hydra

Pipe network analysis of hydraulics network.

According to given property of pipes and boundary conditions, hydra can solve the flowrate (steady state) on each pipe in a hydraulic network.

Currently, only CPU version has the function to obtain equations from graph automatically.

The GPU version for the course project is in cuda folder.

How to use this?

• Linux

  make cpu
  ./test_cpu.exe ./data/network1.dat
  

• Windows (MinGW-win64)

  mingw32-make
  ./test_cpu.exe ./data/network1.dat
  

Todos

• Use cuSolverSP to solve for large scale network. (As we test, bicg is difficult to converge in large scale case, though it is suitable to be parallelized.)
• Use to +-1RMQ to solve LCA

How does hydra work?

Kirshoff's first law (Law of conservation, node equations)

Based on conservation of flow rate, summation of flow rate on a node should be zero. Thus, we can obtain node equations for every node (if boundary condition is pressure, then source nodes are excluded.).

Kirshoff's second law (loop equations)

To satisfy Kirshoff's second law, total presure drop (head loss) of any loop should be zero. Thus, we can obtain loop equations for every loop (we only include independent loops to solve unknowns.).

Head loss in pipe

The pressure drop of a pipe can be calculated according to

• Darcy–Weisbach equation

  

or

• More general form

  

  (r is combination of the coefficients, n is usually between 1.5 to 2)

  (p.s. n = 1 => Hagen-Poiseuille Equation; n = 1.85 => Hazen-Williams equation; n = 2.0 => Darcy-Weisbach equation)

• Actual equation used

  Becaues the sign of each term should correspond to the direction of flow, the head loss and its derivative become

  

Independent loops searching

1. Construct BFS spanning tree
2. For every pipe that is not included in the spanning tree, use the two endpoints of that pipe as starting point and end point.
3. Then find lowest common ancestor of the starting point and end point, the travelled path is the loop we want to find. (connect the endpoints also)

Use Newton's method to solve system of non-linear equations

We have to solve system of non-linear equations.

We use incidence matrix to store coefficients (+r, -r, +1, -1) and a vector to store constants (bounary conditions) in node equations and loop equations.

We compute residual matrix R and its jacobian J at every x.

Solver for updates in Newton's method

We implement Gaussian ellimination (with partial pivoting) and Biconjugate gradient method to solve.

For large scale network (number of pipe > 300), we thus suggest to use Gaussian ellimination.

Boundary condition

We support two kind of boundary conditions:

1. Flowrate at node

nNeq = nN - 1
nLeq = nE - nNeq
(nN: #nodes, nE: #edges (unknows), nNeq: #node_eq, nLeq: #loop_eq)

• One node equation should be excluded because it is dependent to the others.

1. Pressure at node

nNeq = nN - nN0
nLeq = nL + nN0 - 1

• Additional (number of sources - 1) loops pass through different sources should be included.
• Node equation on source nodes should be excluded.

Special cases that can generate equivalent network

Currently, we have support the function for converting following conditions into equivalent network, so you have to convert these conditions by yourself.

1. parallel pipes between two nodes

We only support single pipe between two nodes. But you still can convert it to equivalent structure and retrive the actual results based on it.

1. long serial pipes between two nodes

1. multiple sources on one node

Just sum them up.

1. multiple separated networks

Just put them into different network.dat.

Example: network4.dat

Contact

[email protected]