Multiscale Finite Element Approach Research Articles

A two level approach for simulating Bose-Einstein condensates by localized orthogonal decomposition

In this work, we consider the numerical computation of ground states and dynamics of single-component Bose-Einstein condensates (BECs). The corresponding models are spatially discretized with a multiscale finite element approach known as Localized Orthogonal Decomposition (LOD). Despite the outstanding approximation properties of such a discretization in the context of BECs, taking full advantage of it without creating severe computational bottlenecks can be tricky. In this paper, we therefore present two fully-discrete numerical approaches that are formulated in such a way that they take special account of the structure of the LOD spaces. One approach is devoted to the computation of ground states and another one for the computation of dynamics. A central focus of this paper is also the discussion of implementation aspects that are very important for the practical realization of the methods. In particular, we discuss the use of suitable data structures that keep the memory costs economical. The paper concludes with various numerical experiments in 1D, 2D and 3D that investigate convergence rates and approximation properties of the methods and which demonstrate their performance and computational efficiency, also in comparison to spectral and standard finite element approaches.

Computational assessment of effect of porosity on thermal conductivity of 3D-printed PEEK/carbon-fiber composite

The three-dimensional (3D) printing of polymer matrix composites has attracted considerable attention. However, the addition of filler materials to the extruded polymer matrix results in inner voids within the beads along with larger voids, called intervoids, which are generated between the beads during the 3D printing process. These voids degrade the thermal transport properties of 3D-printed polyether ether ketone (PEEK) and PEEK/carbon-fiber composites. The effects of these pores (i.e., their number, size, and location) on the thermal conductivity of 3D-printed PEEK and a PEEK/carbon-fiber composite were investigated using a multiscale finite element approach. The presence of the carbon fibers mitigated the effect of the pores on the thermal conductivity of the PEEK/carbon-fiber composite, particularly along the longitudinal direction. Moreover, the inner voids had a dominant effect on the PEEK and PEEK/carbon-fiber composite. Finally, the orientation of the intervoids had no effect on the thermal conductivity of the printed materials.

Global-to-local simulation of the thermal history in the laser powder bed fusion process based on a multiscale finite element approach

Global-to-local simulation of the thermal history in the laser powder bed fusion process based on a multiscale finite element approach

The dynamic recrystallization behavior of the Ti-5.5Mo-7.2Al-4.5Zr-2.6Sn-2.1Cr titanium alloy during hot rolling based on macro-meso multiscale crystal plasticity finite element approach

The understanding of dynamic recrystallization under complex thermo-mechanical coupling is still a technical challenge. Therefore, in this work, combining the Johnson-Mehl-Avrami-Kolmogorov recrystallization grain model, a macro-meso multiscale crystal plasticity finite element approach was innovatively proposed. The isothermal hot compression test was performed to obtain the true stress-strain curves of Ti-5.5Mo-7.2Al-4.5Zr-2.6Sn-2.1Cr titanium alloy (strain rates: 0.001 s−1, 0.01 s−1, 0.1 s−1, 1 s−1, and 10 s−1; temperatures: 650 ℃, 700 ℃, 750 ℃, 800 ℃, and 850 ℃). Afterward, corresponding fitted parameters can be used to update the volume fraction and size of recrystallized grains of each sub-step. The macro simulation results showed that the surface layer had prominent strain partitioning and lower equivalent stress after hot rolling (thickness reduction: 60 %). For the region of interest, the calculated volume fraction and equivalent circle diameter of recrystallized grains were 2.45% and ∼1 µm, respectively, which were in good agreement with the experimental results.

Temperature-dependent multiscale modeling of graphene sheet under finite deformation

The homogenized constitutive models that have been utilized to simulate the behavior of nanostructures are typically based on the Cauchy–Born hypothesis, which seeks the fundamental properties of material via relating atomistic information to an assumed homogeneous deformation field. It is well known that temperature has a profound effect on the validity and size-dependency of the Cauchy-Born hypothesis in finite deformations. In this study, a temperature-related Cauchy–Born formulation is established for graphene sheets and its performance is examined through a direct comparison between the continuum-based constitutive model and molecular dynamics analysis. It is demonstrated that if the temperature increases, the validity surfaces shrink, which is not highly dependent on the size of specimen. At finite strains, graphene sheet exhibits material softening and hardening, which are manifested in the elastic constants. The developed constitutive model is used to study the effect of applied deformations at various temperature levels on the elastic constants of graphene sheets. Finally, a novel multiscale finite element approach is developed for the analysis of graphene sheets in finite strain regime based on the proposed atomistic-continuum model. The method can be used efficiently in the simulation of large specimens where the application of molecular dynamics requires considerable computational efforts. The efficiency and robustness of the proposed multiscale approach are shown through several numerical examples.

