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Journal articles on the topic 'Multiscale homogenization'

Author: Grafiati

Published: 4 June 2021

Last updated: 8 June 2024

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1

NGUYEN, VINH PHU, MARTIJN STROEVEN, and LAMBERTUS JOHANNES SLUYS. "MULTISCALE CONTINUOUS AND DISCONTINUOUS MODELING OF HETEROGENEOUS MATERIALS: A REVIEW ON RECENT DEVELOPMENTS." Journal of Multiscale Modelling 03, no. 04 (December 2011): 229–70. http://dx.doi.org/10.1142/s1756973711000509.

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This paper reviews the recent developments in the field of multiscale modelling of heterogeneous materials with emphasis on homogenization methods and strain localization problems. Among other topics, the following are discussed (i) numerical homogenization or unit cell methods, (ii) continuous computational homogenization for bulk modelling, (iii) discontinuous computational homogenization for adhesive/cohesive crack modelling and (iv) continuous-discontinuous computational homogenization for cohesive failures. Different boundary conditions imposed on representative volume elements are described. Computational aspects concerning robustness and computational cost of multiscale simulations are presented.

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2

Lukkassen, Dag, Annette Meidell, and Peter Wall. "Multiscale homogenization of monotone operators." Discrete & Continuous Dynamical Systems - A 22, no. 3 (2008): 711–27. http://dx.doi.org/10.3934/dcds.2008.22.711.

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3

Sokolov, Alexander Pavlovich, and Anton Yurievich Pershin. "Computer-Aided Design of Composite Materials Using Reversible Multiscale Homogenization and Graph-Based Software Engineering." Key Engineering Materials 779 (September 2018): 11–18. http://dx.doi.org/10.4028/www.scientific.net/kem.779.11.

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In this work, a new software for computer-aided design of composite materials with predefined thermomechanical properties is presented in case of incomplete input data. The mathematical basis of underlying computational method of the properties identification is a modified method of multiscale homogenization named reversible multiscale homogenization method. The system has a modular architecture and includes software implementation of the reversible multiscale homogenization method based on a new technique of construction of software implementations of complex computational methods. The latter was named «Graph-based software engineering» (GBSE) and is based on category and graph theories. The corresponding numerical and experimental results were obtained and compared. The expediency of GBSE approach is discussed for the case of the development of complex computational methods required when solving the applied problems of the design of new heterogeneous materials.

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4

Hedjazian, N., Y. Capdeville, and T. Bodin. "Multiscale seismic imaging with inverse homogenization." Geophysical Journal International 226, no. 1 (March 27, 2021): 676–91. http://dx.doi.org/10.1093/gji/ggab121.

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Summary Seismic imaging techniques such as elastic full waveform inversion (FWI) have their spatial resolution limited by the maximum frequency present in the observed waveforms. Scales smaller than a fraction of the minimum wavelength cannot be resolved, and only a smoothed, effective version of the true underlying medium can be recovered. These finite-frequency effects are revealed by the upscaling or homogenization theory of wave propagation. Homogenization aims at computing larger scale effective properties of a medium containing small-scale heterogeneities. We study how this theory can be used in the context of FWI. The seismic imaging problem is broken down in a two-stage multiscale approach. In the first step, called homogenized FWI (HFWI), observed waveforms are inverted for a smooth, fully anisotropic effective medium, that does not contain scales smaller than the shortest wavelength present in the wavefield. The solution being an effective medium, it is difficult to directly interpret it. It requires a second step, called downscaling or inverse homogenization, where the smooth image is used as data, and the goal is to recover small-scale parameters. All the information contained in the observed waveforms is extracted in the HFWI step. The solution of the downscaling step is highly non-unique as many small-scale models may share the same long wavelength effective properties. We therefore rely on the introduction of external a priori information, and cast the problem in a Bayesian formulation. The ensemble of potential fine-scale models sharing the same long wavelength effective properties is explored with a Markov chain Monte Carlo algorithm. We illustrate the method with a synthetic cavity detection problem: we search for the position, size and shape of void inclusions in a homogeneous elastic medium, where the size of cavities is smaller than the resolving length of the seismic data. We illustrate the advantages of introducing the homogenization theory at both stages. In HFWI, homogenization acts as a natural regularization helping convergence towards meaningful solution models. Working with fully anisotropic effective media prevents the leakage of anisotropy induced by the fine scales into isotropic macroparameters estimates. In the downscaling step, the forward theory is the homogenization itself. It is computationally cheap, allowing us to consider geological models with more complexity (e.g. including discontinuities) and use stochastic inversion techniques.

