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22 pages, 2386 KiB

Open AccessArticle

Micro–Macro Relationships in the Simulation of Wave Propagation Phenomenon Using the Discrete Element Method

by Jerzy Rojek, Nikhil Madan and Szymon Nosewicz

Materials 2019, 12(24), 4241; https://doi.org/10.3390/ma12244241 - 17 Dec 2019

Cited by 6 | Viewed by 2952

Abstract

The present work is aimed to investigate the capability of the discrete element method (DEM) to model properly wave propagation in solid materials, with special focus on the determination of elastic properties through wave velocities. Reference micro–macro relationships for elastic constitutive parameters have [...] Read more.

The present work is aimed to investigate the capability of the discrete element method (DEM) to model properly wave propagation in solid materials, with special focus on the determination of elastic properties through wave velocities. Reference micro–macro relationships for elastic constitutive parameters have been based on the kinematic hypothesis as well as obtained numerically by simulation of a quasistatic uniaxial compression test. The validity of these relationships in the dynamic analysis of the wave propagation has been checked. Propagation of the longitudinal and shear wave pulse in rectangular sample discretized with discs has been analysed. Wave propagation velocities obtained in the analysis have been used to determine elastic properties. Elastic properties obtained in the dynamic analysis have been compared with those determined by simulation of the quasistatic compression test. Full article

(This article belongs to the Special Issue Multi-scale Modeling of Materials and Structures)

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10 pages, 3512 KiB

Open AccessArticle

3D Viscoplastic Finite Element Modeling of Dislocation Generation in a Large Size Si Ingot of the Directional Solidification Stage

by Maohua Lin, Xinjiang Wu, Xinqin Liao, Min Shi, Disheng Ou and Chi-Tay Tsai

Materials 2019, 12(17), 2783; https://doi.org/10.3390/ma12172783 - 29 Aug 2019

Cited by 3 | Viewed by 2410

Abstract

Growing very large size silicon ingots with low dislocation density is a critical issue for the photovoltaic industry to reduce the production cost of the high-efficiency solar cell for affordable green energy. The thermal stresses, which are produced as the result of the [...] Read more.

Growing very large size silicon ingots with low dislocation density is a critical issue for the photovoltaic industry to reduce the production cost of the high-efficiency solar cell for affordable green energy. The thermal stresses, which are produced as the result of the non-uniform temperature field, would generate dislocation in the ingot. This is a complicated thermal viscoplasticity process during the cooling process of crystal growth. A nonlinear three-dimensional transient formulation derived from the Hassen-Sumino model (HAS) was applied to predict the number of dislocation densities, which couples the macroscopic viscoplastic deformation with the microscopic dislocation dynamics. A typical cooling process during the growth of very large size (G5 size: 0.84 m × 0.84 m × 0.3 m) Si ingot is used as an example to validate the developed HAS model and the results are compared with those obtained from qualitatively critical resolved shear stress model (CRSS). The result demonstrates that this finite element model not only predicts a similar pattern of dislocation generation with the CRSS model but also anticipate the dislocation density quantity generated in the Si ingot. A modified cooling process is also employed to study the effect of the cooling process on the generation of the dislocation. It clearly shows that dislocation density is drastically decreased by modifying the cooling process. The results obtained from this model can provide valuable information for engineers to design a better cooling process for reducing the dislocation density produced in the Si ingot under the crystal growth process. Full article

(This article belongs to the Special Issue Multi-scale Modeling of Materials and Structures)

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15 pages, 6449 KiB

Open AccessFeature PaperArticle

Analytical Thermal Modeling of Metal Additive Manufacturing by Heat Sink Solution

by Jinqiang Ning, Daniel E. Sievers, Hamid Garmestani and Steven Y. Liang

Materials 2019, 12(16), 2568; https://doi.org/10.3390/ma12162568 - 12 Aug 2019

Cited by 34 | Viewed by 5882

Abstract

Metal additive manufacturing can produce geometrically complex parts with effective cost. The high thermal gradients due to the repeatedly rapid heat and solidification cause defects in the produced parts, such as cracks, porosity, undesired residual stress, and part distortion. Different techniques were employed [...] Read more.

