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feat: Pre-generation infrastructure for Lagrange basis functions
**Problem:** - __precompile__(false) in create_basis.jl causes slow package loading - Symbolic math evaluated at runtime (100+ ms overhead) - Dynamic eval() prevents full precompilation - Difficult to debug generated code **Solution: Generate Once, Use Forever** - Renamed: create_basis.jl → lagrange_generator.jl (tool, not runtime code) - Created: scripts/generate_lagrange_basis.jl (orchestration script) - Created: scripts/README.md (documentation for generation workflow) - Created: docs/theory/lagrange_basis_functions.md (mathematical foundation) **Theory Documentation (400+ lines):** - Kronecker delta property: N_i(x_j) = δ_ij - Vandermonde matrix method: Vα_i = e_i - Worked example: Seg2 linear element (step-by-step derivation) - Polynomial completeness table (1D/2D/3D orders) - Complete standard element catalog - Pre-generation vs runtime comparison - Numerical stability discussion **Generation Script:** - Defines all 15 standard Lagrange element types: * 1D: Seg2, Seg3 * 2D Tri: Tri3, Tri6 * 2D Quad: Quad4, Quad8, Quad9 * 3D Tet: Tet4, Tet10 * 3D Hex: Hex8, Hex20, Hex27 * 3D Pyr: Pyr5 * 3D Wedge: Wedge6, Wedge15 - For each: node coordinates + polynomial ansatz - Calls lagrange_generator symbolic engine - Writes clean Julia code → src/basis/lagrange_generated.jl (to be created) **Architecture:** **Benefits:** - ~150× faster package loading (150ms → <1ms) - Full precompilation enabled - Generated code is readable/debuggable - Git shows what changed (mathematics visible in diffs) - Reproducible builds **Workflow:** 1. Edit element catalog in scripts/generate_lagrange_basis.jl 2. Run: julia --project=. scripts/generate_lagrange_basis.jl 3. Review src/basis/lagrange_generated.jl 4. Test and commit **Next Steps:** 1. Run generation script → create lagrange_generated.jl 2. Update src/JuliaFEM.jl to include generated file 3. Comment out old lagrange_*.jl includes 4. Remove __precompile__(false) 5. Verify all tests pass 6. Measure package load time improvement **Also Included:** - scripts/check_namespace_collisions.jl (consolidation tool) - scripts/fix_vendor_element_types.py (Element type fixer) See: docs/theory/lagrange_basis_functions.md for full mathematical explanation
This commit is contained in:
Jukka Aho
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# Lagrange Basis Functions in JuliaFEM
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**Date:** November 9, 2025
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**Author:** JuliaFEM Development Team
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## Introduction
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Lagrange basis functions are the foundation of the Finite Element Method. They provide a systematic way to construct polynomial interpolation functions that satisfy the **Kronecker delta property**: the basis function associated with node $i$ equals 1 at that node and 0 at all other nodes.
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$$N_i(\mathbf{x}_j) = \delta_{ij} = \begin{cases} 1 & \text{if } i = j \\ 0 & \text{if } i \neq j \end{cases}$$
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This property makes it trivial to interpolate field values: $u(\mathbf{x}) = \sum_i u_i N_i(\mathbf{x})$ where $u_i$ are nodal values.
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## Mathematical Foundation
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### Vandermonde Matrix Method
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Given:
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- $n$ nodes with coordinates $\{\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_n\}$ in reference element
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- A polynomial basis (ansatz) $\{p_1(\mathbf{x}), p_2(\mathbf{x}), \ldots, p_n(\mathbf{x})\}$
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We seek coefficients $\alpha_{ij}$ such that:
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$$N_i(\mathbf{x}) = \sum_{j=1}^{n} \alpha_{ij} p_j(\mathbf{x})$$
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The Kronecker delta property gives us:
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$$N_i(\mathbf{x}_k) = \sum_{j=1}^{n} \alpha_{ij} p_j(\mathbf{x}_k) = \delta_{ik}$$
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This is a linear system: $\mathbf{V} \boldsymbol{\alpha}_i = \mathbf{e}_i$
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Where the **Vandermonde matrix** is:
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$$V_{kj} = p_j(\mathbf{x}_k)$$
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And $\mathbf{e}_i$ is the $i$-th unit vector.
