Abstract X-ray tomography can be used to measure displacement fields from in situ or ex situ mechanical tests. The present work aims to quantify displacement uncertainties resulting from different reconstruction steps implemented in computed tomography. Two samples made of polyamide and aluminum alloy were imaged in situ by acquiring radiograph sets corresponding to two tomographic turns. From these data sets, the standard displacement uncertainties were assessed via digital volume correlation (DVC) and projection-based DVC (P-DVC). Careful calibration of the computed tomography geometry, accounting for beam hardening and correcting dark-and flat-field variations, was crucial to reach sub-decivoxel uncertainties.

ABSTRACT Fluid‐filled phase‐field fracture simulations require robust, scalable solvers that can handle strongly nonlinear, non‐smooth mechanics and tightly coupled flow on locally refined meshes. In this work, we develop an adaptive finite element framework for quasi‐static, fluid‐filled phase‐field fractures that combines semi‐smooth Newton methods, fixed‐stress iterative coupling, and matrix‐free geometric multigrid preconditioning. The geomechanics subproblem, with displacement and phase field as primary variables, is solved by a semi‐smooth combined Newton method based on a primal‐dual active set formulation. The resulting linear systems are treated with the generalized minimal residual method and matrix‐free geometric multigrid preconditioners on locally refined meshes, with a local smoothing approach and careful treatment in the active set updates. The pressure equation is coupled to the geomechanics system via a fixed‐stress iterative scheme which iterates until both subproblem residuals are sufficiently minimized, and each flow subproblem is likewise preconditioned by matrix‐free geometric multigrid on the same adaptive meshes. In addition, we introduce an enhanced fracture‐width computation, inspired by and extending existing phase‐field aperture formulas, to obtain more accurate and mesh‐robust width fields consistent with the regularized fracture profile. The overall computational framework is demonstrated on a set of two‐ and three‐dimensional benchmark problems, highlighting its robustness, efficiency, and accuracy for simulating fluid‐filled phase‐field fractures with local mesh refinement.

While 1D analytical solutions of the variational phase-field fracture method have been extensively reported and provide critical insights into crack nucleation and damage evolution criteria, their counterparts for pressurized fractures have received scant attention. This work presents a systematic study of a family of variational phase-field models for pressurized fracture. We derive both homogeneous and localized analytical solutions for a 1D bar under uniaxial tension. We find that in these models, the material’s strength depends on two key parameters: the internal length scale l0 and the crack surface pressure p. We show how to modulate l0 to correctly capture the material strength, similar to classical methods. A key finding is that, unlike in classical models, solely modulating the internal length scale l0 to capture material strength becomes insufficient under non-zero Poisson’s ratio. By testing different models, we identify one (the AT1-I2 model) that is not affected by this issue. All analytical findings and the identified model’s efficacy are validated through two-dimensional finite element simulations.

PurposeThis study presents a new variational model for non-Newtonian fluids based on Hamilton's principle with an internal variable describing spatial and temporal variations of the viscosity.Design/methodology/approachThe internal variable evolves in response to local flow conditions, enabling more refined and dynamic representation of complex non-Newtonian behaviors. The Type 1 model, originally introduced by Junker and Wick (2025) [P. Junker and T. Wick, " Space-Time Modeling and Numerical Simulations of Non-Newtonian Fluids Using Internal Variables," International Journal for Numerical Methods in Fluids 97, no. 12 (2025)] is revisited in the present work within an unified variational framework, and extended by introducing a new Type 2 model. Both models are derived from distinct free energy potentials: the Type 1 model describes viscosity evolution through the interaction between velocity and displacement gradients, while the Type 2 model captures viscosity variations driven by the magnitude of the strain.FindingsBoth formulations are capable of reproducing shear-thinning and shear-thickening behaviors within a unified and thermodynamically consistent setting. Unlike conventional constitutive laws expressed as nonlinear algebraic relations between stress and the rate of strain, the proposed framework naturally incorporates the evolution of the viscosity both in space and time. Simulations in two and three spatial dimensions demonstrate that the proposed models effectively capture a wide range of non-Newtonian flow characteristics.Originality/valueThe variational approach based on Hamilton's principle, incorporating an internal variable, is novel. This allows for a wider range of non-Newtonian fluid flows models, which can be further studied in engineering and applied mathematics.

In this contribution, space-time concepts are used to design Galerkin finite element schemes in time and space for the discretization of partial differential equations. These include single PDEs, and coupled systems of PDEs, and linear and nonlinear models. The extension of several prior studies is a unified abstract formulation from which specific applications can be derived. Therein, goal functionals and error representations are specified demonstrating the abstract formalisms.

We study a fluid-poroelasticity interaction (FPSI) problem coupling the unsteady Stokes equations with the fully dynamic Biot system. A major challenge in such problems is to design partitioned schemes that remain robust in locking-related parameter regimes while preserving the physical interface coupling structure.To address this issue, we introduce two auxiliary variables and reformulate the Biot system as a four-field problem consisting of a dynamic Stokes-like system coupled with a diffusion equation. Crucially, this reformulation preserves the original interface conditions. Based on Robin-Robin transmission conditions with explicitly lagged interface data, we construct a fully decoupled scheme in which the fluid and poroelastic subproblems can be solved independently and in parallel at each time step, without sub-iterations.We prove that the resulting method is unconditionally stable and derive optimal-order error estimates in the H^1-norm. The analysis further shows that the scheme is robust with respect to extreme poroelastic parameters and avoids the locking effects inherent in standard formulations. Numerical experiments confirm the theoretical convergence results and demonstrate the locking-robust performance of the proposed method.

