On Stable Extremals of the Potential Energy Functional

2021 ◽  
Vol 62 (3) ◽  
pp. 482-488
Author(s):  
N. M. Poluboyarova
Keyword(s):  
Potential Energy ◽  
Energy Functional ◽  
Potential Energy Functional

Principles of minimum potential energy and complementary energy

2020 ◽  
pp. 228-248
Author(s):  
Lallit Anand ◽  
Sanjay Govindjee
Keyword(s):  
Boundary Condition ◽  
Potential Energy ◽  
Displacement Field ◽  
Mixed Boundary ◽  
Energy Functional ◽  
The Body ◽  
Minimum Potential Energy ◽  
Complementary Energy ◽  
Minimum Potential ◽  
Potential Energy Functional

With the displacement field taken as the only fundamental unknown field in a mixed-boundary-value problem for linear elastostatics, the principle of minimum potential energy asserts that a potential energy functional, which is defined as the difference between the free energy of the body and the work done by the prescribed surface tractions and the body forces --- assumes a smaller value for the actual solution of the mixed problem than for any other kinematically admissible displacement field which satisfies the displacement boundary condition. This principle provides a weak or variational method for solving mixed boundary-value-problems of elastostatics. In particular, instead of solving the governing Navier form of the partial differential equations of equilibrium, one can search for a displacement field such that the first variation of the potential energy functional vanishes. A similar principle of minimum complementary energy, which is phrased in terms of statically admissible stress fields which satisfy the equilibrium equation and the traction boundary condition, is also discussed. The principles of minimum potential energy and minimum complementary energy can also be applied to derive specialized principles which are particularly well-suited to solving structural problems; in this context the celebrated theorems of Castigliano are discussed.

Two Kinds of Finite Element Variables Based on B-Spline Wavelet on Interval for Curved Beam

2019 ◽  
Vol 11 (02) ◽  
pp. 1950017 ◽  
Author(s):  
Yanfei He ◽  
Xingwu Zhang ◽  
Jia Geng ◽  
Xuefeng Chen ◽  
Zengguang Li
Keyword(s):  
Finite Element ◽  
Potential Energy ◽  
Free Vibration Analysis ◽  
Energy Functional ◽  
Curved Beam ◽  
Spline Wavelet ◽  
B Spline ◽  
Generalized Potential ◽  
Potential Energy Functional ◽  
Two Kinds Of Variables

Curved beam structure has been widely used in engineering, due to its good load-bearing and geometric characteristics. More common methods for analyzing and designing this structure are the finite element methods (FEMs), but these methods have many disadvantages. Fortunately, the multivariable wavelet FEMs can solve these drawbacks. However, the multivariable generalized potential energy functional of curved beam, used to construct this element, has not been given in previous literature. In this paper, the generalized potential energy functional for curved beam with two kinds of variables is derived initially. On this basis, the B-spline wavelet on the interval (BSWI) is used as the interpolation function to construct the wavelet curved beam element with two kinds of variables. In the end, several typical numerical examples of thin to thick curved beams are given, which show that the present element is more effective in static and free vibration analysis of curved beam structures.

On the minimum of the potential energy functional at a critical point

Meccanica ◽  
1974 ◽  
Vol 9 (4) ◽  
pp. 256-264
Author(s):  
Giovanni Romano
Keyword(s):  
Potential Energy ◽  
Critical Point ◽  
Energy Functional ◽  
Potential Energy Functional

scholarly journals Magnetic motion of spherical frictional charged particles on the unit sphere

2019 ◽  
Vol 65 (5 Sept-Oct) ◽  
pp. 496 ◽  
Author(s):  
Talat Körpınar ◽  
Ridvan Cem Demirkol
Keyword(s):  
Magnetic Field ◽  
Charged Particle ◽  
Potential Energy ◽  
Frictional Force ◽  
Poynting Vector ◽  
Positive Curvature ◽  
Charged Particles ◽  
Energy Functional ◽  
Ordinary Space ◽  
Potential Energy Functional

Mathematically, the sphere unit S² is described to be a 2-sphere in an ordinary space with a positive curvature. In this study, we aim to present the manipulation of a spherical charged particle in a continuous motion with a magnetic field on the sphere S² while it is exposed to a frictional force. In other words, we effot to derive the exact geometric characterization for the spherical charged particle under the influence of a frictional force field on the unit 2-sphere. This approach also helps to discover some physical and kinematical characterizations belonging to the particle such as the magnetic motion, the torque, the potential energy functional, and the Poynting vector.

