On One Approach to Simulation of Stiff Dynamic Systems

The power-law dependence of the quality factor Q on frequency (i.e. Q = Q 0 | f / f 0 | γ ), which is called the power-law frequency-dependent Q for brevity, is a common phenomenological description of seismic wave attenuation in the interior of the Earth. The wave equation in differential form is essential to forward and inverse modelling of dissipative seismic waveforms in an accurate and efficient manner. However, all existing methods are seemingly unable to explicitly incorporate the exponent parameter γ into the wave equation in differential form. This drawback apparently limits the development of gradient-based inverse methods for spatially varying γ via the wave equation. In this paper, we use a weighting function method to derive the viscoacoustic wave equations for the power-law frequency-dependent Q , which explicitly involve the parameters γ and Q 0 . A critical step in this method is to construct a dissipative model, for which the complex modulus is expressed as an N -order series in terms of the weighting function. Numerical examples are used to illustrate the accuracy of the dissipative model, the effect of γ on waveforms, and the application of the newly proposed wave equations in viscoacoustic wavefield modelling for the power-law frequency-dependent Q .

Evolutionary approaches are widely applied in solving various types of problems. The paper considers the application of EvolODE and EvolODES approaches to the identification of dynamic systems. EvolODE helps to obtain a model in the form of an ordinary differential equation without restrictions on the type of the equation. EvolODES searches for a model in the form of an ordinary differential equation system. The algorithmic basis of these approaches is a modified genetic programming algorithm for finding the structure of ordinary differential equations and differential evolution to optimize the values of numerical constants used in the equation. Testing for these approaches on problems in the form of ordinary differential equations and their systems was conducted. The influence of noise present in the data and the sample size on the model error was considered for each of the approaches. The symbolic accuracy of the resulting equations was studied. The proposed approaches make it possible to obtain models in symbolic form. They will provide opportunities for further interpretation and application.

Hybrid finite element methods (HFEM) as used in the present paper refer to a coupling of the finite element method (FEM) applied to Maxwell's equations in differential form and the method of moments (MOM) applied to Maxwell's equations in integral form. This coupling allows the strengths of each method to supplant the other. The finite element method models penetrable materials and complicated geometry, while the method of moments provides the exact radiation boundary condition where the finite element mesh is terminated. In the paper a three-dimensional HFEM implementation is presented. As in Boyse and Seidl (1991), the finite element formulation is coupled to a highly efficient BOR MOM code which provides the exact near field radiation condition. The scalar and vector potential formulation completely eliminates spurious modes when using nodal based elements, allowing iterative methods to rapidly converge and the computation of very accurate internal fields. However, calculations involving models with sharp corners initially produced very inaccurate results. A new element (Boyse et al., 1992 and Bardi et al., 1993) was used along corners and edges which combines both nodal and edge basis functions. The addition of these new elements produces very accurate results while retaining conventional nodal elements throughout most of the model.

In this paper, three size-dependent formulations are developed for the analysis of Timoshenko nanobeams with various end conditions based on the nonlocal and strain gradient theories. The nonlocal governing equations are presented based on the stress-driven model of Eringen’s theory. First, a strain gradient Timoshenko beam model is developed. The governing equations of the integral stress-driven model, and then those of differential stress-driven model together with associated constitutive boundary conditions are obtained in the next step. With the aim of addressing the static bending and free vibration problems, the nonlocal governing equations in integral form are directly solved by constructing matrix differential and integral operators. Furthermore, the governing equations in differential form together with constitutive boundary conditions are discretized and solved via the mentioned operators. It is shown that there is a good agreement between the results obtained from solving the integral and differential governing equations of stress-driven nonlocal models. Several comparative studies are also conducted for the bending and vibration analyses of nanobeams based on the strain gradient and stress-driven nonlocal models. The results reveal that in both models, increasing the nonlocal/length scale parameter has a stiffening effect on the response of the system. However, the stiffening effect corresponding to the strain gradient model is more pronounced than that corresponding to the stress-driven nonlocal model.

