The present paper proposes a Bayesian framework for inverse problems that seamlessly integrates optimization and inversion to enable rapid surrogate modeling, accurate parameter inference, and rigorous uncertainty quantification. Bayesian optimization is employed to adaptively construct accurate Gaussian process surrogate models using a minimal number of high-fidelity model evaluations, strategically focusing sampling in regions of high predictive uncertainty. The trained surrogate model is then leveraged within a Bayesian inversion scheme to infer optimal parameter values by combining prior knowledge with observed quantities of interest, resulting in posterior distributions that rigorously characterize epistemic uncertainty. The framework is theoretically grounded, computationally efficient, and particularly suited for engineering applications in which high-fidelity models – whether arising from numerical simulations or physical experiments – are computationally expensive, analytically intractable, or difficult to replicate, and data availability is limited. Furthermore, the combined use of Bayesian optimization and inversion outperforms their separate application, highlighting the synergistic benefits of unifying the two approaches. The performance of the proposed Bayesian framework is demonstrated on a suite of one- and two-dimensional analytical benchmarks, including the Mixed Gaussian-Periodic, Lévy, Griewank, Forrester, and Rosenbrock functions, which provide a controlled setting to assess surrogate modeling accuracy, parameter inference robustness, and uncertainty quantification. The results demonstrate the framework's effectiveness in efficiently solving inverse problems while providing informative uncertainty quantification and supporting reliable engineering decision-making at reduced computational cost.

We propose a physics-augmented neural network (PANN) framework for finite strain incompressible viscoelasticity within the generalized standard materials theory. The formulation is based on the multiplicative decomposition of the deformation gradient and enforces unimodularity of the inelastic deformation part throughout the evolution. Invariant-based representations of the free energy and the dual dissipation potential by (partially) monotonic and fully input-convex neural networks ensure thermodynamic consistency, objectivity, and material symmetry by construction. The evolution of the internal variables during training is handled by solving the evolution equations using an implicit exponential time integrator. In addition, a trainable gate layer combined with lp regularization automatically identifies the required number of internal variables during training. The PANN is calibrated with synthetic and experimental data, showing excellent agreement for a wide range of deformation rates and different load paths. We also show that the proposed model achieves excellent interpolation as well as plausible and accurate extrapolation behaviors. In addition, we demonstrate consistency of the PANN with linear viscoelasticity by linearization of the full model.

Polyconvex constitutive modeling is attractive as it guarantees stability of numerical simulations and can improve the generalization behavior of material models. However, in certain applications, polyconvex formulations perform poorly in reproducing the underlying ground truth material response, which can effectively preclude their practical use. In this work, we address this issue and investigate the limitations of polyconvex constitutive modeling. The main contributions of this paper are as follows: (1) We analyze the theoretical reasons why polyconvexity may, in some cases, impose overly restrictive constraints that limit the achievable accuracy of constitutive models. Thereby, we provide analytical ellipticity guarantees for two non-polyconvex Mooney-Rivlin type potentials. (2) We investigate the practical limitations of polyconvex physics-augmented neural network constitutive models using two representative formulations: models using structural tensor-based invariants and models using signed singular values. Their performance is evaluated on datasets obtained from homogenized microstructured materials, and their predictive capabilities are assessed in finite element simulations. (3) Overall, we provide an overview of benefits, limitations, and mitigation strategies of polyconvex constitutive modeling.

In this work, we develop a neural network-based, data-driven, decoupled multiscale scheme for the modeling of structured magnetically soft magnetorheological elastomers (MREs). On the microscale, sampled magneto-mechanical loading paths are imposed on a representative volume element containing spherical particles and an elastomer matrix, and the resulting boundary value problem is solved using a mixed finite element formulation. The computed microscale responses are homogenized to construct a database for the training and testing of a macroscopic physics-augmented neural network model. The proposed model automatically detects the material's preferred direction during training and enforces key physical principles, including objectivity, material symmetry, thermodynamic consistency, and the normalization of free energy, stress, and magnetization. Within the range of the training data, the model enables accurate predictions of magnetization, mechanical stress, and total stress. For larger magnetic fields, the model yields plausible results. Finally, we apply the model to investigate the magnetostrictive behavior of a macroscopic spherical MRE sample, which exhibits contraction along the magnetic field direction when aligned with the material's preferred direction.

