Time-Domain MAS-SIBC Method

• Time-Domain MAS-SIBC Method is a meshless formulation that combines auxiliary sources with standard impedance boundary conditions to model transient electromagnetic scattering using only boundary unknowns.
• It employs a causal convolution with a t⁻¹/² memory kernel and a sequential time-marching scheme to efficiently simulate EM interactions in conductive media under moderate conductivity regimes.
• The method extends to metasurface applications by reducing the GSTC framework, offering validated performance against analytical and numerical benchmarks across diverse geometries.

Time-domain MAS-SIBC is a meshless transient electromagnetic formulation that combines the Method of Auxiliary Sources (MAS) with the Standard Impedance Boundary Condition (SIBC) to model scattering from conductive boundaries through boundary-only unknowns rather than volumetric field solves. In the formulation developed for infinitely long cylinders of moderate conductivity, fictitious filamentary sources radiate from an offset auxiliary contour and are chosen so that the total tangential fields on the physical contour satisfy a causal time-domain SIBC (Kouroublakis et al., 7 Aug 2025). In the related metasurface literature, the same construction appears as a specialization of a broader MAS-GSTC framework: when only the electric sheet susceptibility or admittance is retained and the magnetic and magnetoelectric terms are set to zero, the time-domain GSTC enforcement reduces to a time-domain MAS-SIBC configuration (Kouroublakis et al., 8 May 2026).

1. Definition, regime of validity, and problem class

The method is formulated for two-dimensional scattering by infinite cylinders aligned with the $z$-axis, with cross section $C$ a smooth closed curve. The exterior region is vacuum, the cylinder is non-magnetic with $\mu=\mu_0$, and the conductivity $\sigma$ is finite but moderate. Both $\mathrm{TM}_z$ and $\mathrm{TE}_z$ polarizations are treated. The tested excitations are transient plane waves with Gaussian envelope and transient cylindrical waves from line sources. The numerical study includes circular, elliptical, super-circular, rounded-triangular, and inverted-elliptical cross sections, as well as a planar boundary representing a lossy half plane (Kouroublakis et al., 7 Aug 2025).

Its validity is tied to the first-order SIBC assumption

$\sigma \gg \omega_{\max}\epsilon,$

where $\omega_{\max}$ is the maximum angular frequency in the incident pulse spectrum. In that regime, internal fields decay rapidly normal to the surface, the interaction is dominated by surface effects, and the formulation avoids computing interior fields. The numerical examples use RF conductivities in the range $5\text{--}15~\mathrm{S/m}$, with representative values $\sigma=5,10,11,15~\mathrm{S/m}$. The cited application domain includes carbon-based composites, conductive polymers, and doped dielectrics used in wearable electronics, electromagnetic interference shielding, and biomedical sensors (Kouroublakis et al., 7 Aug 2025).

MAS is advantageous in this setting because the physical boundary condition is enforced directly on $C$0, while the fields are represented analytically by radiation from auxiliary sources on an offset contour $C$1. This boundary-only character is also emphasized in the metasurface formulation, where the sheet is modeled by boundary conditions and only the metasurface line is discretized (Kouroublakis et al., 8 May 2026).

2. Time-domain impedance boundary condition

The frequency-domain starting point is the Leontovich boundary condition

$C$2

with outward unit normal $C$3 on $C$4. In the regime $C$5, the adopted surface impedance is

$C$6

For time-domain conversion, the formulation rewrites the condition as

$C$7

Applying the inverse Fourier transform and imposing causality yields

$C$8

with

$C$9

The time-domain SIBC is therefore a causal convolution between a $\mu=\mu_0$0 memory kernel and the time derivative of the tangential magnetic field (Kouroublakis et al., 7 Aug 2025).

In the metasurface setting, the same causal-convolution perspective is expressed through impedance-type GSTCs. For dispersive sheets with frequency-dependent admittance and impedance dyadics, the inverse Fourier or Laplace transform produces causal kernels $\mu=\mu_0$1 and $\mu=\mu_0$2, so that

$\mu=\mu_0$3

The SIBC special case is obtained by setting $\mu=\mu_0$4. In the paper’s explicit reduction, the time-domain MAS-GSTC becomes MAS-SIBC by setting $\mu=\mu_0$5 and retaining only the electric block $\mu=\mu_0$6 (Kouroublakis et al., 8 May 2026).

3. MAS field representation and boundary enforcement

MAS, also described here as the Method of Fundamental Solutions, represents the scattered field by fictitious filamentary sources placed on an auxiliary contour $\mu=\mu_0$7 inside the conducting region. In the combined-sources formulation for $\mu=\mu_0$8,

$\mu=\mu_0$9

where $\sigma$0 is the parent surface current density on $\sigma$1, $\sigma$2, and $\sigma$3 is the outward normal on $\sigma$4. The $\sigma$5 case is obtained by swapping the roles of electric and magnetic sources (Kouroublakis et al., 7 Aug 2025).

