arXiv subjects

Oliver Weeger

Publications and source records attributed to Oliver Weeger.

At least 19 recordsLinked to original sources

arXiv2026

On limitations of polyconvexity

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.

cs.CE↗

arXiv2026

Advances in polyconvex anisotropic hyperelasticity

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.

cs.CE↗

arXiv2026

The cross-sectional warping problem for hyperelastic beams: An efficient formulation in Voigt notation

Beam theory has traditionally been restricted to small elastic strains and rigid cross-sections. Relaxing these assumptions within closed-form analytical frameworks remains challenging. In contrast, the cross-sectional warping problem provides a computational approach that enables the derivation of general, nonlinear constitutive relations for beam models, thereby overcoming both limitations. In this work, we reinterpret the cross-sectional warping problem for hyperelastic beams and propose a fully material formulation in terms of the Green-Lagrange strain and the second Piola-Kirchhoff stress tensors. Owing to the symmetry of these tensors, the formulation can be expressed efficiently in Voigt notation and is thus particularly well-suited for straightforward numerical implementation. We demonstrate the validity of this alternative formulation in numerical examples, including the computation of the effective beam stiffness, for which we derive the sensitivities of the warping displacement. To promote reproducibility, we accompany this article with an open-access repository containing the isogeometric finite element implementation and all numerical examples presented herein, enabling other researchers to readily reproduce and build upon our results.

cs.CE↗

arXiv2026

Immersed boundary-conformal isogeometric methods for magnetostatics

Isogeometric analysis was proposed to bridge the gap between computer-aided design and numerical discretization. However, standard multi-patch isogeometric analysis mandates conformal discretizations across patch interfaces, posing challenges for multi-material domain problems. In the context of electric machines, this requirement becomes evident in a large number of patches needed to represent machines consisting of several domains and materials. In this work, we adopt, extend, and evaluate three non-conformal discretization strategies for magnetostatic problems: a fully immersed approach, the union with non-conformal patches, and the union with conformal layers. In all three methods, boundary-conformal high-order quadrature rules are employed for integration over trimmed boundary and interface elements. In the two union approaches, material regions are, as far as possible, represented by independent non-conformal spline patches that are embedded within a background patch and coupled weakly through Nitsche's method. In the latter framework, critical interfaces are additionally surrounded by conformal layers that enable the strong imposition of boundary conditions and improved resolution of interface behavior. The proposed approaches are assessed through several magnetostatic benchmark problems and an industrial application. The numerical results show that the union methods achieve highly accurate solutions, while the fully immersed approach struggles with discontinuities in field gradients across material interfaces. Nevertheless, these methods significantly reduce the geometric preprocessing effort compared to conventional, conformal multi-patch analysis and require substantially fewer patches. Overall, this demonstrates that our immersed boundary-conformal isogeometric framework possesses great potential for efficient simulation of complex electromagnetic devices.

cs.CE↗

arXiv2025

Machine Learning with Physics Knowledge for Prediction: A Survey

This survey examines the broad suite of methods and models for combining machine learning with physics knowledge for prediction and forecast, with a focus on partial differential equations. These methods have attracted significant interest due to their potential impact on advancing scientific research and industrial practices by improving predictive models with small- or large-scale datasets and expressive predictive models with useful inductive biases. The survey has two parts. The first considers incorporating physics knowledge on an architectural level through objective functions, structured predictive models, and data augmentation. The second considers data as physics knowledge, which motivates looking at multi-task, meta, and contextual learning as an alternative approach to incorporating physics knowledge in a data-driven fashion. Finally, we also provide an industrial perspective on the application of these methods and a survey of the open-source ecosystem for physics-informed machine learning.