We propose a precise and efficient physics-augmented neural network (PANN) to model strain-induced crystallization in rubbery polymers. We demonstrate that the model can be flexibly employed for both unfilled and filled natural rubber (NR). The approach is based on a two potential framework, similar to the concept of generalized standard materials (GSMs). To describe the material behavior, neural network-based free energy and dissipation potentials are employed. The evolution of crystallinity is derived from the two potentials. To ensure boundedness of the crystallinity, a novel constrained GSM-type evolution problem is proposed. To this end, two additional Lagrange multipliers together with the corresponding Karush-Kuhn-Tucker conditions are introduced. As a result, it is guaranteed that crystallinity can be interpreted as a variable of concentration type. The neural network-based potentials ensure all physically desirable properties by construction. Most importantly, objectivity, material symmetry and thermodynamic consistency are automatically fulfilled. In addition, an alternative derivation of the governing model equations in time-discrete form is presented based on an incremental variational framework, which also serves as the basis for a finite element implementation. We demonstrate the predictive capability of the PANN using three different experimental data sets from literature, considering both stress and crystallinity evolution at material point level as well as the corresponding field distributions in a notched specimen. Moreover, we show that model parameterization is also possible when experimental crystallinity data is not available, still enabling suitable stress predictions.

We propose an unsupervised learning framework for calibrating a physics-augmented neural network (PANN) for small-strain viscoelasticity via full-field data. It only requires quantities that are directly accessible in real experiments for training, namely global reaction forces and surface displacements. The underlying PANN is embedded in the generalized standard materials theory, in which two scalar-valued potentials render the constitutive model thermodynamically consistent by construction, while invariant-based representations of the free energy and the dual dissipation potential additionally ensure material symmetry. Considering a thin specimen under the plane stress assumption, we formulate a constrained optimization problem based on the equilibrium gap method in combination with quasi-Newton optimizers and automatic differentiation. Thereby, the unknown out-of-plane strain follows from the plane stress condition and the evolution of the internal variables is captured by an implicit time integration scheme. The resulting system of nonlinear equations is solved via a local Newton iteration at quadrature point and time step. To drastically reduce the computational cost of training, the backward adjoint method is employed to compute the gradient of the target loss, instead of backpropagating through all Newton iteration steps. The proposed framework is demonstrated for synthetic data, including noisy displacements and forces, showing excellent agreement across a wide range of deformation rates and load paths.

Establishing structure-property linkages in polycrystalline materials requires representative two- (2D) and three- (3D) dimensional microstructural inputs for full-field simulations. A core objective of microstructure characterization and reconstruction is the generative synthesis of 2D and 3D microstructures that reflect a target statistical ensemble using limited 2D data as a reference. This work introduces an orientation-based differentiable microstructure characterization and reconstruction framework, implemented in MCRpy, to perform reconstructions of voxelized images. Unit quaternions in combination with symmetrized hyperspherical harmonics are utilized to derive a continuous, symmetry-invariant representation of crystallographic orientations to overcome the numerical singularities and discontinuities associated with traditional Euler-based methods. The descriptor-based reconstructions are driven by a set combining two-point spatial correlations, a novel hybrid three-point variogram, and a mean variation regularizer to capture both global texture and local interfacial topology. The framework's efficiency is demonstrated by reconstructing 3D realizations from 2D orientation data of an aluminum alloy after thermo-mechanical processing, successfully recovering both morphological features and crystallographic distribution. Systematic benchmarking indicates that second-order gradient-based optimization, utilizing the L-BFGS-B algorithm, effectively navigates the complex loss landscape to generate high-fidelity realizations with minimal residuals. This methodology provides a versatile, open-source framework for the digital synthesis of polycrystalline representative volume elements to facilitate the rapid development of microstructure-informed materials design workflows.

