arXiv subjects

Ulrich Römer

Publications and source records attributed to Ulrich Römer.

At least 19 recordsLinked to original sources

arXiv2026

A Regularized Ensemble Kalman Filter for Stochastic Phase Field Models of Brittle Fracture

The phase-field approach to brittle fracture provides a continuum framework for modeling crack initiation and propagation without explicit representation of discrete crack surfaces, provided the spatial discretization is fine enough to resolve the regularization length scale. However, uncertain local material parameters due to material defects can strongly influence simulation results, such as crack paths and remaining structural strength. At the same time, the ability to continuously monitor structures using sensors allows complementing modeling predictions with, e.g., displacement measurements. In this contribution, we connect these two complementary sources of information and present a Bayesian inference procedure that allows updating the current model state with incoming sensor data. We construct a Bayesian prior for the model state (both displacements and phase-field) and employ an ensemble Kalman filter (EnKF) to perform the update. In the EnKF, the update is computed by performing a Kalman shift on each ensemble member. Since the standard EnKF may produce assimilated states that violate common modeling assumptions, we present a phase field-based regularization technique as a proximal step correction toward model-consistent updates. 1D and 2D numerical examples demonstrate the performance and accuracy of the proposed method and show that the updated state matches the ground truth reasonably well. Unlike traditional Bayesian inversion techniques, which have already been applied to brittle fracture, we infer not the model parameters but the model state, i.e., the displacement field and the phase-field. Although only displacements are observed, the strong correlation between both fields also allows inference of the posterior phase-field.

cs.CE↗

arXiv2026

Uncertainty quantification in model discovery by distilling interpretable material constitutive models from Gaussian process posteriors

Constitutive model discovery refers to the task of identifying an appropriate model structure, usually from a predefined model library, while simultaneously inferring its material parameters. The data used for model discovery are measured in mechanical tests and are thus inevitably affected by noise which, in turn, induces uncertainties. Previously proposed methods for uncertainty quantification in model discovery either require the selection of a prior for the material parameters, are restricted to linear coefficients of the model library or are limited in the flexibility of the inferred parameter probability distribution. We therefore propose a partially Bayesian framework for uncertainty quantification in model discovery that does not require prior selection for the material parameters and also allows for the discovery of constitutive models with inner-non-linear parameters: First, we augment the available stress-deformation data with a Gaussian process. Second, we approximate the parameter distribution by a normalizing flow, which allows for modeling complex joint distributions. Third, we distill the parameter distribution by matching the distribution of stress-deformation functions induced by the parameters with the Gaussian process posterior. Fourth, we perform a Sobol' sensitivity analysis to obtain a sparse and interpretable model. We demonstrate the capability of our framework for both isotropic and experimental anisotropic data.

cs.LG↗

arXiv2025

Statistical reduced order modelling for the parametric Helmholtz equation

Predictive modeling involving simulation and sensor data at the same time, is a growing challenge in computational science. Even with large-scale finite element models, a mismatch to the sensor data often remains, which can be attributed to different sources of uncertainty. For such a scenario, the statistical finite element method (statFEM) can be used to condition a simulated field on given sensor data. This yields a posterior solution which resembles the data much better and additionally provides consistent estimates of uncertainty, including model misspecification. For frequency or parameter dependent problems, occurring, e.g. in acoustics or electromagnetism, solving the full order model at the frequency grid and conditioning it on data quickly results in a prohibitive computational cost. In this case, the introduction of a surrogate in form of a reduced order model yields much smaller systems of equations. In this paper, we propose a reduced order statFEM framework relying on Krylov-based moment matching. We introduce a data model which explicitly includes the bias induced by the reduced approximation, which is estimated by an inexpensive error indicator. The results of the new statistical reduced order method are compared to the standard statFEM procedure applied to a ROM prior, i.e. without explicitly accounting for the reduced order bias. The proposed method yields better accuracy and faster convergence throughout a given frequency range for different numerical examples.

