Material constraints, lagrange multipliers, and compatibility. Applications to rod and shell theories

1991 ◽  
Vol 116 (3) ◽  
pp. 257-299 ◽  
Author(s):  
Stuart S. Antman ◽  
Randall S. Marlow
Keyword(s):  
Lagrange Multipliers ◽  
Material Constraints

Applying the Method of Lagrange Multipliers to Derive an Estimator for Unsampled Soil Properties

2014 ◽  
Vol 11 (1) ◽  
pp. 15
Author(s):  
Set Foong Ng ◽  
Pei Eng Ch’ng ◽  
Yee Ming Chew ◽  
Kok Shien Ng
Keyword(s):  
Soil Properties ◽  
Lagrange Multipliers ◽  
Unbiased Estimator ◽  
Soil Property ◽  
Minimum Variance ◽  
Distance Method ◽  
True Value ◽  
A Site ◽  
Method Of Lagrange Multipliers ◽  
Inverse Distance

Soil properties are very crucial for civil engineers to differentiate one type of soil from another and to predict its mechanical behavior. However, it is not practical to measure soil properties at all the locations at a site. In this paper, an estimator is derived to estimate the unknown values for soil properties from locations where soil samples were not collected. The estimator is obtained by combining the concept of the ‘Inverse Distance Method’ into the technique of ‘Kriging’. The method of Lagrange Multipliers is applied in this paper. It is shown that the estimator derived in this paper is an unbiased estimator. The partiality of the estimator with respect to the true value is zero. Hence, the estimated value will be equal to the true value of the soil property. It is also shown that the variance between the estimator and the soil property is minimised. Hence, the distribution of this unbiased estimator with minimum variance spreads the least from the true value. With this characteristic of minimum variance unbiased estimator, a high accuracy estimation of soil property could be obtained.

Determination of eigenvectors with Lagrange multipliers

2021 ◽  
Author(s):  
Wooyong Han ◽  
Dong-Won Jung ◽  
Jungil Lee ◽  
Chaehyun Yu
Keyword(s):  
Lagrange Multipliers

Staggered solution of fluid-porous-media interaction using the method of local Lagrange multipliers

PAMM ◽  
2014 ◽  
Vol 14 (1) ◽  
pp. 473-474 ◽  
Author(s):  
Seyedmohammad Zinatbakhsh ◽  
Wolfgang Ehlers
Keyword(s):  
Porous Media ◽  
Lagrange Multipliers ◽  
Media Interaction

The Determination of Maximum Range for an Unpowered Atmospheric Vehicle

1964 ◽  
Vol 68 (638) ◽  
pp. 111-116 ◽  
Author(s):  
D. J. Bell
Keyword(s):  
Calculus Of Variations ◽  
Lagrange Multipliers ◽  
Digital Computer ◽  
Necessary Conditions ◽  
Lagrange Equations ◽  
Initial Portion ◽  
End Conditions ◽  
Maximum Range ◽  
Isothermal Atmosphere

SummaryThe problem of maximising the range of a given unpowered, air-launched vehicle is formed as one of Mayer type in the calculus of variations. Eulers’ necessary conditions for the existence of an extremal are stated together with the natural end conditions. The problem reduces to finding the incidence programme which will give the greatest range.The vehicle is assumed to be an air-to-ground, winged unpowered vehicle flying in an isothermal atmosphere above a flat earth. It is also assumed to be a point mass acted upon by the forces of lift, drag and weight. The acceleration due to gravity is assumed constant.The fundamental constraints of the problem and the Euler-Lagrange equations are programmed for an automatic digital computer. By considering the Lagrange multipliers involved in the problem a method of search is devised based on finding flight paths with maximum range for specified final velocities. It is shown that this method leads to trajectories which are sufficiently close to the “best” trajectory for most practical purposes.It is concluded that such a method is practical and is particularly useful in obtaining the optimum incidence programme during the initial portion of the flight path.

A Phenomenological Theory of Socioeconomic Systems with Spatial Interactions

1982 ◽  
Vol 14 (7) ◽  
pp. 869-888 ◽  
Author(s):  
P F Lesse
Keyword(s):  
Lagrange Multipliers ◽  
Average Cost ◽  
Phenomenological Theory ◽  
Traffic Flows ◽  
Spatial Interactions ◽  
Practical Application ◽  
Density Distributions ◽  
Expected Values ◽  
Probability Density Distributions ◽  
The Relationship

This paper deals with a class of models which describe spatial interactions and are based on Jaynes's principle. The variables entering these models can be partitioned in four groups: (a) probability density distributions (for example, relative traffic flows), (b) expected values (average cost of travel), (c) their duals (Lagrange multipliers, traffic impedance coefficient), and (d) operators transforming probabilities into expected values. The paper presents several dual formulations replacing the problem of maximizing entropy in terms of the group of variables (a) by equivalent extreme problems involving groups (b)-(d). These problems form the basis of a phenomenological theory. The theory makes it possible to derive useful relationships among groups (b) and (c). There are two topics discussed: (1) practical application of the theory (with examples), (2) the relationship between socioeconomic modelling and statistical mechanics.

scholarly journals CONTACT PROBLEMS FOR NONLINEARLY ELASTIC MATERIALS: WEAK SOLVABILITY INVOLVING DUAL LAGRANGE MULTIPLIERS

2010 ◽  
Vol 52 (2) ◽  
pp. 160-178 ◽  
Author(s):  
A. MATEI ◽  
R. CIURCEA
Keyword(s):  
Lagrange Multipliers ◽  
Variational Equation ◽  
Small Deformation ◽  
Contact Problems ◽  
Point Theory ◽  
The Body ◽  
Elastic Materials ◽  
Weak Formulation ◽  
Nonlinearly Elastic ◽  
Weak Solvability

AbstractA class of problems modelling the contact between nonlinearly elastic materials and rigid foundations is analysed for static processes under the small deformation hypothesis. In the present paper, the contact between the body and the foundation can be frictional bilateral or frictionless unilateral. For every mechanical problem in the class considered, we derive a weak formulation consisting of a nonlinear variational equation and a variational inequality involving dual Lagrange multipliers. The weak solvability of the models is established by using saddle-point theory and a fixed-point technique. This approach is useful for the development of efficient algorithms for approximating weak solutions.

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