A Generalized Expression for the Energy Density of Electromagnetic Waves in Media with Strong Temporal Dispersion

1972 ◽  
Vol 27 (7) ◽  
pp. 1094-1097 ◽  
Author(s):  
Dan Anderson
Keyword(s):  
Energy Density ◽  
Electromagnetic Waves ◽  
Standard Form ◽  
Higher Order ◽  
Chirp Pulse ◽  
Higher Order Terms ◽  
Simple Medium ◽  
Derivatives Of ◽  
Classical Derivation ◽  
Temporal Dispersion

Abstract An expression is derived for the energy density of an electromagnetic wave of rather general form in a lossless, homogeneous and isotropic medium exhibiting temporal dispersion. The result is a generalization of the standard form and takes into account higher order derivatives of the di-electric and permeability functions as well as of the electric and magnetic field amplitudes of the wave. The classical derivation of the expression for the energy density given by BRILLOUIN1, is dis-cussed in some detail and, finally, a qualitative example of a chirp pulse in a simple medium is given, which illustrates the importance of higher order terms

Differentiation of Mass and Stiffness Matrices for High Order Sensitivity Calculations in Finite Element-Based Equilibrium Problems

1993 ◽  
Vol 115 (4) ◽  
pp. 829-832 ◽  
Author(s):  
J. E. Bernard ◽  
S. K. Kwon ◽  
J. A. Wilson
Keyword(s):  
Finite Element ◽  
Design Variable ◽  
Equilibrium Problems ◽  
Higher Order ◽  
High Order ◽  
Stiffness Matrices ◽  
Higher Order Terms ◽  
Fixed Boundaries ◽  
Cubic Polynomials ◽  
Derivatives Of

Extension of sensitivity methods to include higher order terms depends on the ability to compute higher order derivatives of the mass and stiffness matrices. This paper presents a method based on the use of cubic polynomials to fit mass and stiffness matrices across a range of interest of the design variable. The method is illustrated through an example which uses Pade´ approximants to expand the solution to a statics problem. The design variable is the thickness of one part of a plate with fixed boundaries. The solution gives a very good approximation over fivefold change in the value of the design variable.

scholarly journals The Ostrogradsky Prescription for BFV Formalism

1997 ◽  
Vol 12 (27) ◽  
pp. 1991-2004
Author(s):  
Khazret S. Nirov
Keyword(s):  
Higher Order ◽  
Gauge Invariant ◽  
Primary Constraint ◽  
Brst Charge ◽  
Higher Order Terms ◽  
Derivatives Of ◽  
Constraint Surface

Gauge-invariant systems of a general form with higher order time derivatives of gauge parameters are investigated within the framework of the BFV formalism. Higher order terms of the BRST charge and BRST-invariant Hamiltonian are obtained. It is shown that the identification rules for Lagrangian and Hamiltonian BRST ghost variables depend on the choice of the extension of constraints from the primary constraint surface.

Differentiation of Mass and Stiffness Matrices for High Order Sensitivity Calculations in Finite Element-Based Equilibrium Problems

1990 ◽  
Author(s):  
J. E. Bernard ◽  
S. K. Kwon ◽  
J. A. Wilson
Keyword(s):  
Finite Element ◽  
Design Variable ◽  
Equilibrium Problems ◽  
Higher Order ◽  
High Order ◽  
Stiffness Matrices ◽  
Higher Order Terms ◽  
Fixed Boundaries ◽  
Cubic Polynomials ◽  
Derivatives Of

Abstract Extension of sensitivity methods to include higher order terms depends on the ability to compute higher order derivatives of the mass and stiffness matrices. This paper presents a method based on the use of cubic polynomials to fit mass and stiffness matrices across a range of interest of the design variable. The method is illustrated through an example which uses Padé approximants to expand the solution to a statics problem. The design variable is the thickness of one part of a plate with fixed boundaries. The solution gives a very good approximation over five fold change in the value of the design variable.

scholarly journals Higher Order Elastic Constants, Gruneisen Parameters and Lattice Thermal Expansion of Lithium Niobate

2006 ◽  
Vol 3 (3) ◽  
pp. 122-133 ◽  
Author(s):  
Thresiamma Philip ◽  
C. S. Menon ◽  
K. Indulekha
Keyword(s):  
Energy Density ◽  
Lithium Niobate ◽  
Elastic Constants ◽  
Elastic Stiffness ◽  
Higher Order ◽  
Second Order ◽  
Third Order ◽  
Third Order Elastic Constants ◽  
Gruneisen Parameters ◽  
Derivatives Of