cs.CE↗

arXiv2025

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↗

arXiv2025

Knowledge-Based Aerospace Engineering -- A Systematic Literature Review

The aerospace industry operates at the frontier of technological innovation while maintaining high standards regarding safety and reliability. In this environment, with an enormous potential for re-use and adaptation of existing solutions and methods, Knowledge-Based Engineering (KBE) has been applied for decades. The objective of this study is to identify and examine state-of-the-art knowledge management practices in the field of aerospace engineering. Our contributions include: 1) A SWARM-SLR of over 1,000 articles with qualitative analysis of 164 selected articles, supported by two aerospace engineering domain expert surveys. 2) A knowledge graph of over 700 knowledge-based aerospace engineering processes, software, and data, formalized in the interoperable Web Ontology Language (OWL) and mapped to Wikidata entries where possible. The knowledge graph is represented on the Open Research Knowledge Graph (ORKG), and an aerospace Wikibase, for reuse and continuation of structuring aerospace engineering knowledge exchange. 3) Our resulting intermediate and final artifacts of the knowledge synthesis, available as a Zenodo dataset. This review sets a precedent for structured, semantic-based approaches to managing aerospace engineering knowledge. By advancing these principles, research, and industry can achieve more efficient design processes, enhanced collaboration, and a stronger commitment to sustainable aviation.

cs.CE↗

arXiv2025

Mechanical State Estimation with a Polynomial-Chaos-Based Statistical Finite Element Method

The Statistical Finite Element Method (statFEM) offers a Bayesian framework for integrating computational models with observational data, thus providing improved predictions for structural health monitoring and digital twinning. This paper presents an efficient sampling-free statFEM tailored for non-conjugate, non-Gaussian prior probability densities. We assume that constitutive parameters, modeled as weakly stationary random fields, are the primary source of uncertainty and approximate them using Karhunen-Loève (KL) expansion. The resulting stochastic solution field, i.e., the displacement field, is a non-stationary, non-Gaussian random field, which we approximate via Polynomial Chaos (PC) expansion. The PC coefficients are determined through projection using Smolyak sparse grids. Additionally, we model the measurement noise as a stationary Gaussian random field and the model misspecification as a mean-free, non-stationary Gaussian random field, which is also approximated using KL expansion. The coefficients of the KL expansion are treated as hyperparameters. The PC coefficients of the stochastic posterior displacement field are computed using the Gauss-Markov-Kálmán filter, while the hyperparameters are determined by maximizing the marginal likelihood. We demonstrate the efficiency and convergence of the proposed method through one- and two-dimensional elastostatic problems.

cs.CE↗

arXiv2025

Gaussian Processes enabled model calibration in the context of deep geological disposal

Deep geological repositories are critical for the long-term storage of hazardous materials, where understanding the mechanical behavior of emplacement drifts is essential for safety assurance. This study presents a surrogate modeling approach for the mechanical response of emplacement drifts in rock salt formations, utilizing Gaussian Processes (GPs). The surrogate model serves as an efficient substitute for high-fidelity mechanical simulations in many-query scenarios, including time-dependent sensitivity analyses and calibration tasks. By significantly reducing computational demands, this approach facilitates faster design iterations and enhances the interpretation of monitoring data. The findings indicate that only a few key parameters are sufficient to accurately reflect in-situ conditions in complex rock salt models. Identifying these parameters is crucial for ensuring the reliability and safety of deep geological disposal systems.

physics.geo-ph↗

arXiv2024

Rational kernel-based interpolation for complex-valued frequency response functions

This work is concerned with the kernel-based approximation of a complex-valued function from data, where the frequency response function of a partial differential equation in the frequency domain is of particular interest. In this setting, kernel methods are employed more and more frequently, however, standard kernels do not perform well. Moreover, the role and mathematical implications of the underlying pair of kernels, which arises naturally in the complex-valued case, remain to be addressed. We introduce new reproducing kernel Hilbert spaces of complex-valued functions, and formulate the problem of complex-valued interpolation with a kernel pair as minimum norm interpolation in these spaces. Moreover, we combine the interpolant with a low-order rational function, where the order is adaptively selected based on a new model selection criterion. Numerical results on examples from different fields, including electromagnetics and acoustic examples, illustrate the performance of the method, also in comparison to available rational approximation methods.

cs.CE↗

arXiv2024

Uncertainty Analysis of Limit Cycle Oscillations in Nonlinear Dynamical Systems with the Fourier Generalized Polynomial Chaos Expansion

