Numerical Homogenization Method Research Articles

Experimental and Numerical Investigation of Polymer-Based 3D-Printed Lattice Structures with Largely Tunable Mechanical Properties Based on Triply Periodic Minimal Surface.

Triply periodic minimal surfaces (TPMSs) have demonstrated significant potential in lattice structure design and have been successfully applied across multiple industrial fields. In this work, a novel lattice structure with tunable anisotropic properties is proposed based on two typical TPMS types, and their mechanical performances are studied both experimentally and numerically after being fabricated using a polymer 3D printing process. Initially, adjustments are made to the original TPMS lattice structures to obtain honeycomb lattice structures, which are found to possess significant anisotropy, by utilizing numerical homogenization methods. Based on this, a continuous self-twisting deformation is proposed to change the topology of the honeycomb lattice structures to largely tune the mechanical properties. Quasi-static compression experiments are conducted with different twisting angles, and the results indicate that self-twisting can affect the mechanical properties in specific directions of the structure, and also enhance the energy absorption capacity. Additionally, it mitigates the risk of structural collapse and failure during compression while diminishing structural anisotropy. The proposed self-twisting strategy, based on honeycomb lattice structures, has been proven valuable in advancing the investigation of lattice structures with largely tunable mechanical properties.

Multiscale simulation and prediction of mechanical properties for carbon fiber reinforced polymer orthotropic laminate

AbstractA multiscale analysis method based on numerical homogenization is presented for predicting the mechanical properties of carbon fiber reinforced polymer (CFRP) orthogonally symmetric laminate. Initially, based on the structural characteristics of CFRP orthotropic symmetric laminate, the relational equations for calculating the mechanical engineering constants from the stiffness coefficient are proposed. Subsequently, the finite element model (FEM) of orthotropic laminate is established. Corresponding subroutines are developed through ABAQUS‐Python to extract and compute the simulation results and obtain the mechanical properties of the laminate. Next, the macroscale mechanical properties are estimated by combining the micro‐scale mechanical model, classical laminate theory (CLT), and volume averaging method. Ultimately, the accuracy of the proposed equations and the feasibility of the numerical homogenization method are demonstrated by comparing the results of numerical simulation predictions, theoretical predictions, and experiments.Highlights Present the relational equations for calculating the mechanical engineering constants of CFRP orthotropic symmetric laminate from the stiffness coefficient. Establish the FEM of orthotropic laminate and develop corresponding subroutines to obtain the mechanical properties of the laminate. Analyze the mechanical properties of orthotropic laminate through microscopic modeling and CLT.

A series of auxetic metamaterials with negative thermal expansion based on L-shaped microstructures

Mechanical metamaterials are facing various multi-functional requirements. In this study, a series of auxetic metamaterials with enhanced negative thermal expansion and improved stiffness are proposed. These metamaterials consist of four novel L-shaped microstructures connected in various clockwise or counterclockwise directions. Two length ratios and four angle parameters are utilized to define the geometry of these metamaterials. Both analytical analysis and validated numerical homogenization methods are employed to determine their effective thermoelastic properties, including effective Young's modulus, Poisson's ratio, and coefficient of thermal expansion. It is demonstrated that tailoring these microstructural geometries can modulate the effective thermoelastic properties of these metamaterials accordingly. By appropriately tailoring the geometric parameters, these metamaterials exhibit increased effective stiffness as well as more pronounced auxeticity and negative thermal expansion compared to traditional triangle and trapezoid microstructures. Notably, the in-plane effective anisotropic or isotropic Poisson's ratio and coefficient of thermal expansion can be adjustable over a broader range from negative to positive values. High stiffness, High strength, and near-zero thermal expansion can be achieved concurrently in these metamaterials, which is very conducive for structural designing in thermal environments. These metamaterials with a coupled design of stiffness, thermal expansion, and Poisson's ratio hold significant potential for advancing applications in satellite platforms, space structures, composite sandwiches, and precision equipment.

