Linear Memory DAE Arena & Lowering

The Differential Algebraic Equations (DAE) representation in ModelScript is implemented via DAEBuilder in @modelscript/runtime (packages/runtime/src/dae/wasm_dae.ts).

Unlike traditional compilers that construct linked expression trees using heap-allocated objects, DAEBuilder operates strictly on flat, contiguous typed arrays (Struct-of-Arrays) in WebAssembly linear memory.

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Arena Struct-of-Arrays (SoA) Layout

The arena maintains three primary contiguous arrays:

┌───────────────────────────────────────────────────────────────────┐
│                      DAEBuilder Linear Memory                     │
├─────────────────┬──────────────────┬──────────────────────────────┤
│ Buffer          │ Element Size     │ Contents                     │
├─────────────────┼──────────────────┼──────────────────────────────┤
│ varData         │ 32 bytes         │ Name, type, variability,     │
│                 │                  │ causality, start value, flags│
├─────────────────┼──────────────────┼──────────────────────────────┤
│ exprData        │ 16 bytes         │ OpKind, data1, leftExprId,   │
│                 │                  │ rightExprId                  │
├─────────────────┼──────────────────┼──────────────────────────────┤
│ eqData          │ 24 bytes         │ EqKind, lhsExprId,           │
│                 │                  │ rhsExprId, flags             │
└─────────────────┴──────────────────┴──────────────────────────────┘

Every variable, expression, and equation is assigned a persistent 32-bit integer handle:

• VarId: Handle to a variable in varData
• ExprId: Handle to an expression node in exprData
• EqId: Handle to an equation in eqData

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Lowering Pipeline

Flattening a high-level Modelica model into linear DAE form proceeds in two coordinated layers:

Layer 1: Components & Hierarchy

Reads Salsa queries (instantiate, classInstance, variability) to allocate variables and parameters in DAEBuilder:

• Scalar variables (Real, Integer, Boolean, String)
• Array dimensions and indices
• Start values, nominals, and min/max boundaries

Layer 2: Equations & Statements

Walks the linear CST of the class body (db.cstNode(classId)):

• Lowers binary, unary, and call expressions directly into ExprId nodes in linear memory.
• Emits equality equations (lhs = rhs), initial equations, and algorithm statements.

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Expression Lowering Rules

To match OpenModelica's canonical output, the flattener applies specific normalization rules:

1. Real Coercion in Binary Operations: If one operand of an arithmetic operator is Real and the other is Integer, the integer operand is automatically cast to Real using castToRealExpr().
2. Negation Distribution: Expressions of the form -(a * b) are rewritten canonically to (-a) * b.
3. Binding Literal Kind Matching: When a binding expression is lowered, addIntLiteral() is used for VarType.Integer and addRealLiteral() for VarType.Real (e.g. parameter Real R = 100 becomes 100.0).

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Optimization & Transformation Passes

Once lowered to the arena, equations pass through four high-performance transformations:

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1. Constant Folding (foldArenaConstants): Recursively simplifies compile-time constant arithmetic expressions directly in the arena.
2. Alias Elimination (eliminateArenaAliases): Detects trivial equality equations ($x=y$) generated by connectors and replaces all references with a single canonical variable handle.
3. Pantelides Index Reduction: Differentiates high-index algebraic constraints to reduce the DAE index to 1 or 0 for stable numerical integration.
4. Tarjan BLT Partitioning: Constructs the bipartite incidence graph between variables and equations, computes maximum matching, and finds Strongly Connected Components (SCC) using Tarjan's algorithm.