Multiscale Finite Element Research Articles

CECM: A continuous empirical cubature method with application to the dimensional hyperreduction of parameterized finite element models

We propose a method for finding optimal quadrature/cubature rules with positive weights for parameterized functions in 1D, 2D or 3D spatial domains. The method takes as starting point the values of the functions at the Gauss points of a finite element (FE) mesh of the spatial domain for a representative sample of input parameters, and then construct an elementwise continuous orthogonal basis for such functions using the truncated Singular Value Decomposition (SVD) along with element polynomial fitting. To avoid possible memory bottlenecks in computing the SVD, we propose a Sequential Randomized SVD (SRSVD) in which the matrix is provided in a column-partitioned format, and which uses randomization to accelerate the processing of each individual block. After computing the basis functions, the method determines an exact integration rule for such functions, featuring as many points as functions, and in which the points are selected among the Gauss points of the FE mesh. Finally, the desired optimal rule is obtained by an sparsification process in which the algorithm zeroes one weight at a time while readjusting the positions and weights of the remaining points so that the constraints of the problem are satisfied. We apply this methodology to multivariate polynomials in cartesian domains to demonstrate that the method is indeed able to produce optimal rules – i.e., Gauss product rules –, and to a 2-parameters, 3D sinusoidal–exponential function to illustrate the use of the SRSVD in scenarios in which the standard SVD cannot handle the operation because of memory limitations. Lastly, the fact that the method does not require the analytical expression of the integrand functions – just their values at the FE Gauss points – makes it suitable for dealing with the so-called hyperreduction of parameterized finite element models. We exemplify this by showing its performance in the derivation of low-dimensional surrogate models in the context of the multiscale FE method. The Matlab source codes of both the CECM and the SRSVD, along with the scripts for launching the numerical tests, are openly accessible in the public repository https://github.com/Rbravo555/CECM-continuous-empirical-cubature-method.

Numerical multiscale analysis of 3D printed short fiber composites parts: Filament anisotropy and toolpath effects

AbstractThe aim of the present investigation is the development of a numerical model able to adequately represent the effect of several variables, associated to the fused deposition modeling (FDM) procedure, on the mechanical behavior of 3D printed parts. Specifically, 3D printed carbon short‐fiber reinforced thermoplastic parts are numerically analyzed. Previous experimental results have proven that this kind of parts show a global anisotropic behavior, in terms of classical mechanical parameters as stiffness. Thus, special emphasis is done in analyzing the effect of the raster angle / toolpath (inherent to FDM) and the internal microstructure of the deposited filaments (due to the presence of the short fibers). Multiscale finite element models are used to represent the linear elastic behavior at macro scale. The numerical models are also able to include the effect of porosity. Based on experimental results of 3D printed composite parts with 100% infill and different raster angles, elastic transversely isotropic properties are estimated for the individual deposited filaments using a reverse engineering procedure. Obtained results show that for an adequate modeling of FDM composite parts, anisotropic properties of the filament must be taken into account, even when quasi‐isotropic printing parameters are used (“cross‐ply” configurations). Finally, additional numerical analyses of some parameters associated to the FDM technique are done. Specifically, the effect of porosity related to the infill pattern and percentage on the global (macro) apparent stiffness is analyzed.

EMsFEM based concurrent topology optimization method for hierarchical structure with multiple substructures

An efficient concurrent topology optimization method based on the Extended Multiscale Finite Element Method (EMsFEM) is proposed to design the hierarchical structure with multiple substructures in this paper. Firstly, the mathematical description model for the topology of hierarchical structures and the interpolation model for material are established at both the macro and substructural levels. Then the concurrent topology optimization formulation for hierarchical structures is proposed based on EMsFEM. After that, the sensitivity analysis scheme of the objective function and constraint functions is given, based on which the Method of Moving Asymptotes (MMA) is employed to update design variables. The assumption of separation of length scales is not required in the proposed method, which avoids the disconnection between adjacent substructures. Besides, without limiting the topology and spatial distribution of substructures in advance, the concurrent topology optimization for hierarchical structures with multiple substructures can be realized using this method. The provided numerical examples verify the correctness and effectiveness of the proposed sensitivity analysis method as well as the feasibility, effectiveness and scalability of the proposed concurrent topology optimization method for hierarchical structures. Finally, the influence of the initial substructure configurations and the number of substructure types on the final configuration and mechanical performance of hierarchical structures are explored with a series of numerical examples.

