Continuum Models and Discrete Systems

We present a probabilistic learning inference that assimilates data (target set) into a parameterized large stochastic computational model resulting from discretizing a stochastic boundary value problem (BVP). A target is imposed on a vector-valued random quantity of interest (QoI), observed as the stochastic solution of the BVP. The probabilistic inference estimates the posterior probability model, which is constrained both by the second-order moment of the random residue of the BVP stochastic equations and the target set composed of statistical moments of the QoI. We assume that evaluating a single realization of the BVP is computationally expensive, so the training dataset comprises only a few points differing from big data approaches. The presented application contributes to three-dimensional stochastic homogenization of heterogeneous linear elastic media, specifically when the mesoscale and macroscale are not separated.

Mechanics and physics of random media strongly suggest that stochastic PDEs and stochastic finite element methods require mesoscale tensor-valued random fields (TRFs) of constitutive laws with locally anisotropic fluctuations. Such models are also useful when there is interest in fields of dependent quantities (velocity, strain, stress...) that need to be constrained by the balance laws (of mass, momentum...); examples are irrotational and solenoidal TRFs. In this article, we review the canonical forms of general correlation structures of second-order, mean-square continuous, wide-sense homogeneous and isotropic TRFs of ranks 1, ... , 4 in 3d. Besides “conventional” covariances, this approach can be used to construct TRFs with fractal and Hurst (long-range memory) characteristics. The current research extends the earlier work on scalar-valued RFs (including random processes) in vibration problems, rods and beams with random properties under random loadings, as well as elastodynamics, wavefronts, fracture, homogenization of random media, and contact mechanics.

This paper examines the feasibility of performing simulations of Gaussian random fields at all points of compact domains of \(\mathbb {R}^d\). The simulations produced by the Karhunen-Loève method are compared with those obtained using the standard turning bands method. The interest of combining both methods is also tested.

In view of predicting the overall thermal conductivity of ceramics with a granular microstructure exhibiting complex networks of pores preferably located at the interfaces between granules, a simple microstructural model has been developed to capture the dominant features of such a specific morphology. Generated 3D virtual networks of pores are obtained as the intersection of a Voronoi tessellation, representing the granules, and a boolean model of monodisperse overlapping spheres [1]. The two non-dimensional parameters fully determining such networks have been first adjusted so as to reproduce at best some morphological properties extracted from 2D optical images of various real samples [2]. To further validate this model, several small samples of the same materials have in addition been imaged in 3D by synchrotron X-ray micro-computed tomography, with a specific setup adapted to the very strong attenuation of these ceramics. The consistency of the results measured in 2D on cross-sections of tomography images with those measured on optical microscopy images has been first checked. Next, qualitative comparisons between observed 3D pore networks and simulated ones with parameters adjusted from 2D observations have been complemented by more quantitative ones.

Epithelial tissues consist of cells attached to each other by junctions that may move collectively. We review results obtained by numerical simulations of vertex models in which cells are polygons that tile a region of the plane. In particular, we present results for wound healing assays, in which cells migrate to an empty space, and for antagonistic migration assays, in which cells from one tissue invade an adjacent tissue comprising cells of different type. We discuss coarse-graining the active vertex model to obtain continuum equations at macroscopic scales and review a selection of results on the anisotropy transition of homogeneous phases, their stability and the effects of shear.

The paper discusses a long-standing problem of exact bounds for the effective properties of multimaterial composites. The suggested bounds refine Hashin-Shtrikman’s bounds in the region of parameters where the last ones are loose. They are derived by the description of gradient fields in the components of optimal structures. We show that the gradient fields vary in restricted domains; taking this into account, we modify the Translation method to obtain new exact bounds. These bounds are multifaced: the active or passive inequality constraints lead to algebraically different expressions and topologically different optimal structures. The bounds for the energy and optimal structures are explicitly computed for three-material conducting and elastic composites (one material is void). However, a unified approach to deriving a set of constraints in the supporting fields is yet to be developed.

We investigate the influence of crystallographic twins on the elastoplastic response of \(\gamma \)-TiAl intermetallics via full-field FFT-based computations. We first introduce a hierarchical stochastic model, which is used to simulate synthetic polycrystalline microstructures containing twin grains with certain morphologies, and apply it to generate representative volume elements. Second, we develop a Fourier-based method with regularization for solving the effective and local mechanical response of polycrystalline media using the Méric-Cailletaud crystal plasticity constitutive law. Numerical results show that, across configurations of twinning, the corresponding average effective response is similar. Although differences were quantified, the effect of twins on the yield stress is negligible in practice (less than \(1\%\)).

