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Journal articles on the topic 'Variational principles'

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Author: Grafiati

Published: 4 June 2021

Last updated: 28 July 2024

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1

HUANG, YONG-CHANG, XI-GUO LEE, and MING-XUE SHAO. "UNIFIED EXPRESSIONS OF ALL INTEGRAL VARIATIONAL PRINCIPLES." Modern Physics Letters A 21, no. 14 (May 10, 2006): 1107–15. http://dx.doi.org/10.1142/s0217732306019232.

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In terms of the quantitative causal principle, this paper obtains a general variational principle, gives unified expressions of the general, Hamilton, Voss, Hölder, Maupertuis–Lagrange variational principles of integral style, the invariant quantities of the general, Voss, Hölder, Maupertuis–Lagrange variational principles are given, finally the Noether conservation charges of the general, Voss, Hölder, Maupertuis–Lagrange variational principles are deduced, and the intrinsic relations among the invariant quantities and the Noether conservation charges of all the integral variational principles are achieved.

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2

Zhang, Qi-hao, and Dian-kui Liu. "Some Problems in the Theory of Nonconservative Elasticity." Transactions of the Canadian Society for Mechanical Engineering 10, no. 1 (March 1986): 28–33. http://dx.doi.org/10.1139/tcsme-1986-0005.

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This study develops the general quasi-variational principles for nonconservative problems in the theory of elasticity such as the quasi-potential energy principle, the quasi-complementary energy principle, the generalized quasi-variational principle and quasi-Hamilton principle. The application of these quasi-variational principles to finite element analysis is also discussed and illustrated with some examples. The total variational principle for nonconservative systems of two variables is also studied.

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3

Zhu, Jiang, Lei Wei, Yeol Je Cho, and Cheng Cheng Zhu. "Vectorial Ekeland Variational Principles and Inclusion Problems in Cone Quasi-Uniform Spaces." Abstract and Applied Analysis 2012 (2012): 1–19. http://dx.doi.org/10.1155/2012/310369.

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Some new vectorial Ekeland variational principles in cone quasi-uniform spaces are proved. Some new equivalent principles, vectorial quasivariational inclusion principle, vectorial quasi-optimization principle, vectorial quasiequilibrium principle are obtained. Also, several other important principles in nonlinear analysis are extended to cone quasi-uniform spaces. The results of this paper extend, generalize, and improve the corresponding results for Ekeland's variational principles of the directed vectorial perturbation type and other generalizations of Ekeland's variational principles in the setting ofF-type topological space and quasi-metric spaces in the literatures. Even in usual real metric spaces, some of our results are new.

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4

Xue, Shou Yi. "Study on Parametric Variational Principles in Elasticity." Applied Mechanics and Materials 501-504 (January 2014): 2475–78. http://dx.doi.org/10.4028/www.scientific.net/amm.501-504.2475.

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A parametric variational principle is deduced according to the equivalent integral form of all the controlling equations and boundary conditions in elasticity, and by adjusting the parameters, all kinds of variational principles put forward past and some new variational principles can be gained, which means that the method above is more clear in concept, and more concise.

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5

He, Ji-Huan. "Lagrange crisis and generalized variational principle for 3D unsteady flow." International Journal of Numerical Methods for Heat & Fluid Flow 30, no. 3 (September 20, 2019): 1189–96. http://dx.doi.org/10.1108/hff-07-2019-0577.

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Purpose A three-dimensional (3D) unsteady potential flow might admit a variational principle. The purpose of this paper is to adopt a semi-inverse method to search for the variational formulation from the governing equations. Design/methodology/approach A suitable trial functional with a possible unknown function is constructed, and the identification of the unknown function is given in detail. The Lagrange multiplier method is used to establish a generalized variational principle, but in vain. Findings Some new variational principles are obtained, and the semi-inverse method can easily overcome the Lagrange crisis. Practical implications The semi-inverse method sheds a promising light on variational theory, and it can replace the Lagrange multiplier method for the establishment of a generalized variational principle. It can be used for the establishment of a variational principle for fractal and fractional calculus. Originality/value This paper establishes some new variational principles for the 3D unsteady flow and suggests an effective method to eliminate the Lagrange crisis.

