Lagrange crisis and generalized variational principle for 3D unsteady flow

2019 ◽  
Vol 30 (3) ◽  
pp. 1189-1196 ◽  
Author(s):  
Ji-Huan He
Keyword(s):  
Variational Principle ◽  
Unsteady Flow ◽  
Lagrange Multiplier ◽  
Unknown Function ◽  
Inverse Method ◽  
Variational Principles ◽  
Multiplier Method ◽  
Generalized Variational Principle ◽  
Content Type ◽  
3D Unsteady Flow

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.

A FRACTAL VARIATIONAL THEORY FOR ONE-DIMENSIONAL COMPRESSIBLE FLOW IN A MICROGRAVITY SPACE

Fractals ◽  
2020 ◽  
Vol 28 (02) ◽  
pp. 2050024 ◽  
Author(s):  
JI-HUAN HE
Keyword(s):  
Compressible Flow ◽  
Lagrange Multiplier ◽  
Variational Formulation ◽  
Inverse Method ◽  
Variational Principles ◽  
Microgravity Condition ◽  
Multiplier Method ◽  
Variational Theory ◽  
One Dimensional ◽  
The One

The semi-inverse method is adopted to establish a family of fractal variational principles of the one-dimensional compressible flow under the microgravity condition, and Cauchy–Lagrange integral is successfully derived from the obtained variational formulation. A suitable application of the Lagrange multiplier method is also elucidated.

scholarly journals Generalized Variational Principle for Long Water-Wave Equation by He's Semi-Inverse Method

2009 ◽  
Vol 2009 ◽  
pp. 1-5 ◽  
Author(s):  
Weimin Zhang
Keyword(s):  
Water Wave ◽  
Inverse Method ◽  
Variational Principles ◽  
Nonlinear Partial Differential Equations ◽  
Nonlinear Partial Differential Equation ◽  
Variational Formula ◽  
Generalized Variational Principle ◽  
Partial Differential ◽  
Water Wave Problem ◽  
Mathematics And Physics

Variational principles for nonlinear partial differential equations have come to play an important role in mathematics and physics. However, it is well known that not every nonlinear partial differential equation admits a variational formula. In this paper, He's semi-inverse method is used to construct a family of variational principles for the long water-wave problem.

scholarly journals On the semi-inverse method and variational principle

2013 ◽  
Vol 17 (5) ◽  
pp. 1565-1568 ◽  
Author(s):  
Xue-Wei Li ◽  
Ya Li ◽  
Ji-Huan He
Keyword(s):  
Heat Conduction ◽  
Variational Principle ◽  
Inverse Method ◽  
Generalized Variational Principle ◽  
Open Forum ◽  
One Dimensional ◽  
History Of

In this Open Forum, Liu et al. proved the equivalence between He-Lee 2009 variational principle and that by Tao and Chen (Tao, Z. L., Chen, G. H., Thermal Science, 17(2013), pp. 951-952) for one dimensional heat conduction. We confirm the correction of Liu et al.?s proof, and give a short remark on the history of the semi-inverse method for establishment of a generalized variational principle.

scholarly journals Thermodynamic Foundation of Generalized Variational Principle

2021 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
Variational Principle ◽  
Multiplier Method ◽  
Generalized Variational Principle ◽  
Lagrangian Multiplier ◽  
Stress Strain ◽  
Strain Relation ◽  
Stress Strain Relation ◽  
Lagrangian Multiplier Method ◽  
Open Question

One open question remains regarding the theory of the generalized variational principle, that is, why the stress-strain relation still be derived from the generalized variational principle while the Lagrangian multiplier method is applied in vain? This study shows that the generalized variational principle can only be understood and implemented correctly within the framework of thermodynamics. As long as the functional has one of the combination $A(\epsilon_{ij})-\sigma_{ij}\epsilon_{ij}$ or $B(\sigma_{ij})-\sigma_{ij}\epsilon_{ij}$, its corresponding variational principle will produce the stress-strain relation without the need to introduce extra constraints by the Lagrangian multiplier method. It is proved herein that the Hu-Washizu functional $\Pi_{HW}[u_i,\epsilon_{ij},\sigma_{ij}]$ and Hu-Washizu variational principle comprise a real three-field functional.