Research on Multiscale Modeling and Experiment of CFRP Milling.

High-quality milling of carbon fiber reinforced polymer (CFRP) composites is of great importance for the high-performance manufacturing of structures made of this hard-to-machine material. In this paper, a multiscale finite element (FE) model, considering the thermal-mechanical coupling effect, was developed to simulate the milling process and reveal its material removal mechanism. The corresponding milling experiments were conducted to validate the simulated cutting forces and temperature, which were in good agreement with the experiment results. In the macroscale model, the Hashin failure criteria were used to estimate the failure of the composites. In the microscale model, the fibers, matrix, and the fiber-matrix interface were modeled separately, to investigate the mechanisms of material removal behavior during milling, among fiber breakage, matrix cracking, and fiber-matrix debonding. Based on the macroscale numerical and experimental results, the higher cutting speed was demonstrated to improve the surface quality of CFRP milling. According to the results from the microscale model, the material removal mechanism varies depending on the orientation of the fibers and can be divided into four stages. The outcome of this work provides guidelines to further investigate optimal manufacturing parameters for the milling of CFRP composites and their cutting mechanisms.

Convergence of the CEM-GMsFEM for compressible flow in highly heterogeneous media

This paper presents and analyses a Constraint Energy Minimization Generalized Multiscale Finite Element Method (CEM-GMsFEM) for solving single-phase non-linear compressible flows in highly heterogeneous media. The construction of CEM-GMsFEM hinges on two crucial steps: First, the auxiliary space is constructed by solving local spectral problems, where the basis functions corresponding to small eigenvalues are captured. Then the basis functions are obtained by solving local energy minimization problems over the oversampling domains using the auxiliary space. The basis functions have exponential decay outside the corresponding local oversampling regions. The convergence of the proposed method is provided, and we show that this convergence only depends on the coarse grid size and is independent of the heterogeneities. An online enrichment guided by a posteriori error estimator is developed to enhance computational efficiency. Several numerical experiments on a three-dimensional case to confirm the theoretical findings are presented, illustrating the performance of the method and giving efficient and accurate numerical solutions.

Moving mesh method with variational multiscale finite element method for convection–diffusion–reaction equations

Moving mesh method with variational multiscale finite element method for convection–diffusion–reaction equations

Empirical Interscale Finite Element Method (EIFEM) for modeling heterogeneous structures via localized hyperreduction

This work proposes a special type of Finite Element (FE) technology – the Empirical Interscale FE method – for modeling heterogeneous structures in the small strain regime, for both dynamic and static analyses. The method combines a domain decomposition framework, where interface conditions are established through “fictitious” frames, with dimensional hyperreduction at subdomain level. Similar to other multiscale FE methods, the structure is assumed to be partitioned into coarse-scale elements, each of these elements is equipped with a fine-scale subgrid, and the displacements of the boundaries of the coarse-scale elements are described by standard polynomial FE shape functions. The distinguishing feature of the proposed method is the employed “interscale” variational formulation, which directly relates coarse-scale nodal internal forces with fine-scale stresses, thereby avoiding the typical nested local/global problems that appear, in the nonlinear regime, in other multiscale methods. This distinctive feature, along with hyperreduction schemes for nodal internal and external body forces , greatly facilitate the implementation of the proposed formulation in existing FE codes for solid elements. Indeed, one only has to change the location and weights of the integration points, and to replace a few polynomial-based FE matrices with “empirical” operators, i.e., derived from the information obtained in appropriately chosen computational experiments. We demonstrate that the elements resulting from this formulation are not afflicted by volumetric locking when dealing with nearly-incompressible materials, and that they can handle non-matching fine-scale grids as well as curved structures. Last but not least, we show that, for periodic structures, this method converges upon mesh refinement to the solution delivered by classical first-order computational homogenization. Thus, although the method does not presuppose scale separation, it can represent solutions in this limiting case by taking sufficiently small coarse-scale elements.