A key challenge in material theory is the formulation of models that satisfy all common mechanical constitutive conditions while retaining sufficient flexibility. In this context, several important modeling aspects remain unresolved for polyconvex anisotropic hyperelasticity. We address some of these challenges and apply our results for physics-augmented neural network (PANN) constitutive modeling. The main contributions of this paper are as follows: (1) We propose a new constitutive modeling framework for polyconvex anisotropic hyperelasticity based on triclinic invariants and group symmetrization, and apply it to a PANN model. For finite symmetry groups, this PANN model fulfills all common mechanical constitutive conditions a priori. (2) We propose a new group symmetrization-based method for the construction of polyconvex invariants for finite symmetry groups. Based on this, we derive a new integrity basis for a tetragonal symmetry group and a new functional basis for a cubic symmetry group. To the best of our knowledge, these are the first polyconvex integrity or functional bases for symmetry groups characterized by structural tensors of order higher than two. (3) We benchmark different polyconvex PANN constitutive models with highly nonlinear homogenization data of cubic metamaterials. (4) We provide an introduction to the construction of polyconvex invariants, integrity and functional bases and review established polyconvex bases alongside the newly derived ones.

A key challenge in material theory is the formulation of models that satisfy all common mechanical constitutive conditions while retaining sufficient flexibility. In this context, several important modeling aspects remain unresolved for polyconvex anisotropic hyperelasticity. We address some of these challenges and apply our results for physics-augmented neural network (PANN) constitutive modeling. The main contributions of this paper are as follows: (1) We propose a new polyconvex PANN constitutive model for anisotropic hyperelasticity based on triclinic invariants and group symmetrization. For finite symmetry groups, this model fulfills all common mechanical constitutive conditions a priori. (2) We propose a group symmetrization-based method for the construction of polyconvex invariants for finite symmetry groups. Based on this, we derive a new integrity basis for a tetragonal symmetry group and a new functional basis for a cubic symmetry group. To the best of our knowledge, these are the first polyconvex integrity or functional bases for symmetry groups characterized by structural tensors of order higher than two. (3) We provide an extensive introduction to the construction of polyconvex integrity and functional bases, which form the basis of polyconvex invariant-based constitutive models. We discuss polyconvex bases for triclinic, isotropic, transversely isotropic, monoclinic, rhombic, tetragonal, and cubic symmetry groups. (4) We benchmark the polyconvex PANN constitutive models with highly nonlinear homogenization data of cubic metamaterials.

We present minimal integrity bases for all common anisotropies in hyperelasticity via the structural tensor concept, which can be used to formulate any algebraic invariant function in the elements of the respective bases. Hence, the provided minimal integrity bases are of great interest for formulating a concise but general anisotropic material model. Our work covers results for the 11 types of anisotropy that arise from the classical 7 crystal systems, as well as findings for 4 additional non-crystal anisotropies derived from the cylindrical, spherical, and icosahedral symmetry systems. By using well-known results from literature about structural tensors, isotropic invariants, and isotropic extension, functional bases are directly determined. A simple analytical-numerical approach is employed to identify polynomial relations between the invariants of these functional bases, thereby enabling the construction of functional bases of reduced cardinality. After that, we show that the determined reduced functional bases are also minimal integrity bases by identifying polynomial relations to known integrity bases from literature. Furthermore, fundamental concepts from invariant theory, including the Hironaka decomposition of invariant rings and the closely related Hilbert series, are employed to further validate the results. Alongside the presented findings, this article also aims to provide an introductory overview of the complex field of modeling anisotropic materials, especially for researchers with an engineering background.

