Numerical Homogenization Technique Research Articles

Homogenization theory and nonlinearities in Darcy's law

The theory of homogenization of differential equations has become an open field of research in several areas of the exact sciences and has proved to be a powerful tool for understanding the global behavior of heterogeneous materials. Despite knowing that the deduction of Darcy’s law through the Navier-Stokes equations has been debated for decades, many questions remain open, mainly regarding more complex boundary conditions, cases involving multiphase flows and the numerical homogenization techniques. It is known today that Darcy’s law is presented in the form of a linear relationship only for a range of hydraulic gradient and that this range overlaps the range of laminar flow through soil voids. Therefore, it is proposed in this work to understand the loss of linearity in Darcy’s law, based on the theory of homogenization, modifying and exploring the limit results obtained by Allaire in 1991.

Application of Artificial Neural Networks to Numerical Homogenization of the Precast Hollow-Core Concrete Slabs

The main goal of this work is to combine the usage of the numerical homogenization technique for determining the effective properties of representative volume elements with artificial neural networks. The effective properties are defined according to the classical laminate theory. The purpose is to create and train a rapid surrogate model for the quick calculation of the mechanical properties of hollow concrete slabs. First, the homogenization algorithm was implemented, which determines membrane, bending and transverse shearing properties of a given parametrized hollow-core precast slab reinforced with steel bars. The algorithm uses the finite element mesh but does not require a formal solution of the finite element method problem. Second, the learning and training artificial intelligence framework was created and fed with a dataset obtained by optimal Latin hypercube sampling. In the study, a multilayer perceptron type of artificial neural network was used. This allows for obtaining rapid calculations of the effective properties of a particular hollow-core precast slab by using a surrogate model. In the paper, it has been proven that such a model, obtained via complex numerical calculations, gives a very accurate estimation of the properties and can be used in many practical tasks, such as optimization problems or computer-aided design decisions. Above all, the efficient setup of the artificial neural network has been sought and presented.

Effect of plant short fibers on the mechanical properties of carbon fiber reinforced epoxy matrix by using FEM based numerical homogenization technique

In this study, we utilized the material design library in Ansys Product 2022 R1 to analyze the impact of short plant fibers on the mechanical properties of epoxy resin. We employed the representative volume element (RVE) approach and examined different volume fractions up to 50%. The technique employed in this study was multi-scale modeling based on the finite element method (FEM) for numerical homogenization. Subsequently, we used the same software to develop a five-layer laminated composite. The samples were based on epoxy resin enriched with short plant fibers at fractions of 20% and 30%, as well as pure epoxy resin. The laminated composites were reinforced with carbon fibers. The objective of this study was to analyze the mechanical response of each sample during a tensile test, focusing on multiple parameters. The sought-after results included total deformation, directional deformation, equivalent and elastic stress (Von-mises). The samples containing epoxy resin enriched with 30% of short hemp fibers exhibited remarkable mechanical strength compared to the other samples. These results suggest that the incorporation of short hemp fibers into the matrix led to a significant improvement in the mechanical strength of the composite.

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.

Metasurface characterization based on eigenmode analysis and averaging of electromagnetic fields

A fully numerical homogenization technique for the retrieval of the effective surface susceptibilities of a periodic composite metasurface is developed in this work. We utilize the so-called dual field-flux finite element formulation scheme to accurately calculate the eigenmodes of a composite periodic metasurface, a scheme that possesses a crucial advantage: The capability of evaluating all field components and their derivatives accurately and with the same order of approximation is a requirement for our proposed technique. Next, we derive the generalized sheet transition condition equations for a general bianisotropic metasurface, which correlate the field components on either sides of the metasurface with its surface susceptibilities. At this point, we establish a new field averaging scheme for the acquisition of the average field components of the modes supported by the metasurface. Combining this computational information, we derive a set of linear algebraic equations based on the GSTCs and the average electromagnetic fields and numerically solve it, so as to obtain the effective surface susceptibilities of a bianisotropic metasurface. Comparison with the results of other techniques in the literature shows very good agreement, relatively to the resonance behavior of the returned values and their position at the frequency spectrum. The advantages that distinguish the proposed technique over other related methods are its foundation on the intrinsic modal information of the eigenmodes supported by the metasurface and its independence of any wave excitation schemes or involvement of analytical polarizability calculations.

