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Relevant bibliographies by topics / Lagrangian functions

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Academic literature on the topic 'Lagrangian functions'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 August 2024

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Journal articles on the topic "Lagrangian functions"

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Musielak, Zdzislaw E., Niyousha Davachi, and Marialis Rosario-Franco. "Special Functions of Mathematical Physics: A Unified Lagrangian Formalism." Mathematics 8, no. 3 (March 9, 2020): 379. http://dx.doi.org/10.3390/math8030379.

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Lagrangian formalism is established for differential equations with special functions of mathematical physics as solutions. Formalism is based on either standard or non-standard Lagrangians. This work shows that the procedure of deriving the standard Lagrangians leads to Lagrangians for which the Euler–Lagrange equation vanishes identically, and that only some of these Lagrangians become the null Lagrangians with the well-defined gauge functions. It is also demonstrated that the non-standard Lagrangians require that the Euler–Lagrange equations are amended by the auxiliary conditions, which is a new phenomenon in the calculus of variations. The existence of the auxiliary conditions has profound implications on the validity of the Helmholtz conditions. The obtained results are used to derive the Lagrangians for the Airy, Bessel, Legendre and Hermite equations. The presented examples clearly demonstrate that the developed Lagrangian formalism is applicable to all considered differential equations, including the Airy (and other similar) equations, and that the regular and modified Bessel equations are the only ones with the gauge functions. Possible implications of the existence of the gauge functions for these equations are discussed.

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Xuegang, Yu. "Hyperbolic Lagrangian functions." Applied Mathematics and Mechanics 19, no. 12 (December 1998): 1189–95. http://dx.doi.org/10.1007/bf02456640.

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Ambot, Amelia A. P., Herry F. Lalus, and Hartoyo Yudhawardana. "ANALISIS LAGRANGIAN NULL NONSTANDAR DAN FUNGSI GAUGE UNTUK HUKUM INERSIA NEWTON : SEBUAH REVIEW." JOURNAL ONLINE OF PHYSICS 9, no. 1 (November 2, 2023): 6–14. http://dx.doi.org/10.22437/jop.v9i1.25909.

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This paper describes a review of a journal entitled 'Nonstandard Null Lagrangians and Gauge Functions for Newtonian Law of Inertia' which discusses Nonstandard Lagrangian Null solutions for Newton's Law of Inertia. The purpose of this study is to present in detail the Lagrangian Formalism method for generating Nonstandard Lagrangian Null and its Gauge function for Newton's Law of Inertia, as well as the role of action invariant in generating Lagrangian Null and Exact Gauge functions, by deriving a one-dimensional oscillator arm using the basic Lagrangian equations. The Nonstandard Null Lagrangian is derived from the Nonstandard Lagrangian, then the two Lagrangian Null are entered into the action invariant to make it Exact, after the Nonstandard Lagrangian Null is declared Exact, it is substituted into which represents Newton's Law of Inertia. The results of this research show that an Exact Nonstandard Lagrangian Null can be generated by making the Gauge Function Invariant, so that the first Nonstandard Lagrangian Null and Gauge Function for Newton's Law of Inertia are obtained.

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Musielak, Z. E., N. Davachi, and M. Rosario-Franco. "Lagrangians, Gauge Functions, and Lie Groups for Semigroup of Second-Order Differential Equations." Journal of Applied Mathematics 2020 (June 22, 2020): 1–11. http://dx.doi.org/10.1155/2020/3170130.

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A set of linear second-order differential equations is converted into a semigroup, whose algebraic structure is used to generate novel equations. The Lagrangian formalism based on standard, null, and nonstandard Lagrangians is established for all members of the semigroup. For the null Lagrangians, their corresponding gauge functions are derived. The obtained Lagrangians are either new or generalization of those previously known. The previously developed Lie group approach to derive some equations of the semigroup is also described. It is shown that certain equations of the semigroup cannot be factorized, and therefore, their Lie groups cannot be determined. A possible solution of this problem is proposed, and the relationship between the Lagrangian formalism and the Lie group approach is discussed.

