Numerical Homogenization Research Articles

Numerical investigation of the effective mechanical properties of architected structures: A comparative study of flexural stiffness, homogenization, and elastic anisotropy

Abstract The mechanical behavior of architected structures is influenced by various parameters, including the topology of their unit cells. This anisotropic nature requires the determination of the mechanical properties under different loading scenarios. This study employs numerical investigation to characterize the influence of topology on the mechanical properties of eight architected structures, focusing on effective elastic properties and anisotropic elastic behavior. The analyzed topologies encompass four based on struts (lattices) and four based on triply periodic minimal surfaces (TPMS), comprising Sheet and Network phases. Initially, beams composed of architected structures are subjected to flexure, with Euler-Bernoulli and Tymoshenko’s theories utilized in a first numerical approach to determine their effective properties. Subsequently, a numerical homogenization method along with the Voigt-Reuss-Hill scheme is employed in a second approach. A more substantial influence of topology on the effective properties is observed in low relative densities. The study revealed that for a relative density of 10%, the appropriate selection of the topology increases the stiffness of a structure by up to ~126%. The EBT approach underestimated the stiffness by up to ~26% due to neglecting the impact of shear on beam deflection. The tensorial anisotropy index revealed up to ~27% higher anisotropy compared to the Zener index. These findings provide a valuable numerical tool for the comparison and selection of architected structures suitable for diverse applications.

Both Effective Elastic Properties and Voids Interaction Energy Estimations within the Context of 3D Nano-Porous Materials

This study aims to estimate elastic properties as well as voids interaction energy within the context of 3D nano-porous materials exhibiting spherical or cylindrical voids, with or without the effect of surface energy. For this purpose, numerical homogenization is applied, based on the finite element method (FEM) in combination with the level-set technique. Surface energy is approximated by explicitly taking into account its contribution to overall stiffness in the variational formulation. Several cases are then investigated to appreciate the interaction energy between multiple 3D nanovoids on the actual behavior of nanomaterials. As expected, and like surface energy, the results show that interaction energy keeps affecting the effective behavior by increasing the size of RVE as well as the number of voids inside. This influence is more marked the more cylindrical the voids and the more closely they are adjacent to each other.

Fluid-filled lattices for responsive orthotic insoles

Abstract Orthotic insoles are essential for alleviating discomfort and preventing injuries in the foot caused by high peak pressures in the plantar tissue. Traditional orthotic insoles, often prescribed to address these issues, are designed based on the static foot shape and pressure measurements, lacking responsiveness to dynamic movements. This study explores the behaviour of fluid-filled lattices for improving the functionality of orthotic insoles, focusing on energy dissipation and pressure redistribution capabilities. Using numerical homogenisation, the research integrates hyperelastic and permeability models to simulate the behaviour of Solid–Liquid Composites. Experimental tests validated these models, examining the influence of fluid viscosity and structural variations on energy dissipation and pressure distribution. Results show that fluid-filled lattices provide enhanced energy dissipation and reduce peak pressures by evening out the pressure distribution compared to non-fluid-filled samples. These findings highlight the potential of fluid-filled lattices to improve the performance and comfort of orthotic insoles.

Designing spongy-bone-like cellular materials: Matched topology and anisotropy

Bone is a natural material with properties such as high specific stiffness and strength. These exceptional mechanical properties are attributed to the meso-scale structure and elastic anisotropy of spongy bone. Replicating the topological traits and mechanical properties of spongy bone presents a novel opportunity to develop high-performance cellular materials. To achieve this, we propose an innovative framework for designing biomimetic cellular materials that match the trabecular structure and elastic anisotropy of spongy bone. This framework introduces a forward-flow design process that utilizes gradient-based feature tuning on a low-dimensional feature vector, transforming the complex inverse design problem into an efficient iterative process. A key innovation in our approach is the use of a pre-trained generative model, SliceGAN, to reconstruct 3D unit cells from 2D micro-CT images. This significantly reduces the cost and time associated with traditional layer-by-layer CT scans typically required for 3D training data. Numerical homogenization is then used to determine the effective elastic stiffness matrix, and a Fourier neural operator is trained to predict these matrices efficiently, greatly enhancing the computational efficiency of the design process. Using this framework, we successfully designed unit cells with topological traits and elastic anisotropy that closely approximate those of natural spongy bone. This opens new avenues for developing spongy-bone-mimetic cellular materials with exceptional mechanical properties. Moreover, the framework's versatility allows it to be extended to the design of other bio-inspired cellular materials.

