text:
\section{Introduction}
\IEEEPARstart{M}{etasurfaces} are electrically thin layers with embedded subwavelength-sized scatterers \cite{HollawayMetasurfaceReview,MetasurfaceReview}. They are two-dimensional (2D) reductions of three-dimensional (3D) metamaterials \cite{MetamaterialsSolymar,CalozMTM,3DMTMZedler} and offer much richer funtionalities than traditional frequency selectice surfaces \cite{MunkFSS}. The advantages of metasurfaces over metamaterials include lower loss, lighter weight, and easier fabrication. Applications of metasurfaces include polarization transformers \cite{PfeifferGrbicPolControl}, generalized refraction \cite{NYuLightProp}, broadband absorbers \cite{MetasurfaceBroadbandAbsorber}, spatial waveguides \cite{SpatialWaveguideMetasurface}, remotely-controlled spatial processors \cite{AchouriMetasurfaceEPJ}, aberration free lens \cite{AberrationFreeLens},\cite{AietaAchromaticMS}, flat optical components \cite{FlatOpticsYu}, LED efficiency enhancers \cite{LEDEnhancerMS}, and spatial isolators \cite{SpatialIsolatorMS}. Metasurfaces achieve these functionalities by creating discontinuities in the electromagnetic fields. Such discontinuities can be modeled by generalized sheet transition conditions (GSTCs) \cite{HollawayMetasurfaceReview}, \cite{IdemenBook}. Therefore, it is important to develop numerical modeling of GSTCs for the study of metasurfaces.

The modeling of GSTCs in the finite difference method has been recently reported in \cite{YousefFDFD}. In this paper, we present the modeling of GSTCs in the finite element method (FEM), which is one of the most widely used numerical methods to simulate electromagnetic boundary-value problems (BVP) \cite{JinFEMBook,JinCEMBook}. The FEM is particularly suited for solving practical engineering problems given its ability to model complex geometries with adaptive tetrahedral meshes. At present, commercial electromagnetic simulation softwares can model several boundary conditions, such as perfect electric conductor (PEC), perfect magnetic conductor (PMC), periodic boundary condition (PBC), standard impedance boundary condition (SIBC), radiation boundary condition (RBC), and perfectly matched layer (PML). Recently, a formulation to incorporate a generalized impedance boundary condition (GIBC) has also been proposed \cite{FEMGIBC}. However, no commercial CAD tools have yet incorporated the modeling of GSTCs. Since the FEM is the computational method used in the most frequently used CAD tools and since metasurfaces have become increasingly prominent in electromagnetic engineering, there is clearly a need for the modeling of GSTCs in the FEM framework. It should be noted that physical metasurfaces have a finite subwavelength thickness. Simulating such structures directly would result in very dense meshes around the metasurfaces and hence compromise the simulation efficiency. By replacing a physical metasurface by an equivalent GSTC, the burden of mesh generation can be reduced significantly and the simulation efficiency can be enhanced considerably. This is particularly important in simulation scenarios where multiple metasurfaces are involved or when repetitive simulations are required for optimization \cite{achouri2015general}.

The organization of the paper is as follows. Section II recalls the GSTC metasurface synthesis equations. This is followed by the FEM-GSTC formulation in 1D and 2D domains in Sections III and IV, respectively. Section V presents some simulation results, and Section VI extends the method to simulate bianisotropic metasurfaces. Conclusions are provided in Section VII.

\section{Metasurface synthesis equations}
For a metasurface placed perpendicular to the  $z$ direction of a cartesian coordinate system,
GSTCs in their most general form are given as \cite{IdemenBook},\cite{achouri2015general}
\begin{subequations}
\label{GSTCeqns}
\begin{align}
\hat{z} \times \Delta \vec{H} = j\omega \vec{P}_{||} - \hat{z} \times \nabla_{||}M_{z}, \label{GSTC1} \\
\hat{z} \times \Delta \vec{E} = -j\omega \mu \vec{M}_{||} - \hat{z} \times \nabla_{||}\left(\frac{P_{z}}{\epsilon}\right), \label{GSTC2}
\end{align}
\end{subequations}
where $\nabla_{||} = \hat{x} {\partial/ \partial x} + \hat{y}{\partial/ \partial y}$, $\vec{P}$ and $\vec{M}$ are the electric and magnetic polarization densities, respectively, and
$\epsilon$ and $\mu$ are the permittivity and permeability of the surrounding medium. Moreover, $\Delta \vec{E}$ and $\Delta \vec{H}$ represent respectively the differences between $\vec{E}$ and $\vec{H}$ on the two sides of the metasurface. For a problem of reflection and transmission, $\Delta \vec{\psi} = \vec{\psi}^{\mathrm{tr}} - (\vec{\psi}^{\mathrm{ref}} + \vec{\psi}^{\mathrm{inc}})$ with the superscripts ``tr,'' ``ref,'' and ``inc'' denoting the transmitted, reflected, and incident fields, respectively. Throughout this work, the normal components of the polarization densities are assumed to be zero, i.e.\ $P_{z} = M_{z} = 0$ \cite{YousefFDFD},\cite{achouri2015general}.
The polarization densities can be expressed as
\begin{subequations} \label{PMEqn}
\begin{equation} \label{Peqn1}
\vec{P} = \epsilon\overline{\overline{\chi}}_{\mathrm{ee}}\vec{E}_{\mathrm{av}} + \sqrt{\epsilon \mu}\ \overline{\overline{\chi}}_{\mathrm{em}}\vec{H}_{\mathrm{av}}
\end{equation}
\begin{equation} \label{Meqn1}
\vec{M} = \overline{\overline{\chi}}_{\mathrm{mm}}\vec{H}_{\mathrm{av}} + \sqrt{\epsilon \mu}\ \overline{\overline{\chi}}_{\mathrm{me}}\vec{E}_{\mathrm{av}}
\end{equation}
\end{subequations}
where $\overline{\overline{\chi}}_{\mathrm{ee}},\overline{\overline{\chi}}_{\mathrm{mm}},\overline{\overline{\chi}}_{\mathrm{em}}$, and $\overline{\overline{\chi}}_{\mathrm{me}}$ are the electric/magnetic (first e/m subscripts) susceptibility tensors describing the response to the electric/magnetic (second e/m subscripts) excitations, and the subscript ``av'' denotes the average of the fields on both sides of the metasurface, $\vec{\psi}_{\mathrm{av}} = [(\vec{\psi}^{\mathrm{inc}} + \vec{\psi}^{\mathrm{ref}}) + \vec{\psi}^{\mathrm{tr}}]/2$. Substituting (\ref{PMEqn}) into (\ref{GSTCeqns}) results in the following metasurface synthesis equations:
\begin{subequations}
\label{GSTCexpandedeqns}
\begin{equation} \label{GSTCexpandedeqns1}
\begin{split}
\begin{pmatrix}
-\Delta H_{y} \\
\Delta H_{x}
\end{pmatrix}
 = & j\omega\epsilon
\begin{pmatrix}
\chi_{\mathrm{ee}}^{xx} & \chi_{\mathrm{ee}}^{xy} \\
\chi_{\mathrm{ee}}^{yx} & \chi_{\mathrm{ee}}^{yy}
\end{pmatrix}
\begin{pmatrix}
E_{x,\mathrm{av}} \\
E_{y,\mathrm{av}}
\end{pmatrix}
\\ & + j\omega\sqrt{\epsilon \mu}
\begin{pmatrix}
\chi_{\mathrm{em}}^{xx} & \chi_{\mathrm{em}}^{xy} \\
\chi_{\mathrm{em}}^{yx} & \chi_{\mathrm{em}}^{yy}
\end{pmatrix}
\begin{pmatrix}
H_{x,\mathrm{av}} \\
H_{y,\mathrm{av}}
\end{pmatrix} ,
\end{split}
\end{equation}
\begin{equation}
\label{GSTCexpandedeqns2}
\begin{split}
\begin{pmatrix}
\Delta E_{y} \\
-\Delta E_{x}
\end{pmatrix}
  = & j\omega\mu
\begin{pmatrix}
\chi_{\mathrm{mm}}^{xx} & \chi_{\mathrm{mm}}^{xy} \\
\chi_{\mathrm{mm}}^{yx} & \chi_{\mathrm{mm}}^{yy}
\end{pmatrix}
\begin{pmatrix}
H_{x,\mathrm{av}} \\
H_{y,\mathrm{av}}
\end{pmatrix}
\\ & + j\omega\sqrt{\epsilon \mu}
\begin{pmatrix}
\chi_{\mathrm{me}}^{xx} & \chi_{\mathrm{me}}^{xy} \\
\chi_{\mathrm{me}}^{yx} & \chi_{\mathrm{me}}^{yy}
\end{pmatrix}
\begin{pmatrix}
E_{x,\mathrm{av}} \\
E_{y,\mathrm{av}}
\end{pmatrix}
\end{split}
\end{equation}
\end{subequations}
which are applicable for a general bianisotropic metasurface. In the case of a mono-anisotropic metasurface, $\overline{\overline{\chi}}_{\mathrm{em}} = \overline{\overline{\chi}}_{\mathrm{me}} = 0$. Through out this paper, we have assumed, for simplicity but without loss of generality,  that the cross-polarization terms (i.e. tensor elements with either $xy$ or $yx$ as the superscript) in all the four susceptibility tensors are zero.