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ZHANG, LI-WEI, YONG-HONG REN, YUE WU, and XIAN-TAO XIAO. "A CLASS OF NONLINEAR LAGRANGIANS: THEORY AND ALGORITHM." Asia-Pacific Journal of Operational Research 25, no. 03 (June 2008): 327–71. http://dx.doi.org/10.1142/s021759590800178x.

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This paper establishes a theory framework of a class of nonlinear Lagrangians for solving nonlinear programming problems with inequality constraints. A set of conditions are proposed to guarantee the convergence of nonlinear Lagrangian algorithms, to analyze condition numbers of nonlinear Lagrangian Hessians as well as to develop the dual approaches. These conditions are satisfied by well-known nonlinear Lagrangians appearing in literature. The convergence theorem shows that the dual algorithm based on any nonlinear Lagrangian in the class is locally convergent when the penalty parameter is less than a threshold under a set of suitable conditions on problem functions and the error bound solution, depending on the penalty parameter, is also established. The paper also develops the dual problems based on the proposed nonlinear Lagrangians, and the related duality theorem and saddle point theorem are demonstrated. Furthermore, it is shown that the condition numbers of Lagrangian Hessians at optimal solutions are proportional to the controlling penalty parameters. We report some numerical results obtained by using nonlinear Lagrangians.

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Musielak, Zdzislaw E. "Nonstandard Null Lagrangians and Gauge Functions for Newtonian Law of Inertia." Physics 3, no. 4 (October 4, 2021): 903–12. http://dx.doi.org/10.3390/physics3040056.

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New null Lagrangians and gauge functions are derived and they are called nonstandard because their forms are different than those previously found. The invariance of the action is used to make the Lagrangians and gauge functions exact. The first exact nonstandard null Lagrangian and its gauge function for the law of inertia are obtained, and their physical implications are discussed.

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El-Nabulsi, R. A. "Nonstandard fractional exponential Lagrangians, fractional geodesic equation, complex general relativity, and discrete gravity." Canadian Journal of Physics 91, no. 8 (August 2013): 618–22. http://dx.doi.org/10.1139/cjp-2013-0145.

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Nonstandard Lagrangians are generating functions of different equations of motion. They have gained increasing importance in many different fields. In fact, nonstandard Lagrangians date back to 1978, when Arnold entitled them “non-natural” in his classic book, Mathematical Methods of Classical Mechanics (Springer, New York. 1978). In applied mathematics, most dynamical equations can be obtained by using generating Lagrangian functions (e.g., power-law and exponential Lagrangians), which has been shown by mathematicians, who have also demonstrated that there is an infinite number of such functions. Besides this interesting field, the topic of fractional calculus of variations has gained growing importance because of its wide application in different fields of science. In this paper, we generalize the fractional actionlike variational approach for the case of a nonstandard exponential Lagrangian. To appreciate this new approach, we explore some of its main consequences in Einstein’s general relativity. Some results are revealed and discussed accordingly mainly the transition from general relativity to complex relativity and emergence of a discrete gravitational coupling constant.

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Chen, Bang-Yen. "Jacobi's elliptic functions and Lagrangian immersions." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 126, no. 4 (1996): 687–704. http://dx.doi.org/10.1017/s0308210500023003.

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First, we establish a sharp inequality between the squared mean curvature and the scalar curvature for a Lagrangian submanifold in a nonflat complex-space-form. Then, by utilising the Jacobi's elliptic functions en and dn, we introduce three families of Lagrangian submanifolds and two exceptional Lagrangian submanifolds Fn, Ln in nonflat complex-space-forms which satisfy the equality case of the inequality. Finally, we obtain the complete classification of Lagrangian submanifolds in nonflat complex-space-forms which satisfy this basic equality.

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Jeffrey, Lisa. "Chern–Simons gauge theory and symplectic quantum mechanics." Canadian Journal of Physics 93, no. 9 (September 2015): 971–73. http://dx.doi.org/10.1139/cjp-2014-0563.