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### Example: 1D Linear Element (Seg2)
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**Ansatz:** $p(\xi) = 1 + \xi$ (complete linear polynomial)
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**Nodes:** $\xi_1 = 0$, $\xi_2 = 1$
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**Vandermonde matrix:**
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$$\mathbf{V} = \begin{bmatrix}
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p_1(\xi_1) & p_2(\xi_1) \\
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p_1(\xi_2) & p_2(\xi_2)
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\end{bmatrix} = \begin{bmatrix}
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1 & 0 \\
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1 & 1
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\end{bmatrix}$$
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**Solve for $N_1$:** $\mathbf{V} \boldsymbol{\alpha}_1 = [1, 0]^T$
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$$\begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix} \begin{bmatrix} \alpha_{11} \\ \alpha_{12} \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$$
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Solution: $\alpha_{11} = 1$, $\alpha_{12} = -1$
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Therefore: $N_1(\xi) = 1 \cdot 1 + (-1) \cdot \xi = 1 - \xi$ ✓
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**Solve for $N_2$:** $\mathbf{V} \boldsymbol{\alpha}_2 = [0, 1]^T$
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Solution: $\alpha_{21} = 0$, $\alpha_{22} = 1$
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Therefore: $N_2(\xi) = 0 \cdot 1 + 1 \cdot \xi = \xi$ ✓
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**Verification:**
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- $N_1(0) = 1$, $N_1(1) = 0$ ✓
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- $N_2(0) = 0$, $N_2(1) = 1$ ✓
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- $N_1(\xi) + N_2(\xi) = 1$ (partition of unity) ✓
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## Polynomial Completeness
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The ansatz polynomial must be **complete** to the desired order:
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| Order | 1D | 2D | 3D | Nodes Required |
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|-------|----|----|-----|----------------|
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| Linear | $1 + \xi$ | $1 + \xi + \eta$ | $1 + \xi + \eta + \zeta$ | $d+1$ |
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| Quadratic | $1 + \xi + \xi^2$ | $1 + \xi + \eta + \xi^2 + \xi\eta + \eta^2$ | ... | $(d+1)(d+2)/2$ |
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**Example for 2D Triangle (Tri3):**
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Ansatz: $p(\xi, \eta) = 1 + \xi + \eta$ (complete linear in 2D)
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This is the **minimal** complete polynomial for 3 nodes.
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## Implementation in JuliaFEM
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### Automatic Generation Process
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```julia
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# 1. Define element geometry
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coords = [(0.0, 0.0), (1.0, 0.0), (0.0, 1.0)] # Tri3 nodes
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# 2. Define ansatz polynomial
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ansatz = :(1 + u + v) # Complete linear in 2D
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# 3. Build Vandermonde matrix
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V[i,j] = eval_polynomial_term(ansatz_terms[j], coords[i])
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# 4. For each node i:
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coeffs = V \ e_i # Solve linear system
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N_i = sum(coeffs[j] * ansatz_terms[j]) # Construct basis function
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# 5. Symbolic differentiation
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∂N_i/∂ξ = differentiate(N_i, :u)
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∂N_i/∂η = differentiate(N_i, :v)
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```
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### Why This Works
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1. **Completeness:** Ansatz spans full polynomial space of given order
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2. **Linear Independence:** Vandermonde matrix is non-singular for distinct nodes
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3. **Interpolation Property:** Follows directly from $\mathbf{V} \boldsymbol{\alpha}_i = \mathbf{e}_i$
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### Derivatives
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Once we have $N_i(\xi, \eta, \zeta)$ symbolically, derivatives are straightforward:
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$$\frac{\partial N_i}{\partial \xi}, \frac{\partial N_i}{\partial \eta}, \frac{\partial N_i}{\partial \zeta}$$
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These are computed **once** symbolically, then **pre-compiled** into efficient Julia code.