Mean-field homogenization of liquid metal elastomer composites: comparative study and exact dilute solution of core–shell inclusion

Liquid metal-elastomer composites (LMECs) have gathered significant attention for their potential applications in various functional stretchable devices, with inclusion sizes ranging from micrometers to nanometers. These composites exhibit exceptional properties, such as high electric permittivity and thermal conductivity, surpassing those of the elastomer matrix, thus enabling a broader range of applications without compromising the material's stretchability. To investigate the diverse effective elastic and functional properties of LMECs, micromechanics-based homogenization method based on Eshelby's inclusion solution are invaluable. However, the extreme contrast in elastic constants among the phases in LMECs, particularly for nanosized inclusions where a considerable amount of stiff metal oxide forms around the inclusions, can lead to critical failure in predicting effective properties if inadequate homogenization approach is employed. In this study, we present multiple mean-field homogenization approaches applicable to LMECs with core-shell morphology, namely: (i) multi-phase, (ii) sequential, (iii) pseudo-grain, and (iv) direct approaches. We compare the accuracy of the models concerning effective elastic, thermal, and dielectric properties, evaluated against numerical homogenization results and compared with reported experimental data. Specifically, we highlight homogenization scheme utilizing exact field solutions of dilute core-shell inclusion, emphasizing the importance of accurately capturing the field in the micromechanics of LMECs. Furthermore, we demonstrate that widely utilized interphase model could not properly resolve the core-shell morphology and thus should be avoided. This comprehensive assessment provides critical insights into the proper homogenization strategies for designing advanced LMECs with precise prediction of effective properties.

Estimation and validation of elastic constants in fused filament fabrication 3D printing: From mesoscale to macroscale

Evaluating the mechanical properties of 3D-printed parts is a cumbersome task. This study poses an approach based on computational homogenization to estimate the elastic constants of fused filament fabrication 3D-printed parts. A whole methodology for characterization and experimental validation is necessary to improve finite element numerical models.Samples are characterized both mechanically and geometrically. To improve the characterization, novel algorithms based on micro-computed tomography images and image segmentation techniques are implemented. Thereafter, elastic constants are estimated, informed by the characterization results. The method’s effectiveness is assessed through a deep comparison, based on the digital image correlation technique, between different experimental samples and finite element models.Results show that the numerical estimation provided in this work is accurate enough to develop realistic finite element models, including anisotropy of the structures. However, defects and voids developed during 3D printing are an important source of error for these numerical models.This work provides an estimation and validation of elastic constants in 3D-printed parts, including geometrical characterization, numerical homogenization and experimental validation. The results of this work provide valuable information encompassing techniques to improve the characterization of 3D-printed parts, tendencies on elastic constants, and main sources of error.

Design of Steel-Cu composites for enhancing thermal properties of plastic processing tools by using a numerical model of the microstructure

Limitationsof conventional materials used for plastic processing tools, particularly related to thermal conductivity, significantly influence production efficiency, with the cooling time of moulded parts appearing as one of the key factors. Therefore, this study focuses on designing Steel-Cu composites with the aid of a numerical model of the composite's microstructure for enhancing their overall thermal properties. We present a novel procedure for designing such composites, which facilitates the identification of the optimal shape for the phases to improve overall thermal conductivity. We have developed a data-driven homogenization model to aid the optimization process and ensure time-efficient solutions. The data has been generated through numerical homogenization based on the finite element method. The proposed shape optimization procedure has been applied to determine the optimal shape of the Cu phase across various volume fractions. We found out that the optimal shape of the Cu phase is not constant but depends on its volume fraction. Emphasizing the practical utility of the proposed procedure, it is noteworthy that once the data-driven model is established, it enables a time-efficient optimization of the Cu phase shape for arbitrary volume fractions of phases. Consequently, this may expedite the decision-making process for material manufacturers.