Architected materials are designed with specific configurations that offer enhanced properties, making them ideal for addressing various challenges in materials science, architecture, aerodynamics, and mechanical engineering. Their unique quality, coupled with the ability to tailor mechanical properties in every direction, renders them highly suitable for industries like aerospace, automotive, marine, and construction. However, the application of architected materials depends on the development of accurate models to understand the complex relationship between microstructure characteristics and macroscopic behavior. Despite the proposal and discussion of numerous analytical and numerical methods in recent years, very few studies have derived explicit formulas for effective mechanical properties. This paper contributes to this underexplored area by presenting a mathematical formulation and modeling technique for the effective elastic moduli of a three-dimensional orthotropic hexatruss lattice. The analytical relations we have established, validated through comprehensive experimental tests on a 3D-printed lattice, demonstrate the significant impact of lattice parameters on macroscopic properties. Practically, our results could simplify the process of parametric optimization for architectured materials, offering a less resource-intensive approach to optimization since parameter changes do not necessitate lattice regeneration.

Subjecting thermoplastic-based laminates to high thermal energy, such as a flame, leads to the gradual deterioration of the matrix, involving solid-state changes and significant fluctuations in thermomechanical properties. Despite these changes and the presence of substantial temperature gradients, the laminates can still bear mechanical loads, even after the matrix has melted. When temperatures exceed the melting point, the primary mechanism of matrix thermal decomposition involves the formation of voids. While these voids weaken the material from a mechanical perspective, they also act as thermal insulators, protecting the matrix on the side facing away from the heat source. To accurately describe the evolution of thermomechanical properties whereas one face of the laminates is exposed to a flame, it is crucial to understand the kinetics of void formation. Experimental investigations revealed that porosity content and swelling are strongly dependent on the time and temperature of thermal exposure. A mesoscopic Finite Element model was thus developed, representing porosities at a structural level. The formation of voids and the associated swelling were simulated using a probabilistic approach, guiding the progressive transformation of elements into voids based on their thermal state.

A new solution scheme for homogenization of composites is studied in this paper. This scheme introduced in Sab et al. [10] is obtained by adapting Eyre-Milton’s one [1]. It can efficiently handle the case of infinitely double-contrasted composites containing both pores and infinitely rigid inclusions. While a very simple micro structure was previously considered by the authors, a much more realistic micro structure is studied here, and more insights on the role of the reference medium introduced by the Lippmann-Schwinger equation are given.

Starting from the finite element equations of a periodic beam lattice, a continuum theory is derived via Taylor series expansions. The continualised equations of motion are of the micropolar type and thus contain stiffness and inertia contributions in terms of translational as well as rotational degrees of freedom. Scrutiny of the underlying energy functionals and of the model’s response to excitation via harmonic waves confirms that the continualised model is unstable. In order to stabilise the model, Padé approximation is applied to the rotational equation of motion. As a result of this process, micro-inertia terms appear in the model that are expressed as the spatial gradients of the standard rotational inertia terms. It is demonstrated by means of an analysis of dispersive waves that the Padé approximation not only stabilises the model but also improves the accuracy with which the continuum model approximates the original discrete model.

In this paper we introduce a mathematical model designed to analyse the interaction between gyroscopic forces and the gravitational field and which has been essential in studying the dynamic response of several classes of chiral metamaterials. For an elementary cell, which includes either a normal or inverted gyroscopic pendulum, we provide the full classification of its transient oscillations linked to the sufficiently high rate of spin. The characteristics of physical chirality in the presence of gravity cover a range of physical phenomena at different scales. The theoretical work is accompanied by the illustrative examples.

The macroscopic permittivity tensors of a film-shaped metamaterial medium are calculated for the case when the charge carrier concentration \(N_0\) of each constituent of the composite metamaterial varies across the film thickness. The influence of an applied transverse voltage on the surface plasmon resonances and on the optical properties of such a system are studied. It is shown that the macroscopic effective permittivity tensor, as well as other optical properties of the metamaterial, are extremely sensitive to the applied transverse voltage. This can be used to construct fast switches and other optical devices.

A method for designing variational principles for the dynamics of a possibly dissipative and non-conservatively forced chain of particles is demonstrated. Some qualitative features of the formulation are discussed.

Programmable metamaterials (PMs) belong to the class of metamaterials offering controllable and variable physical properties. As metamaterials, they are architectured, cellular materials and exhibit exotic behavior, but are more specifically tailored for engineering purposes: PMs can be applied wherever a custom-fit reaction to environmental and operating conditions of a material is required. Instead of a homogeneous layout of unit cells, often considered for common metamaterials, we construct PMs by an individual distribution of varying unit cells. In order to tackle this customization of material response, a computational framework similar to topology or material optimization is proposed. Beyond that, our work is based on a multiscale approach and a surrogate material model, allowing a broad range of applications with different classes of unit cells and target functions under finite strains. In this contribution, we present the complete simulation chain: from a parametrized unit cell to the final model of the programmable material, ready to be manufactured. Numerical results with different unit cells are shown and compared to fully resolved simulations as well as to experiments.

