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The development of new type of hydrogels featuring enhanced properties is a recent topic and of particular interest for a wide range of industrial applications, especially in biomedical sector. The present contribution focuses on the mutliscale modeling of hydrogels that are treated by the enzymatic mineralization and thus enriched with calcium phosphate. More specifically, the calcium phosphate forms spherical or honeycomb structures in the hydrogel matrix, which significantly improves effective material properties such as stiffness and strength. The chosen multiscale finite element method (FEM) homogenization strategy uses the Hill–Mandel macrohomogeneity condition for bridging two scales: the macroscopic boundary value problem (BVP) simulates the specimen behavior, whereas the microscopic BVP investigates the representative volume element (RVE) depicting the heterogeneous multiphase microstructure. The approach proposed uses the Ogden model to simulate the hydrogel and the neo‐Hooke model for the calcium phosphate phase. It varies the RVE type and the macroscopic tests in order to study the influence of the microstructure on the effective behavior and uses experimental data to determine missing microscopic material parameters. Chosen numerical examples demonstrate the applicability of the numerical tool for the estimation of the optimal microscopic arrangement of phases. [ABSTRACT FROM AUTHOR]

The 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. [ABSTRACT FROM AUTHOR]

An optimal adaptive multiscale finite element method(AMsFEM) for numerical solutions of flow problems modelled by the Oldroyd B model is developed. Complex flows experience instabilities due to a phenomena known as the high Weissenberg number problem. The stabilisers are terms in-cooperated into the variational formulation when applying the finite element method. For the selected few stabilisers, numerical experiments are performed to study the convergence of the solutions. These demonstrate that adaptive strategies reduce the computational load of flow simulation. A best performing combination of choice of stabiliser and adaptive strategy is suggested.

Abstract The 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.

The randomness and multi-level structures inherent to porous composites with open-pore make it difficult to establish equivalent geometric models at different scales for multi-scale simulations. This paper presents a combined experimental and simulation approach to the preparation, structural analysis and multiscale simulation of quartz fibre-reinforced phenolic composites. The elementalized open-pore porous models with the Knudsen effect, the random fibre yarn models and the random fibre felt models have been established and assembled into a composite structural model after homogenization. The thermal conductivity parameters of the porous model are calculated and transferred to the fibre yarn and fibre felt models for simulation. Thereafter, the thermal conductivity parameters of the three models are transferred to the composite structure model and simulated to obtain its equivalent thermal conductivity. The experimental and simulation results demonstrate that the introduction of the Knudsen effect can reduce the simulation error of the composite structure model by an order of magnitude. In combination with the random contact characteristics of the yarns, the sequential multiscale finite element heat transfer simulation with an error of 0.5 % can be achieved. [Display omitted] • The porous structures are connected by shortest connection path algorithm. • Knudsen effect has been introduced into the open-pore porous model for the first time. • Multi-scale simulation was finished based on three most fundamental material parameters. [ABSTRACT FROM AUTHOR]

We study the numerical approximation of advection–diffusion equations with highly oscillatory coefficients and possibly dominant advection terms by means of the Multiscale Finite Element Method (MsFEM). The latter method is a now classical, finite element type method that performs a Galerkin approximation on a problem-dependent basis set, itself precomputed in an offline stage. The approach is implemented here using basis functions that locally resolve both the diffusion and the advection terms. Variants with additional bubble functions and possibly weak inter-element continuity are proposed. Some theoretical arguments and a comprehensive set of numerical experiments allow to investigate and compare the stability and the accuracy of the approaches. The best approach constructed is shown to be adequate for both the diffusion- and advection-dominated regimes, and does not rely on an auxiliary stabilization parameter that would have to be properly adjusted. [ABSTRACT FROM AUTHOR]