In engineering, simulations play a vital role in predicting the behavior of a nonlinear dynamical system. In order to enhance the reliability of predictions, it is essential to incorporate the inherent uncertainties that are present in all real-world systems. Consequently, stochastic predictions are of significant importance, particularly during design or reliability analysis. In this work, we concentrate on the stochastic prediction of limit cycle oscillations, which typically occur in nonlinear dynamical systems and are of great technical importance. To address uncertainties in the limit cycle oscillations, we rely on the recently proposed Fourier generalized Polynomial Chaos expansion (FgPC), which combines Fourier analysis with spectral stochastic methods. In this paper, we demonstrate that valuable insights into the dynamics and their variability can be gained with a FgPC analysis, considering different benchmarks. These are the well-known forced Duffing oscillator and a more complex model from cell biology in which highly non-linear electrophysiological processes are closely linked to diffusive processes. With our spectral method, we are able to predict complicated marginal distributions of the limit cycle oscillations and, additionally, for self-excited systems, the uncertainty in the base frequency. Finally we study the sparsity of the FgPC coefficients as a basis for adaptive approximation.

cs.CE↗

arXiv2024

Partitioned Surrogates and Thompson Sampling for Multidisciplinary Bayesian Optimization

The long runtime associated with simulating multidisciplinary systems challenges the use of Bayesian optimization for multidisciplinary design optimization (MDO). This is particularly the case if the coupled system is modeled in a partitioned manner and feedback loops, known as strong coupling, are present. This work introduces a method for Bayesian optimization in MDO called "Multidisciplinary Design Optimization using Thompson Sampling", abbreviated as MDO-TS. Instead of replacing the whole system with a surrogate, we substitute each discipline with such a Gaussian process. Since an entire multidisciplinary analysis is no longer required for enrichment, evaluations can potentially be saved. However, the objective and associated uncertainty are no longer analytically estimated. Since most adaptive sampling strategies assume the availability of these estimates, they cannot be applied without modification. Thompson sampling does not require this explicit availability. Instead, Thompson sampling balances exploration and exploitation by selecting actions based on optimizing random samples from the objective. We combine Thompson sampling with an approximate sampling strategy that uses random Fourier features. This approach produces continuous functions that can be evaluated iteratively. We study the application of this infill criterion to both an analytical problem and the shape optimization of a simple fluid-structure interaction example.

cs.CE↗

arXiv2024

Efficient low rank model order reduction of vibroacoustic problems under stochastic loads

This contribution combines a low-rank matrix approximation through Singular Value Decomposition (SVD) with second-order Krylov subspace-based Model Order Reduction (MOR), in order to efficiently propagate input uncertainties through a given vibroacoustic model. The vibroacoustic model consists of a plate coupled to a fluid into which the plate radiates sound due to a turbulent boundary layer excitation. This excitation is subject to uncertainties due to the stochastic nature of the turbulence and the computational cost of simulating the coupled problem with stochastic forcing is very high. The proposed method approximates the output uncertainties in an efficient way, by reducing the evaluation cost of the model in terms of DOFs and samples by using the factors of the SVD low-rank approximation directly as input for the MOR algorithm. Here, the covariance matrix of the vector of unknowns can efficiently be approximated with only a fraction of the original number of evaluations. Therefore, the approach is a promising step to further reducing the computational effort of large-scale vibroacoustic evaluations.

cs.CE↗

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↗

arXiv2023

Inferring Displacement Fields from Sparse Measurements Using the Statistical Finite Element Method

A well-established approach for inferring full displacement and stress fields from possibly sparse data is to calibrate the parameter of a given constitutive model using a Bayesian update. After calibration, a (stochastic) forward simulation is conducted with the identified model parameters to resolve physical fields in regions that were not accessible to the measurement device. A shortcoming of model calibration is that the model is deemed to best represent reality, which is only sometimes the case, especially in the context of the aging of structures and materials. While this issue is often addressed with repeated model calibration, a different approach is followed in the recently proposed statistical Finite Element Method (statFEM). Instead of using Bayes' theorem to update model parameters, the displacement is chosen as the stochastic prior and updated to fit the measurement data more closely. For this purpose, the statFEM framework introduces a so-called model-reality mismatch, parametrized by only three hyperparameters. This makes the inference of full-field data computationally efficient in an online stage: If the stochastic prior can be computed offline, solving the underlying partial differential equation (PDE) online is unnecessary. Compared to solving a PDE, identifying only three hyperparameters and conditioning the state on the sensor data requires much fewer computational resources. This paper presents two contributions to the existing statFEM approach: First, we use a non-intrusive polynomial chaos method to compute the prior, enabling the use of complex mechanical models in deterministic formulations. Second, we examine the influence of prior material models (linear elastic and St.Venant Kirchhoff material with uncertain Young's modulus) on the updated solution. We present statFEM results for 1D and 2D examples, while an extension to 3D is straightforward.