On effective thermal conductivity of heterogeneous material: From periodic unit inclusion cell to large representative volume element

The custom design of a composite material requires the knowledge of its effective behavior which depends on several parameters such as the properties of the constituents, their distributions, shapes and sizes. In thermal conductivity, all these parameters have a direct and significant influence on the effective thermal conductivity (ETC) of composites. These parameters act differently on the behavior of the composite according to its use as much as a good conductor by improving its ETC or by reducing it by considering it as an insulator. The main goal of this work is an extensive study of all possible situations of ETC by a numerical homogenization technique based on the Finite Elements Method. For this purpose, the two main categories of composites, namely the periodic composites represented by the unit cell and the random RVE, are considered. For each category of composite, different contrasts, different particles’ volume fractions, aspect ratios, positions, and orientations are studied. In order to ensure the simulation results are effective and representative, two boundary conditions, namely Uniform Temperature Gradient Boundary Conditions (UGT) and Periodic Thermal Boundary Conditions (PBC), were imposed on all considered microstructures. The results of this study constitute a synthesis of the effective thermal conductivities of all the possible situations which can be summarized as follows: On the one hand, the reinforcements shape effect is very important irrespective of the nature of the periodic and random composites and on the other hand, the two types of composites do not have the same behavior as that for the reinforcements with a circular or almost circular shape. This study also emphasizes the situations where the ETC can be estimated analytically and when homogenization is necessary. From a practical point of view, and on the basis of the obtained results, this study made it possible to show clearly in which situation the composite becomes more conducting and in the opposite case when it becomes more insulating.

Characterization of Pultruded Glass-Fiber Reinforced Polymers with Two-Step Homogenization

The aim of this work is to determine effective elastic properties of pultruded Glass Fiber Reinforced Polymer using micro-CT in conjunction with a two-step numerical homogenization technique. The two-step homogenization involves the segmentation of the material’s layers, which was made here by means of a machine learning approach. The segmentation was validated through the comparison between the phase’s volume fractions of samples obtained from the segmented images and laboratory tests. Further, a standard accuracy analysis in a 10-fold cross validation was performed. The samples’ effective axial Young’s modulus obtained by our numerical homogenization were compared to results obtained from experimental tests. For both the experimental tests and the image-based numerical analysis we considered samples extracted from the same profile. The two-step methodology allowed the homogenization of large volumes of the composite corresponding to the whole thickness of the profile, imaged with a high resolution. In addition to the axial effective Young’s modulus, our methodology was also able to successfully provide all the other elastic properties along the three orthogonal directions, even the ones that are arduous to be obtained in laboratory setups.

Surrogate optimization of a lattice foot orthotic

Background:Additive manufacturing enables to print patient-specific Foot Orthotics (FOs). In FOs featuring lattice structures, the variation of the cell’s dimensions provides a locally variable stiffness to meet the therapeutic needs of each patient. In an optimization problem, however, using explicit Finite Element (FE) simulation of lattice FOs with converged 3D elements is computationally prohibitive. This paper presents a framework to efficiently optimize the cell’s dimensions of a honeycomb lattice FO for flat foot condition. Methods:We built a surrogate based on shell elements whose mechanical properties were computed by the numerical homogenization technique. The model was submitted to a static pressure distribution of a flat foot and it predicted the displacement field for a given set of geometrical parameters of the honeycomb FO. This FE simulation was considered as a black-box and a derivative-free optimization solver was employed. The cost function was defined based on the difference between the predicted displacement by the model against a therapeutic target displacement. Results:Using the homogenized model as a surrogate significantly accelerated the stiffness optimization of the lattice FO. The homogenized model could predict the displacement field 78 times faster than the explicit model. When 2000 evaluations were required in an optimization problem, the computational time was reduced from 34 days to 10 hours using the homogenized model rather than explicit model. Moreover, in the homogenized model, there was no need to re-create and re-mesh the insole’s geometry in each iteration of the optimization. It was only required to update the effective properties. Conclusion:The presented homogenized model can be used as a surrogate within an optimization framework to customize cell’s dimensions of honeycomb lattice FO in a computationally efficient manner.

