient Proximal Galerkin discretizations, providing a rigorous and robust solution that avoids the parameter sensitivity and convergence issues associated with traditional penalty methods.
All methodologies developed in this thesis are implemented in PhaseFieldX, an open-source software package designed to ensure reproducibility and to facilitate future research in the field.
RESUMEN
La mecánica de fractura es esencial para predecir el fallo de materiales y estructuras. El método de Fractura por Campo de Fase (PFF) ha emergido como potente herramienta computacional, modelando las grietas como bandas de daño difuso y superando limitaciones de enfoques discretos clásicos. Esta formulación elimina el seguimiento explícito de grietas y maneja topologías complejas como ramificación y fusión. Pese a su éxito, enfrenta importantes retos numéricos y físicos, especialmente en la imposición de restricciones de desigualdad en algunos modelos, la medición precisa de cantidades geométricas distorsionadas por la localización de deformaciones, y el alto coste computacional de simulaciones de fatiga ciclo a ciclo.
Para abordar el desafío computacional de la fatiga donde las simulaciones ciclo a ciclo imponen costes prohibitivos, esta tesis introduce un marco de análisis de fatiga usando solo carga monótona. Mantiene la flexibilidad topológica del método empleando un solucionador especializado controlado por energía que traza la trayectoria completa de equilibrio de la grieta, capturando inestabilidades como el "snap-back". Con una única simulación cuasi-estática, se extrae numéricamente la tasa de flexibilidad de la estructura respecto al área de la grieta, conectando los resultados con la Mecánica de Fractura Elástica Lineal (LEFM) e integrando la ley de Paris. Esto permite predecir la vida a fatiga reduciendo drásticamente el esfuerzo computacional.
Para garantizar la precisión de las cantidades geométricas extraídas, se aborda la sobreestimación del área de grieta debida a la localización numérica de deformaciones, que también afecta a fuerzas y desplazamientos. Se propone el Método de Corrección de Doble Gradiente (DGCM), basado en la equipartición de energía, donde las contribuciones de energía del campo de fase y su gradiente se igualan cuando el parámetro de escala tiende a cero. Como los artefactos numéricos afectan principalmente al campo de fase, dejando inalterado al gradiente, el área se aproxima con precisión como el doble de la energía del gradiente. Este método es independiente de la malla y aplicable a todo el dominio, incluyendo problemas en 3D.
Finalmente, para mejorar la robustez de los modelos de campo de fase con restricciones, se presenta el algoritmo de Punto Proximal de Variable Latente (LVPP). En simulaciones PFF, el campo de fase debe acotarse, típicamente entre 0 (sin daño) y 1 (completamente dañado), y cumplir una condición de irreversibilidad para que el daño no disminuya. Aunque algunos modelos estándar respetan sus márgenes de manera natural, otras regularizaciones exigen que estas restricciones se impongan de forma explícita. El marco LVPP reformula estos problemas de minimización con restricciones de desigualdad como una secuencia de problemas de punto de silla sin restricciones usando una variable latente auxiliar. Esto permite usar discretizaciones de Proximal Galerkin, aportando una solución rigurosa y robusta que evita la sensibilidad y falta convergencia de los métodos de penalización.
Todas las metodologías desarrolladas en esta tesis se han implementado en PhaseFieldX, un paquete de software de código abierto diseñado para garantizar la reproducibilidad y facilitar la investigación futura en el campo.