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Svanstedt, Nils. "Multiscale stochastic homogenization of monotone operators." Networks & Heterogeneous Media 2, no. 1 (2007): 181–92. http://dx.doi.org/10.3934/nhm.2007.2.181.

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6

Imkeller, Peter, N. Sri Namachchivaya, Nicolas Perkowski, and Hoong C. Yeong. "A Homogenization Approach to Multiscale Filtering." Procedia IUTAM 5 (2012): 34–45. http://dx.doi.org/10.1016/j.piutam.2012.06.005.

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7

Heida, Martin, Ralf Kornhuber, and Joscha Podlesny. "Fractal Homogenization of Multiscale Interface Problems." Multiscale Modeling & Simulation 18, no. 1 (January 2020): 294–314. http://dx.doi.org/10.1137/18m1204759.

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8

Yoshimura, Akinori, Anthony M. Waas, and Yoshiyasu Hirano. "Multiscale homogenization for nearly periodic structures." Composite Structures 153 (October 2016): 345–55. http://dx.doi.org/10.1016/j.compstruct.2016.06.002.

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9

Chen, Zhangxin. "Multiscale methods for elliptic homogenization problems." Numerical Methods for Partial Differential Equations 22, no. 2 (2006): 317–60. http://dx.doi.org/10.1002/num.20099.

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10

Sakata, Seiichiro, and Fumihiro Ashida. "Stochastic Microscopic Stress Analysis of a Composite Material via Multiscale Analysis Considering Microscopic Uncertainty." Key Engineering Materials 452-453 (November 2010): 277–80. http://dx.doi.org/10.4028/www.scientific.net/kem.452-453.277.

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This paper discusses a stochastic microscopic stress analysis of a composite material for a microscopic random variation. The stochastic stress analysis is performed via a stochastic homogenization and multiscale analysis. The homogenization method is employed for the multiscale analysis and the Monte-Carlo simulation or perturbation-based method can be employed for the stochastic analysis. In this paper, outline of the analysis and some numerical results are provided.

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11

Agić, Ante. "Multiscale Modeling Electrospun Nanofiber Structures." Materials Science Forum 714 (March 2012): 33–40. http://dx.doi.org/10.4028/www.scientific.net/msf.714.33.

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The carbon nanotube (CNT) structure is a promising building block for future nanocomposite structures. Mechanical properties of the electrospun butadiene elastomer reinforced with CNT are analyzed by multiscale method. Effective properties of the fiber at microscale determined by homogenization procedure using modified shear-lag model, while on the macro scale effective properties for the point-bonded stochastic fibrous network determined by volume homogenization procedure using multilevel finite element. Random fibrous network was generated according experimentally determined stochastic quantificators. Influence of CNT reinforcement on elastic modulus of electrospun sheet on macroscopic level is determined.

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12

Bufford, Laura, Elisa Davoli, and Irene Fonseca. "Multiscale homogenization in Kirchhoff's nonlinear plate theory." Mathematical Models and Methods in Applied Sciences 25, no. 09 (May 28, 2015): 1765–812. http://dx.doi.org/10.1142/s0218202515500451.

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The interplay between multiscale homogenization and dimension reduction for nonlinear elastic thin plates is analyzed in the case in which the scaling of the energy corresponds to Kirchhoff's nonlinear bending theory for plates. Different limit models are deduced depending on the relative ratio between the thickness parameter h and the two homogenization scales ε and ε2.

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13

Brcic, Marino, Marko Canadija, and Josip Brnic. "Multiscale Modeling of Nanocomposite Structures with Defects." Key Engineering Materials 577-578 (September 2013): 141–44. http://dx.doi.org/10.4028/www.scientific.net/kem.577-578.141.

