Jan N. Fuhg

Kshitiz Upadhyay, Jan N. Fuhg, Nikolaos Bouklas et al.

A novel data-driven constitutive modeling approach is proposed, which combines the physics-informed nature of modeling based on continuum thermodynamics with the benefits of machine learning. This approach is demonstrated on strain-rate-sensitive soft materials. This model is based on the viscous dissipation-based visco-hyperelasticity framework where the total stress is decomposed into volumetric, isochoric hyperelastic, and isochoric viscous overstress contributions. It is shown that each of these stress components can be written as linear combinations of the components of an irreducible integrity basis. Three Gaussian process regression-based surrogate models are trained (one per stress component) between principal invariants of strain and strain rate tensors and the corresponding coefficients of the integrity basis components. It is demonstrated that this type of model construction enforces key physics-based constraints on the predicted responses: the second law of thermodynamics, the principles of local action and determinism, objectivity, the balance of angular momentum, an assumed reference state, isotropy, and limited memory. The three surrogate models that constitute our constitutive model are evaluated by training them on small-size numerically generated data sets corresponding to a single deformation mode and then analyzing their predictions over a much wider testing regime comprising multiple deformation modes. Our physics-informed data-driven constitutive model predictions are compared with the corresponding predictions of classical continuum thermodynamics-based and purely data-driven models. It is shown that our surrogate models can reasonably capture the stress-strain-strain rate responses in both training and testing regimes, and provide improvements in terms of prediction accuracy, generalizability to multiple deformation modes, and compatibility with limited data.

L. River Spencer, William D. Meador, Adrian Buganza Tepole et al.

Calibrating thermomechanical material models from experiments is challenging because deformation, temperature, and force responses are strongly coupled, while measurements are usually restricted to specimen surfaces. We present a full-field calibration framework for coupled finite-strain thermomechanical material models using boundary displacement, reaction-force data, and temperature. The forward model is formulated as a near-incompressible thermo-hyperelastic problem with thermomechanical coupling derived from a Helmholtz free energy, and the inverse problem is posed as a PDE-constrained optimization problem with weighted observation terms for the available data streams. Reduced gradients are computed with adjoint sensitivities that are obtained by automatic differentiation, enabling gradient-based calibration of nonlinear transient thermomechanical systems. The formulation is first verified on synthetic examples involving uniform thermal preconditioning and localized transient rod contact, where the ground-truth parameters are recovered from full-field measurements and force observations. The same workflow is then applied to experimental thermomechanical data by first calibrating a hyperelastic mechanical baseline from cyclic equibiaxial loading and subsequently identifying thermal expansion and directional shrinkage parameters from surface-temperature and boundary-force histories. The results demonstrate that coupled thermomechanical parameters can be inferred from experimentally accessible surface data without requiring volumetric observations.

Jan N. Fuhg, Craig M. Hamel, Kyle Johnson et al.

The development of accurate constitutive models for materials that undergo path-dependent processes continues to be a complex challenge in computational solid mechanics. Challenges arise both in considering the appropriate model assumptions and from the viewpoint of data availability, verification, and validation. Recently, data-driven modeling approaches have been proposed that aim to establish stress-evolution laws that avoid user-chosen functional forms by relying on machine learning representations and algorithms. However, these approaches not only require a significant amount of data but also need data that probes the full stress space with a variety of complex loading paths. Furthermore, they rarely enforce all necessary thermodynamic principles as hard constraints. Hence, they are in particular not suitable for low-data or limited-data regimes, where the first arises from the cost of obtaining the data and the latter from the experimental limitations of obtaining labeled data, which is commonly the case in engineering applications. In this work, we discuss a hybrid framework that can work on a variable amount of data by relying on the modularity of the elastoplasticity formulation where each component of the model can be chosen to be either a classical phenomenological or a data-driven model depending on the amount of available information and the complexity of the response. The method is tested on synthetic uniaxial data coming from simulations as well as cyclic experimental data for structural materials. The discovered material models are found to not only interpolate well but also allow for accurate extrapolation in a thermodynamically consistent manner far outside the domain of the training data. Training aspects and details of the implementation of these models into Finite Element simulations are discussed and analyzed.