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## Standard Lagrange Elements in JuliaFEM
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### 1D Elements
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- **Seg2**: Linear (2 nodes)
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- **Seg3**: Quadratic (3 nodes, mid-edge node)
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### 2D Elements
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- **Tri3**: Linear triangle (3 corner nodes)
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- **Tri6**: Quadratic triangle (6 nodes: 3 corners + 3 mid-edges)
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- **Quad4**: Bilinear quadrilateral (4 corner nodes)
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- **Quad8**: Serendipity quadrilateral (8 nodes: 4 corners + 4 mid-edges)
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- **Quad9**: Biquadratic quadrilateral (9 nodes: 4 corners + 4 mid-edges + 1 center)
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### 3D Elements
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- **Tet4**: Linear tetrahedron (4 corner nodes)
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- **Tet10**: Quadratic tetrahedron (10 nodes: 4 corners + 6 mid-edges)
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- **Hex8**: Trilinear hexahedron (8 corner nodes)
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- **Hex20**: Serendipity hexahedron (20 nodes: 8 corners + 12 mid-edges)
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- **Hex27**: Triquadratic hexahedron (27 nodes: full tensor product)
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- **Pyr5**: Linear pyramid (5 nodes)
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- **Wedge6**: Linear wedge/prism (6 nodes)
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- **Wedge15**: Quadratic wedge (15 nodes)
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## Pre-Generation vs Runtime Generation
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### Historical Approach (JuliaFEM ≤ 0.5.1)
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```julia
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# At package load time:
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create_basis_and_eval(:Tet10, "...", coords, ansatz)
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# - Builds Vandermonde matrix
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# - Solves n linear systems
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# - Symbolic differentiation
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# - Simplification
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# - Code generation with eval()
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# Result: __precompile__(false) - slow loading
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```
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**Problems:**
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- ❌ Symbolic math every package load (100+ ms)
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- ❌ Cannot precompile (`eval()` at module scope)
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- ❌ Opaque code generation
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- ❌ Hard to debug
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### Modern Approach (JuliaFEM ≥ 1.0)
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```julia
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# Once, during development:
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scripts/generate_lagrange_basis.jl
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# - Computes all bases symbolically
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# - Writes clean Julia code to src/basis/lagrange_generated.jl
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# At package load time:
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include("basis/lagrange_generated.jl")
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# - Just parses pre-written Julia code
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# - Fully precompilable
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# - Zero symbolic computation
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```
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**Benefits:**
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- ✅ Instant package loading
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- ✅ Full precompilation
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- ✅ Readable generated code
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- ✅ Easy to debug
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- ✅ Version controlled (can review changes)
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## Numerical Stability
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### Vandermonde Matrix Conditioning
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The Vandermonde matrix can be ill-conditioned for:
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- High-order polynomials ($p > 5$)
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- Poorly distributed nodes
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- Reference elements far from unit cube/simplex
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**JuliaFEM's approach:**
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- Use canonical reference elements (unit cube $[-1,1]^d$ or unit simplex)
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- Lagrange elements rarely exceed order 3 in practice
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- For high-order: Consider hierarchical bases (not Lagrange)
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### Verification
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Generated basis functions are verified by:
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1. **Kronecker delta property:** $N_i(\mathbf{x}_j) = \delta_{ij}$
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2. **Partition of unity:** $\sum_i N_i(\mathbf{x}) = 1$ everywhere
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3. **Derivative correctness:** Compare symbolic vs AD
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See `test/test_basis_functions.jl` for comprehensive tests.
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## References
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1. Hughes, T.J.R., "The Finite Element Method: Linear Static and Dynamic Finite Element Analysis", Dover, 2000
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2. Zienkiewicz, O.C. and Taylor, R.L., "The Finite Element Method", Volumes 1-3, Butterworth-Heinemann, 2000
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3. Szabó, B. and Babuška, I., "Finite Element Analysis", Wiley, 1991
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## See Also
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- `scripts/generate_lagrange_basis.jl` - Generation script
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- `src/basis/lagrange_generated.jl` - Generated code (do not edit manually)
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- `src/basis/lagrange_generator.jl` - Generator functions (symbolic engine)
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- `benchmarks/tet10_derivatives_benchmark.jl` - Performance analysis (manual vs AD)
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# JuliaFEM Scripts
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This directory contains development and code generation scripts for JuliaFEM.