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A method for the numerical modeling of mechanical behavior of nanocomposite materials reinforced with the carbon nanotubes, based on the computational homogenization as a multiscale method, is presented. The matrix reinforcement interactions, based on the weak van der Waals forces are incorporated into the multiscale model and are represented by the nonlinear rod elements. The reinforcements, i.e. carbon nanotubes, are modeled as a space frame structure, using beam finite elements. Computational homogenization and representative volume element (RVE) are the basis of the presented numerical model of the nanocomposites. Nanoscale model is based on beam and non-linear rod finite elements. An algorithm is developed for the analysis of the presented nanostructure, and for the purpose of the software verification, examples, i.e. models of the nanocomposite material are presented. Also, the nanocomposite model with various vacancy defects in the reinforcement, i.e. nanotube, has been prepared and the obtained results are compared and discussed.Keywords Nanocomposite materials · Carbon nanotubes · Multiscale modelling · Computational homogenization

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14

Tan, Wee Chin, and Viet Ha Hoang. "Sparse tensor product finite element method for nonlinear multiscale variational inequalities of monotone type." IMA Journal of Numerical Analysis 40, no. 3 (March 28, 2019): 1875–907. http://dx.doi.org/10.1093/imanum/drz011.

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Abstract We study an essentially optimal finite element (FE) method for locally periodic nonlinear multiscale variational inequalities of monotone type in a domain $D\subset{\mathbb{R}}^d$ that depend on a macroscopic and $n$ microscopic scales. The scales are separable. Using multiscale convergence we deduce a multiscale homogenized variational inequality in a tensorized domain in the high-dimensional space ${\mathbb R}^{(n+1)d}$. Given sufficient regularity on the solution the sparse tensor product FE method is developed for this problem, which attains an essentially equal (i.e., it differs by only a logarithmic factor) level of accuracy to that of the full tensor product FE method, but requires an essentially optimal number of degrees of freedom which is equal to that for solving a problem in ${{\mathbb{R}}}^d$ apart from a logarithmic factor. For two-scale problems we deduce a new homogenization error for the nonlinear monotone variational inequality. A numerical corrector is then constructed with an explicit error in terms of the homogenization and the FE errors. For general multiscale problems we deduce a numerical corrector from the FE solution of the multiscale homogenized problem, but without an explicit error as such a homogenization error is not available.

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15

Ramírez-Torres, Ariel, Raimondo Penta, Reinaldo Rodríguez-Ramos, and Alfio Grillo. "Effective properties of hierarchical fiber-reinforced composites via a three-scale asymptotic homogenization approach." Mathematics and Mechanics of Solids 24, no. 11 (May 25, 2019): 3554–74. http://dx.doi.org/10.1177/1081286519847687.

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The study of the properties of multiscale composites is of great interest in engineering and biology. Particularly, hierarchical composite structures can be found in nature and in engineering. During the past decades, the multiscale asymptotic homogenization technique has shown its potential in the description of such composites by taking advantage of their characteristics at the smaller scales, ciphered in the so-called effective coefficients. Here, we extend previous works by studying the in-plane and out-of-plane effective properties of hierarchical linear elastic solid composites via a three-scale asymptotic homogenization technique. In particular, the approach is adjusted for a multiscale composite with a square-symmetric arrangement of uniaxially aligned cylindrical fibers, and the formulae for computing its effective properties are provided. Finally, we show the potential of the proposed asymptotic homogenization procedure by modeling the effective properties of musculoskeletal mineralized tissues, and we compare the results with theoretical and experimental data for bone and tendon tissues.

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16

Liu, Xiong, and Wenming He. "A New Estimate for the Homogenization Method for Second-Order Elliptic Problem with Rapidly Oscillating Periodic Coefficients." Journal of Function Spaces 2021 (June 19, 2021): 1–6. http://dx.doi.org/10.1155/2021/8036814.

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In this paper, we will investigate a multiscale homogenization theory for a second-order elliptic problem with rapidly oscillating periodic coefficients of the form ∂ / ∂ x i a i j x / ε , x ∂ u ε x / ∂ x j = f x . Noticing the fact that the classic homogenization theory presented by Oleinik has a high demand for the smoothness of the homogenization solution u 0 , we present a new estimate for the homogenization method under the weaker smoothness that homogenization solution u 0 satisfies than the classical homogenization theory needs.

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17

Lesičar, Tomislav, Zdenko Tonković, and Jurica Sorić. "Measure of Nonlocal Response in Multiscale Gradient Modeling." Key Engineering Materials 713 (September 2016): 297–300. http://dx.doi.org/10.4028/www.scientific.net/kem.713.297.

