Bernhard Eidel

Andreas Fischer, Bernhard Eidel

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$^2$.

Aagashram Neelakandan, Karsten Albe, Bernhard Eidel

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.

Bernhard Eidel, Charlotte Kuhn

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.

Bernhard Eidel

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.

Bernhard Eidel, Andreas Fischer

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$^2$.