arXiv subjects

Bernhard Eidel

Publications and source records attributed to Bernhard Eidel.

16 recordsLinked to original sources

arXiv2026

Atomistic-Continuum Coupling by Homogenization

Classical atomistic simulations based on interatomic potentials resolve lattice instabilities, defect nucleation, and microstructure evolution with high fidelity, but their accessible system sizes remain far below those required for micrometer-scale structural analyses. We develop a two-scale atomistic-continuum framework that couples a nonlinear finite-element boundary-value problem at the microscale to periodic molecular-statics cell problems at quadrature points. The scale transition is formulated by computational homogenization in the sense of Hill-Mandel energy equivalence. Instead of prescribing a continuum constitutive law on the lower scale, the atomistic cell is driven directly by the continuum deformation and returns volume-averaged stresses in work-conjugate form together with effective tangent moduli. Numerical examples for single-crystalline copper show pronounced tension-compression asymmetry, abrupt instability-driven defect nucleation, rapid stabilization under reversed cyclic loading, and localized elastic-plastic transition in cantilever bending. In all these strongly nonlinear scenarios, the coarse-scale Newton solver remains robust and recovers near-quadratic convergence in its final iterations. The two-scale framework thus extends potential-based atomistic modeling to structural length scales that are inaccessible to direct atomistic simulation in the present quasi-static, athermal setting.

math.NA↗

arXiv2021

Deep Convolutional Neural Networks Predict Elasticity Tensors and their Bounds in Homogenization

In the present work, 3D convolutional neural networks (CNNs) are trained to link random heterogeneous, two-phase materials of arbitrary phase fractions to their elastic macroscale stiffness thus replacing explicit homogenization simulations. In order to reduce the uncertainty of the true stiffness of the synthetic composites due to unknown boundary conditions (BCs), the CNNs predict beyond the stiffness for periodic BC the upper bound through kinematically uniform BC, and the lower bound through stress uniform BC. This work describes the workflow of the homogenization-CNN, from microstructure generation over the CNN design, the operations of convolution, nonlinear activation and pooling as well as training and validation along with backpropagation up to performance measurements in tests. Therein the CNNs demonstrate the predictive accuracy not only for the standard test set but also for samples of the real, two-phase microstructure of a diamond-based coating. The CNN that covers all three boundary types is virtually as accurate as the separate treatment in three different nets. The CNNs of this contribution provide through stiffness bounds an indicator of the proper RVE size for individual snapshot samples. Moreover, they enable statistical analyses for the effective elastic stiffness on ensembles of synthetical microstructures without costly simulations.

cond-mat.mtrl-sci↗

arXiv2021

Computational Homogenization of Concrete in the Cyber Size-Resolution-Discretization (SRD) Parameter Space

Micro- and mesostructures of multiphase materials obtained from tomography and image acquisition are an ever more important database for simulation analyses. Huge data sets for reconstructed 3d volumes typically as voxel grids call for criteria and measures to find an affordable balance of accuracy and efficiency. The present work shows for a 3d mesostructure of concrete in the elastic deformation range, how the computational complexity in analyses of numerical homogenization can be reduced at controlled errors. Reduction is systematically applied to specimen size S, resolution R, and discretization D, which span the newly introduced SRD parameter space. Key indicators for accuracy are (i) the phase fractions, (ii) the homogenized elasticity tensor, (iii) its invariance with respect to the applied boundary conditions and (iv) the total error as well as spatial error distributions, which are computed and estimated. Pre-analyses in the 2d SRD parameter sub-space explore the transferability to the 3d case. Beyond the concrete specimen undergoing elastic deformations in the present work, the proposed concept enables accuracy-efficiency balances for various classes of heterogeneous materials in different deformation regimes and thus contributes to build comprehensive digital twins of materials with validated attributes.

cs.CE↗

arXiv2020

From image data towards microstructure information -- accuracy analysis at the digital core of materials