Jan N. Fuhg, Reese E. Jones, Nikolaos Bouklas

Data-driven constitutive modeling with neural networks has received increased interest in recent years due to its ability to easily incorporate physical and mechanistic constraints and to overcome the challenging and time-consuming task of formulating phenomenological constitutive laws that can accurately capture the observed material response. However, even though neural network-based constitutive laws have been shown to generalize proficiently, the generated representations are not easily interpretable due to their high number of trainable parameters. Sparse regression approaches exist that allow to obtaining interpretable expressions, but the user is tasked with creating a library of model forms which by construction limits their expressiveness to the functional forms provided in the libraries. In this work, we propose to train regularized physics-augmented neural network-based constitutive models utilizing a smoothed version of $L^{0}$-regularization. This aims to maintain the trustworthiness inherited by the physical constraints, but also enables interpretability which has not been possible thus far on any type of machine learning-based constitutive model where model forms were not assumed a-priory but were actually discovered. During the training process, the network simultaneously fits the training data and penalizes the number of active parameters, while also ensuring constitutive constraints such as thermodynamic consistency. We show that the method can reliably obtain interpretable and trustworthy constitutive models for compressible and incompressible hyperelasticity, yield functions, and hardening models for elastoplasticity, for synthetic and experimental data.

L. River Spencer, Manuel K. Rausch, Chad M. Landis et al.

Magneto-active elastomers exhibit large, nonlinear deformations under combined mechanical loading and magnetic fields, and their effective behavior is strongly governed by microstructural heterogeneity. Predictive modeling of these materials is challenging because their response involves strong magneto-mechanical coupling, large deformations, and the nearly incompressible behavior of elastomeric matrices. Existing multiscale approaches often rely on staggered strategies or formulations that do not robustly treat near-incompressibility in strongly coupled settings. This work presents a fully coupled computational homogenization framework for nearly incompressible magnetoelastic composites in which the mechanical deformation and magnetostatic fields are solved monolithically on a representative volume element (RVE). The microscale problem uses a mixed finite-element discretization with Lagrangian displacement degrees of freedom and a N'ed'elec-based magnetic vector potential, enabling a curl-conforming representation of magnetic induction together with periodic boundary constraints for both mechanical and magnetic fields. Near-incompressibility is treated using J-bar stabilization, in which the volumetric response is controlled by the cell-averaged dilatation while the isochoric response is evaluated using a scaled deformation gradient. The constitutive behavior is derived from an additive free-energy decomposition with hyperelastic, vacuum magnetic, and saturation-type magnetization contributions. The resulting formulation enables robust three-dimensional RVE simulations of heterogeneous magneto-elastic composites with complex particle distributions under large deformations and strong coupling. Numerical examples show how particle interactions, microstructural arrangement, and inclusion compressibility influence deformation patterns and the effective magneto-mechanical response.

Jan N. Fuhg, Nikolaos Bouklas, Reese E. Jones

Data-driven constitutive modeling frameworks based on neural networks and classical representation theorems have recently gained considerable attention due to their ability to easily incorporate constitutive constraints and their excellent generalization performance. In these models, the stress prediction follows from a linear combination of invariant-dependent coefficient functions and known tensor basis generators. However, thus far the formulations have been limited to stress representations based on the classical Rivlin and Ericksen form, while the performance of alternative representations has yet to be investigated. In this work, we survey a variety of tensor basis neural network models for modeling hyperelastic materials in a finite deformation context, including a number of so far unexplored formulations which use theoretically equivalent invariants and generators to Finger-Rivlin-Ericksen. Furthermore, we compare potential-based and coefficient-based approaches, as well as different calibration techniques. Nine variants are tested against both noisy and noiseless datasets for three different materials. Theoretical and practical insights into the performance of each formulation are given.

Aryan Tyagi, Alex de Beer, Tiangang Cui et al.