Metal additive manufacturing can produce geometrically complex parts with effective cost. The high thermal gradients due to the repeatedly rapid heat and solidification cause defects in the produced parts, such as cracks, porosity, undesired residual stress, and part distortion. Different techniques were employed for temperature investigation. Experimental measurement and finite element method-based numerical models are limited by the restricted accessibility and expensive computational cost, respectively. The available physics-based analytical model has promising short computational efficiency without resorting to finite element method or any iteration-based simulations. However, the heat transfer boundary condition cannot be considered without the involvement of finite element method or iteration-based simulations, which significantly reduces the computational efficiency, and thus the usefulness of the developed model. This work presents an explicit and closed-form solution, namely heat sink solution, to consider the heat transfer boundary condition. The heat sink solution was developed from the moving point heat source solution based on heat transfer of convection and radiation. The part boundary is mathematically discretized into many heats sinks due to the non-uniform temperature distribution, which causes non-uniform heat loss. The temperature profiles, thermal gradients, and temperature-affected material properties are calculated and presented. Good agreements were observed upon validation against experimental molten pool measurements. Full article

(This article belongs to the Special Issue Multi-scale Modeling of Materials and Structures)

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11 pages, 5756 KiB

Open AccessArticle

Efficient Calculation Methods for the Diffusion Coefficient of Interstitial Solutes in Dilute Alloys

by Xiaoshuang Wang, Jürgen Faßbender and Matthias Posselt

Materials 2019, 12(9), 1491; https://doi.org/10.3390/ma12091491 - 8 May 2019

Cited by 3 | Viewed by 2884

Abstract

In the example of oxygen diffusion in dilute ferritic iron alloys it is shown that the calculation of the diffusion coefficient can be separated into a contribution related to the migration in the interaction region between oxygen and the substitutional solute and a [...] Read more.

In the example of oxygen diffusion in dilute ferritic iron alloys it is shown that the calculation of the diffusion coefficient can be separated into a contribution related to the migration in the interaction region between oxygen and the substitutional solute and a part related to diffusion in pure body centered cubic (bcc) Fe. The corresponding diffusion times are determined by analytical expressions using Density-Functional-Theory (DFT) data for the respective binding energies. The diffusion coefficient in the interaction region must be determined by atomistic kinetic Monte Carlo (AKMC) simulations with DFT values for the migration barriers as input data. In contrast to previous calculations, AKMC simulation must only be performed for one concentration of the substitutional solute, and the obtained results can be employed to obtain data for other concentrations in a very efficient manner. This leads to a tremendous decrease of computational efforts. Under certain conditions it is even possible to use analytical expressions where merely DFT data for the binding energies are needed. The limits of applicability of the presented calculation procedures are discussed in detail. The methods presented in this work can be generalized to interstitial diffusion in other host materials with small concentrations of substitutional solutes. Full article

(This article belongs to the Special Issue Multi-scale Modeling of Materials and Structures)

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17 pages, 13564 KiB

Open AccessArticle

Modeling and Testing of the Sandwich Composite Manhole Cover Designed for Pedestrian Networks

by Calin Itu, Camelia Cerbu and Teofil-Florin Galatanu

Materials 2019, 12(7), 1114; https://doi.org/10.3390/ma12071114 - 3 Apr 2019

Cited by 3 | Viewed by 5030

Abstract

This research concerning the topic, pursues the design, manufacturing, analysis and testing of the manhole cover that may be used in pedestrian networks. Although there are certain commercially available manhole covers made of glass-reinforced composites, there are a few papers published related to [...] Read more.

This research concerning the topic, pursues the design, manufacturing, analysis and testing of the manhole cover that may be used in pedestrian networks. Although there are certain commercially available manhole covers made of glass-reinforced composites, there are a few papers published related to the modelling, simulation and mechanical testing of such parts. Herein, the manhole cover is made of the sandwich composite. The novelty of this kind of cover is to use an oriented strand board (OSB) as the core between two sides containing layers, which are reinforced with glass fibers. The OSB core leads to the increase of the stiffness-weight ratio. The paper describes the materials corresponding to the layers of the composite cover, geometry of the cover, technology used to manufacture the bending specimens and cover tested. Specimens made of materials that correspond to each layer of the cover, are tested in bending in order to determine their mechanical properties (flexural strength and flexural modulus). Bending tests and testing of the cover are also described. The composite manhole cover is also analysed by the finite element method to obtain the state of stresses and strains. The strains of the manhole cover are experimentally measured by using the tensometric method. Finally, comparing the strains with the strains experimentally measured validates the numerical simulation model. Full article

