Application Of Periodic Boundary Conditions Research Articles

Multiscale modelling of particulate composites with spherical inclusions

AbstractThis paper presents a novel and effective strategy for modelling three-dimensional periodic representative volume elements (RVE) of particulate composites. The proposed method aims to generate an RVE that can represent the microstructure of particulate composites with hollow spherical inclusions for homogenization (e.g., deriving the full-field effective elastic properties). The RVE features periodic and randomised geometry suitable for the application of periodic boundary conditions in finite element analysis. A robust algorithm is introduced following the combined theories of Monte Carlo and collision driven molecular dynamics to pack spherical particles in random spatial positions within the RVE. This novel technique can achieve a high particle-matrix volume ratio of up to 50% while still maintaining geometric periodicity across the domain and random distribution of inclusions within the RVE. Another algorithm is established to apply periodic boundary conditions (PBC) to precisely generate full field elastic properties of such microstructures. Furthermore, a user-friendly automatic ABAQUS CAE plug-in tool ‘Gen_PRVE’ is developed to generate three-dimensional RVE of any spherical particulate composite or porous material. Gen_PRVE provides users with a great deal of flexibility to generate Representative Volume Elements (RVEs) with varying side dimensions, sphere sizes, and periodic mesh resolutions. In addition, this tool can be effectively utilized to conduct a rapid mesh convergence study, an RVE size sensitivity study, and investigate the impact of inclusion/matrix volume fraction on the solution. Lastly, examples of these applications are presented.

Characterization of mechanical equivalent properties for node enhanced graded lattice structure

Abstract Considering the stress concentration at the rod connection of traditional body-centered cubic (BCC) lattice structure, a node enhanced BCC (NBCC) lattice structure was proposed. In recent years, graded lattice structure has gradually attracted attention due to their unique mechanical properties. In this paper, two different graded NBCC lattice structures were designed and their static mechanical properties were evaluated through quasi-static compression experiments and simulations. Moreover, homogenization theory was applied to calculate the equivalent modulus of lattice structure. This method was extended from uniform lattice structure to graded lattice structure by improving the application of periodic boundary conditions. The obtained results were in good agreement with the experimental and simulation data. The quasi-static compression tests demonstrated that NBCC can effectively reduce the stress concentration and improve the load-bearing capacity about 25% compared to BCC. The graded lattice structures exhibit varying mechanical properties depending on their design and present better mechanical performance in the anisotropic direction. Finally, the relative relationship between the equivalent elastic modulus and the equivalent shear modulus in the lattice structure was studied. The prediction formula for equivalent shear modulus was extended based on the Gibson-Ashby formula.

Thermal–mechanical metamaterial analysis and optimization using an Abaqus plugin

This paper demonstrates how commercial finite-element software and optimization algorithms can be combined to fully explore the design space of thermal–mechanical metamaterials to reveal trends and new insight. This is achieved by developing an Abaqus plugin (EasyPBC) that automates the application of periodic boundary conditions and computes effective elastic and thermal expansion properties for 2D and 3D problems. Abaqus is then linked to an optimizer to fully explore the design space and optimal trade-off between thermal and mechanical properties for two example metamaterials. The first example is a auxetic 2D star-shaped metamaterial, where the proposed approach is used to create a design envelope for Poisson’s ratio and thermal expansion coefficient by solving a series of constrained optimization problems. The second example is a 3D metamaterial based on an octet truss, with additional members to expand the design space. A multi-objective optimization problem is solved to find the optimal trade-off between Young’s modulus and thermal expansion coefficient in a prescribed direction. The results of both examples expand our knowledge about the range of properties for these metamaterials, and designs for optimal trade-off between thermal and mechanical properties.

An exploration on the numerical investigation of 3D laminated composites with carbon fibre tow modelling using SwiftComp module in ABAQUS