Phase-field approaches to fracture, initially designed as regularization of the Griffith model of brittle fracture, are now commonly viewed as gradient-damage models whose regularization length becomes a material property driving crack nucleation. One weakness of this approach is that the strength surface cannot be arbitrary: its shape is dictated by the elastic energy, and its magnitude by the regularization length. We focus on the antiplane version of the model introduced by Bourdin, Marigo, Maurini and Zolesi (arXiv:2506.22558), which handles crack propagation along unknown paths and nucleation governed by an arbitrary convex strength surface by degrading the strength instead of the stiffness. It can be interpreted as a regularization of softening plasticity in which localization bands obey an equivalent cohesive law set by the strength domain and the toughness, while the role of the regularization length, when small compared to the elasto-cohesive length, is purely numerical. Strength, stiffness, and toughness thus become independent material data, and limit analysis, perfect plasticity, cohesive fracture, and brittle fracture merge into a single variational framework. We derive closed-form solutions for a simple shear problem, propose a numerical scheme combining alternate minimization and conic programming, and numerically verify the equivalent cohesive law, its independence of the regularization, and the size effect governed by the elasto-cohesive length. A"surfing"simulation highlights the structure of the propagating crack while a re-entrant V-notch is used to show how the model bridges small-scale yielding, cohesive fracture, and brittle fracture without a priori hypotheses.
Evolving crack fields in structural health monitoring and fracture assessment must often be inferred from sparse mechanical measurements rather than dense full-field observations. We develop CNN2D--ConvGRU, a convolutional-recurrent framework for measurement-conditioned reconstruction of time-dependent phase-field brittle fracture. At each load step, the model maps a fixed-length history of phase and displacement fields and sparse current-step displacement measurements to the current full-field state. New measurements are assimilated during sequential deployment, making the framework a state-inference surrogate rather than an autonomous time integrator. It reproduces crack paths, damage evolution, and bulk displacement response, with the largest errors near propagating crack tips and steep displacement gradients and some drift at late stages. Without retraining, weights learned on a $256 \times 256$ raster are evaluated on a $512 \times 512$ raster of the same physical domain and finite-element discretization using a proportionally refined measurement grid. This empirical raster-and-sensing transfer preserves the principal damage topology and global damage evolution, although fine-scale displacement errors increase near crack tips. Comparisons with alternative spatial and temporal architectures show that CNN2D--ConvGRU offers a favorable balance between reconstruction accuracy and computational cost. Relative to repeated finite-element solutions, sequential reconstruction achieves mean speedups of $175\times$ on CPU and $253\times$ on GPU. These results demonstrate efficient full-field fracture-state reconstruction from sparse observations while retaining the spatial structure and history dependence of phase-field fracture.
Pressure vessel analysis in the chemical, nuclear, and new-energy industries requires solving the same elasticity problem across many materials, geometries, and loads, where mesh quality and repeated solving govern both accuracy and cost. The finite element method (FEM) cannot amortise this repeated cost and fails on degenerate meshes, while the neural operators meant to replace it still need labelled data that FEM must generate. This paper proposes FEVessel, an adaptation of the Pretrained Finite Element Method (PFEM) to three-dimensional (3D) pressure vessels, and validates four capabilities across the two limitations above. FEVessel i) encodes each vessel as a point cloud with coordinate, material, and load channels, ii) pretrains a Transolver operator on the total potential energy instead of FEM labels, and iii) warm-starts iterative solvers with its prediction. A single model generalises across material, geometry, and boundary conditions at a $1.35\%$ relative displacement error, and its $2.07\%$ strain error is about $4.7$ times lower than that of a supervised Fourier neural operator ($9.72\%$), whose structured grid cannot preserve the through-thickness strain. Its warm start cuts algebraic multigrid iterations from $195$ to $18$, a $9.2\times$ end-to-end wall-clock speedup at the $10^{-3}$ engineering tolerance. The model transfers across mesh resolutions without retraining, holding about $3\%$ error at only $30\%$ of the training point density. On inverted and sliver meshes where FEM fails, the error remains below $3.66\%$. To our knowledge, this is the first systematic study of mesh-independent solution on industrially relevant 3D pressure vessels with degenerate meshes. Because training needs no labels, FEVessel works exactly where FEM cannot supply any, removing manual mesh repair from the analysis pipeline.
Yipin Sun, Yizheng Wang, Yuzhou Lin et al.· 0 citations
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