On the use of a meshless solver for PDEs governing electromagnetic transients

Abstract

In this paper some key elements of the Smoothed Particle Hydrodynamics methodology suitably reformulated for analyzing electromagnetic transients are investigated. The attention is focused on the interpolating smoothing kernel function which strongly influences the computational results. Some issues are provided by adopting the polynomial reproducing conditions. Validation tests involving Gaussian and cubic B-spline smoothing kernel functions in one and two dimensions are reported.

Introduction

In the last two decades the meshless methods have known a great success in the simulation of a wide problems variety as a valid computational alternative to grid methods. They share common features such as the avoidance of the use of grids, but are different in functions approximation and computational processes.

The numerical technique known as Smoothed Particle Hydrodynamics (SPH) [4], [5], [6], [10], [11] is a meshless method and its attractiveness and popularity is due to the evaluation of unknown field functions and relative differential operators by means of an integral representation based on a suitable interpolating function. The integral representation is discretized by using a set of particles scattered in the problem domain.

The appropriate choice of the smoothing kernel function is a crucial task before performing any computation using the SPH solver. The smoothing kernel function is of remarkable importance since it not only determines the interpolating pattern, but also defines the width of the influence area of a particle determined by a parameter h called as smoothing length. The choice of this parameter is a key variable for the kernel’s worth. In fact, the smoothing kernel function should have a certain degree of consistency which can be expressed by its ability to reproduce the polynomials in both integral and discrete formulations [2], [3], [7]. For each smoothing kernel function only a set of h values, related to the interspacing particles, verifying the polynomial reproducing conditions must be taken into account. In this paper an analysis of the smoothing length h values is carried out by adopting two bell-shaped smoothing kernel functions widely used in literature [7], [9], [11]. Namely, Gaussian and cubic B-spline smoothing kernel functions are considered in one and two dimensions. Validation tests are performed by considering the SPH method suitably reformulated for solving the partial differential equations (PDEs) governing electromagnetic transients [1]. The particle expressions of the Maxwell’s curl equations are provided in one and two dimensions in free space. By working with curl equations the derivatives of the unknown field functions components must verify the consistency conditions. Various simulations are reported with different values of the smoothing length h by considering the transient propagation of a time and space variable pulse. The 2-D model is proposed for a transverse electric wave.

The paper is organized as follows. In Section 2 the background of SPH method is assumed and an analysis of the fundamental issues is reported; namely, the discrete constant and linear consistency conditions are investigated and a set of h values is determined. In Section 3 the meshless formulation of the Maxwell’s curl equations is provided in one and two dimensions. Simulation results referred to canonical case studies are reported.