Two-scale Finite Element Research Articles

Two-Scale Finite Element Approximation of a Homogenized Plate Model

Two-Scale Finite Element Approximation of a Homogenized Plate Model

Experimental and numerical study of the thermomechanical properties of flexible self-reinforced polyimide composite membrane

Polyimide membranes are favored in the aerospace field for their excellent comprehensive properties, but new application requirements demand higher strength, modulus and thermal expansion properties. Here, self-reinforced polyimide composite membranes (SRPICM) with varying fiber tow arrangement densities were fabricated by unidirectional reinforcement of polyimide fiber tows to enhance the thermomechanical properties of the membranes while maintaining the characteristics of low thickness and flexibility. A micro-scale representative volume element model with interface, and a macro-scale model containing cracks were developed based on the membranes' morphology to investigate the tensile and thermal expansion behavior of SRPICM. Experimental and numerical analyses demonstrated that fiber tows significantly improved the longitudinal tensile properties of SRPICM, with maximum increases in modulus and strength to 34.02 GPa and 1.26 GPa, respectively, over 13 times those of pure PI membranes. Further, the test results, combined with the two-scale finite element model revealed the evolution of longitudinal and transverse tensile deformation and thermal expansion behavior of SRPICM. The validity of the two-scale model was confirmed by experimental results, attributed to the practical consideration of interfacial bonding, prefabricated cracks and thermal residual stress effects during initial modeling. Notably, SRPICM with 10 fiber tows/20 mm arrangement density exhibited excellent longitudinal tensile properties (modulus and strength of 14.01 GPa and 470.89 MPa, respectively) and a longitudinal thermal expansion close to zero (0.03 μm/m°C), making it an ideal material for aerospace applications.

Two-scale finite element simulations for UT inspection of micro-structured materials in CIVA

Ultrasonic testing (UT) of highly heterogeneous structures is an important issue for many industrial applications in civil engineering for concrete inspection as well as in nuclear and oil & gas, where it applies to cast steels and weld joints. In these materials, performance is limited by the phenomenon of multiple wave scattering at the microstructure scale. In this context, the use of effective simulation tools is particularly useful both for improving interpretation of UT and for optimizing or demonstrating the performance of inspections. However, the scales of heterogeneity to be taken into account are often smaller than fractions of a tenth of a wavelength. In 3D simulations, it is therefore difficult, if not impossible, to combine computational performances with the refinement of the modelled interactions. To meet this challenge, CEA is developing an original approach based on two applications of the finite element (FE) method implemented in CIVA. The first application uses micro-scale heterogeneous material data on small representative elemental volumes (REVs) to feed a second code dedicated to macro-scale with effective medium characteristic. This second code relies on a high-order finite element model that is very efficient at the scale of the inspected industrial component. In this communication we present the main features of these FE tools and illustrate their performances on different application cases for welded parts and concrete

Concurrent Topology Optimization of Multi-Scale Composite Structures Subjected to Dynamic Loads in the Time Domain

This paper presents a concurrent topology optimization of multi-scale composite structures subjected to general time-dependent loads for minimizing dynamic compliance. A three-field density-based method is adopted to implement the concurrent topological design, with macroscopic effective properties of the microstructure evaluated through energy-based homogenization method (EBHM). Transient response is obtained from the two-scale finite element analysis with the HHT-α approach as an implicit time integration procedure. Design sensitivities are formulated employing the adjoint variable method (AVM) based on two main philosophies: “discretize-then-differentiate” and “differentiate-then-discretize” approaches, respectively. The method of moving asymptotes is adopted to update the design variables at two scales. Several benchmark examples are presented to demonstrate that the “discretize-then-differentiate” AVM attains consistent sensitivities in an inherent manner such that the resulting optimal topology is more efficient when compared with the “differentiate-then-discretize” AVM. Moreover, the potential of the proposed method for concurrent dynamic topology optimization problems under general time-dependent loads is also highlighted.