cs.LG↗

arXiv2025

Physics-based Machine Learning for Computational Fracture Mechanics

This study introduces a physics-based machine learning framework for modeling both brittle and ductile fractures. Unlike physics-informed neural networks, which solve partial differential equations by embedding physical laws as soft constraints in loss functions and enforcing boundary conditions via collocation points, our framework integrates physical principles, such as the governing equations and constraints, directly into the neural network architecture. This approach eliminates the dependency on problem-specific retraining for new boundary value problems, ensuring adaptability and consistency. By embedding constitutive behavior into the network's foundational design, our method represents a significant step toward unifying material modeling with machine learning for computational fracture mechanics. Specifically, a feedforward neural network is designed to embed physical laws within its architecture, ensuring thermodynamic consistency. Building on this foundation, synthetic datasets generated from finite element-based phase-field simulations are employed to train the proposed framework, focusing on capturing the homogeneous responses of brittle and ductile fractures. Detailed analyses are performed on the stored elastic energy and the dissipated work due to plasticity and fracture, demonstrating the capability of the framework to predict essential fracture features. The proposed physics-based machine learning framework overcomes the shortcomings of classical machine learning models, which rely heavily on large datasets and lack guarantees of physical principles. By leveraging its physics-integrated design, the physics-based machine learning framework demonstrates exceptional performance in predicting key properties of brittle and ductile fractures with limited training data.

math.NA↗

arXiv2025

Stable Port-Hamiltonian Neural Networks

In recent years, nonlinear dynamic system identification using artificial neural networks has garnered attention due to its broad potential applications across science and engineering. However, purely data-driven approaches often struggle with extrapolation and may yield physically implausible forecasts. Furthermore, the learned dynamics can exhibit instabilities, making it difficult to apply such models safely and robustly. This article introduces stable port-Hamiltonian neural networks, a machine learning architecture that incorporates physical biases of energy conservation and dissipation while ensuring global Lyapunov stability of the learned dynamics. Through illustrative and real-world examples, we demonstrate that these strong inductive biases facilitate robust learning of stable dynamics from sparse data, while avoiding instability and surpassing purely data-driven approaches in accuracy and physically meaningful generalization. Furthermore, the model's applicability and potential for data-driven surrogate modeling are showcased on multi-physics simulation data.

cs.LG↗

arXiv2024

Physics-augmented neural networks for constitutive modeling of hyperelastic geometrically exact beams

We present neural network-based constitutive models for hyperelastic geometrically exact beams. The proposed models are physics-augmented, i.e., formulated to fulfill important mechanical conditions by construction, which improves accuracy and generalization. Strains and curvatures of the beam are used as input for feed-forward neural networks that represent the effective hyperelastic beam potential. Forces and moments are received as the gradients of the beam potential, ensuring thermodynamic consistency. Normalization conditions are considered via additional projection terms. Symmetry conditions are implemented by an invariant-based approach for transverse isotropy and a more flexible point symmetry constraint, which is included in transverse isotropy but poses fewer restrictions on the constitutive response. Furthermore, a data augmentation approach is proposed to improve the scaling behavior of the models for varying cross-section radii. Additionally, we introduce a parameterization with a scalar parameter to represent ring-shaped cross-sections with different ratios between the inner and outer radii. Formulating the beam potential as a neural network provides a highly flexible model. This enables efficient constitutive surrogate modeling for geometrically exact beams with nonlinear material behavior and cross-sectional deformation, which otherwise would require computationally much more expensive methods. The models are calibrated and tested with data generated for beams with circular and ring-shaped hyperelastic deformable cross-sections at varying inner and outer radii, showing excellent accuracy and generalization. The applicability of the proposed point symmetric model is further demonstrated by applying it in beam simulations. In all studied cases, the proposed model shows excellent performance.

cs.CE↗

arXiv2024

Digital twin inference from multi-physical simulation data of DED additive manufacturing processes with neural ODEs