Due to the outstanding performance, composite materials are widely used and analyzing their properties and designing them based on performance has become a crucial task in the field of many manufacturing industries. Composite materials possess complex multiscale structures, and traditional fine-scale finite element modeling and analysis may lead to severe computational resource challenges. To overcome this difficulty, breakthroughs in key technologies of multiscale accelerated analysis algorithms are required. In this study, an innovative approach based on artificial intelligence and multiscale finite element method is presented. This approach involves partitioning the entire composite material structure into coarse grids that resemble homogenous structures of similar size, providing results consistent to fine-grid finite element analysis. By utilizing CNN for image feature recognition and employing the CGAN adversarial method, coarse-grid equivalent stiffness matrices and multiscale shape functions from completely random microstructures of composite materials can be obtained. Consequently, this enables a rapid response process from microstructure to low-resolution grid to high-resolution physical field, with remarkably accurate physical field results. Moreover, compared to traditional fine-grid finite element methods, this approach significantly reduces memory usage and computation time. This method is applicable to composite materials with varying shaped inclusions, different component properties, and diverse geometric distributions, allowing these materials to perform high-fidelity finite element calculations on coarse grids and predict their mechanical behavior. Furthermore, this breakthrough opens avenues for accelerating the optimization design of composite materials with diverse mechanical functionalities, by employing a bottom-up approach. [ABSTRACT FROM AUTHOR]

A cornerstone of computational solid mechanics in the context of digital transformation are databases for microstructures obtained from advanced tomography techniques. Uniform discretizations of pixelized images in 2D are the raw‐data point of departure for simulation analyses. This paper proposes the concept of a unified error analysis for image‐based microstructure representations in uniform resolution along with adaptively coarsened discretizations. The analysis distinguishes between a resolution error due to finite, possibly coarsened image resolution and a discretization error, and investigates their quantitative relation, spatial distributions and their impacts on the simulation results both on the microscale and the macroscale in the context of computational homogenization. The assessment of accuracy and efficiency is carried out for an exemplary two‐phase material. Beyond the example considered here the concept is a rational tool in the transformation of raw image data into microstructure information adapted to particular simulation needs and endows the digital twin of real microstructures with validated characteristics for reliable, predictive simulations. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we present new upscaled HDG methods for Brinkman equations in the context of high-contrast heterogeneous media. The a priori error estimates are derived in terms of both fine and coarse scale parameters that depend on the high-contrast coefficient weakly. Due to the heterogeneity of the problem, a huge global system will be produced after the numerical discretization of HDG method. Thanks to the upscaled structure of the proposed methods, we are able to reduce the huge global system onto the skeleton of the coarse mesh only while still capturing important fine scale features of this problem. The finite element space over the coarse mesh is irrelevant to the fine scale computation. This feature makes our proposed method very attractive. Several numerical examples are presented to support our theoretical findings. [ABSTRACT FROM AUTHOR]

Abstract: The present contribution focuses on the application of the multiscale finite element method to the modeling of actin networks that are embedded in the cytosol. These cell components are of particular importance with regard to the cell response to external stimuli. The homogenization strategy chosen uses the Hill‐Mandel macrohomogeneity condition for bridging 2 scales: the macroscopic scale that is related to the cell level and the microscopic scale related to the representative volume element. For the modeling of filaments, the Holzapfel‐Ogden β‐model is applied. It provides a relationship between the tensile force and the caused stretches, serves as the basis for the derivation of the stress and elasticity tensors, and enables a novel finite element implementation. The elements with the neo‐Hookean constitutive law are applied for the simulation of the cytosol. The results presented corroborate the main advantage of the concept, namely, its flexibility with regard to the choice of the representative volume element as well as of macroscopic tests. The focus is particularly placed on the study of the filament orientation and of its influence on the effective behavior. [ABSTRACT FROM AUTHOR]

