Methods for Solving the Viscoelasticity Equations for Cylinder and Sphere Problems

The method of partial overlapping regions is demonstrated as a modification of the alternating Schwarz's technique. The presented approach uses overlapping region concept to reduce a boundary-value problem for the scalar or vector Helmholtz's equation to a Fredholm's second kind integral equation which can be easily algebraized by means of Galerkin's technique. Applications of the method to printed-circuit transmission line analysis and antenna array analysis are presented and discussed.

Scalar and vector Helmholtz equations and their applications in electromagnetics are discussed. Relevance of the Helmholtz equation to wave motion is explained. Derivation involving Maxwell's equations and auxiliary scalar and vector potentials is presented. Basic analytical solution techniques for homogeneous and nonhomogeneous cases are introduced and illustrated in simple problems such as guided waves and radiation from a finite‐length dipole. Numerical methods and the operator approach to investigation of the Helmholtz equation are briefly outlined.

Trace formulas are derived for the coefficients of a matrix differential equation representing wave propagation in a medium which supports N types of waves. In general these waves are coupled to each other but are assumed to become uncoupled as lzl -> ?o. The two coefficients of the ordinary differential equation, N x N matrices, are assumed known for z 0. A point source is located in the region z 0. The unique features of this work are a derivation of the trace formula in which the Jost functions are not needed, an alternate measurement method based on an impedance concept and a trace equation based on the impedance concept. Introduction. Motivated by the work of Trubowitz (1), trace formula methods were introduced by Deift and Trubowitz (2) to study one-dimensional inverse quantum mechanical scattering problems on the line. The trace formula expresses the potential as a functional of the reflection coefficient. Greene (3) used the same techniques to study the one-dimensional acoustical wave equation with the angular frequency playing the role of the quantum mechanical energy. Stickler and Deift (4) examined the acoustic problem of a point source in an inhomogeneous half space bounded by a pressure release surface. The sound speed was recovered from a measurement of the normal component of the acoustic velocity at the pressure release surface. The role of the quantum mechanical energy was played by a vertical wave number, which does not correspond to either a spatial or temporal derivative. Using again the analogy between vertical wave number (the wave number in the direction of the coefficients' variation) and the quantum mechanical energy, Stickler (5) showed how the trace formula methods could be used to recover the sound speed when it approaches different values at plus and minus infinity. In a quantum mechanical context this means that the potential approaches zero, say at minus infinity, but some positive constant, not necessarily known, at plus infinity. (In the Deift-Trubowitz analysis the potential approached zero in both limits.) Stickler (6) also discussed this same problem in an electromagnetic context. Tomei (7) has used the trace methods to recover the coefficients of a linear, third order equation associated with the Boussinesq equation. In the present paper the trace method is extended to a one-dimensional vector Helmholtz equation which supports N types of waves and for which the waves are coupled to each other except at infinity. The trace method yields a functional relation- ship between the coefficients of the Helmholtz equation, a reflection coefficient described below and assumed known and a specific solution of the vector Helmholtz equation with known and prescribed initial conditions. These relationships are called the trace formulas and are derived here. The coefficients of the vector Helmholtz differential equation can be replaced by these trace formulas and this results in a system of coupled, nonlinear ordinary differential equations with known initial condi- tions. Once this solution is determined, the coefficients can be recovered. The existence of a solution of this coupled system is not given.

With their high degree of specificity and investigator control, in vitro disease models provide a natural complement to in vivo models. Especially in organs such as the brain, where anatomical limitations make in vivo experiments challenging, in vitro models have been increasingly used to mimic disease pathology. However, brain mimetic models may not fully replicate the mechanical environment in vivo, which has been shown to influence a variety of cell behaviors. Specifically, many disease models consider only the linear elastic modulus of brain, which describes the stiffness of a material with the assumption that mechanical behavior is independent of loading rate. Here, we characterized porcine brain tissue using a modified stress relaxation test, and across a panel of viscoelastic models, showed that stiffness depends on loading rate. As such, the linear elastic modulus does not accurately reflect the viscoelastic properties of native brain. Among viscoelastic models, the Maxwell model was selected for further analysis because of its simplicity and excellent curve fit (R2 = 0.99 ± 0.0006). Thus, mechanical response of native brain and hydrogel mimetic models was analyzed using the Maxwell model and the linear elastic model to evaluate the effects of strain rate, time post mortem, region, tissue type (i.e., bulk brain vs white matter), and in brain mimetic models, hydrogel composition, on observed mechanical properties. In comparing the Maxwell and linear elastic models, linear elastic modulus is consistently lower than the Maxwell elastic modulus across all brain regions. Additionally, the Maxwell model is sensitive to changes in viscosity and small changes in elasticity, demonstrating improved fidelity. These findings demonstrate the insufficiency of linear elastic modulus as a primary mechanical characterization for brain mimetic materials and provide quantitative information toward the future design of materials that more closely mimic mechanical features of brain.

SPHEREPACK 3.0 is a collection of FORTRAN programs that facilitates computer modeling of geophysical processes. The package contains programs for computing certain common differential operators including divergence, vorticity, gradients, and the Laplacian of both scalar and vector functions. Programs are also available for inverting these operators. For example, given divergence and vorticity, the package can be used to compute the velocity components. The Laplacian can also be inverted and therefore the package can be used to solve both the scalar and vector Helmholtz equations. Use of the package is illustrated by three sample programs that solve the scalar Helmholtz equation, the time-dependent linear advection equation, and the time-dependent nonlinear shallow-water equations. Accurate solutions are obtained via the spectral method that uses both scalar and vector spherical harmonic transforms that are available to the user. The package also contains utility programs for computing the associated Legendre functions, Gauss points and weights, and multiple fast Fourier transforms. Programs are provided for both equally spaced and Gauss distributed latitudinal points as well as programs that transfer data between these grids.