A vectorizable potential energy functional for reactive scattering

1987 ◽  
Vol 72 (4) ◽  
pp. 253-264 ◽  
Author(s):  
Ernesto Garcia ◽  
Luigi Ciccarelli ◽  
Antonio Lagan�
Keyword(s):  
Potential Energy ◽  
Energy Functional ◽  
Reactive Scattering ◽  
Potential Energy Functional

Stability of the critical equilibrium of a compressed column on a Winkler foundation

2021 ◽  
pp. 108128652110615
Author(s):  
Mingzhi Gao ◽  
Ming Jin
Keyword(s):  
Potential Energy ◽  
Energy Functional ◽  
Exact Expression ◽  
Second Order ◽  
Winkler Foundation ◽  
Critical Equilibrium ◽  
Order Variation ◽  
The Stability ◽  
Potential Energy Functional ◽  
Theoretical Results

In this paper, the critical equilibrium of a simply supported compressed column on a Winkler foundation is analyzed based on Koiter’s theory. The exact expression of the potential energy functional is presented. By the Fourier series of the disturbance deflection, the second-order variation of the potential energy is expressed as a quadratic form. At critical equilibrium, the second-order variation of the potential energy is semi-positive definite, so that the stability of the critical equilibrium is determined by the sign of the fourth-order variation or sixth-order variation. It can be seen that only in two small ranges of elastic-foundation stiffness is the corresponding critical state stable and the bifurcation equilibrium upward. Then, the theoretical results of this paper are compared with previous experimental and theoretical results.

Analysis of Orthotropic Thin Rectangular Simply Supported Plate Subjected to both In-Plane Compression and Lateral Loads

2020 ◽  
Vol 48 ◽  
pp. 63-77
Author(s):  
D.O. Onwuka ◽  
O.M. Ibearugbulem ◽  
Duke Bertram
Keyword(s):  
Yield Strength ◽  
Potential Energy ◽  
Elastic Stability ◽  
Lateral Load ◽  
Energy Functional ◽  
Total Potential Energy ◽  
Lateral Loads ◽  
Total Potential ◽  
Plane Compression ◽  
Potential Energy Functional

This study presents the analysis of thin rectangular orthotropic plate, simply supported at all edges (SSSS) subjected to both in-plane compression and lateral loads. The total potential energy functional was used in the analysis. The general variation of the total potential energy functional was done and the governing equation was obtained. The solution of the direct integration of the governing equation gave the deflection of the plate as a product of the coefficient of deflection and an orthogonal polynomial shape function. The expression for the coefficient of deflection was obtained by the direct variation of the total potential energy functional. This was used to derive the equation for the Lateral load parameter of an orthotropic thin rectangular plate carrying both in-plane compression and lateral loads based on the maximum deflection condition and also based on the elastic stability (yield strength) condition. The peculiar deflection equation for the SSSS plate was obtained using the formulated polynomial shape function. Numerical examples were carried out to determine the lateral load parameters corresponding to various plate thickness and permissible deflection for orthotropic thin SSSS plate carrying both in-plane compression and lateral load. In the same way, the lateral load parameters using the elastic stability condition (yield strength) were obtained for yield strength of 275 MPa, 355 MPa and 410 MPa

scholarly journals On instability of extremals of potential energy functional

2018 ◽  
Vol 10 (3) ◽  
pp. 77-85
Author(s):  
N M Poluboyarova
Keyword(s):  
Potential Energy ◽  
Energy Functional ◽  
Potential Energy Functional

scholarly journals Progress in calculating the potential energy surface of H 3 +

2012 ◽  
Vol 370 (1978) ◽  
pp. 5001-5013 ◽  
Author(s):  
Ludwik Adamowicz ◽  
Michele Pavanello
Keyword(s):  
Potential Energy ◽  
Potential Energy Surface ◽  
Energy Surface ◽  
Molecular Geometry ◽  
Computational Procedure ◽  
Energy Functional ◽  
Basis Functions ◽  
Vibrational States ◽  
Energy Gradient ◽  
Explicitly Correlated

The most accurate electronic structure calculations are performed using wave function expansions in terms of basis functions explicitly dependent on the inter-electron distances. In our recent work, we use such basis functions to calculate a highly accurate potential energy surface (PES) for the H ion. The functions are explicitly correlated Gaussians, which include inter-electron distances in the exponent. Key to obtaining the high accuracy in the calculations has been the use of the analytical energy gradient determined with respect to the Gaussian exponential parameters in the minimization of the Rayleigh–Ritz variational energy functional. The effective elimination of linear dependences between the basis functions and the automatic adjustment of the positions of the Gaussian centres to the changing molecular geometry of the system are the keys to the success of the computational procedure. After adiabatic and relativistic corrections are added to the PES and with an effective accounting of the non-adiabatic effects in the calculation of the rotational/vibrational states, the experimental H rovibrational spectrum is reproduced at the 0.1 cm −1 accuracy level up to 16 600 cm −1 above the ground state.

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