Modelling Disappearance of Ethylene from Controlled Storage Atmospheres Exposed to Ultraviolet Radiation

On the expanded Maxwell’s equations for moving charged media system – General theory, mathematical solutions and applications in TENG

This paper analyses corona discharge in ambient air flow associated with laboratory-scaled wire-to-plate electrostatic precipitator (WPESP). The corona discharge is analysed by a combined iterative computational technique based on the finite element method (FEM) and charge simulation method (CSM). The phenomenon is mathematically described by Maxwell's equations in differential form. A finite element method is used to solve the Poisson's equation and the charge simulation method is used to satisfy the current continuity condition. Measurement method of the positive dc corona current density and electric field, taking into account the air flow velocity, has been introduced. The computed results are compared with experimental results to test the effectiveness of this approach.

Growth equations: A general equation and a survey of special cases

We give a revised presentation of the Yakubovskii–Faddeev formalism based on a systematic study of the N-body system chain structure. Completeness properties of the corresponding equations in differential form are considered. The expressions of physical and spurious solutions are given in terms of the N-body asymptotic partition Hamiltonians eigenvectors.

The modification of classical homogeneous nucleation theory to account for scavenging of molecular clusters by a preexisting aerosol is considered. A mathematical model proposed by Friedlander for an aerosol in the free-molecular regime is used to derive a new criterion for the importance of scavenging in such a system. This is done by writing the governing cluster balance equations in differential form and expressing the solution as a perturbation series in terms of the aerosol surface area. It is found that scavenging has an increasingly important effect on the nucleation rate as the vapor-phase supersaturation ratio decreases or the cluster surface energy increases.

Dynamic processes of quartic autocatalysis chemical reaction in Williamson nanofluid flow over a parabolic surface

This paper proposes a circuit network analogue for the solution of Maxwell's equations in differential form for two dimensional domains. First an equivalence is established among the electric and magnetic field and the corresponding voltage and current in the equivalent network. The proposed equivalent circuit cell consists of resistances, inductances and capacitances. Independent and controlled voltage/current sources are used to implement circuit models for radiating sources and boundary conditions. Standard circuit techniques and simulators are used to solve the resulting network in terms of voltages, currents and their corresponding field's variables. In the paper, some applications are given about the analysis of linear and nonlinear two dimensional shielding structures. The results and CPU time are compared with those given by a dedicated finite differences code. The advantages of this approach are shown and its limitations discussed.

The finite‐element or finite‐difference techniques are well known for the solution of Maxwell's equation in differential form. But terminating the mesh accurately at a finite distance from the body in the case of an open problem is a major challenge. Though there have been several techniques presented [1, 2], this new approach allows for the terminating surface to encapsulate the body very tightly. In this proposed method, the finite‐element technique is used for open‐region problems, whereas an integral equation solution approach using the Green's function is applied to enforce the radiation condition. At each iteration cycle, field sources within the domain are evaluated and their potential at the terminating surface is calculated. The proposed method leads to increased computational efficiency of the finite‐element method. It can be generalized for the case of inhomogeneous and nonlinear media, for static and dynamic fields. Typical numerical results are presented for the solution of the Laplace equation to illustrate the accuracy of the technique. © 1995 John Wiley & Sons. Inc.

In this introductory chapter we will explain the physical model and derive the boundary value problems which we will investigate in this monograph. We begin by formulating Maxwell’s equations in differential form as our starting point. In this monograph we consider exclusively linear media; that is, the constitutive relations are linear. The restriction to special cases leads to the analogous equations in electrostatics or magnetostatics and, assuming periodic time dependence when going into the frequency domain, to time-harmonic fields.KeywordsHelmholtz EquationMaxwell SystemReference ProblemHertz PotentialSpherical Wave FunctionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We provide a study of quantum chromodynamics with the technique of Dyson-Schwinger equations in differential form. In this way, we are able to approach the non-perturbative limit and recover, with some approximations, the 't Hooft limit of the theory. Quark mass in the propagator term goes off-shell at low-energies signaling confinement. A condition for such occurrence in the theory is provided.