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Metallic materials may show an ultra-fine lamellar morphology leading to desirable macroscopic mechanical properties. In this paper, an analytical method for modeling the size-dependent mechanical behavior of material systems with lamellar microstructure is proposed. The main contribution of this manuscript is the combination of the exact elastic localization relations for a periodic laminate having inhomogeneous phases with a gradient plasticity constitutive model. This allows a new formulation of the equations governing the evolution of the plastic deformation as a closed system of equations. The yield functions form a system of integro-differential equations that can be solved analytically. This new framework allows to model the size-dependent mechanical behavior of multiphase lamellar materials where the subdomains are elastoplastic single crystals. This is demonstrated using a two-phase laminate. In addition, bounds for the qualitative distributions of the plastic slip and the back stress are derived using a dimensionless quantity. The kinematic hardening due to the back stress depends on the lamella width and the slip system orientation. The back stress vanishes not only for increasing lamella widths but also in slip systems where the slip plane normal is parallel to the lamination direction.

In this article, the influence of large deformations on the effective permeability of a bicontinuous porous material is investigated. On the fine-scale, Neo–Hooke hyperelasticity is considered for the solid skeleton. In a third medium approach, we model the pore space as filled with a softer material of the same type. Fluid flow through the deformed pores is expressed in terms of a Stokes' flow model. The influence of the pore pressure on the deformation is neglected, resulting in a one-way coupling which allows for a sequential solution of the two physical problems. The framework of Variationally Consistent Homogenization is used to derive a two-scale formulation based on a Representative Volume Element (RVE) characterizing the microstructure. Finally, a two-step procedure to compute the deformation dependent permeability is established: Firstly, the deformation of the RVE for a given macroscale deformation gradient is computed. Secondly, sensitivities for the fluid flow through the deformed RVE are computed and used to determine the effective permeability tensor. A numerical study is conducted for sets of RVEs with the same material parameters and different porosity. For the case of uniaxial compression, a significant influence of the deformation on the effective permeability is observed: For a macroscale compression of $��20�\%�$, the effective permeability orthogonal to the compression direction is reduced by almost $��60�\%�$.

We present a workflow for the inverse design of architected materials with targeted effective mechanical properties. The approach leverages a low-dimensional descriptor space to represent the topology and morphology of complex mesostructures, enabling efficient navigation within the design space. Data for training is generated through numerical homogenization using a Fast Fourier Transform (FFT)-based solver, providing high-fidelity mappings from structural descriptors on the mesoscale to effective properties. A neural network (NN)-based surrogate model is trained to approximate this mapping. The inverse design task is then formulated as an optimization problem over the descriptor space, where gradient-based optimizers are applied to identify the descriptors, the inputs of the surrogate. We focus on the case of anisotropic linear elasticity and demonstrate the method using spinodoid architected materials, which offer tunable anisotropy and a low-dimensional descriptor space. The framework is validated for the inverse design targeting the anisotropic stiffness of a femoral bone sample. In addition, we propose a method to determine the anisotropy class of a given stiffness tensor. This enables a quantitative evaluation of how closely the bone's anisotropy class can be approximated by spinodoids. We analyze the influence of the optimization loss function on the inverse design outcome by comparing results across different losses. Ultimately, a logarithmic loss function is chosen, as it enables simultaneous optimization of the stiffness and compliance.

The characteristics of microstructure morphology of micro-heterogeneous materials may vary over the macroscopic length scale and thus result in macroscopically distributed, uncertain material properties. Hence, multiscale approaches for the structural analysis of such materials, for example, in terms of the $��{�\text{FE}�}^{�2�}�$-method, should not be based on a single representative volume element. In this contribution a method is proposed, which considers different, artificial statistically similar volume elements at each macroscopic integration point which mimic the microstructure variability of the real material as a random field. For this purpose, the microstructure variation of the real material is quantified first in terms of the distribution of a scalar measure containing deviations of statistical measures of higher order, and then this distribution is used to construct a set of artificial microstructures to be used as volume elements within the multiscale simulation. To avoid manual discretization of the large amount of statistically similar volume elements, the finite cell method is combined with concurrent computational homogenization following the $��{�\text{FE}�}^{�2�}�$-method. The proposed method is demonstrated for two examples, a simpler tensile experiment for testing purposes and a simplified, idealized deep drawing process.

