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Full Equations (FEQ) Model for the Solution of the Full, Dynamic Equations of Motion for One-Dimensional Unsteady Flow in Open Channels and Through Control Structures

U.S. GEOLOGICAL SURVEY WATER-RESOURCES INVESTIGATIONS REPORT 96-4240

10.4 Example: Four Branches with Loop


A four-branch network of channels with a loop is shown in figure 36.

In this example, each branch has a single computational element to limit the number of equations that must be presented. The junction conditions are again the simple requirement that the water-surface elevations in all the branches at a junction should be the same. This is the simplest condition, but yet the most demanding, in establishing nonzero diagonal elements in the
Jacobian matrix for the network. The NMCI (section 13.6) fragment in FEQ describing stream-system connections is the following:

[... additional input values that are not needed for the discussion of this example; blanks indicate no input in these columns]

Table

If node U1 is specified as the initial boundary node, the pattern of the Jacobian matrix is the following:

Columns

Table

There are 71 nonzero or potentially nonzero elements in the Jacobian matrix. This number is called the length of the profile of the matrix. Had the boundary node been D4 instead of U1, the profile length would have been 66. If the user does not request an explicit choice for the initial boundary node, then all possible boundary nodes are attempted in FEQ simulation and the node that minimizes the length of the profile of the matrix is selected. A valid boundary node has the boundary condition of flow as a function of time or flow as a function of elevation. The matrix cannot be formed in FEQ simulation from a boundary where water-surface elevation is specified as a function of time. To study a case where water-surface elevation is a function of time at all upstream boundaries, the user can add a dummy branch with zero flow to the system so that the matrix may be formed.


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