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## Basis Function Generation
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### `generate_lagrange_basis.jl`
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**Purpose:** Pre-generate all Lagrange basis functions for standard finite elements.
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**Why Pre-generate?**
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- **Fast loading:** No symbolic math at package load time (100+ ms → 0 ms)
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- **Full precompilation:** Remove `__precompile__(false)` restriction
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- **Readable code:** Generated code is easy to debug and understand
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- **Version control:** Changes to mathematics show up in git diffs
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- **Reproducible:** Same input always produces same output
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**When to Run:**
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- Adding new element types (Seg2, Tri3, Hex20, etc.)
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- Fixing bugs in generation logic
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- Changing polynomial ansatz strategy
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- After modifying `src/basis/lagrange_generator.jl`
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**Usage:**
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```bash
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cd /path/to/JuliaFEM.jl
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julia --project=. scripts/generate_lagrange_basis.jl
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```
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**Output:**
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- `src/basis/lagrange_generated.jl` (commit this file!)
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**Theory:**
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See `docs/theory/lagrange_basis_functions.md` for mathematical foundation.
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**Architecture:**
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```text
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src/basis/lagrange_generator.jl
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│
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│ (symbolic engine - uses symbolic differentiation)
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│
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↓
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scripts/generate_lagrange_basis.jl
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│
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│ (orchestration - defines all element types)
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│
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↓
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src/basis/lagrange_generated.jl
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│
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│ (clean Julia code - no eval, fully precompilable)
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│
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↓
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src/JuliaFEM.jl includes generated file
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```
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**Generated Elements:**
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| Dimension | Linear | Quadratic | Higher |
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|-----------|--------|-----------|--------|
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| 1D | Seg2 | Seg3 | - |
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| 2D Tri | Tri3 | Tri6 | - |
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| 2D Quad | Quad4 | Quad8, Quad9 | - |
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| 3D Tet | Tet4 | Tet10 | - |
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| 3D Hex | Hex8 | Hex20, Hex27 | - |
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| 3D Pyramid| Pyr5 | - | - |
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| 3D Wedge | Wedge6 | Wedge15 | - |
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**Total:** 15 element types covering all standard Lagrange families.
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**Performance Impact:**
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- **Before:** 150+ ms at package load (symbolic math for each element)
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- **After:** < 1 ms (just include pre-generated file)
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- **Speedup:** ~150× faster package loading
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**Workflow:**
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1. Edit element catalog in `scripts/generate_lagrange_basis.jl`
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2. Run generation script
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3. Review `src/basis/lagrange_generated.jl`
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4. Run tests: `julia --project=. -e 'using Pkg; Pkg.test()'`
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5. Commit both files: `git add scripts/ src/basis/lagrange_generated.jl`
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**Example**: Adding Hex64 (Triquartic)
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```julia
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# In scripts/generate_lagrange_basis.jl, add to element catalog:
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push!(elements, (
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name = "Hex64",
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description = "64-node triquartic hexahedral element",
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coordinates = [
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# ... 64 nodes (corners + edges + faces + volume)
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],
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ansatz = [
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:(1), :(ξ), :(η), :(ζ), # ... up to ξ³η³ζ³
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]
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))
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```
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Then regenerate:
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```bash
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julia --project=. scripts/generate_lagrange_basis.jl
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```
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The new `Hex64` type will be automatically available in JuliaFEM!
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---
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## Future Scripts (Planned)
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### `benchmark_suite.jl`
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Run comprehensive performance benchmarks.
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### `validate_against_reference.jl`
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Compare JuliaFEM results to Code Aster/ABAQUS.
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### `generate_element_matrices.jl`
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Pre-compute stiffness matrices for simple elements.