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6

Pan, Yuhua, Yuanfeng Wang, and Li Su. "ESTABLISHMENT OF DYNAMIC EQUATIONS FOR DAMPED SYSTEMS BASED ON QUASI-VARIATIONAL PRINCIPLES." Transactions of the Canadian Society for Mechanical Engineering 40, no. 5 (December 2016): 859–70. http://dx.doi.org/10.1139/tcsme-2016-0070.

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In this paper, quasi-variational principles for non-conservative damped systems are studied. A Hamiltontype quasi-variational principle for non-conservative systems in analytical mechanics and a quasi-variational principle of potential energy in non-conservative elastodynamics systems are proposed in simplified forms respectively, by using the direct variational integral method. On the basis of the standard linear solid model for viscoelastic materials, the dynamic equations of exponentially damped systems are established through the proposed quasi-variational principles. A distinction between the internal damping described by exponential damping and the external damping described by viscous one in a vibrating structure is according to different physical mechanisms, which gives some indication of the correct mechanism of damping.

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7

Auchmuty, Giles. "Variational principles for variational inequalities." Numerical Functional Analysis and Optimization 10, no. 9-10 (January 1989): 863–74. http://dx.doi.org/10.1080/01630568908816335.

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8

Tessarotto, Massimo, and Claudio Cremaschini. "The Principle of Covariance and the Hamiltonian Formulation of General Relativity." Entropy 23, no. 2 (February 10, 2021): 215. http://dx.doi.org/10.3390/e23020215.

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The implications of the general covariance principle for the establishment of a Hamiltonian variational formulation of classical General Relativity are addressed. The analysis is performed in the framework of the Einstein-Hilbert variational theory. Preliminarily, customary Lagrangian variational principles are reviewed, pointing out the existence of a novel variational formulation in which the class of variations remains unconstrained. As a second step, the conditions of validity of the non-manifestly covariant ADM variational theory are questioned. The main result concerns the proof of its intrinsic non-Hamiltonian character and the failure of this approach in providing a symplectic structure of space-time. In contrast, it is demonstrated that a solution reconciling the physical requirements of covariance and manifest covariance of variational theory with the existence of a classical Hamiltonian structure for the gravitational field can be reached in the framework of synchronous variational principles. Both path-integral and volume-integral realizations of the Hamilton variational principle are explicitly determined and the corresponding physical interpretations are pointed out.

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9

Street, O. D., and D. Crisan. "Semi-martingale driven variational principles." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 477, no. 2247 (March 2021): 20200957. http://dx.doi.org/10.1098/rspa.2020.0957.

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Spearheaded by the recent efforts to derive stochastic geophysical fluid dynamics models, we present a general framework for introducing stochasticity into variational principles through the concept of a semi-martingale driven variational principle and constraining the component variables to be compatible with the driving semi-martingale. Within this framework and the corresponding choice of constraints, the Euler–Poincaré equation can be easily deduced. We show that the deterministic theory is a special case of this class of stochastic variational principles. Moreover, this is a natural framework that enables us to correctly characterize the pressure term in incompressible stochastic fluid models. Other general constraints can also be incorporated as long as they are compatible with the driving semi-martingale.

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10

Cremaschini, Claudio, and Massimo Tessarotto. "Unconstrained Lagrangian Variational Principles for the Einstein Field Equations." Entropy 25, no. 2 (February 12, 2023): 337. http://dx.doi.org/10.3390/e25020337.

