arXiv subjects

Paul Steinmann

Publications and source records attributed to Paul Steinmann.

At least 19 recordsLinked to original sources

arXiv2026

Unveiling the Multiphysics Complexity: An Isogeometric Framework for Inducing Bifurcation and Tracing Post-Buckling Paths in Electroelastic Thin Shells

Electroelastic shells are widely used in soft actuators, sensors, and energy harvesters owing to their large electrically induced deformations. However, the accurate simulation of their complex nonlinear multiphysics coupling, including bifurcation and post-buckling responses, remains challenging. This work presents an isogeometric Kirchhoff-Love shell formulation for the nonlinear analysis of electroelastic thin structures undergoing finite deformations. The formulation incorporates geometrically nonlinear kinematics, Maxwell-stress-induced electromechanical coupling, material incompressibility, and initial prestretch. Catmull--Clark subdivision surfaces are employed to ensure the C1 continuity required by Kirchhoff--Love shell theory. Consistent tangent operators are derived analytically, and a static condensation procedure is introduced to satisfy the plane-stress constraint. To trace bifurcation and post-buckling equilibrium paths, a staged Newton--Raphson algorithm with arc-length continuation and eigenmode perturbation is adopted. Numerical examples involving spherical membranes, prestretched circular plates, and toroidal membranes demonstrate the capability of the proposed framework to accurately capture large deformations, symmetry-breaking instabilities, and post-buckling responses under coupled electromechanical loading.

math.NA↗

arXiv2026

Learning finite viscoelasticity with DAVIS: A supervised framework for generalized standard materials

This work revisits the recently proposed data-adaptive viscoelasticity (DAVIS) framework, a spline-based formulation of finite viscoelasticity within the generalized standard materials setting. DAVIS enables a data-driven representation of equilibrium and non-equilibrium constitutive functions while retaining thermodynamic consistency and supporting parameter identification via finite element model updating. The present contribution focuses on improving the robustness and identifiability of non-equilibrium branches in generalized Maxwell-type models. To this end, two extensions of the original formulation are introduced. First, the spline representation is reformulated in terms of curvature-based variables, which is especially convenient to enforce monotonicity and convexity constraints by construction through a smooth parameter mapping. Second, the adaptation of interpolation domains is decoupled from the inner parameter identification by means of a staggered, block-alternating strategy: spline coefficients are optimized for fixed domain endpoints, while the endpoints are updated in an outer loop based on smooth statistics of sampled invariants. This separation alleviates an inherent scaling ambiguity between interpolation domains and spline coefficients that can impair conditioning in viscoelastic inverse problems. The underlying constitutive model remains the finite strain viscoelasticity framework of Reese and Govindjee. The proposed identification strategy is assessed for homogeneous uniaxial loading-unloading tests, which facilitates the study of identifiability and robustness of non-equilibrium branches.

cs.CE↗

arXiv2026

Learning ultra-compressible hyperelasticity with splines: Constitutive asymmetries and non-unique representations

Highly compressible solids, such as foams, exhibit complex responses, including pronounced tension-compression asymmetry. Capturing such behaviors within unified hyperelastic frameworks remains challenging. Invariant-based hyperelastic models are commonly identified from standard tests such as homogeneous uniaxial tension/compression and simple shear, implicitly assuming a unique energy representation. Here we show that this assumption is fundamentally violated and that, oftentimes, the choice of which term should prevail is just a matter of taste. Using spline-based strain-energy density functions as a data-adaptive tool and stress-strain experimental data for elastomeric foams, we expose this non-uniqueness, often hidden in low-parameter formulations. Our framework captures the volumetric deformation of ultra-light foams used in racing shoes using homogeneous experimental data from tension, compression, and shear. We formulate an overly rich ansatz of separable and non-separable energies in the ($\bar{I}_1$, $\bar{I}_2$, $J$) space \`a la Money-Rivlin. These constructs, defined by multiplicative decompositions, resemble classical invariant-based models while generalizing them to a data-driven spline representation. This serves two purposes: (i) to capture the response under complex volumetric deformation modes and (ii) to allow non-uniqueness in the identification problem to emerge naturally. We find that a coupling term between isochoric and volumetric deformation, such as $\Psi(\bar{I}_1,J)$ or $\Psi(\bar{I}_2,J)$, is essential and that additional coupling terms help but are not fully necessary; rather, they pronounce the non-uniqueness. As a consequence, different models may be indistinguishable on available data. Importantly, these challenges are not specific to splines but extend to traditional and neural network-based models.

