arXiv subjects

Felix Fritzen

Publications and source records attributed to Felix Fritzen.

13 recordsLinked to original sources

arXiv2025

Robust inverse material design with physical guarantees using the Voigt-Reuss Net

We propose a spectrally normalized surrogate for forward and inverse mechanical homogenization with hard physical guarantees. Leveraging the Voigt-Reuss bounds, we factor their difference via a Cholesky-like operator and learn a dimensionless, symmetric positive semi-definite representation with eigenvalues in $[0,1]$; the inverse map returns symmetric positive-definite predictions that lie between the bounds in the L\"owner sense. In 3D linear elasticity on an open dataset of stochastic biphasic microstructures, a fully connected Voigt-Reuss net trained on $>\!7.5\times 10^{5}$ FFT-based labels with 236 isotropy-invariant descriptors and three contrast parameters recovers the isotropic projection with near-perfect fidelity (isotropy-related entries: $R^2 \ge 0.998$), while anisotropy-revealing couplings are unidentifiable from $SO(3)$-invariant inputs. Tensor-level relative Frobenius errors have median $\approx 1.7\%$ and mean $\approx 3.4\%$ across splits. For 2D plane strain on thresholded trigonometric microstructures, coupling spectral normalization with a differentiable renderer and a CNN yields $R^2>0.99$ on all components, subpercent normalized losses, accurate tracking of percolation-induced eigenvalue jumps, and robust generalization to out-of-distribution images. Treating the parametric microstructure as design variables, batched first-order optimization with a single surrogate matches target tensors within a few percent and returns diverse near-optimal designs. Overall, the Voigt-Reuss net unifies accurate, physically admissible forward prediction with large-batch, constraint-consistent inverse design, and is generic to elliptic operators and coupled-physics settings.

cs.LG↗

arXiv2025

Microstructural Insights into Fast Ion Transport in Solid Electrolytes via Multiscale Modeling

Improving solid electrolytes is critical for high-performance all-solid-state batteries, yet the microstructural features that enable fast ion transport remain poorly understood. Here, we use multiscale modeling to resolve polycrystalline ion transport from atomic-scale hopping at grain boundaries to continuum-scale percolation, thereby providing insights into realistic solid-electrolyte microstructures. Accurate lightweight machine-learning potentials -- developed via closed-loop active learning for exemplar argyrodites Li$_6$PS$_5$X, X $\in$ {Cl, Br, I} -- are employed to integrate molecular dynamics with finite element simulations. We find that diffusion barriers of the anion-ordered bulk scale linearly with anion radius. Grain boundaries exert opposite effects depending on the bulk: enhancing ion diffusion in low-diffusivity phases but suppressing it in fast-diffusing ones. Li$_6$PS$_5$I exhibits non-Arrhenius transport behavior consistent with experimental observations. Our results clarify the pivotal role of grain boundaries in ion transport and guide a priori microstructural design of advanced solid electrolytes.

cond-mat.mtrl-sci↗

arXiv2025

Accelerating Conjugate Gradient Solvers for Homogenization Problems with Unitary Neural Operators

Rapid and reliable solvers for parametric partial differential equations (PDEs) are needed in many scientific and engineering disciplines. For example, there is a growing demand for composites and architected materials with heterogeneous microstructures. Designing such materials and predicting their behavior in practical applications requires solving homogenization problems for a wide range of material parameters and microstructures. While classical numerical solvers offer reliable and accurate solutions supported by a solid theoretical foundation, their high computational costs and slow convergence remain limiting factors. As a result, scientific machine learning is emerging as a promising alternative. However, such approaches often lack guaranteed accuracy and physical consistency. This raises the question of whether it is possible to develop hybrid approaches that combine the advantages of both data-driven methods and classical solvers. To address this, we introduce UNO-CG, a hybrid solver that accelerates conjugate gradient (CG) solvers using specially designed machine-learned preconditioners, while ensuring convergence by construction. As a preconditioner, we propose Unitary Neural Operators as a modification of Fourier Neural Operators. Our method can be interpreted as a data-driven discovery of Green's functions, which are then used to accelerate iterative solvers. We evaluate UNO-CG on various homogenization problems involving heterogeneous microstructures and millions of degrees of freedom. Our results demonstrate that UNO-CG enables a substantial reduction in the number of iterations and is competitive with handcrafted preconditioners for homogenization problems that involve expert knowledge. Moreover, UNO-CG maintains strong performance across a variety of boundary conditions, where many specialized solvers are not applicable, highlighting its versatility and robustness.