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This paper deals with the problem of establishing a systematic theoretical formulation of variational principles for the continuum gravitational field dynamics of classical General Relativity (GR). In this reference, the existence of multiple Lagrangian functions underlying the Einstein field equations (EFE) but having different physical connotations is pointed out. Given validity of the Principle of Manifest Covariance (PMC), a set of corresponding variational principles can be constructed. These are classified in two categories, respectively, referred to as constrained and unconstrained Lagrangian principles. They differ for the normalization properties required to be satisfied by the variational fields with respect to the analogous conditions holding for the extremal fields. However, it is proved that only the unconstrained framework correctly reproduces EFE as extremal equations. Remarkably, the synchronous variational principle recently discovered belongs to this category. Instead, the constrained class can reproduce the Hilbert–Einstein formulation, although its validity demands unavoidably violation of PMC. In view of the mathematical structure of GR based on tensor representation and its conceptual meaning, it is therefore concluded that the unconstrained variational setting should be regarded as the natural and more fundamental framework for the establishment of the variational theory of EFE and the consequent formulation of consistent Hamiltonian and quantum gravity theories.

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11

He, Ji-Huan. "Hamilton Principle and Generalized Variational Principles of Linear Thermopiezoelectricity." Journal of Applied Mechanics 68, no. 4 (October 19, 2000): 666–67. http://dx.doi.org/10.1115/1.1352067.

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12

Yuan, Xiao, Suguru Endo, Qi Zhao, Ying Li, and Simon C. Benjamin. "Theory of variational quantum simulation." Quantum 3 (October 7, 2019): 191. http://dx.doi.org/10.22331/q-2019-10-07-191.

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The variational method is a versatile tool for classical simulation of a variety of quantum systems. Great efforts have recently been devoted to its extension to quantum computing for efficiently solving static many-body problems and simulating real and imaginary time dynamics. In this work, we first review the conventional variational principles, including the Rayleigh-Ritz method for solving static problems, and the Dirac and Frenkel variational principle, the McLachlan's variational principle, and the time-dependent variational principle, for simulating real time dynamics. We focus on the simulation of dynamics and discuss the connections of the three variational principles. Previous works mainly focus on the unitary evolution of pure states. In this work, we introduce variational quantum simulation of mixed states under general stochastic evolution. We show how the results can be reduced to the pure state case with a correction term that takes accounts of global phase alignment. For variational simulation of imaginary time evolution, we also extend it to the mixed state scenario and discuss variational Gibbs state preparation. We further elaborate on the design of ansatz that is compatible with post-selection measurement and the implementation of the generalised variational algorithms with quantum circuits. Our work completes the theory of variational quantum simulation of general real and imaginary time evolution and it is applicable to near-term quantum hardware.

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13

ANTONEVICH, A. B., V. I. BAKHTIN, and A. V. LEBEDEV. "Ont-entropy and variational principle for the spectral radii of transfer and weighted shift operators." Ergodic Theory and Dynamical Systems 31, no. 4 (June 25, 2010): 995–1042. http://dx.doi.org/10.1017/s0143385710000210.

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AbstractThe paper deals with the variational principles for evaluation of the spectral radii of transfer and weighted shift operators associated with a dynamical system. These variational principles have been the matter of numerous investigations and the principal results have been achieved in the situation when the dynamical system is either reversible or a topological Markov chain. As the main summands, these principles contain the integrals over invariant measures and the Kolmogorov–Sinai entropy. In the paper we derive the variational principle for anarbitrarydynamical system. It gives the explicit description of the Legendre dual object to the spectral potential. It is shown that in general this principle contains not the Kolmogorov–Sinai entropy but a new invariant of entropy type—thet-entropy.

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14

Tessarotto, Massimo, and Claudio Cremaschini. "Background Independence and Gauge Invariance in General Relativity Part 1—The Classical Theory." Symmetry 14, no. 10 (October 6, 2022): 2083. http://dx.doi.org/10.3390/sym14102083.