(This article belongs to the Special Issue Multi-scale Modeling of Materials and Structures)

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19 pages, 4090 KiB

Open AccessArticle

A Finite Element Model for Dynamic Analysis of Triple-Layer Composite Plates with Layers Connected by Shear Connectors Subjected to Moving Load

by Hoang-Nam Nguyen, Tan-Y. Nguyen, Ke Van Tran, Thanh Trung Tran, Truong-Thinh Nguyen, Van-Duc Phan and Thom Van Do

Materials 2019, 12(4), 598; https://doi.org/10.3390/ma12040598 - 16 Feb 2019

Cited by 36 | Viewed by 3841

Abstract

Triple-layered composite plates are created by joining three composite layers using shear connectors. These layers, which are assumed to be always in contact and able to move relatively to each other during deformation, could be the same or different in geometric dimensions and [...] Read more.

Triple-layered composite plates are created by joining three composite layers using shear connectors. These layers, which are assumed to be always in contact and able to move relatively to each other during deformation, could be the same or different in geometric dimensions and material. They are applied in various engineering fields such as ship-building, aircraft wing manufacturing, etc. However, there are only a few publications regarding the calculation of this kind of plate. This paper proposes novel equations, which utilize Mindlin’s theory and finite element modelling to simulate the forced vibration of triple-layered composite plates with layers connected by shear connectors subjected to a moving load. Moreover, a Matlab computation program is introduced to verify the reliability of the proposed equations, as well as the influence of some parameters, such as boundary conditions, the rigidity of the shear connector, thickness-to-length ratio, and the moving load velocity on the dynamic response of the composite plate. Full article

(This article belongs to the Special Issue Multi-scale Modeling of Materials and Structures)

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17 pages, 16292 KiB

Open AccessArticle

Atomic-Approach to Predict the Energetically Favored Composition Region and to Characterize the Short-, Medium-, and Extended-Range Structures of the Ti-Nb-Al Ternary Metallic Glasses

by Bei Cai, Jianbo Liu, Jiahao Li, Menghao Yang and Baixin Liu

Materials 2019, 12(3), 432; https://doi.org/10.3390/ma12030432 - 31 Jan 2019

Cited by 2 | Viewed by 3639

Abstract

Ab initio calculations were conducted to assist the construction of the n-body potential of the Ti-Nb-Al ternary metal system. Applying the constructed Ti-Nb-Al interatomic potential, molecular dynamics and Monte Carlo simulations were performed to predict a quadrilateral composition region, within which metallic glass [...] Read more.

Ab initio calculations were conducted to assist the construction of the n-body potential of the Ti-Nb-Al ternary metal system. Applying the constructed Ti-Nb-Al interatomic potential, molecular dynamics and Monte Carlo simulations were performed to predict a quadrilateral composition region, within which metallic glass was energetically favored to be formed. In addition, the amorphous driving force of those predicted possible glassy alloys was derived and an optimized composition around Ti₁₅Nb₄₅Al₄₀ was pinpointed, implying that this alloy was easier to be obtained. The atomic structure of Ti-Nb-Al metallic glasses was identified by short-, medium-, and extended-range analysis/calculations, and their hierarchical structures were responsible to the formation ability and unique properties in many aspects. Full article

(This article belongs to the Special Issue Multi-scale Modeling of Materials and Structures)

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8 pages, 2579 KiB

Open AccessArticle

Crystallization Behavior of Al₇₀Fe_12.5V_12.5Nb₅ Amorphous Alloy Formed by Mechanical Alloying

by Xuan Liu, Xingfu Wang, Yongli Si, Xiaokang Zhong and Fusheng Han

Materials 2019, 12(3), 383; https://doi.org/10.3390/ma12030383 - 26 Jan 2019

Cited by 6 | Viewed by 2916

Abstract

In this study, the formation and crystallization of the Al₇₀Fe_12.5V_12.5Nb₅ amorphous alloys has been investigated. The addition of Nb enhances the supercooled liquid region and glass forming ability of the Al-Fe-V amorphous alloys. The Al₇₀ [...] Read more.