\section{GSTCs in 1D FEM}
Consider a 1D BVP where $z$ is the only dimension along which material and field variations exist. The computational domain extends from $z=0$ to $z=L$ and a GSTC surface is located at $z=z_{m}$, where $0 < z_{m} < L$. Assuming $E_{x}(z)$ and $H_{y}(z)$ as the field quantities, the scalar wave equation for $E_{x}(z)$ is given by
\begin{equation}
\frac{d}{dz}\left[\frac{1}{\mu_{r}(z)}\frac{dE_{x}}{dz}\right] + k_{o}^{2}\epsilon_{r}(z)E_{x}(z) = 0.
\end{equation}
The FEM domain discretization along with element numbers and global node numbers is shown in Fig.\ \ref{fig_1DFEM}, where $E_{x1}$ is located at $z=0$ and $E_{x8}$ is located at $z=L$. Nodes 4 and 5 are placed on either side of the GSTC surface.
\begin{figure}[!h]
\centering
\includegraphics[width=3.5in]{FEM1D_pdf.pdf}
\caption{1D FEM-GSTC computational domain: $E_{x}$ nodes.}
\label{fig_1DFEM}
\end{figure}
The electric field in the element $e$ can be expressed by interpolating the nodal electric field values using linear basis functions \cite{JinFEMBook} as
\begin{equation}
E_{x}^{e}(z) = \sum_{j=1}^{2} E_{xj}^{e}N_{j}^{e}(z),
\end{equation}
where $N_{j}^{e}(z)$ is the linear basis function. Using Galerkin's method yields the following linear systems of equations  for the unknown nodal values:
\begin{subequations}
\label{ExFEMeqns}
\begin{equation}
\begin{bmatrix}
K_{11}^{1} & K_{12}^{1} & 0 & 0 \\
K_{12}^{1} & K_{22}^{1} + K_{11}^{2} & K_{12}^{2} & 0 \\
0 & K_{12}^{2} & K_{22}^{2} + K_{11}^{3} & K_{12}^{3} \\
0 & 0 & K_{12}^{3} & K_{22}^{3}
\end{bmatrix}
\begin{bmatrix}
E_{x1} \\
E_{x2} \\
E_{x3} \\
E_{x4}
\end{bmatrix}
=
\begin{bmatrix}
b_{1} \\
0 \\
0 \\
b_{4}
\end{bmatrix},
\end{equation}
\begin{equation}
\begin{bmatrix}
K_{11}^{4} & K_{12}^{4} & 0 & 0 \\
K_{12}^{4} & K_{22}^{4} + K_{11}^{5} & K_{12}^{5} & 0 \\
0 & K_{12}^{5} & K_{22}^{5} + K_{11}^{6} & K_{12}^{6} \\
0 & 0 & K_{12}^{6} & K_{22}^{6}
\end{bmatrix}
\begin{bmatrix}
E_{x5} \\
E_{x6} \\
E_{x7} \\
E_{x8}
\end{bmatrix}
=
\begin{bmatrix}
b_{5} \\
0 \\
0 \\
b_{8}
\end{bmatrix}
.
\end{equation}
\end{subequations}
The matrix elements are calculated by using
\begin{equation}
K^{e}_{ij} = \int_{z^{e}_{1}}^{z^{e}_{2}}\left[\frac{1}{\mu_{r}}\frac{dN^{e}_{i}}{dz}
\frac{dN^{e}_{j}}{dz} - k_{o}^{2}\epsilon_{r}N^{e}_{i}N^{e}_{j}\right] dz,
\label{Keijeqn}
\end{equation}
where $\epsilon_{r}$ and $\mu_{r}$ are the relative permittivity and relative permeability of the medium within the element. The elements of the right-hand-side vectors in (\ref{ExFEMeqns}) are given by
\begin{subequations}
\begin{align}
b_{1} &= -\frac{1}{\mu_{r}}\frac{dE_{x}}{dz}\bigg|_{z=0}, \\
b_{4} &= \frac{1}{\mu_{r}}\frac{dE_{x}}{dz}\bigg|_{z=z_{m}^{-}}, \\
b_{5} &= -\frac{1}{\mu_{r}}\frac{dE_{x}}{dz}\bigg|_{z=z_{m}^{+}}, \\
b_{8} &= \frac{1}{\mu_{r}}\frac{dE_{x}}{dz}\bigg|_{z=L}.
\end{align}
\end{subequations}
The $b_{1}$ and $b_{8}$ can be evaluated by applying the first-order ABC \cite{JinFEMBook}. Such an ABC will excite the computational domain with a plane wave propagating along the $+z$ direction and absorb all the outgoing waves. For the case of an incident wave represented by $e^{-jkz}$, $b_{1}$ and $b_{8}$ are given by
\begin{subequations}
\begin{align}
b_{1} &= \frac{-jk}{\mu_{r}} E_{x1} + \frac{2jk}{\mu_{r}},\\
b_{8} &= \frac{-jk}{\mu_{r}} E_{x8}.
\end{align}
\end{subequations}
The $b_{4}$ and $b_{5}$ can be used to incorporate the metasurface synthesis equations (\ref{GSTCexpandedeqns}). From the Maxwell-Faraday equation,
\begin{subequations}
\begin{align}
b_{4} &= -j\omega\mu_{o}H_{y}\big|_{z=z_{m}^{-}},\\
b_{5} &= j\omega\mu_{o}H_{y}\big|_{z=z_{m}^{+}}.
\end{align}
\end{subequations}
For a mono-isotropic metasurface, equation (\ref{GSTCexpandedeqns}) reduces to
\begin{subequations}
\label{MSsynthesisEqn}
\begin{align}
E_{x}|_{z=z_{m}^{+}} - E_{x}|_{z=z_{m}^{-}} &= \frac{-j\omega \mu \chi_{\mathrm{mm}}^{yy}}{2}
\left(H_{y}|_{z=z_{m}^{+}} + H_{y}|_{z=z_{m}^{-}}\right), \\
H_{y}|_{z=z_{m}^{+}} - H_{y}|_{z=z_{m}^{-}} &= \frac{-j\omega \epsilon \chi_{\mathrm{ee}}^{xx}}{2}
\left(E_{x}|_{z=z_{m}^{+}} + E_{x}|_{z=z_{m}^{-}}\right).
\end{align}
\end{subequations}
Inverting the above equations yields the magnetic field components on either side of the GSTC surface in terms of the electric field components as
\begin{equation}
\begin{bmatrix}
j\omega\mu H_{y}|_{z_{m}^{+}} \\
-j\omega\mu H_{y}|_{z_{m}^{-}}
\end{bmatrix}
=
\begin{bmatrix}
A & B \\
B & A
\end{bmatrix}
\begin{bmatrix}
E_{x}|_{z_{m}^{+}} \\
E_{x}|_{z_{m}^{-}}
\end{bmatrix}
\end{equation}
with $A$ and $B$ given by
\begin{subequations}
\begin{equation}
A = \frac{k^{2}\chi_{\mathrm{ee}}^{xx}\chi_{\mathrm{mm}}^{yy}-4}{4\chi_{\mathrm{mm}}^{yy}},
\end{equation}
\begin{equation}
B = \frac{k^{2}\chi_{\mathrm{ee}}^{xx}\chi_{\mathrm{mm}}^{yy}+4}{4\chi_{\mathrm{mm}}^{yy}}.
\end{equation}
\end{subequations}
With these, $b_{4}$ and $b_{5}$ can be written in terms of the electric field values adjacent to the GSTC surface as
\begin{subequations}
\label{1DGSTCEqn}
\begin{align}
b_{4} = \frac{B}{\mu_{r}} E_{x5} + \frac{A}{\mu_{r}} E_{x4}, \\
b_{5} = \frac{A}{\mu_{r}} E_{x5} + \frac{B}{\mu_{r}} E_{x4}.
\end{align}
\end{subequations}
The above two equations incorporate GSTC in 1D FEM. In the code, the locations of the nodes corresponding to $E_{x4}$ and $E_{x5}$ can be at $z = z_{m}$. However, $E_{x4}$ should be used to calculate the field solution in element $3$ and $E_{x5}$ should be used to calculate the field solution in element $4$.