A cornerstone of computational solid mechanics in the context of digital transformation are databases for microstructures obtained from advanced tomography techniques. Uniform discretizations of pixelized images in 2D are the raw-data point of departure for simulation analyses. This paper proposes the concept of a unified error analysis for image-based microstructure representations in uniform resolution along with adaptively coarsened discretizations. The analysis distinguishes between a modeling error due to finite, possibly coarsened image resolution and a discretization error, investigates their quantitative relation, spatial distributions and their impacts on the simulation results both on the microscale and the macroscale in the context of computational homogenization. The assessment of accuracy and efficiency is carried out for an exemplary two-phase material. Beyond the example considered here the concept is a rational tool in the transformation of raw image data into microstructure information adapted to particular simulation needs and endows the digital twin of real microstructures with validated characteristics for reliable, predictive simulations.

math.NA↗

arXiv2020

Exploring metamaterials' structures through the relaxed micromorphic model: switching an acoustic screen into an acoustic absorber

While the design of always new metamaterials with exotic static and dynamic properties is attracting deep attention in the last decades, little effort is made to explore their interactions with other materials. This prevents the conception of (meta-)structures that can enhance metamaterials' unorthodox behaviours and that can be employed in real engineering applications. In this paper, we give a first answer to this challenging problem by showing that the relaxed micromorphic model with zero static characteristic length can be usefully applied to describe the refractive properties of simple meta-structures for extended frequency ranges and for any direction of propagation of the incident wave. Thanks to the simplified model's structure, we are able to efficiently explore different configurations and to show that a given meta-structure can drastically change its overall refractive behaviour when varying the elastic properties of specific meta-structural elements. In some cases, changing the stiffness of a homogeneous material which is in contact with a metamaterial's slab, reverses the structure's refractive behaviour by switching it from an acoustic screen (total reflection) into an acoustic absorber (total transmission). The present paper clearly indicates that, while the study and enhancement of the intrinsic metamaterials' properties is certainly of great importance, it is even more challenging to enable the conception of meta-structures that can eventually boost the use of metamaterials in real-case applications.

physics.app-ph↗

arXiv2019

A Nonlinear Finite Element Heterogeneous Multiscale Method for the Homogenization of Hyperelastic Solids and a Novel Staggered Two-Scale Solution Algorithm

In this paper we address three aspects of nonlinear computational homogenization of elastic solids by two-scale finite element methods. First, we present a nonlinear formulation of the finite element heterogeneous multiscale method FE-HMM in a Lagrangean formulation that covers geometrical nonlinearity and, more general, hyperelasticity. Second, a-priori estimates of FE-HMM, which exist so far only for the fully linear elastic case in solid mechanics, are assessed in the regime of nonlinear elasticity. The measured convergence rates agree fairly well with those of the fully linear regime. Third, we revise the standard solution algorithm of FE$^2$ which is a staggered scheme in terms of a nested loop embedding the full solution of the micro problem into one macro solution iteration step. We demonstrate that suchlike staggered scheme, which is typically realized by a nested two-level Newton algorithm, can safely and efficiently be replaced by direct alternations between micro and macro iterations. The novel algorithmic structure is exemplarily detailed for the proposed nonlinear FE-HMM, its efficiency is substantiated by a considerable speedup in numerical tests.

math.NA↗

arXiv2019

Error Analysis for Quadtree-Type Mesh-Coarsening Algorithms Adapted to Pixelized Heterogeneous Microstructures