math.NA↗

arXiv2025

A collapsed interface approach to resolve grain boundaries in finite element simulations of polycrystalline diffusion

Atomic diffusion affects the properties of various engineering materials, which predominantly occur in the polycrystalline state. A rigorous description of polycrystalline diffusion must therefore account for crystallographic defects, especially grain boundaries (GBs), whose structure and volume fraction - and hence the effective grain size - govern mass transport. Experiments and atomistic simulations consistently show that GBs can accelerate diffusion by up to several orders of magnitude and that fluxes along and across the interface are generally anisotropic. Conventional mesoscale models either neglect GBs or invoke idealized analytical corrections. Fully resolved finite-element meshes are accurate but computationally infeasible when nanometer-thin GB layers are involved. We introduce a collapsed-interface finite element that integrates the GB thickness analytically and embeds the result in a two-dimensional surface element. The formulation (i) treats in-plane and through-plane diffusivity independently, (ii) couples to the surrounding grain matrix without the need for mesh manipulations, and (iii) parametrizes both grain size and GB volume fraction via simple affine scalings, allowing systematic variation without remeshing. Effective diffusivity tensors are extracted by linear computational homogenization. The new finite element reproduces three-dimensional GB transport phenomena - channeled fluxes, concentration discontinuities - at a fraction of the computational cost of explicit models. Parametric studies spanning multiple orders of magnitude in GB diffusivity reveal four distinct diffusion regimes and quantify their impact on the overall response. The framework thus connects atomistic data and continuum predictions, providing an efficient tool for diffusion-driven design and optimization of polycrystalline materials.

cond-mat.mtrl-sci↗

arXiv2025

Spectral Normalization and Voigt-Reuss net: A universal approach to microstructure-property forecasting with physical guarantees

Heterogeneous materials are crucial to producing lightweight components, functional components, and structures composed of them. A crucial step in the design process is the rapid evaluation of their effective mechanical, thermal, or, in general, constitutive properties. The established procedure is to use forward models that accept microstructure geometry and local constitutive properties as inputs. The classical simulation-based approach, which uses, e.g., finite elements and FFT-based solvers, can require substantial computational resources. At the same time, simulation-based models struggle to provide gradients with respect to the microstructure and the constitutive parameters. Such gradients are, however, of paramount importance for microstructure design and for inverting the microstructure-property mapping. Machine learning surrogates can excel in these situations. However, they can lead to unphysical predictions that violate essential bounds on the constitutive response, such as the upper (Voigt-like) or the lower (Reuss-like) bound in linear elasticity. Therefore, we propose a novel spectral normalization scheme that a priori enforces these bounds. The approach is fully agnostic with respect to the chosen microstructural features and the utilized surrogate model. All of these will automatically and strictly predict outputs that obey the upper and lower bounds by construction. The technique can be used for any constitutive tensor that is symmetric and where upper and lower bounds (in the L\"owner sense) exist, i.e., for permeability, thermal conductivity, linear elasticity, and many more. We demonstrate the use of spectral normalization in the Voigt-Reuss net using a simple neural network. Numerical examples on truly extensive datasets illustrate the improved accuracy, robustness, and independence of the type of input features in comparison to much-used neural networks.

cs.LG↗

arXiv2024

Atomistic modeling of bulk and grain boundary diffusion in solid electrolyte Li$_6$PS$_5$Cl using machine-learning interatomic potentials

Li$_6$PS$_5$Cl is a promising candidate for the solid electrolyte in all-solid-state Li-ion batteries. In applications, this material is in a polycrystalline state with grain boundaries (GBs) that can affect ionic conductivity. While atomistic modeling provides valuable information on the impact of GBs on Li diffusion, such studies face either high computational cost (\textit{ab initio} methods) or accuracy limitations (classical potentials) as challenges. Here, we develop a quality-level-based active learning scheme for efficient and systematic development of \textit{ab initio}-based machine-learning interatomic potentials, specifically moment tensor potentials (MTPs), for large-scale, long-time, and high-accuracy simulations of complex atomic structures and diffusion mechanisms as encountered in solid electrolytes. Based on this scheme, we obtain MTPs for Li$_6$PS$_5$Cl and investigate two tilt GBs, $\Sigma3(1\bar{1}2)[110]$, $\Sigma3(\bar{1}11)[110]$, and one twist GB, $\Sigma5(001)[001]$. All three GBs exhibit low formation energies of less than \SI{20}{meV/\angstrom\textsuperscript{2}}, indicating their high stability in polycrystalline Li$_6$PS$_5$Cl. Using the MTPs, diffusion coefficients of the anion-ordered and anion-disordered bulk, as well as the three GBs, are obtained from molecular dynamics simulations of atomistic models. At \SI{300}{\kelvin}, the GB diffusion coefficients fall between the ones of the anion-ordered bulk structure (\SI{0.012e-7}{cm^2/s}, corresponding ionic conductivity about \SI{0.2}{mS/cm}) and the anion-disordered bulk structure (\SI{50}{\percent} Cl/S-anion disorder; \SI{2.203e-7}{cm^2/s}, about \SI{29.8}{mS/cm}) of Li$_6$PS$_5$Cl. Experimental data fall between the Arrhenius-extrapolated diffusion coefficients of the investigated atomic structures.