Over the past decade, the previous hysteretic recognition of the fire threat to bridges has provoked significant progress in understanding the fire performance of bridges. However, despite the growing maritime shipment of combustibles, scarce studies have contributed to evaluating the safety of steel cable-stayed bridges subjected to under-girder ship fires. Limited existing knowledge is still based upon the less precise temperature curve method and flame geometry modeling. This study developed an advanced numerical method to more accurately evaluate the response of steel cable-stayed bridges subjected to ship fires. First, the computational fluid dynamics (CFD) approach was used to reproduce realistic fire conditions. An efficient thermal boundary, adiabatic surface temperature, was then transferred to the thermal finite element model of affected girder segments for calculating the temperature propagation. Lastly, this paper used the multiscale structural finite element method to incorporate the local performance variation of exposed girders into the response of the entire bridge. The proposed method was validated by predicting a thermomechanical response of the steel beam exposed to realistic underneath fires, agreeing well with the data from two large-scale structural fire experiments by NIST. Based on the method, the response of a typical steel cable-stayed bridge subjected to ship fires was simulated. Results show that fires can introduce local damage to the girder segments above the fire and alter the internal force distribution of the entire bridge. The bending moment of the girder sections over the fire significantly increases, and the flexural characteristic of pylons is also altered. Both thermal convection and radiation contribute to the temperature increment, and the latter dominates at the locations above fires. Adopting temperature curves can overestimate or underestimate the fire impact, introducing uncertainties to the safety of steel cable-stayed bridges in ship fires and can cause unrealistic stress concentration on the border of heated and unheated domains of exposed floors. The developed advanced method is more competent in capturing the inhomogeneous temperature and stress distribution. • CFD fire models were coupled with FEM for steel cable-stayed bridges in ship fires. • Thermal radiation governs the heat absorbed by the girder floor exposed to ship fires. • Temperatures of the girder floor propagate from the central region to the periphery. • Ship fires bring local damage to the girder and alter the entire structural behavior. • CFD-FEM method is more competent than traditional temperature-curve methods. [ABSTRACT FROM AUTHOR]

This paper applies topology optimisation to the design of structures with periodic and layered microstructural details without length scale separation, i.e. considering the complete macroscopic structure and its response, while resolving all microstructural details, as compared to the often used homogenisation approach. The approach takes boundary conditions into account and ensures connected and macroscopically optimised microstructures regardless of the difference in micro- and macroscopic length scales. This results in microstructures tailored for specific applications rather than specific properties. Manufacturability is further ensured by the use of robust topology optimisation. Dealing with the complete macroscopic structure and its response is computationally challenging as very fine discretisations are needed in order to resolve all microstructural details. Therefore, this paper shows the benefits of applying a contrast-independent spectral preconditioner based on the multiscale finite element method (MsFEM) to large structures with fully-resolved microstructural details. It is shown that a single preconditioner can be reused for many design iterations and used for several design realisations, in turn leading to massive savings in computational cost. The density-based topology optimisation approach combined with a Heaviside projection filter and a stochastic robust formulation is used on various problems, with both periodic and layered microstructures. The presented approach is shown to allow for the topology optimisation of very large problems in Matlab , specifically a problem with 26 million displacement degrees of freedom in 26 hours using a single computational thread. [ABSTRACT FROM AUTHOR]

SUMMARY This paper presents a general coupling extended multiscale FEM (GCEMs) for solving the coupling problem of elasto-plastic consolidation of heterogeneous saturated porous media. In the GCEMs, the numerical multiscale base functions for the solid skeleton and fluid phase of the coupling system are all constructed on the basis of the equivalent stiffness matrix of the unit cell, which not only contain the interaction between the solid and fluid phases but also consider the time effect. Furthermore, in order to improve the computational accuracy for two-dimensional problems, a multi-node coarse element strategy for the GCEMs is proposed, and a two-scale iteration algorithm for the elasto-plastic consolidation analysis is developed. Some one-dimensional and two-dimensional homogeneous and heterogeneous numerical examples are carried out to validate the proposed method through the comparison with the coupling multiscale FEM and standard FEM. Numerical results show that the newly developed GCEMs can almost preserve the same convergent property as the standard FEM and also possesses the advantages of high computational efficiency. In addition, the GCEMs can be easily applied to other coupling multifield and multiphase transient problems. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We introduce a finite element method for numerical upscaling of second order elliptic equations with highly heterogeneous coefficients. The method is based on a mixed formulation of the problem and the concepts of the domain decomposition and the hybrid discontinuous Galerkin methods. The method utilizes three different scales: (1) the scale of the partition of the domain of the problem, (2) the scale of partition of the boundaries of the subdomains (related to the corresponding space of Lagrange multipliers), and (3) the fine-grid scale that is assumed to resolve the scale of the heterogeneous variation of the coefficients. Our proposed method gives a flexible framework that (1) couples independently generated multiscale basis functions in each coarse patch, (2) provides a stable global coupling independent of local discretization, physical scales, and contrast, and (3) allows avoiding any constraints [Arbogast et al., Multiscale Model. Simul., 6 (2007), pp. 319-346] on coarse spaces. In this paper, we develop and study a multiscale HDG method that uses polynomial and homogenization-based multiscale spaces. These coarse spaces are designed for problems with scale separation. In our subsequent paper, we plan to extend our flexible HDG framework to more challenging multiscale problems with nonseparable scales and high contrast and consider enriched coarse spaces that use appropriate local spectral problems. [ABSTRACT FROM AUTHOR]

