Numerical Homogenization Research Articles

Discontinuous Galerkin and C0-IP finite element approximation of periodic Hamilton–Jacobi–Bellman–Isaacs problems with application to numerical homogenization

In the first part of the paper, we study the discontinuous Galerkin (DG) and C0 interior penalty (C0-IP) finite element approximation of the periodic strong solution to the fully nonlinear second-order Hamilton–Jacobi–Bellman–Isaacs (HJBI) equation with coefficients satisfying the Cordes condition. We prove well-posedness and perform abstract a posteriori and a priori analyses which apply to a wide family of numerical schemes. These periodic problems arise as the corrector problems in the homogenization of HJBI equations. The second part of the paper focuses on the numerical approximation to the effective Hamiltonian of ergodic HJBI operators via DG/C0-IP finite element approximations to approximate corrector problems. Finally, we provide numerical experiments demonstrating the performance of the numerical schemes.

Shell-to-Beam Numerical Homogenization of 3D Thin-Walled Perforated Beams.

Determining the geometric characteristics of even complex cross-sections of steel beams is not a major challenge nowadays. The problem arises when openings of various shapes and sizes appear at more or less regular intervals along the length of the beam. Such alternations cause the beam to have different stiffnesses along its length. It has different bending and shear stiffnesses at the opening point and in the full section. In this paper, we present a very convenient and easy-to-implement method of determining the equivalent stiffness of a beam with any cross-section (open or closed) and with any system of holes along its length. The presented method uses the principles of the finite element method (FEM), but does not require any formal analysis, i.e., solving the system of equations. All that is needed is a global stiffness matrix of the representative volumetric element (RVE) of the 3D representation of a beam modeled with shell finite elements. The proposed shell-to-beam homogenization procedure is based on the strain energy equivalence, and allows for precise and quick determination of all equivalent stiffnesses of a beam (flexural and shear). The results of the numerical homogenization procedure were compared with the existing analytical solution and experimental results of various sections. It has been shown that the results obtained are comparable with the reference results.

Multiscale Elliptic PDE Upscaling and Function Approximation via Subsampled Data

There is an intimate connection between numerical upscaling of multiscale PDEs and scattered data approximation of heterogeneous functions: the coarse variables selected for deriving an upscaled equation (in the former) correspond to the sampled information used for approximation (in the latter). As such, both problems can be thought of as recovering a target function based on some coarse data that are either artificially chosen by an upscaling algorithm or determined by some physical measurement process. The purpose of this paper is then to study, under such a setup and for a specific elliptic problem, how the lengthscale of the coarse data, which we refer to as the subsampled lengthscale, influences the accuracy of recovery, given limited computational budgets. Our analysis and experiments identify that reducing the subsampling lengthscale may improve the accuracy, implying a guiding criterion for coarse-graining or data acquisition in this computationally constrained scenario, especially leading to direct insights for the implementation of the Gamblets method in the numerical homogenization literature. Moreover, reducing the lengthscale to zero may lead to a blow-up of approximation error if the target function does not have enough regularity, suggesting the need for a stronger prior assumption on the target function to be approximated. We introduce a singular weight function to deal with it, both theoretically and numerically. This work sheds light on the interplay of the lengthscale of coarse data, the computational costs, the regularity of the target function, and the accuracy of approximations and numerical simulations.

Numerical Modeling and Experimental Investigation of Effective Elastic Properties of the 3D Printed Gyroid Infill

A numerical homogenization approach is presented for the effective elastic moduli of 3D printed cellular infills. A representative volume element of the infill geometry is discretized using either shell or solid elements and analyzed using the finite element method. The elastic moduli of the bulk cellular material are obtained through longitudinal and shear deformations of a representative volume element under periodic boundary conditions. The method is used to analyze the elastic behavior of gyroid infills for varying infill densities. The approach is validated by comparing the Young’s modulus and Poisson’s ratio with those obtained from compression experiments. Results indicate that although the gyroid infill exhibits cubic symmetry, it is nearly isotropic with a low anisotropy index. The numerical predictions are used to develop semi-empirical equations of the effective elastic moduli of gyroid infills as a function of infill density in order to inform design and topology optimization workflows.