cond-mat.mtrl-sci↗

arXiv2023

Hybrid machine-learned homogenization: Bayesian data mining and convolutional neural networks

Beyond the generally deployed features for microstructure property prediction this study aims to improve the machine learned prediction by developing novel feature descriptors. Therefore, Bayesian infused data mining is conducted to acquire samples containing characteristics inexplicable to the current feature set, and suitable feature descriptors to describe these characteristics are proposed. The iterative development of feature descriptors resulted in 37 novel features, being able to reduce the prediction error by roughly one third. To further improve the predictive model, convolutional neural networks (Conv Nets) are deployed to generate auxiliary features in a supervised machine learning manner. The Conv Nets were able to outperform the feature based approach. A key ingredient for that is a newly proposed data augmentation scheme and the development of so-called deep inception modules. A combination of the feature based approach and the convolutional neural network leads to a hybrid neural network: A parallel deployment of the both neural network archetypes in a single model achieved a relative rooted mean squared error below 1%, more than halving the error compared to prior models operating on the same data. The hybrid neural network was found powerful enough to be extended to predict variable material parameters, from a low to high phase contrast, while allowing for arbitrary microstructure geometry at the same time.

cs.LG↗

arXiv2022

FFT-based Homogenization at Finite Strains using Composite Boxels (ComBo)

Computational homogenization is the gold standard for concurrent multi-scale simulations (e.g., FE2) in scale-bridging applications. Experimental and synthetic material microstructures are often represented by 3D image data. The computational complexity of simulations operating on such three-dimensional high-resolution voxel data comprising billions of unknowns induces the need for algorithmically and numerically efficient solvers. The inability of voxelized 3D geometries to capture smooth material interfaces accurately, along with the necessity for complexity reduction, motivates a special local coarse-graining technique called composite voxels [Kabel,M. et al. (2015)]. Composite voxels condense multiple fine-scale voxels into a single voxel obeying a theory-inspired constitutive model by employing laminate theory. Composite voxels enhance local field quality at a modest computational cost. Our contribution comprises the generalization towards composite boxels (ComBo) that are nonequiaxed, a feature that can pay off for materials with a preferred direction. A novel image-based normal detection algorithm is devised which improves the accuracy by around 30\% against the orientation cf. [Kabel,M. et al. (2015) ]. Further, the use of ComBo for finite strain simulations is studied in detail. An efficient implementation is proposed, and an essential back-projection algorithm preventing physically inadmissible states is developed, which improves robustness. Various examples show the efficiency of ComBo and the proposed algorithmic enhancements for nonlinear mechanical problems. The general usability is emphasized by examining and comparing the performance of myriad Fast Fourier Transform (FFT) based solvers including a detailed description of the new Doubly-Fine Material Grid (DFMG). All of the employed schemes benefit from the ComBo discretization.

math.NA↗

arXiv2020

On the generation of periodic discrete structures with identical two-point correlation

Strategies for the generation of periodic discrete structures with identical two-point correlation are developed. Starting from a pair of root structures, which are not related by translation, phase inversion or axis reflections, child structures of arbitrary resolution (i.e., pixel or voxel numbers) and number of phases (i.e., material phases/species) can be generated by means of trivial embedding based phase extension, application of kernels and/or phase coalescence, such that the generated structures inherit the two-point-correlation equivalence. Proofs of the inheritance property are provided by means of the Discrete Fourier Transform theory. A Python 3 implementation of the results is offered by the authors through the Github repository https://github.com/DataAnalyticsEngineering/EQ2PC in order to make the provided results reproducible and useful for all interested readers. Examples for the generation of structures are demonstrated, together with applications in the homogenization theory of periodic media.