The scattered fields are given by exact two-dimensional retarded-potential integrals. For $\sigma$6, the scattered electric field component $\sigma$7 and the scattered magnetic field $\sigma$8 are expressed as integrals over $\sigma$9 and retarded time $\mathrm{TM}_z$0, where $\mathrm{TM}_z$1 and $\mathrm{TM}_z$2. The formulation explicitly includes both the electric-current contribution and the magnetic-current contribution arising from the combined-source blend (Kouroublakis et al., 7 Aug 2025).

After time and space discretization, the parent current derivative is expanded at source points $\mathrm{TM}_z$3 with unknown amplitudes $\mathrm{TM}_z$4, and a causal temporal basis is used with retarded onset determined by the nearest testing point. Evaluated at testing points $\mathrm{TM}_z$5 on $\mathrm{TM}_z$6, the discrete scattered fields are written in terms of precomputed weights $\mathrm{TM}_z$7 and $\mathrm{TM}_z$8, with retarded delays entering through

$\mathrm{TM}_z$9

The total tangential fields at the boundary are then constrained by the discrete time-domain SIBC at each testing point. This produces a convolutional collocation system in which the unknown source amplitudes are advanced sequentially in time (Kouroublakis et al., 7 Aug 2025).

The metasurface formulation follows the same structural pattern. There, scattered or transmitted fields are expanded by auxiliary sources on two fictitious lines, one per half-space, and GSTC or SIBC enforcement is imposed at collocation points on the metasurface line. The discrete fields again take the form of analytic retarded-potential weights $\mathrm{TE}_z$0 and $\mathrm{TE}_z$1, and the difference between the full GSTC and its SIBC specialization lies only in the kernel set retained in the boundary operator (Kouroublakis et al., 8 May 2026).

4. Discrete time marching, source placement, and numerical realization

The time-domain SIBC is discretized with a backward difference for $\mathrm{TE}_z$2, giving

$\mathrm{TE}_z$3

In matrix form, the resulting system is block-lower triangular and block-Toeplitz: $\mathrm{TE}_z$4 with

$\mathrm{TE}_z$5

This structure supports sequential time marching. An explicit formulation is possible when, in the first interval, only the nearest source contributes to each testing point and that source is closer to that testing point than to any other. The paper states that an implicit formulation with $\mathrm{TE}_z$6, solved by least squares at each time step, is typically favored for stability and reduced late-time errors (Kouroublakis et al., 7 Aug 2025).

The practical discretization is tied to the incident-pulse spectrum. For $\mathrm{TE}_z$7, the paper gives $\mathrm{TE}_z$8 and Nyquist interval $\mathrm{TE}_z$9. The studied cases use $\sigma \gg \omega_{\max}\epsilon,$0 for the circular cylinder, while other geometries use $\sigma \gg \omega_{\max}\epsilon,$1 between $\sigma \gg \omega_{\max}\epsilon,$2 and $\sigma \gg \omega_{\max}\epsilon,$3. The SIBC kernel samples are computed via IFFT of $\sigma \gg \omega_{\max}\epsilon,$4 over $\sigma \gg \omega_{\max}\epsilon,$5, $\sigma \gg \omega_{\max}\epsilon,$6, and $\sigma \gg \omega_{\max}\epsilon,$7 (Kouroublakis et al., 7 Aug 2025).

Source placement depends on geometry. For smooth convex shapes, the auxiliary contour is obtained by simple scaling, such as $\sigma \gg \omega_{\max}\epsilon,$8 for the circle and $\sigma \gg \omega_{\max}\epsilon,$9 for the ellipse. For super-circular, rounded-triangular, and inverted-elliptical contours, selected matching points are shifted inward along outward normals by a fixed distance $\omega_{\max}$0, and every fifth matching point provides a source location. For the planar problem, odd numbers of matching points and sources are placed symmetrically, with empirical spacings

$\omega_{\max}$1

The related metasurface work reports a similar marching-on-in-time procedure, but emphasizes that $\omega_{\max}$2 is chosen by pulse bandwidth rather than by a CFL constraint because MAS does not discretize space (Kouroublakis et al., 8 May 2026).

5. Validation, benchmark geometries, and observed numerical behavior

The method is validated against analytical frequency-domain solutions for canonical shapes and against COMSOL Multiphysics followed by IFFT for general geometries. Across all tested shapes and for both $\omega_{\max}$3 and $\omega_{\max}$4, the reported agreement is excellent. Observation examples are given in forward, normal, and backward directions at $\omega_{\max}$5 and $\omega_{\max}$6. For the circular $\omega_{\max}$7 plane-wave case, a PEC comparison is included to emphasize the distinction between finite-conductivity and perfectly conducting responses (Kouroublakis et al., 7 Aug 2025).