An Online Generalized Multiscale Finite Element Method for Dual-continuum Unsaturated Filtration Problem in Domains with Rough Boundaries

An Online Generalized Multiscale Finite Element Method for Dual-continuum Unsaturated Filtration Problem in Domains with Rough Boundaries

Online Coupled Generalized Multiscale Finite Element Method for the Poroelasticity Problem in Three-Dimensional Media

Online Coupled Generalized Multiscale Finite Element Method for the Poroelasticity Problem in Three-Dimensional Media

Multiscale finite element method combined with local preconditioning for solving Euler equations

In this work we present a nonlinear multiscale finite element method combined with local preconditioning for solving compressible Euler equations in conservative variables. The formulations are based on the strategy of separating scales, in which it is the core of the variational multiscale (finite element) methodology. The subgrid scale space is defined using bubble functions that vanish on the boundary of the elements, allowing to use a local Schur complement to define the resolved scale problem. The resulting numerical procedure allows the fine scales to depend on time. The formulation proposed added artificial viscosity isotropically in all scales of the discretization. Due to the fact that, density-based schemes suffer with undesirable effects of low speed flow including low convergence rate and loss of accuracy, local preconditioning is applied to the set of equations in the continuous case. We evaluate the multiscale formulation with local preconditioning in the low Mach number comparing with the non-preconditioned case. The experiments show that density-based schemes combined with local preconditioning yields good results.

Enhancement effect of two-tiered heterostructure in piezoelectric filler-dispersed polymer composites

The effect of two-tiered heterostructure was verified by computational simulations and prototype experiments as a new approach for improving the performance of piezoelectric (PE) filler-dispersed polymer materials. Multiscale finite element analysis (FEA) based on asymptotic homogenization theory was applied to systematically investigate the fundamental properties of 0–3 composites in the first level and the relationship between macroscopic piezoelectricity and connectivity of the constituent phases in the second level. Simulations indicated that macroscopic piezoelectricity can be improved by controlling the PE filler content of the constituent phases in the two-tiered heterostructure, and that the connectivity of the PE phases in the direction of the applied electric field can effectively enhancing piezoelectricity. Furthermore, the two-tiered heterostructure discovered by multiscale FEA was fabricated by the melt-casting, demonstrating a piezoelectricity improvement of approximately 480 %.

Accelerated ultraviolet aging behavior and numerical simulation of ramie/carbon fiber reinforced polyethylene terephthalate glycol hybrid composites

Multi-fibers reinforced polymer hybrid composites can be designed to achieve excellent comprehensive performance through maximizing fibers physical and chemical properties. The accelerated ultraviolet (UV) aging behavior and mechanical degradation of natural ramie fiber (R) and carbon fiber (C) reinforced polyethylene terephthalate glycol (PETG) hybrid composites were investigated to broaden the industrial applications and understand the performance of ramie plant fiber products. The results show that the color and thermal properties of PETG matrix after UV aging have changed. The flexural properties of R/C laminated composites vary with the material types and layup sequences. The flexural modulus of the unaged [C/R/C/R]s and [C/C/R/R]s composites increase 1.2 and 1.0 times as compared with the unaged [R/C/R/C]s and [R/R/C/C]s. After 28 days of UV aging, the flexural strength retention rates of the [R/R/R/R]s, [R/R/C/C]s, [R/C/R/C]s, [C/R/C/R]s, [C/C/R/R]s and [C/C/C/C]s composites were 77.1 %, 79.2 %, 78.8 %, 81.7 %, 83 % and 87.4 %, respectively. The fracture strains of the R/C composites increase significantly after UV aging. The multiscale finite element model of the R/C composites is established to study the UV aging mechanism and predict the mechanical degradation. The knowledge gained in this work will be beneficial for the design and engineering application of plant fiber reinforced polymer hybrid composite materials.