The modelling and analysis of composites, at the microscale and macroscale levels, saw a tremendous shift due to the growth of the computational power and mathematical representation of the reinforcement. This study was conducted to assess the importance of modelling fibre tow in the composite at the macroscale level compared to the homogenisation of a microscale unit cell model using SwiftComp GUI in ABAQUS. One microscale unit cell model and three macroscale laminates (CLCCA, CLSwC, and UT) models were created. The microscale model was subjected to homogenisation. The models were then subjected to tensile and shear loading in ABAQUS. The value of young's modulus in the longitudinal and transverse direction by SwiftComp was 12.91% and 5.41%, respectively higher than that by the CCA method. In contrast, Poisson's ratio was underpredicted by 2.35%. Due to the over-prediction of elastic moduli by SwiftComp GUI module, it is not recommended to homogenize the unit cell. The difference in the young's modulus and shear modulus is less than 7% for the UT model compared with the CLCCA model. In conclusion, the CL modelling approach with numerical methods such as CCA is favourable as it provides acceptable results with relative ease of meshing, application of periodic boundary conditions, and lower computational time. On the other hand, the UT model can study the fibre–matrix behaviour by specifying the contact property between the fibre and matrix instead of assuming the bond to be perfect. Hence, it is favourable to model tow and matrix separately when reviewing the failure mechanism on any unbalanced or unsymmetrical laminated composite.

Automatic application of periodic boundary conditions in meso analysis of composite solid propellant

In the meso numerical simulation of composite solid propellant, the periodic boundary conditions are difficult to be applied due to the complexity of the model and the number of element nodes, which makes it not widely used in the meso analysis. Therefore, based on the periodic boundary condition theory, this paper adopts ABAQUS_Python secondary development, periodic boundary conditions were applied to the meso numerical model of composite solid propellant by Python script under three working conditions of two-dimensional principal axis direction and plane shear direction. The results show that the deformation and stress distribution of the numerical model of composite solid propellant conform to the characteristics of periodic boundary conditions.

Development of an ABAQUS™ plug-in to evaluate the fourth-order elasticity tensor of a periodic material via homogenization by the asymptotic expansion method

This paper presents the development of an ABAQUS™ plug-in for estimation of the effective properties of heterogeneous three-dimensional periodic media using the asymptotic homogenization method (AHM). The developed plug-in is user-friendly and requires only basic skills in ABAQUS™. All aspects of the plug-in development are presented, including the solutions found to implement this approach within the finite element package in ABAQUS™. The application of the load cases is performed by using a Uexpan subroutine written in the Fortran language. The application of periodic boundary conditions, as well as the post-processing, are made in Python script. All codes are connected by an ABAQUS™ plug-in and can be called directly by the graphical user interface (GUI) of the software. In order to use the plug-in, the user must only generate the geometry, define the properties of the materials, and mesh the model. The remaining calculations are done automatically. Several case studies are presented and compared with the results found in the literature. Additionally, several element types were evaluated in order to test the versatility of the implemented method. The results show an excellent agreement with the results found in the literature. All of the implemented codes are provided as supplementary material to this paper.

Effect of Including Periodic Boundary Condition on the Fatigue Behaviour of Cancellous Bone

Trabecular bone consists of complex webbing of plates and struts, in which the properties vary across anatomical sites. The substantial constraint is the reduction on discretization error will reduce time in computation. So it is significant to consider carefully the boundary condition effects when utilizing such a complex multiaxial loading mode. Additionally, multiaxial loading gives distinct effects towards boundary condition compare to uniaxial whereas percentage prediction of fatigue failure is lower and applying of periodic boundary reflect a more precise real loading condition. 3D models of trabecular samples were constructed for FE simulations. The response of the models towards simulated mechanical loading was investigated. Preparation of the models begins with 3D reconstruction of micro-CT stacked images, follows by segmentation, meshing and refurbishing process. The resistance of trabecular bone deformation to loading in both uniaxial and multiaxial modes improved the fatigue life and failure with application of periodic boundary conditions.

Hierarchical multi-scale modelling of flax fibre/epoxy composite by means of general anisotropic viscoelastic-viscoplastic constitutive models: Part I – Micromechanical model

A main problem for the development of comprehensive continuum mechanical models is the lack of experimental data. To overcome this problem, a non-associative viscoelastic-viscoplastic constitutive model which takes into account combined isotropic-kinematic hardening effects for epoxy matrix as well as a three dimensional non-associative anisotropic nonlinear elastic/plastic constitutive model for flax fibre is developed. The constitutive models are assigned to a representative volume element (RVE) of a unidirectional flax fibre/epoxy composite. Upon application of periodic boundary conditions (PBCs), different loading scenarios are subjected to the RVE and the mechanical response of the composite is studied. Based on the performed micromechanical analyses, a continuum model is developed for the unidirectional flax fibre/epoxy composite in Part II of this paper.