Multiscale analysis of composite structures with goal-oriented mesh adaptivity and reduced order homogenization

We present a multiscale finite element approach for composite structure analysis, applying reduced order homogenization to obtain an accurate surrogate model for microscale RVE computations and establishing macroscale mesh adaptivity towards an error control of a user-defined quantity of interest. By means of the so-called nonuniform transformation field analysis, reduced variables are deduced from a space–time decomposition of inelastic strain fields. Considering the macroscopic dissipation power as a volume-averaged microscopic dissipation power, closed-form macroscale constitutive relations are established, thus resulting into a reduced order homogenization problem. For ease of error estimate, we propose a multifield formulation for the reduced order problem. Based on duality techniques, a backward-in-time dual problem is derived from a Lagrange method, leading to error representations aiming at the quantity of interest. Accordingly, an efficient error estimator combined with a patch recovery technique avoiding nonlinear computations is proposed. The effectiveness of the resulting adaptive algorithm is illustrated by several numerical examples w.r.t. a prototype model.

Numerical modeling of residual stresses during vibratory peening of a 3-stage Blisk – a multi-scale discrete element and finite element approach

Vibratory-based manufacturing processes such as polishing, and peening are widely used for surface enhancement applications. Vibratory peening is an excellent process to introduce residual stresses, improve fatigue life, and enhance the surface quality of metallic components. This research aims to understand the combined peening-polishing mechanism in the vibratory peening process of a gas turbine engine’s 3-stage blisk using a novel horizontal vibratory peening method. The research was aimed to understand the common understanding that the deeper the component in the vibratory machine, the higher would be the media-component interaction effect and better would be the peening effect. However, contrary to the hypothesis, the experimental findings showed significant peening (60.0 % higher than bottom) in the coupons near the top surface of the media in the trough. The DEM results revealed that the media-component normal relative velocity (Vn) and normal contact energy (Enc) contributed to the peening effect and the contact force (Fc), tangential contact energy (Etc) contributed to the polishing effect. The DEM results were used as input conditions for FEM to predict the residual stresses from peening. The FEM simulated residual stress trend matches the experimental findings, with differences not exceeding ∼ 40 % for most cases. The combined DEM-FEM approach provides an opportunity to understand the complex behavior of media-component interaction in a vibratory process. With this understanding, vibratory peening all three stages of a blisk simultaneously reduces the overall production cost and time.

A multi-scale finite element approach for the mechanical behavior analysis of 3D braided composite structures

With the development of composite processing technology, three-dimensional braided composites have been widely used in automotive, aerospace and other high-tech industries. Due to specific yarn distribution in microstructure, three-dimensional braided composites have excellent mechanical properties. To further optimize or design their mechanical performance, it is important to accurately estimate the effective mechanical properties of braided performs with different braiding parameters such as braiding angle, pitch length and fiber volume fraction. To this end, the microstructure (i.e. yarn pattern) of braided perform generated by 1 × 1 four-step braiding process was first analyzed in detail with the three cell model, which was established from the simulation of braiding process. Secondly, multiphase finite element method was employed and introduced into asymptotic expansion homogenization framework to predict the equivalent elastic modulus and microscopic stress of three-dimensional braided composites. Finally, the calculated equivalent engineering elastic modulus was proved to be in good agreement with experimental measurements of braided composite samples. Thanks to the multi-scale approach, the microscopic stress, i.e., local stress state of the composite perform under macroscopic load can be obtained simultaneously, which can be used in further study of damage and failure behavior.

