Moritz Flaschel

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Flaschel

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Moritz

ORCID

0000-0002-1365-9272

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Publications1 - 10 of 15

Calibration of material parameters based on 180° and 90° ferroelectric domain wall properties in Ginzburg–Landau–Devonshire phase field models
Flaschel, Moritz; De Lorenzis, Laura (2020)
Archive of Applied Mechanics
Ferroelectric phase field models based on the Ginzburg–Landau–Devonshire theory are characterized by a large number of material parameters with problematic physical interpretation. In this study, we systematically address the relationship between these parameters and the main properties of ferroelectric domain walls. A variational approach is used to derive closed form solutions for the polarization fields at the phase transition regions as well as for the propagation velocities of the domain walls. Introducing a modified set of material parameters, which appropriately scales different contributions to the free energy, we are able to accurately calibrate these parameters based on domain wall thickness and energy of both 180∘ and 90∘ domain walls. Moreover, the mobility parameter appearing in the Ginzburg–Landau evolution equation can be accurately calibrated based on the propagation velocity of the domain walls.
Automated Discovery of Material Models in Continuum Solid Mechanics
Flaschel, Moritz (2023)
The mathematical description of the mechanical behavior of solid materials at the continuum scale is one of the oldest and most challenging tasks in solid mechanics and material science. The process of finding adequate mathematical formulae to describe the material characteristics is called material modeling and it is traditionally a time-consuming, intuition-driven and error-prone task. The new advances in data-driven and machine-learning-based methods promise an automation of the material model discovery process. In this work, the novel method EUCLID (Efficient Unsupervised Constitutive Law Identification and Discovery) is discussed, which aims to mitigate the two main weak spots of the state-of-the-art machine-learning-based methods for material modeling; their data inefficiency and their physical uninterpretability. Instead of deploying a supervised training process informed by many labeled stress-strain data pairs, which are difficult to acquire experimentally, at the core of EUCLID stands an unsupervised and physics-driven training process that is informed by full-field displacement and reaction force data. In this way, all data needed for the material characterization can be acquired from a single experiment. Further, instead of describing the material behavior by uninterpretable black-box models, EUCLID discovers a suitable material model from a potentially large set of candidate models using sparse regression. A sparsity promoting regularization term ensures that the discovered model is parsimonious, i.e., it entails a small number of material parameters, function evaluations and internal variables, thus increasing the physical interpretability and efficiency of the model. By posing certain restrictions on the material model candidates, it is ensured that the discovered models fulfill physical requirements that are well-known in the field. In this thesis, EUCLID is studied in the context of hyperelastic materials, elastoplastic materials and generalized standard materials. The method is verified by numerical tests, proving that EUCLID infers from indirect data appropriate material models that are encoded by parsimonious mathematical expressions. Finally, the current challenges and future perspectives of EUCLID are critically discussed.
Automated discovery of generalized standard material models with EUCLID
Flaschel, Moritz; Kumar, Siddhant; De Lorenzis, Laura (2023)
Computer Methods in Applied Mechanics and Engineering
We extend the scope of our recently developed approach for unsupervised automated discovery of material laws (denoted as EUCLID) to the general case of a material belonging to an unknown class of constitutive behavior. To this end, we leverage the theory of generalized standard materials, which encompasses a plethora of important constitutive classes including elasticity, viscosity, plasticity and arbitrary combinations thereof. We show that, based only on full-field kinematic measurements and net reaction forces, EUCLID is able to automatically discover the two scalar thermodynamic potentials, namely, the Helmholtz free energy and the dissipation potential, which completely define the behavior of generalized standard materials. The a priori enforced constraint of convexity on these potentials guarantees by construction stability and thermodynamic consistency of the discovered model; balance of linear momentum acts as a fundamental constraint to replace the availability of stress–strain labeled pairs; sparsity promoting regularization enables the automatic selection of a small subset from a possibly large number of candidate model features and thus leads to a parsimonious, i.e., simple and interpretable, model. Importantly, since model features go hand in hand with the correspondingly active internal variables, sparse regression automatically induces a parsimonious selection of the few internal variables needed for an accurate but simple description of the material behavior. A fully automatic procedure leads to the selection of the hyperparameter controlling the weight of the sparsity promoting regularization term, in order to strike a user-defined balance between model accuracy and simplicity. By testing the method on synthetic data including artificial noise, we demonstrate that EUCLID is able to automatically discover the true hidden material model from a large catalogue of constitutive classes, including elasticity, viscoelasticity, elastoplasticity, viscoplasticity, isotropic and kinematic hardening.