Structural reliability evaluation for composites constitutes a fundamentally high-dimensional multiscale problem, as microscale material uncertainties must propagate to the macroscale and can be quantified as high-dimensional random fields. Conventional approaches are computationally intractable, as they rely on repeatedly solving coupled partial differential equation systems across scales while contending with the exponential complexity inherent in high-dimensional uncertainty quantification. This work introduces a scalable and physically consistent framework that addresses both bottlenecks simultaneously in the case of separation of scales and (anisotropic) linear elasticity. In particular, we couple a physics-augmented Voigt--Reuss Neural Network (VRNN) with the Deep Inverse Rosenblatt Transport (DIRT) method to estimate the posterior probability of structural failure. The VRNN is used to resolve the computationally expensive FE$^2$ scheme by providing a near-instantaneous evaluation of the homogenized stiffness tensor that is guaranteed to be symmetric, positive-definite, and strictly bounded within the Voigt--Reuss limits, enabling fast evaluation of the homogenized responses. The DIRT method constructs a sequence of functional tensor train approximations to efficiently store an approximation of the high-dimensional optimal importance sampling distribution for estimating the probability of failure. This mitigates the curse of dimensionality arising from the Karhunen--Loève expansion of the random fields. The framework is demonstrated on a three-dimensional heterogeneous benchmark problem, where the uncertainty in the microscale material properties is characterized by a Bayesian posterior distribution obtained from limited strain observations. Our results show that the proposed framework can provide low-variance estimates of failure probabilities in dimensions up to 150.

L. River Spencer, Reagan A. Cardoza, Vijay K. Dubey et al.

Ultrasound imaging tasks such as calibration, inverse parameter estimation, and acquisition design require models that are physically grounded, efficient, and differentiable with respect to meaningful material and system parameters. While full-wave solvers offer high fidelity, they are often too expensive for iterative optimization, and existing ray-based methods have mostly been limited to forward simulation. In this work, we present a fully differentiable end-to-end ultrasound simulation framework based on full-path Monte Carlo ray tracing. Building on UltraRay, the method propagates gradients from image-space losses back through acoustic transport, beamforming, and post-processing, enabling gradient-based optimization over scene and acquisition parameters. The framework combines differentiable ray transport in Mitsuba 3/Dr.Jit with a custom differentiable bridge through the ultrasound image-formation pipeline. Forward examples reproduce expected geometric image features and capture more complex anatomical structures. In inverse problems, the method recovers known parameters in a simulated-reference setting and identifies effective parameters that improve agreement between simulated and experimental B-mode images in a simulation-to-real setting. Finite-difference comparisons further support the consistency of the computed gradients. Overall, this work provides a practical foundation for differentiable, physics-based ultrasound simulation and optimization.

Asghar A. Jadoon, Aryan Tyagi, L. River Spencer et al.

Multiscale topology optimization (TO) of hyperelastic materials remains computationally prohibitive due to the repeated solution of microscale boundary value problems. In this work, we present a concurrent multiscale topology optimization framework that overcomes this limitation by leveraging physics-augmented neural networks (PANNs) as surrogate constitutive models. The proposed approach enables the simultaneous optimization of macroscale material distribution and microscale descriptors, within a unified nonlinear finite strain setting. The surrogate models are constructed using input-specific neural networks (ISNNs) that enforce key physical principles directly within the architecture, including convexity and material symmetry through invariant-based representations and structural tensors. This ensures thermodynamic consistency and numerical stability while accurately representing homogenized anisotropic hyperelastic responses. The trained PANNs replace the microscale boundary value problem and provide efficient evaluations of stresses and consistent tangent moduli using analytical first and second derivatives of the neural network, enabling tractable large-scale multiscale optimization. The framework is demonstrated on representative microstructures exhibiting transversely isotropic, cubic anisotropic, and nearly incompressible isotropic behavior. The results show that the proposed method captures complex multiscale interactions and enables physically meaningful spatial tailoring of material properties, while significantly reducing computational cost compared to classical FE$^2$ approaches. These findings establish PANNs as a powerful tool for high-fidelity multiscale topology optimization of nonlinear anisotropic materials.

Adrian Buganza Tepole, Asghar Jadoon, Manuel Rausch et al.

