Asymptotic Homogenization Method Research Articles

Dynamics of ribbed plates with inner resonance: Analytical homogenized models and experimental validation

This paper deals with the theoretical, numerical and experimental behavior of periodic orthogonally ribbed plates. It extends the paper (Fossat et al., 2018) in which a comprehensive homogenized model has been established for flexural and torsional motion of periodic 1D-ribbed plates. New theoretical results describing the out-of-plane behavior of cellular plates involving inner resonance phenomena, are derived using an asymptotic approach. In this aim, the out-of-plane model of beam grids accounting for local bending and torsion is first established through the asymptotic method of homogenization of periodic discrete media. Then, the coupling between the beam grid and the internal plates (fully or partially connected to it) is detailed. This lead to an explicit analytical formulation of the equivalent plate model whose effective parameters arise from the geometry and mechanical properties of the unit cell. The unconventional features of the flexural wave dispersion are shown to be straightforwardly related to inner-resonance phenomena. These theoretical results are successfully compared to numerical computations conducted using WFEM. Furthermore, experiments performed on two prototypes of ribbed plates evidence the ability of the homogenized model to describe their complex dynamic behavior. The latter is characterized by the co-existence of a dynamic regime at both the micro-scale of the period and the macro-scale of the whole structure, that results in an inhomogeneous kinematics where the plate and beam displacements differ at the leading order. These unique features depart from the usual assumptions retained in plate mechanics and generates the observed non-conventional features. In conclusion, it is stressed that the study yields design rules to tailor cellular panels having specific atypical features in a given frequency range.

Magneto-electro-elastic node-based smoothed point interpolation method for micromechanical analysis of natural frequencies of nanobeams

As an addition to the traditional finite element method (FEM), the magneto-electro-elastic node-based smoothed point interpolation method (MEE-NS-PIM) with asymptotic homogenization method (AHM) is presented to solve the micromechanical problems of MEE nanobeams, which overcomes the deficiency of FEM and improves the accuracy of the calculation results. Firstly, the basic equations of MEE medium are derived. Secondly, AHM is adopted to calculate the property parameters of MEE materials under microcosmic situations, and the AHM model is illustrated. Then, the relative formulations of the discretized system used to calculate the frequency of MEE nanostructures are deduced based on MEE-NS-PIM. Moreover, several numerical examples are calculated, and the results of MEE-NS-PIM are compared with those of FEM, which proves the convergence, precision, and effectiveness of MEE-NS-PIM. Therefore, MEE-NS-PIM combined with AHM can be used to analyze the microcosmic MEE coupling problems and obtain a more accurate and reliable solution for MEE micromechanics.

Physical interpretation of asymptotic expansion homogenization method for the thermomechanical problem

For the thermomechanical problem of periodical composite structures, mathematical formulation of asymptotic expansion homogenization (AEH) method up to any expansion is derived. This is done by constructing a full decoupling form for expansion terms of each order. The weighted residual method is utilized to convert the mathematical formulations into matrix forms for convenient use in the standard finite element method. Elastic and thermal characteristic displacements are compared to elastic and thermal quasi displacements, respectively. Then, the self-balancing property of the quasi load corresponding to quasi displacements, dimensional analysis method, and geometric intuitiveness method of the quasi load are utilized to reveal the physical interpretations of elastic and thermal characteristic displacements and expansion terms of each order. The necessity of the second-order expansion term for the micro analysis of periodical composite structures is emphasized. Numerical results validate the formulations and physical interpretation of AEH for the thermomechanical problem and lay the foundation for its application.

Buckling surrogate-based optimization framework for hierarchical stiffened composite shells by enhanced variance reduction method