\section{GSTCs in 2D FEM}
The 2D computational domain considered in this work is shown in Fig.\ \ref{fig_2DFEM}, where a finite-sized GSTC surface is located at $z=z_{m}$.
\begin{figure}[!h]
\centering
\includegraphics[width=3.2in]{FEM2D_pdf.pdf}
\caption{2D FEM computational domain.}
\label{fig_2DFEM}
\end{figure}
Assuming a $\mathrm{TE}$ polarization ($E_{x},E_{z},H_{y}$), the wave equation for $H_{y}(x,z)$ reads
\begin{equation}
\label{waveEqn2DFEM}
\begin{split}
\frac{\partial}{\partial x}\left[\frac{1}{\epsilon_{r}(x,z)}\frac{\partial H_{y}}{\partial x}\right] & + \frac{\partial }{\partial z}\left[\frac{1}{\epsilon_{r}(x,z)} \frac{\partial H_{y}}{\partial z}\right] \\
& + k_{o}^{2}\mu_{r}(x,z) H_{y} = 0.
\end{split}
\end{equation}
Applying Galerkin's method to (\ref{waveEqn2DFEM}) results in a system of equations for nodal values of $H_{y}$, which can be written as
\begin{equation}
\begin{bmatrix}
K_{11} & K_{12} & \cdots & K_{1N} \\
K_{21} & K_{22} & \cdots & K_{2N} \\
\vdots & \ddots & \cdots & \vdots \\
K_{N1} & K_{N2} & \cdots & K_{NN}
\end{bmatrix}
\begin{bmatrix}
H_{y1} \\
H_{y2} \\
\vdots \\
H_{yN}
\end{bmatrix}
=
\begin{bmatrix}
g_{1} \\
g_{2} \\
\vdots \\
g_{N}
\end{bmatrix} ,
\end{equation}
where $N$ is the total number of nodes in the computational domain. The calculation of the elements of the stiffness matrix, $K_{ij}$, is same as that of 2D FEM without any GSTC surfaces \cite{JinFEMBook}. The elements $g_{j}$ are non-zero only if the corresponding node (i.e. node $j$) lies on the boundary or if the node lies on a surface (or curve) where there is a field discontinuity \cite{JinFEMBook}. Similar to the 1D case, a GSTC surface is modeled by placing nodes both above and below the GSTC surface as shown in Fig.\ \ref{fig_GSTC2DFEM}. It should be noted that the nodes above and below the GSTC surface do not necessarily need to have different locations. But the nodes above the GSTC surface should be used to calculate the field inside the triangular elements which are located above the GSTC surface and the same applies for the nodes below the GSTC surface.
\begin{figure}[!h]
\centering
\includegraphics[width=3.5in]{GSTC2DFEM_1_pdf.pdf}
\caption{Double nodes around the GSTC surface.}
\label{fig_GSTC2DFEM}
\end{figure}
Consider Fig.\ \ref{fig_GSTC2DFEM}, where the dashed line represents the metasurface. The expression for $g_{b}$ (i.e $g_{j}$ for node $b$) can be written as \cite{JinFEMBook}
\begin{equation}
\label{gb1}
\begin{split}
g_{b} = \int_{0}^{l_{ab}}\left(-\hat{z}\right) \cdot \left[\frac{1}{\epsilon_{r}}\frac{\partial H_{y}}{\partial z} \hat{z}\right] \frac{\xi}{l_{ab}} d\xi  \\
 \quad\quad\quad + \int_{0}^{l_{bc}}\left(-\hat{z}\right) \cdot \left[\frac{1}{\epsilon_{r}}\frac{\partial H_{y}}{\partial z}\hat{z}\right] \left(1-\frac{\xi}{l_{bc}}\right) d\xi
\end{split}
\end{equation}
where the first integral corresponds to edge $ab$ and the second integral corresponds to edge $bc$. The contribution to $g_{b}$ of other edges which are connected to node $b$ vanishes because of field continuity. Further, note that $-\hat{z}$ is the unit normal vector pointing outwards from elements $e1$ and $e3$ for edges $ab$ and $bc$, respectively, ${\xi}/{l_{ab}}$ is the value of 2D linear basis function $N^{e1}_{s}(x,z)$ along edge $ab$, where $s$ is the local node number of global node $b$ in element $e1$, and $1 - {\xi}/{l_{bc}}$ is the value of 2D linear basis function $N^{e3}_{t}(x,z)$ along edge $bc$, where $t$ is the local node number of global node $b$ in element $e3$.
By using Maxwell's equations, (\ref{gb1}) reduces to
\begin{equation}
\begin{split}
g_{b} = \int_{0}^{l_{ab}} j\omega \epsilon_{o}E_{x}\big|_{z_{m}^{+}}  \frac{\xi}{l_{ab}} d\xi  \\
\quad\quad\quad+ \int_{0}^{l_{bc}}j\omega \epsilon_{o}E_{x}\big|_{z_{m}^{+}}  \left(1-\frac{\xi}{l_{bc}}\right) d\xi,
\label{gb2}
\end{split}
\end{equation}
where $E_{x}\big|_{z_{m}^{+}}$ is a function of $\xi$.
The metasurface synthesis equations in (\ref{MSsynthesisEqn}) can be used to obtain expressions for $E_{x}|_{z_{m}^{+}}$ and $E_{x}|_{z_{m}^{-}}$ in  terms of $H_{y}|_{z_{m}^{+}}$ and
$H_{y}|_{z_{m}^{-}}$ as
\begin{equation}
\begin{bmatrix}
j\omega\epsilon E_{x}|_{z_{m}^{+}} \\
-j\omega\epsilon E_{x}|_{z_{m}^{-}}
\end{bmatrix}
=
\begin{bmatrix}
C & D \\
D & C
\end{bmatrix}
\begin{bmatrix}
H_{y}|_{z_{m}^{+}} \\
H_{y}|_{z_{m}^{-}}
\end{bmatrix}
\end{equation}
where the coefficients $C$ and $D$ are given by
\begin{subequations}
\begin{equation}
C = \frac{k^{2}\chi_{\mathrm{ee}}^{xx}\chi_{\mathrm{mm}}^{yy}-4}{4\chi_{\mathrm{ee}}^{xx}},
\end{equation}
\begin{equation}
D = \frac{k^{2}\chi_{\mathrm{ee}}^{xx}\chi_{\mathrm{mm}}^{yy}+4}{4\chi_{\mathrm{ee}}^{xx}}.
\end{equation}
\end{subequations}
In the segment $ab$, $E_{x}\big|_{z_{m}^{+}}$ will be denoted as $E_{x}^{ab}(\xi)$ and in the segment $bc$, $E_{x}\big|_{z_{m}^{+}}$ will be denoted as $E_{x}^{bc}(\xi)$. Following this notation, we obtain
\begin{subequations}
\label{Eyeqns}
\begin{align}
j\omega\epsilon_{o} E_{x}^{ab}(\xi) = \frac{C_{p}}{\epsilon_{r}}H_{y}^{ab}(\xi)
+ \frac{D_{p}}{\epsilon_{r}}H_{y}^{de}(\xi) \label{Exabeqn}\\
j\omega\epsilon_{o} E_{x}^{bc}(\xi) = \frac{C_{r}}{\epsilon_{r}}H_{y}^{bc}(\xi)
+ \frac{D_{r}}{\epsilon_{r}}H_{y}^{ef}(\xi) \label{Exbceqn}
\end{align}
\end{subequations}
where it has been assumed that the edges $ab$ and $bc$ are short enough such that the coefficients $C$ and $D$ are approximately constant on these segments. Therefore, $C_{p},D_{p},C_{r},$ and $D_{r}$ are the values of the coefficients $C$ and $D$ at the edge midpoints $p$ and $r$, respectively, as shown in Fig.\ 3. Since linear finite elements are used, linear interpolation can be used to obtain the values of $H_{y}$ at the edges:
\begin{subequations}
\label{Hyeqns}
\begin{align}
H_{y}^{ab}(\xi) &= H_{ya}\left(1-\frac{\xi}{l_{ab}}\right) + H_{yb}\frac{\xi}{l_{ab}}\ , \ 0 \le \xi \le l_{ab}, \label{Hyabeqn}\\
H_{y}^{de}(\xi) &= H_{yd}\left(1-\frac{\xi}{l_{ab}}\right) + H_{ye}\frac{\xi}{l_{ab}}\ , \ 0 \le \xi \le l_{ab}, \label{Hydeeqn}\\
H_{y}^{bc}(\xi) &= H_{yb}\left(1-\frac{\xi}{l_{bc}}\right) + H_{yc}\frac{\xi}{l_{bc}}\ , \ 0 \le \xi \le l_{bc}, \label{Hybceqn}\\
H_{y}^{ef}(\xi) &= H_{ye}\left(1-\frac{\xi}{l_{bc}}\right) + H_{yf}\frac{\xi}{l_{bc}}\ , \ 0 \le \xi \le l_{bc}. \label{Hyefeqn}
\end{align}
\end{subequations}
In the above equations, it is assumed that $l_{ab} = l_{de}$ and $l_{bc} = l_{ef}$, which is equivalent of having an identical mesh just above and below the GSTC surface. Substituting (\ref{Hyeqns}) into (\ref{Eyeqns}) and followed by further substitution into the integrals in (\ref{gb2}), an expression for $g_{b}$ can be obtained as
\begin{equation}
\begin{split}
g_{b} = \frac{C_{p}l_{ab}}{6\epsilon_{r}}H_{ya} + \frac{D_{p}l_{ab}}{6\epsilon_{r}}H_{yd} +
\left(\frac{C_{p}l_{ab}}{3\epsilon_{r}} + \frac{C_{r}l_{bc}}{3\epsilon_{r}}\right)H_{yb}  \\
+ \left(\frac{D_{p}l_{ab}}{3\epsilon_{r}} + \frac{D_{r}l_{bc}}{3\epsilon_{r}}\right)H_{ye} +
\frac{C_{r}l_{bc}}{6\epsilon_{r}}H_{yc} + \frac{D_{r}l_{bc}}{6\epsilon_{r}}H_{yf}.
\end{split}
\end{equation}
By following the same procedure, an expression for $g_{e}$ can be obtained as
\begin{equation}
\begin{split}
g_{e} = \frac{C_{p}l_{de}}{6\epsilon_{r}}H_{yd} + \frac{D_{p}l_{de}}{6\epsilon_{r}}H_{ya} +
\left(\frac{C_{p}l_{de}}{3\epsilon_{r}} + \frac{C_{r}l_{ef}}{3\epsilon_{r}}\right)H_{ye} \\
+ \left(\frac{D_{p}l_{ab}}{3\epsilon_{r}} + \frac{D_{r}l_{bc}}{3\epsilon_{r}}\right)H_{yb} +
\frac{C_{r}l_{ef}}{6\epsilon_{r}}H_{yf} + \frac{D_{r}l_{ef}}{6\epsilon_{r}}H_{yc}.
\end{split}
\end{equation}
The evaluation of $g_{b}$ and $g_{e}$ for every node above and below the GSTC surface completes the incorporation of GSTC into 2D FEM.

\section{Simulation Results}
This section presents simulation examples to validate FEM-GSTC in both 1D and 2D domains. The metasurface susceptibilities are synthesized by using (\ref{MSsynthesisEqn}). For vacuum on either side of the metasurface, the susceptibilites are obtained as
\begin{subequations}
\label{MetasurfaceSynthExample}
\begin{equation}
\chi_{\mathrm{ee}}^{xx} = \frac{2}{j\omega\epsilon_{o}}\left[\frac{H_{y}^{\mathrm{inc}} + H_{y}^{\mathrm{ref}} - H_{y}^{\mathrm{tr}}}{E_{x}^{\mathrm{inc}} + E_{x}^{\mathrm{ref}} + E_{x}^{\mathrm{tr}}}\right],
\end{equation}
\begin{equation}
\chi_{\mathrm{mm}}^{yy} = \frac{2}{j\omega\mu_{o}}\left[\frac{E_{x}^{\mathrm{inc}} + E_{x}^{\mathrm{ref}} - E_{x}^{\mathrm{tr}}}{H_{y}^{\mathrm{inc}} + H_{y}^{\mathrm{ref}} + H_{y}^{\mathrm{tr}}}\right].
\end{equation}
\end{subequations}
The simulation frequency is 5 GHz. For both 1D and 2D codes, a simple first-order analytical ABC is used, although the use of the second-order  ABC or PML would  result in more accurate results.

\subsection{1D Example}
For validating 1D FEM-GSTC, a fully absorbing metasurface is simulated, for which the reflected and transmitted fields are zero in (\ref{MetasurfaceSynthExample}). The susceptibilities for such a metasurface are thus found as $\chi_{\mathrm{ee}}^{xx} = \chi_{\mathrm{mm}}^{yy} = {2}/{jk_{o}}$, whose negative imaginary nature indicate dissipation \footnote{This may be easily verified by setting $\epsilon_\text{r}=1+\chi_\text{ee,im}^{xx}$. We have then $k_z=nk_0=\sqrt{\epsilon_\text{r}}k_0=\sqrt{1+j\chi_\text{ee,im}^{xx}}k_0\overset{|\chi_\text{ee,im}^{xx}|\ll1}{=}(1+j\chi_\text{ee,im}^{xx}/2)k_0$. So a negative $\chi_\text{ee,im}^{xx}$ implies a decaying wave along $+z$ direction}. The total length of the computational domain is $20\lambda$. The metasurface is located at $z_{m}=10\lambda$. A plane wave with the electric field of magnitude 1 V/m is incident on the metasurface from the left.  The simulation results are plotted in Fig.\ \ref{fig_fullAbs1D},
\begin{figure}[!h]
\centering
\includegraphics[width=3.7in]{FullAbs_1D_pdf.pdf}
\caption{1D FEM-GSTC: Fully absorbing metasurface.}
\label{fig_fullAbs1D}
\end{figure}
where it can be seen that the transmitted field to the right of the metasurface is zero, as specified. On the left side of the metasurface, only the incident wave is present, corresponding to unity-magnitude quadrature real and imaginary phasor parts. If a reflected wave were present, one would observe a partly standing-wave pattern with varying field magnitude.

\subsection{2D Examples}
The dimension of the computational domain used for the two 2D FEM-GSTC examples is $26\lambda \times 26\lambda$. The first-order ABC is used on all of the four boundaries of the rectangular computational domain. A finite-sized GSTC surface is located at $z_{m} = 13\lambda$ with a dimension along the $x$-axis of $20\lambda$. A plane wave multiplied by a Gaussian profile (Gaussian variation along the $x$ direction) is incident on the metasurface from below.

