Asymptotic homogenization of a problem for a wave equation on a microperiodic medium

Authors

DOI:

https://doi.org/10.5902/2179460X87229

Keywords:

Wave equation, Asymptotic homogenization method, Formal asymptotic solution

Abstract

The asymptotic homogenization method is a mathematical technique that allows studying the physical properties of a microheterogeneous, periodic medium characterized by rapidly oscillating coefficients through a homogeneous medium that is asymptotically equivalent to the micro-heterogeneous medium. The method involves constructing a two-scale formal asymptotic solution of the original problem, and by applying mathematical formalism, a problem is formulated over a homogenized medium known as the homogenized problem. Utilizing maximum principles, it is proven that the solution to the homogenized problem is an asymptotic expansion of the solution to the original problem. This work aims to apply this method to a problem for a hyperbolic equation and demonstrate the proximity between the solutions.

Downloads

Download data is not yet available.

Author Biographies

Douglas Machado da Silva, Universidade Federal de Pelotas

Holds a degree in Mathematics Degree - Daytime from the Federal University of Pelotas (2021). He is currently a master's student in Mathematical Modeling at the Federal University of Pelotas.

Leslie Darien Pérez Fernández, Universidade Federal de Pelotas

Has a bachelor's degree in Mathematics from the University of Havana (2001), a master's degree in Mathematics from the University of Havana (2006), and a PhD in Mathematics from the Institute of Cybernetics, Mathematics, and Physics (2010 - recognized by the National Commission of Scientific Degrees of Cuba). I received the Annual Prize from the Cuban Academy of Sciences for Scientific Research Results in 2017 (as a collaborator), 2009 (as the main author), and 2006 (as a co-author), as well as the Prize from the Nuclear Energy and Advanced Technologies Agency of the Ministry of Science, Technology, and Environment of Cuba for Outstanding Scientific-Technical Results in 2010 and 2008. Since March 2013, I have been a professor in the Department of Mathematics and Statistics at the Institute of Physics and Mathematics at the Federal University of Pelotas (DME-IFM-UFPel), and since July 2015, I have been a permanent faculty member of the Graduate Program in Mathematical Modeling (PPGMMat-IFM-UFPel). I am a member of research groups in Optimization, Control, and Nonlinear Analysis (UFPel), Computational Mechanics and Numerical Methods (UFS - Federal University of Sergipe), and Mathematical and Computational Modeling in the Mechanics of Continuous Media (UFF - Federal Fluminense University). My interests lie in the modeling and simulation of multiscale physical and biological phenomena and related mathematical methods, mainly asymptotic and variational homogenization methods

Alexandre Molter, Universidade Federal de Pelotas

Has a bachelor's degree in Mathematics Education from the University of Vale do Rio dos Sinos (2001), a master's degree in Mathematical Modeling from the Regional University of the Northwestern State of Rio Grande do Sul (2004), a PhD in Mechanical Engineering from the Federal University of Rio Grande do Sul (2008), and completed a postdoctoral fellowship at the Federal University of Rio Grande do Sul (2009). Currently, I am an Associate Professor in the Department of Mathematics at the Federal University of Pelotas. My expertise lies in the field of Mathematics, with a focus on Mathematical Modeling, primarily working on topics such as dynamical systems, control, and optimization

Julián Bravo Castillero, Universidad Nacional Autónoma de México

Dr. Julián Bravo Castillero is a Research Professor at the National Autonomous University of Mexico, at the Institute of Applied Mathematics and Systems Research, in its academic unit in the state of Yucatán, Mexico, located in Mérida, Yucatán. He was a Full Professor and Research Professor at the Faculty of Mathematics and Computer Science at the University of Havana, Cuba, and one of the leaders of the Solid Mechanics Group at that institution. His research is related to the development of mathematical homogenization methods and their applications, primarily in determining effective laws for composite materials and wave propagation in heterogeneous structures for the mathematical modeling of new materials, bone biomechanics, transport phenomena, sensors, and actuators, among others. Dr. Castillero has over thirty years of experience in this line of research and has published over 140 scientific articles in high-impact specialized journals. He is a co-author of six papers awarded by the Cuban Academy of Sciences in 1996, 2002, 2006, 2009, 2013, and 2017, and in 2007, he received the Pablo Miquel Merino National Mathematics Prize, awarded by the Cuban Society of Mathematics and Computer Science. In 2014 and 2015, he received the University of Havana Rector's Prize for co-authoring results of major significance and originality. He has conducted teaching and research internships at various institutions in Germany, Brazil, Colombia, Cuba, Spain, France, and Mexico. Currently, he is a member of the National Researchers System (level 2) of CONACYT, Mexico.

References

Allaire, G., Bal, G. (1999). Homogenization of the criticality spectral equation in neutron transport. Mathematical Modelling and

Numerical Analysis, 33(4), 721–746.

Auriault, J., Boutin, C., Geindreau, C. (2009). Homogenization of Coupled Phenomena in Heterogenous Media, 1º edn. Wiley.

Bakhvalov, N., Panasenko, G. (1989). Homogenisation: Averaging Processes in Periodic Media: Mathematical Problems in the

Mechanics of Composite Material, 1º edn. Springer Netherlands.

BendsØe, M. P., Sigmund, O. (2003). Topology Optimization: Theory, Methods and Applications, 1º edn. Springer.