An improved ordered SIMP approach for multiscale concurrent topology optimization with multiple microstructures

Cellular structures are often used to improve the stiffness, fatigue strength, damage tolerance, and other superior functioning properties of structures in engineering. This paper presents a novel multiscale concurrent topology optimization method to optimize structures which are periodically filled with multiple microstructures and connected by solid interfaces. This method generates better structural performance and does not need to preprocess the initial design domain. It enables to save the computational resources, and improves the freedom of structural optimization design to a certain extent. Solid interface layers are built to connect different microstructure blocks, and hence the full degrees of design freedom of microstructures can be further explored. At the macroscale, a novel piecewise projection and a series of gradient-based filtering operations are proposed to distinguish the microstructure blocks and interface layers, respectively. Then, an improved ordered solid isotropic material with penalization (SIMP) method is proposed to optimize the spatial distribution of different microstructures with an affordable computational cost. At the microscale, the microstructures are generated by the numerical homogenization method. Microstructures with different volume constraints are treated as different materials, and the volume fraction limit value of the microstructure corresponds to the design variable of the improved ordered SIMP method. Finally, the compliance minimization problem under the constraint of material volume fractions is investigated, and sensitivity analysis is derived. Several 2D and 3D numerical examples are provided to demonstrate the effectiveness of the proposed method.

Design of multiphase auxetic metamaterials by a parametric color level set method

This paper presents a systematic optimization design method for the multiphase auxetic metamaterials with different deformation mechanisms in both 2D and 3D scenarios. In this method, the parametric color level set (PCLS) is developed to describe different material phases in the microstructures, in which at most 2L materials phases can be precisely represented by only L description functions without any overlaps. Furthermore, clear and smooth material interfaces can be guaranteed in the design, and multiple material usage constraints can be efficiently handled by the well-established gradient-based algorithm. The shape derivative theory is introduced to analyze the design sensitivities for the optimization problem. The effective elasticity properties of the multiphase composites are evaluated by the numerical homogenization method under periodic boundary conditions. Various symmetric conditions are defined and enforced to induce the re-entrant and chiral patterns in both 2D and 3D metamaterials. Several numerical examples are provided to demonstrate the features of the proposed method in tailoring different types of multiphase auxetics. The multiphase metamaterials with two and more material phases are discussed. It is shown that the presented design method can be used to devise both the re-entrant and chiral auxetic metamaterials with excellent auxetic and stiffness properties.

Two-scale modeling of nano-clay-filled shape memory polymers

Shape memory polymers (SMPs) are introduced as polymers that have the ability to return to their early programmed shape after exposure to an external stimulus. Enhancement of the material with nano-clay filler has improved its thermomechanical properties and increased the range of its applications in many fields of industry. Due to the tiny size of filler and the heterogeneous nature of the material structure at different scale levels, characterizing the material’s thermomechanical flow using conventional experimental equipment is a far-fetched task. Furthermore, providing one numerical model that is able to simulate the material thermomechanical behavior by including all the effects of the lower scale material structure is also very hard. In this study, a two-scale modeling approach is developed by a combination of the numerical homogenization scheme, 3D Representative Volume Element (RVE) concept, and finite element method. The effects of the filler weight fractions on the overall effective elastic constants as well as the material flow under a finite deformation are investigated. The resulting elastic constants and the stress–strain curves show a fairly good agreement with the analytical results. Furthermore, all the investigated results provide a deep understanding of the material behavior and a starting point for the next higher scale level modeling approaches.