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The implications of the principles of general and manifest covariance, together with those of the objectivity principle, are considered for the purpose of establishing a DeDonder–Weyl-type Hamiltonian variational formulation for classical general relativity. Based on the analysis of the Einstein–Hilbert variational principle, it is shown that only synchronous variational principles permit the construction of fully 4−tensor Lagrangian and Hamiltonian theories of this type. In addition, the possible validity of an extended Hamiltonian formulation in which Lagrangian variables include also the Ricci tensor is investigated and shown to occur provided the classical cosmological constant is non-vanishing.

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15

Marotti de Sciarra, Francesco. "Some Variational Principles for Coupled Thermoelasticity." Journal of Engineering 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/516462.

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The nonlinear thermoelasticity of type II proposed by Green and Naghdi is considered. The thermoelastic structural model is formulated in a quasistatic range, and the related thermoelastic variational formulation in the complete set of state variables is recovered. Hence a consistent framework to derive all the variational formulations with different combinations of the state variables is provided, and a family of mixed variational formulations, with different combinations of state variables, is provided starting from the general variational formulation. A uniqueness condition is provided on the basis of a suitable variational formulation.

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16

Pedregal, Pablo. "Nonlocal variational principles." Nonlinear Analysis: Theory, Methods & Applications 29, no. 12 (December 1997): 1379–92. http://dx.doi.org/10.1016/s0362-546x(96)00185-x.

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17

Mikhailov, M. D. "Generalized variational principles." Communications in Applied Numerical Methods 6, no. 5 (July 1990): 369–73. http://dx.doi.org/10.1002/cnm.1630060506.

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18

Baolian, Fu. "Coupled variational principles and generalized coupled variational principles in photoelasticity." Applied Mathematics and Mechanics 17, no. 1 (January 1996): 39–44. http://dx.doi.org/10.1007/bf00131293.

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19

Brajerčík, J., and D. Krupka. "Variational principles for locally variational forms." Journal of Mathematical Physics 46, no. 5 (May 2005): 052903. http://dx.doi.org/10.1063/1.1901323.

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20

He, J. H. "Generalized Hellinger-Reissner Principle." Journal of Applied Mechanics 67, no. 2 (January 16, 1998): 326–31. http://dx.doi.org/10.1115/1.1303826.

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By the semi-inverse method of establishing variational principles, the Hellinger-Reissner principle can be obtained straightforwardly from energy trial-functionals without using Lagrange multipliers, and a family of generalized Hellinger-Reissner principles with an arbitrary constant are also obtained, some of which are unknown to us at the present time. The present theory provides a straightforward tool to search for various variational principles directly from governing equations and boundary conditions. [S0021-8936(00)00702-9]

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21

Liu, Hong-Yan, Lei Zhao, and Hong-Yan Tu. "A short remark on Stewart 1962 variational principle for laminar flow in a uniform duct." Thermal Science 20, no. 1 (2016): 359–61. http://dx.doi.org/10.2298/tsci140321063l.

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This paper concludes that Stewart 1962 variational principle for laminar flow in a uniform duct is for a differential-difference. Some generalized variational principles are elucidated with or without Stewart?s discrete treatment.

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22

Mang, H., P. Helnwein, and R. H. Gallagher. "On the Nonuniqueness of Solutions Obtained With Simplified Variational Principles." Journal of Applied Mechanics 63, no. 3 (September 1, 1996): 820–27. http://dx.doi.org/10.1115/1.2823368.

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The attempt to satisfy subsidiary conditions in boundary value problems without additional independent unknowns in the form of Lagrange multipliers has led to the development of so-called “simplified variational principles.” They are based on using the Euler-Lagrange equations for the Lagrange multipliers to express the multipliers in terms of the original variables. It is shown that the conversion of a variational principle with subsidiary conditions to such a simplified variational principle may lead to the loss of uniqueness of the solution of a boundary value problem. A particularly simple form of the geometrically nonlinear theory of bending of beams is used as the vehicle for this proof. The development given in this paper is entirely analytical. Hence, the deficiencies of the investigated simplified variational principle are fundamental.