This contribution deals with the application of the inverse homogenization method to the determination of geometrical properties of cancellous bone. The approach represents a combination of an extended version of the Marquardt-Levenberg method with the multiscale finite element method. The former belongs to the group of gradient-based optimization strategies, while the latter is a numerical homogenization method, suitable for the modeling of materials with a highly heterogeneous microstructure. The extension of the Marquardt-Levenberg method is concerned with the selection strategy for distinguishing the global minimum from the plethora of local minima. Within the numerical examples, the bone is modeled as a biphasic viscoelastic medium and three different representative volume elements are taken into consideration. Different models enable the simulation of the bone either as a purely isotropic or as a transversally anisotropic medium. Main geometrical properties of trabeculae are determined from data on effective shear modulus but alternative schemes are also possible. [ABSTRACT FROM AUTHOR]

The model proposed in this paper is based on the fact that the reflection might have a significant contribution to the attenuation of the acoustic waves propagating through the cancellous bone. The numerical implementation of the mentioned effect is realized by the development of a new representative volume element that includes an infinitesimally thin 'transient' layer on the contact surface of the bone and the marrow. This layer serves to model the amplitude transformation of the incident wave by the transition through media with different acoustic impedances and to take into account the energy loss due to the reflection. The proposed representative volume element together with the multiscale finite element is used to simulate the wave propagation and to evaluate the attenuation coefficient for samples with different effective densities in the dependence of the applied excitation frequency. The obtained numerical values show a very good agreement with the experimental results. Moreover, the model enables the determination of the upper and the lower bound for the attenuation coefficient. [ABSTRACT FROM AUTHOR]

Abstract: This contribution deals with the modeling of viscoelastic and shrinkage effects accompanying the curing of polymers at multiple length scales. For the modeling of viscous effects, the deformation at the microlevel is decomposed into an elastic and a viscoelastic part, and a corresponding energy density consisting of equilibrium and non-equilibrium parts is proposed. The former is related to the total deformation; it has the form of a convolution integral and depends on the time evolution of the material parameters. The non-equilibrium part depends on the elastic part of deformations only. The material parameters are constant in time, thus an integral form is not necessary. In contrast to the viscous effects, the modeling of shrinkage effects does not require any further extension of the expression for the energy density, but an additional decomposition of the deformation into a shrinkage and a mechanical part. Since the material compressibility is taken into consideration, a multifield formulation is applied for the equilibrium as well as for the non-equilibrium energy part. As a final aspect, the paper includes a study of macroheterogenous polymers for whose modeling the multiscale finite element method is applied. Within this numerical approach, a macroscopic body is treated as a homogeneous body whose effective properties are evaluated on the basis of the simulations which are carried out at the level of the representative volume element. The application of the model proposed is illustrated on the basis of examples studying the influence of individual parameters on the stress state as well as the influence of the volume fraction of different phases at the microscale on the effective material behavior. [Copyright &y& Elsevier]

Abstract: In many cases, the microstructure of composite materials is not known and cannot directly be accessed such that an inverse analysis is necessary for its investigation. This approach requires the implementation of two tools: an optimization method for the minimization of the error problem and a mechanical approach for the solution of the direct problem, i.e. the simulation of composite materials. Our particular choice deals with the combination of the Levenberg–Marquardt method with the multiscale finite element method. The numerical examples are concerned with the investigation of the elastic parameters for two-phase materials. Emphasis is placed on the discussion of convergence and sensitivity with respect to the initial guess. [Copyright &y& Elsevier]

Abstract: This paper provides a continuum mechanical model for the curing of polymers, including the incompressibility effects arising at the late stages of the process. For this purpose, the free energy density functional is split into a deviatoric and a volumetric part, and a multifield formulation is inserted. An integral formulation of the functional is used to depict the time-dependent material behavior. The model is also coupled with the multiscale finite element method, a numerical approach serving for the modeling of heterogeneous materials with a highly oscillatory microstructure. The effects of the proposed extensions are illustrated on the basis of several numerical examples concerned with the study of the influence of Poisson’s ratio on the curing process and the behavior of the microheterogeneous polymers. [Copyright &y& Elsevier]