Study of interwall interaction during the pull separation of ultra-long double-walled carbon nanotubes under lateral loading

The successful fabrication of ultra-long carbon nanotubes (CNTs) makes CNTs ideal candidates for bearing systems due to the long sliding distance with low interwall interaction. However, it is difficult to measure the pull-out forces of ultra-long CNTs along their axes in real-time because of the difficulty of performing long-distance pull-out experiments. In this study, we proposed a lateral loading method to measure the interwall interaction force of ultra-long CNTs. By using a theoretical model and multi-scale finite element simulations, we could predict the interwall interaction force from the pull-out force in lateral loading. This pull-out force depends on the length of CNTs, and a critical length divides the pull-out force into two categories: maximum pull-out force due to local buckling and steady pull-out force due to sliding. In order to correctly measure the interwall interaction force by lateral loading, for the DWCNTs with the outer tube diameter of 0.955 nm, we have to make sure the lengths are greater than this critical value (54.2 nm) as experimental subjects.

Coupled dynamic and piezoelectric response of RPEHs embedded in road under traffic load via multiscale finite element models

Road Piezoelectric Energy Harvesting Systems (RPEHs) effectively collect and convert mechanical energy from road surface into electrical energy for intelligent traffic guidance and monitoring. To accurately evaluate the energy harvesting efficiency of RPEHs under traffic loads and road conditions, a multiscale finite element model of traffic load-road-RPEHs is developed. The model encompasses a encapsulated stacked piezoelectric transducer embedded in an large-scale road model. This approach addresses the limitations of previous studies that neglected the effect of RPEHs structure on the dynamic and piezoelectric response. The large-scale model provides the dynamic response of RPEHs for an independent small-scale model of the stacked piezoelectric transducer to enable coupled dynamic and piezoelectric behavior. Furthermore, a piezoelectric output theory considering the loss impedance is formulated. Combining this theory with the open-circuit voltage obtained from the multiscale models, we determine the output voltage and power of RPEHs under various conditions, such as vehicle speeds, axle weights, burial depths. The results demonstrate that the loss impedance undergoes significant changes under different conditions, exerting a substantial impact on the output of RPEHs, which should be considered during analysis. RPEHs positioned closer to the road surface produce higher output but also exhibit increased stress concentration. The pavement modulus is inversely proportional to the piezoelectric output, while the subgrade modulus has a comparatively smaller impact due to the stiffness of RPEHs.

A simple machine learning-based framework for faster multi-scale simulations of path-independent materials at large strains

Coupled multi-scale finite element analyses have gained traction over the last years due to the increasing available computational resources. Nevertheless, in the pursuit of accurate results within a reasonable time frame, replacing these high-fidelity micromechanical simulations with reduced-order data-driven models has been explored recently by the modelling community. In this work, two classes of machine learning models are trained for a porous hyperelastic microstructure to predict (i) whether the microscopic equilibrium problem is likely to fail and (ii) the stress–strain response. The former may be used to identify critical macroscopic points where one may fall back to the high-fidelity analysis and possibly apply convergence bowl-widening techniques. For the latter, both a linear regression with polynomial features and artificial Neural Networks have been used, and the required stress–strain derivatives for solving the equilibrium problem have been derived analytically. A weight regularisation is introduced to stabilise the tangent operator and several strategies are discussed for imposing null stresses in undeformed configurations for both regression models. The regression techniques, here analysed exclusively in the context of porous hyperelastic materials, evidence very promising prospects to accelerate multi-scale analyses of solids under large deformation.

Determination of the Effective Electrical Conductivity of a Fluid–Saturated Core from Computed Tomography Data

Abstract—This paper proposes a technique for determining the effective specific electrical conductivity of rock samples when their digital models are used. A modified algorithm for reconstructing the internal structure of the sample from the core’s nondestructive imaging data can be used to construct a relevant discrete model that approximates the pore space with a high degree of accuracy. Unlike existing approaches, the reconstructed discrete geometric model of a heterogeneous medium sample is hierarchical and oriented to the application of parallel computational schemes of multiscale finite element methods for a forward mathematical simulation of electromagnetic processes. The paper presents the results of solving the problem of determining the effective specific electrical conductivity of fluid–saturated rock samples and compares them with the data from laboratory experiments.