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23

Ván, P., and R. Kovács. "Variational principles and nonequilibrium thermodynamics." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 378, no. 2170 (March 30, 2020): 20190178. http://dx.doi.org/10.1098/rsta.2019.0178.

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Variational principles play a fundamental role in deriving the evolution equations of physics. They work well in the case of non-dissipative evolution, but for dissipative systems, the variational principles are not unique and not constructive. With the methods of modern nonequilibrium thermodynamics, one can derive evolution equations for dissipative phenomena and, surprisingly, in several cases, one can also reproduce the Euler–Lagrange form and symplectic structure of the evolution equations for non-dissipative processes. In this work, we examine some demonstrative examples and compare thermodynamic and variational techniques. Then, we argue that, instead of searching for variational principles for dissipative systems, there is another viable programme: the second law alone can be an effective tool to construct evolution equations for both dissipative and non-dissipative processes. This article is part of the theme issue ‘Fundamental aspects of nonequilibrium thermodynamics’.

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24

Song, Hai Yan, Zhen Gong Zhou, Li Fu Liang, and Zong Min Liu. "Generalized Variational Principles of Quasi-Static Electro-Magneto-Thermo-Elasticity." Key Engineering Materials 419-420 (October 2009): 153–56. http://dx.doi.org/10.4028/www.scientific.net/kem.419-420.153.

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Finite element method is an approximate computational method which is widely used and continuously developed. Variational principles are the foundation of finite element method. In this paper, the generalized variational principles of quasi-static electro-magneto-thermo-elasticity are established using variational integral method. And these variational principles can be degenerated to be variational principles of simple coupling properties materials, which offer theoretical support for finite element method of coupling problems of electro-magneto-thermo-elastic.

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25

Fan, Tao, and Hai Yan Song. "Generalized Quasi-Variational Principles and Application in Nonlinear Non-Conservative Elasto-Dynamics." Key Engineering Materials 385-387 (July 2008): 577–80. http://dx.doi.org/10.4028/www.scientific.net/kem.385-387.577.

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The generalized quasi-variational principles with two kinds of variables of time initial value problem were established in nonlinear non-conservative elasto-dynamics. Then, the analytic solution of time initial value problem of a typical non-conservative elasto-dynamics was studied by applying the obtained quasi-complementary variational principle.

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26

LOPEZ, C. "VARIATIONAL CALCULUS, SYMMETRIES AND REDUCTION." International Journal of Geometric Methods in Modern Physics 03, no. 03 (May 2006): 577–90. http://dx.doi.org/10.1142/s0219887806001260.

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Variational Principles and Symmetries are both fundamental ingredients of the most relevant (either classical or modern) physical theories, and of many interesting models in Pure and Applied Mathematics. While the variational principle determines the law (differential and sometimes algebraic equations) of the dynamics, the symmetries give way to reduction. Noether's theorem connects some symmetries with constants of the motion in the framework of Lagrangian (point symmetries) or Hamiltonian (canonical symmetries) systems, becoming a powerful tool in the reduction process. This entanglement between variational principles and symmetries is the source of a very rich machinery that holds outstanding mathematical beauty and physical interpretations; however, it sometimes generates misunderstandings about the different roles played by each of these geometric ingredients in the framework of the theory. The aim of this paper is to review some basic facts about the geometric anatomy and physiology of classical dynamical systems ruled by a variational principle and subject to symmetries, distilling the inherited properties of the system according to its cause. A brief discussion of some illustrative examples accompany the exposition to enhance the main ideas.

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27

Zhong, Xing-Fu, and Zhi-Jing Chen. "Variational principles for topological pressures on subsets." Nonlinearity 36, no. 2 (January 5, 2023): 1168–91. http://dx.doi.org/10.1088/1361-6544/acadca.