Pixel- and voxel-based representations of microstructures obtained from tomographic imaging methods is an established standard in computational materials science. The corresponding highly resolved, uniform discretitization in numerical analysis is adequate to accurately describe the geometry of interfaces and defects in microstructures and, therefore, to capture the physical processes in these regions of interest. For the defect-free interior of phases and grains however, the high resolution is in view of only weakly varying field properties not necessary such that mesh-coarsening in these regions can improve efficiency without severe losses of accuracy in simulations. The present work proposes two different variants of adaptive, quadtree-based mesh-coarsening algorithms applied to pixelized images that serves the purpose of a preprocessor for consecutive finite element analyses, here, in the context of numerical homogenization. Error analysis is carried out on the microscale by error estimation which itself is assessed by true error computation. A modified stress recovery scheme for a superconvergent error estimator is proposed which overcomes the deficits of the standard recovery scheme for nodal stress computation in cases of interfaces with stiffness jump. By virtue of error analysis the improved efficiency by the reduction of unknowns is put into relation to the increase of the discretization error. This quantitative analysis sets a rational basis for decisions on favorable meshes having the best trade-off between accuracy and efficiency as will be underpinned by various examples.

math.NA↗

arXiv2019

Effective description of anisotropic wave dispersion in mechanical band-gap metamaterials via the relaxed micromorphic model

In this paper the relaxed micromorphic material model for anisotropic elasticity is used to describe the dynamical behavior of a band-gap metamaterial with tetragonal symmetry. Unlike other continuum models (Cauchy, Cosserat, second gradient, classical Mindlin-Eringen micromorphic etc.), the relaxed micromorphic model is endowed to capture the main microscopic and macroscopic characteristics of the targeted metamaterial, namely, stiffness, anisotropy, dispersion and band-gaps. The simple structure of our material model, which simultaneously lives on a micro-, a meso- and a macroscopic scale, requires only the identification of a limited number of frequency-independent and thus truly constitutive parameters, valid for both static and wave-propagation analyses in the plane. The static macro- and micro- parameters are identified by numerical homogenization in static tests on the unit-cell level in [30]. The remaining inertia parameters for dynamical analyses are calibrated on the dispersion curves of the same metamaterial as obtained by a classical Bloch-Floquet analysis for two wave directions. We demonstrate via polar plots that the obtained material parameters describe very well the response of the structural material for all wave directions in the plane, thus covering the complete panorama of anisotropy of the targeted metamaterial.

physics.app-ph↗

arXiv2019

Identification of scale-independent material parameters in the relaxed micromorphic model through model-adapted first order homogenization

We rigorously determine the scale-independent short range elastic parameters in the relaxed micromorphic generalized continuum model for a given periodic microstructure. This is done using both classical periodic homogenization and a new procedure involving the concept of apparent material stiffness of a unit-cell under affine Dirichlet boundary conditions and Neumann's principle on the overall representation of anisotropy. We explain our idea of "maximal" stiffness of the unit-cell and use state of the art first order numerical homogenization methods to obtain the needed parameters for a given tetragonal unit-cell. These results are used in the accompanying paper [16] to describe the wave propagation including band-gaps in the same tetragonal metamaterial.

physics.app-ph↗

arXiv2018

Estimating the Effective Elasticity Properties of a Diamond/$β$-SiC Composite Thin Film by 3D Reconstruction and Numerical Homogenization

The main aim of the present work is to estimate the effective elastic stiffnesses of a two-phase diamond/$β$-SiC composite thin film that is fabricated by chemical vapor deposition. The parameters of linear elasticity are determined by numerical homogenization. The database is sparse since for the 3D volume of interest only two micrographs displaying the phase distributions in perpendicular planes are available; micrographs each of a cross-section and the surface of the thin film. A representative volume element (RVE) is reconstructed by an optimization software and by means of identified material symmetries in 2D of the specimen. The elastic homogenization results indicate that the two-phase diamond/$β$-SiC composite exhibits the behavior of transverse isotropy, for which the set of six independent material parameters is identified.

cond-mat.mtrl-sci↗

arXiv2018

Convergence and Error Analysis of FE-HMM/FE$^2$ for Energetically Consistent Micro-Coupling Conditions in Linear Elastic Solids