A multi-scale finite element approach to mechanical performance of polyurethane/CNT nanocomposite foam

Manufacturing and testing of polymer nanocomposites reinforced with nanoparticles is not cost effective due to high costs of nanomaterials and nanoparticles dispersion problems, and is very time consuming. Therefore, predicting the mechanical properties of these materials using finite element methods is a suitable solution. Accordingly, in the present study, multi-scale finite element method considering the random distribution of carbon nanotubes (CNT) in polyurethane foam is developed to investigate the mechanical behavior of these foams. Experimentally examinations under uniaxial compressive load and SEM tests are performed on samples of polyurethane foam reinforced with 2 wt% and 4 wt% carbon nanotubes. In order to consider the realistic assumptions, the interface between the nanotubes with the matrix is simulated using the cohesive zone model, whose parameters are obtained by calibrating the finite element model results with the experimental results of the polyurethane/CNT nanocomposite foam. The results show that the multi-scale finite element model developed in this study shows the good agreement with experimental results. Moreover, with this method simulation of the mechanical behavior of polyurethane foam reinforced with carbon nanotubes with real conditions becomes possible. Hence, it is easy to determine the impact of effective parameters on mechanical properties of these materials. Experimental results show that in comparison with pure foam, by adding 2 wt% and 4 wt% carbon nanotubes to polyurethane foam, the compressive strength of foam is enhanced by 49 % and 82 %, respectively.

Assessment of the multifactorial causes of atypical femoral fractures using a novel multiscale finite element approach.

Atypical femoral fracture (AFF), which is a low energy fracture in the subtrochanteric or diaphysis region of the femur, has multifactorial causes that span macro- to microscale mechanisms including femoral geometry, cortical bone composition and structure. However, the extent of individual and combined influence of these factors on AFF is still not well understood. As a result, the aim of this study is to develop a multiscale fracture mechanics-based finite element modeling framework that is capable of quantifying the individual and combined influence of macroscale femoral geometrical properties as well as cortical bone microscale material properties and structure on AFF. In this study, three different femoral geometries with two different cortical bone microstructures, and two different material property distributions were investigated by first determining the critical AFF locations in the femur using macroscale stress analysis and then performing coupled macro-microscale fracture simulations. The simulation results showed that femoral geometry led to substantial differences in crack growth independent of cortical microstructure and tissue level material properties. The results suggest that multiple femoral geometrical properties, including neck-shaft angle and curvature, may contribute to the fracture behavior at AFF sites rather than a single macroscale geometrical feature. Osteonal area had a significant effect on microcrack propagation at AFF sites independent of microscale material property distribution and femoral geometry. In addition, cortical bone tissue level material heterogeneity improved the fracture resistance independent of femoral geometry and cortical microstructure. In summary, the computational approach developed in this study identified the individual, combined, and relative influence of multiscale factors on AFF risk. The new framework developed in this study could help identify the governing multiscale mechanisms of AFF and bring additional insight into the possible association of long-term bisphosphate treatment with AFF.

Accelerating 2D frequency-domain full-waveform inversion via fast wave modeling using a model reduction technique

Full-waveform inversion (FWI) is widely used to infer earth structures and rock properties. In FWI, most of the computation arises from the repeated simulations of wave propagation. Although frequency-domain implementations have several advantages, solving the Helmholtz equation incurs a major computational cost associated with the inversion of large matrices. Hence, we have used a new model reduction technique called the generalized multiscale finite-element method (GM FEM) to perform this task rapidly for forward and backward simulations. This in turn leads to the acceleration of the FWI. In addition, the multiscale finite-element approach allows flexible, adaptive selection of modeling parameters (i.e., grid size, number of basis functions) for different target frequencies, providing further speed up. We apply this frequency-domain, multiscale FEM approach to the Marmousi-2 model, and the FWI results indicated how varying the number of basis functions can control the trade-off between the accuracy and computational speed. In addition, we introduced FWI examples applied to field data from the Gulf of Mexico. These field data examples indicate that applying our multiscale FWI with a relatively small number of basis functions can quickly construct a macrovelocity model using low frequencies. We also evaluate a strategy to optimize the FWI procedure by using frequency-adaptive multiscale basis functions based on the target frequency group. In general, we can reduce the run time by up to 30% through the application of GM FEM wave modeling in FWI.

Experimental and numerical approach to study mechanical and fracture properties of high-density polyethylene carbon nanotubes composite

This work investigates the elasto-plastic and fracture properties of high-density polyethylene (HDPE) carbon nanotubes (CNTs) composite using a multi-scale finite element (MSFE) approach. The composites consist of CNTs, which are randomly embedded within the HDPE matrix throughout the volume. CNT reinforcement is numerically modelled by considering elastic properties while the elasto-plastic constitutive law is used for the matrix (HDPE) composition. Elastic properties of composites are estimated at meso-scale by numerical modelling of a representative volume element (RVE). Hollomon’s (power-hardening law) model has been employed to get the plastic properties of the polymer composite. Equivalent consecutive properties are predicted at meso-scale, further, these properties have been used at macro-scale simulation for fracture analysis. A detailed study of stress intensity factors has been presented for the different volume fraction of CNTs in the composite composition. The presented numerical results are supported by experimental investigations.

