arXiv subjects

Sebastian Resch-Schopper

Publications and source records attributed to Sebastian Resch-Schopper.

3 recordsLinked to original sources

arXiv2026

Uncertainty-Aware Calculation of Analytical Gradients of Matrix-Interpolatory Reduced-Order Models for Efficient Structural Optimization

This paper presents an adaptive sampling algorithm tailored for the optimization of parametrized dynamical systems using projection-based model order reduction. Unlike classical sampling strategies, this framework does not aim for a small approximation error in the global sense but focuses on identifying and refining promising regions early on while reducing expensive full order model evaluations. The algorithm is tested on two models: a Timoshenko beam and a Kelvin cell, which ought to be optimized in terms of the system output in the frequency domain. For that, different norms of the transfer function are used as the objective function, while up to two geometrical parameters form the vector of design variables. The sampled full order models are reduced using the iterative rational Krylov algorithm and reprojected into a global basis. Subsequently, the models are parametrized by performing sparse Bayesian regression on matrix entry level of the reduced operators. Thompson sampling is carried out using the posterior distribution of the polynomial coefficients in order to account for uncertainties in the trained regression models. The strategy deployed for sample acquisition incorporates a gradient-based search on the parametrized reduced order model, which involves analytical gradients obtained via adjoint sensitivity analysis. By adding the found optimum to the sample set, the sample set is iteratively refined. Results demonstrate robust convergence towards the global optimum but highlight the computational cost introduced by the gradient-based optimization. The probabilistic extensions seamlessly integrate into existing matrix-interpolatory reduction frameworks and enable the analytical calculation of gradients under uncertainty.

cs.CE↗

arXiv2025

Consistent Parametric Model Order Reduction by Matrix Interpolation for Varying Underlying Meshes

Parametric model order reduction (pMOR) is a powerful tool for accelerating finite element (FE) simulations while maintaining parametric dependencies. For geometric parameters, pMOR by matrix interpolation is a well-suited approach because it does not require an affine representation of the parametric dependency, which is often not available for geometric parameters. However, the method requires that the underlying FE mesh has the same number of degrees of freedom and the same topology for all parameter configurations. This requirement can be difficult or even impossible to achieve for large parameter ranges or when automatic meshing is used. In this work, we propose a novel framework for pMOR by matrix interpolation for varying underlying meshes. The key idea is to understand the sampled reduced bases as continuous displacement fields that can be represented in different discretizations. By using mesh morphing and basis interpolation, the sampled reduced bases described in varying meshes can all be represented in terms of one reference mesh. This not only allows for performing pMOR by matrix interpolation, but also enables comparing the subspaces that the reduced bases span, which is important to detect strong changes that could lead to inconsistencies in the reduced operators. For mesh morphing, two strategies, namely morphing by spring analogy with elastic hardening and radial basis function morphing, were implemented and tested. Numerical experiments on a beam-shaped plate and a plate with a hole for one- and two-dimensional parameter spaces show that the proposed framework achieves high accuracy for both morphing methods and performs significantly better than two existing approaches for pMOR by matrix interpolation for varying underlying meshes.

math.NA↗

arXiv2024

Inconsistency Removal of Reduced Bases in Parametric Model Order Reduction by Matrix Interpolation using Adaptive Sampling and Clustering

Parametric model order reduction by matrix interpolation allows for efficient prediction of the behavior of dynamic systems without requiring knowledge about the underlying parametric dependency. Within this approach, reduced models are first sampled and then made consistent with each other by transforming the underlying reduced bases. Finally, the transformed reduced operators can be interpolated to predict reduced models for queried parameter points. However, the accuracy of the predicted reduced model strongly depends on the similarity of the sampled reduced bases. If the local reduced bases change significantly over the parameter space, inconsistencies are introduced in the training data for the matrix interpolation. These strong changes in the reduced bases can occur due to the model order reduction method used, a change of the system's dynamics with a change of the parameters, and mode switching and truncation. In this paper, individual approaches for removing these inconsistencies are extended and combined into one general framework to simultaneously treat multiple sources of inconsistency. For that, modal truncation is used for the reduction, an adaptive sampling of the parameter space is performed, and eventually, the parameter space is partitioned into regions in which all local reduced bases are consistent with each other. The proposed framework is applied to a cantilever Timoshenko beam and the Kelvin cell for one- to three-dimensional parameter spaces. Compared to the original version of parametric model order reduction by matrix interpolation and an existing method for inconsistency removal, the proposed framework leads to parametric reduced models with significantly smaller errors.

math.DS↗