A cornerstone of numerical homogenization is the equivalence of the microscopic and the macroscopic energy densities, which is referred to as Hill-Mandel condition. Among these coupling conditions, the cases of periodic, linear displacement and constant traction conditions are most prominent in engineering applications. While the stiffness hierarchy of these coupling conditions is a theoretically established and numerically verified result, very little is known about the numerical errors and convergence properties for each of them in various norms. The present work addresses these aspects both on the macroscale and the microscale for linear as well as quadratic finite element shape functions. The analysis addresses aspects of (i) regularity and how its loss affects the convergence behavior on both scales compared with the a priori estimates, of (ii) error propagation from micro to macro and of (iii) optimal micro-macro mesh refinement strategy. For constant traction conditions two different approaches are compared. The performance of a recovery-type error estimation based on superconvergence is assessed. All results of the present work are valid for both the Finite Element Heterogeneous Multiscale Method FE-HMM and for FE$^2$.

math.NA↗

arXiv2017

The Heterogeneous Multiscale Finite Element Method for the Homogenization of Linear Elastic Solids and a Comparison with the FE$^2$ Method

The Heterogeneous Multiscale Finite Element Method (FE-HMM) is a two-scale FEM based on asymptotic homogenization for solving multiscale partial differential equations. It was introduced in [W. E and B. Engquist, \emph{Commun. Math. Sci.}, 1 (2003), 87--132]. The objective of the present work is an FE-HMM formulation for the homogenization of linear elastic solids in a geometrical linear frame, and doing so, for the first time, of a vector-valued field problem. A key ingredient of FE-HMM is that macrostiffness is estimated by stiffness sampling on heterogeneous microdomains in terms a of modified quadrature formula, which implies an equivalence of energy densities of the microscale with the macroscale. Beyond this coincidence with the Hill-Mandel macrohomogeneity condition, which is the cornerstone of the FE$^2$ method, we elaborate a conceptual comparison with the latter method. After developing an algorithmic framework we (i) assess the existing a priori convergence estimates for the micro- and macro-errors in various norms, (ii) verify optimal strategies in uniform micro-macro mesh refinements based on the estimates, (iii) analyze superconvergence properties of FE-HMM, and (iv) compare the numerical results of FE-HMM with those of FE$^2$.

math.NA↗

arXiv2016

Geometry of logarithmic strain measures in solid mechanics

We consider the two logarithmic strain measures\[ω_{\rm iso}=\|\mathrm{dev}_n\log U\|=\|\mathrm{dev}_n\log \sqrt{F^TF}\|\quad\text{ and }\quad ω_{\rm vol}=|\mathrm{tr}(\log U)|=|\mathrm{tr}(\log\sqrt{F^TF})|\,,\]which are isotropic invariants of the Hencky strain tensor $\log U$, and show that they can be uniquely characterized by purely geometric methods based on the geodesic distance on the general linear group $\mathrm{GL}(n)$. Here, $F$ is the deformation gradient, $U=\sqrt{F^TF}$ is the right Biot-stretch tensor, $\log$ denotes the principal matrix logarithm, $\|.\|$ is the Frobenius matrix norm, $\mathrm{tr}$ is the trace operator and $\mathrm{dev}_n X$ is the $n$-dimensional deviator of $X\in\mathbb{R}^{n\times n}$. This characterization identifies the Hencky (or true) strain tensor as the natural nonlinear extension of the linear (infinitesimal) strain tensor $\varepsilon=\mathrm{sym}\nabla u$, which is the symmetric part of the displacement gradient $\nabla u$, and reveals a close geometric relation between the classical quadratic isotropic energy potential \[μ\,\|\mathrm{dev}_n\mathrm{sym}\nabla u\|^2+\fracκ{2}\,[\mathrm{tr}(\mathrm{sym}\nabla u)]^2=μ\,\|\mathrm{dev}_n\varepsilon\|^2+\fracκ{2}\,[\mathrm{tr}(\varepsilon)]^2\]in linear elasticity and the geometrically nonlinear quadratic isotropic Hencky energy\[μ\,\|\mathrm{dev}_n\log U\|^2+\fracκ{2}\,[\mathrm{tr}(\log U)]^2=μ\,ω_{\rm iso}^2+\frac\kappa2\,ω_{\rm vol}^2\,,\]where $μ$ is the shear modulus and $κ$ denotes the bulk modulus. Our deduction involves a new fundamental logarithmic minimization property of the orthogonal polar factor $R$, where $F=R\,U$ is the polar decomposition of $F$. We also contrast our approach with prior attempts to establish the logarithmic Hencky strain tensor directly as the preferred strain tensor in nonlinear isotropic elasticity.