Single-test evaluation of directional elastic properties of anisotropic structured materials
Boddapati, Jagannadh; Flaschel, Moritz; Kumar, Siddhant; et al. (2023)
Journal of the Mechanics and Physics of Solids
When the elastic properties of structured materials become direction-dependent, the number of their descriptors increases. For example, in two-dimensions, the anisotropic behavior of materials is described by up to 6 independent elastic stiffness parameters, as opposed to only 2 needed for isotropic materials. Such high number of parameters expands the design space of structured materials and leads to unusual phenomena, such as materials that can shear under uniaxial compression. However, an increased number of properties descriptors and the coupling between shear and normal deformations render the experimental evaluation of material properties more challenging. In this paper, we propose a methodology based on the virtual fields method to identify six separate stiffness tensor parameters of two-dimensional anisotropic structured materials using just one tension test, thus eliminating the need for multiple experiments, as it is typical in traditional methods. The approach requires no stress data and uses full-field displacement data and global force data. We show the accuracy of our method using synthetic data generated from finite element simulations as well as experimental data from additively manufactured specimens.
Discovering plasticity models without stress data
Flaschel, Moritz; Kumar, Siddhant; De Lorenzis, Laura (2022)
npj Computational Materials
We propose an approach for data-driven automated discovery of material laws, which we call EUCLID (Efficient Unsupervised Constitutive Law Identification and Discovery), and we apply it here to the discovery of plasticity models, including arbitrarily shaped yield surfaces and isotropic and/or kinematic hardening laws. The approach is unsupervised, i.e., it requires no stress data but only full-field displacement and global force data; it delivers interpretable models, i.e., models that are embodied by parsimonious mathematical expressions discovered through sparse regression of a potentially large catalog of candidate functions; it is one-shot, i.e., discovery only needs one experiment. The material model library is constructed by expanding the yield function with a Fourier series, whereas isotropic and kinematic hardening is introduced by assuming a yield function dependency on internal history variables that evolve with the plastic deformation. For selecting the most relevant Fourier modes and identifying the hardening behavior, EUCLID employs physics knowledge, i.e., the optimization problem that governs the discovery enforces the equilibrium constraints in the bulk and at the loaded boundary of the domain. Sparsity promoting regularization is deployed to generate a set of solutions out of which a solution with low cost and high parsimony is automatically selected. Through virtual experiments, we demonstrate the ability of EUCLID to accurately discover several plastic yield surfaces and hardening mechanisms of different complexity.
Automated identification of linear viscoelastic constitutive laws with EUCLID
Marino, Enzo; Flaschel, Moritz; Kumar, Siddhant; et al. (2023)
Mechanics of Materials
We extend EUCLID, a computational strategy for automated material model discovery and identification, to linear viscoelasticity. For this case, we perform a priori model selection by adopting a generalized Maxwell model expressed by a Prony series, and deploy EUCLID for identification. The methodology is based on four ingredients: i. full-field displacement and net force data; ii. a very wide material model library — in our case, a very large number of terms in the Prony series; iii. the linear momentum balance constraint; iv. the sparsity constraint. The devised strategy comprises two stages. Stage 1 relies on sparse regression; it enforces momentum balance on the data and exploits sparsity-promoting regularization to drastically reduce the number of terms in the Prony series and identify the material parameters. Stage 2 relies on k-means clustering; starting from the reduced set of terms from stage 1, it further reduces their number by grouping together Maxwell elements with very close relaxation times and summing the corresponding moduli. Automated procedures are proposed for the choice of the regularization parameter in stage 1 and of the number of clusters in stage 2. The overall strategy is demonstrated on artificial numerical data, both without and with the addition of noise, and shown to efficiently and accurately identify a linear viscoelastic model with five relaxation times across four orders of magnitude, out of a library with several hundreds of terms spanning relaxation times across seven orders of magnitude.
Discovering non-associated pressure-sensitive plasticity models with EUCLID
Xu, Haotian; Flaschel, Moritz; De Lorenzis, Laura (2025)
Advanced Modeling and Simulation in Engineering Sciences