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Realistic description of heterogeneous material behavior demands more accurate modeling at multiple scales. Multiscale scheme employing second-order homogenization requires C1 continuity at the macrolevel, while classical continuum is usually kept at the microlevel (C1-C0 homogenization). However, due to C1-C0 transition, consistency of macroscale variables is violated. This research proposes a new second-order homogenization scheme employing C1 continuity at both scales. Discretization is performed by the C1 finite element and Aifantis gradient elasticity theory. A new gradient boundary conditions are derived. The relation between the Aifantis length scale and the RVE size has been found. The new procedure is tested on a benchmark example. After successful development of the C1-C1 multiscale scheme, the next step is an extension to consistent scaling of the microscale strain localization towards a macroscopic fracture.

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Lesičar, Tomislav, Zdenko Tonković, and Jurica Sorić. "Boundary Conditions in a Multiscale Homogenization Procedure." Key Engineering Materials 577-578 (September 2013): 297–300. http://dx.doi.org/10.4028/www.scientific.net/kem.577-578.297.

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This paper is concerned with a second-order multiscale computational homogenization scheme for heterogeneous materials at small strains. A special attention is directed to the macro-micro transition and the application of the generalized periodic boundary conditions on the representative volume element at the microlevel. For discretization at the macrolevel the C1 plane strain triangular finite element based on the strain gradient theory is derived, while the standard C0 quadrilateral finite element is used on the RVE. The implementation of a microfluctuation integral condition has been performed using several numerical integration techniques. Finally, a numerical example of a pure bending problem is given to illustrate the efficiency and accuracy of the proposed multiscale homogenization approach.

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19

Flodén, Liselott, Anders Holmbom, Marianne Olsson Lindberg, and Jens Persson. "Homogenization of Parabolic Equations with an Arbitrary Number of Scales in Both Space and Time." Journal of Applied Mathematics 2014 (2014): 1–16. http://dx.doi.org/10.1155/2014/101685.

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The main contribution of this paper is the homogenization of the linear parabolic equation∂tuε(x,t)-∇·(a(x/εq1,...,x/εqn,t/εr1,...,t/εrm)∇uε(x,t))=f(x,t)exhibiting an arbitrary finite number of both spatial and temporal scales. We briefly recall some fundamentals of multiscale convergence and provide a characterization of multiscale limits for gradients, in an evolution setting adapted to a quite general class of well-separated scales, which we name by jointly well-separated scales (see appendix for the proof). We proceed with a weaker version of this concept called very weak multiscale convergence. We prove a compactness result with respect to this latter type for jointly well-separated scales. This is a key result for performing the homogenization of parabolic problems combining rapid spatial and temporal oscillations such as the problem above. Applying this compactness result together with a characterization of multiscale limits of sequences of gradients we carry out the homogenization procedure, where we together with the homogenized problem obtainnlocal problems, that is, one for each spatial microscale. To illustrate the use of the obtained result, we apply it to a case with three spatial and three temporal scales withq1=1,q2=2, and0<r1<r2.

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20

Li, Beibei, Cheng Liu, Xiaoyu Zhao, Jinrui Ye, and Fei Guo. "Multiscale Study of the Effect of Fiber Twist Angle and Interface on the Viscoelasticity of 2D Woven Composites." Materials 16, no. 7 (March 28, 2023): 2689. http://dx.doi.org/10.3390/ma16072689.

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Time and temperature affect the viscoelasticity of woven composites, and thus affect their long-term mechanical properties. We develop a multiscale method considering fiber twist angle and interfaces to predict viscoelasticity. The multiscale approach is based on homogenization theory and the time–temperature superposition principle (TTSP). It is carried out in two steps. Firstly, the effective viscoelasticity properties of yarn are calculated using microscale homogenization; yarn comprises elastic fibers, interface, and a viscoelastic matrix. Subsequently, the effective viscoelasticity properties of woven composites are computed by mesoscale homogenization; it consists of homogenized viscoelastic yarns and matrix. Moreover, the multiscale method is verified using the Mechanics of Structure genome (MSG) consequence. Finally, the effect of temperature, fiber twist angle, fiber array, and coating on either the yarn’s effective relaxation stiffness or the relaxation moduli of the woven composite is investigated. The results show that increased temperature shortens the relaxation time of viscoelastic woven composites, and fiber twist angle affects tensors in the relaxation stiffness matrix of the yarn; the coating affects the overall mechanical properties of woven composites as well.