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We describe the relation between the Chern–Simons gauge theory partition function and the partition function defined using the symplectic action functional as the Lagrangian. We show that the partition functions obtained using these two Lagrangians agree, and we identify the semiclassical formula for the partition function defined using the symplectic action functional. We also compute the semiclassical formulas for the partition functions obtained using the two different Lagrangians: the Chern–Simons functional and the symplectic action functional.

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Obaidullah, U., and Sameerah Jamal. "pp-wave potential functions: A complete study using Noether symmetries." International Journal of Geometric Methods in Modern Physics 18, no. 07 (March 18, 2021): 2150108. http://dx.doi.org/10.1142/s0219887821501085.

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In this paper, we examine the functional forms of the potentials [Formula: see text] that emerge from the Lagrangian of the pp-wave spacetime. To facilitate this investigation, Noether symmetries are employed as well as their linear combinations and subalgebras. We exploit the geometric fact that Noether point symmetries of geodesic Lagrangians are generated from the Homothetic algebra of spacetimes. Thus, we provide a complete analysis of the potentials of this spacetime, which are split into 14 isometry categories.

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Dissertations / Theses on the topic "Lagrangian functions"

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Brown, Jeffrey W. "Lagrangian field observations of rip currents." Access to citation, abstract and download form provided by ProQuest Information and Learning Company; downloadable PDF file, 133 p, 2008. http://proquest.umi.com/pqdweb?did=1633772921&sid=6&Fmt=2&clientId=8331&RQT=309&VName=PQD.

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Lai, Shih-yih. "Lagrangian relaxation and discrete optimization problems /." Digital version accessible at:, 1999. http://wwwlib.umi.com/cr/utexas/main.

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Hare, Warren L. "The Quadratic Sub-Lagrangian of prox-regular functions." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2000. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp01/MQ60128.pdf.

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Yolcu, Türkay. "Parabolic systems and an underlying Lagrangian." Atlanta, Ga. : Georgia Institute of Technology, 2009. http://hdl.handle.net/1853/29760.

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Thesis (Ph.D)--Mathematics, Georgia Institute of Technology, 2010.
Committee Chair: Gangbo, Wilfrid; Committee Member: Chow, Shui-Nee; Committee Member: Harrell, Evans; Committee Member: Swiech, Andrzej; Committee Member: Yezzi, Anthony Joseph. Part of the SMARTech Electronic Thesis and Dissertation Collection.

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Zhou, Long. "Mesoscale variability and Lagrangian statistics in the tropical North Atlantic /." View online ; access limited to URI, 2009. http://digitalcommons.uri.edu/dissertations/AAI3401129.

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Uygun, Adnan. "Network interdiction by Lagrangian relaxation and branch-and-bound." Thesis, Monterey, Calif. : Springfield, Va. : Naval Postgraduate School ; Available from National Technical Information Service, 2002. http://library.nps.navy.mil/uhtbin/hyperion-image/02Jun%5FUygun.pdf.

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Song, Qiang. "Non-euler-lagrangian pareto-optimality conditions for dynamic multiple-criterion decision problems." Diss., Georgia Institute of Technology, 1997. http://hdl.handle.net/1853/24920.

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Yolcu, Türkay. "Parabolic systems and an underlying Lagrangian." Diss., Georgia Institute of Technology, 2009. http://hdl.handle.net/1853/29760.

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In this thesis, we extend De Giorgi's interpolation method to a class of parabolic equations which are not gradient flows but possess an entropy functional and an underlying Lagrangian. The new fact in the study is that not only the Lagrangian may depend on spatial variables, but also it does not induce a metric. Assuming the initial condition is a density function, not necessarily smooth, but solely of bounded first moments and finite "entropy", we use a variational scheme to discretize the equation in time and construct approximate solutions. Moreover, De Giorgi's interpolation method is revealed to be a powerful tool for proving convergence of our algorithm. Finally, we analyze uniqueness and stability of our solution in L¹.

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Silverberg, Jon P. "On Lagrangian meshless methods in free-surface flows." Thesis, (1.7 MB), 2005. http://edocs.nps.edu/AR/topic/theses/2005/Jan/05Jan_Silverberg.pdf.

