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.

Curvature-dependent energies: a geometric and analytical approach

2017 ◽  
Vol 147 (3) ◽  
pp. 449-503 ◽  
Author(s):  
Emilio Acerbi ◽  
Domenico Mucci
Keyword(s):  
Bounded Variation ◽  
Analytical Approach ◽  
Total Curvature ◽  
Representation Formula ◽  
Energy Functional ◽  
Continuous Functions ◽  
Explicit Representation ◽  
Functions Of Bounded Variation ◽  
One Dimensional ◽  
Relaxed Energy

We consider the total curvature of graphs of curves in high-codimension Euclidean space. We introduce the corresponding relaxed energy functional and prove an explicit representation formula. In the case of continuous Cartesian curves, i.e. of graphs cu of continuous functions u on an interval, we show that the relaxed energy is finite if and only if the curve cu has bounded variation and finite total curvature. In this case, moreover, the total curvature does not depend on the Cantor part of the derivative of u. We treat the wider class of graphs of one-dimensional functions of bounded variation, and we prove that the relaxed energy is given by the sum of the length and total curvature of the new curve obtained by closing the holes in cu generated by jumps of u with vertical segments.

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.

ACCURATE FINITE DIFFERENCE REPRESENTATION OF NEUMANN BOUNDARY DATA FOR TIME-HARMONIC WAVE PROPAGATION

1996 ◽  
Vol 04 (04) ◽  
pp. 425-432 ◽  
Author(s):  
ISAAC HARARI
Keyword(s):  
Finite Difference ◽  
Boundary Data ◽  
Truncation Error ◽  
Harmonic Wave ◽  
Neumann Boundary ◽  
Local Truncation Error ◽  
Matrix Equations ◽  
Order Accuracy ◽  
Time Harmonic ◽  
Global Accuracy

Finite difference stencils for inhomogeneous Neumann boundary conditions in acoustic problems with arbitrary source distributions are constructed and analyzed. Boundary stencils are compatible with corresponding interior stencils, preserving symmetry of matrix equations without degrading global accuracy. Higher-order accuracy is attained within the compact support of lower-order methods. Results are verified by local truncation error analysis.

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.

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.

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