Physics-informed machine learning in asymptotic homogenization of elliptic equations

We apply physics-informed neural networks (PINNs) to first-order, two-scale, periodic asymptotic homogenization of the property tensor in a generic elliptic equation. The problem of lack of differentiability of property tensors at the sharp phase interfaces is circumvented by making use of a diffuse interface approach. Periodic boundary conditions are incorporated strictly through the introduction of an input-transfer layer (Fourier feature mapping), which takes the sine and cosine of the inner product of position and reciprocal lattice vectors. This, together with the absence of Dirichlet boundary conditions, results in a lossless boundary condition application. Consequently, the sole contributors to the loss are the locally-scaled differential equation residuals. We use crystalline arrangements that are defined via Bravais lattices to demonstrate the formulation's versatility based on the reciprocal lattice vectors. We also show that considering integer multiples of the reciprocal basis in the Fourier mapping leads to improved convergence of high-frequency functions. We consider applications in one, two, and three dimensions, including periodic composites, composed of embeddings of monodisperse inclusions in the form of disks/spheres, and stochastic monodisperse disk arrangements.</p

Physics-informed machine learning in asymptotic homogenization of elliptic equations

We apply physics-informed neural networks (PINNs) to first-order, two-scale, periodic asymptotic homogenization of the property tensor in a generic elliptic equation. The problem of lack of differentiability of property tensors at the sharp phase interfaces is circumvented by making use of a diffuse interface approach. Periodic boundary conditions are incorporated strictly through the introduction of an input-transfer layer (Fourier feature mapping), which takes the sine and cosine of the inner product of position and reciprocal lattice vectors. This, together with the absence of Dirichlet boundary conditions, results in a lossless boundary condition application. Consequently, the sole contributors to the loss are the locally-scaled differential equation residuals. We use crystalline arrangements that are defined via Bravais lattices to demonstrate the formulation’s versatility based on the reciprocal lattice vectors. We also show that considering integer multiples of the reciprocal basis in the Fourier mapping leads to improved convergence of high-frequency functions. We consider applications in one, two, and three dimensions, including periodic composites, composed of embeddings of monodisperse inclusions in the form of disks/spheres, and stochastic monodisperse disk arrangements

Generative reconstruction of 3D volume elements for Ti-6Al-4V basketweave microstructure by optimization of CNN-based microstructural descriptors

We present a methodology for the generative reconstruction of 3D Volume Elements (VE) for numerical multiscale analysis of Ti-6Al-4V processed by Additive Manufacturing (AM). The basketweave morphology, which is typically dominant in AM-processed Ti-6Al-4V, is analyzed in conventional Electron Backscatter Diffusion (EBSD) micrographs. Prior \b{eta}-grain reconstruction is performed to obtain the out-of-plane orientation of the observed grains leveraging Burgers orientation relationship. Convolutional Neural Network (CNN) - based microstructure descriptors are extracted from the 2D data, and used for cross-section-based optimization of pixel values on orthogonal planes in 3D, using the Microstructure Characterization and Reconstruction (MCR) implementation MCRpy [16]. In order to utilize MCRpy, which performs best for binary systems, the basketweave microstructure, which consists of up to twelve distinct grain orientations, is decomposed into several separate two-phase systems. Our reconstructions capture key characteristics of the titanium basketweave morphology and show qualitative resemblance to experimentally obtained 3D data. The preservation of volume fraction during assembly of the reconstruction remains an unadressed challenge at this stage

Experimental and computational study of ductile fracture in small punch tests

A unified experimental-computational study on ductile fracture initiation and propagation during small punch testing is presented. Tests are carried out at room temperature with unnotched disks of different thicknesses where large-scale yielding prevails. In thinner specimens, the fracture occurs with severe necking under membrane tension, whereas for thicker ones a through thickness shearing mode prevails changing the crack orientation relative to the loading direction. Computational studies involve finite element simulations using a shear modified Gurson-Tvergaard-Needleman porous plasticity model with an integral-type nonlocal formulation. The predicted punch load-displacement curves and deformed profiles are in good agreement with the experimental results

Elastostatics of star-polygon tile-based architectured planar lattices

We showed a panoptic view of architectured planar lattices based on star-polygon tilings. Four star-polygon-based lattice sub-families were investigated numerically and experimentally. Finite element-based homogenization allowed computation of Poisson's ratio, elastic modulus, shear modulus, and planar bulk modulus. A comprehensive understanding of the range of properties and micromechanical deformation mechanisms was developed. By adjusting the star angle from $0^\circ$ to the uniqueness limit ($120^\circ$ to $150^\circ$), our results showed an over 250-fold range in elastic modulus, over a 10-fold range in density, and a range of $-0.919$ to $+0.988$ for Poisson's ratio. Additively manufactured lattices showed good agreement in properties. The additive manufacturing procedure for each lattice is available on www.fullcontrol.xyz/#/models/1d3528. Three of the four sub-families exhibited in-plane elastic isotropy. One showed high stiffness with auxeticity at low density with a primarily axial deformation mode as opposed to bending deformation for the other three lattices. The range of achievable properties, demonstrated with property maps, proves the extension of the conventional material-property space. Lattice metamaterials with Triangle-Triangle, Kagome, Hexagonal, Square, Truncated Archimedean, Triangular, and Truncated Hexagonal topologies have been studied in the literature individually. We show that all these structures belong to the presented overarching lattices