Lippmann–Schwinger Spectrum, Composite Materials Eigenstates and Their Role in Computational Homogenization

Stiffness and strength evaluation of lattice-based mechanical metamaterials by decoupled two-scale analysis

Anisotropic strain gradient thermoelasticity for cellular structures: Plate models, homogenization and isogeometric analysis

The computation of Green’s function is a basic and time-consuming task in realizing seismic imaging using integral operators because the function is the kernel of the integral operators and because every image point functions as the source point of Green’s function. If the perturbation theory is used, the problem of the computation of Green’s function can be transformed into one of solving the Lippmann–Schwinger (L–S) equation. However, if the velocity model under consideration has large scale and strong heterogeneity, solving the L–S equation may become difficult because only numerical or successive approximate (iterative) methods can be used in this case. In the literature, one of these methods is the generalized successive over-relaxation (GSOR) iterative method, which can effectively solve the L–S equation and obtain the desired convergent iterative series. However, the GSOR iterative method may encounter slow convergence when calculating the high-frequency Green’s function. In this paper, we propose a new scheme that utilizes the GSOR iterative with a precondition to solve the complex wavenumber L–S equation in a slightly attenuated medium. The complex wavenumber with imaginary components localizes the energy of the background Green’s function and reduces its singularity by enabling exponential decay. Introduction of the preconditioning operator can further improve the convergence speed of the GSOR iterative series. Then, we provide a preconditioned generalized successive over-relaxation (pre-GSOR) iterative format. Our numerical results show that if an appropriate damping factor and a proper preconditioning operator are selected, the method presented here outperforms the GSOR iterative for the real wavenumber L–S equation in terms of the convergence speed, accuracy, and adaptation to high frequencies.

A novel double‐higher‐order entire‐domain volume integral equation (VIE) technique for efficient analysis of electromagnetic structures with continuously inhomogeneous dielectric materials is presented. The technique takes advantage of large curved hexahedral discretization elements—enabled by double‐higher‐order modeling (higher‐order modeling of both the geometry and the current)—in applications involving highly inhomogeneous dielectric bodies. Lagrange‐type modeling of an arbitrary continuous variation of the equivalent complex permittivity of the dielectric throughout each VIE geometrical element is implemented, in place of piecewise homogeneous approximate models of the inhomogeneous structures. The technique combines the features of the previous double‐higher‐order piecewise homogeneous VIE method and continuously inhomogeneous finite element method (FEM). This appears to be the first implementation and demonstration of a VIE method with double‐higher‐order discretization elements and conformal modeling of inhomogeneous dielectric materials embedded within elements that are also higher (arbitrary) order (with arbitrary material‐representation orders within each curved and large VIE element). The new technique is validated and evaluated by comparisons with a continuously inhomogeneous double‐higher‐order FEM technique, a piecewise homogeneous version of the double‐higher‐order VIE technique, and a commercial piecewise homogeneous FEM code. The examples include two real‐world applications involving continuously inhomogeneous permittivity profiles: scattering from an egg‐shaped melting hailstone and near‐field analysis of a Luneburg lens, illuminated by a corrugated horn antenna. The results show that the new technique is more efficient and ensures considerable reductions in the number of unknowns and computational time when compared to the three alternative approaches.

In this contribution, we derive a consistent variational formulation for computational homogenization methods and show that traditional FE and IGA approaches are special discretization and solution techniques of this most general framework. This allows us to enhance dramatically the numerical analysis as well as the solution of the arising algebraic system. In particular, we expand the dimension of the continuous system, discretize the higher dimensional problem consistently and apply afterwards a discrete null‐space matrix to remove the additional dimensions. A benchmark problem, for which we can obtain an analytical solution, demonstrates the superiority of the chosen approach aiming to reduce the immense computational costs of traditional FE and IGA formulations to a fraction of the original requirements. Finally, we demonstrate a further reduction of the computational costs for the solution of general nonlinear problems.

