Chapter 5 Computational Study of Multiscale Homogenization for Surface Stress
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NH-PINN: Neural homogenization-based physics-informed neural network for multiscale problems
Physics-informed neural network (PINN) is a data-driven approach to solving equations. It is successful in many applications; however, the accuracy of the PINN is not satisfactory when it is used to solve multiscale equations. Homogenization approximates a multiscale equation by a homogenized equation without multiscale property; it includes solving cell problems and the homogenized equation. The cell problems are periodic, and we propose an oversampling strategy that significantly improves the PINN accuracy on periodic problems. The homogenized equation has a constant or slow dependency coefficient and can also be solved by PINN accurately. We hence proposed a 3-step method, neural homogenization based PINN (NH-PINN), to improve the PINN accuracy for solving multiscale problems with the help of homogenization.
97 MATHEMATICS AND COMPUTING↗
Thermal conductivity of cement paste: Influence of macro-porosity
Thermal conductivity of cement paste is significantly influenced by the presence of macro-pores within it. These pores (pores with size greater than 6 μm, measured using microCT) which may be filled with air and/or water significantly alters the thermal conductivity of the resulting material. A combination of experiments, molecular dynamic investigations of major constituents of the cement paste, multiscale homogenization techniques, and inclusion of thermal conductivity of the pores containing either air or water has been utilized in this study to reveal the influence of pores in the calculation of thermal conductivity of the composite material.
36 MATERIALS SCIENCE↗
Assessment of effective elastic constants of U-10Mo fuel: A multiscale modeling and homogenization study
The significant microstructural changes that U-Mo fuel undergoes during operation degrades its mechanical properties and structural integrity. Microstructural evolution entails the formation, evolution, and redistribution of porosity in conjunction with grain refinement. In the present paper, we employ numerical approaches to assess the impact of the various microstructural features—grains, nanoscale intragranular fission gas bubbles, and mesoscale intergranular voids—on the degradation of elastic constants. Phase-field microstructure models are combined with the asymptotic expansion homogenization technique in order to derive the effective elastic constants as a function of porosity and fission density. Here the results are verified and compared against theoretical bounds. Using this approach, elastic degradation in operating nuclear fuels can be quantified when the distributions of microstructural features are known.
11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
Correlations of multiscale structural evolution and homogeneous flows in metallic glass ribbons
Studying the flow behavior is critical to understand the deformation mechanism of amorphous solids. However, detecting the basic flow events in amorphous solids is challenging. Here, by simultaneous SAXS/WAXS, elementary flow carriers in wound metallic glasses are identified from flow-induced structural heterogeneities with a radius of gyration of 2.5~3.5 nm. Their size increases and morphology changes from sphere-like to rod-like under flow. Moreover, the atomic structure exhibits an unusual change to a more disordered state during winding/annealing at the temperature of ~0.8 T g . This work provides an atomic-to-nanoscale description of the flow carriers of amorphous solids during deformation. Impact Statement The elementary flow carriers in metallic glasses have a radius of gyration of 2.5~3.5 nm. During flow, their size increases and morphology changes from sphere-like to rod-like.
36 MATERIALS SCIENCE↗
Micrometer: Micromechanics transformer for predicting full field mechanical responses of heterogeneous materials
Predicting mechanical responses of heterogeneous materials across scales remains a significant challenge. Traditional computational methods often struggle with complex and multiscale nature of these materials, limiting their effectiveness in real-world applications. Here, in this paper, we introduce Micrometer, a vision transformer based deep learning model designed to predict full field mechanical responses of heterogeneous materials, bridging the gap between computer vision and solid mechanics problems. We show that Micrometer, trained on a large-scale high-resolution dataset of 2D fiber-reinforced composites, can achieve state-of-the-art performance in predicting microscale strain fields across a wide range of material properties and loading conditions. Our model demonstrates accuracy and computational efficiency in applications such as computational homogenization and multiscale modeling, reducing computational time by up to two orders of magnitude compared to conventional numerical solvers while maintaining less than 1 % errors in predicting macroscale stress fields. Furthermore, we showcase Micrometer’s adaptability through transfer learning experiments on new materials with limited data, highlighting its potential to tackle diverse scenarios in computational solid mechanics. These results represent a significant step towards AI-driven innovation in materials science, addressing the limitations of traditional numerical methods and paving the way for more efficient simulations of heterogeneous materials across various industrial applications.