Bensoussan, A., Lions, J., Papanicolau, G. (1978). Asymptotic Analysis for Periodic Structures, 1º edn. North Holland.

Capdville, Y., Guillot, L., Marigo, J. (2010a). 1-d non-periodic homogenization for the seismic wave equation. Geophysical

Journal International, 181(2), 897–910.

Capdville, Y., Guillot, L., Marigo, J. (2010b). 2-d non-periodic homogenization to upscale elastic media for p-sv waves.

Geophysical Journal International, 182(2), 903–922.

Ciblac, T., Morel, J. (2014). Sustainable Masonry, 1º edn. Wiley.

Ciouranescu, D., Donato, P. (1999). An Introduction to Homogenization, 1º edn. Oxford University Press.

Costa, C., Pérez-Fernández, L. D., Bravo-Castillero, J. (2018). Pollutant dispersion modeling via mathematical homogenization and

integral transform-based multilayer methods. Em: Towards Mathematics, Computers and Environment: A Disasters Perspective,

pp. 59–82.

Da, D. (2019). Topology Optimization Design of Heterogeneous Materials and Structures, 1º edn. Wiley.

Desbrun, M., Donaldson, R., Owhadi, H. (2013). Modeling across scales: Discrete geometric structures in homogenization and

inverse homogenization. Em: Multiscale Analysis and Nonlinear Dynamics: From Genes to the Brain, pp. 19–64.

Dimitrienko, Y. (1997). Heat mass transport and thermal stresses in porous charring materials. Transport in Porous Media, 27(2),

–170.

Dormieux, L., Kondo, D. (2016). Micromechanics of Fracture and Damage, 1º edn. Wiley.

Lima, E. (2018). Análise real: Funções de uma variável, 12º edn. IMPA.

Ng, C. (2006). Dispersion in steady and oscillatory flows through a tube with reversible and irreversible wall reactions. Proceedings

of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 462(2066), 481–515.

Parnel, W., Grimal, Q. (2009). The influence of mesoscale porosity on cortical bone anisotropy. Journal of the Royal Society

Interface, 6(30), 97–109.

Pérez-Fernández, L. D., Beck, A. T. (2014). Failure detection in umbilical cables via electroactive elements - a mathematical

homogenization approach. International Journal of Modeling and Simulation for the Petroleum Industry, 8, 34–39.

Tartar, L. (2009). The General Theory of Homogenization: A Personalized Introduction, 1º edn. Springer.

Torquato, S. (2002). Random Heterogeneus Materials: Microstructure and Macroscopic properties, 1º edn. Springer.

Weston, M. (2007). Nuclear Reactor Physics, 1º edn. Wiley.

Downloads

Published

2024-11-07

How to Cite

Silva, D. M. da, Fernández, L. D. P., Molter, A., & Bravo Castillero, J. (2024). Asymptotic homogenization of a problem for a wave equation on a microperiodic medium. Ciência E Natura, 46(esp. 1), e87229. https://doi.org/10.5902/2179460X87229

Issue

Vol. 46 No. esp. 1 (2024): ERMAC e ENMC

Section

Special Edition 1

License

Copyright (c) 2024 Ciência e Natura

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

To access the DECLARATION AND TRANSFER OF COPYRIGHT AUTHOR’S DECLARATION AND COPYRIGHT LICENSE click here.

 

Ethical Guidelines for Journal Publication

The Ciência e Natura journal is committed to ensuring ethics in publication and quality of articles. 

Conformance to standards of ethical behavior is therefore expected of all parties involved: Authors, Editors, Reviewers, and the Publisher.

In particular, 

Authors: Authors should present an objective discussion of the significance of research work as well as sufficient detail and references to permit others to replicate the experiments. Fraudulent or knowingly inaccurate statements constitute unethical behavior and are unacceptable. Review Articles should also be objective, comprehensive, and accurate accounts of the state of the art. The Authors should ensure that their work is entirely original works, and if the work and/or words of others have been used, this has been appropriately acknowledged. Plagiarism in all its forms constitutes unethical publishing behavior and is unacceptable. Submitting the same manuscript to more than one journal concurrently constitutes unethical publishing behavior and is unacceptable. Authors should not submit articles describing essentially the same research to more than one journal. The corresponding Author should ensure that there is a full consensus of all Co-authors in approving the final version of the paper and its submission for publication.

Editors: Editors should evaluate manuscripts exclusively on the basis of their academic merit. An Editor must not use unpublished information in the editor's own research without the express written consent of the Author. Editors should take reasonable responsive measures when ethical complaints have been presented concerning a submitted manuscript or published paper.

Reviewers: Any manuscripts received for review must be treated as confidential documents. Privileged information or ideas obtained through peer review must be kept confidential and not used for personal advantage. Reviewers should be conducted objectively, and observations should be formulated clearly with supporting arguments, so that Authors can use them for improving the paper. Any selected Reviewer who feels unqualified to review the research reported in a manuscript or knows that its prompt review will be impossible should notify the Editor and excuse himself from the review process. Reviewers should not consider manuscripts in which they have conflicts of interest resulting from competitive, collaborative, or other relationships or connections with any of the authors, companies, or institutions connected to the papers.

Most read articles by the same author(s)