Generalized variational principles of the Benney-Lin equation arising in fluid dynamics

2021 ◽  
Author(s):  
Kang-Jia Wang ◽  
Jian-Fang Wang
Keyword(s):  
Fluid Dynamics ◽  
Variational Principle ◽  
Differential Equations ◽  
Inverse Method ◽  
Variational Principles ◽  
Nonlinear Partial Differential Equations ◽  
Solution Structures ◽  
Generalized Variational Principles ◽  
Insight Into ◽  
Derivation Process

Abstract Variational principle is important since it can not only reveal the possible solution structures of the equation but also provide the conservation laws in an energy form. Unfortunately, not all the differential equations can find their variational forms. In this work, the Benney-Lin equation is studied and its two different generalized variational principles are successfully established by using the semi-inverse method. The derivation process is given in detail. The finding in this work is expected to give a insight into the study of the nonlinear partial differential equations arising in fluid dynamics.

Generalized Variational Principle of Plates on Elastic Foundation

1991 ◽  
Vol 58 (4) ◽  
pp. 1001-1004 ◽  
Author(s):  
R. L. Yuan ◽  
L. S. Wang
Keyword(s):  
Integral Equation ◽  
Variational Principle ◽  
Potential Energy ◽  
Elastic Foundation ◽  
Lagrange Multiplier ◽  
Generalized Variational Principle ◽  
Minimum Potential Energy ◽  
Minimum Potential ◽  
Plates On Elastic Foundation

The theory of variational principle is enhanced by using the Lagrange multiplier to establish a generalized variational principle for plates on an elastic foundation. In the first part of this paper, the principle of minimum potential energy is introduced in which the integral equation is employed as the variational constrained condition. In the second part, it is shown that the generalized variational principle with two variational functions can be established. This represents, to the authors’ knowledge, the first treatment of the variational principle with these types of equations.

Application of 2D base force element method with complementary energy principle for arbitrary meshes

2014 ◽  
Vol 31 (4) ◽  
pp. 691-708 ◽  
Author(s):  
Yijiang Peng ◽  
Nana Zong ◽  
Lijuan Zhang ◽  
Jiwei Pu
Keyword(s):  
Lagrange Multiplier ◽  
Lagrange Multiplier Method ◽  
Multiplier Method ◽  
Complementary Energy ◽  
Energy Principle ◽  
Content Type ◽  
Complementary Energy Principle ◽  
Force Element ◽  
2D Model ◽  
Element Method

Purpose – The purpose of this paper is to present a two-dimensional (2D) model of the base force element method (BFEM) based on the complementary energy principle. The study proposes a model of the BFEM for arbitrary mesh problems. Design/methodology/approach – The BFEM uses the base forces given by Gao (2003) as fundamental variables to describe the stress state of an elastic system. An explicit expression of element compliance matrix is derived using the concept of base forces. The detailed formulations of governing equations for the BFEM are given using the Lagrange multiplier method. The explicit displacement expression of nodes is given. To verify the 2D model, a program on the BFEM using MATLAB language is made and a number of examples on arbitrary polygonal meshes and aberrant meshes are provided to illustrate the BFEM. Findings – A good agreement is obtained between the numerical and theoretical results. Based on the studies, it is found that the 2D formulation of BFEM with complementary energy principle provides reliable predictions for arbitrary mesh problems. Research limitations/implications – Due to the use of Lagrange multiplier method, there are more basic unknowns in the control equation. The proposed method will be improved in the future. Practical implications – This paper presents a new idea and a new numerical method, and to explore new ways to solve the problem of arbitrary meshes. Originality/value – The paper presents a 2D model of the BFEM using the complementary energy principle for arbitrary mesh problems.

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