The first example considers a generalized refracting metasurface with no reflection \cite{achouri2015general}. The metasurface transforms a normally incident plane wave to a plane wave propagating at $\pi/4$ radians to the metasurface normal. The susceptibilities are synthesized using (\ref{MetasurfaceSynthExample}) and explicitly given, plotted and interpreted in \cite{AchouriComparison}. These monoisotropic susceptibilities essentially correspond to a phase-gradient metasurface with loss (and hence negative imaginary susceptibility). The simulation results are plotted in Figs.\ \ref{fig_GenRefReal} and \ref{fig_GenRefMag}.
\begin{figure}[!h]
\centering
\includegraphics[width=3.6in]{GenRefReal_pdf.pdf}
\caption{Generalized refraction metasurface: Real part of $H_{y}$.}
\label{fig_GenRefReal}
\end{figure}
\begin{figure}[!h]
\centering
\includegraphics[width=3.6in]{GenRefMag_pdf.pdf}
\caption{Generalized refraction metasurface: Magnitude  of $H_{y}$.}
\label{fig_GenRefMag}
\end{figure}
From these figures, the expected refraction at $\pi/4$ radians is clearly observed. In Fig.\ \ref{fig_GenRefReal}, it can be seen that the metasurface does not create any reflections, as specified. Note that the slight standing-wave pattern observed at the top right corner of Fig.\ 6 is due to the reflections from the absorbing boundary of the computational domain, where the first-order ABC was used. If the second-order ABC or a PML is used, these reflections can be reduced. Since the goal of this work is to implement GSTC in FEM, this issue is not further studied in this case. In Fig.\ \ref{fig_GenRefMag}, weak scattering can be seen at the left end of the Gaussian beam. This could be due to the fact that the susceptibilities were synthesized for the case of plane waves on either side of the metasurface, whereas in the simulation, a plane wave modulated by a transverse Gaussian profile is used. Such a wave is not an exact solution to the vector wave equation. The COMSOL simulation results for a generalized refraction metasurface are reported in \cite{YousefFDFD}. In the COMSOL simulations, the metasurface was represented as a thin slab of a subwavelength thickness. The COMSOL simulation results in \cite{YousefFDFD} showed unspecified refracted beams. Similar to the FDTD-GSTC \cite{YousefFDFD}, the FEM-GSTC does not result in these spurious refracted beams.

The second example is of a fully absorbing metasurface, which is simply the bi-dimensional counterpart of Sec. V.A. The simulation results are shown in Figs.\ \ref{fig_FullyAbsReal} and \ref{fig_FullyAbsMag}.As specified, zero transmission and reflection can be verified in these figures. The COMSOL simulations for the same problem are reported in \cite{YousefFDFD}, where the results showed a partial transmission of the incident beam.
\begin{figure}[!h]
\centering
\includegraphics[width=3.6in]{FullyAbsReal_pdf.pdf}
\caption{Fully absorbing metasurface: Real part of $H_{y}$.}
\label{fig_FullyAbsReal}
\end{figure}
\begin{figure}[!h]
\centering
\includegraphics[width=3.6in]{FullyAbsMag_pdf.pdf}
\caption{Fully absorbing metasurface: Magnitude of $H_{y}$.}
\label{fig_FullyAbsMag}
\end{figure}

\section{FEM-GSTC for Bianisotropic Metasurfaces}
The FEM-GSTC described in the previous sections can be extended to model bianisotropic metasurfaces. In this section, we consider bianisotropic metasurfaces where the off-diagonal terms of all the four susceptibility tensors are zero, i.e. $\chi_{\mathrm{ee}}^{xy} = \chi_{\mathrm{ee}}^{yx} = 0$, $\chi_{\mathrm{em}}^{xy} = \chi_{\mathrm{em}}^{yx} = 0$, $\chi_{\mathrm{mm}}^{xy} = \chi_{\mathrm{mm}}^{yx} = 0$, and  $\chi_{\mathrm{me}}^{xy} = \chi_{\mathrm{me}}^{yx} = 0$ in (\ref{GSTCexpandedeqns}). In this case, equation (\ref{GSTCexpandedeqns}) becomes
\begin{subequations}
\label{bianisoeqns}
\begin{equation}
\begin{split}
E_{x}\big|_{z_{m}^{+}} - E_{x}\big|_{z_{m}^{-}} = -\frac{j\omega\mu\chi_{\mathrm{mm}}^{yy}}{2}\left(H_{y}\big|_{z_{m}^{+}} + H_{y}\big|_{z_{m}^{-}}\right) \\
- \frac{j\omega\sqrt{\mu\epsilon}\chi_{\mathrm{me}}^{yy}}{2}\left(E_{y}\big|_{z_{m}^{+}} + E_{y}\big|_{z_{m}^{-}}\right),
\end{split}
\end{equation}
\begin{equation}
\begin{split}
H_{y}\big|_{z_{m}^{+}} - H_{y}\big|_{z_{m}^{-}} = -\frac{j\omega\epsilon\chi_{\mathrm{ee}}^{xx}}{2}\left(E_{x}\big|_{z_{m}^{+}} + E_{x}\big|_{z_{m}^{-}}\right) \\
- \frac{j\omega\sqrt{\mu\epsilon}\chi_{\mathrm{em}}^{xx}}{2}\left(H_{x}\big|_{z_{m}^{+}} + H_{x}\big|_{z_{m}^{-}}\right),
\end{split}
\end{equation}
\begin{equation}
\begin{split}
E_{y}\big|_{z_{m}^{+}} - E_{y}\big|_{z_{m}^{-}} = \frac{j\omega\mu\chi_{\mathrm{mm}}^{xx}}{2}\left(H_{x}\big|_{z_{m}^{+}} + H_{x}\big|_{z_{m}^{-}}\right) \\
+ \frac{j\omega\sqrt{\mu\epsilon}\chi_{\mathrm{me}}^{xx}}{2}\left(E_{x}\big|_{z_{m}^{+}} + E_{x}\big|_{z_{m}^{-}}\right),
\end{split}
\end{equation}
\begin{equation}
\begin{split}
H_{x}\big|_{z_{m}^{+}} - H_{x}\big|_{z_{m}^{-}} = \frac{j\omega\epsilon\chi_{\mathrm{ee}}^{yy}}{2}\left(E_{y}\big|_{z_{m}^{+}} + E_{y}\big|_{z_{m}^{-}}\right) \\
+ \frac{j\omega\sqrt{\mu\epsilon}\chi_{\mathrm{em}}^{yy}}{2}\left(H_{y}\big|_{z_{m}^{+}} + H_{y}\big|_{z_{m}^{-}}\right).
\end{split}
\end{equation}
\end{subequations}
It may be observed from the coupling between field components in these equations that the diagonal elements of the $\overline{\overline{\chi}}_{\mathrm{em}}$ and $\overline{\overline{\chi}}_{\mathrm{me}}$ tensors result in a gyrotropic (chiral) metasurface.