Improving mechanical properties of lattice structures using nonuniform hollow struts

In contrast to constant-strut lattices, the introduction of innovative strut designs has the potential to enhance the mechanical properties of lattice structures. This study presents a Bézier curve-based nonuniform section design for body-centered cubic lattices with hollow struts (BCCH). Periodic boundary conditions are applied to the unit cells, and a finite element (FE) numerical homogenization method is employed to assess their elastic properties. A comprehensive dataset is generated through the FE model, which is subsequently divided into training, validation, and testing sets. The coordinates of the Bézier curve control points serve as inputs to deep learning networks, which are trained on the training dataset. Relative density and elastic properties are treated as two distinct networks, with the validation set utilized to prevent overfitting. The objective function consists of two weighted components: one aims to maximize the relative Young's modulus, while the other ensures that the relative density achieves a specified value. An evolutionary algorithm is employed to optimize the objective function, with variations in the control point coordinates constrained to specific ranges. By leveraging the fast inference ability of the deep learning model, the stiffness and orientation-dependent mechanical properties can be efficiently tailored. Our results demonstrate that the optimized structures demonstrate superior stiffness (+92.8 %) and distributed stress field compared to the benchmark lattice. The design method also enables tailoring of specific mechanical properties, including isotropic elasticity. 3D-printed lattice designs were fabricated and compression tests confirmed agreement with simulation results in terms of stiffness. Additionally, the optimized designs exhibit superior strength (+99.6 %) and toughness compared to the benchmark lattices.

Estimating equivalent elastic properties of frozen clay soils using an XFEM-based computational homogenization

This study addresses the challenge of estimating the elastic properties of heterogeneous frozen clay soils by introducing a comprehensive approach that combines analytical and numerical models. The frozen clay soil is treated as a mixture composed of frozen clay-water composites and nonclay mineral inclusions. An inversion algorithm is employed to deduce the elastic properties of the matrix (clay-water composites) of two artificially frozen sandy clay samples with known temperature-dependent elastic properties. Subsequently, a two-dimensional numerical simulation using the eXtended Finite Element Method (XFEM) is conducted to carry out numerical homogenization by considering the imperfect bond among frozen clay-water composites and nonclay minerals. The numerical homogenization model offers insights into the temperature-dependent behavior of the interface stiffness parameter. The numerical homogenization results are compared with conventional numerical homogenization approaches like the FEM, which rigidly defines the bonding between inclusions and the matrix. The comparison indicates that the neglect of imperfect bonds among clay-water composites and nonclay minerals will lead to unrealistic outcomes in cases with a high fraction of inclusions. This integrated approach advances the understanding and prediction of elastic properties of frozen clay soils by considering their heterogeneous nature.

Tailoring Solid-State Battery Performance through Computationally Tuning the Composition of LLZO-LCO Composite Cathodes

The composite cathode consisting of garnet-type solid-electrolyte Li7La3Zr2O12 (LLZO) and active material LiCoO2 (LCO) is one of the most promising solutions for the cathode design of all-solid-state lithium batteries. The transport properties of Li within the LLZO-LCO composite cathode are known to be sensitively dependent on the composition and microstructural features. This study focuses on computationally refining the performance of solid-state batteries by modifying the porosity and volume fractions of the LLZO-LCO composite cathode while considering the variations in the microstructural features. Various 2D and 3D microstructures were synthesized using a stochastic stacking particle model and the corresponding effective diffusivity of Li is calculated using a numerical homogenization method, which allows for a systematic assessment of the impact of composition on lithium transport within the microstructure. By identifying the optimal ratio between LLZO and LCO phases and porosity, this research yields crucial insights for improving the performance of solid-state batteries by tailoring the composite cathode.Figure 1: A) An example 3-dimensional microstructure with 20.3% percent vacuum (white space) and 39.25% percent LCO (yellow) and 40.45% LLZO (orange). B) The highest effective diffusivity (purple) is found when the LLZO percentage is highest. Once the vacuum percentage is 30% or higher, minimal effective diffusivity (red) is observed.This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. Figure 1

On exploration of directional extreme mechanical attributes and energy absorption of bending-dominated and buckling-induced negative Poisson’s ratio metamaterials

