11.3.4 The ISAT Algorithm

ISAT is a powerful tool that enables realistic chemistry to be incorporated in multi-dimensional flow simulations by accelerating the chemistry calculations. Typical speed-ups of 100-fold are common. This power is apparent if one considers that with a 100-fold speed-up, a simulation that would take three months without ISAT can be run in one day.

At the start of an ANSYS FLUENT simulation using ISAT, the ISAT table is empty. For the first reaction step, Equation  11.3-8 is integrated with a stiff ODE solver. This is called Direct Integration (DI). The first table entry is created and consists of:

• the initial composition $\phi^0$

• the mapping $\phi^1$

• the mapping gradient matrix $A = \partial \phi^1 / \partial \phi^0$

• a hyper-ellipsoid of accuracy

The next reaction mapping is calculated as follows: The initial composition vector for this particle is denoted $\phi^0_q$, where the subscript $q$ denotes a query. The existing table (consisting of one entry at this stage) is queried by interpolating the new mapping as

The mapping gradient is hence used to linearly interpolate the table when queried. The ellipsoid of accuracy (EOA) is the elliptical space around the table point $\phi^0$ where the linear approximation to the mapping is accurate to the specified tolerance, $\epsilon_{\rm tol}$.

If the query point $\phi^1_q$ is within the EOA, then the linear interpolation by Equation  11.3-9 is sufficiently accurate, and the mapping is retrieved. Otherwise, a direct integration (DI) is performed and the mapping error $\epsilon = \vert B (\phi^1_{DI}-\phi^1_q)\vert$ is calculated (here, $B$ is a scaling matrix). If this error is smaller than the specified error tolerance ( $\epsilon < \epsilon_{tol}$), then the original interpolation $\phi^1_q$ is accurate and the EOA is grown so as to include $\phi^0_q$. If not, a new table entry is added.

Table entries are stored as leaves in a binary tree. When a new table entry is added, the original leaf becomes a node with two leaves--the original leaf and the new entry. A cutting hyper-plane is created at the new node, so that the two leaves are on either side of this cutting plane. A composition vector $\phi^0_q$ will hence lie on either side of this hyper-plane.

The ISAT algorithm is summarized as follows:

1.   The ISAT table is queried for every composition vector during the reaction step.

2.   For each query $\phi^0_q$ the table is traversed to identify a leaf whose composition $\phi^0$ is close to $\phi^0_q$.

3.   If the query composition $\phi^0_q$ lies within the EOA of the leaf, then the mapping $\phi^1_q$ is retrieved using interpolation by Equation  11.3-9. Otherwise, Direct Integration (DI) is performed and the error $\epsilon$ between the DI and the linear interpolation is measured.

4.   If the error $\epsilon$ is less than the tolerance, then the ellipsoid of accuracy is grown and the DI result is returned. Otherwise, a new table entry is added.

At the start of the simulation, most operations are adds and grows. Later, as more of the composition space is tabulated, retrieves become frequent. Since adds and grows are very slow whereas retrieves are relatively quick, initial ANSYS FLUENT iterations are slow but accelerate as the table is built.