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Thesis (Master of Engineering in Ocean Engineering)--University of California at Berkeley, 2004.
"January 2005." Description based on title screen as viewed on May 25, 2010. DTIC Descriptor(s): Fluid Dynamics, Lagrangian Functions, Equations Of Motion, Acceleration, Formulations, Grids, Continuum Mechanics, Gaussian Quadrature, Derivatives (Mathematics), Compact Disks, Boundary Value Problems, Polynomials, Interpolation, Pressure, Operators (Mathematics). DTIC Identifier(s): Multimedia (CD-Rom), Moving Grids, Meshless Discretization, Lifs (Lagrange Implicit Fraction Step), Lagrangian Dynamics, Meshless Operators, Mlip (Multidimensional Lagrange Interpolating Polynomials), Flux Boundary Conditions, Radial Basis Functions Includes bibliographical references (58-59).

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Almesallmy, Mohammed. "Experimental and analytical investigation of inertial propulsion mechanisms and motion simulation of rigid-multi-body mechanical systems /." View online ; access limited to URI, 2006. http://0-digitalcommons.uri.edu.helin.uri.edu/dissertations/3239899.

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Books on the topic "Lagrangian functions"

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Brizard, Alain Jean. An introduction to Lagrangian mechanics. Hackensack, NJ: World Scientific, 2008.

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Brizard, Alain Jean. An introduction to Lagrangian mechanics. Hackensack, NJ: World Scientific, 2008.

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Deriglazov, Alexei. Classical mechanics: Hamiltonian and Lagrangian Formalism. Berlin: Springer Verlag, 2010.

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Rubinstein, Robert. Effects of helicity on Lagrangian and Eulerian time correlations in turbulence. Hampton, VA: Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, 1998.

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Abasov, T. M. Modifit︠s︡irovannye funkt︠s︡ii Lagranzha v zadachakh otyskanii︠a︡ sedlovykh tochek. Baku: Ėlm, 1989.

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Golʹshteĭn, E. G. Modifit͡s︡irovannye funkt͡s︡ii Lagranzha: Teorii͡a︡ i metody optimizat͡s︡ii. Moskva: "Nauka," Glav. red. fiziko-matematicheskoĭ lit-ry, 1989.

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Akademi͡ia nauk SSSR. Vychislitelʹnyĭ ͡tsentr, ed. Modifi͡tsirovannye funk͡tsii Lagranzha v nelineĭnom programmirovanii. Moskva: V͡TS An SSSR, 1988.

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Mušicki, Đorđe. Degenerate systems in generalized mechanics. Beograd: Matematički Institut, 1992.

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Mušicki, Đorđe. Degenerate systems in generalized mechanics. Beograd: Matematički Institut, 1992.

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Ghandehari, Mostafa. Snell's law in normed linear planes. Arlington: Dept. of Mathematics, University of Texas at Arlington, 1997.

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Book chapters on the topic "Lagrangian functions"

1

Grandy, Walter T. "Lagrangian Formulation and Green Functions." In Relativistic Quantum Mechanics of Leptons and Fields, 167–203. Dordrecht: Springer Netherlands, 1991. http://dx.doi.org/10.1007/978-94-011-3302-9_6.

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Salvaña, Mary Lai O., and Marc G. Genton. "Lagrangian Spatio-Temporal Nonstationary Covariance Functions." In Advances in Contemporary Statistics and Econometrics, 427–47. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-73249-3_22.

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Oh, Yong-Geun. "Critical Points of Potential Functions and Floer Cohomology." In Lagrangian Floer Theory and Its Deformations, 223–74. Singapore: Springer Nature Singapore, 2024. http://dx.doi.org/10.1007/978-981-97-1798-9_6.

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Schapira, Pierre. "Constructible functions, Lagrangian cycles and computational geometry." In The Gelfand Mathematical Seminars, 1990–1992, 189–202. Boston, MA: Birkhäuser Boston, 1993. http://dx.doi.org/10.1007/978-1-4612-0345-2_12.

