The Inflation and Contact Constraint of a Rectangular Mooney Membrane

1974 ◽  
Vol 41 (4) ◽  
pp. 979-984 ◽  
Author(s):  
W. W. Feng ◽  
J. T. Tielking ◽  
P. Huang
Keyword(s):  
Minimum Energy ◽  
Optimization Theory ◽  
Energy Functional ◽  
Contact Boundary ◽  
Variable Metric ◽  
Contact Constraint ◽  
Rigid Constraint ◽  
Independent Functions ◽  
Potential Energy Functional ◽  
Variable Metric Method

This paper presents a minimum energy solution for the deformed configuration of an edge-bonded rectangular membrane loaded with uniform pressure and contacting a frictionless rigid constraint. A technique borrowed from optimization theory is employed to derive a potential energy functional which contains the contact constraint condition with no increase in the number of independent functions. This energy functional is minimized by a series of geometrically admissible, continuous, coordinate functions with constant coefficients determined by the Ritz procedure. The variable-metric method, as generalized by Fletcher and Powell, is used to find the coefficients in the energy minimizing series solutions. The results presented show the contact boundary and the distortion of a square gridwork laid on the undeformed membrane.

The energy and helicity of knotted magnetic flux tubes

1995 ◽  
Vol 451 (1943) ◽  
pp. 609-629 ◽  
Keyword(s):  
Magnetic Field ◽  
Magnetic Energy ◽  
Minimum Energy ◽  
Circular Cross Section ◽  
Parametric Representation ◽  
Energy Functional ◽  
Helical Axis ◽  
Torus Knots ◽  
Flux Tubes ◽  
Kink Mode

Magnetic relaxation of a magnetic field embedded in a perfectly conducting incompressible fluid to minimum energy magnetostatic equilibrium states is considered. It is supposed that the magnetic field is confined to a single flux tube which may be knotted. A local non-orthogonal coordinate system, zero-framed with respect to the knot, is introduced, and the field is decomposed into toroidal and poloidal ingredients with respect to this system. The helicity of the field is then determined; this vanishes for a field that is either purely toroidal or purely poloidal. The magnetic energy functional is calculated under the simplifying assumptions that the tube is axially uniform and of circular cross-section. The case of a tube with helical axis is first considered, and new results concerning kink mode instability and associated bifurcations are obtained. The case of flux tubes in the form of torus knots is then considered and the ‘ground-state’ energy function ͞m ( h ) (where h is an internal twist parameter) is obtained; as expected, ͞m ( h ), which is a topological invariant of the knot, increases with increasing knot complexity. The function ͞m ( h ) provides an upper bound on the corresponding function m ( h ) that applies when the above constraints on tube structure are removed. The technique is applicable to any knot admitting a parametric representation, on condition that points of vanishing curvature are excluded.

Boundary Perturbations due to the Presence of Small Cracks

2015 ◽  
Author(s):  
Habib Ammari ◽  
Elie Bretin ◽  
Josselin Garnier ◽  
Hyeonbae Kang ◽  
Hyundae Lee ◽  
...  
Keyword(s):  
Asymptotic Formula ◽  
Neumann Boundary ◽  
Representation Formula ◽  
Fundamental Solutions ◽  
Energy Functional ◽  
Finite Hilbert Transform ◽  
Time Harmonic ◽  
Elastic Potential Energy ◽  
Potential Energy Functional ◽  
Boundary Perturbations

This chapter considers the perturbations of the displacement (or traction) vector that are due to the presence of a small crack with homogeneous Neumann boundary conditions in an elastic medium. It derives an asymptotic formula for the boundary perturbations of the displacement as the length of the crack tends to zero. Using analytical results for the finite Hilbert transform, the chapter derives an asymptotic expansion of the effect of a small Neumann crack on the boundary values of the solution. It also derives the topological derivative of the elastic potential energy functional and proves a useful representation formula for the Kelvin matrix of the fundamental solutions of Lamé system. Finally, it gives an asymptotic formula for the effect of a small linear crack in the time-harmonic regime.

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.

scholarly journals Random perturbation of the projected variable metric method for nonsmooth nonconvex optimization problems with linear constraints

2011 ◽  
Vol 21 (2) ◽  
pp. 317-329 ◽  
Author(s):  
Abdelkrim El Mouatasim ◽  
Rachid Ellaia ◽  
Eduardo de Cursi
Keyword(s):  
Nonconvex Optimization ◽  
Optimization Problems ◽  
Random Perturbation ◽  
Linear Constraints ◽  
Nonconvex Optimization Problems ◽  
Variable Metric ◽  
Lipschitz Continuous ◽  
Nonsmooth Nonconvex Optimization ◽  
Variable Metric Method ◽  
Set Of Points

Random perturbation of the projected variable metric method for nonsmooth nonconvex optimization problems with linear constraintsWe present a random perturbation of the projected variable metric method for solving linearly constrained nonsmooth (i.e., nondifferentiable) nonconvex optimization problems, and we establish the convergence to a global minimum for a locally Lipschitz continuous objective function which may be nondifferentiable on a countable set of points. Numerical results show the effectiveness of the proposed approach.

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