Synthesis, characterization, and modelling the behavior of in-situ ZrO2 nanoparticles dispersed epoxy nanocomposite

The cured epoxy has poor toughness owing to highly cross-linked nature. The present work reports inclusion of ZrO2 nanoparticles with epoxy via ultrasonic mixing process for improving the toughness. The developed nanocomposites are characterized using mechanical tests, fractographic analysis and dynamic mechanical thermal analysis (DMTA). The mode-1 fracture toughness (K1C), fracture energy (G1C), storage modulus (E') and glass transition temperature (Tg) for different filler concentration are evaluated. A theoretical explanation for improved ZrO2-epoxy nanocomposite behaviour is reported by analyzing the fracture parameters and morphology of fractured samples. A micromechanical finite element (FE) model is constituted for predicting macroscopic elastic properties of composites using numerical homogenization scheme. The model predicted location of fracture initiation under triaxial and uniaxial loading conditions. The SEM micrographs and modelling results collectively explained the fracture mechanism. The experimentation and modelling results are compared.

The family of elastically isotropic stretching-dominated cubic truss lattices

We present a family of elastically isotropic stretching-dominated cubic truss lattices. Their Young’s, shear, bulk moduli approach nearly 1/3 of the theoretical Hashin-Shtrikman upper bounds at low density limit, which are the maximum isotropic stiffness that truss lattices can achieve. The proposed lattices are designed by combination of two elementary anisotropic lattices based on theoretical criteria for elastically isotropic stretching-dominated truss lattices. The elementary truss lattices are identified by exhaustive search of the Euclidean Patterns in Non-Euclidean Tilings database. Matrix analysis formulas for periodic pin-jointed assemblies are developed to prove that all combined lattices exhibit stretching-dominated behaviors under arbitrary macroscopic loads, i.e., their corresponding periodic pin-jointed assemblies have no macroscopic strain-producing mechanisms. The stiffness of the combined lattices is further evaluated through C3D10 solid element based finite element models and numerical homogenization. Numerical results show good agreement with theoretical calculations at low densities, thus validating the stretching-dominated behavior and elastic isotropy. The lattices exhibit different elastic moduli and anisotropy at moderate relative densities due to their different joint connections, among which the sqc1 + sqc5432 lattice possesses both nearly isotropic stiffness and relatively high elastic moduli at 30% relative density. This work presents a systematic method to design elastically isotropic cubic truss lattices with optimal isotropic stiffness, and reveals that the optimal designs are infinite.

Parametric optimization of the effective thermal conductivity for a three-phase particle-filled composite

Functionalized materials, consisting of a particle-filled polymer matrix, enable a considerable increase in thermal conductivity compared to standard plastics. This is particularly important in electromobility applications, e.g. in encapsulated circuit boards, to ensure sufficient thermal management. In this paper, a method for the optimization of the effective thermal conductivity (ETC) by considering electrically insulating composites is presented. To show the diversity of microstructural design options by using different filler/matrix compositions, the method is introduced by taking a particle-filled plastic with three phases and high contrast in the conductivities of the individual phases as example. Particles with different sizes and shapes in the micrometer range are considered. The task of setting a target ETC is formulated as a parameter optimization problem. The ETC is determined by microstructure simulation and numerical homogenization where virtually created representative volume elements (RVEs) are used. To study the influence of the different design variables, metamodeling and sensitivity analysis techniques are introduced. It turns out, that only the filler volume fraction, the aspect ratio and the proportions of the individual fillers are sensitive parameters. After neglecting non-sensitive parameters, global optimization in the reduced design space using an evolutionary algorithm is performed. The novelty of the paper is that several fillers with different microstructural parameters, such as diameter, aspect ratio and shape, are included as design variables in the structural optimization. Since the evaluation of the objective function of the structural optimization problem is very expensive, a cheaper surrogate model is introduced in the paper. This approach allows an efficient computation of microstructural realizations with a target ETC and can be extended to other microstructures and effective properties as well.