In this study, the formation and crystallization of the Al₇₀Fe_12.5V_12.5Nb₅ amorphous alloys has been investigated. The addition of Nb enhances the supercooled liquid region and glass forming ability of the Al-Fe-V amorphous alloys. The Al₇₀Fe_12.5V_12.5Nb₅ amorphous alloy exhibits two distinct crystallization steps and a large supercooled liquid region at more than 100 K. Kissinger and Ozawa analyses showed that the two activation energies for crystallization (E_x) were estimated to be 366.3 ± 23.9 and 380.5 ± 23.9 kJ/mol. Large supercooled liquid regions are expected to gain an application field of Al-based amorphous alloys. Full article

(This article belongs to the Special Issue Multi-scale Modeling of Materials and Structures)

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18 pages, 2254 KiB

Open AccessArticle

Discrete Element Modeling of Intermetallic Matrix Composite Manufacturing by Powder Metallurgy

by Szymon Nosewicz, Jerzy Rojek, Marcin Chmielewski and Katarzyna Pietrzak

Materials 2019, 12(2), 281; https://doi.org/10.3390/ma12020281 - 16 Jan 2019

Cited by 8 | Viewed by 2923

Abstract

This paper presents a numerical and experimental analysis of manufacturing of intermetallic ceramic composites by powder metallurgy techniques. The scope of the paper includes the formulation and development of an original numerical model of powder metallurgy of two-phase material within the framework of [...] Read more.

This paper presents a numerical and experimental analysis of manufacturing of intermetallic ceramic composites by powder metallurgy techniques. The scope of the paper includes the formulation and development of an original numerical model of powder metallurgy of two-phase material within the framework of the discrete element method, simulations of powder metallurgy processes for different combinations of process parameters, and a verification of the numerical model based on own experimental results. Intermetallic-based composite NiAl–Al

_{2}

O

_{3}

has been selected as representative material for experimental and numerical studies in this investigation. Special emphasis was given to the interactions between the intermetallic and ceramic particles by formulating the special model for adhesive contact bond. In order to properly represent a real microstructure of a two-phase sintered body, a discrete element specimen was generated using a special algorithm. Numerical validation showed the correct numerical representation of a sintered two-phase composite specimen. Finally, micromechanical analysis was performed to explain the macroscopic behavior of the sintered sample. The evolution of the coordination number, a number of equilibrium contacts, and the distribution of the cohesive neck size with respect to time are presented. Full article

(This article belongs to the Special Issue Multi-scale Modeling of Materials and Structures)

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10 pages, 3753 KiB

Open AccessArticle

Dual Toroidal Dipole Resonance Metamaterials under a Terahertz Domain

by Shuang Wang, Song Wang, Quan Li, Xiaoli Zhao and Jianyu Zhu

Materials 2018, 11(10), 2036; https://doi.org/10.3390/ma11102036 - 19 Oct 2018

Cited by 14 | Viewed by 4163

Abstract

We proposed and fabricated a flexible, planar, U-shape-modified structure metamaterial (MM) that was composed of two metallic pattern layers separated by a polyimide layer, where each metallic pattern layer consists of two U-shaped split ring resonators (USRRs). The coupling effect between the two [...] Read more.

We proposed and fabricated a flexible, planar, U-shape-modified structure metamaterial (MM) that was composed of two metallic pattern layers separated by a polyimide layer, where each metallic pattern layer consists of two U-shaped split ring resonators (USRRs). The coupling effect between the two USRRs in the same metallic layer was vital to the formation of dual toroidal dipole (TD) resonances. The measured and simulated results showed that both low quality factor (Q) (~1.82) and high Q (~10.31) TD resonances were acquired synchronously at two different frequencies in the MMs by adjusting the distance between the two coplanar USRRs. With the interaction of the USRRs, the energy levels of the USRRs were split into inductance-capacitance (LC)-induced TD resonance at low frequency and dipole-induced TD resonance at high frequency. Thus, the electric multipole interaction played an important role in determining the energy level of the TD resonance. The better strength of the high frequency TD resonance can be confined to an electromagnetic field inside a smaller circular region, and thus, a higher Q was obtained. In order to investigate the TD mechanism more in depth, the power of the electric dipole, magnetic dipole, electric circular dipole, and TD were quantitatively calculated. Dual TD MMs on a freestanding substrate will have potential applications in functional terahertz devices for practical applications. Full article

(This article belongs to the Special Issue Multi-scale Modeling of Materials and Structures)

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11 pages, 1730 KiB

Open AccessArticle

Mechanical and Thermal Conductivity Properties of Enhanced Phases in Mg-Zn-Zr System from First Principles

by Shuo Wang, Yuhong Zhao, Huijun Guo, Feifei Lan and Hua Hou

Materials 2018, 11(10), 2010; https://doi.org/10.3390/ma11102010 - 17 Oct 2018

Cited by 10 | Viewed by 3531

Abstract

In this paper, the mechanical properties and minimum thermal conductivity of ZnZr, Zn₂Zr, Zn₂Zr₃, and MgZn₂ are calculated from first principles. The results show that the considered Zn-Zr intermetallic compounds are effective strengthening phases compared to [...] Read more.