Composite materials↗
Multiscale Progressive Failure Analysis of 3D Woven Composites
Application of three-dimensional (3D) woven composites is growing as an alternative to the use of ply-based composite materials. However, the design, analysis, modeling, and optimization of these materials is more challenging due to their complex and inherently multiscale geometries. Herein, a multiscale modeling procedure, based on efficient, semi-analytical micromechanical theories rather than the traditional finite element approach, is presented and applied to a 3D woven carbon–epoxy composite. A crack-band progressive damage model was employed for the matrix constituent to capture the globally observed nonlinear response. Realistic microstructural dimensions and tow-fiber volume fractions were determined from detailed X-ray computed tomography (CT) and scanning electron microscopy data. Pre-existing binder-tow disbonds and weft-tow waviness, observed in X-ray CT scans of the composite, were also included in the model. The results were compared with experimental data for the in-plane tensile and shear behavior of the composite. The tensile predictions exhibited good correlations with the test data. While the model was able to capture the less brittle nature of the in-plane shear response, quantitative measures were underpredicted to some degree.
3D woven composites↗
A Catalyst-Like System Enables Efficient Perovskite Solar Cells
High-quality perovskite films are essential for achieving high performance of optoelectronic devices; However, solution-processed perovskite films are known to suffer from compositional and structural inhomogeneity due to lack of systematic control over the kinetics during the formation. Here, in this work, the microscopic homogeneity of perovskite films is successfully enhanced by modulating the conversion reaction kinetics using a catalyst-like system generated by a foaming agent. The chemical and structural evolution during this catalytic conversion is revealed by a multimodal synchrotron toolkit with spatial resolutions spanning many length scales. Combining these insights with computational investigations, a cyclic conversion pathway model is developed that yields exceptional perovskite homogeneity due to enhanced conversion, having a power conversion efficiency of 24.51% for photovoltaic devices. This work establishes a systematic link between processing of precursor and homogeneity of the perovskite films.
14 SOLAR ENERGY↗
A mean field homogenization model for the mechanical response of ceramic matrix composites
SiC/SiC composites offer exceptional mechanical stability at high temperatures and under irradiation. These ceramic matrix composites are therefore strong candidate materials for future nuclear energy applications. Their mechanical response, which exhibits pseudo-plasticity, is mediated by matrix cracking, fiber debonding, and fiber pull-out due to slip. Here, this study introduces a mechanistic model for the behavior of unidirectionally reinforced SiC/SiC composites. Specifically a mean field homogenization approach is proposed to account for all deformation and degradation modes during mechanical deformation. The homogenization scheme relies on a Mori Tanaka method that is extended to consider the effects of the coating’s elasto-plastic response on the development of micromechanical fields. Further, the model proposed introduces a method to effectively account for the role of localized damage (i.e., cracks) on mechanical fields within both the fiber and the matrix. Upon validating the model against experimental data, the roles of interface sliding, coating dimensions and intrinsic elastic response, as well as of microstructure (e.g. porosity, fiber volume fraction) are discussed.