Symmetrized Two-Scale Finite Element Discretizations for Partial Differential Equations with Symmetric Solutions

Abstract In this paper, some symmetrized two-scale finite element methods are proposed for a class of partial differential equations with symmetric solutions. With these methods, the finite element approximation on a fine tensor-product grid is reduced to the finite element approximations on a much coarser grid and a univariant fine grid. It is shown by both theory and numerics including electronic structure calculations that the resulting approximations still maintain an asymptotically optimal accuracy. By symmetrized two-scale finite element methods, the computational cost can be reduced further by a factor of 𝑑 approximately compared with two-scale finite element methods when Ω = ( 0 , 1 ) d \Omega=(0,1)^{d} . Consequently, symmetrized two-scale finite element methods reduce computational cost significantly.

Two-scale modeling of particle crack initiation and propagation in particle-reinforced composites by using microscale analysis based on Voronoi cell finite element model

In this study, a two-scale finite element model has been developed for the elastic analysis of composite materials. The coarse finite element mesh was assumed for the macroscale analysis, while Voronoi cell finite element model (VCFEM) was used for the microscale analysis of heterogeneous domains. The adaptive coupling between the two scales in non-critical and critical regions was accomplished through the master–slave nodes method. Two scale elements were connected together through the use of shared nodes and the order of the edge displacement interpolation. A new extended VCFEM was proposed to simulate particle damage generation, and a remeshing algorithm was constructed to simulate particle crack generation and propagation in particle-reinforced composites. For crack propagation analysis, four new combined Voronoi elements were developed and analyzed at the microscale. The obtained results for the stresses, stress intensity factors, and crack path were compared with those calculated using the commercial calculation software Abaqus. The two sets of results are in very good agreement, confirming the accuracy of the proposed method used for determining the stress concentrations and also for coarse-mesh discretization.

Two‐scale FE‐FFT‐based thermo‐mechanically coupled modeling of elasto‐viscoplastic polycrystalline materials at finite strains

AbstractDue to the general pursuit of technological advancement, structural components need to meet increasingly higher standards. In order to optimize the performance behavior of the used materials, detailed knowledge of the overall as well as microscopic material behavior under certain mechanical and thermal loading conditions is required. Hence, we present a two‐scale finite element (FE) and fast Fourier transformation (FFT)‐based method incorporating finite strains and a thermo‐mechanically coupled constitutive model for elasto‐viscoplastic polycrystalline materials. Assuming that the length scale of the microscale is sufficiently smaller compared to the length scale of the macroscale, we consider the macroscopic and microscopic boundary value problem as two coupled subproblems. The macroscopic boundary value problem is solved utilizing the finite element method. In each macroscopic integration point, the microscopic boundary value problem is embedded as a periodic unit cell whose solution fields are computed utilizing fast Fourier transforms and a Newton‐Krylov solver. The scale transition is performed by defining the macroscopic quantities via the volume averages of their microscopic counterparts. In order to demonstrate the use of the proposed framework, we predict the macroscopic and microscopic fields of a polycrystalline material within a numerical example using an efficient and accurate FE‐FFT‐based two‐scale method.

Multiscale analysis on the anisotropic thermal conduction of laminated fabrics by finite element method

In this paper, we present a novel “Two-scale finite element method” (tFEM) based on virtual fiber models to investigate the heat conduction behavior of stacked quartz woven fabrics and predict their anisotropic thermal conductivity. Considering the random fiber distribution and the twisted characteristic of yarns, the yarn-scale model was established. Furthermore, the single-layer fabric was composed of interwoven virtual yarns, which are stacked to be the multi-laminate fabric-scale model. Both the yarn-scale model and the fabric-scale model were combined with air matrix to form the two-phase composite model. Hot-Disk thermal constant analyzer was used to measure the anisotropic thermal conductivity of yarns and woven fabrics. Excellent agreement between simulations and experiments is obtained, which indicates the multiscale finite element models in this paper is accurate. The innovations of the study are that not only the effects of fiber numbers and diameters on the thermal conductivity of twist yarns and quartz ply-fabrics are analyzed, but also the detailed heat flux distribution and temperature distribution between fibers and the air are simulated and analyzed. Moreover, the isotropic thermal conductivity of the yarn radial direction and the anisotropic thermal conductivity of quartz fabrics are illustrated.