A digital twin is a virtual representation that accurately replicates its physical counterpart, fostering bi-directional real-time data exchange throughout the entire process lifecycle. For Laser Directed Energy Deposition of Wire (DED-LB/w) additive manufacturing processes, digital twins may help to control the residual stress design in build parts. This study focuses on providing faster-than-real-time and highly accurate surrogate models for the formation of residual stresses by employing neural ordinary differential equations. The approach enables accurate prediction of temperatures and altered structural properties like stress tensor components. The developed surrogates can ultimately facilitate on-the-fly re-optimization of the ongoing manufacturing process to achieve desired structural outcomes. Consequently, this building block contributes significantly to realizing digital twins and the first-time-right paradigm in additive manufacturing.

cs.CE↗

arXiv2024

Multiscale topology optimization of functionally graded lattice structures based on physics-augmented neural network material models

We present a new framework for the simultaneous optimiziation of both the topology as well as the relative density grading of cellular structures and materials, also known as lattices. Due to manufacturing constraints, the optimization problem falls into the class of NP-complete mixed-integer nonlinear programming problems. To tackle this difficulty, we obtain a relaxed problem from a multiplicative split of the relative density and a penalization approach. The sensitivities of the objective function are derived such that any gradient-based solver might be applied for the iterative update of the design variables. In a next step, we introduce a material model that is parametric in the design variables of interest and suitable to describe the isotropic deformation behavior of quasi-stochastic lattices. For that, we derive and implement further physical constraints and enhance a physics-augmented neural network from the literature that was formulated initially for rhombic materials. Finally, to illustrate the applicability of the method, we incorporate the material model into our computational framework and exemplary optimize two-and three-dimensional benchmark structures as well as a complex aircraft component.

cs.CE↗

arXiv2024

TwinLab: a framework for data-efficient training of non-intrusive reduced-order models for digital twins

Purpose: Simulation-based digital twins represent an effort to provide high-accuracy real-time insights into operational physical processes. However, the computation time of many multi-physical simulation models is far from real-time. It might even exceed sensible time frames to produce sufficient data for training data-driven reduced-order models. This study presents TwinLab, a framework for data-efficient, yet accurate training of neural-ODE type reduced-order models with only two data sets. Design/methodology/approach: Correlations between test errors of reduced-order models and distinct features of corresponding training data are investigated. Having found the single best data sets for training, a second data set is sought with the help of similarity and error measures to enrich the training process effectively. Findings: Adding a suitable second training data set in the training process reduces the test error by up to 49% compared to the best base reduced-order model trained only with one data set. Such a second training data set should at least yield a good reduced-order model on its own and exhibit higher levels of dissimilarity to the base training data set regarding the respective excitation signal. Moreover, the base reduced-order model should have elevated test errors on the second data set. The relative error of the time series ranges from 0.18% to 0.49%. Prediction speed-ups of up to a factor of 36,000 are observed. Originality: The proposed computational framework facilitates the automated, data-efficient extraction of non-intrusive reduced-order models for digital twins from existing simulation models, independent of the simulation software.

cs.CE↗

arXiv2024

Polyconvex neural network models of thermoelasticity

Machine-learning function representations such as neural networks have proven to be excellent constructs for constitutive modeling due to their flexibility to represent highly nonlinear data and their ability to incorporate constitutive constraints, which also allows them to generalize well to unseen data. In this work, we extend a polyconvex hyperelastic neural network framework to thermo-hyperelasticity by specifying the thermodynamic and material theoretic requirements for an expansion of the Helmholtz free energy expressed in terms of deformation invariants and temperature. Different formulations which a priori ensure polyconvexity with respect to deformation and concavity with respect to temperature are proposed and discussed. The physics-augmented neural networks are furthermore calibrated with a recently proposed sparsification algorithm that not only aims to fit the training data but also penalizes the number of active parameters, which prevents overfitting in the low data regime and promotes generalization. The performance of the proposed framework is demonstrated on synthetic data, which illustrate the expected thermomechanical phenomena, and existing temperature-dependent uniaxial tension and tension-torsion experimental datasets.