math.DG↗

arXiv2015

Is there an order-barrier $p\leq2$ for time integration in computational elasto-plasticity?

This paper is devoted to the question, whether there is an order barrier $p\leq2$ for time integration in computational elasto-plasticity. In the analysis we use an implicit Runge-Kutta (RK) method of order $p=3$ for integrating the evolution equations of plastic flow within a nonlinear finite element framework. We show that two novel algorithmic conditions are necessary to overcome the order barrier, (i) total strains must have the same order in time as the time integrator itself, (ii) accurate initial data must be calculated via detecting the elastic-plastic switching point (SP) in the predictor step. Condition (i) is for a \emph{consistent} coupling of the global boundary value problem (BVP) with the local initial value problems (IVP) via displacements/strains. Condition (ii) generates consistent initial data of the IVPs. The third condition, which is not algorithmic but physical in nature, is that (iii) the total strain path in time must be smooth such that condition (i) can be fulfilled at all. This requirement is met by materials showing a sufficiently smooth elastic-plastic transition in the stress-strain curve. We propose effective means to fulfil conditions (i) and (ii). We show in finite element simulations that, if condition (iii) is additionally met, the present method yields the full, theoretical convergence order 3 thus overcoming the barrier $p\leq 2$ for the first time. The observed speed-up for a 3rd order RK method is considerable compared with Backward Euler.

math.NA↗

arXiv2014

The axiomatic deduction of the quadratic Hencky strain energy by Heinrich Hencky

The introduction of the quadratic Hencky strain energy based on the logarithmic strain tensor log V is a milestone in the development of nonlinear elasticity theory in the first half of the 20th century. Since the original manuscripts are written in German, they are not easily accessible today. However, we believe that the deductive approach taken by Hencky deserves to be rediscovered today. In this work we have gathered parts of the original contributions "Über die Form des Elastizitätsgesetzes bei ideal elastischen Stoffen", "Welche Umstände bedingen die Verfestigung bei der bildsamen Verformung von festen isotropen Körpern?" and "Das Superpositionsgesetz eines endlich deformierten relaxationsfähigen elastischen Kontinuums und seine Bedeutung für eine exakte Ableitung der Gleichungen für die zähe Flüssigkeit in der Eulerschen Form" which center around this deductive approach. We tried to provide, for the first time, a faithful translation into English. All footnotes are our addition.

math.HO↗

arXiv2013

A Riemannian approach to strain measures in nonlinear elasticity

The isotropic Hencky strain energy appears naturally as a distance measure of the deformation gradient to the set SO(n) of rigid rotations in the canonical left-invariant Riemannian metric on the general linear group GL(n). Objectivity requires the Riemannian metric to be left-GL(n)-invariant, isotropy requires the Riemannian metric to be right-O(n)-invariant. The latter two conditions are satisfied for a three-parameter family of Riemannian metrics on the tangent space of GL(n). Surprisingly, the final result is basically independent of the chosen parameters. In deriving the result, geodesics on GL(n) have to be parametrized and a novel minimization problem, involving the matrix logarithm for non-symmetric arguments, has to be solved.

math.CA↗