# Roller Numerical Methods

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 {\displaystyle {\frac {\partial (S_{sr}c_{j})}{\partial x}}{\bigg |}_{i,j}=\left\{{\begin{aligned}&{\frac {(S_{sr}c_{j})_{i,j}-(S_{sr}c_{j})_{i,j-1}}{\delta x_{i,j-1}}},{\text{for }}c_{i,j}>0\\&{\frac {(S_{sr}c_{j})_{i,j+1}-(S_{sr}c_{j})_{i,j}}{\delta x_{i,j}}},{\text{for }}c_{i,j}<0\end{aligned}}\right.} (1)
where ${\displaystyle S_{sr}=2E_{sr}}$ and i and j indicate the position along either the rows or columns, and ${\displaystyle \delta x_{i,j}}$ is the cell-center distance between adjacent cells in the jth direction and at position i. The second-order upwind scheme is given by
 {\displaystyle {\frac {\partial (S_{sr}c_{j})}{\partial x_{j}}}{\bigg |}_{i,j}=\left\{{\begin{aligned}&{\frac {3(S_{sr}c_{j})_{i,j}-4(S_{sr}c_{j})_{i,j-1}+(S_{sr}c_{j})_{i,j-2}}{\delta x_{i,j}+\delta x_{i,j-1}}},{\text{for }}c_{i,j}>0\\&{\frac {-3(S_{sr}c_{j})_{i,j}+4(S_{sr}c_{j})_{i,j+1}-(S_{sr}c_{j})_{i,j+2}}{\delta x_{i,j}+\delta x_{i,j+1}}},{\text{for }}c_{i,j}<0\end{aligned}}\right.} (2)
The surface roller calculation is achieved by setting the initial roller energy and time-stepping until the steady-state solution is reached. For simplicity, an explicit Euler scheme is used as follows {{Equation ${\displaystyle (S_{sr})^{n+1}=(S_{sr})^{n}+\Delta t_{sr}\left(-D_{r}+f_{e}D_{br}-{\frac {\partial (S_{sr}c_{j})}{\partial x}}\right)^{n}}$ |3}}
where ${\displaystyle \Delta _{sr}}$ is the surface roller time step and is determined as ${\displaystyle \Delta t_{sr}={\text{0.5 max}}(\Delta x_{j}/c){\text{ where }}\Delta x_{j}}$ is the cell size in the ${\displaystyle j^{th}}$ direction. The steady-state solution is typically reached after ~40-80 time steps and takes about 1-2 seconds to run on a desktop personal computer.