Numerical analysis of steady-state heat conduction behavior in high-temperature superconductor Roebel cable

In this work, the steady-state heat transfer characteristics of REBCO tape and impregnated Roebel cable are studied numerically. By establishing three-dimensional finite element models of tape and impregnated cable, the temperature-dependent equivalent thermal conductivities of REBCO tapes and impregnated Roebel cables are estimated based on numerical homogenization technique. Effects of different geometric and material parameters are discussed. The steady-state heat transfer characteristics of a transposed tape in Roebel cable are further investigated. The equivalent thermal conductivities of REBCO tape or impregnated cable show strong anisotropy. Various sensitivities to geometric and material parameters are observed in equivalent thermal conductivities. The total heat flux distribution of the transposed tape is closely related to the Roebel angle and the curvature radius at outer corner or inner corner of this region. The corresponding structure and components should be optimized in actual application.

Modeling of the Pore Shape Effect on the Effective Young's Modulus of Lotus-Type Porous Materials by a Numerical Homogenization Technique

Lotus-type porous materials (LTPMs) are considered as a new category of engineering materials. They are porous materials characterized by long, straight, unidirectional cylindrical pores, and are obtained via unidirectional solidification from a melt under hydrogen and argon atmospheres. The anisotropic pore morphology of lotus-type materials results in the anisotropy of their mechanical and physical properties. This study aims at investigating the effect of cross-sectional pore shapes on the effective Young's modulus (EYM) of LTPMs. The representative volume element-based finite element homogenization method was used to compute the effective bulk and shear moduli. Subsequently, the EYM was deduced from the effective bulk and shear moduli. The numerical results of the circular pores were validated by comparing them with experimental results. Because the results indicated that the EYM is extremely sensitive to the variation in the pore shapes, a formula for estimating the EYM of LTPMs by considering the pore shapes was developed and validated.

Computationally efficient model to predict the deformations of a cellular foot orthotic

BackgroundFoot orthotics (FOs) are frequently prescribed to provide comfortable walking for patients. Finite element (FE) simulation and 3D printing pave the way to analyse, optimize and fabricate functionally graded lattice FOs where the local stiffness can vary to meet the therapeutic needs of each individual patient. Explicit FE modelling of lattice FOs with converged 3D solid elements is computationally prohibitive. This paper presents a more computationally efficient FE model of cellular FOs. MethodThe presented FE model features shell elements whose mechanical properties were computed from the numerical homogenization technique. To verify the results, the predictions of the homogenized models were compared to the explicit model's predictions when the FO was under a static pressure distribution of a foot. To validate the results, the predictions were also compared with experimental measurements when the FO was under a vertical displacement at the medial longitudinal arch. ResultsThe verification procedure showed that the homogenized model was 46 times faster than the explicit model, while their relative difference was less than 8% to predict the local minimum of out-of-plane displacement. The validation procedure showed that both models predicted the same contact force with a relative difference of less than 1%. The predicted force-displacement curves were also within a 90% confidence interval of the experimental measurements having a relative difference smaller than 10%. In this case, using the homogenized model reduced the computational time from 22 h to 22 min. ConclusionThe presented homogenized model can be therefore employed to speed up the FE simulation to predict the deformations of the cellular FOs.

Electromagnetic parameter retrieval technique utilizing eigenvalue analysis and field averaging

In this paper, a fully numerical homogenization technique is presented for the retrieval of the effective constitutive parameters of a periodic composite medium. Based on the eigensolution returned by the utilized field-flux FEM formulation, a proper field averaging process of the field components is proposed over specific paths of the periodic unit cell of an arbitrary structure. Subsequently, the constitutive relations for every eigenmode supported by the medium are derived, forming a linear system of equations. The solution of this system returns a set of fifteen effective constitutive parameters, including the magnetoelectric coefficients, which account for possible cross-polarization effects, causing bianisotropy.