The second and third-order elastic constants and pressure derivatives of second- order elastic constants of trigonal LiNbO3(lithium niobate) have been obtained using the deformation theory. The strain energy density estimated using finite strain elasticity is compared with the strain dependent lattice energy density obtained from the elastic continuum model approximation. The second-order elastic constants and the non-vanishing third-order elastic constants along with the pressure derivatives of trigonal LiNbO3are obtained in the present work. The second and third-order elastic constants are compared with available experimental values. The second-order elastic constant C11which corresponds to the elastic stiffness along the basal plane of the crystal is less than C33which corresponds to the elastic stiffness tensor component along thec-axis of the crystal. The pressure derivatives, dC'ij/dp obtained in the present work, indicate that trigonal LiNbO3is compressible. The higher order elastic constants are used to find the generalized Gruneisen parameters of the elastic waves propagating in different directions in LiNbO3. The Brugger gammas are evaluated and the low temperature limit of the Gruneisen gamma is obtained. The results are compared with available reported values.

A C0 three-node triangular element based on preprocessing approach for thick sandwich plates

2017 ◽  
Vol 21 (6) ◽  
pp. 1820-1842
Author(s):  
Wu Zhen ◽  
Ma Rui ◽  
Chen Wanji
Keyword(s):  
Transverse Shear ◽  
Equilibrium Equation ◽  
Three Dimensional ◽  
Higher Order ◽  
Triangular Element ◽  
Shear Stresses ◽  
Sandwich Plates ◽  
Transverse Shear Stress ◽  
Transverse Shear Stresses ◽  
Derivatives Of

This paper will try to overcome two difficulties encountered by the C0 three-node triangular element based on the displacement-based higher-order models. They are (i) transverse shear stresses computed from constitutive equations vanish at the clamped edges, and (ii) it is difficult to accurately produce the transverse shear stresses even using the integration of the three-dimensional equilibrium equation. Invalidation of the equilibrium equation approach ought to attribute to the higher-order derivations of displacement parameters involved in transverse shear stress components after integrating three-dimensional equilibrium equation. Thus, the higher-order derivatives of displacement parameters will be taken out from transverse shear stress field by using the three-field Hu–Washizu variational principle before the finite element procedure is implemented. Therefore, such method is named as the preprocessing method for transverse shear stresses in present work. Because the higher-order derivatives of displacement parameters have been eliminated, a C0 three-node triangular element based on the higher-order zig-zag theory can be presented by using the linear interpolation function. Performance of the proposed element is numerically evaluated by analyzing multilayered sandwich plates with different loading conditions, lamination sequences, material constants and boundary conditions, and it can be found that the present model works well in the finite element framework.

Using the Multi-Parameter Fracture Mechanics for more Accurate Description of Stress/Displacement Crack Tip Fields

2013 ◽  
Vol 586 ◽  
pp. 237-240 ◽  
Author(s):  
Lucie Šestáková
Keyword(s):  
Stress Distribution ◽  
Fracture Mechanics ◽  
Boundary Conditions ◽  
Displacement Field ◽  
Singular Term ◽  
Crack Tip ◽  
Higher Order ◽  
Precise Description ◽  
Accurate Knowledge ◽  
Higher Order Terms

Most of fracture analyses often require an accurate knowledge of the stress/displacement field over the investigated body. However, this can be sometimes problematic when only one (singular) term of the Williams expansion is considered. Therefore, also other terms should be taken into account. Such an approach, referred to as multi-parameter fracture mechanics is used and investigated in this paper. Its importance for short/long cracks and the influence of different boundary conditions are studied. It has been found out that higher-order terms of the Williams expansion can contribute to more precise description of the stress distribution near the crack tip especially for long cracks. Unfortunately, the dependences obtained from the analyses presented are not unambiguous and it cannot be strictly derived how many of the higher-order terms are sufficient.

Expansions of the mildly relativistic dispersion function for nearly perpendicular electron cyclotron ray tracing

1999 ◽  
Vol 61 (1) ◽  
pp. 121-128 ◽  
Author(s):  
I. P. SHKAROFSKY
Keyword(s):  
Ray Tracing ◽  
Computer Programs ◽  
Higher Order ◽  
Electron Cyclotron ◽  
Dispersion Function ◽  
Relativistic Dispersion ◽  
Plasma Dispersion ◽  
Derivatives Of ◽  
Cyclotron Harmonic ◽  
Experienced Difficulty

To trace rays very close to the nth electron cyclotron harmonic, we need the mildly relativistic plasma dispersion function and its higher-order derivatives. Expressions for these functions have been obtained as an expansion for nearly perpendicular propagation in a region where computer programs have previously experienced difficulty in accuracy, namely when the magnitude of (c/vt)2 (ω−nωc)/ω is between 1 and 10. In this region, the large-argument expansions are not yet valid, but partial cancellations of terms occur. The expansion is expressed as a sum over derivatives of the ordinary dispersion function Z. New expressions are derived to relate higher-order derivatives of Z to Z itself in this region of concern in terms of a finite series.

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