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Di Pillo, Gianni, and Stefano Lucidi. "On Exact Augmented Lagrangian Functions in Nonlinear Programming." In Nonlinear Optimization and Applications, 85–100. Boston, MA: Springer US, 1996. http://dx.doi.org/10.1007/978-1-4899-0289-4_7.

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Mishchenko, A. S., V. E. Shatalov, and B. Yu Sternin. "The Asymptotics of Integrals of Rapidly Oscillating Functions with a Complex Phase." In Lagrangian Manifolds and the Maslov Operator, 208–31. Berlin, Heidelberg: Springer Berlin Heidelberg, 1990. http://dx.doi.org/10.1007/978-3-642-61259-6_6.

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Di Pillo, Gianni, and Stefano Lucidi. "Erratum to: On Exact Augmented Lagrangian Functions in Nonlinear Programming." In Nonlinear Optimization and Applications, 369. Boston, MA: Springer US, 1996. http://dx.doi.org/10.1007/978-1-4899-0289-4_25.

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Skiadas, Christos H. "A Lagrangian Approach for the Selection of Growth Functions in Forecasting." In Advances in Stochastic Modelling and Data Analysis, 189–94. Dordrecht: Springer Netherlands, 1995. http://dx.doi.org/10.1007/978-94-017-0663-6_12.

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Gass, Saul I., and Carl M. Harris. "Lagrangian function." In Encyclopedia of Operations Research and Management Science, 437. New York, NY: Springer US, 2001. http://dx.doi.org/10.1007/1-4020-0611-x_512.

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Rubinov, Alexander, and Xiaoqi Yang. "Augmented Lagrangians." In Lagrange-type Functions in Constrained Non-Convex Optimization, 173–220. Boston, MA: Springer US, 2003. http://dx.doi.org/10.1007/978-1-4419-9172-0_5.

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Conference papers on the topic "Lagrangian functions"

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Gracey, John. "The static potential in the Gribov-Zwanziger Lagrangian." In International Workshop on QCD Green’s Functions, Confinement and Phenomenology. Trieste, Italy: Sissa Medialab, 2010. http://dx.doi.org/10.22323/1.087.0016.

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Cao, Lu, and Weisheng Chen. "Distributed continuous-time optimization based on Lagrangian functions." In 2014 33rd Chinese Control Conference (CCC). IEEE, 2014. http://dx.doi.org/10.1109/chicc.2014.6895931.

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Noack, Matti, Johann Reger, and Jerome Jouffroy. "Adaptive Velocity Estimation for Lagrangian Systems using Modulating Functions." In 2023 IEEE International Conference on Mechatronics (ICM). IEEE, 2023. http://dx.doi.org/10.1109/icm54990.2023.10101935.

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Gu, Linxia, and Ashok V. Kumar. "A Lagrangian Uniform-Mesh Finite Element Method Applied to Problems Governed by Poisson’s Equation." In ASME 2007 International Mechanical Engineering Congress and Exposition. ASMEDC, 2007. http://dx.doi.org/10.1115/imece2007-41282.

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A method is presented for the solution of Poisson’s Equations using a Lagrangian formulation. The interpolation functions are the Lagrangian operation of those used in the classical finite element method, which automatically satisfy boundary conditions exactly even though there are no nodes on the boundaries of the domain. The integration is introduced in an implicit way by using approximated step functions. Classical surface integration terms used in the weak form are unnecessary due to the interpolation function in the Lagrangian formulation. Furthermore, the Lagrangian formulation simplified the connection between the mesh and the solid structures, thus providing a very easy way to solve the problems without a conforming mesh.

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ABOLHASSANI, JAMSHID, IDEEN SADREHAGHIGHI, SURENDRA TIWARI, and ROBERT SMITH. "Applications of Lagrangian blending functions for grid generation around airplane geometries." In 28th Aerospace Sciences Meeting. Reston, Virigina: American Institute of Aeronautics and Astronautics, 1990. http://dx.doi.org/10.2514/6.1990-9.