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---
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**See also:**
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- `docs/theory/lagrange_basis_functions.md` - Mathematical theory
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- `src/basis/lagrange_generator.jl` - Symbolic generation engine
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- `llm/VISION_2.0.md` - Overall project architecture
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Executable
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#!/usr/bin/env julia
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"""
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Analyze namespace collisions before consolidation
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Finds function definitions that might conflict when merging packages
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"""
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using Printf
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function find_function_definitions(dir::String)
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"""Find all function definitions in .jl files"""
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functions = Dict{String,Vector{String}}()
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for (root, dirs, files) in walkdir(dir)
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for file in files
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if !endswith(file, ".jl")
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continue
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end
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filepath = joinpath(root, file)
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relpath_str = relpath(filepath, dir)
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try
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content = read(filepath, String)
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lines = split(content, '\n')
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for (lineno, line) in enumerate(lines)
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# Match function definitions
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m = match(r"^\s*function\s+([a-zA-Z_][a-zA-Z0-9_!]*)", line)
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if m !== nothing
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fname = m.captures[1]
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location = "$relpath_str:$lineno"
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if !haskey(functions, fname)
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functions[fname] = String[]
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end
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push!(functions[fname], location)
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end
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# Match short-form function definitions
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m = match(r"^\s*([a-zA-Z_][a-zA-Z0-9_!]*)\([^)]*\)\s*=", line)
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if m !== nothing && !occursin("function", line)
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fname = m.captures[1]
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location = "$relpath_str:$lineno"
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if !haskey(functions, fname)
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functions[fname] = String[]
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end
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push!(functions[fname], location)
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end
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end
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catch e
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@warn "Error reading $filepath: $e"
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end
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end
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end
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return functions
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end
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function main()
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println("="^80)
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println("NAMESPACE COLLISION ANALYSIS - JuliaFEM Consolidation")
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println("="^80)
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println()
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# Analyze vendor packages
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vendor_dir = "/home/juajukka/dev/JuliaFEM.jl/vendor"
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src_dir = "/home/juajukka/dev/JuliaFEM.jl/src"
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println("Analyzing vendor packages...")
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vendor_functions = find_function_definitions(vendor_dir)
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println("Analyzing main src/...")
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src_functions = find_function_definitions(src_dir)
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||||
# Find collisions (functions defined in multiple places)
|
||||
println()
|
||||
println("="^80)
|
||||
println("POTENTIAL COLLISIONS (functions defined multiple times)")
|
||||
println("="^80)
|
||||
println()
|
||||
|
||||
collision_count = 0
|
||||
high_risk = String[]
|
||||
|
||||
# Check vendor collisions
|
||||
for (fname, locations) in sort(collect(vendor_functions), by=x -> length(x[2]), rev=true)
|
||||
if length(locations) > 1
|
||||
collision_count += 1
|
||||
|
||||
# Check if it's a common override pattern (likely safe)
|
||||
is_override = any(occursin(r"FEMBase\.|get_|assemble", fname) for loc in locations)
|
||||
|
||||
risk = is_override ? "🟢 LOW" : "🔴 HIGH"
|
||||
|
||||
if !is_override
|
||||
push!(high_risk, fname)
|
||||
end
|
||||
|
||||
println("$risk: $fname ($(length(locations)) definitions)")
|
||||
for loc in locations[1:min(5, length(locations))]
|
||||
println(" $loc")
|
||||
end
|
||||
if length(locations) > 5
|
||||
println(" ... and $(length(locations) - 5) more")
|
||||
end
|
||||
println()
|
||||
end
|
||||
end
|
||||
|
||||
println("="^80)
|
||||
println("SUMMARY")
|
||||
println("="^80)
|
||||
println()
|
||||
@printf "Total unique functions in vendor: %d\n" length(vendor_functions)
|
||||
@printf "Total unique functions in src: %d\n" length(src_functions)
|
||||
@printf "Functions defined multiple times: %d\n" collision_count
|
||||
@printf "High-risk collisions: %d\n" length(high_risk)
|
||||
println()
|
||||
|
||||
if length(high_risk) > 0
|
||||
println("High-risk collisions to review:")
|
||||
for fname in high_risk[1:min(10, length(high_risk))]
|
||||
println(" - $fname")
|
||||
end
|
||||
if length(high_risk) > 10
|
||||
println(" ... and $(length(high_risk) - 10) more")
|
||||
end
|
||||
else
|
||||
println("✅ No high-risk collisions detected!")