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Abstract In this paper, we investigate the relations between various types of topological pressures and different versions of measure-theoretical pressures. We extend Feng–Huang’s variational principle for packing entropy to packing pressure and obtain two new variational principles for Pesin–Pitskel and packing pressures respectively. We show that various types of Katok pressures for an ergodic measure with respect to a potential function are equal to the sum of measure-theoretic entropy of this measure and the integral of the potential function. Moreover, we obtain Billingsley type theorem for packing pressure, which indicates that packing pressure can be determined by measure-theoretic upper local pressure of measures, and a variational principle for packing pressure of the set of generic points for any invariant ergodic Borel probability measure.

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28

Milton, Graeme W., Pierre Seppecher, and Guy Bouchitté. "Minimization variational principles for acoustics, elastodynamics and electromagnetism in lossy inhomogeneous bodies at fixed frequency." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465, no. 2102 (October 10, 2008): 367–96. http://dx.doi.org/10.1098/rspa.2008.0195.

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The classical energy minimization principles of Dirichlet and Thompson are extended as minimization principles to acoustics, elastodynamics and electromagnetism in lossy inhomogeneous bodies at a fixed frequency. This is done by building upon the ideas of Cherkaev and Gibiansky, who derived minimization variational principles for quasistatics. In the absence of free current, the primary electromagnetic minimization variational principles have a minimum, which is the time-averaged electrical power dissipated in the body. The variational principles provide constraints on the boundary values of the fields when the moduli are known. Conversely, when the boundary values of the fields have been measured, then they provide information about the values of the moduli within the body. This should have applications to electromagnetic tomography. We also derive saddle-point variational principles that correspond to the variational principles of Gurtin, Willis and Borcea.

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29

Zhukovskaya, Zukhra Tagirovna, and Sergey Evgenyevich Zhukovskiy. "ON GENERALIZATIONS AND APPLICATIONS OF VARIATIONAL PRINCIPLES OF NONLINEAR ANALYSIS." Tambov University Reports. Series: Natural and Technical Sciences, no. 123 (2018): 377–85. http://dx.doi.org/10.20310/1810-0198-2018-23-123-377-385.

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There are considered some classes of functions to which variational principles of nonlinear are applicable. In particular, it is shown that the Bishop-Phelps variational principle is applicable to some unbounded below functions. The properties of locally Lipschitzian mappings are investigated. Conditions for a mapping that is pseudo-Lipschitzian at every point of its graph to be Lipschitzian are derived.

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30

Rauch, Marc. "Multivariate variational principles for topological pressure." Stochastics and Dynamics 15, no. 03 (May 22, 2015): 1550016. http://dx.doi.org/10.1142/s0219493715500161.

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We prove two variational principles for continuous real-valued functions defined on Cartesian products of a discrete dynamical system. They involve nonadditive topological pressure for a certain class of potentials, and can be seen as extension to the classical variational principles.

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31

Zhang, Guohua. "Variational principles of pressure." Discrete & Continuous Dynamical Systems - A 24, no. 4 (2009): 1409–35. http://dx.doi.org/10.3934/dcds.2009.24.1409.

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32

Kondo, Kyohei. "Variational Principles of Elastica." JOURNAL OF THE JAPAN SOCIETY FOR AERONAUTICAL AND SPACE SCIENCES 52, no. 603 (2004): 167–74. http://dx.doi.org/10.2322/jjsass.52.167.

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33

Micchelli, Ch A. "Curves from variational principles." ESAIM: Mathematical Modelling and Numerical Analysis 26, no. 1 (1992): 77–93. http://dx.doi.org/10.1051/m2an/1992260100771.

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34

Finet, Catherine, Lucas Quarta, and Christophe Troestler. "Vector-valued variational principles." Nonlinear Analysis: Theory, Methods & Applications 52, no. 1 (January 2003): 197–218. http://dx.doi.org/10.1016/s0362-546x(02)00103-7.

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35

Pasicki, Lech. "Transitivity and variational principles." Nonlinear Analysis: Theory, Methods & Applications 74, no. 16 (November 2011): 5678–84. http://dx.doi.org/10.1016/j.na.2011.05.054.