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Ma, Yuanwei, Dezhong Wang, Zhilong Ji, and Nan Qian. "Dynamic Correcting Dispersion Parameters of Lagrangian Puff Model in Atmospheric Tracer Experiments." In 2014 22nd International Conference on Nuclear Engineering. American Society of Mechanical Engineers, 2014. http://dx.doi.org/10.1115/icone22-30347.

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In atmospheric dispersion models of nuclear accident, the empirical dispersion coefficients were obtained under certain experiment conditions, which is different from actual conditions. This deviation brought in the great model errors. A better estimation of the radioactive nuclide’s distribution could be done by correcting coefficients with real-time observed value. This reverse problem is nonlinear and sensitive to initial value. Genetic Algorithm (GA) is an appropriate method for this correction procedure. Fitness function is a particular type of objective function to achieving the set goals. To analysis the fitness functions’ influence on the correction procedure and the dispersion model’s forecast ability, four fitness functions were designed and tested by a numerical simulation. In the numerical simulation, GA, coupled with Lagrange dispersion model, try to estimate the coefficients with model errors taken into consideration. Result shows that the fitness functions, in which station is weighted by observed value and by distance far from release point, perform better when it exists significant model error. After performing the correcting procedure on the Kincaid experiment data, a significant boost was seen in the dispersion model’s forecast ability.

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Gopalan, Balaji, Edwin Malkiel, and Joseph Katz. "Lagrangian Motion of Slightly Buoyant Droplets and Fluid Particles in Isotropic Turbulence." In ASME/JSME 2007 5th Joint Fluids Engineering Conference. ASMEDC, 2007. http://dx.doi.org/10.1115/fedsm2007-37538.

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The diffusion of slightly buoyant diesel oil droplets in isotropic turbulence is studied using high speed in-line digital holographic cinematography. Diesel fuel droplets with specific gravity 0.85 are injected into a 50×50×70 mm3 sample volume located at the central portion of a nearly isotropic turbulence facility. The turbulence in the sample volume is fully characterized using 2D PIV. Probability density functions of the Lagrangian droplet velocity are very close to a Gaussian distribution, which justifies the use of Taylor’s [1] model to calculate diffusion parameters. Similar to Friedman & Katz [2] data, our current results confirm that the mean rise velocity of diesel droplets becomes higher than the quiescent rise velocity at high turbulence levels. For most of the present droplet sizes and turbulence level, the rms of the horizontal droplet velocity fluctuations exceeds that of the horizontal fluid velocity fluctuations. The rms values of the vertical droplet velocity fluctuations are higher than those of the fluid only for the highest turbulence level. The droplet to fluid velocity rms ratio in both directions increases with turbulence level, but decreases with increasing droplet size. Assuming Fickian diffusion, Lagrangian auto-correlation functions of 22,000 droplet tracks are used for calculating the diffusion coefficient as functions of droplet size and turbulence level. Using all the data, we show that the diffusion coefficient scaled by quiescent rise velocity and the turbulence integral length scale is a monotonically increasing function of the turbulence level normalized by the droplet quiescent rise velocity.

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Le Lostec, Nechtan, Philippe Villedieu, and Olivier Simonin. "Comparison Between Grad’s and Quadrature-Based Methods of Moments for the Numerical Simulation of Unsteady Particle-Laden Flows." In ASME 2009 Fluids Engineering Division Summer Meeting. ASMEDC, 2009. http://dx.doi.org/10.1115/fedsm2009-78360.

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We present here a new method of moments for the numerical simulation of particle-laden flows. The closure needed in Eulerian methods relies on writing the kinetic descriptor, the velocity destribution function, as a sum of delta-functions instead of the one-delta-function or close-to-Maxwellian assumption in existing methods. The closure velocity distribution function parameters are computed from the transported moments using a quadrature method. Simulation results are compared to those of a close-to-Maxwellian-based Eulerian method and those of a reference Lagrangian simulation, considering only transport and drag of particles in a Taylor-Green fluid flow. For a particular Stokes number of 1 the velocity distribution function is far from equilibrium and particle trajectory crossing is an important feature. We find that the quadrature-based method performs better than the close-to-equilibrium-based method, giving moment profiles closer to those of the Lagrangian reference simulation. However significant differences still remain between quadrature-based and Lagrangian methods results.