This work numerically explores the directional mechanical characteristics, including the effective anisotropy, multiaxial yield surface, long-frequency stress phase wave propagation, buckling resistance, and energy absorption of bending-dominated (BE) and buckling-induced (BU) auxetic metamaterials, considering the influence of the relative density and negative Poisson’s ratio, for the first time. The accuracy of numerical homogenization in determining the effective Young’s modulus and wave analytical model calculating the wave velocity is confirmed by compared with experimental and numerical data, respectively, with a maximum percent difference of 14.6% for all comparisons found. The results show that BU auxetic structures possess unique mechanical properties, displaying their potential use for fabrication of metamaterials of zero-Poisson’s ratio and tunable strength allowing high strength-to-weight ratios. The anisotropy of BE structures is found to be more extreme compared to their BU counterparts. Interestingly, alignment of multiaxial yield surface of auxetic metamaterials is observed to be opposite to that of nonauxetic metamaterials, resulting in poor fitting against extended Hill’s criterion. The BU auxetic structure demonstrates a better impact wave resistance, but lower buckling resistance in y-direction compared with BE auxetic structure, and more. This work provides in-depth useful mechanical information on the structures before employing them in practical applications.

Computational Homogenization for Inverse Design of Surface-based Inflatables

Surface-based inflatables are composed of two thin layers of nearly inextensible sheet material joined together along carefully selected fusing curves. During inflation, pressure forces separate the two sheets to maximize the enclosed volume. The fusing curves restrict this expansion, leading to a spatially varying in-plane contraction and hence metric frustration. The inflated structure settles into a 3D equilibrium that balances pressure forces with the internal elastic forces of the sheets. We present a computational framework for analyzing and designing surface-based inflatable structures with arbitrary fusing patterns. Our approach employs numerical homogenization to characterize the behavior of parametric families of periodic inflatable patch geometries, which can then be combined to tessellate the sheet with smoothly varying patterns. We propose a novel parametrization of the underlying deformation space that allows accurate, efficient, and systematical analysis of the stretching and bending behavior of inflated patches with potentially open boundaries. We apply our homogenization algorithm to create a database of geometrically diverse fusing patterns spanning a wide range of material properties and deformation characteristics. This database is employed in an inverse design algorithm that solves for fusing curves to best approximate a given input target surface. Local patches are selected and blended to form a global network of curves based on a geometric flattening algorithm. These fusing curves are then further optimized to minimize the distance of the deployed structure to target surface. We show that this approach offers greater flexibility to approximate given target geometries compared to previous work while significantly improving structural performance.

CNN-based prediction of microstructure-derived random property fields of composite materials

The simulation of random spatial variation in the mechanical properties of composite materials using random fields derived from their microstructure can be computationally demanding, since it often requires numerical homogenization of a large number of stochastic volume elements (SVEs). These SVEs are usually extracted from an initial image of the composite microstructure by using a moving window technique and their homogenized properties constitute the discrete data points of the random field at the centroid of each SVE. By replacing the expensive and highly repetitive homogenization procedure with a convolutional neural network (CNN), it is possible to perform nearly instant computations of random property fields. In this work, a CNN is proposed, that takes as input an SVE image and returns its apparent mechanical properties, and can therefore be applied along with the moving window technique. Training is performed on randomly generated SVEs with varying volume fractions and inclusion positions, which are representative of the SVEs obtained during processing of the initial large composite image. It is shown that the proposed CNN can accurately predict the mechanical properties of SVEs and it is subsequently applied for the prediction of random property fields derived from computer simulated and real microstructure images. Results show that, with the proposed methodology, it is possible to make accurate predictions of random fields within a few seconds, a procedure which could potentially require hours of computation time with the finite element based approach.