In this paper, the mechanical properties and minimum thermal conductivity of ZnZr, Zn₂Zr, Zn₂Zr₃, and MgZn₂ are calculated from first principles. The results show that the considered Zn-Zr intermetallic compounds are effective strengthening phases compared to MgZn₂ based on the calculated elastic constants and polycrystalline bulk modulus B, shear modulus G, and Young’s modulus E. Meanwhile, the strong Zn-Zr ionic bondings in ZnZr, Zn₂Zr, and Zn₂Zr₃ alloys lead to the characteristics of a higher modulus but lower ductility than the MgZn₂ alloy. The minimum thermal conductivity of ZnZr, Zn₂Zr, Zn₂Zr₃, and MgZn₂ is 0.48, 0.67, 0.68, and 0.49 W m⁻¹ K⁻¹, respectively, indicating that the thermal conductivity of the Mg-Zn-Zr alloy could be improved as the precipitation of Zn atoms from the α-Mg matrix to form the considered Zn-Zr binary alloys. Based on the analysis of the directional dependence of the minimum thermal conductivity, the minimum thermal conductivity in the direction of [110] can be identified as a crucial short limit for the considered Zn-Zr intermetallic compounds in Mg-Zn-Zr alloys. Full article

(This article belongs to the Special Issue Multi-scale Modeling of Materials and Structures)

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9 pages, 1965 KiB

Open AccessArticle

Cellular Automaton Simulation of the Growth of Anomalous Eutectic during Laser Remelting Process

by Lei Wei, Yongqing Cao, Xin Lin and Weidong Huang

Materials 2018, 11(10), 1844; https://doi.org/10.3390/ma11101844 - 27 Sep 2018

Cited by 3 | Viewed by 2964

Abstract

Anomalous eutectic morphologies were observed during laser remelting of a Ni-Sn powder bed, where it was sandwiched between a lamellar eutectic at the bottom of melt pool. That is, the anomalous eutectic growth mechanism can be divided into two processes: one is the [...] Read more.

Anomalous eutectic morphologies were observed during laser remelting of a Ni-Sn powder bed, where it was sandwiched between a lamellar eutectic at the bottom of melt pool. That is, the anomalous eutectic growth mechanism can be divided into two processes: one is the lamellar to anomalous transition (LAT); the other is the anomalous to lamellar transition (ALT). The thermal distribution at the bottom of melt pool is simulated by the finite difference method. It is found that the cooling rate at the bottom of melt pool is a linear function of time. A cellular automaton (CA) model is developed to simulate the anomalous growth. Simulation results show that the mechanism of the LAT is that one phase overgrows the other followed by subsequent nucleating of the other phase. The mechanism of the ALT is the competitive growth between the anomalous and lamellar eutectic; as the cooling rate increased, the lamellar eutectic is more competitive. Full article

(This article belongs to the Special Issue Multi-scale Modeling of Materials and Structures)

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24 pages, 4662 KiB

Open AccessArticle

A Study of Strain-Driven Nucleation and Extension of Deformed Grain: Phase Field Crystal and Continuum Modeling

by Ling-yi Kong, Ying-jun Gao, Qian-qian Deng, Zhi-rong Luo and Yu-jiang Lu

Materials 2018, 11(10), 1805; https://doi.org/10.3390/ma11101805 - 23 Sep 2018

Cited by 12 | Viewed by 3301

Abstract

The phase-field-crystal (PFC) method is used to investigate migration of grain boundary dislocation and dynamic of strain-driven nucleation and growth of deformed grain in two dimensions. The simulated results show that the deformed grain nucleates through forming a gap with higher strain energy [...] Read more.