Optimization of the numerical homogenization method for cellular structures manufactured by Selective Laser Melting

Rapid development of additive technologies presents enormous opportunities. associated with the production of complex geometries. For this reason. the field of 3D printing has become an ideal technique for producing cellular structures. The paper addresses the issue. related to the determination of the minimum number of unit cells. at which the structure is characterized by a constant effective Young's modulus. Currently. it is not possible to state unequivocally what number of individual cells is needed. For this reason. the paper proposes an algorithm by which the minimum number of unit cells can be determined. The parameters of the structure. such as the relative density and the topology of a single cell. have an important influence. It is worth noting that the base material also plays an important role. Various methods are used for numerical analysis of cellular structures. such as finite element-based analysis. numerical homogenization and beam-based analysis. Nevertheless. each of them has its own advantages as well as disadvantages. This paper proposes to optimize the numerical homogenization of cellular structures. using models consisting of a minimum number of unit cells and a 3D representation of actual printed models of cellular structures. Based on these optimizations. better agreement with experiment was obtained.

Simulation of a Composite with a Polyhydroxybutyrate (PHB) Matrix Reinforced with Cylindrical Inclusions: Prediction of Mechanical Properties.

Biocomposite development, as a sustainable alternative to fossil-derived materials with diverse industrial applications, requires expediting the design process and reducing production costs. Simulation methods offer a solution to these challenges. The main aspects to consider in simulating composite materials successfully include accurately representing microstructure geometry, carefully selecting mesh elements, establishing appropriate boundary conditions representing system forces, utilizing an efficient numerical method to accelerate simulations, and incorporating statistical tools like experimental designs and re-regression models. This study proposes a comprehensive methodology encompassing these aspects. We present the simulation using a numerical homogenization technique based on FEM to analyze the mechanical behavior of a composite material of a polyhydroxybutyrate (PHB) biodegradable matrix reinforced with cylindrical inclusions of flax and kenab. Here, the representative volume element (RVE) considered the geometry, and the numerical homogenization method (NHM) calculated the macro-mechanical behavior of composites. The results were validated using the asymptotic homogenization method (AHM) and experimental data, with error estimations of 0.0019% and 7%, respectively. This model is valuable for predicting longitudinal and transverse elastic moduli, shear modulus, and Poisson's coefficient, emphasizing its significance in composite materials research.

A black-box optimization strategy for customizable global elastic deformation behavior of unit cell-based tri-anti-chiral metamaterials

Metamaterials are a class of materials with a distinctive unit cell-based periodic architecture, often resulting in unique mechanical properties. The potential of metamaterials can be further improved by using gradients of unit cell parameters and thereby creating a specific distribution of material properties in a part. This design freedom comes with the challenge of finding new suitable design and optimization strategies. In this study, a Finite Element simulation-based optimization framework to create a predefined deformation behavior of unit cell-based metamaterials is presented. The framework consists of a numerical homogenization method to create an efficient linear elastic material representation, an interpolation scheme for a fast correlation of unit cell parameters with homogenized material properties and a black-box based optimization part. The framework is tested on a tri-anti-chiral metamaterial with additional geometric parameters and a newly developed transition unit cell. Several specified lateral deformations of simple 2D rectangular test specimens under tensile load are used as primary optimization targets and the homogenized results are validated against Finite Element simulations of fully modeled tri-anti-chiral structures. In addition, the effect of the design space and the number of design variables is studied and several combinations of optimization objectives and constraints are tested.

Simplified Numerical Homogenization Method for the Plasticity Problem with Isotropic Hardening

Simplified Numerical Homogenization Method for the Plasticity Problem with Isotropic Hardening

Effective Superelastic Domains of Schwarz Primitive Shape Memory Alloy Lattices Subjected to Cyclic Loading

This work investigates the evolution of effective phase transformation thresholds in superelastic shape memory alloy (SMA) lattice architectures subjected to cyclic loading. The investigation is carried out by means of finite element analysis and numerical homogenization methods, considering Schwarz primitive triply periodic minimal surface (TPMS) SMA unit cells subjected to periodic boundary conditions. The results are reported for relative densities between 10% and 50%. It is shown that, in all the cases considered, the loading surfaces governing both forward and reverse phase transformations have an ellipsoidal shape that can be reasonably well represented using an extended Hill's criterion that accounts for the influence of hydrostatic pressure. Moreover, by considering polynomial expressions of the coefficients of Hill's criterion in terms of martensite volume fraction and cumulated fraction, the evolution of the loading surfaces as a consequence of functional fatigue is reasonably well approximated. The results suggest that an extended Hill's criterion with variable coefficients can well be used to simulate effective phase transformation and functional fatigue in architected TPMS SMAs.

