Coupling systems

Coupling systems#

This tutorial assumes XCOMPUTE-CLIENT connected to a live session of XCOMPUTE-SERVER, and with one or more numerical systems with two or more regions.

Theory#

Boundary conditions are useful for providing closure to numerical systems, but are limited to asserting values that are known a priori. As systems increase in complexity, such a static approach makes it impossible to gain a global solution.

Coupling interpolates across regions to providing closure, providing a dynamic replacement for typical boundary conditions.This approach enables teams to gain insight into the complex interaction of systems.

A thought experiment: the estimated measure between two nearby regions should be some sort of average between the two regions. As the regions come together in contact, the instantaneous value where they meet should be the same value, as a single point in space-time must converge to a singular value (within some tolerance).

Interpolation#

For coupling nodes of region A to nodes of region B, our mathematical approach utilizes a transition matrix T with dimensions size(A) by size(B). The connection strength between A and B is determined by the specific sampling method, which is then normalized to be Markov-like.

The flow of information from A to B is represented by T while the reverse flow from B to A is represented by T^T. The bias parameter $\alpha$ controls the duplex strength with range [0,1], defaulting to 0.5:

u(A,i+1) = (1-$\alpha$)*u(A,i) + $\alpha$*T(i)*u(B,i)

and

u(B,i+1) = (1-$\alpha$)*u(B,i) + $\alpha$*T(i)^T*u(A,i)

In Practice#

Two Regions#

Coupling is always asserted between two regions (usually surfaces), which can either be defined manually by the user or assigned dynamically based on spatial proximity.

  • intra-system: two regions in the same system, owned by the same system

  • inter-system: two regions in different systems, owned by the parent system

Sampling Mode#

A variety of approaches are required to couple regions for different situations. Currently, supported sampling modes include:

  • 1:1 - direct nodal mapping, only for abstract systems

  • nearest - sparse mapping to closest node

  • distance - dense mapping based on 1/r^2

  • view_factor- dense mapping based on integrated sterradian surface values

  • contact - sparse mapping based on overlapping nodes and elements

Properties#

Most properties that are in-common between coupled systems can be interpolated, but usually, only specific variables should be interpolated.

Examples:

  • given a thermal-fluid interaction, perhaps Temperature is the only PropertyKey required for interpolation.

  • given a fluid-structure interaction, perhaps Displacement and Pressure would be required for interpolation.

Flow Control#

  • forward - permit information flow from A to B

  • reverse - permit information flow from B to A

  • $\alpha$ - relative duplex strength

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