A complete implementation methodology for Asymptotic Homogenization using a finite element commercial software: preprocessing and postprocessing

Homogenization techniques are very important in describing the mechanical behavior of high performance composite materials. Among the different techniques applied to this modeling, the RVE (Representative Volume Element) and AH (Asymptotic Homogenization) play an important role. In this context, the present work aims to present a complete implementation methodology for Asymptotic Homogenization using ABAQUS® software. Although recent work have provided proper methodologies to describe the mechanical behavior of composite materials based on their constituents, the authors verified that gaps still exist. In contrast, this paper presents all the required steps in details. The application of periodic boundary conditions, which is not straightforward on ABAQUS® software, is described for all the load cases to be solved. The computation of required data from software output files as well as the calculation of the homogenized matrix are also described. The expanded equations to obtain the stresses at microscopic level using finite element commercial software, which could not be seen in recent literature, are detailed and the required emphasis is given. This paper also presents examples to validate and evaluate the obtained results.

Homogenization of Mechanical Properties of Unidirectional Fibre Reinforced Composites with Matrix and Interface Defects: A Finite Element Approach

Fiber Reinforced Composites find increasing applications in the areas of Aerospace, Military Armours, Bullet-proof vests, etc. As the composites are composed of two different constituents, there arises a need to determine the effective properties of the homogenous composites. Experimental determination of the effective properties is very expensive considering the amount of experiments that are required to be conducted, the time and cost to be incurred for each experiment, and the permutations and combinations of the optimal fiber volume fraction. The effective properties are essential for modeling of composites with reference to real-time applications. The micro-mechanics approach reduces most of the above mentioned complexities and helps in accurately evaluating the effective properties. In the presented paper, the properties like Young’s Modulus, Poisson Ratio, and Shear modulus of a healthy (defect free) composite is obtained by modeling a Representative Volume Element (RVE) using the commercial Finite Element Analysis (FEA) solver – Abaqus, with application of Periodic Boundary Conditions (PBC). The presented research focuses on Fiber-Reinforced Metal-Matrix composites like AA2024-Al2O3 and the Ceramic-Matrix composites like ZrB2-SiC. In general, defects in composites arise during the manufacturing process. Matrix Crack, Interfacial De-bonding and Fiber Crack are the major defects which degrade the mechanical properties of composites. This paper presents the modeling of Interfacial de-bonding using the Cohesive-Zone Modelling (CZM) technique for every 90° variation in the fiber-matrix interface and the subsequent evaluation of the corresponding homogenous properties. Matrix Crack is modelled as a matrix defect with a ‘V’ notch for varying a/w ratios. For every variation in matrix crack, the corresponding properties are estimated. Numerical evaluation of the individual effects of interfacial de-bonding and fully grown matrix cracks are followed by the modelling of the coupled effects.

Prediction of effective elastic properties of a polypropylene component by an enhanced multiscale simulation of the injection molding process

In the injection molding process of semi-crystalline polymers, the melt undergoes a complex deformation and cooling history which results in an inhomogeneous distribution of spherulites in the component. To evaluate these inhomogeneities in an isotactic polypropylene (α-iPP) part, an integrated multi-scale simulation approach has been developed in Laschet et al. (2017a). This approach combines a macroscale mold filling and heat transfer analysis with a crystallization model at the microscale and a two-scale homogenization scheme in order to calculate microstructure dependent elastic properties of the PP component. This approach is extended here by adding some features improving the local Hooke matrix predictions. A link between Molecular Dynamics (MD) simulations and the homogenization scheme is achieved in order to obtain unknown properties of the pure amorphous and crystalline phases. Next, the accuracy of the crystallization model is improved by considering not only isothermal but also athermal nucleation. The spherulite growth is now evaluated by a continuous cellular growth algorithm. Then, several numerical improvements have been integrated in the homogenization tool. The main ones are the implementation of stabilized mixed finite elements to handle accurately the quasi-incompressibility of the amorphous phase and the application of periodic boundary conditions via a master-slave projection algorithm. Eventually, the distribution of effective mechanical properties over the component thickness is compared for different crystallization scenarios with measured elastic moduli.