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21

Kamiński, Marcin. "Wavelet-based homogenization of unidirectional multiscale composites." Computational Materials Science 27, no. 4 (June 2003): 446–60. http://dx.doi.org/10.1016/s0927-0256(03)00046-6.

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22

Svanstedt, Nils. "Multiscale stochastic homogenization of convection-diffusion equations." Applications of Mathematics 53, no. 2 (January 2008): 143–55. http://dx.doi.org/10.1007/s10492-008-0017-x.

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23

Ferreira, Rita, and Irene Fonseca. "Reiterated Homogenization in $BV$ via Multiscale Convergence." SIAM Journal on Mathematical Analysis 44, no. 3 (January 2012): 2053–98. http://dx.doi.org/10.1137/110826205.

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24

Lesičar, Tomislav, Zdenko Tonković, and Jurica Sorić. "Second-Order Computational Homogenization Approach Using Higher-Order Gradients at Microlevel." Key Engineering Materials 665 (September 2015): 181–84. http://dx.doi.org/10.4028/www.scientific.net/kem.665.181.

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Realistic description of heterogeneous material behavior demands more accurate modeling at macroscopic and microscopic scales. To observe strain localization phenomena and material softening occurring at the microstructural level, an analysis on the microlevel is unavoidable. Multiscale techniques employing several homogenization schemes can be found in literature. Widely used second-order homogenization requiresC1continuity at the macrolevel, while standardC0continuity has usually been hold at microlevel. However, due to theC1-C0transition macroscale variables cannot be defined fully consistently. The present contribution is concerned with a multiscale second-order computational homogenization employingC1continuity at both scales under assumptions of small strains and linear elastic material behavior. All algorithms derived are implemented into the FE software ABAQUS. The numerical efficiency and accuracy of the proposed computational strategy is demonstrated by modeling three point bending test of the notched specimen.

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25

Zhang, Nan, Shuai Gao, Meili Song, Yang Chen, Xiaodong Zhao, Jianguo Liang, and Jun Feng. "A Multiscale Study of CFRP Based on Asymptotic Homogenization with Application to Mechanical Analysis of Composite Pressure Vessels." Polymers 14, no. 14 (July 11, 2022): 2817. http://dx.doi.org/10.3390/polym14142817.

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The application of composites is increasingly extensive due to their advanced properties while the analysis still remains complex on different scales. In this article, carbon fiber reinforced polymer (CFRP) is modeled via asymptotic homogenization employing a representative volume element (RVE) with periodic boundary conditions. A multiscale mechanical model of CFRP is established to bridge the microscopic model, mesoscopic model, and macroscopic model. According to asymptotic homogenization, the coefficients of the material constitutive equation are calculated with volume-averaged stress and strain. Using the homogenized materials properties of CFRP, the tensile experiments of composite layers with the layout of [(0∘/60∘/0∘/−60∘)4] are carried out to validate asymptotic homogenization method. The results indicated that the asymptotic homogenization approach can be used to calculate the homogenized elastic moduli and Poisson’s ratio of the whole structure, where the numerical results are basically consistent with test data. The sequent homogenized CFRP laminate model is applied to the mechanical analysis of type III composite pressure vessels, whereby burst pressure is accurately predicted. This work might shed some light on multiscale analysis of composite pressure vessels.

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26

Miedzińska, Danuta, Robert Panowicz, and Przemysław Jóźwicki. "Multiscale Modelling Method for Chosen Functionally Graded Material." Solid State Phenomena 199 (March 2013): 593–98. http://dx.doi.org/10.4028/www.scientific.net/ssp.199.593.