The role of machine learning for insight into the material behavior of lattices: A surrogate model based on data from finite element simulation

In recent years, lattice structures have gained extensive attention due to their unique properties and great potential in various fields. Their effective material properties can be obtained by several means, e.g., numerical homogenization methods, finite element (FE) simulations, and experiments. Numerical homogenization methods involve certain assumptions, and thus, their applications are limited to those lattice structures satisfying all assumptions. On the contrary, although the FE simulations and experiments can be performed for all lattice structures, they are more time-consuming and tedious than the numerical homogenization methods, especially for lattice structures with complex geometry. In this study, a surrogate model is presented to address all the drawbacks mentioned above. More precisely, the effective elastic modulus of two-dimensional lattice structures with triangular unit cells is predicted by a surrogate model trained with the random forest algorithm. The dataset is generated from FE simulations. Several geometric details along with the elastic modulus of the parent material are selected as input features. These input features are varied to cover a wide range of lattice structures encountered in engineering applications. The correlation between input features is investigated to test their independence. Furthermore, the contribution of each input feature to the prediction of the surrogate models is also thoroughly examined. The accuracy of the proposed surrogate model is evaluated through comparisons with FE simulations, numerical homogenization methods, and experiments for untrained datasets.

Predicting mechanical heterogeneity in glassy polymer nanocomposites via an inverse computational approach based on atomistic molecular simulations and homogenization methods

Probing the mechanical behavior of the region formed between a nanoparticle reinforcement and a polymer matrix in a polymer nanocomposite structure, denoted as the “interphase”, is a main challenge as such regions are difficult to investigate by experimental methods. Here, we accurately characterize the heterogeneous mechanical behavior of polymer nanocomposites, focusing on polymer/nanofiller interphases via a combination of nanomechanical simulations and numerical homogenization techniques. Initially, the global mechanical performance of a glassy poly(ethylene oxide) polymer nanocomposite reinforced with silica nanoparticles is studied using detailed atomistic molecular dynamics simulations for 1.9% and 12.7% silica volume fractions. Next, the polymer/silica interphase thickness is identified by probing the polymer atom-based density distribution profile in the vicinity of the nanofiller at equilibrium. On the basis of this thickness, the interphase is subdivided to check the position-dependent change in mechanical properties. Then, using continuum mechanics and atomistic simulations, we proceed to compute the effective Young’s modulus and Poisson’s ratio of the polymer/nanoparticle interphase as function of the distance from the nanoparticle. In the last step, an inverse numerical homogenization model is proposed to predict the mechanical properties of the interphase on the basis of a comparison criteria with the data from MD. The results were found to be acceptable, raising the possibility of accurately and efficiently predicting interfacial properties in nanostructured materials.

Two-scale asymptotic homogenization analysis of piezoelectric composite materials in generalized curvilinear coordinates

This paper presents a comprehensive study on applying the two-scale asymptotic homogenization method to derive the effective properties of piezoelectric composite materials characterized by generalized periodicity in curvilinear coordinates. The methodology involves formulating the homogenization problem in a general curvilinear framework suitable for materials with complex geometric configurations. By systematically deriving the homogenized equations and effective coefficients, the paper provides a robust theoretical foundation for understanding the macroscopic behavior of piezoelectric composites under various loading conditions. The model’s versatility is demonstrated through its application to specific curvilinear coordinate systems, including the helix coordinate system and cylindrical coordinates with wavy geometry. These examples highlight the potential of the proposed approach in analyzing and designing piezoelectric materials with intricate geometries. The state-of-the-art numerical homogenization methods have also validated the proposed homogenization method, ensuring its accuracy and robustness.

Learning-based multi-continuum model for multiscale flow problems

Multiscale problems can usually be approximated through numerical homogenization by an equation with some effective parameters that can capture the macroscopic behavior of the original system on the coarse grid to speed up the simulation. However, this approach usually assumes scale separation and that the heterogeneity of the solution can be approximated by the solution average in each coarse block. For complex multiscale problems, the computed single effective properties/continuum might be inadequate. In this paper, we propose a novel learning-based multi-continuum model to enrich the homogenized equation and improve the accuracy of the single continuum model for multiscale problems with some given data. Without loss of generalization, we consider a two-continuum case. The first flow equation keeps the information of the original homogenized equation with an additional interaction term. The second continuum is newly introduced, and the effective permeability in the second flow equation is determined by a neural network. The interaction term between the two continua aligns with that used in the Dual-porosity model but with a learnable coefficient determined by another neural network. The new model with neural network terms is then optimized using trusted data. We discuss both direct back-propagation and the adjoint method for the PDE-constraint optimization problem. Our proposed learning-based multi-continuum model can resolve multiple interacted media within each coarse grid block and describe the mass transfer among them, and it has been demonstrated to significantly improve the simulation results through numerical experiments involving both linear and nonlinear flow equations.