A new method for assessing anisotropy in fused deposition modeled parts using computed tomography data

Voids in fused deposition modeled (FDM) parts are assumed to be a key driver for their anisotropic behavior. However, these assumptions are based on investigations of voids using only 2D data (microscopy images). This paper presents a new method to measure such voids by analyzing 3D-data of from X-ray computed tomography (CT), and application of this data for assessment of mechanical parameters. The article is divided into three parts, where the first part elaborates on a proposed method to assess and characterize the void geometry throughout uniaxial printed FDM parts using CT-data. The second part presents an investigation of the void configurations in samples manufactured using different process parameters, aiming to understand how variation in extrusion rate and compensation for non-linear dynamic extrusion behavior affects the void sizes. The third part displays how the information regarding void sizes could be further related to global mechanical properties, using a multiscale finite element approach. The present method of CT-data analysis gives a clear overview of the spatial variation of the void geometry, and findings from the investigated samples suggest that the size of voids have a large non-random spatial variation, highly dependent on the turning points of the toolpaths, and also significantly affected by accumulation of excess material. Printing at a low extrusion rate increases the void sizes considerably, while implementation of an extrusion dynamics compensation algorithm was found to have low impact on the void sizes. The multiscale finite element approach predicts anisotropic elastic behavior, significantly more compliant in the vertical and transversal direction, relative to the printing direction of the infill. It also predicts a non-isotropic strain energy density throughout the specimen, where the location and magnitude of the most energy dense locations vary significantly for different directions of loading, which implicates an anisotropic behavior in terms of failure, in accordance with literature.

Frequency analysis of carbon and silicon nanosheet with surface effects

• FE modeling of surface effects in Si, Graphene–Silicon and SWCNT-Silicon nanosheet . • Numerical analysis of HACNTs on silicon substrate and its effect on frequency. • Development of analytical model to find frequency of HACNTs arrays on silicon. Silicon-based microelectromechanical system (MEMS) and nanoelectromechanical systems (NEMS) have been used to design and fabricate sensitive sensors and actuators. Recent research trends show that graphene and carbon nanotubes (CNTs) have been used to change the surface properties of silicon-based MEMS and NEMS to improve different mechanical, optical and electrical properties of silicon-based composites. In this paper, we focus on analyzing the vibrational characteristics of silicon-based devices when the surface of silicon is coated with single-layer graphene and horizontally aligned carbon nanotubes (HACNTs). To perform the analysis, we use multi-scale finite element approach for developing graphene–silicon nanocomposites (GSNCs) and carbon nanotube-silicon nanocomposites (CSNC) composites in which interface layer of silicon with graphene or CNT is modeled using bonded contact element. Subsequently, we performed modal analysis to find the first transverse mode frequency of GSNC and CSNC composites for beam with smaller as well as longer lengths. The numerical model is compared with classical beam theory with and without surface effect. For GSNCs composites, we take a fixed-free case with lengths in the range of (20 Å–120 Å) and (400 Å–2000 Å), respectively. For CSNC composites, CNT diameter is varied from (5 Å–30 Å) for single walled nanotube . Subsequently, we analyze the influence of HACNTs-on-silicon on its vibrational characteristics. The analysis presented in the paper demonstrate that GSNCs offer a higher bending stiffness compared to single layer graphene (SLGs) and isolated silicon nanosheet which lead to higher natural frequency. A similar trend is found in the case of HACNTs on silicon NS when the number of tubes increases.

A hierarchical multiscale finite element approach for modelling the heat affected zone of welded joints

AbstractIn the present work a hierarchical multiscale approach, based on finite element method, is applied in order to characterize the heat affected zone (HAZ) of a welded joint with parent material S460M. The metallurgical constituents of particular regions of the HAZ was determined experimentally and, regarding the multiscale approach, a first‐order computational homogenization method was implemented. In this context, effective stress‐strain‐relations in particular spots of the HAZ are determined applying prescribed displacement boundary conditions. The obtained relations were considered in subsequent finite element calculations of butt‐welded joints under tensile loading, allowing a comparison of the numerical strain states to experimental results. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Probabilistic sensitivity analysis to understand the influence of micromechanical properties of wood on its macroscopic response