Failure Analysis of Triaxially Braided Composite Under Tension, Compression and Shear Loading

In the current study, continuum damage mechanics-based progressive failure analysis was performed to predict the initiation and progression of failure in triaxially braided textile composite unit cell under tensile, compressive and shear loading conditions. Geometric and finite element modeling was done at mesoscale unit cell level, with repeating nature of the unit cell considered through the application of periodic boundary condition. Analysis result was first validated with experimentally obtained stress–strain curve for uniaxial tension loading. Then, failure prediction was extended for tension, compression and shear failure loading in different directions. Depending on the out-of-plane boundary condition, single ply and infinitely stacked symmetric and antisymmetric plies were also considered to investigate the effect of stacking sequence on the in-plane properties. Detailed examination of failure modes at different damage initiation and propagation stages was presented. After full mechanical properties were obtained from mesoscale virtual testing, macroscale failure analysis of laterally loaded triaxially braided composite cylinder was performed using predicted properties. The result was then compared with test result for validation of the method used.

The scaled boundary finite element method for computational homogenization of heterogeneous media

SummaryMaterials exhibit macroscopic properties that are dependent on the underlying components at lower scales. Computational homogenization using the finite element method (FEM) is often used to determine the effective mechanical properties based on the microstructure. However, the use of FEM might suffer from several difficulties such as mesh generation, application of periodic boundary conditions or computations in presence of material interfaces, and further discontinuities. In this paper, we present an alternative approach for computational homogenization of heterogeneous structures based on the scaled boundary finite element method (SBFEM). Based on quadtrees, we are applying a simple meshing strategy to create polygonal elements for arbitrary complex microstructures by using a relatively small number of elements. We show on selected numerical examples that the proposed computational homogenization technique represents a suitable alternative to classical FEM approaches capable of avoiding some of the mentioned difficulties while accurately and effectively calculating the macroscopic mechanical properties. An example of a two‐scale semiconcurrent coupling between FEM and SBFEM is presented, illustrating the complementarity of both approaches.

Consistent application of periodic boundary conditions in implicit and explicit finite element simulations of damage in composites

This paper presents an implementation-dedicated analysis of Periodic Boundary Conditions (PBCs) for Finite Element (FE) models incorporating highly non-linear effects due to plasticity and damage. This research addresses fiber-reinforced composite materials modeled at micros-scale level using a Representative Volume Element (RVE), where its overall mechanical response is obtained via homogenization techniques. For the sake of clearness, a unidirectional ply with randomly distributed fibers RVE model is assumed. PBCs are implemented for implicit and explicit FE solvers, where conformal and non-conformal meshes can be used. The influence of applying PBCs in the reliability of the mechanical response under tension and shear loading is assessed. Furthermore, the Poisson effect and the consistency of damage and fiber debonding propagation through the periodic boundaries are reported as well as their impact on the homogenized results. Likewise, numerical aspects like computational performance and accuracy are evaluated comparing implicit- versus explicit-based solutions.

Periodic boundary condition and its numerical implementation algorithm for the evaluation of effective mechanical properties of the composites with complicated micro-structures

To evaluate the effective mechanical properties of the composites with complicated micro-structures, the RVE based FE homogenization method with the periodic boundary condition is introduced and implemented in this paper, and the emphasis is on the periodic boundary condition and its numerical implementation algorithm. The pre-processing (such as the generation of geometry model and application of periodic boundary condition), FE analysis and post-processing (such as the average of stress and strain and stress contouring of the surface nodes) concerning the evaluation of the effective mechanical properties of the composites with complicated micro-structures are conducted in the FE package ABAQUS through the Python Interface. Numerical results show that the proposed numerical implementation algorithm of the periodic boundary condition guarantees the stress and strain continuities and uniaxial deformation constraint of the RVEs for the composites with complicated micro-structures. Compared with the Halpin-Tsai model and two-step M-T/Voigt mean-field homogenization method, the RVE based FE homogenization method with the periodic boundary condition is verified to accurately predict the effective elastic properties and elasto-plastic responses of the composites with the complicated micro-structures.

An extended model of the quantum free-electron laser

Previous models of the quantum regime of operation of the Free Electron Laser (QFEL) have performed an averaging and the application of periodic boundary conditions to the coupled Maxwell - Schrodinger equations over short, resonant wavelength intervals of the interaction. Here, an extended, one-dimensional model of the QFEL interaction is presented in the absence of any such averaging or application of periodic boundary conditions, the absence of the latter allowing electron diffusion processes to be modeled throughout the pulse. The model is used to investigate how both the steady-state (CW) and pulsed regimes of QFEL operation are affected. In the steady-state regime it is found that the electrons are confined to evolve as a 2-level system, similar to the previous QFEL models. In the pulsed regime Coherent Spontaneous Emission (CSE) due to the shape of the electron pulse current distribution is shown to be present in the QFEL regime for the first time. However, unlike the classical case, CSE in the QFEL is damped by the effects of quantum diffusion of the electron wavefunction. Electron recoil from the QFEL interaction can also cause a diffusive drift between the recoiled and non-recoiled parts of the electron pulse wavefunction, effectively removing the recoiled part from the primary electron-radiation interaction.