Limit analysis of unidirectional porous media

We propose an effective approach for designing a multi-lattice structure (MLS) that simultaneously considers local densities and the fillet-joint shape of struts to express practical equivalent material properties. The density of each cell is optimized by changing the strut diameter and fillet-joint radius according to loading conditions. The equivalent material properties of MLS, such as elastic modulus and shear modulus, are calculated based on a computational homogenization method. Finite element analyses of the full-shape and homogenized lattice model under external compressive load are conducted to evaluate the equivalent material properties. We also designed an optimized three-point bending structure using the proposed method. Based on the results of the topology optimization, three types of lattices with different relative densities are designed in a local zone considering local deformation modes. The result of this work shows that multi-lattice bending structure has about 86.9 % higher strength than that of a uniform BCC lattice structure with the same weight.

Homogenization methods can be used to predict the effective macroscopic properties of materials that are heterogenous at micro- or fine-scale. Among existing methods for homogenization, computational homogenization is widely used in multiscale analyses of structures and materials. Conventional computational homogenization suffers from long computing times, which substantially limits its application in analyzing engineering problems. The neural networks can be used to construct fully decoupled approaches in nonlinear multiscale methods by mapping macroscopic loading and microscopic response. Computational homogenization methods for nonlinear material and implementation of offline multiscale computation are studied to generate data set. This article intends to model the multiscale constitution using feedforward neural network (FNN) and recurrent neural network (RNN), and appropriate set of loading paths are selected to effectively predict the materials behavior along unknown paths. Applications to two-dimensional multiscale analysis are tested and discussed in detail.

An ${\cal H}^{2}$ -matrix based linear complexity direct matrix solution is developed for the volume integral equation (VIE) based broadband full-wave extraction of general 3-D circuits. Such circuits are in general electrically small or moderate, but contain arbitrarily shaped lossy conductors immersed in inhomogeneous dielectrics with ports located anywhere in the physical layout of the circuit. In the proposed direct solver, we first develop a well-conditioned VIE formulation without sacrificing the rigor and the advantages of the prevailing formulations. This formulation facilitates a robust direct solution of good accuracy even with a rank-1 representation. We then overcome the numerical challenge of solving the resultant highly unstructured system matrix mixed with both square and rectangular dense and sparse matrices by developing a fast linear complexity direct solution. This direct solution is capable of inverting dense matrices involving over 2 million unknowns in less than 1 h on a single CPU core running at 3 GHz. Numerical simulations of large-scale 3-D circuits and comparisons with state-of-the-art linear complexity iterative VIE solvers have demonstrated the accuracy, efficiency, and linear complexity of the proposed direct VIE solver.

Computational second-order homogenization of materials with effective anisotropic strain-gradient behavior

The integral equations describing forward scattering problems are generally solved numerically by iterative methods. In this work, we apply iterative deep neural network to solve both surface integral equation (SIE) and volume integral equation (VIE). The Green's function describing wave physics is involved in both training and prediction stage. The trained neural network can predict either the surface current density on conducting targets from SIE, or the total electromagnetic (EM) field in dielectric volume from VIE. We validate this method with two numerical examples, one for SIE and the other for VIE. This work may provide us a new perspective of incorporating wave physics into neural networks for fast EM computation.

We consider a static problem for statistically homogeneous matrix linear bond-based peridynamic composite materials (CMs). Even for locally elastic composites subjected to inhomogeneous loading, the effective deformations are described by a nonlocal (either the differential or integral) operator relating a statistical average of stresses in the point being considered with a statistical average of strains in the vicinity of this point. The basic hypotheses of locally elastic micromechanics are generalized to their peridynamic counterparts (see for details [1]). The method is based on estimation of a perturbator introduced by one inclusion that is, in fact, the solutions of the basic problem for one inclusion inside the infinite peristatic matrix subjected to the body force. The statistical averages of both the displacements and stresses are estimated by summation of these perturbators for all possible locations of inclusions (in so doing, renormalizing procedure is not required). It allows us to estimate not only the effective nonlocal constitutive equation but also to evaluate the statistical field averages (inhomogeneous and nonlocal) inside the phases at the fine scale that is critically important for advanced modelling (e.g. for any nonlinear phenomena). In particular, in the generalized effective field method (EFM) proposed, the effective field is evaluated from self-consistent estimations by the use of closing of a corresponding integral equation in the framework of the quasi-crystalline approximation. In so doing, the classical effective field hypothesis is relaxed, and the hypothesis of the ellipsoidal symmetry of the random structure of CMs is not used. Numerical results for the estimation of effective displacements are obtained for 1D statistically homogeneous CM bar with the prescribed self equilibrated body forces.