Accurate constitutive models of soft materials are crucial for understanding their mechanical behavior and ensuring reliable predictions in the design process. To this end, scientific machine learning research has produced flexible and general material model architectures that can capture the behavior of a wide range of materials, reducing the need for expert-constructed closed-form models. The focus has gradually shifted towards embedding physical constraints in the network architecture to regularize these over-parameterized models. Two popular approaches are input convex neural networks (ICNN) and neural ordinary differential equations (NODE). A related alternative has been the generalization of closed-form models, such as sparse regression from a large library. Remarkably, all prior work using ICNN or NODE uses the invariants of the Cauchy-Green tensor and none uses the principal stretches. In this work, we construct general polyconvex functions of the principal stretches in a physics-aware deep-learning framework and offer insights and comparisons to invariant-based formulations. The framework is based on recent developments to characterize polyconvex functions in terms of convex functions of the right stretch tensor $\mathbf{U}$, its cofactor $\text{cof}\mathbf{U}$, and its determinant $J$. Any convex function of a symmetric second-order tensor can be described with a convex and symmetric function of its eigenvalues. Thus, we first describe convex functions of $\mathbf{U}$ and $\text{cof}\mathbf{U}$ in terms of their respective eigenvalues using deep Holder sets composed with ICNN functions. A third ICNN takes as input $J$ and the two convex functions of $\mathbf{U}$ and $\text{cof}\mathbf{U}$, and returns the strain energy as output. The ability of the model to capture arbitrary materials is demonstrated using synthetic and experimental data.

Asghar A. Jadoon, D. Thomas Seidl, Reese E. Jones et al.

The black-box nature of neural networks limits the ability to encode or impose specific structural relationships between inputs and outputs. While various studies have introduced architectures that ensure the network's output adheres to a particular form in relation to certain inputs, the majority of these approaches impose constraints on only a single set of inputs. This paper introduces a novel neural network architecture, termed the Input Specific Neural Network (ISNN), which extends this concept by allowing scalar-valued outputs to be subject to multiple constraints. Specifically, the ISNN can enforce convexity in some inputs, non-decreasing monotonicity combined with convexity with respect to others, and simple non-decreasing monotonicity or arbitrary relationships with additional inputs. The paper presents two distinct ISNN architectures, along with equations for the first and second derivatives of the output with respect to the inputs. These networks are broadly applicable. In this work, we restrict their usage to solving problems in computational mechanics. In particular, we show how they can be effectively applied to fitting data-driven constitutive models. We then embed our trained data-driven constitutive laws into a finite element solver where significant time savings can be achieved by using explicit manual differentiation using the derived equations as opposed to automatic differentiation. We also show how ISNNs can be used to learn structural relationships between inputs and outputs via a binary gating mechanism. Particularly, ISNNs are employed to model an anisotropic free energy potential to get the homogenized macroscopic response in a decoupled multiscale setting, where the network learns whether or not the potential should be modeled as polyconvex, and retains only the relevant layers while using the minimum number of inputs.

Vijay K. Dubey, Collin E. Haese, Osman Gültekin et al.

Surrogate models for the rapid inference of nonlinear boundary value problems in mechanics are helpful in a broad range of engineering applications. However, effective surrogate modeling of applications involving the contact of deformable bodies, especially in the context of varying geometries, is still an open issue. In particular, existing methods are confined to rigid body contact or, at best, contact between rigid and soft objects with well-defined contact planes. Furthermore, they employ contact or collision detection filters that serve as a rapid test but use only the necessary and not sufficient conditions for detection. In this work, we present a graph neural network architecture that utilizes continuous collision detection and, for the first time, incorporates sufficient conditions designed for contact between soft deformable bodies. We test its performance on two benchmarks, including a problem in soft tissue mechanics of predicting the closed state of a bioprosthetic aortic valve. We find a regularizing effect on adding additional contact terms to the loss function, leading to better generalization of the network. These benefits hold for simple contact at similar planes and element normal angles, and complex contact at differing planes and element normal angles. We also demonstrate that the framework can handle varying reference geometries. However, such benefits come with high computational costs during training, resulting in a trade-off that may not always be favorable. We quantify the training cost and the resulting inference speedups on various hardware architectures. Importantly, our graph neural network implementation results in up to a thousand-fold speedup for our benchmark problems at inference.

Lehu Bu, Zhaohan Yu, Shaoting Lin et al.