cs.CE↗

arXiv2026

Data-adaptive spline surfaces for non-separable hyperelastic energy functions

Invariant-based models for incompressible isotropic hyperelasticity are typically formulated as functions of the first and second invariants, $W = W(\bar{I}_1, \bar{I}_2)$. A widely used class of models employs separable representations of the form $W(\bar{I}_1, \bar{I}_2) = W_1(\bar{I}_1) + W_2(\bar{I}_2)$, which enable efficient calibration and straightforward enforcement of modeling constraints. However, this decomposition implicitly restricts the coupling between the invariants and may limit the achievable accuracy for complex material responses. Fully coupled data-driven approaches overcome this limitation but often require nonlinear optimization and large parameter sets. In this contribution, we propose a compact alternative: a bivariate B-spline surface defined directly on the physically admissible invariant domain. By aligning the approximation space with physically realizable states, all model parameters contribute meaningfully to the constitutive response. We utilize homogeneous deformation modes to perform a calibration directly from analytical stress relations, eliminating the need for finite element model updating. Owing to the linear dependence of the spline representation on its coefficients, the resulting parameter identification problem reduces to a constrained linear least-squares problem. This enables fast, robust, and initialization-independent calibration, which makes parameter identification practically instantaneous. The results demonstrate that the proposed model improves accuracy compared to separable approaches while requiring only mild regularization in weakly sampled regions. The combination of computational efficiency and the linear structure of a highly expressive spline surface makes the approach particularly attractive for applications requiring repeated calibration, such as uncertainty quantification or interactive material characterization.

cs.CE↗

arXiv2026

Simulation Platform To Evaluate Inversion Techniques For Magnetic Resonance Elastography Data

Magnetic Resonance Elastography (MRE) has become an essential tool in assessing the mechanical properties of soft tissues in-vivo, prompting significant progress in new inversion algorithms. This creates a need for a benchmarking framework to promote uniformity and accessibility. To address this, we introduce a comprehensive in-silico dataset acquired by solving the forward Finite Element calculations of shear wave propagation in a linear visco-elastic material. This dataset aims to serve as a platform for evaluating inversion schemes by providing data that can be used as input with known mechanical properties to these methods. It includes simulations on homogeneous cuboidal domains of varying spatial and temporal resolution, and an extension to more physiological variations, including material inhomogeneity and internal arterial pulsation. We present a comprehensive case study using simulated data as an input to a direct inversion (DI) scheme, which allows for an expedient local inversion into the underlying material parameters. When aiming to reconstruct the parameters describing the linear visco-elastic material behavior via DI, we find that due to compromised convergence properties of frequency-domain stencils, stemming from truncation and subtractive cancellation errors, the reconstruction accuracy depends non-monotonically on the spatial and temporal resolution of the measurement grid. For inhomogeneous domains, the reconstruction was successful with notable interface boundaries. In the presence of pressurized vascular inclusions, a general stiffening of the domain was noted, as the recovered shear modulus was higher than the one assumed in forward modeling. Our study highlights the potential of this dataset as a vital benchmarking tool for advancing the development and refinement of MRE techniques, contributing to more accurate and reliable assessment of soft tissue mechanics.

math.NA↗

arXiv2026

A Convex Route to Thermoelasticity: Learning Internal Energy and Dissipation

We present a physics-based neural network framework for the discovery of constitutive models in fully coupled thermomechanics. In contrast to classical formulations based on the Helmholtz energy, we adopt the internal energy and a dissipation potential as primary constitutive functions, expressed in terms of deformation and entropy. This choice avoids the need to enforce mixed convexity--concavity conditions and facilitates a consistent incorporation of thermodynamic principles. In this contribution, we focus on materials without preferred directions or internal variables. While the formulation is posed in terms of entropy, the temperature is treated as the independent observable, and the entropy is inferred internally through the constitutive relation, enabling thermodynamically consistent modeling without requiring entropy data. Thermodynamic admissibility of the networks is guaranteed by construction. The internal energy and dissipation potential are represented by input convex neural networks, ensuring convexity and compliance with the second law. Objectivity, material symmetry, and normalization are embedded directly into the architecture through invariant-based representations and zero-anchored formulations. We demonstrate the performance of the proposed framework on synthetic and experimental datasets, including purely thermal problems and fully coupled thermomechanical responses of soft tissues and filled rubbers. The results show that the learned models accurately capture the underlying constitutive behavior. All code, data, and trained models are made publicly available via https://doi.org/10.5281/zenodo.19248596.

cs.CE↗

arXiv2026

Deblurring structural edges in variable thickness topology optimization via density-gradient-informed projection