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36

Behera, N. "Variational principles in evolution." Bulletin of Mathematical Biology 58, no. 1 (January 1996): 175–202. http://dx.doi.org/10.1016/0092-8240(95)00316-9.

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37

Lazzari, B. "Variational principles in electromagnetism." IMA Journal of Applied Mathematics 65, no. 1 (August 1, 2000): 45–95. http://dx.doi.org/10.1093/imamat/65.1.45.

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38

Ng, S. K., and W. L. Chan. "Stochastic complementary variational principles." International Journal of Engineering Science 23, no. 6 (January 1985): 633–39. http://dx.doi.org/10.1016/0020-7225(85)90133-8.

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39

Cresson, Jacky. "Non-differentiable variational principles." Journal of Mathematical Analysis and Applications 307, no. 1 (July 2005): 48–64. http://dx.doi.org/10.1016/j.jmaa.2004.10.006.

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40

Kazantsev, V. P. "Variational principles of magnetostatics." Soviet Physics Journal 30, no. 7 (July 1987): 624–28. http://dx.doi.org/10.1007/bf00895233.

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41

Behera, Narayan. "Variational principles in evolution." Bulletin of Mathematical Biology 58, no. 1 (January 1996): 175–202. http://dx.doi.org/10.1007/bf02458287.

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42

Chandrasekar, M. "Thermodynamical Modeling of Boundary Layer Flow With Suction and Injection." Journal of Applied Mechanics 65, no. 3 (September 1, 1998): 764–68. http://dx.doi.org/10.1115/1.2789121.

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The variational principle developed by Gyarmati embodying the principles of non-equilibrium thermodynamics is employed to investigate the laminar boundary layer effect on wedge flows with suction and injection. The velocity function is assumed as a simple third-degree polynomial and the variational principle is formulated. The hydrodynamical boundary layer thickness is derived as the Euler-Lagrange equation of the variational principle. The velocity profiles and skin friction values were computed for various values of suction and injection parameter and wedge angle parameter. The comparison of the present solution with an available exact solution establishes the fact that the accuracy is remarkable.

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43

Wei-zang, Chien. "Variational principles and generalized variational principles for nonlinear elasticity with finite displacement." Applied Mathematics and Mechanics 9, no. 1 (January 1988): 1–12. http://dx.doi.org/10.1007/bf02017881.

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44

Teichtmeister, S., and M. A. Keip. "A Variational Framework for the Thermomechanics of Gradient-Extended Dissipative Solids – with Applications to Diffusion, Damage and Plasticity." Journal of Elasticity 148, no. 1 (January 2022): 81–126. http://dx.doi.org/10.1007/s10659-022-09884-6.

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AbstractThe paper presents a versatile framework for solids which undergo nonisothermal processes with irreversibly changing microstructure at large strains. It outlines rate-type and incremental variational principles for the full thermomechanical coupling in gradient-extended dissipative materials. It is shown that these principles yield as Euler equations essentially the macro- and micro-balances as well as the energy equation. Starting point is the incorporation of the entropy and entropy rate as canonical arguments into constitutive energy and dissipation functions, which additionally depend on the gradient-extended mechanical state and its rate, respectively. By means of (generalized) Legendre transformations, extended variational principles with thermal as well as mechanical driving forces can be constructed. On the thermal side, a rigorous distinction between the quantity conjugate to the entropy and the quantity conjugate to the entropy rate is essential here. Formulations with mechanical driving forces are especially suitable when considering possibly temperature-dependent threshold mechanisms. With regard to variationally consistent incrementations, we suggest an update scheme which renders the exact form of the intrinsic dissipation and is highly suitable when considering adiabatic processes. It is shown that this proposed numerical algorithm has the structure of an operator split. To underline the broad applicability of the proposed framework, we set up three model problems as applications: Cahn-Hilliard diffusion coupled with temperature evolution, where we propose a new variational principle in terms of the species flux vector, as well as thermomechanics of gradient damage and gradient plasticity. In a numerical example we study the formation of a cross shear band.