Multifunctional mechanical metamaterials with tunable double-negative isotropic properties

This research was focused on innovative design of lattice metamaterials that can exhibit tunable double-negative mechanical properties and elastic isotropy simultaneously. A discrete topology optimization method using a multi-material ground structure was developed to create microlattices exhibiting both negative thermal expansion coefficient and negative Poisson’s ratio in a single integrated design, while maintaining elastic isotropy. First, the numerical homogenization method with beam elements was used to estimate the effective thermal and elastic properties of a microlattice. Second, the topological design, subject to required geometric constraints, was formulated as a mixed integer programming problem to discover a series of multi-material microlattices that present customized isotropic values of negative thermal expansion coefficient and negative Poisson’s ratio. Finally, several three-dimensional multi-material microstructures were produced by altering either the cross-sections or constituent materials of struts to demonstrate their tunable mechanical properties.

An asymptotic homogenization model for evaluating the mechanical properties of random fiber reinforced composites with high volume fraction

In this study, we propose an asymptotic homogenization strategy based on local-global RVEs to predict the elastic properties of random fiber reinforced composites (RFCs) with high fiber volume fraction (HFVF). During the calculation, RFCs were divided into a set of local pseudo units with various material properties determined by its spatial orientation of the short fibers. The macroscopic mechanical properties of the RFCs with HFVF were asymptotically determined by integrating transformation matrices, periodic boundary conditions, and numerical homogenization method. The results show that the proposed models were in good agreement with the experimental tests of short fiber reinforced magnesium matrix (Cfs/Mg) composites. When the length ratio of the RVE to the local pseudo L/Ls was >10, the overall orientation of the fibers tended to be completely random, and the results gradually converged. With an increase in the fiber volume fraction, the effective elastic properties of the Cfs/Mg composites grew linearly. Additionally, the proposed asymptotic homogenization framework can also be extended to evaluate other macroscopic physical properties of RFCs such as thermal and electric conductivities.

Effect of the shape and the distribution of cells on the effective thermal conductivity of polyurethane foam

AbstractTo design foams with specific properties, it is important to be able to predict their effective properties. The objective of this study is to compare and quantify the effect of the geometrical parameters on the thermal properties. Then the study uses both analytical and numerical homogenization methods to connect micro‐scale parameters to the macro‐scale thermal behavior. Several statistical studies of the morphology have been used to model the different categories of PU foams. The foam is modeled by a periodic structure of open, closed, or mixed cells. For each type of cell, different idealized shapes of a single cell are used, while keeping a constant porosity for each category. In the case of mixed cells, the effect of the relative positioning of cells is investigated. The results show that the effective thermal conductivity is considerably impacted by the cell shape. Moreover, the geometrical modeling of the struts, in the heuristic modeling, has also an important effect. Furthermore, the relative position of gas cavities, in the spherical representation, influences the effective thermal conductivity. However, the distribution of open and closed cells, in the case of mixed cell foam, has a limited effect.

Topology optimization design of graded infills for 3D curved volume by a conformal sweeping method

Swept volumes are widely existed in various significant engineering applications. However, an effective design method for the lightweight swept volume with porous infills is lacking. This paper develops a novel topology optimization method for the design of curved volumes filled with spatially-varying microstructures. The infill optimization formulation for the curved swept volume is established to concurrently optimize both the microstructural geometry and the macroscopic distribution of the graded infills, aiming at minimizing the structural compliance subject to a global volume constraint. A conformal sweeping approach is developed to map the curved volume into a regular parameterization​ cube along the sweeping trajectory. In this fashion, the graded infills defined in a regular unit cell can be first filled into this cubic parameterization space, and then reversely mapped into the original volume to conformally match the curved geometry. A numerical homogenization method for parallelepiped microstructures is introduced to evaluate several sampling microstructures. The local level sets approach is used to create a family of connectable microstructures, while a polynomial interpolation is employed to rapidly interpolate the graded property of the generated microstructures based on the properties of sampling microstructures. Two different sweeping volume types are investigated. Several examples with respect to the infill optimization design of curved swept volumes are provided to illustrate the characteristics of the proposed method.

