arXiv subjects

Stefan Hartmann

Publications and source records attributed to Stefan Hartmann.

11 recordsLinked to original sources

arXiv2026

Boundary-Level-Constrained Refinement for Suppressing Trimming-Induced High-Frequency Outliers in Explicit Isogeometric Analysis

In explicit dynamics, the critical time step is governed by the maximum eigenfrequency of the semi-discrete system. In Isogeometric Analysis (IGA), open knot vectors introduce a characteristic spectral boundary effect that can lead to high-frequency outliers and restrict the admissible time step. In the row-sum-lumped setting considered here, extending the computational patch and trimming away the exterior boundary functions mitigates this time-step penalty, but does not necessarily prevent boundary-adjacent basis functions from governing the maximum eigenfrequency. We show that reduced-support basis functions adjacent to the trimming boundary can remain critical spectral contributors even for knot-exact trimming, i.e., in the absence of small cut cells. Hence, small cut cells are not required for trimming-induced time-step penalties. Knot-exact and arbitrary trimming represent different degrees of support reduction within the same underlying mechanism. To control this effect, we propose the Boundary-Level-Constrained Refinement (BLCR) strategy, a local refinement constraint for LR- and THB-splines. BLCR constrains the refinement level of basis functions whose support intersects the trimming boundary relative to that of the refined interior. In all configurations investigated in this work, this constraint ensures that the maximum eigenfrequency is governed by untrimmed interior basis functions rather than by refined trimmed functions. Consequently, the trimming-induced high-frequency outliers governing the critical time step are suppressed, yielding a larger admissible time step than globally refined trimmed B-splines with the same interior resolution.[...]

math.NA↗

arXiv2026

Projection-based coupling of infrared thermography and stereocorrelation-based digital image correlation

Full-field measurement techniques such as digital image correlation and infrared thermography are prevalent in experimental solid mechanics. Digital image correlation is used to analyze surface deformation, while infrared thermography quantifies surface temperature fields. However, sophisticated procedures are necessary to express both datasets in the same Lagrangian frame, especially when analyzing non-flat surfaces. In this study, we propose an external projection-based coupling that uses the pinhole camera model to relate two-dimensional temperature data measured by infrared thermography to three-dimensional point coordinates from stereocorrelation-based digital image correlation. Unlike existing multiview approaches, we utilize two independently calibrated industrial-grade systems and augment the experimental evaluation with the pinhole camera model. The projection matrix of the camera model is calibrated using a single image of a reference object. Through this projection, temperature fields are accurately represented at material points. Our method is particularly suited for, but not restricted to, curved surfaces and straightforward to embed in existing experimental protocols, as the image registration is kept as is. Additionally, we propose using radial basis functions as a global interpolation ansatz in both space and time to compute in-plane temperature gradients and even temperature rates on curved surfaces, thereby providing an extensive and information-rich full-field dataset.

cs.CV↗

arXiv2026

Uncertainty Quantification in Calibration and Simulation of Thermo-Chemical Curing of Epoxy Resins

Curing of epoxy resins poses a particular challenge in terms of modeling, experimental investigation, and numerical implementation, as it is a thermo-chemo-mechanical process. Several constitutive relations are required to model these processes, yielding numerous material parameters. The calibration of the constitutive relations must be performed using multiple steps, wherein uncertainties unavoidably propagate. In this study, we investigate the propagation of uncertainties during both the multi-step calibration procedure and the numerical simulation of curing processes with the identified parameters. For both, we employ the first-order second-moment method, which is carefully evaluated through coverage tests and by comparing it to the Monte Carlo method as a reference. It is demonstrated that the first-order second-moment method efficiently yields reasonable results, although providing only a first-order approximation of the highly nonlinear stochastic model response.

cs.CE↗

arXiv2025

A Neural Operator based Hybrid Microscale Model for Multiscale Simulation of Rate-Dependent Materials