The surrogate-based optimization of hierarchical stiffened composite shells against buckling is a typical multimodal and multivariables optimization problem. To improve the computational efficiency and global optimizing ability of the surrogate-based optimization of hierarchical stiffened composite shells, an enhanced variance reduction method based on Latinized partially stratified sampling and multifidelity analysis methods is proposed in this paper and then integrated into the surrogate-based optimization framework. In the offline step of the optimization framework, candidate pairing strategies of design variables are generated by Latinized partially stratified sampling and compared by performing priori optimizations based on the low-fidelity analysis method, and the optimal pairing strategy is therefore determined. On the basis of the optimal pairing strategy, the surrogate-based optimization is carried out using the high-fidelity analysis method in the online step. With less computational cost in the offline step, the proposed enhanced variance reduction method overcomes the limitation of Latinized partially stratified sampling that the optimal pairing strategy is not obvious in complex problems. Then, extensive optimization examples are carried out to verify the efficiency and effectiveness of the proposed optimization framework. Given an approximate computational cost, the optimal buckling result of the proposed framework using enhanced variance reduction method increases by 18.2% than that of the traditional framework based on Latin hypercube sampling. In particular, the advantage of enhanced variance reduction method in the space-filling ability is highlighted in comparison to Latin hypercube sampling. When achieving an approximate global optimal solution, the proposed framework reduces the total computational cost by 76.3% than the traditional framework. Finally, the numerical implementation of asymptotic homogenization method is used herein for the accurate prediction of effective stiffness coefficients of the initial design and optimal results. Through comparison, it is concluded that the high axial stiffness and bending stiffness are the main mechanism for the high load-carrying capacity of optimal results.

A Coupling Electromechanical Cell-Based Smoothed Finite Element Method Based on Micromechanics for Dynamic Characteristics of Piezoelectric Composite Materials

Coupling electromechanical cell-based smoothed finite element method (CSFEM) with the asymptotic homogenization method (AHM) is presented to overcome the overstiffness of FEM. This method could accurately simulate the dynamic responses and electromechanical coupling effects of piezoelectric composite material (PCM) structures. Firstly, the efficient performances for active compounds of round cross-section fibers are calculated based on AHM. Secondly, in the CSFEM, electromechanical multi-physic-field FEM is coupled with gradient smoothing technique. CSFEM returns the nearly exact stiffness of continuum structures, which auto discretes the elements in complex areas more readily and thus remarkably reduces the numerical errors. Static and dynamic characteristics of four PCM structures are investigated using CSFEM with AHM. Results are compared with analytical solution and those of FEM, which proves that CSFEM with AHM is more accurate and reliable than the standard FEM when solving problems of complex structures. Additionally, CSFEM could provide results of higher accuracy even using distorted meshes. Therefore, such method is a robust tool for analyzing mechanical properties of PCM structures.

Simple closed-form property expressions of a metafluid composed of a hexagonal array of transversely isotropic elastic fibres embedded in an ideal fluid

A hexagonal array of parallel circular cylinders embedded in a matrix is considered. The fibers are filled with a transversely isotropic (TI) elastic material whereas the matrix material is an ideal fluid. The asymptotic homogenization method, applied to a similar problem where the matrix is also a TI elastic material, produces formulae for the effective TI elastic composite. By a limit procedure when the shear modulii of matrix tend to zero, the formulae yield very simple expressions for the effective composite for the above problem, namely, it is a TI metafluid. The particular case of isotropic fibres leads to an isotropic metafluid.

Multiscale analysis for predicting the constitutive tensor effective coefficients of layered composites with micro and macro failures

The aim of the present work consists of predicting the effective coefficients of constitutive tensor for layered composites with delamination in macro-scale considering the influence of the debonding between fiber and matrix in micro-scale. Thus, a multiscale methodology is proposed to solve this problem. Firstly, the effective coefficients for the micro-scale level are calculated via the Finite Element Method (FEM), considering different degrees of micro failure. By using the micro-scale level homogenization results, the effective coefficients for layered composites are calculated by the Finite Element Method, as well as by Asymptotic Homogenization Method (AHM), considering different extensions of delamination (failures in macro-scale). Results show that the macro-scale analyses are affected by the micro failure, with the most influence of the micro failure in the coefficients C11*, C12* and C66*. In addition, when the thickness of the adhesive between layers increases, the effective coefficients decrease with the macro failure (delamination). Comparisons between homogenized and heterogeneous numerical models show that for almost all effective coefficients, there are excellent convergences. Only for the values of C12*, there are relevant divergences for specific limit cases. Finally, the macro-scale results obtained via FEM and AHM are compared to evaluate the advantageous and disadvantageous of the proposed multiscale methodology.