Thermoplastic composite materials have the potential to meet the high-performance demands of the automotive and aviation industries in terms of cost and cycle time due to their rapid consolidation and forming capabilities. To take advantage of the rapid forming capabilities of thermoplastic materials, the processes may include external heating of the semi-finished products, followed by the forming process, including the compaction and solidification in isothermal molding tools. In the forming process, the laminate is continuously cooled whereby a non-isothermal crystallization process occurs in semi-crystalline polymers, which governs the phase transition from liquid to solid. As semi-crystalline thermoplastics can only be formed above their recrystallization temperature, the solidification phase sets the limits for the processing window. In order to predict the manufacturing process limitations and boundaries, it is necessary to build up a holistic understanding of the solidification behavior. Therefore, this study aims to identify how temperature and crystallinity affect the formability of thermoplastic composites in industrial process conditions. By analyzing a CF/ PA6 tape material as well as its neat PA6 polymer using differential scanning calorimetry (DSC) and fast scanning calorimetry (FSC), the necessity and difficulty of scanning fiber-matrix composite samples is evaluated. Using a modified Nakamura-Ziabicki model, the measured relative degree of crystallinity (DoC) is fitted across a wide range of constant cooling rates for both the composite tape material and the neat matrix material. The presence of carbon fibers increases the crystallization growth rate, leading to faster crystallization kinetics at all measured cooling rates. The model is implemented into the commercial FE software Abaqus (R) using a HETVAL subroutine for numerical heat transfer simulations. To address the unsuitability of conventional mechanical methods for validating crystallization kinetics at the high cooling rates typical of industrial processes, a novel approach utilizing squeezing flow during compaction in a stamp forming experiment was developed to to validate the numerical simulations. The influence of various processing parameters on the forming process is studied by measuring the internal transient temperature of the laminate as well as the displacement of the testing machine during compaction. The analysis concludes that the compaction limit of the stamp-formed specimen due to recrystallization aligns well with both the measured local maximum in cooling rates and the numerically predicted DoC of 50 %. Consequently, implementing the modified Nakamura-Ziabicki model in the process simulation enables accurate prediction of crystallization kinetics. This enables virtual process studies to be integrated into the design phase, facilitating direct feedback into the composite mold development and resource-efficient process optimization.

Tailoring materials to achieve a desired behavior in specific applications is of significant scientific and industrial interest as design of materials is a key driver to innovation. Overcoming the rather slow and expertise-bound traditional forward approaches of trial and error, inverse design is attracting substantial attention. Targeting a property, the design model proposes a candidate structure with the desired property. This concept can be particularly well applied to the field of architected materials as their structures can be directly tuned. The bone-like spinodoid materials are a specific class of architected materials. They are of considerable interest thanks to their non-periodicity, smoothness, and low-dimensional statistical description. Previous work successfully employed machine learning (ML) models for inverse design. The amount of data necessary for most ML approaches poses a severe obstacle for broader application, especially in the context of inelasticity. That is why we propose an inverse-design approach based on Bayesian optimization to operate in the small-data regime. Necessitating substantially less data, a small initial data set is iteratively augmented by in silico generated data until a structure with the targeted properties is found. The application to the inverse design of spinodoid structures of desired elastic properties demonstrates the framework's potential for paving the way for advance in inverse design.

Virtual sensing methods provide estimates for difficult-to-obtain quantities from readily available sensor data. As a result, they enable economic fatigue monitoring for vehicles in fleet scenarios. This work provides an experimental datasets designed for the parameterization and evaluation of virtual sensing strategies. It consists of acceleration and strain measurement data from a sensor-equipped eBike and includes both regular usage data and specific maneuver measurements. Using these datasets, two data-driven virtual sensing strategies are parameterized. The sensor estimation approach approximates strain time series from acceleration data, which are subsequently used for fatigue analysis. In the reduced representation approach, a low-dimensional representation of acceleration measurements is created through a combination of feature extraction and dimensionality reduction, allowing for a direct fatigue damage regression. A comparison highlights the respective advantages and limitations of the presented modeling strategies.

Forming-induced residual stresses highly influence the performance of metallic engineering components. They offer great potential particularly for increasing fatigue life by targeted introduction of compressive residual stresses in failure-critical areas. However, this only holds true if one can understand and predict the change of residual stresses under cyclic mechanical loading and thus ensure their stability. In the present paper, we introduce a combined experimental and numerical approach for the investigation of residual stress evolution under cyclic mechanical loading. Therefore, a suitable experiment is conceptualized and realized using a 4-point bending setup. The initial plastic deformation of each specimen is followed by a certain number of load cycles and experimental residual stress analyses. From this, a course of residual stresses over the fatigue life is constructed. In order to simulate the determined change in residual stresses, a cyclic plasticity model is proposed that takes into account the nonlinear kinematics due to the large deflection of the beam. A parametrization algorithm is presented, which employs a global optimization strategy using uniaxial stress–strain data from various parametrization experiments. The final comparison of experimental and numerical results shows a qualitative agreement. Their stabilization level after a few thousand load cycles can be predicted.