The behavior of materials is influenced by a wide range of phenomena occurring across various time and length scales. To better understand the impact of microstructure on macroscopic response, multiscale modeling strategies are essential. Numerical methods, such as the $\text{FE}^2$ approach, account for micro-macro interactions to predict the global response in a concurrent manner. However, these methods are computationally intensive due to the repeated evaluations of the microscale. This challenge has led to the integration of deep learning techniques into computational homogenization frameworks to accelerate multiscale simulations. In this work, we employ neural operators to predict the microscale physics, resulting in a hybrid model that combines data-driven and physics-based approaches. This allows for physics-guided learning and provides flexibility for different materials and spatial discretizations. We apply this method to time-dependent solid mechanics problems involving viscoelastic material behavior, where the state is represented by internal variables only at the microscale. The constitutive relations of the microscale are incorporated into the model architecture and the internal variables are computed based on established physical principles. The results for homogenized stresses ($<6\%$ error) show that the approach is computationally efficient ($\sim 100 \times$ faster).

physics.comp-ph↗

arXiv2025

EquiNO: A Physics-Informed Neural Operator for Multiscale Simulations

Multiscale problems are ubiquitous in physics. Numerical simulations of such problems by solving partial differential equations (PDEs) at high resolution are computationally too expensive for many-query scenarios, such as uncertainty quantification, remeshing applications, and topology optimization. This limitation has motivated the development of data-driven surrogate models, where microscale computations are substituted by black-box mappings between macroscale quantities. While these approaches offer significant speedups, they typically struggle to incorporate microscale physical constraints, such as the balance of linear momentum. In this contribution, we propose the Equilibrium Neural Operator (EquiNO), a physics-informed PDE surrogate in which equilibrium is hard-enforced by construction. EquiNO achieves this by projecting the solution onto a set of divergence-free basis functions obtained via proper orthogonal decomposition (POD), thereby ensuring satisfaction of equilibrium without relying on penalty terms or multi-objective loss functions. We compare EquiNO with variational physics-informed neural and operator networks that enforce physical constraints only weakly through the loss function, as well as with purely data-driven operator-learning baselines. Our framework, applicable to multiscale FE$^{\,2}$ computations, introduces a finite element-operator learning (FE-OL) approach that integrates the finite element (FE) method with operator learning (OL). We apply the proposed methodology to quasi-static problems in solid mechanics and demonstrate that FE-OL yields accurate solutions even when trained on restricted datasets. The results show that EquiNO achieves speedup factors exceeding 8000-fold compared to traditional methods and offers a robust and physically consistent alternative to existing data-driven surrogate models.

physics.comp-ph↗

arXiv2024

Deterministic and statistical calibration of constitutive models from full-field data with parametric physics-informed neural networks

The calibration of constitutive models from full-field data has recently gained increasing interest due to improvements in full-field measurement capabilities. In addition to the experimental characterization of novel materials, continuous structural health monitoring is another application that is of great interest. However, monitoring is usually associated with severe time constraints, difficult to meet with standard numerical approaches. Therefore, parametric physics-informed neural networks (PINNs) for constitutive model calibration from full-field displacement data are investigated. In an offline stage, a parametric PINN can be trained to learn a parameterized solution of the underlying partial differential equation. In the subsequent online stage, the parametric PINN then acts as a surrogate for the parameters-to-state map in calibration. We test the proposed approach for the deterministic least-squares calibration of a linear elastic as well as a hyperelastic constitutive model from noisy synthetic displacement data. We further carry out Markov chain Monte Carlo-based Bayesian inference to quantify the uncertainty. A proper statistical evaluation of the results underlines the high accuracy of the deterministic calibration and that the estimated uncertainty is valid. Finally, we consider experimental data and show that the results are in good agreement with a finite element method-based calibration. Due to the fast evaluation of PINNs, calibration can be performed in near real-time. This advantage is particularly evident in many-query applications such as Markov chain Monte Carlo-based Bayesian inference.

cs.LG↗

arXiv2024

Enhancing Multiscale Simulations with Constitutive Relations-Aware Deep Operator Networks