This paper investigates the influence of the uncertainty in different micromechanical properties on the variability of the macroscopic response of cross-laminated timber plates, by means of a probabilistic sensitivity analysis. Cross-laminated timber plates can be modelled using a multiscale finite element approach which although suitable, suffers from high computational cost. Investigating parametric importance can incur considerable time penalty since conventional sensitivity analysis relies on a large number of code evaluations to produce accurate results. In order to address this issue, we build a statistical approximation to the code output and use it to perform sensitivity analysis. We investigate the effect of a collection of parameters on the density and Young’s moduli of wood. Additionally, the influence on the response of cross-laminated timber plates subject to bending, in-plane shear and compression loads is investigated due to its relevance within the engineering community. The presented results provide a practical insight into the importance of each micromechanical parameter, which allows research effort to be focused on important wood properties.

Textile reinforced concrete multiscale mechanical modelling: Application to TRC sandwich panels

The present study establishes a numerical strategy for describing the textile/concrete bond behaviour in textile-reinforced concrete (TRC) composites that separates the cohesive and coulomb friction contributions.The textile-concrete bond approach, validated on an existing pull out test in the literature, has been used to calibrate the textile-concrete bond slip law of an existing TRC tested in tension by an innovative inverse approach thanks to its pull-out mode of failure.The calibrated bond slip law has been used as an input parameter to produce an enhanced TRC multiscale numerical model that is based on the nonlinear behaviour of its constitutive components (concrete, textile, and textile-concrete bond slip law) and takes into account all the damage mechanisms of TRC, which are mainly characterized by matrix cracking and yarn pull-out. The model has been validated on the basis of the previous TRC experimental tensile test.Two approaches have been used to account for the mechanical behaviour of a textile-reinforced concrete sandwich panel under a four-point bending test: a macroscopic 3D finite element approach, which considers the composite TRC with its macro tri-linear stress – strain relation under tensile solicitation, and the proposed TRC multiscale 3D finite element approach which, involves definition of the textile-concrete interaction bond slip law. More accurate results have been achieved with the multiscale approach; furthermore, the experimental mode of failure of the sandwich panel has been captured.

A conservative local multiscale model reduction technique for Stokes flows in heterogeneous perforated domains

In this paper, we present a new multiscale model reduction technique for the Stokes flows in heterogeneous perforated domains. The challenge in the numerical simulations of this problem lies in the fact that the solution contains many multiscale features and requires a very fine mesh to resolve all details. In order to efficiently compute the solutions, some model reductions are necessary. To obtain a reduced model, we apply the generalized multiscale finite element approach, which is a framework allowing systematic construction of reduced models. Based on this general framework, we will first construct a local snapshot space, which contains many possible multiscale features of the solution. Using the snapshot space and a local spectral problem, we identify dominant modes in the snapshot space and use them as the multiscale basis functions. Our basis functions are constructed locally with non-overlapping supports, which enhances the sparsity of the resulting linear system. In order to enforce the mass conservation, we propose a hybridized technique, and uses a Lagrange multiplier to achieve mass conservation. We will mathematically analyze the stability and the convergence of the proposed method. In addition, we will present some numerical examples to show the performance of the scheme. We show that, with a few basis functions per coarse region, one can obtain a solution with excellent accuracy.

Numerical stability analysis and flow simulation of lid-driven cavity subjected to high magnetic field

In this work, we study the flow of a conducting fluid inside a two-dimensional square domain. The problem is solved by using a variational multiscale finite element approach. The study focuses on a high magnetic interaction parameter range and high Reynolds number. Under the imposition of a high magnetic field, the flow gets regularized, but fast transient phenomena take place, which could lead to numerical errors. An expression to compute the maximum time step that guarantees convergence in explicit schemes is proposed and validated through numerical tests.

Compressive failure of composites: A computational homogenization approach

This paper revisits the modeling of compressive failure of long fiber composite materials by considering a multiscale finite element approach. It is well known that this failure follows from a fiber microbuckling phenomenon. Fiber microbuckling is governed by both material and geometrical quantities: the elastoplastic shear behavior of the matrix and the fiber misalignment. Although all these parameters are easily accounted by a finite element analysis at the local level, the failure is also influenced by macrostructural quantities. That is why a multilevel finite element model (FE2) is relevant to describe the compressive failure of composite. Furthermore, fiber local buckling leads to a loss of ellipticity of the macroscopic model, which can be a criterion of failure.