The manufacturing of polyamide 12 (PA12) components by HP’s multi-jet fusion (MJF ™) technology is recently attracting more attention thanks to less residual stress accumulation and very good accuracy in terms of geometry and surface quality. In the present work, four build jobs are produced with 12 samples each, yielding 48 samples in total. In each job, the orientation of the specimens is varied parallel to the build plate, adopting two different specimen orientations. Within each build job the specimens are laterally distributed and arranged at two distinct height levels. Performing tensile tests at two different strain rates, material parameters are derived and analyzed using the Gaussian mixture clustering algorithm and a principal component analysis (PCA) for an analysis of the results. This approach demonstrates the potential of applying unsupervised learning methods to investigate process-property linkages. The Gaussian mixture clustering algorithm is run with two, four, and six clusters, respectively, in the same data set to identify four as a sensible number of clusters for subsequent analysis. The clustering algorithm is able to conclusively link the size of the cross section of the manufactured specimens to their respective positions on the build platform. For other material parameters, such as the Young’s modulus, the elongation at tensile strength, and the yield stress, no clustering with respect to the position in the build chamber is recognizable. The PCA yields continuous information regarding the variation of material parameters with respect to the position of the specimen on the build platform. Based on the PCA results, a different behavior of specimens in the two investigated orientations is identifiable.

For the formulation of machine learning-based material models, the usage of invariants of deformation tensors is attractive, since this can a priori guarantee objectivity and material symmetry. In this work, we consider incompressible, isotropic hyperelasticity, where two invariants I1 and I2 are required for depicting a deformation state. First, we aim at enhancing the understanding of the invariants. We provide an explicit representation of the set of invariants that are admissible, i.e. for which (I1, I2) a deformation state does indeed exist. Furthermore, we prove that uniaxial and equi-biaxial deformation correspond to the boundary of the set of admissible invariants. Second, we study how the experimentally-observed behaviour of different materials can be captured by means of neural network models of incompressible hyperelasticity, depending on whether both I1 and I2 or solely one of the invariants, i.e. either only I1 or only I2, are taken into account. To this end, we investigate three different experimental data sets from the literature. In particular, we demonstrate that considering only one invariant, either I1 or I2, can allow for good agreement with experiments in case of small deformations. In contrast, it is necessary to consider both invariants for precise models at large strains, for instance when rubbery polymers are deformed. Moreover, we show that multiaxial experiments are strictly required for the parameterisation of models considering I2. Otherwise, if only data from uniaxial deformation is available, significantly overly stiff responses could be predicted for general deformation states. On the contrary, I1-only models can make qualitatively correct predictions for multiaxial loadings even if parameterised only from uniaxial data, whereas I2-only models are completely incapable in even qualitatively capturing experimental stress data at large deformations.

We create an data-efficient and accurate surrogate model for structure-property linkages of spinodoid metamaterials with only 75 data points – far fewer than the several thousands used in prior works – and demonstrate its use in multi-objective inverse design. The inverse problem of finding a material microstructure that leads to given bulk properties is of great interest in mechanics and materials science. These inverse design tasks often require a large dataset, which can become unaffordable when considering material behavior that requires more expensive simulations or experiments. We generate a data-efficient surrogate for the mapping between the characteristics of the local material structure and the effective elasticity tensor and use it to inversely design structures with multiple objectives simultaneously. The presented neural network-based surrogate model achieves its data efficiency by inherently satisfying certain requirements, such as equivariance with respect to permutations of structure parameters, which avoids having to learn them from data. The resulting surrogate of the forward model is differentiable, allowing its direct use in gradient-based optimization for the inverse design problem. We demonstrate in three inverse design tasks of varying complexity that this approach yields reliable results while requiring significantly less training data than previous approaches based on neural-network surrogates. This paves the way for inverse design involving nonlinear mechanical behavior, where data efficiency is currently the limiting factor.