Multiscale problems are widely observed across diverse domains in physics and engineering. Translating these problems into numerical simulations and solving them using numerical schemes, e.g. the finite element method, is costly due to the demand of solving initial boundary-value problems at multiple scales. On the other hand, multiscale finite element computations are commended for their ability to integrate micro-structural properties into macroscopic computational analyses using homogenization techniques. Recently, neural operator-based surrogate models have shown trustworthy performance for solving a wide range of partial differential equations. In this work, we propose a hybrid method in which we utilize deep operator networks for surrogate modeling of the microscale physics. This allows us to embed the constitutive relations of the microscale into the model architecture and to predict microscale strains and stresses based on the prescribed macroscale strain inputs. Furthermore, numerical homogenization is carried out to obtain the macroscale quantities of interest. We apply the proposed approach to quasi-static problems of solid mechanics. The results demonstrate that our constitutive relations-aware DeepONet can yield accurate solutions even when being confronted with a restricted dataset during model development.

cs.LG↗

arXiv2024

Reduced and All-at-Once Approaches for Model Calibration and Discovery in Computational Solid Mechanics

In the framework of solid mechanics, the task of deriving material parameters from experimental data has recently re-emerged with the progress in full-field measurement capabilities and the renewed advances of machine learning. In this context, new methods such as the virtual fields method and physics-informed neural networks have been developed as alternatives to the already established least-squares and finite element-based approaches. Moreover, model discovery problems are starting to emerge and can also be addressed in a parameter estimation framework. These developments call for a new unified perspective, which is able to cover both traditional parameter estimation methods and novel approaches in which the state variables or the model structure itself are inferred as well. Adopting concepts discussed in the inverse problems community, we distinguish between all-at-once and reduced approaches. With this general framework, we are able to structure a large portion of the literature on parameter estimation in computational mechanics - and we can identify combinations that have not yet been addressed, two of which are proposed in this paper. We also discuss statistical approaches to quantify the uncertainty related to the estimated parameters, and we propose a novel two-step procedure for identification of complex material models based on both frequentist and Bayesian principles. Finally, we illustrate and compare several of the aforementioned methods with mechanical benchmarks based on synthetic and real data.

cs.CE↗

arXiv2015

From Nanopores to Macropores: Fractal Morphology of Graphite

We present a comprehensive structural characterization of two different highly pure nuclear graphites that compasses all relevant length scales from nanometers to sub-mm. This has been achieved by combining several experiments and neutron techniques: Small Angle Neutron Scattering (SANS), high-resolution Spin Echo SANS (SESANS) and neutron imaging. In this way it is possible to probe an extraordinary broad range of 6 orders of magnitude in length from microscopic to macroscopic length scales. The results reveal a fractal structure that extends from $ \sim $ 0.6 nm to 0.6 mm and has surface and mass fractal dimensions both very close to 2.5, a value found for percolating clusters and fractured ranked surfaces in 3D.

cond-mat.mtrl-sci↗

arXiv2015

Relaxed quaternionic Gabor expansions at critical density

Shifted and modulated Gaussian functions play a vital role in the representation of signals. We extend the theory into a quaternionic setting, using two exponential kernels with two complex numbers. As a final result, we show that every continuous and quaternion-valued signal $f$ in the Wiener space can be expanded into a unique $\ell^2$ series on a lattice at critical density $1$, provided one more point is added in the middle of a cell. We call that a relaxed Gabor expansion.

math.CV↗

arXiv2015

Some results on the lattice parameters of quaternionic Gabor frames

Gabor frames play a vital role not only modern harmonic analysis but also in several fields of applied mathematics, for instances, detection of chirps, or image processing. In this work we present a non-trivial generalization of Gabor frames to the quaternionic case and give new density results. The key tool is the two-sided windowed quaternionic Fourier transform (WQFT). As in the complex case, we want to write the WQFT as an inner product between a quaternion-valued signal and shifts and modulates of a real-valued window function. We demonstrate a Heisenberg uncertainty principle and for the results regarding the density, we employ the quaternionic Zak transform to obtain necessary and sufficient conditions to ensure that a quaternionic Gabor system is a quaternionic Gabor frame. We conclude with a proof that the Gabor conjecture do not hold true in the quaternionic case.

math.FA↗