solve_utils

Solvers for the assembled sparse systems. The module exposes both direct and iterative approaches tailored to the two matrix constructions used by the project.

Key routines

solve_system(A, b, num_nodes, num_edges)

Direct solve using a sparse direct solver. Returns (pressures, flows) where both are dictionaries mapping node/edge ids to values.

solve_small_system(A, b, G, boundary_conditions, …)

Solver specialised for the reduced small system. Handles boundary value injection, reconstructs full node pressure vector and computes edge flows.

iterative_solve_small, solve_iterative_system

Higher-level wrappers that perform iterative updates to account for rheology non-linearities (flow-dependent viscosity), branching-angle corrections and other physics that require updating resistances and re-solving until convergence.

Notes

  • The iterative solvers contain adaptive stepping / relaxation logic to stabilise convergence for non-linear problems. For large graphs and tight tolerances consider using the small-matrix route where possible.

API reference

Function arguments

solve_system
Ascipy.sparse matrix

Assembled system matrix (shape (n+m, n+m)).

bnumpy.ndarray

Right-hand side vector of length (n+m).

num_nodesint

Number of nodes in the original graph (n).

num_edgesint

Number of edges in the original graph (m).

pressures, flowstuple of dict

Dictionaries mapping node ids to pressures and edge ids to flows.

solve_small_system
A, bsparse matrix, ndarray

Reduced system matrix and right-hand side as returned by matrix_builder.create_small_matrices().

Gnetworkx.DiGraph

Graph object used to reconstruct full pressures/flows.

boundary_conditionstuple

Summary of boundary indices and values used to inject boundary values after solving the reduced system.

ill_conditioned, p0, current_p, max_iterations, restartoptional

Internal flags for iterative solves and GMRES restarts. See the source for details. Typically not required for external callers.

pressures, flowstuple of dict

Dictionaries mapping node ids to pressures and edge ids to flows.

iterative solvers

The module also exposes iterative wrappers that accept additional parameters for tolerances and solver strategies (GMRES vs direct solves). See the source for the full argument list; common options include tol, maxiter, alpha (relaxation), and adaptive_stepping.

Examples

# Direct solve using the full (n+m) system
pressures, flows = solve_system(A, b, num_nodes=n, num_edges=m)

# Reduced small-system solve
pressures, flows = solve_small_system(A_small, b_small, G, bc_export)

Cross references

  • For matrix construction see FetoFlow.matrix_builder.

  • Nonlinear iterative solves call FetoFlow.resistance_utils.calculate_viscosity_factor_from_radius() and update graph attributes via FetoFlow.geometry_utils.update_geometry_with_pressures_and_flows().

    Direct solve of assembled system:

    pressures, flows = solve_system(A, b, num_nodes, num_edges)

    Small system solve:

    pressures, flows = solve_small_system(A_small, b_small, G, bc_export)

    Iterative non-linear solve with adaptive stepping:

    p, q = iterative_solve_small(A_small, b_small, G, bc_export, tol=0.01, maxiter=50)

    FetoFlow.matrix_builder.create_matrices(), FetoFlow.matrix_builder.create_small_matrices()