Geometry	Polarization / excitation	Example parameters
Circular cylinder	$\omega_{\max}$8, plane wave	$\omega_{\max}$9, $5\text{--}15~\mathrm{S/m}$0, $5\text{--}15~\mathrm{S/m}$1
Circular cylinder	$5\text{--}15~\mathrm{S/m}$2, line source	$5\text{--}15~\mathrm{S/m}$3, differentiated Gaussian current
Elliptical cylinder	$5\text{--}15~\mathrm{S/m}$4, plane wave	semi-axes $5\text{--}15~\mathrm{S/m}$5, $5\text{--}15~\mathrm{S/m}$6, $5\text{--}15~\mathrm{S/m}$7, $5\text{--}15~\mathrm{S/m}$8, $5\text{--}15~\mathrm{S/m}$9
Super-circular cylinder	$\sigma=5,10,11,15~\mathrm{S/m}$0, plane wave	$\sigma=5,10,11,15~\mathrm{S/m}$1, $\sigma=5,10,11,15~\mathrm{S/m}$2, $\sigma=5,10,11,15~\mathrm{S/m}$3, $\sigma=5,10,11,15~\mathrm{S/m}$4
Rounded-triangular cylinder	$\sigma=5,10,11,15~\mathrm{S/m}$5, plane wave	$\sigma=5,10,11,15~\mathrm{S/m}$6, $\sigma=5,10,11,15~\mathrm{S/m}$7, $\sigma=5,10,11,15~\mathrm{S/m}$8, $\sigma=5,10,11,15~\mathrm{S/m}$9
Inverted-elliptical cylinder	$C$00, plane wave	$C$01, $C$02, $C$03, $C$04
Planar boundary	$C$05, line source	source at $C$06, $C$07, $C$08, $C$09

A central diagnostic is the boundary-condition residual $C$10, defined from the maximum discrepancy between the total boundary electric field and the SIBC convolution term, normalized by a reference amplitude. The reported trend is systematic: $C$11 decreases significantly as $C$12 decreases and as $C$13 and $C$14 increase. Proper placement of $C$15 is likewise important; the cited offsets $C$16 scaling and $C$17 are part of the successful configurations (Kouroublakis et al., 7 Aug 2025).

The related metasurface paper provides additional validation of the SIBC specialization. In its anisotropic graphene and anisotropic black phosphorus examples, only $C$18 is nonzero, so the MAS-GSTC system reduces to a MAS-SIBC-like form. Those time-domain results are validated against FD-COMSOL with IFFT and show excellent agreement in reflected and transmitted fields. The Lorentzian metasurface example then illustrates the departure from SIBC by activating both $C$19 and $C$20 (Kouroublakis et al., 8 May 2026).

6. Relation to GSTC, limitations, and extensions

A recurrent ambiguity concerns the relation between SIBC and GSTC. In the metasurface formulation, SIBC is not a separate boundary formalism but a special case of impedance-type GSTC. The reduction occurs when $C$21 and only $C$22 remains, or equivalently when the magnetic surface current is absent and the tangential magnetic-field jump is related to the average tangential electric field through an admittance kernel. The paper states explicitly that graphene and black phosphorus operate in this regime and can therefore be viewed as time-domain MAS-SIBC implementations (Kouroublakis et al., 8 May 2026).

The cylinder formulation has several explicit limitations. It assumes the first-order SIBC regime $C$23; when this condition is not satisfied, internal fields are no longer negligible and higher-order or full boundary models may be required. Highly dispersive or magnetic materials, with $C$24 or frequency-dependent $C$25 and $C$26, are outside the model scope as presented. The study also notes that extremely sharp corners may deteriorate MAS conditioning; rounded features are preferred, and the tested non-convex geometry is an inverted ellipse rather than a sharply cornered profile (Kouroublakis et al., 7 Aug 2025).

The metasurface formulation lists broader modeling assumptions that clarify the present boundary-only viewpoint: two dimensions, infinite planar sheet, zero thickness, local sheet response, linear time-invariant kernels, and no space-time modulation. It also identifies natural extensions, including three-dimensional MAS-GSTC, nonuniform metasurfaces with local tensors, retention of the full bianisotropic kernel set $C$27, least-squares enforcement, and rational-kernel approximations to reduce history load (Kouroublakis et al., 8 May 2026). For the cylinder SIBC problem, the corresponding extensions identified are higher-order SIBC formulations, broadband dispersive materials, magnetic behavior, three-dimensional generalizations, non-cylindrical scatterers, and hybridization with fast solvers such as FMM for large arrays (Kouroublakis et al., 7 Aug 2025).

Within these boundaries, the time-domain MAS-SIBC method occupies a specific niche: transient analysis of conductive scattering problems in which surface effects dominate, the geometry is efficiently described by a boundary contour, and a causal convolutional impedance law can replace interior volumetric simulation.