Cellier-Elmqvist Algebraic Loop Tearing

Implementation: TornBlock in packages/runtime/src/wasm/structural/tearing.ts

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Academic Citations

• Cellier, F. E. (1979). Combined Continuous/Discrete System Simulation by Use of Digital Computers: Techniques and Tools.
  PhD thesis, ETH Zurich.
  DOI: 10.3929/ethz-a-000179979
• Elmqvist, H., & Otter, M. (1994). "Methods for Tearing Systems of Equations in Object-Oriented Modeling."
  Proceedings of the European Simulation Multiconference (ESM'94), pp. 326–332.
• Mahana, P. N., & Eagan, J. P. (1975). "A tearing algorithm for large systems of equations."
  IEEE Transactions on Circuits and Systems.

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ModelScript Architectural Rationale

Even after BLT partitioning, physical systems containing closed kinematic loops, electrical bridge circuits, or multi-phase fluid balances yield coupled Strongly Connected Components of size $N>1$.

Solving an $N$-dimensional non-linear system via Newton-Raphson requires inverting an $N\times N$ Jacobian matrix at every iteration:

$$\mathcal{O}(N^3)$$

The Cellier-Elmqvist Minimum Degree Tearing algorithm identifies a minimal set of $k$ tearing variables ($k\ll N$) such that assuming known values for these $k$ variables breaks all cyclic dependencies in the block. The remaining $N-k$ inner variables can then be evaluated sequentially via a causal explicit forward substitution chain:

$$\begin{array}{rl}{x}_{\text{inner}}& =g(x_{\text{tear}},u)\text{(explicit chain)}\\ R(x_{\text{tear}})& =f_{\text{residual}}(x_{\text{tear}},x_{\text{inner}})=0\text{(small }k\times k\text{ solve)}\end{array}$$

The non-linear iterative solver is restricted to the tiny $k\times k$ residual system, accelerating simulation speed by orders of magnitude.

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ModelScript Modifications & WASM Architecture

1. Unmanaged WebAssembly Kernel: Implemented as @unmanaged classes directly in linear memory with zero garbage collection pauses.
2. Dynamic Minimum Degree Heuristic: Evaluates vertex degrees in the block incidence graph in memory, prioritizing linear tearing variables over non-linear variables to avoid non-linear solving where possible.
3. Partitioned In-Memory Structures: Splits the block into:
   • $k$ tearing variables (tearVarIndices)
   • $N-k$ explicit causal equations (innerEqIndices)
   • $k$ residual constraint equations (residualEqIndices)
4. Direct Matrix Integration: Emits residual Jacobians directly to the in-WASM LU factorizer (luFactor, luSolve).

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Algorithm Steps

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Upstream & Downstream Pipeline Connections

• Upstream Inputs:
  • Receives coupled SCC blocks of size $>1$ from BltEngine.
• Downstream Consumers:
  • Feeds explicit forward chains and $k\times k$ residual systems to the in-WASM Newton-Raphson solver and Sparse LU Solver.
  • Evaluated inside simulation step loops (SUNDIALS, TR-BDF2, Tsit5).