Consider a 1D metasurface problem similar to that in Section III. For a BVP with variations only along the $z$ direction, there are two independent field modes: $\{E_{x},H_{y}\}$ and $\{E_{y},H_{x}\}$. A bianisotropic metasurface induces a  coupling between $\{E_{x},H_{y}\}$ and $\{E_{y},H_{x}\}$ field modes, as seen in (26). Therefore, simulating a bianisotropic metasurface will require a simultaneous processing of the $x$ and $y$ field components. The same principle would apply to a metasurface with off-diagonal components in $\overline{\overline{\chi}}_{\mathrm{ee}}$ and $\overline{\overline{\chi}}_{\mathrm{mm}}$. Thus both $E_{x}$ and $E_{y}$ need to be assigned to the FEM nodes. The domain discretization and node assignment for the case of 8 elements are shown in Figs.\ \ref{fig_1DFEM} and \ref{fig_1DFEM_Ey} for $E_{x}$ and $E_{y}$, respectively.
\begin{figure}[!h]
\centering
\includegraphics[width=3.5in]{FEM1D_BMP_Ey_pdf.pdf}
\caption{1D FEM-GSTC: $E_{y}$ nodes.}
\label{fig_1DFEM_Ey}
\end{figure}
The solution vector is $\left[E_{x1}\ \cdots \ E_{x8}\ E_{y1}\ \cdots \ E_{y8}\right]^{T}$. As in Section III, a system of equations can be written for the $E_{y}$ components as
\begin{subequations}
\label{EyFEMEqn}
\begin{equation}
\begin{bmatrix}
K_{11}^{1} & K_{12}^{1} & 0 & 0 \\
K_{12}^{1} & K_{22}^{1} + K_{11}^{2} & K_{12}^{2} & 0 \\
0 & K_{12}^{2} & K_{22}^{2} + K_{11}^{3} & K_{12}^{3} \\
0 & 0 & K_{12}^{3} & K_{22}^{3}
\end{bmatrix}
\begin{bmatrix}
E_{y1} \\
E_{y2} \\
E_{y3} \\
E_{y4}
\end{bmatrix}
=
\begin{bmatrix}
b_{9} \\
0 \\
0 \\
b_{12}
\end{bmatrix},
\end{equation}
\begin{equation}
\begin{bmatrix}
K_{11}^{4} & K_{12}^{4} & 0 & 0 \\
K_{12}^{4} & K_{22}^{4} + K_{11}^{5} & K_{12}^{5} & 0 \\
0 & K_{12}^{5} & K_{22}^{5} + K_{11}^{6} & K_{12}^{6} \\
0 & 0 & K_{12}^{6} & K_{22}^{6}
\end{bmatrix}
\begin{bmatrix}
E_{y5} \\
E_{y6} \\
E_{y7} \\
E_{y8}
\end{bmatrix}
=
\begin{bmatrix}
b_{13} \\
0 \\
0 \\
b_{16}
\end{bmatrix}
.
\end{equation}
\end{subequations}
The stiffness matrix elements are given by equation (\ref{Keijeqn}).
The elements of the right-hand-side vectors in (\ref{EyFEMEqn}) are given by
\begin{subequations}
\begin{align}
b_{9} &= -\frac{1}{\mu_{r}}\frac{dE_{y}}{dz}\bigg|_{z=0}, \\
b_{12} &= \frac{1}{\mu_{r}}\frac{dE_{y}}{dz}\bigg|_{z=z_{m}^{-}}, \\
b_{13} &= -\frac{1}{\mu_{r}}\frac{dE_{y}}{dz}\bigg|_{z=z_{m}^{+}}, \\
b_{16} &= \frac{1}{\mu_{r}}\frac{dE_{y}}{dz}\bigg|_{z=L}.
\end{align}
\end{subequations}
Similar to $b_{1}$ and $b_{8}$, the elements $b_{9}$ and $b_{16}$ can be evaluated with the first-order ABC. The other elements $b_{4},b_{5}, b_{12}$, and $b_{13}$ can be evaluated by using Maxwell's equations and GSTCs in (\ref{bianisoeqns}). From the Maxwell-Faraday equation, $b_{4},b_{5}$, $b_{12}$, and $b_{13}$ can be converted to expressions in terms of $H_{x}$ and $H_{y}$ on either side of the metasurface. This is followed by using (\ref{bianisoeqns}) to express $H_{y}|_{z_{m}^{-}}$, $H_{y}|_{z_{m}^{+}}$, $H_{x}|_{z_{m}^{-}}$, and $H_{x}|_{z_{m}^{+}}$ in terms of $E_{y}|_{z_{m}^{-}}$, $E_{y}|_{z_{m}^{+}}$, $E_{x}|_{z_{m}^{-}}$, and $E_{x}|_{z_{m}^{+}}$. The final expressions for $b_{4},b_{5},b_{12}$, and $b_{13}$ are
\begin{equation}
\label{AmatrixEqn}
\begin{split}
\begin{bmatrix}
b_{4} \\
b_{5} \\
b_{12} \\
b_{13}
\end{bmatrix}
&=
\begin{bmatrix}
-j\omega \mu_{o} H_{y}\big|_{z_{m}^{-}} \\
j\omega \mu_{o} H_{y}\big|_{z_{m}^{+}} \\
j\omega \mu_{o} H_{x}\big|_{z_{m}^{-}} \\
-j\omega \mu_{o} H_{x}\big|_{z_{m}^{+}}
\end{bmatrix}
\\
&=
\begin{bmatrix}
A_{1} & A_{2} & A_{3} & A_{4} \\
A_{2} & A_{1} & -A_{4} & -A_{3} \\
A_{5} & A_{6} & A_{7} & A_{8} \\
-A_{6} & -A_{5} & A_{8} & A_{7}
\end{bmatrix}
\begin{bmatrix}
E_{x}\big|_{z_{m}^{+}} \\
E_{x}\big|_{z_{m}^{-}} \\
E_{y}\big|_{z_{m}^{+}} \\
E_{y}\big|_{z_{m}^{-}}
\end{bmatrix}
,
\end{split}
\end{equation}
where
\begin{subequations}
\begin{equation}
A_{1} = \frac{k^{2}\chi_{\mathrm{mm}}^{yy}\left(\chi_{\mathrm{ee}}^{xx}\chi_{\mathrm{mm}}^{xx} - \chi_{\mathrm{em}}^{xx}\chi_{\mathrm{me}}^{xx}\right) + 4\chi_{\mathrm{mm}}^{xx}}{4\mu_{r}\chi_{\mathrm{mm}}^{xx}\chi_{\mathrm{mm}}^{yy}},
\end{equation}
\begin{equation}
A_{2} = \frac{k^{2}\chi_{\mathrm{mm}}^{yy}\left(\chi_{\mathrm{ee}}^{xx}\chi_{\mathrm{mm}}^{xx} - \chi_{\mathrm{em}}^{xx}\chi_{\mathrm{me}}^{xx}\right) - 4\chi_{\mathrm{mm}}^{xx}}{4\mu_{r}\chi_{\mathrm{mm}}^{xx}\chi_{\mathrm{mm}}^{yy}},
\end{equation}
\begin{equation}
A_{3} = -\frac{jk_{o}}{2}\sqrt{\frac{\epsilon_{r}}{\mu_{r}}}\left(\frac{\chi_{\mathrm{em}}^{xx}\chi_{\mathrm{mm}}^{yy} - \chi_{\mathrm{mm}}^{xx}\chi_{\mathrm{me}}^{yy}}{\chi_{\mathrm{mm}}^{xx}\chi_{\mathrm{mm}}^{yy}}\right),
\end{equation}
\begin{equation}
A_{4} = \frac{jk_{o}}{2}\sqrt{\frac{\epsilon_{r}}{\mu_{r}}}\left(\frac{\chi_{\mathrm{em}}^{xx}\chi_{\mathrm{mm}}^{yy} + \chi_{\mathrm{mm}}^{xx}\chi_{\mathrm{me}}^{yy}}{\chi_{\mathrm{mm}}^{xx}\chi_{\mathrm{mm}}^{yy}}\right),
\end{equation}
\begin{equation}
A_{5} = \frac{jk_{o}}{2}\sqrt{\frac{\epsilon_{r}}{\mu_{r}}}\left(\frac{\chi_{\mathrm{mm}}^{xx}\chi_{\mathrm{em}}^{yy} - \chi_{\mathrm{me}}^{xx}\chi_{\mathrm{mm}}^{yy}}{\chi_{\mathrm{mm}}^{xx}\chi_{\mathrm{mm}}^{yy}}\right),
\end{equation}
\begin{equation}
A_{6} = -\frac{jk_{o}}{2}\sqrt{\frac{\epsilon_{r}}{\mu_{r}}}\left(\frac{\chi_{\mathrm{mm}}^{xx}\chi_{\mathrm{em}}^{yy} + \chi_{\mathrm{me}}^{xx}\chi_{\mathrm{mm}}^{yy}}{\chi_{\mathrm{mm}}^{xx}\chi_{\mathrm{mm}}^{yy}}\right),
\end{equation}
\begin{equation}
A_{7} = -\frac{k^{2}\chi_{\mathrm{mm}}^{xx}\left(\chi_{\mathrm{em}}^{yy}\chi_{\mathrm{me}}^{yy} - \chi_{\mathrm{ee}}^{yy}\chi_{\mathrm{mm}}^{yy}\right) - 4\chi_{\mathrm{mm}}^{yy}}{4\mu_{r}\chi_{\mathrm{mm}}^{xx}\chi_{\mathrm{mm}}^{yy}},
\end{equation}
\begin{equation}
A_{8} = -\frac{k^{2}\chi_{\mathrm{mm}}^{xx}\left(\chi_{\mathrm{em}}^{yy}\chi_{\mathrm{me}}^{yy} - \chi_{\mathrm{ee}}^{yy}\chi_{\mathrm{mm}}^{yy}\right) + 4\chi_{\mathrm{mm}}^{yy}}{4\mu_{r}\chi_{\mathrm{mm}}^{xx}\chi_{\mathrm{mm}}^{yy}}.
\end{equation}
\end{subequations}

The FEM-GSTC for bianisotropic metasurfaces is validated by a simple example. As seen in (\ref{bianisoeqns}), such a metasurface is characterized by eight free tensorial susceptibility elements for four equations. This corresponds to an underdetermined problem, allowing for the possibility for the metasurface to perform a double transformation, as noted in [21]. We shall consider here a birefringent metasurface \cite{AchouriComparison} that transforms 1) an $x$-polarized incident wave into a $y$-polarized transmitted wave and 2) a $y$-polarized incident wave into a fully absorbed wave, with zero reflection in both cases. In terms of fields, the first transformation reads
\begin{subequations}
\label{wavetr1}
\begin{equation}
E_{x}^{(1)}\big|_{z_{m}^{-}} = e^{-jk_{o}z_{m}} \ , \ H_{y}^{(1)}\big|_{z_{m}^{-}} = \frac{e^{-jk_{o}z_{m}}}{\eta_{o}},
\end{equation}
\begin{equation}
E_{y}^{(1)}\big|_{z_{m}^{-}} = 0 \ , \ H_{x}^{(1)}\big|_{z_{m}^{-}} = 0,
\end{equation}
\begin{equation}
E_{x}^{(1)}\big|_{z_{m}^{+}} = 0 \ , \ H_{y}^{(1)}\big|_{z_{m}^{+}} = 0,
\end{equation}
\begin{equation}
E_{y}^{(1)}\big|_{z_{m}^{+}} = e^{-jk_{o}z_{m}} \ , \ H_{x}^{(1)}\big|_{z_{m}^{+}} = -\frac{e^{-jk_{o}z_{m}}}{\eta_{o}},
\end{equation}
\end{subequations}
and the second transformation reads
\begin{subequations}
\label{wavetr2}
\begin{equation}
E_{x}^{(2)}\big|_{z_{m}^{-}} = 0 \ , \ H_{y}^{(2)}\big|_{z_{m}^{-}} = 0,
\end{equation}
\begin{equation}
E_{y}^{(2)}\big|_{z_{m}^{-}} = e^{-jk_{o}z_{m}} \ , \ H_{x}^{(2)}\big|_{z_{m}^{-}} = -\frac{e^{-jk_{o}z_{m}}}{\eta_{o}},
\end{equation}
\begin{equation}
E_{x}^{(2)}\big|_{z_{m}^{+}} = 0 \ , \ H_{y}^{(2)}\big|_{z_{m}^{+}} = 0,
\end{equation}
\begin{equation}
E_{y}^{(2)}\big|_{z_{m}^{+}} = 0 \ , \ H_{x}^{(2)}\big|_{z_{m}^{+}} = 0.
\end{equation}
\end{subequations}
By substituting (\ref{wavetr1}) and (\ref{wavetr2}) in (\ref{bianisoeqns}), one obtains a linear system of eight equations for the eight unknown susceptibilities, which can be solved to yield
\begin{subequations}
\begin{equation}
\chi_{\mathrm{ee}}^{xx} = \chi_{\mathrm{ee}}^{yy} = -\frac{2j}{k_{o}},
\end{equation}
\begin{equation}
\chi_{\mathrm{mm}}^{xx} = \chi_{\mathrm{mm}}^{yy} = -\frac{2j}{k_{o}},
\end{equation}
\begin{equation}
\chi_{\mathrm{em}}^{xx} = 0 \ , \  \chi_{\mathrm{em}}^{yy} = \frac{4j}{k_{o}},
\end{equation}
\begin{equation}
\chi_{\mathrm{me}}^{xx} = -\frac{4j}{k_{o}} \ , \  \chi_{\mathrm{me}}^{yy} = 0.
\end{equation}
\end{subequations}
The metasurface is illuminated with two plane waves, $E_{x}^{\mathrm{inc}} = e^{-jk_{o}z}$ and $E_{y}^{\mathrm{inc}} = e^{-jk_{o}z}$. The simulation parameters are same as those in Section V. The metasurface is located at $z_{m}=10\lambda$. The real parts of $E_{x}$ and $E_{y}$ are shown in Fig.\ \ref{fig_bianiso},
\begin{figure}[!h]
\centering
\includegraphics[width=3.5in]{Bianiso_pdf.pdf}
\caption{FEM-GSTC simulation results for a bianisotropic metasurface.}
\label{fig_bianiso}
\end{figure}
where it can be observed that on the right side of the metasurface, there is only an $E_{y}$ component. This is due to the fact that the $x$ polarized wave incident on the metasurface from its left side is transformed to a $y$ polarized wave and the $y$ polarized wave incident on the metasurface from its left side is completely absorbed. It can also be seen that for $z < 10\lambda$, there are no reflections.

\section{Conclusion}
In this paper, we presented the FEM modeling of metasurfaces based on GSTCs, where the discontinuities in electromagnetic fields across a metasurface were modeled by assigning nodes to both sides of the metasurface. We derived the FEM-GSTC formulation in both 1D and 2D domains and extended it to handle more general bianistroptic metasurfaces. We also presented several illustrative examples to validate the FEM-GSTC formulation. Future work includes extension of the method to 3D problems and curved metasurfaces. In 3D problems, the susceptibility tensor elements will be functions of both $x$ and $y$. The extension of 2D FEM-GSTC to 3D is straightforward. In 2D, the line integrals along the element edges were used to calculate the right-hand-side of the FEM system of equations. In 3D, surface integrals on the tetrahedral faces should be used to obtain the right-hand-side of the FEM system of equations. Even though this paper assumed zero off-diagonal elements in the susceptibility tensors, the formulation can be easily extended to non-zero off-diagonal elements. In such a case, the matrix elements in (\ref{AmatrixEqn}) will be more involved. It should be noted that in this paper, the GSTC surface is handled by placing field nodes on either side of the GSTC surface. A similar procedure is used in the FDFD formulation in \cite{YousefFDFD}. However, the advantage of FEM-GSTC is its flexibility in placing these nodes conforming to the metasurface geometry. The rectangular Yee cells of FDFD limits this flexibility. Hence FEM-GSTC is more efficient for simulating arbitrarily shaped metasurfaces or in general arbitrarily shaped spatial electromagnetic field discontinuity.