Experimental and numerical investigation on the ablation mechanism of Al2O3/Al2O3-CMCs under continuous-wave laser irradiation

In this study, the ablation mechanism of Al2O3/Al2O3-CMCs under laser irradiation and the influence of laser parameters on the ablation efficiency were examined. The experimental results show that the diameters of the ablation centre and transition area are both positively correlated to the irradiation energy, and the irradiation power has a more prominent effect on the diameter of the ablation centre. The molten Al2O3 changes the reflection and transmission properties of laser beam, which results in an elliptical ablation pit. A macro-meso two-scale finite element (FE) model was established to simulate the laser ablation process progressively. The FE results show that the difference in thermal conductivity between the Al2O3 fibre and matrix induces a discrepancy in ablation at the mesoscale. The radius and depth of the ablation pit increase rapidly at the initial stage, then slowly increase with time. A good correlation between the experimental and numerical results was achieved.

Automotive optoelectronic components submitted to thermal shock: Impact of component architecture on mechanical reliability

New LED technologies & modules allow glare-free automotive lighting. Valeo solutions include the first high-definition LED lighting system. These LEDs technology endure extreme tests conditions in form of thermal shocks or mechanical vibrations to insure reliability under the lifetime of automotive usage. Strict regulations require lead-free solders. Advanced optimization of LEDs for automotive lighting led to multilayered and miniaturized stack-ups. The first lead-free solders were revealed as a weak part of the electronic components and LED assembly. Enormous efforts have been spent on optimizing the lead-free solder material, design and reflow process. However, component miniaturization can also lead to high thermal gradients. Present work analyses the relation between component architectures and thermal shock induced fatigue damage. First, damage after N1 and N2 thermal shock cycles was characterized experimentally. Genuine sample preparations by a Cross polisher followed by high-resolution SEM observations reveal void growth in the solder material and cracking in the current circuit Layer. A two-scale finite element model reproduces both damage types. Simulation results can help to provide design improvements on the component architectures and modules to avoid (minimize) both damage types.

Multi-objective optimization of a composite orthotropic bridge with RSM and NSGA-II algorithm

To address the problems of pavement damage and fatigue cracking of orthotropic steel deck (OSD) in bridges with traditional asphalt overlay, an innovative composite bridge deck composed of OSD with open ribs and ultra-high performance concrete (UHPC) layer was proposed. This paper firstly investigates the stress responses of fatigue-prone details in the composite bridge deck by refined two-scale finite element analysis. The results show that the rib-to-deck joint can achieve an infinite fatigue life, while the floorbeam detail of rib-to-floorbeam joint indicates finite fatigue life. Then, approximate regression models of stress ranges of fatigue details and structure weight were derived via both the central composite design and response surface method. The analysis of correlation and variance demonstrates that the response surface models are significant and sufficiently accurate to predict the objective responses. And 3-D response surface plots were developed to display the interaction effect between each pair of structural parameters on objective responses. Finally, to improve the fatigue performance for achieving an infinite fatigue life under relatively low structure weight, the multi-objective optimization was executed by an Improved Non-dominated Sorting Genetic Algorithm (NSGA-II). The obtained Pareto front shows that there is a strong competition between the stress range of fatigue-prone detail and structure weight. Under the principle of relatively low structure weight, the obtained structural parameters satisfying infinite fatigue life imply that the deck plate thickness is 10 mm, and the floorbeam spacing is 2.4 m as well as the floorbeam thickness is not less than 12 mm.