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The paper deals with the numerical and experimental analyses of functionally graded material structures which are represented by a surface layer of the steel sample hardened during the laser treatment process. A functionally graded parameter of the researched structure was assumed as the hardness value experimentally measured with the use of a Vickers hardness test method. The microstructure of the tested layer was also analyzed for the Vickers test verification. Two homogenization methods were used for the purpose of layer substitute properties for numerical calculations. The first one was to divide the FGM domain into a number of layers in the direction of material gradation and then apply a numerical homogenization method within each layer. The resulting material model describes the FGM as a composite of homogeneous layers. The second method was based on the Mori-Tanaka homogenization theory and was carried out with the use of Digimat software, which is the nonlinear multi-scale materials and structures modelling platform. Both methods were compared and showed good correspondence.

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27

VERNESCU, BOGDAN. "MULTISCALE ANALYSIS OF ELECTRORHEOLOGICAL FLUIDS." International Journal of Modern Physics B 16, no. 17n18 (July 20, 2002): 2643–48. http://dx.doi.org/10.1142/s0217979202012785.

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We construct a microscale model for a rigid particle suspension in a viscous fluid that includes Maxwell electrostatic forces. Via homogenization techniques we characterize the properties the material exhibits at the macroscale. The change in the effective constitutive equations is due to the highly oscillating electrostatic forces. The material properties are determined by both hydrodynamic and electrostatic particle interactions.

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Almqvist, Andreas, Emmanuel Kwame Essel, John Fabricius, and Peter Wall. "Multiscale homogenization of a class of nonlinear equations with applications in lubrication theory and applications." Journal of Function Spaces and Applications 9, no. 1 (2011): 17–40. http://dx.doi.org/10.1155/2009/432170.

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We prove a homogenization result for monotone operators by using the method of multiscale convergence. More precisely, we study the asymptotic behavior asε→0of the solutionsuεof the nonlinear equationdiv⁡aε(x,∇uε)=div⁡bε, where bothaεandbεoscillate rapidly on several microscopic scales andaεsatisfies certain continuity, monotonicity and boundedness conditions. This kind of problem has applications in hydrodynamic thin film lubrication where the bounding surfaces have roughness on several length scales. The homogenization result is obtained by extending the multiscale convergence method to the setting of Sobolev spacesW01,p(Ω), where1<p<∞. In particular we give new proofs of some fundamental theorems concerning this convergence that were first obtained by Allaire and Briane for the casep=2.

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29

Shah, S. Z. H., Puteri S. M. Megat Yusoff, Saravanan Karuppanan, and Zubair Sajid. "Elastic Constants Prediction of 3D Fiber-Reinforced Composites Using Multiscale Homogenization." Processes 8, no. 6 (June 22, 2020): 722. http://dx.doi.org/10.3390/pr8060722.

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This paper presents a multi-scale-homogenization based on a two-step methodology (micro-meso and meso-macro homogenization) to predict the elastic constants of 3D fiber-reinforced composites (FRC). At each level, the elastic constants were predicted through both analytical and numerical methods to ascertain the accuracy of predicted elastic constants. The predicted elastic constants were compared with experimental data. Both methods predicted the in-plane elastic constants “ E x ” and “ E y ” with good accuracy; however, the analytical method under predicts the shear modulus “ G x y ”. The elastic constants predicted through a multiscale homogenization approach can be used to predict the behavior of 3D-FRC under different loading conditions at the macro-level.

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Pruchnicki, Erick. "Some specific aspects of linear homogenization shell theory." Mathematics and Mechanics of Solids 24, no. 4 (May 27, 2018): 1116–28. http://dx.doi.org/10.1177/1081286518773522.

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In this paper we propose a multiscale linear shell theory for simulating the mechanical response of a highly heterogeneous shell of varying thickness. To resolve this issue, a higher-order stress-resultant shell formulation based on multiscale homogenization is considered. At the macroscopic scale level, we approximate the displacement field by a fourth-order Taylor–Young expansion in thickness. The transition between both the microscopic and the macroscopic scales is obtained through the introduction of a specific Hill–Mandel condition. Since we adopt the standard assumption of small strain which is used in linear elasticity, we can present a variant of the homogenization scheme which is valid for small strain. The nonlinearity of the previous model occurs from the assumption of large rotation of the transverse normal.

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Allaire, Grégoire, and Robert Brizzi. "A Multiscale Finite Element Method for Numerical Homogenization." Multiscale Modeling & Simulation 4, no. 3 (January 2005): 790–812. http://dx.doi.org/10.1137/040611239.