The modeling of the dynamics of thin solid state films dates back to a series of seminal papers by W.W. Mullins, published in the years 1956–1958. In those years, there was much interest in the stability of bulk bi- and poly-crystalline materials, which can be strongly affected by the dynamics of their exterior surface features. In recent years the focus has been extended to include many new materials, which can enable the manufacture of strong and thin films; however, much of the basic underlying physics remains the same. Thus it is not unreasonable in considering thin film stability, to return and study the implications of the original modeling equations as developed by Mullins, namely surface diffusion and mean curvature motion. We follow this approach, focusing on surface diffusion and mean curvature motions, while neglecting possible anisotropy and elastic effects. We show that even within this somewhat limited framework, it is possible to obtain new criteria for the stability of holes in thin films, based on a new notion of an effective radius. This note summarizes results appearing in [7, 18, 19]. Many open questions remain.

In this study, we revisit the Choquet capacity in the framework of convolutional neural networks, in (\(\max ,+\))-algebra. By incorporating a discrete and learnable Choquet capacity model, we enhance the ability to represent the spatial arrangement and density variations in random point processes of convolutional neural networks. To validate the effectiveness of our approach, numerical experiments are conducted on synthetic datasets simulating diverse spatial point patterns of the Neyman-Scott process. When compared to classical convolutional neural networks, the proposed approach exhibits comparable or improved performances in terms of classification. Superior results are also observed in regression problems involving the Neyman-Scott parameter that monitors the point patterns spatial dispersion.

Machine learning offers attractive solutions to challenging image processing tasks. Tedious development and parametrization of algorithmic solutions can be replaced by training a convolutional neural network or a random forest with a high potential to generalize. However, machine learning methods rely on huge amounts of representative image data along with a ground truth, usually obtained by manual annotation. Thus, limited availability of training data is a critical bottleneck. We discuss two use cases: optical quality control in industrial production and segmenting crack structures in 3D images of concrete. For optical quality control, all defect types have to be trained but are typically not evenly represented in the training data. Additionally, manual annotation is costly and often inconsistent. It is nearly impossible in the second case: segmentation of crack systems in 3D images of concrete. Synthetic images, generated based on realizations of stochastic geometry models, offer an elegant way out. A wide variety of structure types can be generated. The within structure variation is naturally captured by the stochastic nature of the models and the ground truth is for free. Many new questions arise. In particular, which characteristics of the real image data have to be met to which degree of fidelity.

In order to enable engineers to make informed decisions about material selection, design, and maintenance, contributing to the safety, reliability, and longevity of components and structures, subjected to cyclic (fatigue) loading, it is essential to have knowledge of the threshold value of the stress intensity factor range and the range of plain fatigue limit of a material. Traditional experimental approaches, although offering an accurate determination of these parameters, are, however, expensive and time-consuming. It is thus evident, that there is a need for an alternative methodology, offering accurate and reliable on one hand, but fast and cheap on the other, predictions of aforementioned parameters. The main focus of this chapter is to analyse the ability of the data driven fuzzy inference system (FIS) approach to serve this goal and predict fatigue parameters of a material, knowing material’s strength (static) characteristics. Results, reported in this chapter, for aluminium alloys and different steels data sets demonstrate capabilities of FIS to estimate, with the high degree of accuracy, the range of the plain fatigue limit and the range of the threshold value of the stress intensity factor, based on provided ultimate tensile strength and yield strength of a metal.

Inspired by biological structural designs observed in nature, this work explores the structure-property relationships in structures under dynamic transverse and longitudinal compression. The two primary structures analyzed are low porosity structures inspired by sheep horns and 2D extruded thin-walled structures inspired by design elements found in bamboo, beetles, and crabs. The low-porosity structures exhibit superior mechanical properties in nature, with porosity ranging from 1–5%, while the thin-walled structures provide insights into the effect of geometric feature interactions leading to high energy absorption during impact. The current work utilizes the gated recurrent unit (GRU) model to predict the mechanical response of the structures during the impact. The inputs used in GRU models were varied to present different techniques to predict stress-strain response for the structures at a given loading condition. The first method utilized the parametric representation of the geometric features, while the other used a combinatorial approach and autoencoders to prepare inputs for the GRU model. The ground-truth data was obtained using finite element simulations with the rate-dependent elastoplastic and Johnson-Cook material models. The trained models allow rapid evaluations of stress-strain response and allow the elimination of poor designs.