11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
Learning constitutive relations using symmetric positive definite neural networks
In this work, we present a new neural-network architecture, called the Cholesky-factored symmetric positive definite neural network (SPD-NN), for modeling constitutive relations in computational mechanics. Instead of directly predicting the stress of the material, the SPD-NN trains a neural network to predict the Cholesky factor of the tangent stiffness matrix, based on which the stress is calculated in incremental form. As a result of this special structure, SPD-NN weakly imposes convexity on the strain energy function, satisfies the second order work criterion (Hill's criterion) and time consistency for path-dependent materials, and therefore improves numerical stability, especially when the SPD-NN is used in finite element simulations. Depending on the types of available data, we propose two training methods, namely direct training for strain and stress pairs and indirect training for loads and displacement pairs. We demonstrate the effectiveness of SPD-NN on hyperelastic, elasto-plastic, and multiscale fiber-reinforced plate problems from solid mechanics. The generality and robustness of SPD-NN make it a promising tool for a wide range of constitutive modeling applications.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Bayesian Deep Operator Learning for Homogenized to Fine-Scale Maps for Multiscale PDE
Not provided.
Mathematics↗
Nanoscopic cross-grain cation homogenization in perovskite solar cells
Multiscale cation inhomogeneity has been a major hurdle in state-of-the-art formamidinium–caesium (FA–Cs) mixed-cation perovskites for achieving perovskite solar cells with optimal power conversion efficiencies and durability. Although the field has attempted to homogenize the overall distributions of FA–Cs in perovskite films from both plan and cross-sectional views, our understanding of grain-to-grain cation inhomogeneity and ability to tailor it—that is, spatially resolving the FA–Cs compositional difference between individual grains down to the nanoscale—are lacking. Here we reveal that as fundamental building blocks of a perovskite film, individual grains exhibit cationic compositions deviating from the prescribed ideal composition, severely limiting the interfacial optoelectronic properties and perovskite layer durability. This performance-limiting nanoscopic factor is linked to thermodynamic-driven morphological grooving, leading to a segmented surface landscape. At the grain triple junctions, grooves form nanoscale groove traps that hinder the mixing of solid-state cations across grains and thus retard inter-grain FA–Cs mixing. By rationally modulating the heterointerfacial energies, we reduced the depth of these nanoscale groove traps by a factor of three, significantly improving cation homogeneity. Perovskite solar cells with shallower nanoscale groove traps demonstrate enhanced power conversion efficiencies (25.62%) and improved stability under various standardized international protocols. In conclusion, our work highlights the significance of resolving surface nano-morphologies for homogeneous properties of perovskites.
77 NANOSCIENCE AND NANOTECHNOLOGY↗
An FFT-based approach for Bloch wave analysis: application to polycrystals
A method based on the Fast Fourier Transform is proposed to obtain the dispersion relation of acoustic waves in heterogeneous periodic media with arbitrary microstructures. The microstructure is explicitly considered using a voxelized Representative Volume Element (RVE). The dispersion diagram is obtained solving an eigenvalue problem for Bloch waves in Fourier space. To this aim, two linear operators representing stiffness and mass are defined through the use of differential operators in Fourier space. The smallest eigenvalues are obtained using the implicitly restarted Lanczos and the subspace iteration methods, and the required inverse of the stiffness operator is done using the conjugate gradient with a preconditioner. The method is used to study the propagation of acoustic waves in elastic polycrystals, showing the strong effect of crystal anistropy and polycrystaline texture on the propagation. It is shown that the method combines the simplicity of classical Fourier series analysis with the versatility of Finite Elements to account for complex geometries proving an efficient and general approach which allows the use of large RVEs in 3D.
97 MATHEMATICS AND COMPUTING↗
Evolution of elastic and thermal properties of cementitious composites containing micro-size lightweight fillers after exposure to elevated temperature
This work studies the evolution of elastic and thermal properties of cementitious composites containing micro-size lightweight fillers after exposure to elevated temperatures (up to 800 °C). A multiscale investigation is carried out to study the thermal degradation of cementitious composite materials at multiple scale levels: at cement past level, the dehydration kinetics of cement hydrates (i.e., C–S–H, CH and aluminate hydrates) and development of microcracks are studied as functions of exposure temperatures. The damage mechanism of the inclusion phases and damages within the interfacial transition zone (ITZ) under elevated temperature are also investigated. In addition, the change of mechanical (i.e., elastic moduli, compressive strengths) and thermal (thermal conductivities) properties are measured as functions of the exposure temperature. Based on the experimental studies, a multiscale thermal degradation model is developed where the thermal degradations of cement paste (including water composition, porosity, and solid phase changes), inclusion phases, and the interfaces are captured through a multiscale sub-stepping homogenization scheme. The thermal degradation model is validated through multiple sets of experimental data. Finally, the influence of key parameters including the particle size distribution of the inclusions, the chemical composition, as well as the effects of ITZ damage are studied.