We extend (EUCLID Efficient Unsupervised Constitutive Law Identification and Discovery)-a data-driven framework for automated material model discovery-to pressure-sensitive plasticity models, encompassing arbitrarily shaped yield surfaces with convexity constraints and non-associated flow rules. The method only requires full-field displacement and boundary force data from one single experiment and delivers constitutive laws as interpretable mathematical expressions. We construct a material model library for pressure-sensitive plasticity models with non-associated flow rules in four steps: (1) a Fourier series describes an arbitrary yield surface shape in the deviatoric stress plane; (2) a pressure-sensitive term in the yield function defines the shape of the shear failure surface and determines plastic deformation under tension; (3) a compression cap term determines plastic deformation under compression; (4) a non-associated flow rule may be adopted to avoid the excessive dilatancy induced by plastic deformations. In contrast to traditional parameter identification methods, EUCLID is equipped with a sparsity promoting regularization to restrain the number of model parameters (and thus modeling features) to the minimum needed to accurately interpret the data, thus achieving a compromise between model simplicity and accuracy. The convexity of the learned yield surface is guaranteed by a set of constraints in the inverse optimization problem. We demonstrate the proposed approach in multiple numerical experiments with noisy data, and show the ability of EUCLID to accurately select a suitable material model from the starting library.
A topology optimisation framework to design test specimens for one-shot identification or discovery of material models
Ghouli, Saeid; Flaschel, Moritz; Kumar, Siddhant; et al. (2025)
Journal of the Mechanics and Physics of Solids
The increasing availability of full-field displacement data from imaging techniques in experimental mechanics is determining a gradual shift in the paradigm of material model calibration and discovery, from using several simple-geometry tests towards a few, or even one single test with complicated geometry. The feasibility of such a “one-shot” calibration or discovery heavily relies upon the richness of the measured displacement data, i.e., their ability to probe the space of the state variables and the stress space (whereby the stresses depend on the constitutive law being sought) to an extent sufficient for an accurate and robust calibration or discovery process. The richness of the displacement data is in turn directly governed by the specimen geometry. In this paper, we propose a density-based topology optimisation framework to optimally design the geometry of the target specimen for calibration of an anisotropic elastic material model. To this end, we perform automatic, high-resolution specimen design by maximising the robustness of the solution of the inverse problem, i.e., the identified material parameters, given noisy displacement measurements from digital image correlation. We discuss the choice of the cost function and the design of the topology optimisation framework, and we analyse a range of optimised topologies generated for the identification of isotropic and anisotropic elastic responses.
Bayesian-EUCLID: Discovering hyperelastic material laws with uncertainties
Joshi, Akshay; Thakolkaran, Prakash; Zheng, Yiwen; et al. (2022)
Computer Methods in Applied Mechanics and Engineering
Within the scope of our recent approach for Efficient Unsupervised Constitutive Law Identification and Discovery (EUCLID), we propose an unsupervised Bayesian learning framework for discovery of parsimonious and interpretable constitutive laws with quantifiable uncertainties. As in deterministic EUCLID, we do not resort to stress data, but only to realistically measurable full-field displacement and global reaction force data; as opposed to calibration of an a priori assumed model, we start with a constitutive model ansatz based on a large catalog of candidate functional features; we leverage domain knowledge by including features based on existing, both physics-based and phenomenological, constitutive models. In the new Bayesian-EUCLID approach, we use a hierarchical Bayesian model with sparsity-promoting priors and Monte Carlo sampling to efficiently solve the parsimonious model selection task and discover physically consistent constitutive equations in the form of multivariate multi-modal probabilistic distributions. We demonstrate and validate the ability to accurately and efficiently recover isotropic and anisotropic hyperelastic models like the Neo-Hookean, Isihara, Gent–Thomas, Arruda–Boyce, Ogden, and Holzapfel models in both elastostatics and elastodynamics. The discovered constitutive models are reliable under both epistemic uncertainties — i.e. uncertainties on the true features of the constitutive catalog – and aleatoric uncertainties – which arise from the noise in the displacement field data, and are automatically estimated by the hierarchical Bayesian model.
A Review on Data-Driven Constitutive Laws for Solids
Fuhg, Jan N.; Padmanabha, Govinda Anantha; Bouklas, Nikolaos; et al. (2025)
Archives of Computational Methods in Engineering
This review article highlights state-of-the-art data-driven techniques to discover, encode, surrogate, or emulate constitutive laws that describe the path-independent and path-dependent response of solids. Our objective is to provide an organized taxonomy to a large spectrum of methodologies developed in the past decades and to discuss the benefits and drawbacks of the various techniques for interpreting and forecasting mechanics behavior across different scales. Distinguishing between machine-learning-based and model-free methods, we further categorize approaches based on their interpretability and on their learning process/type of required data, while discussing the key problems of generalization and trustworthiness. We attempt to provide a road map of how these can be reconciled in a data-availability-aware context. We also touch upon relevant aspects such as data sampling techniques, design of experiment, verification, and validation.

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Moritz Flaschel

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