Modeling of Cementitious Representative Volume Element with Various Water–Cement Ratios

This study uses the finite element method (FEM) to measure the mechanical properties of microstructure-based cementitious representative volume elements (RVEs) with various water–cement ratios (W/Cs) generated by CEMHYD3D. The finite element boundary condition effects that significantly and computationally change the elastic properties are studied and discussed. Various boundary conditions in ABAQUS are applied and compared with the results obtained using the variational asymptotic method for unit cell homogenization (VAMUCH). This comparison is conducted using ANSYS. This study aims to analyze and determine the effect of different boundary conditions in detail on the prediction of the elastic properties of cementitious RVE with various W/Cs and identify the best approach in this regard. Results show that Young’s, shear, and bulk moduli decrease with the increase in W/C and the boundary conditions in ABAQUS influence the outcomes, particularly on bulk modulus and Poisson’s ratio.

Topology optimization of anisotropic magnetic composites in actuators using homogenization design method

This work presents topology optimization of anisotropic magnetic composites in actuators. The magnetic composite consists of two different ferromagnetic materials with high and low magnetic reluctivity, which correspond to matrix and fiber materials, respectively. The magnetic composite is expected to enhance the actuator performance when it is properly designed. The proposed design optimization scheme can find the composite layout, fiber volume fraction, and fiber orientations to achieve an actuator force maximization. In the proposed scheme, four microstructure design variables are assigned at each finite element, and the effective homogenized material property (i.e., magnetic reluctivity) is calculated using the asymptotic homogenization method. The microstructure design variables are then optimized to achieve the optimal distribution of the homogenized material property in a macroscopic scale. In the design result, discrete fiber orientations and volume fractions are achieved by applying the discrete penalization scheme. In addition, the obtained composite design result is visualized using the projection method proposed for a periodic composite design result. The effectiveness of the proposed topology optimization procedures is validated in magnetic actuator design examples. In addition, the design result of anisotropic composite is compared with the isotropic multi-material design result to validate the benefit of anisotropic composite in actuators.

Multiscale asymptotic homogenization analysis of epoxy-based composites reinforced with different hexagonal nanosheets

The present work studies the mechanical properties of epoxy-based composites which are reinforced by hexagonal monolayer nanosheets such as graphene, boron nitride, silicon carbide and aluminum nitride, based on the Kalamkarov’s general asymptotic homogenization composite shell model. This parametric study on total atomic interactions in the unit cell with the asymptotic homogenization method allows clearly establishing effective Young’s modulus, shear modulus, Poisson’s ratios and stiffness coefficients of hexagonal nanosheets and epoxy-based composites reinforced with them as functions of the force constants, bond length, thickness and volume fractions. The present homogenized sandwich-like model has simplified rules which consider interfacial forces and clears the behavior of the matrix-reinforcement interface. The results from the present homogenized model show good agreement with the trustworthy results of other researchers which are available in the literature. Moreover, the present model is useful for modeling nanostructures instead of heavy computational modeling or difficult experiments.

An approach for modeling non-ageing linear viscoelastic composites with general periodicity

The present work deals with the modeling of non-ageing linear viscoelastic composite materials and quasi-periodic microstructure. The stratified functions and the curvilinear coordinates play an important role in the design of different geometrical shapes. The main objective focuses on the application of two-scales Asymptotic Homogenization Method (AHM) to obtain the overall behavior of the viscoelastic composite materials. Although the whole process is based on the analysis of laminated configurations, a multi-step homogenization scheme to estimate the effective properties of a structure reinforced with long rectangular fibers and wavy effects is used. The associated local problems, the homogenized problem and the analytical expressions for the effective coefficients are obtained by using the correspondence principle and the Laplace-Carson transform. Also, the interconnection between the effective relaxation modulus and the effective creep compliance is performed. Finally, the inversion to the original temporal space is calculated. Some comparisons between the proposed approach and Finite Elements Method (FEM) results are displayed.

