pymrm.ibm_recon

One-sided normal-derivative operators for the pymrm immersed boundary method.

This module provides Layer 1 of the generalized immersed-interface coupling: for every wall crossing of an pymrm.ibm.IBM object it constructs, on each side separately, a linear formula for the outward normal derivative at the interface point,

.. math:: q_\text{side} ;=; \left.\frac{\partial c}{\partial n_\text{side}} \right|\Gamma ;\approx; \alpha\text{side}, c_\Gamma^\text{side} ;+; \sum_{j \in S_\text{side}} \gamma_j, c_j ,

where :math:c_\Gamma^\text{side} is the (generally unknown) one-sided interface value and :math:S_\text{side} contains nearby same-side cell centres. The weights are polynomially exact generalized finite-difference (GFD) weights obtained from a minimum-weighted-norm problem. These operators are the building blocks for general interface conditions (conjugate diffusion, partition coefficients, contact resistance, surface reactions) assembled in pymrm.ibm_coupling.

Sign conventions¶

The stored unit normal is n = grad(sdf)/|grad(sdf)| and points from the solid (sdf < 0) into the fluid (sdf >= 0).
Derivatives are outward per side, matching pymrm.coupling.construct_interface_matrices: q_out is taken along -n (out of the fluid domain) and q_in along +n (out of the solid domain). For a healthy reconstruction both alpha_out and alpha_in are positive.

Robustness / length-scale strategy¶

Stencil selection is governed by two caps: a geometric one (radius_factor * rings * h_local) and a physical one (length_scale_factor * length_scale). When too few good points are available inside the cap the polynomial degree is lowered instead of reaching farther: p=2 -> enlarge once -> p=1 -> two-point normal formula -> flagged unresolved. Candidate cells must be flood-fill connected to the cut cell through same-side cells, so points across a thin gap or a neighbouring solid are never used.

View module source on GitHub

Public API¶

Symbol	Type	Summary
`IBMNormalDerivative`	class	One-sided normal-derivative operators and diagnostics per IBM crossing.
`construct_ibm_normal_derivative`	function	Construct one-sided normal-derivative operators for every IBM crossing.
`construct_ibm_normal_derivative_ops`	function	Expand the reconstruction operators to the full field layout.
`gfd_normal_derivative_weights`	function	GFD weights for a directional derivative at a single interface point.
`interface_normals`	function	Unit interface normals (solid -> fluid) at each wall crossing.

`IBMNormalDerivative()`¶

Open dedicated reference page

One-sided normal-derivative operators and diagnostics per IBM crossing.

The outward normal derivative on each side of crossing k is

``q_side[k] = alpha_side[k] * c_gamma_side[k] + (D_side @ c_spatial)[k]``

with c_spatial the field on the spatial grid (flattened, C-order) and the outward per side sign convention described in the module docstring.

Attributes¶

normals (ndarray, shape (n_crossings, ndim_s)) Unit normals, pointing from solid to fluid.
alpha_out, alpha_in (ndarray, shape (n_crossings,)) Weight of the one-sided interface value in q_side.
D_out, D_in (csr_array, shape (n_crossings, n_spatial_cells)) Weights of the same-side cell values in q_side.
degree_out, degree_in (ndarray of int) Polynomial degree actually used: 2 or 1 (GFD), 0 (two-point normal formula), -1 (degenerate axis-direction two-point fallback).
n_stencil_out, n_stencil_in (ndarray of int) Number of cell points in the stencil.
radius_out, radius_in (ndarray) Physical search radius of the accepted stencil.
cond_out, cond_in (ndarray) Conditioning of the GFD moment matrix (1 for two-point formulas).
moment_residual_out, moment_residual_in (ndarray) Max-norm residual of the polynomial moment conditions.
weight_norm_out, weight_norm_in (ndarray) h-scaled 1-norm of the weights, O(1) for a healthy formula.
unresolved_out, unresolved_in (ndarray of bool) True where only the degenerate axis-direction fallback was possible.

Shape information¶

n_crossings : int Number of wall crossings (matches ibm.n_crossings). spatial_shape : tuple Spatial grid shape. shape : tuple Full field shape (spatial + non-spatial axes). axes : tuple of int Which axes of shape are spatial. ns_size : int Product of the non-spatial dimensions (1 when purely spatial). n_cells : int Total number of cells, prod(shape). n_spatial_cells : int Number of spatial cells, prod(spatial_shape). h_ref : ndarray, shape (ndim_spatial,) Median cell-centre spacing along each spatial axis (reference scale for the GFD weights).

Sign conventions¶

Robustness / length-scale strategy¶

Public API¶

IBMNormalDerivative()¶

Attributes¶

Shape information¶

Members¶

D_in¶

D_out¶

alpha_in¶

alpha_out¶

axes¶

cond_in¶

cond_out¶

degree_in¶

degree_out¶

h_ref¶

moment_residual_in¶

moment_residual_out¶

n_cells¶

n_crossings¶

n_spatial_cells¶

n_stencil_in¶

n_stencil_out¶

normals¶

ns_size¶

radius_in¶

radius_out¶

shape¶

spatial_shape¶

unresolved_in¶

unresolved_out¶

weight_norm_in¶

weight_norm_out¶

Parameters¶

Returns¶

construct_ibm_normal_derivative_ops(ibm, recon)¶

Parameters¶

Returns¶

gfd_normal_derivative_weights(x_gamma, x_cells, direction, degree = 2, hvec = None, weight_power = 4, interface_penalty = 1.0)¶

Parameters¶

Returns¶

interface_normals(ibm, sdf, x_c)¶

Parameters¶

Returns¶

`IBMNormalDerivative()`¶

`D_in`¶

`D_out`¶

`alpha_in`¶

`alpha_out`¶

`axes`¶

`cond_in`¶

`cond_out`¶

`degree_in`¶

`degree_out`¶

`h_ref`¶

`moment_residual_in`¶

`moment_residual_out`¶

`n_cells`¶

`n_crossings`¶

`n_spatial_cells`¶

`n_stencil_in`¶

`n_stencil_out`¶

`normals`¶

`ns_size`¶

`radius_in`¶

`radius_out`¶

`shape`¶

`spatial_shape`¶

`unresolved_in`¶

`unresolved_out`¶

`weight_norm_in`¶

`weight_norm_out`¶

`construct_ibm_normal_derivative_ops(ibm, recon)`¶

`gfd_normal_derivative_weights(x_gamma, x_cells, direction, degree = 2, hvec = None, weight_power = 4, interface_penalty = 1.0)`¶

`interface_normals(ibm, sdf, x_c)`¶