Two-scale finite element discretizations for nonlinear eigenvalue problems in quantum physics

Two-scale finite element discretizations for nonlinear eigenvalue problems in quantum physics

A two-scale finite element model for the fatigue design of large welded structures

Weld toes and weld roots of continuously welded structures subjected to cyclic loading are critical zones in terms of the fatigue resistance. The finite element method coupled with a fatigue criterion is commonly used to ensure the correct sizing and fatigue design of welded structures. However, weld geometries are often simplified or idealized to limit computational cost. In this work, a numerical two-scale approach is proposed in order to calculate a non-local multiaxial equivalent stress at the weld toe and the weld root from a global finite element shell model.The influence of the parameters of the proposed model on the stiffness behaviour is investigated for three welded structures and for different loading cases. A comparison in terms of stiffness with other models from the literature is also proposed. The results show that the stiffness behaviour is not affected by the parameters of the proposed approach and that it is the most robust model for the different geometries and loading cases studied. The variation in the non-local multiaxial equivalent stress as a function of the parameters of the proposed approach was also studied. The comparison with full solid finite element models makes it possible to define minimum values for the different parameters studied and validates the potential of the proposed approach for the fatigue design of welded structures.

Second-order two-scale analysis for heating effect of periodical composite structure

The multi-scale analysis method is presented for the heat transfer problems of periodical composite structures under irradiation heating. This kind of physical problem can be described as a class of parabolic equations with small periodical oscillating coefficients equipped with the mixed boundary conditions of both Dirichlet and Neumann type. It needs more expensive cost in both computer's memory and CPU time to solve these problems by the usual finite difference method or finite element method, since it is necessary to partition the considering geometrical region finer to capture the heat conduction behavior at small scale. To reduce the computational complexity, an approximate solution is derived by the asymptotic expansion technique. The convergence of asymptotic solution to exact solution is technically proved as the micro-scale parameter tends to zero. Based on the asymptotic expansion formula, a second-order two-scale finite element method is proposed to fully solve this problem in practice. Numerical results show convergent behavior of the proposed method.

An electro-mechanically coupled computational multiscale formulation for electrical conductors

Motivated by experimental findings on deformation induced microcracks in thin metal films and by their influence on the effective macroscopic electrical conductivity, a computational multiscale formulation for electrical conductors is proposed in this contribution. In particular, averaging theorems for kinematic quantities and for their energetic duals are discussed, an extended version of the Hill–Mandel energy equivalence condition is proposed and suitable boundary conditions for the microscale problem are elaborated. The implementation of the proposed framework in a two-scale finite element environment is shown and representative boundary value problems are studied in two- and three-dimensional settings.

Realistic microstructure-based two-scale finite element analysis of 3D C/C composite for effective thermal expansion coefficient

A real microstructure-based finite element model of 3D-hybrid carbon/carbon (3DH C/C) composite is used for the prediction of effective thermal expansion coefficient over the full temperature range (27–2227 °C). The representative volume elements of the real microstructure of 3DH C/C composite include the realistic features such as cross-sectional distortions of the bundle and irregularly shaped voids. Two-scale homogenization approach has been utilized to determine the effective thermomechanical behavior of the composite. The effective elastic properties of the fiber bundles are firstly obtained using the mean-field homogenization approach also called the Mori–Tanaka approach. Later, the computational homogenization approach is utilized for the prediction of effective coefficients of thermal expansion (CTEs) of bundles and 3DH C/C composite. The effective CTEs obtained for 3DH C/C composite are found in good agreement with those obtained through experiments in the literature.

An efficient two-scale 3D FE model of the bone fibril array: comparison of anisotropic elastic properties with analytical methods and micro-sample testing.