Part-scale build orientation optimization for minimizing residual stress and support volume for metal additive manufacturing: Theory and experimental validation

Laser powder bed metal additive manufacturing (AM) has been widely accepted by the industry to manufacture end-use components with complex geometry to achieve desirable performance (i.e. conformal cooling). However, residual stress and large deformation introduced in the laser AM process lead to severe issues, such as cracks, delamination, and large deformation. These issues result in the stoppage of powder spreading and warpage of the component. To overcome these issues, a novel optimization framework based on fast process modeling is proposed to find the optimal build orientation by minimizing the maximum residual stress and support structure volume. For support generation, a voxel-based methodology is proposed to systematically capture support surfaces from STL file, form support structure, and generate Cartesian mesh for fast process modeling. Instead of using conformal mesh, the voxel-based fictitious domain method is used to calculate the stress distribution in the design domain including the support structure, which is represented by the homogenized model. This can circumvent time-consuming mesh generation for geometrically complex geometry and its support structure during the optimization iterations, thus making it possible to minimize residual stress through orientation optimization based on process modeling. Due to its self-supporting and open-cell nature, lattice structure is employed as the support structure to anchor the overhangs to the substrate to prevent distortion resulting from residual stress. Asymptotic homogenization (AH) method is employed to compute the effective properties of lattice structure, while a multiscale model is proposed to compute the yield strength. In particular, the multi-objective optimization including both the residual stress and support volume is discussed and investigated in this work. Experimental validation is conducted on a realistic component with some geometric complexity. By comparing the component and support structure without build orientation optimization, it is found that the proposed framework can significantly reduce the influence of the residual stress on the printed part, ensure the manufacturability of the design, and decrease the material consumption for the sacrificial support structure simultaneously.

Experimental and theoretical study on thermal properties of porous PDMS

Groups of porous polydimethylsiloxane (p-PDMS) with different porosities were fabricated, and thermal properties of them were measured. Theoretical prediction applying an asymptotic homogenization method is also presented. Good agreements between the experimental results and theoretical ones are achieved. Studies show the thermal properties of p-PDMS evidently alter with the porosity, whereas they are nearly independent on the pore size. A fitted formula of the thermal conductivity with respect to the porosity is suggested. Disturbance due to temperature is also studied experimentally. Results presented here make a preparation for further thermal analysis and structural optimization of flexible electronics based on p-PDMS.

Homogenization of thermo-magneto-electro-elastic multilaminated composites with imperfect contact

The Asymptotic Homogenization Method (AHM) is applied to a family of boundary value problems for linear thermo-magneto-electro-elastic (TMEE) heterogeneous material with rapidly oscillating coefficients and magneto-electro-elastic imperfect contact between the phases. Using matrix notation we show the procedure for constructing a formal asymptotic solution that leads to the homogenized problem, the effective coefficients and the local problems. The effective coefficients are functions of the solutions of the local problems, which, in the case of a laminate, reduce to systems of ordinary differential equations. The methodology is illustrated with an example of a two-phase piezoelectric/piezomagnetic laminate formed by transversally isotropic phases. The analytical expressions of the effective moduli work for any number of phases and show a marked dependence on the values of the magneto-electro-elastic imperfect contact parameters. It is also shown that some moduli satisfy exact relations that allow us to compute them when the values of some other modulus is given. The numerical examples show the emergence of product properties such as the magneto-electric, pyromagnetic and pyroelectric effects.

Topology optimization for concurrent design of layer-wise graded lattice materials and structures

Mechanical properties of hierarchical lattice structures depend not only on their overall shapes and topologies, but also on their microstructural configurations. This paper proposes a new method for concurrent topology optimization of structures composed of layer-wise graded lattice microstructures. Both macroscale design variables representing the distribution of different lattice materials and microscale design variables defining the topologies of the microstructural unit cells are to be simultaneously optimized. This formulation thus integrates the microstructure design into the structural design, instead of pursuing a grey macroscale design and then interpreting the intermediate densities into certain microstructures. The proposed method also enlarges the design space by allowing for graded microstructures. Two new design constraints, namely the structural coverage constraint and the average porosity constraint, are introduced into the proposed optimization formulation to reduce the complexity of the constraints in the layer-wise graded design. The macroscale and microscale designs are linked by using the Asymptotic homogenization method to compute the effective elastic properties of the microstructured materials. Numerical examples show validity of the proposed method. It is also found that layer-wise graded lattice structures outperform those with uniform lattice microstructures in terms of structural stiffness. Finite element simulations of constructed models of the optimized designs suggest that graded lattice structures exhibit higher buckling resistance and ultimate load bearing capacity than single-scale solid material structures or uniform lattice structures under the same material usage.