Mathematical Modelling of Composite Phantoms for the Calibration of Ultrasound Devices

We propose a methodology to narrow down the selection of design parameters of phantoms for calibrating ultrasound devices. For doing so, we model the phantoms as composite materials. Through a mathematical homogenization technique, the global properties of the phantom are given in terms of the volume fractions of its components, their geometric shapes and physical properties. We explore these parameters to determine ranges that yield the desirable values of acoustic impedance and attenuation. The procedure is illustrated for a class of sandwich-like phantoms made of layers of agar of different concentrations with complex elastic moduli. In this example, homogenization provides analytical formulae of the global properties that can be used to identify the layers’ thicknesses and concentrations that yield a low impedance and an average attenuation coefficient. These analytical results are compared with a numerical homogenization method, showing excellent agreement.

Design of a functionally graded porous uncemented acetabular component: Influence of polar gradation.

The functionally graded porous metal-backed (FGPMB) acetabular component has the potential to minimize strain-shielding induced bone resorption, caused by stiffness mismatch of implant and host bone. This study is aimed at a novel design of FGPMB acetabular component, which is based on numerical investigations of the mechanical behavior of acetabular components with regard to common failure scenarios, considering various daily activities and implant-bone interface conditions. Both radial and polar functional gradations were implemented, and the effects of the polar gradation exponent on the failure criteria were evaluated. The relationships between porosity and orthotropic mechanical properties of a tetrahedron-based unit cell were obtained using a numerical homogenization method. Strain-shielding in cancellous bone was relatively lesser for the FGPMB than solid metal-backing. Few nodes around the rim were susceptible to implant-bone interfacial debonding, irrespective of the polar gradation exponent. Although the most favorable bone remodeling predictions were obtained for a polar gradation exponent of 0.1, a sudden change in the porosity was observed near the rim of FGPMB. Bone remodeling patterns were similar for polar gradation exponent of 5.0 and solid metal-backing. Moreover, the volumetric wear was maximum and minimum for polar gradation exponents of 0.1 and 5, respectively. Furthermore, the micromotions of different polar gradation exponents were within a range (20-40 μm) that might facilitate bone ingrowth. Considering common failure mechanisms, the FGPMB having polar gradation exponents in the range of 0.1-0.5 appeared to be a viable alternative to the solid acetabular component, within which a gradation exponent of 0.25 seemed the most appropriate design parameter.

Additive manufacturing of cellular structures: Multiscale simulation and optimization

This paper introduces a multiscale and multi-purpose simulation framework to investigate selective beam melting processes for metallic cellular structures along the complete process chain, up to the topology optimization of components made of cellular materials. Process simulation methods on various length scales are introduced to analyze the relation of process strategies and resulting properties of the cellular materials, as e. g. warpage, residual stresses, material grain structure and surface roughness. Numerical homogenization methods are applied to investigate the influence of the grain structure and the topology of the cellular materials on their mechanical properties. Based on these process-dependent mechanical properties, a two-scale optimization of components made of cellular material is performed, aiming in particular on optimizing the buckling resistance of the structures. The interfaces between the different simulation tools are specified and illustrate the flexibility of the numerical framework. It is shown that consistent simulations of additive manufacturing of cellular materials – from the manufacturing process to the optimized component – provide important insights into process–structure interactions and enable tailored additive manufacturing processes.

Massive and fast fabrication of pentamode devices through a multiphase honeycomb-corrugation configuration

Pentamode material (PM) is a special fluid-like acoustic metamaterial with intrinsic broadband merits. The unit cells of PM acoustic devices were based on single-material in previous reports, whose features make the available fabrication techniques very limited and the corresponding fabrication is not economic. In this investigation, a novel PM configuration composed of multi-materials is proposed to facilitate the designing and fabrication of PM devices. The novel multiphase PM configuration contains three parts: honeycomb-corrugation latticed microstructures, mass adjusting weights and interconnecting phases. The mass adjusting weights are bond to metallic latticed structure via interconnecting materials, while metallic lattices are fabricated by stamping forming technique. Simulated Annealing algorithm integrated with numerical homogenization method is developed for optimization of PM designing. A water impedance matching multiphase PM device based on this novel configuration was designed and verified in this study, and it was revealed from both numerical simulation and experiments that the designed multiphase PM device mimicked the acoustic properties of water below 25 kHz. The results indicate that the multiphase PM configuration exhibits superior performance on many aspects such as withstanding much higher hydrostatic pressure, cutting down manufacturing cost by 90% and reducing construction period by 98% compared with conventional techniques.