36 MATERIALS SCIENCE↗
Multiscale modeling of cruciform dwell tests with the uncertainty-quantified parametrically homogenized constitutive model
Abstract not provided
Maloth, Thirupathi↗
Multiscale design of nonlinear materials using a Eulerian shape optimization scheme
Motivated by recent advances in manufacturing, the design of materials is the focal point of interest in the material research community. One of the critical challenges in this field is finding optimal material microstructure for a desired macroscopic response. This work presents a computational method for the mesoscale-level design of particulate composites for an optimal macroscale-level response. The method relies on a custom shape optimization scheme to find the extrema of a nonlinear cost function subject to a set of constraints. Three key “modules” constitute the method: multiscale modeling, sensitivity analysis, and optimization. Multiscale modeling relies on a classical homogenization method and a nonlinear NURBS-based generalized finite element scheme to efficiently and accurately compute the structural response of particulate composites using a nonconformal discretization. A three-parameter isotropic damage law is used to model microstructure-level failure. An analytical sensitivity method is developed to compute the derivatives of the cost/constraint functions with respect to the design variables that control the microstructure's geometry. The derivation uncovers subtle but essential new terms contributing to the sensitivity of finite element shape functions and their spatial derivatives. Several structural problems are solved to demonstrate the applicability, performance, and accuracy of the method for the design of particulate composites with a desired macroscopic nonlinear stress-strain response.
42 ENGINEERING↗
Tardigrade-examples V0.1.0
Tardigrade-examples is a repository of computational workflows that exercise the Tardigrade software package. The Tardigrade software package is an implementation of Eringen’s micromorphic continuum theory with capabilities to support multiscale material modeling. These capabilities include homogenization through the Micromorphic Filter, calibration of micromorphic material models, and macroscale simulation in Tardigrade-MOOSE. This repository investigates continuum upscaling of various direct numerical simulations (DNS) conducted in Abaqus finite element (FE), Ratel FE, and GEOS material point method (MPM) software. Verification of the upscaling workflow is first investigated by considering DNS of trivial stress states for homogeneous materials, results of which indicate that classical continuum behavior is recovered as expected. DNS of heterogeneous materials are then considered.
Allard, Thomas↗
Results of the Micromorphic Upscaling Workflow for the PSAAP III Year 3 Report
Predicting the mechanical response and failure of heterogeneous materials has proven difficult. Multiscale numerical methods based in higher order continuum theories attempt to bridge the gap between microscale and macroscale structural behavior. Micromorphic continuum theories have shown promise. The Tardigrade software package is an implementation of Eringen’s micromorphic continuum theory with capabilities to support multiscale material modeling workflows. These include homogenization through the Micromorphic Filter, calibration of micromorphic material models, and macroscale simulation in Tardigrade-MOOSE. This work discusses micromorphic upscaling efforts of the University of Colorado Boulder PSAAP III multidisciplinary simulation center (MSC). Verification studies are presented that compare the accuracy of the upscaling workflow with analytical solutions for a trivial stress state and homogeneous material using direct numerical simulations (DNS) conducted in the Ratel finite element method (FEM) and GEOS material point method (MPM) codes. These verification studies consider upscaling using the Micromorphic Filter for a “single filter domain”. Finally, DNS of a heterogeneous composite material is upscaled using a “multiple filter domain” method.
36 MATERIALS SCIENCE↗