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45

Cuesta, Mabel. "Minimax theorems onC1manifolds via Ekeland variational principle." Abstract and Applied Analysis 2003, no. 13 (2003): 757–68. http://dx.doi.org/10.1155/s1085337503303100.

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We prove two minimax principles to find almost critical points ofC1functionals restricted to globally definedC1manifolds of codimension1. The proof of the theorems relies on Ekeland variational principle.

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46

Gladwell, G. M. L. "Follower forces: Leipholz's researches into generalized variational principles." Canadian Journal of Civil Engineering 17, no. 3 (June 1, 1990): 287–93. http://dx.doi.org/10.1139/l90-035.

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Classical variational principles, for conservative systems, are associated with the names of Lagrange and Hamilton. Generalized variational principles for nonconservative systems were introduced by various authors and in various physical contexts in the period 1945 – 1966. This review traces their development and their use, by Leipholz, in the period 1971 – 1986. Key words: follower forces, variational principles, generalized, nonconservative, adjoint systems, stability, divergence, flutter.

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Dokmeci, M. Cengiz, and Gulay Altay. "Hamilton’s principle applied to piezomagnetism and related variational principles." Journal of the Acoustical Society of America 115, no. 5 (May 2004): 2462. http://dx.doi.org/10.1121/1.4782366.

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Dökmeci, M. Cengiz. "Hamilton’s principle and associated variational principles in polar thermopiezoelectricity." Physica A: Statistical Mechanics and its Applications 389, no. 15 (August 2010): 2966–74. http://dx.doi.org/10.1016/j.physa.2010.01.002.

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Altay, Gülay, and M. Cengİz Dökmecİ. "On the equations governing the motion of an anisotropic poroelastic material." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 462, no. 2072 (March 21, 2006): 2373–96. http://dx.doi.org/10.1098/rspa.2006.1665.

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We address Biot's equations governing the motion of an anisotropic fluid-saturated poroelastic material with certain properties. First, we investigate the uniqueness in solutions of the three-dimensional governing equations for the regular region of the poroelastic material and enumerate the conditions sufficient for the uniqueness. Next, by applying Hamilton's principle to the motion of the region, we obtain a variational principle that generates only the Biot–Newton equations and the associated natural boundary conditions. Then, by extending the variational principle for the region with an internal fixed surface of discontinuity through Legendre's transformation, we derive a six-field variational principle that operates on all the poroelastic field variables. The variational principle leads, as its Euler–Lagrange equations, to all the governing equations, including the jump conditions but the initial conditions, as a generalized version of the Hellinger–Reissner variational principle. Moreover, we consider the free vibrations of the region, and we discuss some basic properties of eigenvalues and present a variational formulation by Rayleigh's quotient. This work provides a standard tool with the features of variational principles when numerically solving the governing equations in heterogeneous media with finite element methods, treating the free vibrations and consistently deriving some one-dimensional/two-dimensional equations of the poroelastic region.

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Cao, Xiao-Qun, Si-Hang Xie, Hong-Ze Leng, Wen-Long Tian, and Jia-Le Yao. "Generalized variational principles for the modified Benjamin-Bona-Mahony equation in the fractal space." Thermal Science 28, no. 3 Part A (2024): 2341–49. http://dx.doi.org/10.2298/tsci2403341c.

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Because variational principles are very important for some methods to get the numerical or exact solutions, it is very important to seek explicit variational formulations for the non-linear PDE. At first, this paper describes the modified Benjamin-Bona-Mahony equation in fractal porous media or with irregular boundaries. Then, by designing skillfully the trial-Lagrange functional, variational principles are successfully established for the modified Benjamin-Bona-Mahony equation in the fractal space, respectively. Furthermore, the obtained variational principles are proved correct by minimizing the functionals with the calculus of variations.

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