Numerical simulation of hydro-mechanical coupling in fractured vuggy porous media using the equivalent continuum model and embedded discrete fracture model

In this study, an efficient numerical model is proposed to simulate the hydro-mechanical coupling in the fractured vuggy porous media containing multi-scale fractures and vugs. Specifically, an improved Equivalent Continuum Model (ECM), which is obtained through the asymptotic homogenization method, is applied to model micro fractures and vugs, while macro fractures are modeled explicitly by using the Embedded Discrete Fracture Model (EDFM) and non-matching grids. As an important feature of the explicit representment, the effects of fillings and fluid pressure on preventing macro fractures from closing can be considered. After that, the new mixed space discretization (i.e., Mimetic Finite Difference (MFD) method for flow and stabilized eXtended Finite Element Method (XFEM) for geomechanics) and fully coupled methods are applied to solve the proposed model. The mixed space discretization can deal with complex heterogeneous properties (e.g., full tensor permeability), and yields the benefits of local mass conservation and numerical stability in space. Finally, we demonstrate the accuracy and application of the proposed model to capture the coupled hydro-mechanical impacts of multiscale fractures and vugs on fluid flow in fractured porous media.

Numerical and experimental evaluation of TPMS Gyroid scaffolds for bone tissue engineering

The combination of computational methods with 3D printing allows for the control of scaffolds microstructure. Lately, triply periodic minimal surfaces (TPMS) have been used to design porosity-controlled scaffolds for bone tissue engineering (TE). The goal of this work was to assess the mechanical properties of TPMS Gyroid structures with two porosity levels (50 and 70%). The scaffold stiffness function of porosity was determined by the asymptotic homogenisation method and confirmed by mechanical testing. Additionally, microCT analysis confirmed the quality of the printed parts. Thus, the potential of both design and manufacturing processes for bone TE applications is here demonstrated.

On the quasi-static effective behaviour of poroelastic media containing elastic inclusions

The aim of the present study is to derive the effective quasi-static behaviour of a composite medium, made of a poroelastic matrix containing elastic impervious inclusions. For this purpose, the asymptotic homogenisation method is used. On the local scale, the governing equations include Biot’s model of poroelasticity in the porous matrix and Navier equations in the inclusions, with elastic properties of the same order of magnitude. Biot’s diphasic model of poroelasticity is obtained on the macroscopic scale, but with effective parameters that are strongly impacted by the distribution of inclusions, even at low volume fraction. The impact on fluid flow is strictly geometrical, showing that the inclusions do not play the role of a porous network.

Simplified Formulation of Mechanics of Structure Genome

This paper presents a simplified formulation of the mechanics of a structure genome (MSG). The MSG is a unified theory for multiscale constitutive modeling for all types of composite structures. It is generalized from the previous research works based on the variational asymptotic method (VAM), including the variational asymptotic beam sectional analysis, the variational asymptotic plate and shell analysis, and the variational asymptotic method for unit cell homogenization. In this paper, the original complex formulation of the MSG is simplified and the principle of minimum information loss underpinning the MSG is formalized. The simplified formulation of the MSG is applied to construct the linear elastic Euler–Bernoulli beam model, the Kirchhoff plate model, and three-dimensional Cauchy continuum model. The connections of the MSG to previous VAM-based theories are clearly pointed out.

Asymptotic Expansion Homogenization Analysis Using Two-Phase Representative Volume Element for Non-periodic Composite Materials

Asymptotic expansion homogenization (AEH) method is a well-known approach based on the assumption of the periodicity of microstructures to obtain the homogenized material properties of composite materials. The main advantage of this method is that it can be used as a multiscale simulation tool. A new AEH method is developed in this study to estimate the homogenized elastic properties of non-periodic composite materials using two-phase representative volume elements (RVEs) composed of the inner phase of a non-periodic composite material and the outer phase of a homogenized material. The AEH method is repeatedly applied to the two-phase RVEs to update the homogenized elastic properties of non-periodic composite materials.