PH-Net: Parallelepiped microstructure homogenization via 3D Convolutional Neural Networks

Microstructures are attracting academic and industrial interest because of 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 are considered a more efficient choice but the microstructures are limited to cubic shapes, therefore are unsuitable for periodic microstructures with a more general shape, e.g., parallelepipeds. This paper introduces a fine-designed 3D convolutional neural network (CNN) for fast homogenization of parallelepiped microstructures, named PH-Net. Superior to existing data-driven methods, PH-Net predicts the local displacements of microstructures under specified macroscopic strains instead of direct homogeneous material, empowering 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 hundreds of times faster than numerical homogenization and even supports online computing. Moreover, it does not require a labeled dataset and thus the training process is much faster than current deep learning methods. Because it can predict local displacement, PH-Net provides both homogeneous material properties and microscopic mechanical properties, e.g., strain and stress distribution, and yield strength. We also designed a set of physical experiments using 3D printed materials to verify the prediction accuracy of PH-Net.

Fourier Transform approach to numerical homogenization of periodic media containing sharp insulating and superconductive cracks

In the present paper, we propose a new FFT based numerical homogenization method to compute the effective conductivity of fractured media involving insulating and superconductive cracks. The governing Lippmann Schwinger (LS) equations based on temperature jump and heat flux jump distribution localized on crack lines are formulated and the geometry of sharp cracks is described by exact expressions of form factors. To solve the LS equations, the bi conjugate gradient stabilized method is used and leads to fast convergence. Numerical examples in 2D and 3D show a good accuracy when compared with standard Finite Element Method solutions.

The Mechanical Properties Prediction of Poly [(3-hydroxybutyrate)-co-(3-hydroxyvalerate)] (PHBV) Biocomposites on a Chosen Example.

This paper aims to experimentally determine the properties of the poly [(3-hydroxybutyrate)-co-(3-hydroxyvalerate)]-(PHBV)-30% hemp fiber biocomposite, which is important in terms of numerical simulations of product manufacturing, and to evaluate the mechanical properties by means of micromechanical modeling. The biocomposite was manufactured using a single-screw extruder. Specimens for testing were produced by applying the injection molding technology. Utilizing the simulation results of the plastic flow, carried out by the Moldflow Insight 2016 commercial software and the results of experimental tests, the forecasts of selected composite mechanical properties were performed by means of both numerical and analytical homogenization methods. For this purpose, the Digimat software was applied. The necessary experimental data to perform the calculations for the polymer matrix, fibers, and the biocomposite were obtained by rheological and thermal studies as well as elementary mechanical tests. In the paper, the method of determining selected properties of the biocomposite and the method of forecasting its other properties are discussed. It shows the dependence of the predicted, selected properties of the biocomposite on the filler geometry assumed in the calculations and the homogenization method adopted for the calculations. The results of the work allow for the prediction of properties of the PHBV biocomposites-hemp fiber for any amount of filler used. Moreover, the results allow for the estimation of the usefulness of homogenization methods for the prediction of properties of the PHBV-hemp fiber biocomposites. Furthermore, it was found that for the developed and tested biocomposites, the most effective possibility of mechanical properties prediction is using the Mori-Tanaka homogenization model, which unfortunately has some limitations.

Elastic Analysis of Three-Layer Concrete Slab Based on Numerical Homogenization with an Analytical Shear Correction Factor

Sandwich structures are widely used in construction, as well as in the aviation, spaceship, and electronics industries. The interesting result, among others, is the fact that individual layers can be freely selected to meet the planned requirements. In the case of sandwich structures in construction, they must meet the requirements of load-bearing capacity, thermal, and acoustic insulation, and additionally, they must be resistant to biological and chemical corrosion. The paper presents calculation algorithms for Hoff’s three-layer panels. In the first case, the well-known and proven method of finite differences in variation terms was used, assuming actual geometrical and material parameters. In the second case, the numerical homogenization method of the layered panel was used, replacing the stiffnesses of individual layers with a homogeneous equivalent plate with substitute stiffness corrected in shearing by an analytically derived shear correction factor. A comparative analysis of the results of the calculations with the use of both approaches was carried out. A good agreement between the displacement values and the calculated cross-sectional forces was obtained. On this basis, it can be assumed that the static analysis of a slab by simplified methods using numerical homogenization with an analytical shear correction factor is appropriate and can be applied to layer structures.