cond-mat.soft↗

arXiv2024

Nonlinear electro-elastic finite element analysis with neural network constitutive models

In the present work, the applicability of physics-augmented neural network (PANN) constitutive models for complex electro-elastic finite element analysis is demonstrated. For the investigations, PANN models for electro-elastic material behavior at finite deformations are calibrated to different synthetically generated datasets, including an analytical isotropic potential, a homogenised rank-one laminate, and a homogenised metamaterial with a spherical inclusion. Subsequently, boundary value problems inspired by engineering applications of composite electro-elastic materials are considered. Scenarios with large electrically induced deformations and instabilities are particularly challenging and thus necessitate extensive investigations of the PANN constitutive models in the context of finite element analyses. First of all, an excellent prediction quality of the model is required for very general load cases occurring in the simulation. Furthermore, simulation of large deformations and instabilities poses challenges on the stability of the numerical solver, which is closely related to the constitutive model. In all cases studied, the PANN models yield excellent prediction qualities and a stable numerical behavior even in highly nonlinear scenarios. This can be traced back to the PANN models excellent performance in learning both the first and second derivatives of the ground truth electro-elastic potentials, even though it is only calibrated on the first derivatives. Overall, this work demonstrates the applicability of PANN constitutive models for the efficient and robust simulation of engineering applications of composite electro-elastic materials.

cs.CE↗

arXiv2023

Magneto-active composites with locally tailored stiffness produced by laser powder bed fusion

Additive manufacturing technologies enable the production of complex and bioinspired shapes using magneto-responsive materials, which find diverse applications in soft robotics. Particularly, the development of composites with controlled gradients in mechanical properties offers new prospects for advancements in magneto-active materials. However, achieving such composites with gradients typically involves complex multi-material printing procedures. In this study, a single-step laser powder bed fusion (LPBF) process is proposed that enables precise local adjustments of the mechanical stiffness within magneto-active composites. By utilizing distinct laser parameters in specific regions of a composite containing thermoplastic polyurethane and atomized magnetic powder derived from hard magnetic Nd-Fe-B, the stiffness of the composite can be modified within the range of 2 to 22 MPa. Various magneto-responsive actuators with locally tailored stiffness are fabricated and their magnetic performance is investigated. The enhanced response exhibited by actuators with locally adjusted mechanical properties in comparison to their homogeneous counterparts with identical geometries is shown. As a demonstration of a biomedical application, a magnetically responsive stent with localized adjustment is presented with the ability to meet specific requirements in terms of geometry and local stiffness based on an individual's anatomy and disease condition. The proposed method presents an approach for creating functionally graded materials using LPBF, not only for magneto-active materials but also for several other structural and functional materials.

physics.app-ph↗

arXiv2023

A robust finite strain isogeometric solid-beam element

In this work, an efficient and robust isogeometric three-dimensional solid-beam finite element is developed for large deformations and finite rotations with merely displacements as degrees of freedom. The finite strain theory and hyperelastic constitutive models are considered and B-Spline and NURBS are employed for the finite element discretization. Similar to finite elements based on Lagrange polynomials, also NURBS-based formulations are affected by the non-physical phenomena of locking, which constrains the field variables and negatively impacts the solution accuracy and deteriorates convergence behavior. To avoid this problem within the context of a Solid-Beam formulation, the Assumed Natural Strain (ANS) method is applied to alleviate membrane and transversal shear locking and the Enhanced Assumed Strain (EAS) method against Poisson thickness locking. Furthermore, the Mixed Integration Point (MIP) method is employed to make the formulation more efficient and robust. The proposed novel isogeometric solid-beam element is tested on several single-patch and multi-patch benchmark problems, and it is validated against classical solid finite elements and isoparametric solid-beam elements. The results show that the proposed formulation can alleviate the locking effects and significantly improve the performance of the isogeometric solid-beam element. With the developed element, efficient and accurate predictions of mechanical properties of lattice-based structured materials can be achieved. The proposed solid-beam element inherits both the merits of solid elements e.g. flexible boundary conditions and of the beam elements i.e. higher computational efficiency.

