Staggered Grid Generation

A grid type option referred to as T-type involves local refinement in a upper central domain of the model and smooth variations of grid resolution in surrounding lower resolution domains. T-type refinement is enabled by setting the grid_type argument equal to :TtypeRefinedGrid (See StaggeredGrid.initialize!).

The high-resolution domain in the T-type grid has basic grid spacings $\Delta x_{hr}$ and $\Delta y_{hr}$ in the $x-$ and $y$-directions, respectively, and extends laterally from x-coordinate $x_o$ to $x_f$ and vertically from the top of the model domain to a depth of $y_f$. The x-indices of the basic grid corresponding to $x_o$ and $x_f$ are $j_{x_o}$ and $j_{x_f}$, respectively, and are calculated as follows:

eq:ttype-bounding-j-indices

\[\begin{split} j_{x_o} & = int\left(\frac{N_{x,b} - N_{x,hr}}{2}\right) + 1\\ j_{x_f} & = j_{x_o} + int\left( \frac{\left(x_f - x_o\right)}{\Delta x_{hr}} \right) \end{split}\]

Starting with $x_{b,j_{x_o}} = x_o$ the x-coordinates within the high resolution domain from $j=j_{x_o}+1$ to $j=j_{x_f}$ are calculated as follows:

eq:ttype-x-coords-highres

\[x_{b,j} = x_{b,j-1} + \Delta x_{hr} \text{.}\]

Starting with index $j=2$ and ending with index $j=j_{x_o}$ the x-coordinates to the left of $x_o$ are calculate using the following equation (Gerya, 2010):

eq:ttype-x-coords-left

\[x_{b,j} = x_{b,j-1} + \Delta x_{hr}(f_{smooth,L})^{j_{x_o} - j + 1}\]

where $f_{smooth,L}$ is a smoothing factor that is calculated iteratively starting with an initial value of $f_{smooth,L} = 1.1$ using the following equation:

eq:ttype-left-smoothing

\[f_{smooth,L} = \left[ 1 + \frac{x_o}{\Delta x_{hr}}\left(1 - \frac{1}{f_{smooth,L}}\right) \right]^{\frac{1}{N_{steps,L}}}\]

where $N_{steps,L} = j_{x_o} - 1$. The x-coordinates to the right of $x_f$ starting at index $j_{x_f}+1$ are calculated using the following equation (Gerya, 2010):

eq:ttype-x-coords-right

\[x_{b,j} = x_{b,j-1} + \Delta x_{hr}(f_{smooth,R})^{j - j_{x_f}}\]

where $f_{smooth,R}$ is a smoothing factor that is calculated iteratively starting with an initial value of $f_{smooth,R} = 1.1$ using the following equation:

eq:ttype-right-smoothing

\[f_{smooth,R} = \left[ 1 + \frac{x_{size} - x_{b,j_{x_f}}}{\Delta x_{hr}}\left(1 - \frac{1}{f_{smooth,R}}\right) \right]^{\frac{1}{N_{steps,R}}}\]

where $N_{steps,R} = \left(N_{x,b} - j_{x_f}\right)$.

The spacing of the basic grid in the x-direction is given by:

eq:x-spacing-basic

\[\Delta x_{b,j} = x_{b,j+1} - x_{b,j} \text{.}\]

where $j$ ranges from $1$ to $N_{x,b}-1$.

Starting with $y_{b,i} = 0$, the y-coordinates of the high-resolution domain from indices $i=2$ to $i=i_{y_f}$ are calculated as follows:

eq:ttype-y-coords-highres

\[y_{b,i} = y_{b,i-1} + \Delta y_{hr} \text{,}\]

where $i_{y_f}$ is the index of the basic grid corresponding to the depth $y_f$ and is calculated as follows:

eq:ttype-y-index-final

\[i_{y_f} = int\left(\frac{y_f}{\Delta y_{hr}}\right) + 1 \text{.}\]

The y-coordinates below the high-resolution domain starting at index $i = i_{y_f}+1$ are calculated using the following equation (Gerya, 2010):

eq:ttype-y-coords-below-hr

\[y_{b,i} = y_{b,i-1} + \Delta y_{hr}(f_{smooth,B})^{i - i_{y_f}}\]

where $f_{smooth,B}$ is a smoothing factor that is calculated iteratively starting with an initial value of $f_{smooth,B} = 1.1$ using the following equation:

eq:ttype-y-smoothing

\[f_{smooth,B} = \left[ 1 + \frac{y_{size} - y_{b,i_{y_f}}}{\Delta y_{hr}}\left(1 - \frac{1}{f_{smooth,B}}\right) \right]^{\frac{1}{\left(N_{y,b} - i_{y_f}\right)}} \text{.}\]

The spacing of the basic grid in the y-direction is given by:

eq:y-spacing-basic

\[\Delta y_{b,i} = y_{b,i+1} - y_{b,i} \text{.}\]

where $i$ ranges from $1$ to $N_{y,b}-1$.

