Time-domain full-band method using orthogonal edge basis functions

Novel 3D orthogonal basis functions for the vector wave equation solution

“…These basis functions guarantee the tangential continuity and the normal discontinuity of the expanded field, but yet do not satisfy (3). To make them orthogonal and preserve the property of tangential continuity, a new group of vector basis function has to be introduced as:…”

“…By upgrading the two-dimension formulation presented in [3]- [4] to three dimensions, the PML (perfectly matched layer) vector wave equation for electric and magnetic fields in a medium with magnetic permeability equals to the free space magnetic permeability µ 0 is given by: and H z and E x , E y , and E z are the x, y and z magnetic and electric field components, respectively, t is the time, c is the free space light speed, n is the refractive index, δ(t) is the Dirac delta function, ū(t) is the step function, and ε is the electric permittivity. The conductivity σ is zero for regions outside the PML and equal to σ max (ρ/d) 2 within the PML, where σ max is the maximum conductivity, ρ is the distance from the initial border of the PML, and d is the PML thickness.…”

“…The summation in (16) indicates that the global matrices [K] s should consider the term exp[-σ y,z /(ε+jω 0 )t n-1 ] of (15) at each time step, as it depends on the geometry (σ y,z and ε vary for each tetrahedral element) and the time step (t n-1 ). However, it was demonstrated in [3]- [4] that, if σ s and ε are kept constant inside the PML regions, (15) can be simplified without affecting the PML field absorption performance, improving processing time performance. By assuming that to simplify the analysis due to the great number of unknowns involved and due to the addition of the auxiliary basis functions in the 3D simulation, (15) can be rewritten as: …”

“…For structures that can be represented or approximated by one (1D) or two (2D) dimensions, several formulations based on finite elements discretezation have been developed. In general, such formulations make use of either the implicit scheme, where calculation stability is always guaranteed [1]- [2], or the explicit scheme [3]- [4], where computational efforts become smaller and analyses are made in large computational windows. However, when the representation or approximation in one or two dimensions is not possible, the use of a three-dimensional (3D) approach is necessary.…”

New set of 3D orthogonal edge basis functions for the vector wave equation solution

“…For structures that can be represented or approximated by one-(1D) or two-dimension (2D) approaches, several formulations based on finite elements discretezation have been developed. In general, such formulations make use of either the implicit scheme, where calculation stability is always guaranteed [1,2], or the explicit scheme [3,4], where computational efforts become smaller and analyses are made in large computational windows. However, when one or two dimensions are no longer applicable to solve a given problem, a three-dimensional (3D) approach is therefore necessary.…”