A new vector volume integral inequality has been derived using calculus of variations techniques [1]. This inequality was used to determine conditions for waveguide cutoff. The use of the inequality requires only the knowledge of the smallest eigenvalue of the vector Helmholtz's equation for the geometry of the waveguide. The inequality can also be of value in applications of Liapunov Stability Theory to systems of partial differential equations. This result is illustrated by an example.

Radiation boundary conditions developed for use with the vector Helmholtz equation in three dimensions are reviewed, including an alternate second-order radiation condition that preserves the symmetry of the matrix operator arising within a numerical implementation. The accuracy of these conditions is systematically tested as a function of the boundary location using outward-propagating spherical harmonics. >

Finite Element Methods for the Solution of 3D Eddy Current Problems

Although some calculations will be specialized for toroidal systems, some conclusions will be valid independent of the coordinate system. The manuscript contains 1. a proof that on the basis of a solution of Maxwell's equations a solution of curl B = const B may be found, 2. a method to transform the vector Helmholtz equation into a scalar one, 3. three solutions of the scalar Helmholtz equation, of Maxwell's equations and of curl B = const B in quasi‐toroidal coordinates, 4. a method to construct solutions of curl B × B = 4πΔp on the basis of solutions of curl B = const B (including two special solutions and a derivation of the toroidal Shafranov, equation).

The “solutions” of boundary-value problems presented in Section 2.8 are merely formal in that the appropriate Green functions must be found, and finding these Green functions is extremely difficult except in certain special cases such as the Rayleigh-Sommerfeld problem. Moreover, even when the appropriate Green functions can be found they are often expressed in the form of superpositions of elementary solutions of the homogeneous Helmholtz equation that are especially suited for dealing with boundaries of a specific shape. One example of a set of such elementary solutions is the plane waves which arise from applying the method of separation of variables to the homogeneous Helmholtz equation using a Cartesian coordinate system and, as we will see below, form a complete set of basis functions for fitting boundary-value data specified on plane surfaces; e.g., for the RS problems. However, the plane waves have limited utility in solving boundary-value problems involving non-planar boundaries such as spherical boundaries. In that case the method of separation of variables is applied to the Helmholtz equation using a spherical polar coordinate system and the so-called multipole fields arise as a set of elementary solutions that form a basis for fitting boundary-value data specified on spherical boundaries. In this chapter we will briefly review the method of separation of variables for the Helmholtz equation and obtain the resulting eigenfunctions for the important cases of Cartesian, spherical polar and cylindrical coordinate systems.

This article focuses on fractional Maxwell model of viscoelastic materials, which are a generalization of classic Maxwell model to noninteger order derivatives. We present and discuss formulations of the fractional order viscoelastic model and give physical interpretations of the model by using viscoelastic functions. We apply the generalized Caputo fractional derivative to viscoelastic models, namely fractional Maxwell model, fractional Kelvin‐Voigt model, and fractional Zener model. The stress relaxation module and creep compliance for each model are derived analytically using generalized Caputo fractional derivative. We analyze effect of α and newly introduced parameter ρ in all these models. The result shows an effect on viscoelastic models using fractional operator.

General three-dimensional solutions are given for the Laplace equation and Helmholtz equation in the cylindrical co-ordinate system, for processes which remain harmonic with time. The expressions for solution which are presented, not only for the magnetic and electrical field variables, but also for the higher-order vector potential, permit an analytical solution. The interrelationships between the field variables, the vector potential, and the higher-order vector potential are discussed. Several examples of engineering applications of the analytical calculation method are given.

A method for 2-D simulation of an axial vircator with a resonance cavity is described. It uses a self-consistent solution of relativistic motion equations and Maxwell equations for scalar and vector potentials in the Coulomb gauge. The vector potential is represented as a superposition of resonator eigenmodes, where each eigenmode corresponds to a damping oscillator. Damping enables considering radiative energy losses from the open end of the resonator. Compliance of the simulation results with both published experimental studies and those performed by our team is demonstrated.

The electro-optical properties of nanowire array solar cells are analyzed with a microscopic simulation model. For the electromagnetic part, the vectorial Helmholtz equation is solved for the three-dimensional structure, and a modal analysis reveals the absorption mechanisms. The electronic part is analyzed with a drift-diffusion approach. It is shown that with a proper design, a III-V nanowire array solar cell can approach the efficiency of a thin-film solar cell, while only a fraction of the active material volume is used.

It is shown that extended irreversible thermodynamics provides a simple and coherent modeling of viscoelastic bodies and dilute polymer solutions. The basic hypothesis underlying the present formalism is to raise, respectively, the inelastic stress tensor or the viscous stress tensor to the status of an independent variable in addition to the standard variables. Concerning viscoelasticity, the Poynting-Thomson, Maxwell, and Kelvin-Voigt models are recovered as particular cases of the formalism. More complicated models, such as Jeffrey's model, can also be obtained. For dilute polymer solutions, one recovers the relaxational spectrum of viscous modes. In particular, our description encompasses the Rouse model.