A multiscale topological design method of geometrically asymmetric porous sandwich structures for minimizing dynamic compliance

Compared with conventional sandwich structures with two identical face sheets, geometrically asymmetric porous sandwich structures (GAPSSs) can achieve better dynamic performances under certain load and boundary condition, since they provide more choices for geometric design. However, current studies regarding the GAPSSs are mainly analytical- and experimental-based methods with predefined face sheet thicknesses and core configurations, few works investigate the GAPSSs by optimization method. This paper presents a multiscale topology optimization method of the GAPSSs, which is capable of designing both thicknesses of two solid face sheets at macroscale and configurations of porous core at microscale for minimizing dynamic compliance. Specifically, at macroscale, a variable thickness sheet method is employed to optimize the thicknesses of two solid face sheets. Then, at microscale, a parametric level-set method integrated with a numerical homogenization approach is applied to topologically optimize the configuration of the periodic unit cell. The method of moving asymptotes is adopted to update the design variables at both scales. Several 2D and 3D numerical examples are illustrated to show the effectiveness and advantages of the proposed method for designing the GAPSSs. The results indicate that the dynamic compliance of the optimized GAPSSs show superior advantages over the conventional sandwich structures with identical face sheet thicknesses and predefined lattice core.

Non-Local Sensitivity Analysis and Numerical Homogenization in Optimal Design of Single-Wall Corrugated Board Packaging.

The optimal selection of the composition of corrugated cardboard dedicated to specific packaging structures is not an easy task. The use of lighter boards saves material, but at the same time increases the risk of not meeting the guaranteed load capacity. Therefore, the answer to the question “in which layer the basis weight of the paper should be increased?” is not simple or obvious. The method proposed here makes it easy to understand which components and to what extent they affect the load-bearing capacity of packages of various dimensions. The use of numerical homogenization allows for a quick transformation of a cardboard sample, i.e., a representative volume element (RVE) into a flat plate structure with effective parameters describing the membrane and bending stiffness. On the other hand, the use of non-local sensitivity analysis makes it possible to find the relationship between the parameters of the paper and the load capacity of the packaging. The analytical procedures presented in our previous studies were used here to determine (1) the edge crush resistance, (2) critical load, and (3) the load capacity of corrugated cardboard packaging. The method proposed here allows for obtaining a comprehensive and hierarchical list of the parameters that play the most important role in the process of optimal packaging design.

Numerical homogenization of a linearly elastic honeycomb lattice structure and comparison with analytical and experimental results

This paper presents a verification and validation analysis of Finite Element (FE) models predicting the mechanical response of linearly elastic honeycomb structures. We have studied three main models, namely: analytical models based on beam theories, explicit FE models where the cell geometry is explicitly meshed with 3D elements, and numerical homogenized FE models where a plate made of a honeycomb structure is meshed with 2D elements whose mechanical properties were predicted from numerical homogenization. We compared the predictions of these simulations against experimental uni-axial tensile tests where we mechanically tested 3D printed honeycomb specimens having a relative density of 40% and made of 13 and 37 cells, respectively. Comparison of the experimentally measured axial stiffness to the numerical predictions revealed that the numerical homogenization models can predict the apparent in-plane stiffness in structures made of 37 cells within 4%, while the discrepancy increases to 70% when 13 cells are considered. To quantify this discrepancy, we also provided a relationship between the number of represented cells and the discrepancy of the numerical homogenized model against the explicit FE models to predict the in-plane stiffness. We believe that these results could be important for the application of the homogenized models in optimization of honeycomb lattice structures whose relative density varies with space.

Stress‐constrained multiscale topology optimization with connectable graded microstructures using the worst‐case analysis

AbstractThis article proposes a stress‐constrained multiscale topology optimization approach with connectable graded microstructures. The proposed method includes two stages. In the first stage, the shape interpolation method is first employed to generate a series of connectable unit cells. Then the effective elasticity tensors are calculated by the numerical homogenization and XFEM. Besides, the worst‐case analysis and stress correction factor are employed to predict the maximum microscopic stress of the unit cells under arbitrary loading conditions. Furthermore, reduced‐order models for the stress correction factor and effective elasticity tensor are built to efficiently predict the mechanical properties of the unit cell with any specified volume fraction. In the second stage, stress‐constrained topology optimization is employed to find the distribution of microstructures by using established reduced‐order models. Except for applying approaches commonly used in the traditional stress‐constrained topology optimization, the moving Heaviside function is also proposed to include the void material into optimization. Finally, a threshold projection scheme is performed to realize the design of multiscale structures. Two numerical examples are presented to validate the proposed method. In addition, because the worst‐case analysis overestimates the structural stress, an evolutionary discrete optimization is employed to further explore the potential of the multiscale structures.