\ifCLASSOPTIONcaptionsoff
  \newpage
\fi

\bibliographystyle{IEEEtran}

redpajama_set_name: RedPajamaArXiv


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Welcome to the June issue of Touch Base, your official alumni newsletter that helps you stay in touch with old classmates, faculty, and staff and stay abreast of developments at your alma mater, The American College of Greece.
This month, we look forward to the longest daylight hours of the year in the northern hemisphere – casting everything in a favorable light – and also look forward to welcoming and congratulating our new ACG alumni after Commencement.
We also take a look at what took place at the graduation anniversaries of six classes: PIERCE 1944 – 1954 – 1964 (May 29th) as well as PIERCE 1974 – 1984 -1994 (May 16th).
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The implementation of services directed to the person and the family aimed at improving the quality of life of every citizen/user;
The activation of processes of socialization for the person to stimulate paths of social solidarity, acceptance of diversity, non-discrimination;
The development of interventions aimed at preventing disadvantage among individual citizens or at-risk groups;
San Saturnino has an ISO 9001 certified organization for quality. In managing its sundry services, the Cooperative makes use of a qualified organizational system and diversified human and professional resources: social workers, psychologists, pedagogues, educators, home health care-givers, childcare workers etc. SANSAT currently employees 130 fulltime and part time staff.
It is organized in four areas:
Elderly Area: San Saturnino manages an important in-home assistance services and a daily centers for older people.
Social Inclusion Area: The services of this area arise from the Cooperative's long history of work and involvement in the sector of social marginalization. In particular, the services of social emergency have allowed us over the years to come to know and analyze the phenomena linked to numerous and heterogeneous expressions of distress in society, observing the difficulties, abandonment and chronic condition of people left to face the most complex aspects of social life on their own.
Children and Teen-agers Area: This Area defined itself as "Services for childhood and adolescence" after various years' work aimed at contributing to the construction of a shared, scientific and up-to- date culture of childhood and adolescence.
Innovative Projects Sector: This Sector was born from the cooperative's propensity towards the logic of Lifelong Learning, which guides its choices and development with the aims of
guaranteeing the permanent interdisciplinary training and supervision of its own members and workers.
fostering the know-how of all the services and workers of the Cooperative by means of developing and managing pilot projects and research, the systemization of innovative models of intervention, all the while guaranteeing their broad diffusion and applicability.
It is the sector in which the European projects and transnational cooperation are developed. The Cooperative's commitment in the European projects and transnational cooperation stems from the cooperative's propensity to constantly develop the know-how of all its services, also through the dialogue with partners from different contexts. This allows the systematization and dissemination of innovative models of intervention, strengthening a multidisciplinary approach that has always been an operational feature of the cooperative through:
the realization of researches and studies able to deepen the analysis of the social demand of the territories aimed at the correct interpretation of needs and the planning of targeted interventions;
the development of operational interventions based on the logic of multi-agency networks at national and transnational levels.
SANSAT works daily to battle discrimination and promote the empowerment of vulnerable groups. SANSAT is an important local reference point in the field because of their wide experience in the development and management of training and educational interventions, their involvement in employee updating and upgrading activities, and their specialism in developing experimental models. SANSAT has developed specific tools for detection, evaluation and enhancement of competences and for the empowering work. SANSAT has extensive experience in planning and management of focus groups and learning paths.
Kilcooley Women's Centre HQ & Rural Development Office
The Grace Centre
Unit 65 North Down Development Organisation
2-4 Balloo Avenue
Bangor BT19 7QT
"The European Commission's support for the production of this publication does not constitute an endorsement of the contents, which reflects the views only of the authors, and the Commission cannot be held responsible for any use which may be made of the information contained therein."
• Kilcooley Women Center, Northern Ireland
• ACDC, Romania
• GEMEENTE 'S-HERTOGENBOSCH, Netherlands
• Tirantes, Netherlands
• Fundacion ESPLAI, Spain
• Cooperativa Sociale San Saturnino Onlus, Italy
• Federación de Asociaciones Gitanas de Cataluña, Spain
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With Hurricane Irene in Jersey, I decided to share one of my favorite indoor workouts, the deck of cards. Sometimes you can't get to a gym, a mat, and you don't have a lot of space. This workout can be done pretty much in a closet and you don't need anything but a deck of cards. A hurricane isn't the only reason why you might have to workout in your home, apartment, or condo…snow storms forced me many times to workout in my apartment. Don't let inclement weather or anything else stop you from getting to your goals, and to reach your goals you have to be prepared for anything. I could have said, "Oh well, it's a state of an emergency. I guess I don't have to workout today." But I didn't and that kind of thing separates you from your competition.
I have to thank Mike Sparacino, who introduced me to this workout during our time at Hofstra, sometime in 2004. He is the same friend that I mentioned in Wrestling with Your Destiny, who introduced me to the rotating grip ball to help with my grip strength. What the workout entails is taking a deck of everyday playing cards and flipping them over and doing the number of push-ups on the card. So if you get a five you do five push-ups, if it's a ten you do ten push-ups and so on. Jacks are 11, Queens-12, Kings-13, and Aces-14. This computes to 420 push-ups. Pretty simple. If you want to make it more challenging, throw the two jokers in and make them 15, or even 25 each.
I modified this original deck of cards workout. I don't like to waste time and doing two push-ups four times in every deck made me feel like I was wasting time. What I ended up doing was any time I got a 2-9 card I flipped over the next card and added them together and then performed that number of push-ups. It moves the workout along and is harder to do. If it is your first time attempting this deck of cards workout, I recommend not doing that. You will be sore enough the next day.
Now as all wrestlers know, we need a lot more developed muscles than just our chest and triceps to be successful. So I added abs. Whatever card I got, I did my push-ups first and then I doubled the number and then did crunches. For example, if I got a queen, I would do 12 push-ups then 24 crunches, or other ab exercise, which I varied throughout.
After a few times of doing the deck of cards 2, I decided to add pull-ups. Now for this addition you do need a pull-up bar. I used a simple one you can pick up for twenty bucks and hang in your doorway. After I did my push-ups and doubled my abs, I halved the original card number and performed that many pull-ups. For example, if an ace came up I did 14 push-ups, 28 abs, and 7 pull-ups before I flipped the next card. If the card you flipped over is odd like a 7, round-up and do 4 pull-ups.
Another workout you can do inside with little space is jumping rope. I have spent many hours indoors with my jump rope to improve my foot speed and coordination.
The next post will be my other go-to workout out when I was stuck inside.
A word of warning: Build up to the ultimate deck of cards workout.
How lOng do you rest betwee each playing card? So after drawing a 10 and doing them, how long before the next card?
As little rest as possible, especially if you're adding in crunches and pull ups. Work up to that though.
Ok so maybe 15 secs first time? And then try and bring it down.
Yeah try that. Let me know how it goes. Good luck.
That works as well. I like my way of doing it but the way you were taught is certainly not wrong, as long as you do it!
This looks awesome! I've got an old pack laying around at the house, this will def. come in handy to get me up and moving while I'm off work–I bet you can fit several mini workouts in during commercials.
That's great. I hope you like the workouts.
I've seen this work out done where each suit is a different exercise, so for example: hearts are push ups, diamonds are jumping jacks, spades are flutter kicks and clubs are sit ups. If you have a hard time doing one exercise 420 times or so, this may help you build up as well.
Also, for my younger or less in shape kids we would do half the number on the card and round up for the odd numbers. It definitely helped some kids that could only do 5 or 6 push ups before they got too tired. As you get stronger go to the whole card value. Hope this helps!
Great alternatives Laura. Thank you for sharing.
I'm 14 how many weeks should I keep this up to get a little more muscle gain?
In times like these I fall back on a wonderful piece of penmanship by Maya:"You may write me down in history with your bitter twisted lies. You may tread me in the very dirt, but still, like dust, I'll rise."


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San Francisco Giants pitcher Madison Bumgarner delivers to home against the Oakland Athletics at AT&T Park on Friday, July 13, 2018. (Kevin N. Hume/S.F. Examiner)
Giants waste stellar effort from Madison Bumgarner in shutout loss to Padres on Opening Day
Wil Myers provides the offense as the Padres blank the Giants and Madison Bumgarner 2-0
Ethan Kassel
With the signing of Manny Machado, promotion of hotshot prospect Fernando Tatis Jr. and more buzz surrounding the franchise than there had been in at least a decade, the San Diego Padres were eager to show the San Francisco Giants and the rest of the NL West what they were all about.
Sure enough, the sellout throng at Petco Park — the fastest Opening Day sellout in the stadium's history — was treated to see a 2-0 victory, rather than see their park become a southern outpost of the Giants' fanbase.
Padres starter Eric Lauer, taking the mound with all the swagger in the world as Eazy-E blared over the speakers, held San Francisco (0-1) to just four hits over six shutout innings, and Wil Myers provided all the offense the hosts needed.
Machado went 0-for-3 with two strikeouts and a double play as he was victimized by Madison Bumgarner, who struck out nine over seven innings, but the Giants' ace was outdone by Lauer and a quartet of Padres relievers.
Bumgarner was one of the lone bright spots for the Giants, who mustered just five hits — all singles. One of those came from the starting pitcher, and Evan Longoria had a pair of base hits, but the visitors would have just one at-bat with runners in scoring position all day. That came in the seventh with Yangervis Solarte pinch-hitting, and Craig Stammen got him to ground out harmlessly.
Stammen was one of three relievers to take the mound in the seventh inning for San Diego (1-0) after Lauer scattered just four hits and a walk over the first six frames. Trey Wingenter allowed a single to Buster Posey but retired Brandon Crawford on a force out. Aaron Loup got pinch-hitter Gerardo Parra to ground out and Stammen not only took care of Solarte, but set the guests down in order in the eighth as well. Kirby Yates pitched a perfect ninth with a pair of strikeouts for the save, striking out Posey to end the game.
Myers took Bumgarner deep to right-center on a colossal 456-foot blast in the bottom of the third, then singled in the Padres' second run in the bottom of the sixth after Ian Kinsler led off the frame with a double.
Tatis Jr. singled to left off of Bumgarner for his first career hit in the second, and reached again on a bunt single in the fourth. The only comparable feat for the Giants was Connor Joe's first career walk, as the utility man — acquired less than a week ago from the Cincinnati Reds — drew a free pass from Lauer in the second.
Joe was replaced by Parra in the seventh and Michael Reed, who batted after him and went 0-for-2, was pinch-hit for by Solarte. Those two spots in the lineup were handled by newcomers for the Giants, as was the bullpen. The one inning that Bumgarner didn't handle was taken care of by Nick Vincent, a former Padre, who allowed a single to Austin Hedges but got Kinsler to bounce into a double play.
Mike Fiers bounces back in A's win, as Khris Davis launches another home run for Anthony Slocumb
SF Preps Baseball: Sacred Heart Cathedral comes out on top in Joe Cronin Game against Mission