Laser-induced inertial cavitation (LIC)-where microscale vapor bubbles nucleate due to a focused high-energy pulsed laser and then violently collapse under surrounding high local pressures-offers a unique opportunity to investigate soft biological material mechanics at extremely high strain rates (>1000 1/s). Traditional rheological tools are often limited in these regimes by loading speed, resolution, or invasiveness. Here we introduce novel machine learning (ML) based microrheological frameworks that leverage LIC to characterize the viscoelastic properties of biological materials at ultra-high strain rates. We utilize ultra-high-speed imaging to capture time-resolved bubble radius dynamics during LIC events in various soft viscoelastic materials. These bubble radius versus time measurements are then analyzed using a newly developed Bubble Dynamics Transformer (BDT), a neural network trained on physics-based simulation data. The BDT accurately infers material viscoelastic parameters, eliminating the need for iterative fitting or complex inversion processes. This enables fast, accurate, and non-contact characterization of soft materials under extreme loading conditions, with significant implications for biomedical applications and materials science.

Jan N. Fuhg, Nikolaos Bouklas

The introduction of Physics-informed Neural Networks (PINNs) has led to an increased interest in deep neural networks as universal approximators of PDEs in the solid mechanics community. Recently, the Deep Energy Method (DEM) has been proposed. DEM is based on energy minimization principles, contrary to PINN which is based on the residual of the PDEs. A significant advantage of DEM, is that it requires the approximation of lower order derivatives compared to formulations that are based on strong form residuals. However both DEM and classical PINN formulations struggle to resolve fine features of the stress and displacement fields, for example concentration features in solid mechanics applications. We propose an extension to the Deep Energy Method (DEM) to resolve these features for finite strain hyperelasticity. The developed framework termed mixed Deep Energy Method (mDEM) introduces stress measures as an additional output of the NN to the recently introduced pure displacement formulation. Using this approach, Neumann boundary conditions are approximated more accurately and the accuracy around spatial features which are typically responsible for high concentrations is increased. In order to make the proposed approach more versatile, we introduce a numerical integration scheme based on Delaunay integration, which enables the mDEM framework to be used for random training point position sets commonly needed for computational domains with stress concentrations. We highlight the advantages of the proposed approach while showing the shortcomings of classical PINN and DEM formulations. The method is offering comparable results to Finite-Element Method (FEM) on the forward calculation of challenging computational experiments involving domains with fine geometric features and concentrated loads.

Jan N. Fuhg, Amelie Fau

Kriging is an efficient machine-learning tool, which allows to obtain an approximate response of an investigated phenomenon on the whole parametric space. Adaptive schemes provide a the ability to guide the experiment yielding new sample point positions to enrich the metamodel. Herein a novel adaptive scheme called Monte Carlo-intersite Voronoi (MiVor) is proposed to efficiently identify binary decision regions on the basis of a regression surrogate model. The performance of the innovative approach is tested for analytical functions as well as some mechanical problems and is furthermore compared to two regression-based adaptive schemes. For smooth problems, all three methods have comparable performances. For highly fluctuating response surface as encountered e.g. for dynamics or damage problems, the innovative MiVor algorithm performs very well and provides accurate binary classification with only a few observation points.

Jan N. Fuhg

The computational effort for the evaluation of numerical simulations based on e.g. the finite-element method is high. Metamodels can be utilized to create a low-cost alternative. However the number of required samples for the creation of a sufficient metamodel should be kept low, which can be achieved by using adaptive sampling techniques. In this Master thesis adaptive sampling techniques are investigated for their use in creating metamodels with the Kriging technique, which interpolates values by a Gaussian process governed by prior covariances. The Kriging framework with extension to multifidelity problems is presented and utilized to compare adaptive sampling techniques found in the literature for benchmark problems as well as applications for contact mechanics. This thesis offers the first comprehensive comparison of a large spectrum of adaptive techniques for the Kriging framework. Furthermore a multitude of adaptive techniques is introduced to multifidelity Kriging as well as well as to a Kriging model with reduced hyperparameter dimension called partial least squares Kriging. In addition, an innovative adaptive scheme for binary classification is presented and tested for identifying chaotic motion of a Duffing's type oscillator.