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Bof, Nicoletta, Ruggero Carli, and Luca Schenato. "On the performance of consensus based versus Lagrangian based algorithms for quadratic cost functions." In 2016 European Control Conference (ECC). IEEE, 2016. http://dx.doi.org/10.1109/ecc.2016.7810280.

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Coster, J. E., N. Stander, and J. A. Snyman. "Trust Region Augmented Lagrangian Methods With Secant Hessian Updating Applied to Structural Optimization." In ASME 1996 Design Engineering Technical Conferences and Computers in Engineering Conference. American Society of Mechanical Engineers, 1996. http://dx.doi.org/10.1115/96-detc/dac-1461.

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Abstract The problem of determining the optimal sizing design of truss structures is considered. An augmented Lagrangian optimization algorithm which uses a quadratic penalty term is formulated. The implementation uses a first-order Lagrange multiplier update and a strategy for progressively increasing the accuracy with which the bound constrained minimizations are performed. The allowed constraint violation is also progressively decreased but at a slower rate so as to prevent ill-conditioning due to large penalty values. Individual constraint penalties are used and only the penalties of the worst violated constraints are increased. The scheme is globally convergent. The bound constrained minimizations are performed using the SBMIN algorithm where a sophisticated trust-region strategy is employed. The Hessian of the augmented Lagrangian function is approximated using partitioned secant updating. Each function contributing to the Lagrangian is individually approximated by a secant update and the augmented Lagrangian Hessian is formed by appropriate accumulation. The performance of the algorithm is evaluated for a number of different secant updates on standard explicit and truss sizing optimization problems. The results show the formulation to be superior to other implementations of augmented Lagrangian methods reported in the literature and that, under certain conditions, the method approaches the performance of the state-of-the-art SQP and SAM methods. Of the secant updates, the symmetric rank one update is superior to the other updates including the BFGS scheme. It is suggested that the individual function, secant updating employed may be usefully applied in contexts where structural analysis and optimization are performed simultaneously, as in the simultaneous analysis and design method. In such cases the functions are partially separable and the associated Hessians are of low rank.

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Reports on the topic "Lagrangian functions"

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Quan, Michael, and Steven Walton. Arbitrary-order Bernstein basis functions for Lagrangian Hydrodynamics. Office of Scientific and Technical Information (OSTI), August 2022. http://dx.doi.org/10.2172/1883100.

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Liu and Nixon. L52305 Probabilistic Analysis of Pipeline Uplift Resistance. Chantilly, Virginia: Pipeline Research Council International, Inc. (PRCI), June 2010. http://dx.doi.org/10.55274/r0000002.

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To investigate the feasibility of probabilistic analyses of the peak uplift resistance in frozen soils by varying parameters that are known to be important for the development of the uplift resistance under the upward movement of a pipe. A buried pipeline will be subjected to a variety of forces, both internal and external, including the interaction of the pipe with the surrounding soil. The soil-pipe interaction in permafrost regions have to account for the behavior of frozen and unfrozen soil, and transitions between the two as the pipeline traverses in a discontinuous permafrost zone. The variations in the properties and behavior of frozen soils are expected to be substantial in three dimensions of the Right-of-Way (ROW) and with time (seasonal fluctuations and changes with the history of pipeline operation). Given the uncertainties with frozen soil properties and the changes in behavior with time and location, a large variation in soil-pipe interaction characteristics can exist. The uplift resistance of a pipeline is one of these soil-pipe interactions that can be impacted by a variation in soil condition and state. A need was identified to outline the use of a probabilistic analysis of pipe uplift resistance in an attempt to capture the magnitude of these variations and uncertainties of frozen soil and the impact on the soil-pipe interaction. The probabilistic analysis allows the designer of a pipeline to consider a range of uplift resistance to a certain confidence level that would represent the likely values that a pipe may be subjected to. The work presented in this report is more focused on the methodology of the probabilistic approach, rather than the analysis itself for a specific design case, even though an example is provided for illustration purposes. A series of numerical simulations using Fast Lagrangian Analysis of Continua (FLAC) were completed varying one parameter with each run to develop a library of peak uplift resistances for a variety of different temperatures, soil properties and pipe parameters. The FLAC model was previously developed for PRC, a summary of this report is provided here to outline important parameters that were used to complete this analysis. The simulations were used to develop a correlation of peak uplift resistance as a function of soil tensile strain limit, modulus of deformation, and creep of frozen soils. Each of these parameters is dependent of the pipeline conditions such as temperature, displacement rate, and effective frozen cover depth. It is noted that the scope of the work was to develop a probabilistic method of estimating peak uplift resistance in frozen soils. Even though some sensitivity analysis were carried out, as outlined later in this report, to assess the impacts of the variable, detailed uncertainty analysis or risk assessment were not performed.