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USA today Editoweb, 12 feb. 2008
Astronaut John Glenn endorses Clinton - Huckabee questions McCain's ability to 'energize' base - FEMA to use trailers after tornadoes - Sugar refinery fire resists air drops - Partially mummified body found in AZ tub.
Astronaut John Glenn endorses Clinton
Democrat Hillary Rodham Clinton's effort to win Ohio's critical presidential primary got a boost Tuesday when former senator and astronaut John Glenn endorsed her candidacy.
Clinton is counting on a win on March 4 in primaries in Ohio and Texas — where a combined 334 Democratic delegates are at stake — to reinvigorate her candidacy. She lost five nomination contests over the weekend to rival Barack Obama, who was also favored to win primaries in Virginia, Maryland and the District of Columbia on Tuesday.
Huckabee questions McCain's ability to 'energize' base
Former Arkansas Gov. Mike Huckabee, showing no signs of giving up in his quest for the GOP nomination, on Tuesday questioned presumptive nominee Sen. John McCain's (Ariz.) ability to turn out conservatives in a general election.
Huckabee, who has repeatedly bested McCain in conservative, Southern states, said he is the candidate winning the "states that are essential to being elected as a Republican."
FEMA to use trailers after tornadoes
Some of the thousands of trailers purchased by the Federal Emergency Management Agency in 2005 after hurricanes tore through the Gulf Coast may finally be put to use to help victims of last week's tornadoes, officials said Tuesday.
The 7,200 trailers stored at the Hope airport will "definitely" be used in Arkansas and Tennessee as housing for victims of last week's tornadoes, officials said.
Sugar refinery fire resists air drops
Frustrated fire officials are turning to a private firefighting company to help fight the stubborn, high-temperature blaze at a sugar refinery that continues despite the dropping of thousands of gallons of water by a helicopter.
The search for two workers missing since an explosion Thursday at the Imperial Sugar Co. will resume once the fire is out and debris from one section of the plant is removed, authorities said. Six workers are confirmed dead.
Partially mummified body found in AZ tub
A partially mummified body was found in a bathtub filled with dirt in an apartment that was stacked to the ceiling with garbage and human waste.
Owners of the small, standalone unit thought it had been abandoned in August, when rent stopped being paid, Phoenix police Sgt. Joel Tranter said Monday.
J. B. / Source Web


In our last SEEDS FOR GROWTH, we began discussing Conflict…that crippler that terrifies and tears us apart. You know what we're talking about, don't you? You probably have some old, unworkable habits you just naturally fall into when anger and conflict arise in your marriage and family relationships. Right? Yeah, we all do. We walk away (either physically or emotionally), we blow up, we blame….It's a long list. But you know the ones you usually use.
We have some good news for you! The way you handle conflict is a learned response. Now, the good news about that is anything learned can be relearned. You are not trapped in your Conflict Method. You can change. God has said that all along. In fact, He commands it. Romans 12:2 — Do not conform any longer to the pattern of this world, but be transformed by the renewing of your mind.
Remember, there are three basic reasons for our conflicts: (1) because of the way we look at our differences, (2) because of our power struggles and (3) because of our faulty communication patterns. These are the things we want to be transformed FROM. Last time, we talked about our DIFFERENCES. This time, let's examine POWER STRUGGLES.
What is a POWER STRUGGLE? The best way to answer this is probably to give you a few every-day examples.
Now, do you think Barb and George are really debating facts? No. They are in a struggle for POWER. They are fighting for CONTROL.
Are Mary and Sam really fighting over THAT shirt? They may think they are, but this fight isn't escalating over a shirt, is it? It's about CONTROL.
That's what POWER STRUGGLES are all about! Power in a marriage relationship is the ability of one partner to influence or change the behavior of the other. If you will notice in the two examples above, power struggles have two sides as inseparable as the two sides of a coin. The desire to control and the fear of being controlled. But please notice one other thing about these examples: neither side wins. The funny thing about power struggles is that the more you struggle for control, the less control you have. In other words, the harder you try, the worse it gets. And the saddest thing about most power struggles is that they are battles that don't even NEED to be fought. Why fight a battle no one can possibly win?
Perhaps you wish your mate would do some things differently, or maybe BE different in some way? Say, you wish your husband or wife would talk more with you. Now, there is nothing wrong with such wishes. But be careful. They can turn into power struggles very easily. Maybe this scenario will sound familiar.
And so it continues in a vicious circle: Joan keeps nagging and complaining, and Ken keeps resisting. They each want to change the other. But somehow, they just aren't seeing what they are doing IS NOT WORKING.
Poor Jeff and Sarah — the harder they try to change the other, the worse it gets.
So…what do you do about power struggles? How can you short circuit them?
The first step sounds so easy, you'll wonder why you didn't think of it yourself. STOP doing what you are doing! Stop playing your part of the Power Struggles game. If what you have been doing is not working, then DO SOMETHING ELSE.
Have you ever seen two puppies playing with a rag, each pulling with all his might on opposite ends? Finally, they lie exhausted, neither having won the rag. What if one of them decided to just let go of his end? Remember what we said at the beginning? The more you struggle for control, the less control you'll have. That goes for puppies…and for people.
does NOT mean your partner's way is WRONG. In every example we have given in this article, each mate had a very real need. Granted, it was pushed upon the other, but it was still O.K. to have that need. Ask the Lord to open your ears to hear your partner's need underneath what he or she is saying.
Loosen your hold on your own point of view. Your partner MAY be better off if he or she would change and be like you want him or her to be. But what's your real motive here? Is it to make a better person of your mate? Or is it to get what you want? Are you fighting battles that don't need to be fought?
Ask God to help you Devise a plan of what YOU are going to do DIFFERENTLY (not what you want your mate to do differently) and commit it to the Lord. Ask him to help you be consistent and persistent in it.
What are some plans that would work? Let's go back to the examples we gave one by one.
(1) Remember Barb and George and their public struggle over details? What if Barb had turned to George after he first interrupted her with "No, Dear, it was 2 and 1/2 years ago," and replied, "Maybe you're right, Honey." She would have taken away his reason to fight for control, without losing anything herself. The power struggle could have been avoided. Proverbs 17:14 wisely advises, Starting a quarrel is like breaching a dam; so drop the matter before a dispute breaks out.
(2) Our next example was Mary and Sam and the infamous shirt. What if Sam, instead of jumping to his own defense, stopped digging in his heels and responded to Mary's initial barrage with, "What is it about this shirt you don't like, Dear?" That might have given Mary a chance to be honest about how she is afraid people will feel sorry for him because of his old clothes. After that, neither of them might feel the need to struggle for control.
(3) How about Joan and her silent husband Ken? Apparently Joan would be happy if he said something…ANYTHING. But what if Ken took it a step further and honestly admitted why he is so uncommunicative. He could simply say, "Yeah, I guess I don't talk much, do I? I'm really tired when I get home from work. I just haven't considered how you've been here alone with the kids all day and need a grownup to talk to." Joan would probably faint. At least she might not feel the need to nag anymore.
(4) Finally, let's look at Jeff and Sarah. It's pretty obvious Sarah is feeling resentful that she has to make supper alone after working outside the home all day. She wants Jeff to show he appreciates her efforts by helping out. And Jeff wants assurance he is still important to his wife. Can they both get what they want? Yes! Sarah could take the first step by saying something like, "Out of all the things your mother does for your father, what's the one thing you would like most for me to do?" (Maybe it's a neck rub after dinner.) Then she could add, "If you will help me with dinner, I will have a little more time and energy, and it would be my pleasure to give you a neck rub after dinner." That would be Jeff's signal to respond with a hearty "All right!" and head straight for the kitchen.
In each of our original examples, both partners were working very hard — emotionally and physically — to intensify a problem they really wanted to get rid of. Why work so hard to get something you DON'T WANT? That doesn't make any sense, does it? But our pride works against us this way. James, the brother of Jesus, hits the "pride nail" on the head when he declares, What causes fights and quarrels among you? Don't they come from your desires that battle within you? You want something but don't get it….Submit yourselves, then, to God. Resist the devil, and he will flee from you….Humble yourselves before the Lord, and he will lift you up (James 4:1-10).
Yes, if the Holy Spirit lives within you, you really can do what seems impossible. You can lay aside your pride and direct your energy into relearning and practicing these new ways of handling power struggles. And finally, you can start winning!
Some photos used with permission under a Creative Commons license.
© 2014 Growing Towards Oneness | Some photos used with permission under a Creative Commons license.


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Gus G – Fearless (AFM Records)
Friday, 20th April 2018 By Matt Coe
Third albums for any artist can define direction to the good (or bad). Axe maven Gus G already has established a foothold on the scene through his main work in melodic power metal act Firewind – but he would be the first to admit that his solo output has been very scattered in terms of song to song recognition for style. Fearless solidifies a specific outlook – probably due to the established talent on the musician front as well as through the songwriting and production consistency.
Instead of peppering the record with various vocalists or rhythm section players, these ten songs stick to a power trio format – vocalist/bassist Dennis Ward already familiar to most for his production work or playing in Pink Cream 69, and drummer Will Hunt a known entity through his percussion expertise with Evanescence. Because of this, the material takes on a balance between his already well-known power riffing and double bass-oriented approach for "Letting Go" or "Don't Tread on Me", and the modern hard rock catchiness that appears on "Chances" or "Mr. Manson" (the latter having a bit of that Ozzy-swing possibly as a nod to Gus' time with Mr. Osbourne). Dennis' bluesy register has all the right AOR-ish touches when called for, showing tender restraint during the verses for the semi-power ballad "Nothing to Say", while being a little larger than life during the Ratt-like "Way Cool Jr." aspects for "Big City".
Gus G as a guitarist also features three instrumentals into the mix of vocal-oriented songs – the title cut throwing around different progressive arpeggio runs while still developing a very catchy mainframe circular riff and "Thrill of The Chase" keeping things more in an 80's Dokken lineage. The bolder risk of tackling an 80's iconic song like "Money for Nothing" from Dire Straits and reinterpreting it through "Danger Zone"-like keyboard work, and slower, heavier main riffs actually does the track justice, because they kept the essence of the vocal melodies to transpose the song to Gus G.'s melodic metal/hard rock expectations. The thoughtful idea of composing these tracks into a tight, focused trio makes Fearless the strongest Gus G solo effort to date – one that will be easily appealing to all melodic metal fans.
Gus G on Facebook
Gus G. – Quantum Leap (AFM Records)
Gus G. – Last of His Kind
Gus G. – Brand New Revolution (Century Media)
Gus G. Releases "Long Way Down" Video
GUS G. Announces West Coast Mini-Tour; "Blame It On Me" Video Posted
Guitarist GUS G.: "Eyes Wide Open" Video Released