Variable thickness topology optimization (VTTO) is a potent methodology for designing high-performance, high-stiffness sheet structures. However, this method frequently encounters two primary challenges: 1) the formation of undesirable low-thickness regions, which present manufacturing difficulties, and 2) the blurring of structural edges. This blurring is an artifact inherent to the regularization filters required for well-posedness. This paper proposes solutions to address both challenges. First, to mitigate low-thickness regions, we introduce a robust, combined approach. This strategy utilizes a SIMP-based penalization and an updated projection method, which effectively suppresses nearly all low-thickness domains. Second, the main contribution of this work is a novel method to deblur structural edges, termed the density-gradient-informed (DGI) projection. This projection utilizes local density gradient information. It selectively applies a strong projection in high-gradient regions (i.e., structural edges) to restore sharpness, while minimally affecting low-gradient regions within the structure's interior. Numerical examples demonstrate that the DGI projection successfully deblurs the structural edges, restoring a distinct solid-void transition, while preserving the internal form. Most importantly, this significant improvement in edge definition is achieved with a negligible impact on the final structural compliance. This establishes the DGI projection as a non-invasive and effective regularization tool for enhancing VTTO designs.

cs.CE↗

arXiv2026

Unsupervised Material Fingerprinting: Ultra-fast hyperelastic model discovery from full-field experimental measurements

Material Fingerprinting is a lookup table-based strategy to discover material models from experimental measurements, which completely avoids the need to solve an optimization problem. In an offline phase, a comprehensive database of simulated material responses, so-called material fingerprints, is generated for a predefined experimental setup. This database can then be used repeatedly in the online phase to discover material models corresponding to experimentally measured observations. To this end, the experimentally measured fingerprint is compared with all fingerprints in the database to identify the closest match. The primary advantage of this strategy is that it does not require solving a continuous optimization problem. This avoids the associated computational costs as well as issues of ill-posedness caused by local minima in non-convex optimization landscapes. Material Fingerprinting has been successfully demonstrated for supervised datasets consisting of stress-strain pairs, as well as for unsupervised datasets involving full-field displacements and net reaction forces. However, to date, there is no experimental validation for the latter approach which is the objective of this work.

cs.CE↗

arXiv2025

Fluctuations, Clustering, and Interaction-Driven Dynamics in Sedimenting Particles at Low Galileo Numbers: A Neural Network Approach

In this study, we investigate the behaviour of sedimenting solid particles and the influence of microscopic particle dynamics on the collective motion of a sedimenting cloud. Departing from conventional direct numerical simulations (DNS), we introduce a novel machine learning framework, the Interaction-Decomposed Neural Network (IDNN), to model hydrodynamic particle interactions. The IDNN acts as a black-box module within a Lagrangian solver, predicting the particle drag force based on the relative positions of the nearest neighbours. This enables the recovery of force fluctuations, capturing effects previously accessible only through DNS. Our results show an increase in collective settling velocity in the dilute regime, consistent with earlier experimental and numerical studies, which we attribute to (i) fluctuations in the streamwise particle force around a value that is lower than the Stokes limit and (ii) the formation of particle clusters sedimenting at enhanced velocities. These fluctuations originate from persistent entrainment and ejection of particles in and out of the long, diffusive wakes generated by upstream particles at low Galileo numbers. Energy spectra of particle velocity fluctuations reveal a scale-dependent transfer of fluctuation energy, analogous to a turbulent-like cascade, with pronounced large-scale fluctuations at higher volume fractions. At very low volume fractions, fluctuation intensity and energy spectrum amplitudes diminish, though hydrodynamic interactions still remain appreciable.

physics.flu-dyn↗

arXiv2025

In Vivo Wideband MR Elastography for Assessing Age-Related Viscoelastic Changes of the Human Brain

Magnetic Resonance Elastography (MRE) noninvasively maps brain biomechanics and is highly sensitive to alterations associated with aging and neurodegenerative disease. Most implementations use a single frequency or a narrow frequency band, limiting the analysis of frequency-dependent viscoelastic parameters. We developed a dual-actuator wideband MRE (5-50 Hz) protocol and acquired wavefields at 13 frequencies in 24 healthy adults (young: 23-39 years; older: 50-63 years). Shear wave speed (SWS) maps were generated as a proxy for stiffness, and SWS dispersion was modeled using Newtonian, Kelvin-Voigt, and power-law rheological models. Whole-brain stiffness declined with age, with the strongest effect observed at low frequencies (5-16 Hz: -0.24%/year; p=0.019) compared with mid (20-35 Hz: -0.12%/year; p=0.030) and high frequencies (40-50 Hz: -0.10%/year; p=0.165). Compared to older brains, younger adults showed 14.3% higher baseline stiffness in the power-law model (p=0.001) and 8.5-9.0% higher viscosity according to the Newtonian and Kelvin-Voigt model (p<0.05). White and cortical gray matter exhibited similar age-related decreases, while deep gray matter showed an increase in the power-law exponent (+0.001/year; p=0.036), suggesting a transition toward more fluid-like properties associated with aging. Wideband MRE revealed frequency-dependent and region-specific biomechanical alterations with aging, with the strongest effects observed at low frequencies. Extending brain MRE into the low frequency regime potentially enhances sensitivity to solid-fluid interactions. Therefore, low frequency MRE may serve as an early biomechanical marker of microstructural brain changes due to aging and neurodegeneration.