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Russo, David, and William A. Jury. Characterization of Preferential Flow in Spatially Variable Unsaturated Field Soils. United States Department of Agriculture, October 2001. http://dx.doi.org/10.32747/2001.7580681.bard.

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Preferential flow appears to be the rule rather than the exception in field soils and should be considered in the quantitative description of solute transport in the unsaturated zone of heterogeneous formations on the field scale. This study focused on both experimental monitoring and computer simulations to identify important features of preferential flow in the natural environment. The specific objectives of this research were: (1) To conduct dye tracing and multiple tracer experiments on undisturbed field plots to reveal information about the flow velocity, spatial prevalence, and time evolution of a preferential flow event; (2) To conduct numerical experiments to determine (i) whether preferential flow observations are consistent with the Richards flow equation; and (ii) whether volume averaging over a domain experiencing preferential flow is possible; (3) To develop a stochastic or a transfer function model that incorporates preferential flow. Regarding our field work, we succeeded to develop a new method for detecting flow patterns faithfully representing the movement of water flow paths in structured and non-structured soils. The method which is based on application of ammonium carbonate was tested in a laboratory study. Its use to detect preferential flow was also illustrated in a field experiment. It was shown that ammonium carbonate is a more conservative tracer of the water front than the popular Brilliant Blue. In our detailed field experiments we also succeeded to document the occurrence of preferential flow during soil water redistribution following the cessation of precipitation in several structureless field soils. Symptoms of the unstable flow observed included vertical fingers 20 - 60 cm wide, isolated patches, and highly concentrated areas of the tracers in the transmission zone. Soil moisture and tracer measurements revealed that the redistribution flow became fingered following a reversal of matric potential gradient within the wetted area. Regarding our simulation work, we succeeded to develop, implement and test a finite- difference, numerical scheme for solving the equations governing flow and transport in three-dimensional, heterogeneous, bimodal, flow domains with highly contrasting soil materials. Results of our simulations demonstrated that under steady-state flow conditions, the embedded clay lenses (with very low conductivity) in bimodal formations may induce preferential flow, and, consequently, may enhance considerably both the solute spreading and the skewing of the solute breakthrough curves. On the other hand, under transient flow conditions associated with substantial redistribution periods with diminishing water saturation, the effect of the embedded clay lenses on the flow and the transport might diminish substantially. Regarding our stochastic modeling effort, we succeeded to develop a theoretical framework for flow and transport in bimodal, heterogeneous, unsaturated formations, based on a stochastic continuum presentation of the flow and a general Lagrangian description of the transport. Results of our analysis show that, generally, a bimodal distribution of the formation properties, characterized by a relatively complex spatial correlation structure, contributes to the variability in water velocity and, consequently, may considerably enhance solute spreading. This applies especially in formations in which: (i) the correlation length scales and the variances of the soil properties associated with the embedded soil are much larger than those of the background soil; (ii) the contrast between mean properties of the two subdomains is large; (iii) mean water saturation is relatively small; and (iv) the volume fraction of the flow domain occupied by the embedded soil is relatively large.

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