The article deals with the contribution of reflection effects to the attenuation properties of cancellous bone. The bone behaviour is simulated by the multiscale finite element method, a numerical homogenization approach, suitable for the modelling of heterogeneous material with a highly oscillatory microstructure. The focus is on the modelling of a novel type of the representative volume element, which apart from the solid framework filled with fluid marrow also includes an infinitesimally thin ‘transition’ layer at the contact of the phases. The mentioned layer is implemented in order to simulate the amplitude transformation of an incident wave and to take the loss of energy caused by the reflection into account. The given numerical examples consider the simulation of wave propagation through a sample while the excitation frequency is varied. The numerical values are compared with the results which are determined without considering the reflection, in order to point out the contribution of the newly introduced phenomenon. [ABSTRACT FROM AUTHOR]

This paper deals with the application of the multiscale finite element method for simulating the cancellous bone. For this purpose, two types of biphasic representative volume elements are proposed. In the first one, the solid frame consists of thin walls simulated by shell elements. On the other hand, the solid phase of the second model is made up of columns consisting of eight-node brick elements. This choice of representative volume elements is motivated by experimental investigations reporting on the existence of plate-like and rode-like types of cancellous bone and possible conversions between them. The proposed representative volume elements are used to calculate effective material tensors and parameters and to investigate their change in terms of increasing porosity, which is typical for osteoporosis. As a first example, changes in the geometry of the representative volume elements are used to explore material anisotropy. In the end, the final example considers wave propagation through the bone treated as a homogenized medium. [ABSTRACT FROM AUTHOR]

Abstract: A multiscale generalised hp-finite element method (MSFEM) for time harmonic wave propagation in bands of locally periodic media of large, but finite extent, e.g., photonic crystal (PhC) bands, is presented. The method distinguishes itself by its size robustness, i.e., to achieve a prescribed error its computational effort does not depend on the number of periods. The proposed method shows this property for general incident fields, including plane waves incident at a certain angle to the infinite crystal surface, and at frequencies in and outside of the bandgap of the PhC. The proposed MSFEM is based on a precomputed problem adapted multiscale basis. This basis incorporates a set of complex Bloch modes, the eigenfunctions of the infinite PhC, which are modulated by macroscopic piecewise polynomials on a macroscopic FE mesh. The multiscale basis is shown to be efficient for finite PhC bands of any size, provided that boundary effects are resolved with a simple macroscopic boundary layer mesh. The MSFEM, constructed by combing the multiscale basis inside the crystal with some exterior discretisation, is a special case of the generalised finite element method (g-FEM). For the rapid evaluation of the matrix entries we introduce a size robust algorithm for integrals of quasi-periodic micro functions and polynomial macro functions. Size robustness of the present MSFEM in both, the number of basis functions and the computation time, is verified in extensive numerical experiments. [Copyright &y& Elsevier]

Strain localization analysis based on finite element method (FEM) usually requires an intensive computation to capture an accurate shear band and the limit stress, especially in the heterogeneous problems, and thus costs much computational resource and time. This paper proposes an adaptive multiscale FEM (AMsFEM) to improve the computational efficiency of strain localization analysis in heterogeneous solids. In the multiscale analysis, h - and p -adaptive strategies are proposed to update the fine and coarse meshes, respectively. The problem of mesh dependence is handled by the Cosserat continuum theory. In the fine-scale adaptive procedure, triangular elements are taken to discretize the fine-scale domain, and the newest vertex bisection is utilized for refinement based on the gradient of displacement. In the coarse-scale adaptive procedure, a multi-node coarse element technique is considered. By introducing a probability density function for each side of a coarse element, the optimal positions of the newly added coarse nodes can be determined. With the proposed adaptive multiscale procedure, the computational DOFs are reduced smartly and massively. Three representative heterogeneous examples demonstrate that the proposed method can accurately capture shear bands, with an improved computational efficiency and robust convergence. • An adaptive multiscale FEM with Cosserat model is proposed for the heterogeneous localization problems. • The h - and p -adaptive strategies are proposed to update the fine and coarse meshes respectively. • Two-scale adaptive computational framework is developed to deal with the nonlinear problems. • The proposed method exhibits high efficiency and accuracy. [ABSTRACT FROM AUTHOR]