The phase-field-crystal (PFC) method is used to investigate migration of grain boundary dislocation and dynamic of strain-driven nucleation and growth of deformed grain in two dimensions. The simulated results show that the deformed grain nucleates through forming a gap with higher strain energy between the two sub-grain boundaries (SGB) which is split from grain boundary (GB) under applied biaxial strain, and results in the formation of high-density ensembles of cooperative dislocation movement (CDM) that is capable of plastic flow localization (deformed band), which is related to the change of the crystal lattice orientation due to instability of the orientation. The deformed grain stores the strain energy through collective climbing of the dislocation, as well as changing the orientation of the original grain. The deformed grain growth (DGG) is such that the higher strain energy region extends to the lower strain energy region, and its area increase is proportional to the time square. The rule of the time square of the DGG can also be deduced by establishing the dynamic equation of the dislocation of the strain-driven SGB. The copper metal is taken as an example of the calculation, and the obtained result is a good agreement with that of the experiment. Full article

(This article belongs to the Special Issue Multi-scale Modeling of Materials and Structures)

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21 pages, 6302 KiB

Open AccessArticle

Numerical Modelling of the Effect of Filler/Matrix Interfacial Strength on the Fracture of Cementitious Composites

by Xiaowei Ouyang, Zichao Pan, Zhiwei Qian, Yuwei Ma, Guang Ye and Klaas Van Breugel

Materials 2018, 11(8), 1362; https://doi.org/10.3390/ma11081362 - 6 Aug 2018

Cited by 9 | Viewed by 3203

Abstract

The interface between filler and hydration products can have a significant effect on the mechanical properties of the cement paste system. With different adhesion properties between filler and hydration products, the effect of microstructural features (size, shape, surface roughness), particle distribution and area [...] Read more.

The interface between filler and hydration products can have a significant effect on the mechanical properties of the cement paste system. With different adhesion properties between filler and hydration products, the effect of microstructural features (size, shape, surface roughness), particle distribution and area fraction of filler on the fracture behavior of a blended cement paste system is supposed to be different, as well. In order to understand the effect of the microstructural features, particle distribution and area fraction of filler on the fracture behavior of a blended cement paste system with either strong or weak filler-matrix interface, microscale simulations with a lattice model are carried out. The results show that the strength of the filler-matrix interface plays a more important role than the microstructural features, particle distribution and area fraction of filler in the crack propagation and the strength of blended cement paste. The knowledge acquired here provides a clue, or direction, for improving the performance of existing fillers. To improve the performance of fillers in cement paste in terms of strength, priority should be given to improving the bond strength between filler particles and matrix, not to modifying the microstructural features (i.e., shape and surface roughness) of the filler. Full article

(This article belongs to the Special Issue Multi-scale Modeling of Materials and Structures)

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Review

Jump to: Research

46 pages, 12186 KiB

Open AccessReview

Computational Multiscale Solvers for Continuum Approaches

by Francisco Montero-Chacón, José A. Sanz-Herrera and Manuel Doblaré

Materials 2019, 12(5), 691; https://doi.org/10.3390/ma12050691 - 26 Feb 2019

Cited by 10 | Viewed by 4623

Abstract

Computational multiscale analyses are currently ubiquitous in science and technology. Different problems of interest—e.g., mechanical, fluid, thermal, or electromagnetic—involving a domain with two or more clearly distinguished spatial or temporal scales, are candidates to be solved by using this technique. Moreover, the predictable [...] Read more.

Computational multiscale analyses are currently ubiquitous in science and technology. Different problems of interest—e.g., mechanical, fluid, thermal, or electromagnetic—involving a domain with two or more clearly distinguished spatial or temporal scales, are candidates to be solved by using this technique. Moreover, the predictable capability and potential of multiscale analysis may result in an interesting tool for the development of new concept materials, with desired macroscopic or apparent properties through the design of their microstructure, which is now even more possible with the combination of nanotechnology and additive manufacturing. Indeed, the information in terms of field variables at a finer scale is available by solving its associated localization problem. In this work, a review on the algorithmic treatment of multiscale analyses of several problems with a technological interest is presented. The paper collects both classical and modern techniques of multiscale simulation such as those based on the proper generalized decomposition (PGD) approach. Moreover, an overview of available software for the implementation of such numerical schemes is also carried out. The availability and usefulness of this technique in the design of complex microstructural systems are highlighted along the text. In this review, the fine, and hence the coarse scale, are associated with continuum variables so atomistic approaches and coarse-graining transfer techniques are out of the scope of this paper. Full article

(This article belongs to the Special Issue Multi-scale Modeling of Materials and Structures)

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