Computational homogenization of concrete in the cyber Size-Resolution-Discretization (SRD) parameter space

Micro- and mesostructures of multiphase materials obtained from tomography and image acquisition are an ever more important database for simulation analyses. Huge data sets for reconstructed 3d volumes typically as voxel grids call for criteria and measures to find an affordable balance of accuracy and efficiency. The present work shows for a 3d mesostructure of concrete in the elastic deformation range, how the computational complexity in analyses of numerical homogenization can be reduced at controlled errors. Reduction is systematically applied to specimen size S, resolution R, and discretization D, which span the newly introduced SRD parameter space. Key indicators for accuracy are (i) the phase fractions, (ii) the homogenized elasticity tensor, (iii) its invariance with respect to the applied boundary conditions and (iv) the total error as well as spatial error distributions, which are computed and estimated. Pre-analyses in the 2d SRD parameter sub-space explore the transferability to the 3d case. Beyond the concrete specimen undergoing elastic deformations in the present work, the proposed concept enables accuracy-efficiency balances for various classes of heterogeneous materials in different deformation regimes and thus contributes to build comprehensive digital twins of materials with validated attributes. • Numerical homogenization analysis for effective elastic properties of concrete specimen. • Comprehensive sampling of the Size-Resolution-Discretization (SRD) parameter space. • Transition from apparent to effective properties invariant to RVE boundary conditions. • Quality of error estimation transferable from 2d to 3d and from small to large volumes. • Considerable efficiency gain in SRD parameter space at controlled errors.

Homogenization procedures for the numerical analysis of eco-composites

Homogenization procedures have been gaining more and more emphasis in the field of numerical analysis of composite materials. A homogenization procedure is based on the assumption that a set of equations or a representative element that can provide a response equivalent to the one provided by the actual material exists. A numerical multiscale homogenization will be shown where the behavior of the whole composite is obtained by a micromechanical study of the material components and their interaction within of the composite microstructure through an RVE model. The proposed formulation uses a non-linear activation function in the structural scale and its task is to predict if a material point in the structure is in linear or non-linear range. Because of this, the developed strategy only solves the microscopic Boundary Value Problem using the RVE for the non-linear integration points found. The procedure will be used in the nonlinear numerical simulaction of an eco-composite sanwich panel subjected to a four point bending test.

TOPOLOGY OPTIMIZATION OF ADDITIVE MANUFACTURING COMPONENTS WITH LATTICE STRUCTURES FOR MULTIPHYSICS APPLICATIONS

This paper aims to optimize additive manufacturing parts containing lattice structures by means of topology optimization. The proposed method relies on numerical homogenization to model the behavior of the mesoscale lattices inside the macroscale component with an equivalent material model. While the topology of the unit cell is fixed a priori, the material constitutive model is considered to be a function of multiple size parameters of the lattice struts and the orientation of the lattice cell. Both the size parameters and the orientation are considered as design fields in the design domain of the optimization in order to optimize the distribution of orthotropic lattice structures while adapting their orientation to the macroscale loads. In addition, the shape of the macroscale part is described by a separate level set function. Besides avoiding the appearance of degenerate lattice structures, the strict control over the macroscale shape permits the inclusion of different manufacturing constraints. The design fields describing the lattice geometry and macroscale shape are parametrized explicitly by filtering sets of independent optimization variables. This explicit design parametrization allows one to simultaneously optimize the shape of the macroscale part and the lattice structures inside the part. Moreover, the optimization problem is expressed and solved as a standard nonlinear programming problem, which allows one to easily adopt the formulation for different design problems. This interesting feature is illustrated in the numerical examples which deal with different physical problems, and several objective and constraint functions.