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, for example, 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: It can be linked to neural networks, kernel methods, or combined schemes. 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 Löwner sense) exist, that is, 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.

Computational homogenization has become an attractive path to determine the macroscopic inelastic behavior of a structure from the knowledge of the behavior of its (usually simpler) constituents, using a Representative Volume Element (RVE) at the microscale. Since the numerical solution of both scales by means of the FE² method is prohibitively expensive, much research effort has been spent on the development of surrogate models, particularly by making use of data-driven machine learning methods. Due to the vast spectrum of proposed surrogate modeling approaches, the decision on the method of choice for potential applicants is difficult. While computational efficiency is usually used as the central performance measure, a careful distinction between on- and offline efforts is important. Moreover, the simulation context should also influence the selection. If, for instance, a thousand simulations are to be carried out for a single microstructure, the choice of approach may be different than if single, or a few, simulations are to be carried out for a thousand different microstructures. In the latter case, the flexibility and adaption effort of the method to new microstructures plays a key role. The present study aims to derive guidelines in this direction, by comparing different such approaches and carefully analyzing their performance in simulating inelastic macroscale behaviors, influenced by the morphology of 3D foam-like microstructures. Four formulations are considered: the Deshpande–Fleck phenomenological macroscale model, a general analytical surrogate model, a hybrid neural network surrogate model, and a hyper-reduced FE² approach. The methods are compared with respect to computational costs and predictive capabilities for two 3D benchmark scenarios.

We consider a local Cahn–Hilliard-type model for tumor growth as well as a nonlocal model where, compared to the local system, the Laplacian in the equation for the chemical potential is replaced by a nonlocal operator. The latter is defined as a convolution integral with suitable kernels parametrized by a small parameter. For sufficiently smooth bounded domains in three dimensions, we prove convergence of weak solutions of the nonlocal model toward strong solutions of the local model together with convergence rates with respect to the small parameter. The proof is done via a Gronwall-type argument and a convergence result with rates for the nonlocal integral operator toward the Laplacian due to Abels and Hurm.

We study signals that are sparse either on the vertices of a graph or in the graph spectral domain. Recent results on the algebraic properties of random integer matrices as well as on the boundedness of eigenvectors of random matrices imply two types of support size uncertainty principles for graph signals. Indeed, the algebraic properties imply uniqueness results if a sparse signal is sampled at any set of minimal size in the other domain. The boundedness properties of eigenvectors imply stable reconstruction by basis pursuit if a sparse signal is sampled at a slightly larger randomly selected set in the other domain.

The evolution of image halftoning, from its analog roots to contemporary digital methodologies, encapsulates a fascinating journey marked by technological advancements and creative innovations. Yet the theoretical understanding of halftoning is much more recent. In this article, we explore various approaches towards shedding light on the design of halftoning approaches and why they work. We discuss both halftoning in a continuous domain and on a pixel grid. We start by reviewing the mathematical foundation of the so-called electrostatic halftoning method, which departed from the heuristic of considering the back dots of the halftoned image as charged particles attracted by the grey values of the image in combination with mutual repulsion. Such an attraction-repulsion model can be mathematically represented via an energy functional in a reproducing kernel Hilbert space allowing for a rigorous analysis of the resulting optimization problem as well as a convergence analysis in a suitable topology. A second class of methods that we discuss in detail is the class of error diffusion schemes, arguably among the most popular halftoning techniques due to their ability to work directly on a pixel grid and their ease of application. The main idea of these schemes is to choose the locations of the black pixels via a recurrence relation designed to agree with the image in terms of the local averages. We discuss some recent mathematical understanding of these methods that is based on a connection to $��\sum �\Delta �$ quantizers, a popular class of algorithms for analog-to-digital conversion.