cs.CE↗

arXiv2019

On-the-fly adaptivity for nonlinear twoscale simulations using artificial neural networks and reduced order modeling

A multi-fidelity surrogate model for highly nonlinear multiscale problems is proposed. It is based on the introduction of two different surrogate models and an adaptive on-the-fly switching. The two concurrent surrogates are built incrementally starting from a moderate set of evaluations of the full order model. Therefore, a reduced order model (ROM) is generated. Using a hybrid ROM-preconditioned FE solver, additional effective stress-strain data is simulated while the number of samples is kept to a moderate level by using a dedicated and physics-guided sampling technique. Machine learning (ML) is subsequently used to build the second surrogate by means of artificial neural networks (ANN). Different ANN architectures are explored and the features used as inputs of the ANN are fine tuned in order to improve the overall quality of the ML model. Additional ANN surrogates for the stress errors are generated. Therefore, conservative design guidelines for error surrogates are presented by adapting the loss functions of the ANN training in pure regression or pure classification settings. The error surrogates can be used as quality indicators in order to adaptively select the appropriate -- i.e. efficient yet accurate -- surrogate. Two strategies for the on-the-fly switching are investigated and a practicable and robust algorithm is proposed that eliminates relevant technical difficulties attributed to model switching. The provided algorithms and ANN design guidelines can easily be adopted for different problem settings and, thereby, they enable generalization of the used machine learning techniques for a wide range of applications. The resulting hybrid surrogate is employed in challenging multilevel FE simulations for a three-phase composite with pseudo-plastic micro-constituents. Numerical examples highlight the performance of the proposed approach.

physics.comp-ph↗

arXiv2019

Data-Driven Microstructure Property Relations

An image based prediction of the effective heat conductivity for highly heterogeneous microstructured materials is presented. The synthetic materials under consideration show different inclusion morphology, orientation, volume fraction and topology. The prediction of the effective property is made exclusively based on image data with the main emphasis being put on the 2-point spatial correlation function. This task is implemented using both unsupervised and supervised machine learning methods. First, a snapshot proper orthogonal decomposition (POD) is used to analyze big sets of random microstructures and thereafter compress significant characteristics of the microstructure into a low-dimensional feature vector. In order to manage the related amount of data and computations, three different incremental snapshot POD methods are proposed. In the second step, the obtained feature vector is used to predict the effective material property by using feed forward neural networks. Numerical examples regarding the incremental basis identification and the prediction accuracy of the approach are presented. A Python code illustrating the application of the surrogate is freely available.

cs.CE↗

arXiv2019

Finite Strain Homogenization Using a Reduced Basis and Efficient Sampling

The computational homogenization of hyperelastic solids in the geometrically nonlinear context has yet to be treated with sufficient efficiency in order to allow for real-world applications in true multiscale settings. This problem is addressed by a problem-specific surrogate model founded on a reduced basis approximation of the deformation gradient on the microscale. The setup phase is based upon a snapshot POD on deformation gradient fluctuations, in contrast to the widespread displacement-based approach. In order to reduce the computational offline costs, the space of relevant macroscopic stretch tensors is sampled efficiently by employing the Hencky strain. Numerical results show speed-up factors in the order of 5-100 and significantly improved robustness while retaining good accuracy. An open-source demonstrator tool with 50 lines of code emphasizes the simplicity and efficiency of the method.

cs.CE↗

arXiv2016

An algorithmic comparison of the Hyper-Reduction and the Discrete Empirical Interpolation Method for a nonlinear thermal problem

A novel algorithmic discussion of the methodological and numerical differences of competing parametric model reduction techniques for nonlinear problems are presented. First, the Galerkin reduced basis (RB) formulation is presented which fails at providing significant gains with respect to the computational efficiency for nonlinear problems. Renown methods for the reduction of the computing time of nonlinear reduced order models are the Hyper-Reduction and the (Discrete) Empirical Interpolation Method (EIM, DEIM). An algorithmic description and a methodological comparison of both methods are provided. The accuracy of the predictions of the hyper-reduced model and the (D)EIM in comparison to the Galerkin RB is investigated. All three approaches are applied to a simple uncertainty quantification of a planar nonlinear thermal conduction problem. The results are compared to computationally intense finite element simulations.

cs.CE↗