Acausal Connector Balancing & Stream Mixing

Implementation: ConnectionSetResolver & IntUnionFind in languages/modelica/src/connections.ts

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Academic Citations

Acausal Physical Modeling & Connection Sets

• Mattsson, S. E., Elmqvist, H., & Otter, M. (1998). "Physical system modeling with Modelica."
  Control Engineering Practice, 6(4), pp. 501–510.
  DOI: 10.1016/S0967-0661(98)00047-1
• Tarjan, R. E. (1975). "Efficiency of a good but not linear set union algorithm."
  Journal of the ACM, 22(2), pp. 215–225.
  DOI: 10.1145/321879.321884 (Disjoint-Set Union-Find)

Stream Connectors & Convective Transport

• Franke, R., Casella, F., Otter, M., Sielemann, M., Elmqvist, H., Mattsson, S. E., & Olsson, H. (2009). "Stream Connectors - An Extension of Modelica for Device-Oriented Modeling of Convective Transport."
  Proceedings of the 7th Modelica Conference, pp. 116–123.
  DOI: 10.3384/ecp09430076

Expandable Connectors & Multi-Domain Bus Modeling

• Otter, M., Elmqvist, H., & Mattsson, S. E. (2005). "Object-oriented modeling of bus systems with Modelica."
  Proceedings of the 4th Modelica Conference, pp. 295–308.
  Paper PDF

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ModelScript Architectural Rationale

Unlike classical block diagrams (such as causal signal-flow block diagrams) where connections represent directional causal signal flows ($y=f(u)$), ModelScript models acausal physical interaction across multi-energy domains:

• Electrical: Voltage $v$ (potential) and Current $i$ (flow).
• Hydraulic & Pneumatic: Pressure $p$ (potential) and Volume/Mass flow rate $q$ (flow).
• Translational Mechanics: Position/Velocity $s,v$ (potential) and Force $f$ (flow).
• Rotational Mechanics: Angle/Angular velocity $\varphi ,\omega$ (potential) and Torque $\tau$ (flow).
• Thermal & Fluid Transport: Temperature/Enthalpy $h$ (stream) and Heat/Mass flow $\dot{m}$ (flow).

In acausal modeling, connection statements connect(a, b) do not dictate computational causality. The compiler must automatically generate:

1. Potential Equality Equations: Enforcing that all across-variables in a connection set share an identical value.
2. Kirchhoff Flow Equations: Enforcing conservation of mass, charge, or energy such that all through-variables sum to zero at the junction.
3. Convective Stream Transport Equations: Resolving thermodynamic mixing when fluid direction reverses ($m_j\gtrless 0$) without introducing singular divisions near zero mass flow.

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Balancing Algorithm & Mathematical Formulation

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1. Disjoint Connection Set Resolution (Union-Find)

The set of all declared connect(c_a, c_b) statements forms an undirected graph. ModelScript partitions all connector references into disjoint equivalence classes using an integer array-based Union-Find with path compression and rank heuristics:

$$\alpha (N)\le 4\text{for all physical models}$$

For each connection set $S=\{c_1,c_2,\dots ,c_m\}$ of size $m$:

2. Potential Variable Equalization

For every non-flow, non-stream variable $v$ declared inside the connector type:

$$v(c_1)=v(c_2)=\cdots =v(c_m)$$

This yields exactly $m-1$ scalar equations. One variable is chosen as the canonical representative, eliminating $m-1$ redundant states during alias elimination.

3. Kirchhoff Flow Variable Balancing

For every flow variable $f$ declared inside the connector type:

$$\sum _{j=1}^{m}f(c_j)=0$$

This yields exactly $1$ scalar conservation equation per connection set. If an inside-outside connector hierarchy boundary is crossed, the outer flow sign is inverted to preserve directional conservation:

$$f_{\text{outer}}=\sum _{j\in \text{inner}}f(c_j)$$

4. Stream Variable Convective Transport

For fluid ports, transport enthalpy depends on flow direction:

$$h_{\text{transported}}=\{\begin{array}{ll}{h}_{\text{upstream}}& \text{if }\dot{m}>0\\ h_{\text{downstream}}& \text{if }\dot{m}<0\end{array}$$

Direct formulation introduces a discontinuous switch that causes Newton solvers to oscillate around $\dot{m}=0$. ModelScript lowers stream variables using Franke's regularized inStream() and actualStream() operators:

$$\text{inStream}(c_i.h)=\frac{\sum _{j\ne i,{\dot{m}}_j>0}{\dot{m}}_j\cdot c_j.h}{\sum _{j\ne i,{\dot{m}}_j>0}{\dot{m}}_j}$$

Near zero flow ($\parallel {\dot{m}}_j\parallel <ϵ$), smooth polynomial regularization is applied to maintain $C^1$-continuity of the residual Jacobian.

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Upstream & Downstream Integration

Pipeline Component	Upstream Dependencies	Downstream Consumers
IntUnionFind	AST connect clauses from Cst	Connection set partition groups
Potential Equalizer	Variable causality & variability from QueryEngine	Alias elimination (eliminateArenaAliases)
Flow Balancer	Connector definitions	DAEBuilder linear equations
Stream Formulator	Fluid connector declarations	PantelidesEngine, Nonlinear tearing loops