This work develops and analyzes a variational-monolithic unfitted finite element formulation of a linear fluid-structure interaction problem in Eulerian coordinates with a fixed interface. The overall discretization is based on a backward Euler scheme in time and finite elements in space. For the spatial discretization we employ a cut finite element method on a mesh consisting of quadrilateral elements. We use a first-order in time formulation of the elasticity equations, inf-sup stable finite elements in the fluid part and Nitsche's method to incorporate the coupling conditions. Ghost penalty terms guarantee the robustness of the approach independently of the way the interface cuts the finite element mesh. The main objective is to establish stability and a priori error estimates. We prove optimal-order error estimates in space and time and substantiate them with numerical tests.

This paper introduces a novel diffraction based thermo-hydraulic-mechanical (THM) model for fracture propagation using a phase-field fracture (PFF) approach. The key innovation of the THM-PFF model lies in its integrated treatment of four solution variables-displacements, phase-field, pressure, and temperature-each governed by a combination of conservation of momentum (mechanics problem), a variational inequality (constrained minimization problem), mass conservation (pressure problem), and energy conservation (temperature problem). This leads to anew formulation of a coupled variational inequality system. A major advancement is the development of an extended fixed-stress algorithm, where displacements, phase-field, pressures, and temperatures are solved in a staggered sequence. An important aspect of this work is the global coupling of pressures and temperatures across the domain using diffraction systems, with diffraction coefficients defined by material parameters weighted by the diffusive phase-field variable. To ensure robust local mass conservation, we employ enriched Galerkin finite elements (EG) for both pressure and temperature diffraction equations. By enriching the continuous Galerkin basis functions with discontinuous piecewise constants, EG accurately represents solution and parameter discontinuities while preserving local mass and energy conservation-crucial aspects for THM problems and realistic behavior. Moreover, the use of a predictor-corrector local mesh adaptivity scheme is employed, allowing the model to handle small phase-field length-scale parameters while maintaining high numerical accuracy and reasonable computational cost. These new model and algorithmic developments represent significant advances in the field and have been substantiated through rigorous numerical tests.

The modeling of fluids is an important field for mechanics of materials. In this work, we demonstrate that Hamilton's principle, which is well-known for the modeling of solids, can also be formulated to derive the Navier-Stokes equations, which paves the way for easy inclusion of complex material constraints. Furthermore, we expand Hamilton's principle to enable the introduction of "internal variables", which describe the space- and time-dependent evolution of the material properties. Hereby, a novel strategy for the modeling of non-Newtonian fluids is given. Eventually, Hamilton's principle inherently enables a space-time formulation with the automatic derivation of the correct formal functional setting, which covers different scales of viscosity through the internal variable. The resulting system is a space-time multiscale model for fluid flow, which is based on an additional partial differential equation. The model constitutes thus a much more adaptive description of the complex processes in non-Newtonian fluid flow as possible for classical models based on algebraic constitutive laws. This also includes a spatially and temporally local evolution of the effective viscosity, depending on the local flow conditions rather than material parameters and resulting in both shear-thinning and shear-thickening behavior. Numerical examples substantiate our proposed setting by some studies from Newtonian flow to non-Newtonian regimes with fading or increasing viscosity.

In this work, we develop a cut-based unfitted finite element formulation for solving nonlinear, nonstationary fluid-structure interaction with contact in Eulerian coordinates. In the Eulerian description fluid flow modeled by the incompressible Navier-Stokes equations remains in Eulerian coordinates, while elastic solids are transformed from Lagrangian coordinates into the Eulerian system. A monolithic description is adopted. For the spatial discretization, we employ an unfitted finite element method with ghost penalties based on inf-sup stable finite elements. To handle contact, we use a relaxation of the contact condition in combination with a unified Nitsche approach that takes care implicitly of the switch between fluid-structure interaction and contact conditions. The temporal discretization is based on a backward Euler scheme with implicit extensions of solutions at the previous time step. The nonlinear system is solved with a semi-smooth Newton's method with line search. Our formulation, discretization and implementation are substantiated with an elastic falling ball that comes into contact with the bottom boundary, constituting a challenging state-of-the-art benchmark.

In this work, a concept for coupling fluid–structure interaction with brittle fracture in elasticity is proposed. The fluid–structure interaction problem is modeled in terms of the arbitrary Lagrangian–Eulerian technique and couples the isothermal, incompressible Navier–Stokes equations with nonlinear elastodynamics using the Saint-Venant Kirchhoff solid model. The brittle fracture model is based on a phase-field approach for cracks in elasticity and pressurized elastic solids. In order to derive a common framework, the phase-field approach is re-formulated in Lagrangian coordinates to combine it with fluid–structure interaction. A crack irreversibility condition, that is mathematically characterized as an inequality constraint in time, is enforced with the help of an augmented Lagrangian iteration. The resulting problem is highly nonlinear and solved with a modified Newton method (e.g., error-oriented) that specifically allows for a temporary increase of the residuals. The proposed framework is substantiated with several numerical tests. In these examples, computational stability in space and time is shown for several goal functionals, which demonstrates reliability of numerical modeling and algorithmic techniques. But also current limitations such as the necessity of using solid damping are addressed.