Wiley: International Journal for Numerical Methods in Engineering: Table of Contents

A Multiscale Mixed Three‐Field Finite Element Formulation Coupled With Phase Field Fracture for Incompressible Rubber‐Like Materials

Wed, 03 Jun 2026 19:30:24 -0700

ABSTRACT

Rubber-like materials possess remarkable properties such as high stretchability, low modulus, and impressive toughness, making them ideal for applications across various emerging domains. Given their growing relevance, understanding the fracture behavior of these materials is crucial for designing applications against failure. Although numerous models have been developed to simulate fracture propagation in these materials, few accurately account for their incompressible nature, which often leads to numerical challenges. This study introduces a multiscale polymer model integrated with the phase field approach, specifically designed to capture the fracture behavior in incompressible rubber-like materials using mixed finite elements. At the microscale, the entropic chain behavior is modeled using non-Gaussian statistics while also accounting for an internal energy due to molecular bond distortions. The non-affine microsphere network model, adapted for damaged systems, is employed to bridge the microscale deformations with those at the macroscale. Furthermore, the phase field approach is utilized at the macroscale to model damage, which is primarily assumed to be caused by the rupture of chain segments. This is coupled with a three-field mixed formulation, and augmented Lagrangian iterations are performed to strongly enforce the incompressibility constraint. The major advantage of this proposed formulation lies in the fact that it does not lead to an increase in the global system size. The numerical implementation of the proposed model using a monolithic scheme is detailed, and three-dimensional simulations are performed to validate the model's performance. The predictions are compared with experimental data to evaluate the potential and reliability of the framework in accurately predicting the fracture behavior of rubber-like materials. Furthermore, the volumetric deformation contours are visualized to demonstrate the effectiveness of the proposed model in strictly enforcing the incompressibility constraint. Moreover, the effect of the phase field regularization parameter on the predicted fracture behavior in specimens having geometric features of different sizes is investigated.

Solid Mechanics Segregated Solver Acceleration With Jacobian‐Free Newton‐Krylov

Sun, 31 May 2026 21:21:39 -0700

ABSTRACT

The segregated algorithm is a common approach for finite volumes solvers in solid mechanics, providing a memory-efficient and straightforward implementation. Due to the inter-coupling of the components through the source terms, it suffers from a slow convergence behavior in specific scenarios, such as geometries with significantly uneven dimensions. Examples of attempts to improve the performance of the segregated algorithm in such cases are available in the literature, for instance, with machine learning or with multigrid acceleration. On the other hand, Newton solvers and Jacobian-Free Newton-Krylov method have been successfully applied as standalone solvers for fluid mechanics applications, or even for multi-physics coupling in nuclear codes. Complementing recent work on Newton-Krylov as a standalone solver for solid mechanics, this paper proposes a novel approach to tackle the slow convergence issue by coupling the segregated algorithm with a Jacobian-Free Newton-Krylov method. The targeted advantage being the performance improvements for pure mechanical simulations with almost no parametrization from the code user. In the article, the method is introduced and benchmarked against the original segregated algorithm and the Anderson acceleration. 2D and 3D steady-state cases are considered, including small and large deformations, from linear to nonlinear mechanical behaviors. The OpenFOAM Fuel Behavior Analysis Tool (OFFBEAT) is a multidimensional fuel performance code developed jointly by the Paul Scherrer Institute (PSI) and the École Polytechnique Fédérale de Lausanne (EPFL). Using the PETSc library, the innovative coupling with the segregated algorithm has been implemented into OFFBEAT to produce the benchmarks. Still relying on the OpenFOAM framework and the segregated algorithm, the proposed method benefits from the validation of the existing code base. The results, obtained for serial executions, exhibit a promising reduction in computational time and number of iterations to convergence, paving the way for further development in solid mechanics solvers and a possible extension to other physics.

A Multimodal Conditional Mixture Model With Distribution‐Level Physics Priors

Jinkyo Han, Bahador Bahmani — Thu, 28 May 2026 19:12:52 -0700

ABSTRACT

Many scientific and engineering systems exhibit intrinsically multimodal behavior arising from latent regime switching and non-unique physical mechanisms. In such settings, learning the full conditional distribution of admissible outcomes in a physically consistent and interpretable manner remains a challenge. While recent advances in machine learning have enabled powerful multimodal generative modeling, their integration with physics-constrained scientific modeling remains nontrivial, particularly when physical structure must be preserved or data are limited. This work develops a physics-informed multimodal conditional modeling framework based on mixture density representations. Mixture density networks (MDNs) provide an explicit and interpretable parameterization of multimodal conditional distributions. We consider both Gaussian and Student's t$$ t $$ component distributions, the latter enabling robustness to heavy-tailed variability. Physical knowledge is embedded through component-specific regularization terms that penalize violations of governing equations or physical laws. This formulation naturally accommodates non-uniqueness and stochasticity while remaining computationally efficient and amenable to conditioning on contextual inputs. The proposed framework is evaluated across a range of scientific problems in which multimodality arises from intrinsic physical mechanisms rather than observational noise, including bifurcation phenomena in nonlinear dynamical systems, stochastic differential equations (SDEs), and atomistic-scale shock dynamics. In addition, the proposed method is compared with a conditional flow matching (CFM) model, a representative state-of-the-art generative modeling approach, demonstrating that MDNs can achieve competitive performance while offering a simpler and more interpretable formulation.

Direct Differentiation and Adjoint Methods in Multiple Minimal Coordinates for Multibody System Sensitivity Analysis

Hongchen Li, Ye Ding — Thu, 28 May 2026 00:08:11 -0700

ABSTRACT

Sensitivity analysis for multibody systems computes the gradient of a system's dynamic-response cost functional with respect to design variables, enabling effective optimization of the system's response characteristics. Developing high-fidelity, reliable, and computationally efficient sensitivity analysis methods is therefore of great importance. This paper proposes two novel approaches using multiple sets of minimal coordinates to achieve efficient sensitivity analysis for general multibody systems. Unlike methods based on Differential-Algebraic Equations (DAEs), the proposed formulations express both the dynamic and adjoint equations as Ordinary Differential Equations (ODEs), ensuring strict satisfaction of position, velocity, and acceleration constraints throughout the computation. In contrast to methods that rely on a single predetermined set of minimal coordinates, this approach computes sensitivities globally without any risk of parameterization singularities. The approaches are validated on two planar multibody systems and compared against existing methods based on index-1 DAE formulation and generalized coordinate partitioning. Simulation results show that the proposed methods are accurate, efficient, and globally valid.

Issue Information

Thu, 28 May 2026 00:05:25 -0700

International Journal for Numerical Methods in Engineering, Volume 127, Issue 11, 15 June 2026.