Investigation of the effect of raster angle, build orientation, and infill density on the elastic response of 3D printed parts using finite element microstructural modeling and homogenization techniques

Although the literature is abundant with the experimental methods to characterize mechanical behavior of parts made by fused filament fabrication 3D printing, less attention has been paid in using computational models to predict the mechanical properties of these parts. In the present paper, a numerical homogenization technique is developed to predict the effect of printing process parameters on the elastic response of 3D printed parts with cellular lattice structures. The development of finite element computational models of printed parts is based on a multi scale approach. Initially, at the micro scale level, the analysis of micro-mechanical models of a representative volume element is used to calculate the effective orthotropic properties. The finite element models include different infill densities and building/raster orientation maintaining the bonded region between the adjacent fibers and layers. The elastic constants obtained by this method are then used as an input for the creation of macro scale finite element models enabling the simulation of the mechanical response of printed samples subjected to the bending, shear, and tensile loads. Finally, the results obtained by the homogenization technique are validated against more realistic finite element explicit microstructural models and experimental measurements. The results show that, providing an accurate characterization of the properties to be fed into the macro scale model, the use of the homogenization technique is a reliable tool to predict the elastic response of 3D printed parts. The outlined approach provides faster iterative design of 3D printed parts, contributing to reducing the number of experimental replicates and fabrication costs.

Natural fiber alfa/epoxy randomly reinforced composite mechanical properties identification

In recent years, there has been a growing interest in the international scientific community to study natural fiber composites. Natural fibers have interesting mechanical features, such as strength, lightness, and elongation at break. Natural fibers are also bio-sourced, biodegradable and recyclable. When used with plastic or recyclable matrices, natural fiber composites, such as alfa fiber, can also be reused and recycled. The growing global environmental and new environmental regulations have encouraged research on new recyclable composites environment friendly. The recyclable composite must satisfy minimal mechanical properties to be widely used in furniture, automotive or construction industries. To produce a natural fiber composite that can be fabricated widely, in this paper, composite materials were fabricated using the Epoxy resin as matrix and the natural Alfa fiber (tipa tenacissima L.) as reinforcement. The Alfa fibers are short and randomly distributed. Molding technique is used to produce the composite. To judge the usage of the composite in structural applications, the mechanical properties of the biobased composite are identified by numerical homogenization and by experiments. The finite elements numerical homogenization technique is based on a representative cell reinforced with short fibers. Tensile tests were performed. The experimental results validate the numerical model. The results show that reinforcing with Alfa fibers with a 10% volume fraction increase the elastic modulus by about 24.16% compared to Epoxy. Composite failure stresses reached 28.01 MPa which is higher than 14.64 MPa the value of the unreinforced Epoxy failure stress experimentally determined. With better arrangement and higher concentrations, such composites can at least be used for secondary structural uses.

Numerical Homogenization of Multi-Layered Corrugated Cardboard with Creasing or Perforation.

The corrugated board packaging industry is increasingly using advanced numerical tools to design and estimate the load capacity of its products. This is why numerical analyses are becoming a common standard in this branch of manufacturing. Such trends cause either the use of advanced computational models that take into account the full 3D geometry of the flat and wavy layers of corrugated board, or the use of homogenization techniques to simplify the numerical model. The article presents theoretical considerations that extend the numerical homogenization technique already presented in our previous work. The proposed here homogenization procedure also takes into account the creasing and/or perforation of corrugated board (i.e., processes that undoubtedly weaken the stiffness and strength of the corrugated board locally). However, it is not always easy to estimate how exactly these processes affect the bending or torsional stiffness. What is known for sure is that the degradation of stiffness depends, among other things, on the type of cut, its shape, the depth of creasing as well as their position or direction in relation to the corrugation direction. The method proposed here can be successfully applied to model smeared degradation in a finite element or to define degraded interface stiffnesses on a crease line or a perforation line.

Experimental and numerical assessment of high-velocity impact behavior of syntactic foam core sandwich structures

In this study, the high-velocity impact behavior of new designed sandwich structures with syntactic foam core and FML skins is investigated by experimental and numerical approaches. An appropriate numerical model is developed in order to modelling of the impact behavior of sandwich structures using the finite element software ABAQUS/Explicit. In this model, two VUMAT subroutine are presented to simulate the constitutive models of GFRP composite and syntactic foam. In addition, a multiscale finite element procedure is employed to predict the effective mechanical properties of woven orthogonal composite from known properties of its constituents. This procedure is implemented the numerical homogenization techniques using Python scripting. High-velocity impact tests are performed using a gas-gun and steel conical projectiles to assess the accuracy of the numerical model. To this end, the numerical model is validated by comparing the high-velocity impact behavior of sandwich structures to that obtained from experimental study and demonstrated to be effective and accurate. The presented numerical model can predict the residual velocity and perforation energy with 88.8% and 97.4% accuracies, respectively. Then the validated numerical model is employed to investigate the effect of projectile shape, projectile mass, and stacking sequence of skins on the impact behavior of the sandwich structure.