physics.med-ph↗

arXiv2025

A novel multi-thickness topology optimization method for balancing structural performance and manufacturability

Topology optimization (TO) in two dimensions often presents a trade-off between structural performance and manufacturability, with unpenalized (variable-thickness) methods yielding superior but complex designs, and penalized (SIMP) methods producing simpler, truss-like structures with compromised performance. This paper introduces a multi-thickness, density-based topology optimization method designed to bridge this gap. The proposed approach guides the design towards a predefined set of discrete, allowable thicknesses by employing a novel multilevel penalization scheme and a multilevel smoothed Heaviside projection. A continuation strategy for the penalization and projection parameters, combined with an adaptive mesh refinement technique, ensures robust convergence and high-resolution geometric features. The method is validated on standard cantilever and MBB beam benchmarks. Results demonstrate that as the number of allowable thicknesses increases, the designs systematically transition from conventional truss-like structures to high-performance, sheet-like structures. Notably, designs with as few as three discrete thickness levels achieve compliance values within 2\% of those from fully unpenalized, variable-thickness optimization, while significantly outperforming standard SIMP results. The method inherently eliminates impractically thin regions and features, both in the out-of-plane and in-plane directions and produces designs well-suited for both additive manufacturing and conventional fabrication using standard-thickness stock materials, thus maximizing both performance and manufacturability.

cs.CE↗

arXiv2025

Configurational-force-driven adaptive refinement and coarsening in topology optimization

The iterative nature of topology optimization, especially in combination with nonlinear state problems, often requires the solution of thousands of linear equation systems. Furthermore, due to the pixelated design representation, the use of a fine mesh is essential to obtain geometrically well-defined structures and to accurately compute response quantities such as the von Mises stress. Therefore, the computational cost of solving a fine-mesh topology optimization problem quickly adds up. To address this challenge, we consider a multi-level adaptive refinement and coarsening strategy based on configurational forces. Configurational forces based on the Eshelby stress predict configurational changes such as crack propagation or dislocation motion. Due to a relaxation in the calculation of (Eshelby) stresses with respect to the design variables, discrete configurational forces increase not only in highly stressed regions, but also in grey transition regions (design boundaries). For this reason they are an ideal criterion for mesh adaptivity in topology optimization, especially when avoiding stress failure is a priority. By using configurational forces for refinement, we obtain a high-resolution structure where the refined mesh is present along the design boundaries as well as in stress-critical regions. At the same time, multilevel coarsening using the same criterion drastically minimizes the computational effort.

cs.CE↗

arXiv2025

Cutting soft materials: how material differences shape the response

Cutting soft materials is a complex process governed by the interplay of bulk large deformation, interfacial soft fracture, and contact forces with the cutting tool. Existing experimental characterizations and numerical models often fail to capture the variety of observed cutting behaviors, especially the transition from indentation to cutting and the roles of dissipative mechanisms. Here, we combine novel experimental cutting tests on three representative materials-a soft hydrogel, elastomer, and food-based materials-with a coupled computational model that integrates soft fracture, adhesion, and frictional interactions. Our experiments reveal material-dependent cutting behaviors, with abrupt or smooth transitions from indentation to crack initiation, followed by distinct steady cutting regimes. The computational model captures these behaviors and shows that adhesion and viscous cohesive forces dominate tangential stresses, while Coulomb friction plays a negligible role due to low contact pressures. Together, these results provide new mechanistic insights into the physics of soft cutting and offer a unified framework to guide the design of soft materials, cutting tools, and cutting protocols, with direct relevance to surgical applications and the engineering of food textures optimized for mastication.