The pressure grid, denoted by subscript $p$, has nodes located at the center of basic grid cells with $(N_{x,b}-1)$ nodes in the $x$-direction and $(N_{y,b}-1)$ nodes in the $y$-direction fig:staggered-grid. The x- and y-coordinates of the pressure grid are $(x_{p,j}, y_{p,i})$ where $j$ is in the $x$-direction ranging from $1$ to $N_{x,b}-1$ and $i$ is in the $y$-direction ranging from $1$ to $N_{y,b}-1$. The pressure grid spacings in the $x$- and $y$-directions are denoted by $\Delta x_{p,j}$ where $j$ ranges from $1$ to $N_{x,b}-2$ and $\Delta y_{p,i}$ where $i$ ranges from $1$ to $N_{y,b}-2$, respectively. Pressure unknown $P_{(i,j)_p}$, deviatoric normal stress $\sigma'_{xx(i,j)_p}$, deviatoric normal strain rate $\dot \epsilon'_{xx(i,j)_p}$ and visco-plastic normal viscosity $\eta_{vp(i,j)_p}$ are defined on the nodes of the pressure grid. The spacing of the pressure grid in the x-direction is given by:

eq:pressure-x-spacing

\[\Delta x_{p,j} = 0.5\left(x_{b,j+2} - x_{b,j}\right)\]

where $j$ ranges from $1$ to $N_{x,b}-2$. The spacing of the pressure grid in the y-direction is given by:

eq:pressure-y-spacing

\[\Delta y_{p,i} = 0.5\left(y_{b,i+2} - y_{b,i}\right)\]

where $i$ ranges from $1$ to $N_{y,b}-2$. Starting with $x_{p,1} = 0.5(x_{b,1}+x_{b,2})$, the x-coordinates of the pressure grid from indices $j=2$ to $j=N_{x,b}-1$ are calculated as follows:

eq:pressure-x-coords

\[x_{p,j} = x_{p,j-1} + \Delta x_{p,j-1} \text{.}\]

Starting with $y_{p,1} = 0.5(y_{b,1}+y_{b,2})$, the y-coordinates of the pressure grid from indices $i=2$ to $i=N_{y,b}-1$ are calculated as follows:

eq:pressure-y-coords

\[y_{p,i} = y_{p,i-1} + \Delta y_{p,i-1} \text{.}\]

The $v_x$ grid, denoted by subscript $v_x$, has nodes located at the center of left- and right-sides of basic-grid cells and outside the model domain in the y-direction with $N_{x,b}$ nodes in the $x$-direction and $(N_{y,b}+1)$ nodes in the $y$-direction fig:staggered-grid. The x- and y-coordinates of the $v_x$ grid are $(x_{b,j}, y_{v_x,i})$ where $j$ ranges from $1$ to $N_{x,b}$ and $i$ ranges from $1$ to $N_{y,b}+1$. The grid spacings in the y-direction of the $v_x$ grid are denoted by $\Delta y_{v_x,i}$ where $i$ ranges from $1$ to $N_{y,b}$. Spacing in the x-direction is equal to $\Delta x_{b,j}$. The unknown for the x-component of velocity $v_{x(i,j)v_x}$ is defined on the nodes of the $v_x$ grid. Starting with $y_{v_x,1} = 0.5(y_{b,1} - 0.5\Delta y_{b,1})$ and ending with $y_{v_x,N_{y,b}+1} = y_{b,N_{y,b}} + 0.5\Delta y_{b,N_{y,b}-1}$, the y-coordinates of the $v_x$ grid from indices $i=2$ to $i=N_{y,b}$ are calculated as follows:

eq:vx-grid-y-coords

\[y_{v_x,i} = 0.5(y_{b,i} + y_{b,i-1}) \text{.}\]

Starting with $\Delta y_{v_x,1} = \Delta y_{b,1}$ and ending with $\Delta y_{v_x,N_{y,b}} = \Delta y_{b,N_{y,b}-1}$, the grid spacings in the y-direction of the $v_x$ grid from indices $i=2$ to $i=N_{y,b}$ are calculated as follows:

eq:vx-y-spacing

\[\Delta y_{v_x,i} = 0.5(y_{b,i+1} - y_{b,i-1})\]

The $v_y$ grid, denoted by subscript $v_y$, has nodes located at the center top and bottom of basic-grid cells and outside the model domain in the x-direction with $N_{x,b}+1$ nodes in the $x$-direction and $(N_{y,b})$ nodes in the $y$-direction fig:staggered-grid. The x- and y-coordinates of the $v_y$ grid are $(x_{j,v_y}, y_{b,i})$ where $j$ ranges from $1$ to $N_{x,b}+1$ and $i$ ranges from $1$ to $N_{y,b}$. The grid spacings in the x-direction of the $v_y$ grid are denoted by $\Delta x_{j,v_y}$ where $j$ ranges from $1$ to $N_{x,b}$. Spacing in the y-direction is equal to $\Delta y_{b,i}$. The unknown for the y-component of velocity $v_{y(i,j)v_y}$ is defined on the nodes of the $v_y$ grid. Starting with $x_{v_y,1} = 0.5(x_{b,1} - 0.5\Delta x_{b,1})$ and ending with $x_{v_y,N_{x,b}+1} = x_{b,N_{x,b}} + 0.5\Delta x_{b,N_{x,b}-1}$, the x-coordinates of the $v_y$ grid from indices $j=1$ to $j=N_{x,b}$ are calculated as follows:

eq:vy-x-coords

\[x_{v_y,j} = 0.5(x_{b,j} + x_{b,j-1}) \text{.}\]

Starting with $\Delta x_{v_y,1} = \Delta x_{b,1}$ and ending with $\Delta x_{v_y,N_{x,b}} = \Delta x_{b,N_{x,b}-1}$, the grid spacings in the x-direction of the $v_y$ grid from indices $j=2$ to $j=N_{x,b}-1$ are calculated as follows:

eq:vy-x-spacing

\[\Delta x_{v_y,j} = 0.5(x_{b,j+1} - x_{b,j-1})\]