Topology optimization of gradient lattice structure under harmonic load based on multiscale finite element method

Topology optimization of gradient lattice structure under harmonic load based on multiscale finite element method

Deep ensemble learning for high-dimensional subsurface fluid flow modeling

The accuracy of Deep Learning (DL) algorithms can be improved by combining several deep learners into an ensemble. This avoids the continuous endeavor required to adjust the architecture of individual networks or the nature of the propagation. This study investigates prediction improvements possible using Deep Ensemble Learning (DEL) to determine four distinct multiscale basis functions in the mixed Generalized Multiscale Finite Element Method (GMsFEM), involving the permeability field as the only input. 376,250 samples were initially generated, filtered down to 367,811 after data pre-processing. A standard Convolutional Neural Network (CNN) named SkiplessCNN and three skip connection-based CNNs named FirstSkipCNN, MidSkipCNN, and DualSkipCNN were developed for the base learners. For each basis function, these four CNNs were combined into an ensemble model using linear regression and ridge regression, separately, as part of the stacking technique. A comparison of the coefficient of determination (R2) and Mean Squared Error (MSE) confirms the effectiveness of all three skip connections in enhancing the performance of the standard CNN, with DualSkip being the most effective among them. Additionally, as evaluated on the testing subset, the combined models meaningfully outperform the individual models for all basis functions. The case that applies linear regression delivers R2 ranging from 0.8456 to 0.9191 and MSE ranging from 0.0092 to 0.0369. The ridge regression case achieves marginally better predictions with R2 ranging from 0.8539 to 0.922, and MSE ranging from 0.009 to 0.0349 because its solution involves more evenly distributed weights.

A novel shell‐based hierarchical multiscale model for studying three‐dimensional four‐directional braided composite shafts

AbstractThe mechanical properties of three‐dimensional (3D) braided composites are related to damage evolution. However, the multi‐scale damage mechanism of 3D braided composite shafts remains unclear. The main purpose of this article is to propose a shell‐based multiscale finite element model for effectively predicting the compressive and torsional properties of 3D four‐directional (3D4d) braided composite shafts. Considering the relation between equivalent ply angles and yarn structures, the multiscale model was developed based on a universal tubular unit cell. The equivalent ply angle was derived through the yarns' local braiding and horizontal orientation angles, and the elastic properties of each equivalent ply were derived through the stiffness averaging method. Both derivations were achieved by creatively utilizing deformation gradient theory and Nanson's formula from continuum mechanics. LaRC04 failure criteria and energy‐based damage evolution law were implemented to characterize the equivalent UD‐ply's damage behaviors. The proposed model was validated experimentally, and good agreements were observed between the failure morphologies and load–displacement curves of FEA and experimental results. The results show that the fiber tension and compression damages are the major failure modes of 3D4d braided composite shafts under compressive and torque loads, respectively. Finally, a parametric study was carried out and compared with that from other literature to clarify the meaning of this article in relevant studies. It provides an efficient and useful tool for the design and manufacturing of 3D4d braided composite shafts with desirable mechanical properties.Highlights A shell‐based hierarchical multiscale model is developed for 3D braided composite shafts. An analyzing method is developed to investigate the tubular yarn structures. A large‐scale torsional experimental scheme is developed.

Multiscale modeling of cutting processes for TiBw/TA15 composites based on the interface model parameters identification method by microcolumn compression

TiBw/TA15 composites are featured by the network-structured distribution of the reinforced phase quasi-continuously. The multiphase microstructure including the TiB ceramic whisker reinforcement, the titanium alloy matrix and the interface plays a significant role in cutting processes. In this paper, multiscale finite element (FE) cutting models in accordance with the structural characteristics of TiBw/TA15 composites are established. Then, the interfacial mechanical parameters of TiBw/TA15 composites are recognized by the microcolumn compression approach, and the thermoplastic equivalent constitutive parameters (ECP) of the TiBw-rich region in the model are obtained by a two-step FE simulation. The cutting model is validated by orthogonal turning experiments. Finally, the formation of segmented chips can be visualized by the cutting models, which is caused by the compressive stress and the TiB whiskers failure. In addition, the mapping relationship between the critical equivalent strain (CES) and the hardened layer depth (HLD) is established. The results indicate that the HLD reduces with the decrease of cutting speed (VC) and depth of cutting (h). The findings in this paper facilitate a further understanding of the TiBw/TA15 composites cutting mechanism.