math.NA↗

arXiv2023

Parametrised polyconvex hyperelasticity with physics-augmented neural networks

In the present work, neural networks are applied to formulate parametrised hyperelastic constitutive models. The models fulfill all common mechanical conditions of hyperelasticity by construction. In particular, partially input-convex neural network (pICNN) architectures are applied based on feed-forward neural networks. Receiving two different sets of input arguments, pICNNs are convex in one of them, while for the other, they represent arbitrary relationships which are not necessarily convex. In this way, the model can fulfill convexity conditions stemming from mechanical considerations without being too restrictive on the functional relationship in additional parameters, which may not necessarily be convex. Two different models are introduced, where one can represent arbitrary functional relationships in the additional parameters, while the other is monotonic in the additional parameters. As a first proof of concept, the model is calibrated to data generated with two differently parametrised analytical potentials, whereby three different pICNN architectures are investigated. In all cases, the proposed model shows excellent performance.

cs.CE↗

arXiv2023

Neural networks meet hyperelasticity: A guide to enforcing physics

In the present work, a hyperelastic constitutive model based on neural networks is proposed which fulfills all common constitutive conditions by construction, and in particular, is applicable to compressible material behavior. Using different sets of invariants as inputs, a hyperelastic potential is formulated as a convex neural network, thus fulfilling symmetry of the stress tensor, objectivity, material symmetry, polyconvexity, and thermodynamic consistency. In addition, a physically sensible stress behavior of the model is ensured by using analytical growth terms, as well as normalization terms which ensure the undeformed state to be stress free and with zero energy. In particular, polyconvex, invariant-based stress normalization terms are formulated for both isotropic and transversely isotropic material behavior. By fulfilling all of these conditions in an exact way, the proposed physics-augmented model combines a sound mechanical basis with the extraordinary flexibility that neural networks offer. Thus, it harmonizes the theory of hyperelasticity developed in the last decades with the up-to-date techniques of machine learning. Furthermore, the non-negativity of the hyperelastic neural network-based potentials is numerically examined by sampling the space of admissible deformations states, which, to the best of the authors' knowledge, is the only possibility for the considered nonlinear compressible models. For the isotropic neural network model, the sampling space required for that is reduced by analytical considerations. In addition, a proof for the non-negativity of the compressible Neo-Hooke potential is presented. The applicability of the model is demonstrated by calibrating it on data generated with analytical potentials, which is followed by an application of the model to finite element simulations. In addition, an adaption of the model to noisy data is shown and its [...]

cs.CE↗

arXiv2023

Advanced discretization techniques for hyperelastic physics-augmented neural networks

In the present work, advanced spatial and temporal discretization techniques are tailored to hyperelastic physics-augmented neural networks, i.e., neural network based constitutive models which fulfill all relevant mechanical conditions of hyperelasticity by construction. The framework takes into account the structure of neural network-based constitutive models, in particular, that their derivatives are more complex compared to analytical models. The proposed framework allows for convenient mixed Hu-Washizu like finite element formulations applicable to nearly incompressible material behavior. The key feature of this work is a tailored energy-momentum scheme for time discretization, which allows for energy and momentum preserving dynamical simulations. Both the mixed formulation and the energy-momentum discretization are applied in finite element analysis. For this, a hyperelastic physics-augmented neural network model is calibrated to data generated with an analytical potential. In all finite element simulations, the proposed discretization techniques show excellent performance. All of this demonstrates that, from a formal point of view, neural networks are essentially mathematical functions. As such, they can be applied in numerical methods as straightforwardly as analytical constitutive models. Nevertheless, their special structure suggests to tailor advanced discretization methods, to arrive at compact mathematical formulations and convenient implementations.

cs.CE↗