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32

Unnikrishnan, V. U., G. U. Unnikrishnan, and J. N. Reddy. "Multiscale Homogenization Based Analysis of Polymeric Nanofiber Scaffolds." Mechanics of Advanced Materials and Structures 15, no. 8 (December 4, 2008): 558–66. http://dx.doi.org/10.1080/15376490802470440.

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33

Daubechies, Ingrid, Olof Runborg, and Jing Zou. "A Sparse Spectral Method for Homogenization Multiscale Problems." Multiscale Modeling & Simulation 6, no. 3 (January 2007): 711–40. http://dx.doi.org/10.1137/060676258.

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34

Hales, J. D., M. R. Tonks, K. Chockalingam, D. M. Perez, S. R. Novascone, B. W. Spencer, and R. L. Williamson. "Asymptotic expansion homogenization for multiscale nuclear fuel analysis." Computational Materials Science 99 (March 2015): 290–97. http://dx.doi.org/10.1016/j.commatsci.2014.12.039.

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35

Holmbom, Anders, Nils Svanstedt, and Niklas Wellander. "Multiscale convergence and reiterated homogenization of parabolic problems." Applications of Mathematics 50, no. 2 (April 2005): 131–51. http://dx.doi.org/10.1007/s10492-005-0009-z.

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36

Amirat, Youcef, Gregory A. Chechkin, and Maxim Romanov. "On multiscale homogenization problems in boundary layer theory." Zeitschrift für angewandte Mathematik und Physik 63, no. 3 (October 9, 2011): 475–502. http://dx.doi.org/10.1007/s00033-011-0167-7.

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37

Temizer, İ. "Multiscale thermomechanical contact: Computational homogenization with isogeometric analysis." International Journal for Numerical Methods in Engineering 97, no. 8 (December 13, 2013): 582–607. http://dx.doi.org/10.1002/nme.4604.

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Henyš, Petr, and Gleb Pokatilov. "The edge smoothed finite element for multiscale homogenization." Engineering Analysis with Boundary Elements 156 (November 2023): 70–77. http://dx.doi.org/10.1016/j.enganabound.2023.07.043.

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39

Pruchnicki, Erick. "On the homogenization of a nonlinear shell." Mathematics and Mechanics of Solids 24, no. 4 (April 25, 2018): 1054–64. http://dx.doi.org/10.1177/1081286518768674.

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In this paper we propose a multiscale finite-strain shell theory for simulating the mechanical response of a highly heterogeneous shell of varying thickness. To resolve this issue, a higher-order stress-resultant shell formulation based on multiscale homogenization is considered. At the macroscopic scale level, we approximate the displacement field by a fifth-order Taylor–Young expansion in thickness. We take account of the microscale fluctuations by introducing a boundary value problem over the domain of a three-dimensional representative volume element (RVE). The geometrical form and the dimensions of the RVE are determined by the representative microstructure of the heterogeneity. In this way, an in-plane homogenization is directly combined with a through thickness stress integration. As a result, the macroscopic stress resultants are the volume averages through RVE of microscopic stress. All microstructural constituents are modeled as first-order continua and three-dimensional continuum, described by the standard equilibrium and the constitutive equations. This type of theory is anxiously awaited.

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40

Karamnejad, Amin, Awais Ahmed, and Lambertus Johannes Sluys. "A Numerical Homogenization Scheme for Glass Particle-Toughened Polymers Under Dynamic Loading." Journal of Multiscale Modelling 08, no. 01 (February 22, 2017): 1750001. http://dx.doi.org/10.1142/s1756973717500019.

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A numerical homogenization scheme is presented to model glass particle-toughened polymer materials under dynamic loading. A constitutive law is developed for the polymer material and validated by comparing the results to experimental test data. A similar constitutive law as that of the polymer material with unknown material parameters is assumed for the glass particle-toughened polymer. A homogenization scheme is used to determine the unknown material parameters from the boundary value problem (BVP) of a representative volume element. Unlike the standard computational homogenization scheme, the proposed numerical homogenization scheme can be used after localization occurs in the material. The proposed multiscale model is then verified against direct numerical simulation.

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41

Souza, Flavio V., and David H. Allen. "Computation of Homogenized Constitutive Tensor of Elastic Solids Containing Evolving Cracks." International Journal of Damage Mechanics 21, no. 2 (March 17, 2011): 267–91. http://dx.doi.org/10.1177/1056789510397071.

