Ali Haydar , Laura Galuppi , Gianni Royer-Carfagni
{"title":"基于通量的热问题弱公式,发展了毕奥特变分原理","authors":"Ali Haydar , Laura Galuppi , Gianni Royer-Carfagni","doi":"10.1016/j.ijengsci.2024.104103","DOIUrl":null,"url":null,"abstract":"<div><p>We propose a weak form of the transient heat equations for solid bodies, as a time-dependent spatial variation of the heat displacement vector field, whose time derivative is the heat flux. This develops the variational principle originally proposed by Biot, inasmuch Fourier’s law is embedded as a holonomic constraint, while energy conservation results from the variation (the vice-versa from Biot). This is a neat formulation because only the heat displacement appears in the variational equations, whereas Biot’s form also involved the unknown temperature field: Fourier’s law is used only <em>a posteriori</em> to recover the temperature. Since the heat displacement is generally more regular than the temperature field, it represents a natural variable in problems with material inhomogeneities, uneven radiation, thermal shocks. The three-dimensional analytical set-up is presented in comparison with Biot’s, for boundary conditions accounting for radiation and convection. A mechanical analogy with the equilibrium of an elastic bar with viscous constraints is suggested for the one-dimensional case. The variational equations are implemented in a finite element code. Numerical experiments on benchmark problems, involving high temperature gradients, confirm the efficiency of the proposed approach in many structural problems.</p></div>","PeriodicalId":14053,"journal":{"name":"International Journal of Engineering Science","volume":null,"pages":null},"PeriodicalIF":5.7000,"publicationDate":"2024-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0020722524000879/pdfft?md5=c4dc5d4819ef2895d9de89f4739566bd&pid=1-s2.0-S0020722524000879-main.pdf","citationCount":"0","resultStr":"{\"title\":\"A neat flux-based weak formulation for thermal problems which develops Biot’s variational principle\",\"authors\":\"Ali Haydar , Laura Galuppi , Gianni Royer-Carfagni\",\"doi\":\"10.1016/j.ijengsci.2024.104103\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We propose a weak form of the transient heat equations for solid bodies, as a time-dependent spatial variation of the heat displacement vector field, whose time derivative is the heat flux. This develops the variational principle originally proposed by Biot, inasmuch Fourier’s law is embedded as a holonomic constraint, while energy conservation results from the variation (the vice-versa from Biot). This is a neat formulation because only the heat displacement appears in the variational equations, whereas Biot’s form also involved the unknown temperature field: Fourier’s law is used only <em>a posteriori</em> to recover the temperature. Since the heat displacement is generally more regular than the temperature field, it represents a natural variable in problems with material inhomogeneities, uneven radiation, thermal shocks. The three-dimensional analytical set-up is presented in comparison with Biot’s, for boundary conditions accounting for radiation and convection. A mechanical analogy with the equilibrium of an elastic bar with viscous constraints is suggested for the one-dimensional case. The variational equations are implemented in a finite element code. Numerical experiments on benchmark problems, involving high temperature gradients, confirm the efficiency of the proposed approach in many structural problems.</p></div>\",\"PeriodicalId\":14053,\"journal\":{\"name\":\"International Journal of Engineering Science\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":5.7000,\"publicationDate\":\"2024-06-20\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S0020722524000879/pdfft?md5=c4dc5d4819ef2895d9de89f4739566bd&pid=1-s2.0-S0020722524000879-main.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"International Journal of Engineering Science\",\"FirstCategoryId\":\"5\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0020722524000879\",\"RegionNum\":1,\"RegionCategory\":\"工程技术\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"ENGINEERING, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"International Journal of Engineering Science","FirstCategoryId":"5","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0020722524000879","RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"ENGINEERING, MULTIDISCIPLINARY","Score":null,"Total":0}
A neat flux-based weak formulation for thermal problems which develops Biot’s variational principle
We propose a weak form of the transient heat equations for solid bodies, as a time-dependent spatial variation of the heat displacement vector field, whose time derivative is the heat flux. This develops the variational principle originally proposed by Biot, inasmuch Fourier’s law is embedded as a holonomic constraint, while energy conservation results from the variation (the vice-versa from Biot). This is a neat formulation because only the heat displacement appears in the variational equations, whereas Biot’s form also involved the unknown temperature field: Fourier’s law is used only a posteriori to recover the temperature. Since the heat displacement is generally more regular than the temperature field, it represents a natural variable in problems with material inhomogeneities, uneven radiation, thermal shocks. The three-dimensional analytical set-up is presented in comparison with Biot’s, for boundary conditions accounting for radiation and convection. A mechanical analogy with the equilibrium of an elastic bar with viscous constraints is suggested for the one-dimensional case. The variational equations are implemented in a finite element code. Numerical experiments on benchmark problems, involving high temperature gradients, confirm the efficiency of the proposed approach in many structural problems.
期刊介绍:
The International Journal of Engineering Science is not limited to a specific aspect of science and engineering but is instead devoted to a wide range of subfields in the engineering sciences. While it encourages a broad spectrum of contribution in the engineering sciences, its core interest lies in issues concerning material modeling and response. Articles of interdisciplinary nature are particularly welcome.
The primary goal of the new editors is to maintain high quality of publications. There will be a commitment to expediting the time taken for the publication of the papers. The articles that are sent for reviews will have names of the authors deleted with a view towards enhancing the objectivity and fairness of the review process.
Articles that are devoted to the purely mathematical aspects without a discussion of the physical implications of the results or the consideration of specific examples are discouraged. Articles concerning material science should not be limited merely to a description and recording of observations but should contain theoretical or quantitative discussion of the results.