The studies of the deformation of physically nonlinear homogeneous and composite materials are systematized. Algorithms to determine the effective elastic properties and stress–strain state of particulate, laminated, fibrous, and laminated fibrous composite materials with physically nonlinear components are outlined, and their deformation patterns are studied. Composites are considered as two-component materials of random structure. Their effective properties are determined using the conditional averaging method. The nonlinear equations that allow for the physical nonlinearity of the components are solved by an iterative method. The relationship between macrostresses and macrostrains is established. Macrostress–macrostrain curves of homogeneous and composite materials are analyzed

The FE computational homogenization method is a predictive multi‐scale method without the need for constitutive assumptions and/or potential function postulates at the macro engineering scale. Instead, the effective micro‐structural responses are extracted directly from a representative volume element (RVE) underlying each macro point. However, the FE method is still computationally too expensive for most practical uses, since the micro‐macro FE coupling is done at each loading step/iteration for the entire domain. To this end, the machine learning method has been utilized in the literature for the offline training of a surrogate model to predict the RVE homogenized response for general loading conditions. In this contribution, the neural network (NN) is incorporated into the macro finite element framework in a non‐intrusive manner. This is termed as the FE‐NN framework, in analogy to the FE method. In general, online simulations in the FE‐NN method is very efficient, with predictions matching closely to those obtained from reference direct numerical simulations (DNS). A bottleneck with the FE‐NN framework, however, is the high computational cost associated with the data generation for offline NN model setup. In this paper, focusing on the FE‐NN multi‐scale framework for non‐linear elastic deformation of heterogeneous materials, a sequential training strategy with knowledge transfer is proposed, to enable an efficient offline microscopic NN model setup. For a given target RVE, we first consider a simplified source RVE, where data can be generated rapidly, for the NN pre‐training of surrogate model. The pre‐trained network parameters are next downloaded to initialize the target NN surrogate model, followed by a fine‐tuning training process, using only a small dataset generated by the computationally expensive high‐fidelity RVE. The efficiency of the proposed sequential learning method over the conventional NN training, as well as, its excellent predictive capability for multi‐scale analyses, are demonstrated for a multi‐phase composite material. The proposed FE‐NN‐KT approach can be implemented easily without complicated pre‐processing procedures, since the data generation only involves the standard extraction of effective responses from RVEs, together with a standard NN architecture.

Functional metal oxide particles are often added to the polymers to prepare flexible functional polymer composites with adequate mechanical properties. ZnO and cellulose nanofibrils (CNF) outstand among these metal oxides and the polymer matrices respectively due to their various advantages. Herein, we in situ prepare ZnO microrods in the presence of CNF, which resultes in a layered composite structure. The ZnO microrods are sandwiched between the CNF layers and strongly bind to highly charged CNF, which provides a better stress transfer during mechanical activity. Digital image correction (DIC) and finite element analysis-based computational homogenization methods are used to investigate the relationship between mechanical properties and composite structure, and the stress transfer to the ZnO microrods. Full-field strain measurements in DIC reveal that the in situ ZnO microrods preparation leads to their homogenous distribution in the CNF matrix unlike other methods, which require external means such as ultrasonication. The computational homogenization technique provides a fairly good insight into the stress transfer between constituents in microstructure as well as a good prediction of macroscopic mechanical properties, which otherwise, would be challenging to be assessed by any ordinary mechanical testing in the layered composites. Finally, we also demonstrate that these composites could be used as physiological motion sensors for human health monitoring.