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The determination of the equivalent (homogenized) constitutive tensor is one of the most important steps in multiscale models as well as in the classical homogenization theory. In this article, a procedure for determining the homogenized instantaneous (tangent) constitutive tensor of elastic materials containing growing cracks is proposed. The primary purpose of this procedure is its use in two-way coupled multiscale finite element algorithms that can model crack formation and propagation at the local microstructure. The procedure is basically developed by relating the local displacement field to the global strain tensor at each location and using first-order homogenization techniques. The finite element formulation is developed and some example problems are presented in order to verify and demonstrate the model capabilities.

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42

Liu, Yang, Haitao Zhao, Shun Dong, Xiaoguang Zhao, Yahui Peng, and Ji’an Chen. "Multiscale model construction and modulus prediction of needle-punched carbon/carbon composites." Polymers and Polymer Composites 31 (February 17, 2023): 096739112311589. http://dx.doi.org/10.1177/09673911231158999.

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In this paper, the modulus prediction of NP-C/C is completed by establishing a multiscale finite element model, and the influence weight analysis of the modulus is carried out by using the analytical homogenization method. Firstly, three Micro-RVE, including unidirectional composites, short fiber reinforced composites and porous matrix, are established to characterize the layers of NP-C/C. Then, the finite element model is used for homogenization and passed upward into Meso-RVE, and the modulus prediction is completed. Finally, the analytical homogenization method is used combined with the MATLAB® script to analyze the influence weight of the modulus by adjusting the parameters. The results show that the average errors of stiffness coefficient prediction values obtained based on the analytical homogenization method are only 6.0% relative to the experimental values, which are lower than 25.3% of the finite element model. The modulus prediction method given in this study does not need to establish a micromechanical model with complex structure, and can be directly completed by computer script when the prediction accuracy is met, which is simple and feasible.

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43

Kornhuber, Ralf, and Harry Yserentant. "Numerical Homogenization of Elliptic Multiscale Problems by Subspace Decomposition." Multiscale Modeling & Simulation 14, no. 3 (January 2016): 1017–36. http://dx.doi.org/10.1137/15m1028510.

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Marques, M., J. Belinha, A. F. Oliveira, M. C. Manzanares Céspedes, and R. N. Jorge. "A multiscale homogenization procedure using the fabric tensor concept." Science and Technology of Materials 30, no. 1 (January 2018): 27–34. http://dx.doi.org/10.1016/j.stmat.2018.01.002.

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Ambrosio, Luigi, and Hermano Frid. "Multiscale Young Measures in almost Periodic Homogenization and Applications." Archive for Rational Mechanics and Analysis 192, no. 1 (May 9, 2008): 37–85. http://dx.doi.org/10.1007/s00205-008-0127-3.

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Bacigalupo, A., L. Morini, and A. Piccolroaz. "Multiscale asymptotic homogenization analysis of thermo-diffusive composite materials." International Journal of Solids and Structures 85-86 (May 2016): 15–33. http://dx.doi.org/10.1016/j.ijsolstr.2016.01.016.

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Abdulle, A., and A. Nonnenmacher. "Adaptive finite element heterogeneous multiscale method for homogenization problems." Computer Methods in Applied Mechanics and Engineering 200, no. 37-40 (September 2011): 2710–26. http://dx.doi.org/10.1016/j.cma.2010.06.012.

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Larsson, Ragnar, and Mats Landervik. "A stress-resultant shell theory based on multiscale homogenization." Computer Methods in Applied Mechanics and Engineering 263 (August 2013): 1–11. http://dx.doi.org/10.1016/j.cma.2013.04.011.

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Hernández, J. A., J. Oliver, A. E. Huespe, M. A. Caicedo, and J. C. Cante. "High-performance model reduction techniques in computational multiscale homogenization." Computer Methods in Applied Mechanics and Engineering 276 (July 2014): 149–89. http://dx.doi.org/10.1016/j.cma.2014.03.011.

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Ben Arous, G�rard, and Houman Owhadi. "Multiscale homogenization with bounded ratios and anomalous slow diffusion." Communications on Pure and Applied Mathematics 56, no. 1 (November 22, 2002): 80–113. http://dx.doi.org/10.1002/cpa.10053.

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