Comparison of periodic mesh and free mesh on the mechanical properties prediction of 3D braided composites

Periodic unit-cells are the premises for applying the periodic boundary conditions in the meso-scale finite element analysis of textile composites. However, due to the extremely complicated microstructure, there is a conflict between high-quality quick mesh generation and efficient application of periodic boundary conditions. A freely generated mesh of the unit-cell combined with more general periodic boundary conditions is assumed to be a more practical approach. In this paper, the general periodic boundary conditions are imposed by establishing linear constraint equations between master surface nodes and slave surface nodes of the unit-cell on ABAQUS software platform. For the same unit-cell model of 3D braided composites with periodic mesh and free mesh, the deformation, stress distribution and the predicted stiffness and strength properties under typical loadings are compared. The numerical results obtained by means of free mesh unit-cell agree well with those using periodic mesh proving the effectiveness and practicability of the new approach. It can be remarked that the general periodic boundary conditions are suitable for the free mesh generation of unit-cells for a complicated microstructure reducing the difficulty of meshing and improving the quality of mesh generation.

Automatic three-dimensional geometry and mesh generation of periodic representative volume elements for matrix-inclusion composites

This paper introduces an efficient method to automatically generate and mesh a periodic three-dimensional microstructure for matrix-inclusion composites. Such models are of major importance in the field of computational micromechanics for homogenization purposes utilizing unit cell models. The main focus of this contribution is on the creation of cubic representative volume elements (RVEs) featuring a periodic geometry and a periodic mesh topology suitable for the application of periodic boundary conditions in the framework of finite element simulations. Our method systematically combines various meshing tools in an extremely efficient and robust algorithm. The RVE generation itself follows a straightforward random sequential absorption approach resulting in a randomized periodic microstructure. Special emphasis is placed on the discretization procedure to maintain a high quality mesh with as few elements as possible, thus, manageable for computer simulations applicable to high volume concentrations, high number of inclusions and complex inclusion geometries. Examples elucidate the ability of the proposed approach to efficiently generate large RVEs with a high number of anisotropic inclusions incorporating extreme aspect ratios but still maintaining a high quality mesh and a low number of elements.

The N-terminal of annexin A1 as a secondary membrane binding site: a molecular dynamics study.

Annexin A1 has been shown to cause membrane aggregation and fusion, yet the mechanism of these activities is not clearly understood. In this work, molecular dynamics simulations were performed on monomeric annexin A1 positioned between two negatively charged monolayers using AMBER's all atom force field to gain insight into the mechanism of fusion. Each phospolipid monolayer was made up of 180 DOPC molecules and 45 DOPG molecules to achieve a 4:1 ratio. The space between the two monolayers was explicitly solvated using TIP3P waters in a rectilinear box. The constructed setup contained up to 0.14 million atoms. Application of periodic boundary conditions to the simulation setup gave the desired effect of two continuous membrane bilayers. Nonbonded interactions were calculated between the N-terminal residues and the bottom layer of phospholipids, which displayed a strong attraction of K26 and K29 to the lipid head-groups. The side-chains of these two residues were observed to orient themselves in close proximity (∼3.5 Å) with the polar head-groups of the phospholipids.

A Gibbs-ensemble based technique for Monte Carlo simulation of electric double layer capacitors (EDLC) at constant voltage.

Current methods for molecular simulations of Electric Double Layer Capacitors (EDLC) have both the electrodes and the electrolyte region in a single simulation box. This necessitates simulation of the electrode-electrolyte region interface. Typical capacitors have macroscopic dimensions where the fraction of the molecules at the electrode-electrolyte region interface is very low. Hence, large systems sizes are needed to minimize the electrode-electrolyte region interfacial effects. To overcome these problems, a new technique based on the Gibbs Ensemble is proposed for simulation of an EDLC. In the proposed technique, each electrode is simulated in a separate simulation box. Application of periodic boundary conditions eliminates the interfacial effects. This in addition to the use of constant voltage ensemble allows for a more convenient comparison of simulation results with experimental measurements on typical EDLCs.