cond-mat.soft↗

arXiv2025

Configurational forces explain echelon cracks in soft materials

Soft fracture in highly deformable solids involves both geometric and constitutive nonlinearities, necessitating advanced theoretical and computational frameworks for its accurate understanding. Tensile fractures subjected to mixed-mode loading deviate from their original planar shape, resulting in echelon crack patterns. When out-of-plane shear is superimposed, a crack front segments into an array of tilted facets. The physical interpretation of echelon cracks is only marginally understood, and it is customarily based on rather limited approaches based on Linear Elastic Fracture Mechanics. Here we investigate mixed-mode I + III fracture within the framework of configurational mechanics. Using the Configurational Force Method, implemented as a post-processing algorithm in a finite-element-based simulation, we compute the configurational forces acting at the crack tip of model fracture geometries prior to propagation. Configurational forces characterize both the magnitude and direction of propagation for maximal energy release rate. Our results reveal the complex interactions between tilted facets and their critical role in shaping the fracture morphology. We also examine the effects of facet coalescence-driven by the growth of the parent crack-where neighboring facets merge into a unified crack front. These findings provide new insights into fracture processes in soft, quasi-brittle materials under mixed-mode loading.

cond-mat.soft↗

arXiv2025

A statistical theory for polymer elasticity: from molecular kinematics to continuum behavior

Predicting the macroscopic mechanical behavior of polymeric materials from the micro-structural features has remained a challenge for decades. Existing theoretical models often fail to accurately capture the experimental data, due to non-physical assumptions that link the molecule kinematics with the macroscopic deformation. In this work, we construct a novel Hamiltonian for chain segments enabling a unified statistical description of both individual macromolecular chains and continuum polymer networks. The chain kinematics, including the stretch and orientation properties, are retrieved by the thermodynamic observables without phenomenological assumptions. The theory shows that the chain stretch is specified by a simple relation via its current spatial direction and the continuum Eulerian logarithmic strain, while the probability of a chain in this spatial direction is governed by the new Hamiltonian of a single segment. The model shows a significantly improved prediction on the hyperelastic response of elastomers, relying on minimal, physically-grounded parameters.

cond-mat.soft↗

arXiv2025

A micromorphic-based artificial diffusion method for stabilized finite element approximation of convection-diffusion problems

We present a novel artificial diffusion method to circumvent the instabilities associated with the standard finite element approximation of convection-diffusion equations. Motivated by the micromorphic approach, we introduce an auxiliary variable, which is related to the gradient of the field of interest, and which leads to a coupled problem. Conditions for well-posedness of the resulting formulation are established. We carry out a comprehensive numerical study to compare the proposed methodology against some well-established approaches in one- and two-dimensional settings. The proposed method outperforms established approaches in general in approximating accurately the solutions to pertinent and challenging problems.

math.NA↗

arXiv2025

A gradient-enhanced approach for stable finite element approximations of reaction-convection-diffusion problems

We develop a micromorphic-based approach for finite element stabilization of reaction-convection-diffusion equations, by gradient enhancement of the field of interest via introducing an auxiliary variable. The well-posedness of the coupled-field approach is established, together with an error estimate. Through a set of 1D and 2D numerical examples the high accuracy and enhanced stability of the approach in approximating solutions associated with complex problems is demonstrated, for situations of varying reactivity and convection.

math-ph↗

arXiv2025

Biaxial characterization of soft elastomers: experiments and data-adaptive configurational forces for fracture

Understanding the fracture mechanics of soft solids remains a fundamental challenge due to their complex, nonlinear responses under large deformations. While multiaxial loading is key to probing their mechanical behavior, the role of such loading in fracture processes is still poorly understood. Here, we present a combined experimental-computational framework to investigate fracture in soft elastomers under equi-biaxial loading. We report original equi-biaxial quasi-static experiments on five elastomeric materials, revealing a spectrum of material and fracture behavior, from brittle-like to highly deformable response with crack tip strains exceeding 150 %. Motivated by these observations, we develop a hybrid computational testbed that mirrors the experimental setup and enables virtual biaxial tests. Central to this framework are two components: a data-adaptive formulation of hyperelastic energy functions that flexibly captures material behavior, and a post-processing implementation of the Configurational Force Method, providing a computationally efficient estimate of the J-integral at the crack tip. Our data-adaptive framework for hyperelastic energy functions proves versatility to capture with high accuracy the hyperelastic behavior observed in the biaxial experiments. This is important because accurately capturing the constitutive behaviour of soft solids is key for a reliable application of the Configurational Force Method to soft solids. In the limit of crack onset, a critical value of the crack tip configurational force allows for a criterion of fracture toughness. Together, our experimental, theoretical, and computational contributions offer a new paradigm for characterizing and designing soft materials with tailored fracture properties.

cond-mat.soft↗