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Key Predictions For Executive Job Search In 2022
Part of the series "Today's True Leadership" by Kathy Caprino
Organizations look for strong candidates who embody traits that help work cultures flourish
By now, most of us have heard the term "The Great Resignation" and many have taken the opportunity and joined this movement, to quit their jobs and pursue new roles or careers that are better for them and more. tailored to their needs, values and desired lifestyle. While lower-level employees and middle managers have left their jobs in greater numbers in this tight labor market, this had not necessarily been the case for CEOs until 2021. But according to a recent Heidrick & Struggles study, In the first half of 2021, there was evidence of an increase in CEO turnover.
In my own career and leadership coaching work with executives around the world, I have observed that as time passed throughout this pandemic, just as we experience it through other times. crisis, people begin to feel the urgency to rebuild their lives. and careers, and don't waste a single extra minute in roles or organizations that no longer fit.
To learn more about some key predictions and trends for executives looking to embark on a new job search, I recently caught up with Dave Opton, founder of ExecuNet, a network of private executives he started in 1988. Today, ExecuNet is an executive career service. organization where C-Suite executives and their direct reports can get trusted advice and personal support to change jobs within the company, build a portfolio of board seats, start a side business, or do business. career in other rewarding new directions. Opton was President of ExecuNet until his retirement in December 2021.
Here's what Opton shares about executive job search trends for 2022:
Kathy Caprino: What have you seen to have changed significantly in the executive labor market since the pandemic?
Dave Opton: Whether executives chose to leave or stay, the pandemic has provided high-level professionals with the opportunity to assess their long-term career goals and priorities. At the executive level, it is not so much the story of the Great Resignation as the Great reassessment. Executives assess their professional lives and reflect on their options for the next big chapter in their careers.
They want to be enthusiastic about the job again. Many executives seem willing to change jobs and locations, or even explore different career paths, if that means finding a job that fulfills them. They are reinventing what their professional life could be.
Caprino: Why do you think these specific changes have occurred in a more significant way during this time of pandemic?
Option: Unexpected events or shocks disrupt our normal routines, force us out of our comfort zones, and lead us to ask ourselves big questions about what matters in life and what is worth doing. It's no wonder, then, that the pandemic has prompted many people to rethink their careers. Many re-examine what work means to them, if and how they are valued, and their quality of life at work. Among other things, they are looking for more freedom, more flexibility and more meaning and happiness.
Caprino: What are your predictions for continuing trends and changes in 2022?
Option: The talent pool will continue to be exceptional and competitive. Surveys of a wide range of executive recruiters show us that demand is strong and will continue to be strong.
The executive recruitment process is settling into a new routine. Video interviews have made it easier to match employers with potential hires, and the new normal will be the hybrid scenario.
However, what businesses are looking for has changed and probably for a long time to come. Companies will prioritize different values in their new leaders. They will continue to seek out what we at ExecuNet call "utility actors" to lead them.: people who can play in several positions because companies do not know where they are going to place the best players in the field.
The pandemic has created all kinds of disruption, from technology to corporate culture and a greater focus on DEI. The next organization that hires you will want to know: do you have the ability to speak to a large stakeholder audience and perform a number of different key functions? It is essential that leaders inspire a culture of belonging to all members of the company and involve them on issues and concerns that affect the long-term value creation of the company.
These difficult times have led to an expansion of empathy, self-discipline, adaptation, communication and innovation among professionals. Your next business will be looking for evidence that you've cared for people – including employees, clients, and customers – handled stress and change successfully, decided and communicated correctly and quickly, and embraced uncertainty.
Caprino: Who will be successful in their search and what must executives do to stand out from the competition and thrive?
Opton: If you are an executive entering the hiring circle, you will need to demonstrate a few essentials. First, you are a leader who can quickly identify real issues and take corrective action quickly.
Second, that you have a deep and clear perception of the world in which we currently live and work. Perceptibility far outweighs competence. In 2022, those who succeed in their search will be those with a reputation for creating value for others, which includes sharing not only your expertise, but also your time, connections, kindness and your compassion.
Caprino: In particular, what has changed in job search and what do you think are the top three dos and don'ts?
Option: If you are an executive looking for a job, you need to understand that your ideal job Is it that exist, or, more importantly, you can create it. It is either out there now or it is possible to build it. Your research begins with identifying your 'next big next', then taking action to attract that role and connecting with hiring managers who want what you have to offer.
To beat your competition in today's competitive marketplace, you have to tell your best story of what you can do and how you do it. Many executives feel like the finalist, not the "winner", as they always come second in interviews. Building a compelling story of your unique worth and learning to talk about yourself with confidence and confidence is essential to your success.
The keys to telling this powerful story of value are:
Clarity: Be clear about what you want and what your highest and best usage is.
Control: Shape the narrative by taking your audience on a journey not through your resume, but rather a journey through the why of what you do. Simon Sinek said it best: "People don't buy what you do; they buy why you do it. Don't let your past achievements and titles tell your story.
Emotion: Your new business will want a leader with a vision, purpose, and passion. Tell your story in a concise and emotional way so that your next employer will feel yours and see the meaning you will create for them.
As for the dos and don'ts, well, don't just point and click on whatever job posting you see. Don't start your job search with the mindset that the first thing you need to do is grab the attention of recruiters.
Instead, start by reclaiming your own attention – around your worth and gifts, talents, experiences, ideas and other traits that make you special and things you can do for others, as well as where and with. who you want to share those gifts next.
Don't delay in trying to find your next big neighbor. Start today. It exists in one key form or another, but you have to attract it. And finally, don't make the mistake of telling yourself that you should do it on your own. It is neither possible nor productive. Most of us have mastered something, but haven't mastered the art of executive job hunting. To do this, turn to other people who can guide, mentor and support you and start building a strong community of support now.
Caprino: A final word for executives who are ready for their following?
Option: Thinking of my own retirement and all that went into creating ExecuNet, I consider my legacy. I realized about 10 years ago that my legacy would be the set of different people who have made things great for our executive members. I also wish I had decided to start a business earlier. I put it off until a business merger affected my future and found myself outside.
Change is the one thing that always happens, it's the only constant. The best thing you can do for your career is to become really good at change.
Take a forward-looking vision and build the strategic framework for your next great future today.
For more information, visit ExecuNet.
Kathy caprino is a career and leadership coach, author and speaker helping professional women build rewarding careers of importance.


Despite the historical feats made by minority groups over the past few decades, their legal success, for the most part, hasn't gone much further than idealistic words on a page. For example, in modern countries like Australia, only twenty percent of senior executives, chief executives, and directors employed by the top 200 companies are culturally diverse.
Embracing cultural diversity offers competitive advantages for both small businesses and large corporations. A proactive human resource strategy is a key component for any modern company. By creating an atmosphere of diversity, your company will fine tune its relationship with culturally diverse consumers. To assist you in achieving this goal, we have put together a list of ways you can turn your company into an example of cultural diversity.
Though you should never promote an employee based solely on his or her cultural identity; they should also not be excluded from leadership positions. You should judge fairly, look at the individual's qualifications, skills and character, and employ the best person for the job. Doing so helps to spread cultural awareness within your company, and looks great from the perspective of a client or investor simply because your clients will be from a diverse background. Promoting a culturally diverse leadership also sends a message to lower level minority employees: glass ceilings do not exist in this company, and hard work and proper qualifications do pay off.
Offering cultural training for new employees, and advanced cultural sensitivity training for manager positions, helps to guarantee and foster a culturally diverse atmosphere. Doing so helps teach employees to identity prejudices and misconceptions that may exist in their mind. It will teach them to respect other cultures, and promote compassion and understanding. This educational approach also fosters intercultural communication skills, and helps to create a more effective dialogue with employees who speak English as a second language.
Recognizing minority holidays is an emphatic method of promoting cultural awareness within your company. Make sure your human resource department incorporates holiday rewards similar to those for mainstream holidays. Be sure to offer the appropriate time off, and recognition of their culture's holiday. Doing so will set your company apart, and build trust and loyalty among employees. Small things can have a big impact.
Cultural diversity is created when all your employees understand and appreciate the numerous benefits of diversity in the workplace. A diverse culture brings new and innovative perspectives, offers fresh ideas, unique personalities, and different planning techniques. A diverse workforce garners mutual dignity and respect, and creates a public reputation of fairness and equality — which certainly will never hurt sales or investment!
It is a new world. Our world is a globally connected environment that demands a broad, and diverse workforce. This new world offers both the opportunity to prove your company is a modern, and culturally diverse place; but it also opens the doors of discrimination and the possibility of trapping minorities under glass ceilings. It can only help your bottom line to promote a culture of diversity, acceptance, and understanding within your company. By doing so, you will not only generate a stronger, more productive work force; but you will also give your company an excellent reputation among investors and customers.


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We require a number of reliable and dedicated employees to cover many aspects of care in the community. We are currently offering both part time and full time work in either of the following areas, *Rhyl-Rhuddlan* and/or *Prestatyn-Meliden-Dyserth*. Zoning our Care Support Workers to one of these areas means there's no major travelling between calls. We keep you to one area and pay travelling time. We will set your working days in discussion with you at interview and they never change so you always know you will be working the shifts you chose at interview.
This allows for consistency for both our clients and carers.
7.30am – 1.30pm and/or 4.00pm – 9.30pm, to include some alternate weekend work.
We require a number of reliable and dedicated employees for various openings covering many aspects of care in the local community. Our valued staff provide various levels of community care, for example:- intimate personal care, meal prep, shopping and domestic calls, supported outings, medication prompts, check calls and sitting services, accompanied hospital visits, pet care and prescription collections.
No experience necessary as we have mandatory in-house, induction which includes the Full Social Care Induction Framework for all new employees and various training programmes that each employee will be involved with on a regular basis, enhancing knowledge and promoting safe working practices. Every carer will be required to also complete their Diploma Level 2 in Health and Social Care Adults, we enrol you within your first 8 months. Uniform and PPE is provided Free. If you've worked for us before and are considering returning to our team, give us a call as we'd be happy to discuss the possibility of any returns.
A full driving licence and access to transport is essential for this role.
Please see the recruitment and training section of our website for more details of what we offer and job specifications.
If you are applying for a position within our company, please take a look at our interview expectations.
When arriving for interview please….
Once you have completed your application, please take time to have a look around our website and familiarise yourself with our company.
Please arrive on time for your allocated interview.
You can park for 2 hours on the Left side of Kinmel Street.
There is a car park at the top of Kinmel Street or around the corner to the left.
What interests you about Mpca Ltd and why have you applied?
Familiarise yourself with the services we provide.
Familiarise yourself with the job description and the role.
Ensure you can substantiate what your application says about you.
Expect to answer questions on how your experience, skills and knowledge would transfer into our company.
If you have been asked to bring anything to the interview, please ensure you do so.
Please call 01745 350923 and ask to speak to Lydia or Karen Hughes or alternatively, apply online using the link at the top of the page.
In line with industry legislation and best practice all applicants must provide 2 reliable references, 1 being your most recent employer and / or last health sector employer, and also possess an acceptable Michael Phillips Care Agency Ltd Enhanced DBS before a final offer of employment is made.
MPCA Ltd will cover the costs of the DBS applications on the understanding that new employees do not leave the company within the first 26 weeks.