Ph-Net: Parallelepiped Microstructure Homogenization Via 3d Convolutional Neural Networks

Microstructures are attracting academic and industrial interests with the rapid development of additive manufacturing. The numerical homogenization method has been well studied for analyzing mechanical behaviors of microstructures; however, it is too time-consuming to be applied to online computing or applications requiring high-frequency calling, e.g., topology optimization. Data-driven homogenization methods emerge as a more efficient choice but limit the microstructures into a cubic shape, which are infeasible to the periodic microstructures with a more general shape, e.g., parallelepiped. This paper introduces a fine-designed 3D convolutional neural network (CNN) for fast homogenization of parallel-shaped microstructures, named PH-Net. Superior to existing data-driven methods, PH-Net predicts the local displacements of microstructures under specified macroscope strains instead of direct homogeneous material, motivating us to present a label-free loss function based on minimal potential energy. For dataset construction, we introduce a shape-material transformation and voxel-material tensor to encode microstructure type,base material and boundary shape together as the input of PH-Net, such that it is CNN-friendly and enhances PH-Net on generalization in terms of microstructure type, base material, and boundary shape. PH-Net predicts homogenized properties with hundreds of acceleration compared to the numerical homogenization method and even supports online computing. Moreover, it does not require a labeled dataset and thus is much faster than current deep learning methods in training processing. Benefiting from predicting local displacement, PH-Net provides both homogeneous material properties and microscopic mechanical properties, e.g., strain and stress distribution, yield strength, etc. We design a group of physical experiments and verify the prediction accuracy of PH-Net.

Design and Mechanical Characterization of an S-Based TPMS Hollow Isotropic Cellular Structure

Cellular structures are regarded as excellent candidates for lightweight-design, load-bearing, and energy-absorbing applications. In this paper, a novel S-based TPMS hollow isotropic cellular structure is proposed with both superior load-bearing and energy-absorbing performances. The hollow cellular structure is designed with Boolean operation based on the Fischer-Koch (S) implicit triply periodic minimal surfaces (TPMS) with different level parameters. The anisotropy and effective elasticity properties of cellular structures are evaluated with the numerical homogenization method. The finite element method is further conducted to analyze the static mechanical performance of hollow cellular structure considering the size effect. The compression experiments are finally carried out to reveal the compression properties and energy-absorption characteristics. Numerical results of the Zener ratio proved that the S-based hollow cellular structure tends to be isotropic, even better than the sheet-based Gyroid TPMS. Compared with the solid counterpart, the S-based hollow cellular structure has a higher elastic modulus, better load-bearing and energy absorption characteristics.

Efficient numerical model for effective thermal conductivity of concrete with aggregates of different morphologies and imperfect interfaces

The existing numerical model for predicting the effective coefficient of thermal conductivity (CTC) of concrete usually ignores its mesostructure such as the three dimensional (3D) irregular aggregate morphology, imperfect interface, phase in voids, and spatial distribution of aggregates. Meanwhile, most numerical models adopt the uniform temperature gradient in the homogenisation process, resulting in the larger model size required for convergence and lower post-processing efficiency. An algorithm to generate concrete with aggregates of different morphologies and imperfect interfaces was proposed. Subsequently, a simple method of imposing the periodic boundary conditions and an efficient post-processing method were proposed, thereby reducing the time consumption in the numerical homogenisation substantially. In addition, based on the mesomechanical analytical model, DIGIMAT, and experimental data, the rationality of the proposed method and established models was validated, and the effect of the 3D mesostructure of concrete on its effective CTC was analysed. Results show that the concrete exhibits imperfect interfaces in its service life, especially in the dried working condition. The concrete gradation, phase in voids, and aggregate orientation have different degrees of influence on the effective CTC of concrete, and the effect of aggregate morphology can be ignored.