We present a hybrid phase-field model of fracture at finite deformation and its application to quasi-incompressible, hyperelastic rubber. The key idea is to combine the predictive capability of the well-established phase-field approach to fracture with a physics-augmented neural network (PANN) that serves as a flexible, high-fidelity model of the response of the bulk material. To this end, recently developed neural network approaches are developed further to better meet specific requirements of the phase-field framework. In particular, a novel architecture for a hyperelastic PANN is presented, that enables a decoupled description of the volumetric and the isochoric response based on a corresponding additive decomposition of the Helmholtz free energy. This is of particular interest when modelling fracture of soft quasi-incompressible solids with the phase-field approach, where a weakening of the incompressibility constraint in fracturing material is required. In addition, such an additive decomposition of the free energy is a prerequisite for the application of several split methods, i.e. decompositions of free energy into degraded and non-degraded portions, which can improve model behaviour under multiaxial stress states. For the formulation of the hybrid model, we define a pseudo-potential, in which the phase-field ansatz for fracture dissipation is combined with a polyconvex PANN model of the isochoric response. The PANN is formulated in polyconvex isochoric invariants. As a result, it can be shown that the PANN fulfils all desirable properties of hyperelastic potentials by construction. In particular, it is proven to be zero and take a global minimum for the undeformed state, which does also hold in case of deviations away from incompressibility. Moreover, a classical mixed displacement-pressure formulation of incompressibility based on the perturbed Lagrangian approach is included. Thereby, a relaxation of the incompressibility constraint in fracturing material is applied. This weakening of incompressibility is shown in to be essential in order to prevent numerical issues in the simulation, which would otherwise arise from the presence of zones showing both negligible isochoric stiffness and very high resistance against volume changes. The model is implemented in the finite element framework FEniCSx and studied by means of several examples. To this end, training and validation of the PANN are performed based on experimental data from the literature, and the hybrid fracture model is then verified against results of fracture experiments.

Solid oxide fuel cells (SOFCs) are a promising technology for producing clean energy in the future. To enhance the efficiency and the lifetime of SOFCs and SOFC stacks, novel materials and innovative material compositions for the constituent layers-namely the cathode, anode, and electrolyte-must be developed. Since the electrodes are of tremendous importance for an efficient and long-term operation, computational methods are required to enable and accelerate the development process. In this work, the microstructure of an SOFC anode is characterised by certain geometrical descriptors, such as the volume fraction, the two-point correlation function, the lineal-path function, the tortuosity, the percolation of each constituent, and the triple phase boundary (TPB). In addition, the effective physical properties are determined by the first-order computational homogenisation method to enable the usage of real anisotropic material parameters in macroscopic fuel cell simulations. Special attention is paid to the mechanical behaviour, including secondary creep. A general framework for creating structure-property relationships for fuel cell electrodes is established. Since it is unfeasible to build large datasets based on experimental tomography images, the microstructure characterisation and reconstruction tool MCRpy is used to generate microstructures based on the geometrical descriptors (i) of the whole three-dimensional tomography image and (ii) of only three two-dimensional slices. A two-step approach is employed with a differentiable microstructure characterisation and reconstruction (DMCR) algorithm and the classical Yeong-Torquato algorithm. The results show a very good agreement of the geometrical and the physical properties between the original tomography image and the reconstructed microstructures. However, to decrease the small differences in the properties, it is necessary to include further geometrical descriptors into the reconstruction process in further research. The created framework can support the development of optimised electrode microstructures using inverse design methods.

We present a data-efficient neural-network model for predicting linear-elastic properties of spinodoid metamaterials from their mesoscale structure. Our machine-learning model requires 75 data points for training, greatly improving data efficiency over previous models that required thousands of training samples. We achieve this by leveraging concepts from geometric learning. Specifically, we exploit physical properties, such as positive semi-definiteness of the elasticity tensor, as well as structural invariances and equivariances of the problem, for example with respect to coordinate axes permutations. The neural network model is designed to exactly fulfill these constraints; it does not have to learn them from data. The resulting model enables data- and compute-efficient inverse design of spinodoid metamaterials. In inverse design, the goal is to find a material mesostructure that leads to desired mechanical properties on the macroscale. Exactly fulfilling physical and structural constraints, the present neural network model remains differentiable. This allows using fast gradient-based optimizers for inverse design. We demonstrate this by inversely designing spinodoid metamaterials that achieve desired linear elastic target properties in three dimensions. Inverse design is treated as a constrained optimization problem over the parameters describing the metamaterial. The results confirm that the present approach requires significantly less training data than previous machine-learning approaches and allows incorporating multiple objectives in the inverse design process. Since the structure of the design space is independent of the target material properties, we hope that such data-efficient models will be useful also for inverse design of spinodoids beyond linear elasticity.