Data-driven Computational Homogenization Using Neural Networks

Fusion of data mining and computational mechanics is a modern approach for the exploitation of available data within rigorous modeling. First steps in this direction have been focused on the usage of neural networks and other soft computing tools as metamodeling tools. This framework seems suitable for numerical homogenization techniques realized within the so-called FE 2 environment, where the lower-level analysis of a detailed representative volume element is replaced by a prediction based on a previously prepared database. Numerically prepared data are used here, although the method can be used with experimental data as well. In this case, the need for a constitutive description of the fine scale is bypassed. Extraction of material properties from the database, required by the upper-level finite element analysis, is based on backpropagation artificial neural networks. The method is applicable to monuments and masonry structural systems. We investigate this approach here for the analysis of masonry structures with elastoplastic behavior. Results indicate a satisfactory comparison with published research.

Topological Design of Microstructures ofMaterials Containing Multiple Phases of Distinct Poisson’s Ratios

A methodology for achieving the maximum bulk or shear modulus in an elastic composite composed of two isotropic phases with distinct Poisson’s ratios is proposed. A topology optimization algorithm is developed which is capable of nding microstructures with extreme properties very close to theoretical upper bounds. The effective mechanical properties of the designed composite are determined by a numerical homogenization technique. The sensitivities with respect to design variables are derived by simultaneously interpolating Young’s modulus and Poisson’s ratio using different parameters. The so-called solid isotropic material with penalization method is developed to establish the optimization formulation. Maximum bulk or shear modulus is considered as the objective function, and the volume fraction of constituent phases is taken as constraints. The method of moving asymptotes is applied to update the design variables. Several 3D numerical examples are presented to demonstrate the effectiveness of the proposed structural optimization method. The effects of key parameters such as Poisson’s ratios and volume fractions of constituent phase on the nal designs are investigated. A series of novel microstructures are obtained from the proposed approach. It is found that the optimized bulk and shear moduli of all the studied composites are very close to the Hashin-Shtrikman-Walpole bounds.

Experimental and numerical study on mechanical behavior and resistance to natural weathering of sugarcane leave reinforced polymer composite

The waste sugarcane dry leaves were utilized as the reinforcement with the epoxy matrix to prepare the sugarcane dry leaves reinforced polymer composites (SDLRPCs). The tensile, compressive, impact and flexural tests were conducted following respective ASTM standards. The proportion of SDL in SDLRPCs for the most effective fiber–matrix interaction was determined. The statistical analysis has confirmed that the error in experimental results was within the specified limits. The resistance to natural weathering of SDLRPCs were evaluated following ASTM D5338-11. The effect of weathering on tensile strength was also investigated. The scanning electron microscopy (SEM) micrographs revealed reasons for fracture of SDLRPC samples during mechanical testing. It was noticed that the biodegradability corresponds with resistance to natural weathering; more biodegradable wood plastic composites were having the lesser resistance to natural weathering. The micromechanical finite element (FE) model coupled with the numerical homogenization technique (NHT) was constituted to predict the elastic properties of SDLRPCs. The micro-scale stress analysis revealed that the von Mises stress for SDLRPC was up to 267% higher than the pure epoxy. The transient/static micro-mechanical models were constituted to predict the tensile, compressive, impact, and flexural behavior of SDLRPCs. The developed models were found in-line with the experimental data.

Analytical Comparison of Two Multiscale Coupling Methods for Nonlinear Solid Mechanics

Abstract The aim of this work is to compare two existing multilevel computational approaches coming from two different families of multiscale methods in a nonlinear solid mechanics framework. A locally adaptive multigrid method and a numerical homogenization technique are considered. Both classes of methods aim to enrich a global model representing the structure’s behavior with more sophisticated local models depicting fine localized phenomena. It is clearly shown that even being developed with different vocations, such approaches reveal several common features. The main conceptual difference relying on the scale separation condition has finally a limited influence on the algorithmic aspects. Hence, this comparison enables to highlight a unified framework for multiscale coupling methods.