In this study, 3D finite element analyses (FEA) are conducted to quantify the orthotropic elastic properties and investigate the load transfer mechanism of bone at the sub-lamellar level. Three finite element (FE) unit cells with periodic boundary conditions are presented to model a two-scale microstructure of bone including a mineralized collagen fibril (MCF), the extrafibrillar matrix (EFM) and the resulting fibril array (FAY) under arbitrary loading. The axial and transverse elastic properties of the FAY computed by FEA are calibrated with unique experimental results on ovine micro-samples showing a coherent fibril orientation. They are then systematically compared with those calculated using analytical methods including the basic Voigt, Reuss and shear-lag models, the Mori-Tanaka scheme and the upper and lower bounds by Hashin and Shtrikman. The predicted axial strain ratios between the two-scales are discussed with respect to a recent small-angle X-ray scattering and wide-angle X-ray diffraction study. Beyond apparent elastic properties, the FE models provide stress distributions at both hierarchical levels, confirm the shear lag mechanisms within the MCF and between MCF and EFM and identify potential damage sites under arbitrary loading conditions. A comprehensive sensitivity analysis shows that mineral volume fraction in the fibril array is the dominant parameter on the axial and transverse elastic moduli, while the MCF volume fraction in FAY is the most sensitive variable for the ratio of axial versus transverse elastic modulus followed by the elastic moduli of hydroxyapatite and collagen. The FE model of the FAY developed and calibrated in the current study represents an anatomically realistic, experimentally validated and computationally efficient basis for investigating the apparent yield, post-yield and failure behaviors of lamellar bone in future research.

FE2 simulations of magnetorheological elastomers: influence of microscopic boundary conditions, microstructures and free space on the macroscopic responses of MREs

In the current work, the response of heterogeneous magnetorheological elastomers (MREs) which are loaded by external magnetic fields in the absence and also in the presence of free space is studied. A fully-coupled two-scale finite element computational homogenization procedure is used to derive the material response at the macro-scale from the averaged response of the underlying micro-scale problem. Different combinations of boundary conditions, that satisfy the Hill-Mandel condition and are based on the primary variables of the magneto-elastic enthalpy and energy functionals are applied to solve the micro-scale boundary value problem. Furthermore, the influences of various microstructures on the macroscopic response of the MREs are investigated. The results indicate that the choice of microscopic boundary conditions and microstructure types can significantly affect the macroscopic responses of MREs.

Two-scale Finite Element Discretizations for Semilinear Parabolic Equations

In this paper, to reduce the computational cost of solving semilinear parabolic equations on a tensor product domain Ω⊂ℝ^<i>d</i> with <i>d</i> = 2 or 3, some two-scale finite element discretizations are proposed and analyzed. The time derivative in semilinear parabolic equations is approximated by the backward Euler finite difference scheme. The two-scale finite element method is designed for the space discretization. The idea of the two-scale finite element method is based on an understanding of a finite element solution to an elliptic problem on a tensor product domain. The high frequency parts of the finite element solution can be well captured on some univariate fine grids and the low frequency parts can be approximated on a coarse grid. Thus the two-scale finite element approximation is defined as a linear combination of some standard finite element approximations on some univariate fine grids and a coarse grid satisfying <i>H</i> = <i>O</i> (<i>h</i>^1/2), where <i>h</i> and <i>H</i> are the fine and coarse mesh widths, respectively. It is shown theoretically and numerically that the backward Euler two-scale finite element solution not only achieves the same order of accuracy in the <i>H</i>¹ (Ω) norm as the backward Euler standard finite element solution, but also reduces the number of degrees of freedom from <i>O</i>(<i>h</i>^-<i>d</i>×<i>τ</i>^-1) to <i>O</i>(<i>h</i>^{-(<i>(d)</i>+1)/2}×<i>τ</i>^-1) where <i>τ</i> is the time step. Consequently the backward Euler two-scale finite element method for semilinear parabolic equations is more efficient than the backward Euler standard finite element method.

Finite element approximation of the Isaacs equation

We propose and analyze a two-scale finite element method for the Isaacs equation. The fine scale is given by the mesh sizehwhereas the coarse scaleεis dictated by an integro-differential approximation of the partial differential equation. We show that the method satisfies the discrete maximum principle provided that the mesh is weakly acute. This, in conjunction with weak operator consistency of the finite element method, allows us to establish convergence of the numerical solution to the viscosity solution asε,h→ 0, andε ≳ (h|logh|)1/2. In addition, using a discrete Alexandrov Bakelman Pucci estimate we deduce rates of convergence, under suitable smoothness assumptions on the exact solution.