Multiscale modeling on the multi-physical behaviors of high temperature superconducting magnets based on a combined global homogenization and local refinement scheme

The safe and stable operation is a crucial issue in the development of high-field high temperature superconducting (HTS) magnets. In this paper, we construct a multiscale model which couples the homogenized global (macroscopic) behavior and the refined local (mesoscopic) characteristics to simulate the coupled electromagnetic-mechanical-thermal behaviors of the HTS magnets. In the model, the numerical homogenization method is adopted to simulate the macroscopic behavior of the magnets and identify the ‘dangerous region’ of the magnet which are prone to damage or quench. Then, a refined local sub-model which coupling with the macroscopic homogenization model is established by considering the microstructure and physical parameters of each components of the HTS tapes in the ‘dangerous region’. Thus, a combined global homogenization and local refinement scheme which balances the computational efficiency and numerical accuracy is developed to simulate the coupled multi-physical behaviors of the HTS magnets including the quench and its propagation. Our results show that the refined local sub-model can simulate the electromagnetic field and the stress-strain at the scale of the tape more accurately. Characteristics, such as the discontinuous stress distribution across the interfaces between different layers and the current shunt from the HTS layer to metallic layers during the quench process of HTS tapes, which are beyond the capability of the homogenization model, have also been well depicted by the refined sub-model.

Sparse Cholesky factorization for solving nonlinear PDEs via Gaussian processes

In recent years, there has been widespread adoption of machine learning-based approaches to automate the solving of partial differential equations (PDEs). Among these approaches, Gaussian processes (GPs) and kernel methods have garnered considerable interest due to their flexibility, robust theoretical guarantees, and close ties to traditional methods. They can transform the solving of general nonlinear PDEs into solving quadratic optimization problems with nonlinear, PDE-induced constraints. However, the complexity bottleneck lies in computing with dense kernel matrices obtained from pointwise evaluations of the covariance kernel, and its partial derivatives, a result of the PDE constraint and for which fast algorithms are scarce. The primary goal of this paper is to provide a near-linear complexity algorithm for working with such kernel matrices. We present a sparse Cholesky factorization algorithm for these matrices based on the near-sparsity of the Cholesky factor under a novel ordering of pointwise and derivative measurements. The near-sparsity is rigorously justified by directly connecting the factor to GP regression and exponential decay of basis functions in numerical homogenization. We then employ the Vecchia approximation of GPs, which is optimal in the Kullback-Leibler divergence, to compute the approximate factor. This enables us to compute ϵ \epsilon -approximate inverse Cholesky factors of the kernel matrices with complexity O ( N log d ⁡ ( N / ϵ ) ) O(N\log ^d(N/\epsilon )) in space and O ( N log 2 d ⁡ ( N / ϵ ) ) O(N\log ^{2d}(N/\epsilon )) in time. We integrate sparse Cholesky factorizations into optimization algorithms to obtain fast solvers of the nonlinear PDE. We numerically illustrate our algorithm’s near-linear space/time complexity for a broad class of nonlinear PDEs such as the nonlinear elliptic, Burgers, and Monge-Ampère equations. In summary, we provide a fast, scalable, and accurate method for solving general PDEs with GPs and kernel methods.

Design of the shell-infill structures using a phase field-based topology optimization method

The design of shell-infill structures has been a focal point in the topology optimization community due to their advantages in energy absorption characteristics, strength-to weight ratio and bucking resistance. This paper introduces a phase field-based topology optimization method for designing shell-infill structures. Interface-related issues can be easily addressed through the phase field function. A coupled topology optimization process is proposed to establish the connection between the shell and infill, facilitating the generation of optimized structures. The shell thickness, infill pattern and infill volume percentage, can be naturally controlled by different model parameters. Additionally, multiscale phase field topology optimization integrates the numerical homogenization method to evaluate the effective elasticity matrix of the microstructural infill. The approach is introduced for a uniform, periodical microstructure layout in the infill region, thereby achieving superior mechanical properties. Numerical results indicate the effectiveness of the proposed method in the design of both 2D and 3D shell-infill structures.