cs.CE↗

arXiv2023

An $hp$-adaptive multi-element stochastic collocation method for surrogate modeling with information re-use

This paper introduces an $hp$-adaptive multi-element stochastic collocation method, which additionally allows to re-use existing model evaluations during either $h$- or $p$-refinement. The collocation method is based on weighted Leja nodes. After $h$-refinement, local interpolations are stabilized by adding and sorting Leja nodes on each newly created sub-element in a hierarchical manner. For $p$-refinement, the local polynomial approximations are based on total-degree or dimension-adaptive bases. The method is applied in the context of forward and inverse uncertainty quantification to handle non-smooth or strongly localised response surfaces. The performance of the proposed method is assessed in several test cases, also in comparison to competing methods.

cs.CE↗

arXiv2022

Assessing the Performance of Leja and Clenshaw-Curtis Collocation for Computational Electromagnetics with Random Input Data

We consider the problem of quantifying uncertainty regarding the output of an electromagnetic field problem in the presence of a large number of uncertain input parameters. In order to reduce the growth in complexity with the number of dimensions, we employ a dimension-adaptive stochastic collocation method based on nested univariate nodes. We examine the accuracy and performance of collocation schemes based on Clenshaw-Curtis and Leja rules, for the cases of uniform and bounded, non-uniform random inputs, respectively. Based on numerical experiments with an academic electromagnetic field model, we compare the two rules in both the univariate and multivariate case and for both quadrature and interpolation purposes. Results for a real-world electromagnetic field application featuring high-dimensional input uncertainty are also presented.

cs.CE↗

arXiv2021

Bayesian Dynamical System Identification With Unified Sparsity Priors And Model Uncertainty

This work is concerned with uncertainty quantification in reduced-order dynamical system identification. Reduced-order models for system dynamics are ubiquitous in design and control applications and recent efforts focus on their data-driven construction. Our starting point is the sparse-identification of nonlinear dynamics (SINDy) framework, which reformulates system identification as a regression problem, where unknown functions are approximated from a sparse subset of an underlying library. In this manuscript, we formulate this system identification method in a Bayesian framework to handle parameter and structural model uncertainties. We present a general approach to enforce sparsity, which builds on the recently introduced class of neuronized priors. We perform comparisons between different variants such as Lasso, horseshoe, and spike and slab priors, which are all obtained by modifying a single activation function. We also outline how state observation noise can be incorporated with a probabilistic state-space model. The resulting Bayesian regression framework is robust and simple to implement. We apply the method to two generic numerical applications, the pendulum and the Lorenz system, and one aerodynamic application using experimental measurements.

eess.SY↗

arXiv2020

Yield Optimization based on Adaptive Newton-Monte Carlo and Polynomial Surrogates

In this paper we present an algorithm for yield estimation and optimization exploiting Hessian based optimization methods, an adaptive Monte Carlo (MC) strategy, polynomial surrogates and several error indicators. Yield estimation is used to quantify the impact of uncertainty in a manufacturing process. Since computational efficiency is one main issue in uncertainty quantification, we propose a hybrid method, where a large part of a MC sample is evaluated with a surrogate model, and only a small subset of the sample is re-evaluated with a high fidelity finite element model. In order to determine this critical fraction of the sample, an adjoint error indicator is used for both the surrogate error and the finite element error. For yield optimization we propose an adaptive Newton-MC method. We reduce computational effort and control the MC error by adaptively increasing the sample size. The proposed method minimizes the impact of uncertainty by optimizing the yield. It allows to control the finite element error, surrogate error and MC error. At the same time it is much more efficient than standard MC approaches combined with standard Newton algorithms.

cs.CE↗

arXiv2020

Magnetic Field Simulation with Data-Driven Material Modeling

This paper developes a data-driven magnetostatic finite-element (FE) solver which directly exploits measured material data instead of a material curve constructed from it. The distances between the field solution and the measurement points are minimized while enforcing Maxwell's equations. The minimization problem is solved by employing the Lagrange multiplier approach. The procedure wraps the FE method within an outer data-driven iteration. The method is capable of considering anisotropic materials and is adapted to deal with models featuring a combination of exact material knowledge and measured material data. Thereto, three approaches with an increasing level of intrusivity according to the FE formulation are proposed. The numerical results for a quadrupole-magnet model show that data-driven field simulation is feasible and affordable and overcomes the need of modeling the material law.

physics.comp-ph↗