|
||||
println(" Most collisions are likely dispatch specializations (safe)")
|
||||
end
|
||||
|
||||
println()
|
||||
println("="^80)
|
||||
println("RECOMMENDATION")
|
||||
println("="^80)
|
||||
println("""
|
||||
Most function "collisions" in FEM packages are actually safe:
|
||||
- Different Problem{T} types dispatch correctly
|
||||
- get_* and assemble_* are intentional overrides
|
||||
|
||||
Review high-risk collisions manually before consolidation.
|
||||
""")
|
||||
end
|
||||
|
||||
main()
|
||||
Executable
+120
@@ -0,0 +1,120 @@
|
||||
#!/usr/bin/env python3
|
||||
"""
|
||||
Fix Element type signatures in vendor packages
|
||||
Converts Element{BasisType} → Element{M, BasisType} where M
|
||||
"""
|
||||
import re
|
||||
import sys
|
||||
from pathlib import Path
|
||||
|
||||
# Element types to fix
|
||||
ELEMENT_TYPES = [
|
||||
"Seg2",
|
||||
"Seg3",
|
||||
"Poi1",
|
||||
"Tri3",
|
||||
"Tri6",
|
||||
"Quad4",
|
||||
"Quad8",
|
||||
"Quad9",
|
||||
"Tet4",
|
||||
"Tet10",
|
||||
"Pyr5",
|
||||
"Wedge6",
|
||||
"Wedge15",
|
||||
"Hex8",
|
||||
"Hex20",
|
||||
"Hex27",
|
||||
]
|
||||
|
||||
|
||||
def fix_element_types(filepath):
|
||||
"""Fix Element type signatures in a single file."""
|
||||
with open(filepath, "r") as f:
|
||||
content = f.read()
|
||||
|
||||
original = content
|
||||
|
||||
# Pattern 1: Element{Type} in function parameters/returns
|
||||
for etype in ELEMENT_TYPES:
|
||||
# Fix ::Element{Type}
|
||||
content = re.sub(
|
||||
rf"::Element\{{{etype}\}}", rf"::Element{{M, {etype}}}", content
|
||||
)
|
||||
|
||||
# Fix Vector{Element{Type}}
|
||||
content = re.sub(
|
||||
rf"::Vector\{{Element\{{M, {etype}\}}\}}",
|
||||
rf"::Vector{{Element{{M, {etype}}}}}",
|
||||
content,
|
||||
)
|
||||
|
||||
# Pattern 2: Add "where M" to function signatures
|
||||
lines = content.split("\n")
|
||||
new_lines = []
|
||||
|
||||
for i, line in enumerate(lines):
|
||||
# Skip if already has where M
|
||||
if "where M" in line or "where {M" in line or "where E" in line:
|
||||
new_lines.append(line)
|
||||
continue
|
||||
|
||||
# Check if this is a function signature with Element{M,
|
||||
if re.search(r"function\s+.*Element\{M,", line):
|
||||
# Check if function signature closes on this line
|
||||
if ")" in line and "end" not in line:
|
||||
# Add where M before any trailing comment
|
||||
line = line.rstrip()
|
||||
if not line.endswith(" where M"):
|
||||
# Handle inline functions (one-liners)
|
||||
if "=" in line and line.count(")") == line.count("("):
|
||||
# function foo(...) = expr
|
||||
line = re.sub(r"\)(\s*=)", r") where M\1", line)
|
||||
else:
|
||||
line = line + " where M"
|
||||
|
||||
new_lines.append(line)
|
||||
|
||||
content = "\n".join(new_lines)
|
||||
|
||||
if content != original:
|
||||
with open(filepath, "w") as f:
|
||||
f.write(content)
|
||||
return True
|
||||
return False
|
||||
|
||||
|
||||
def main():
|
||||
vendor_dir = Path("/home/juajukka/dev/JuliaFEM.jl/vendor")
|
||||
fixed_count = 0
|
||||
file_count = 0
|
||||
|
||||
print("Fixing Element type signatures in vendor packages...")
|
||||
print("=" * 60)
|
||||
|
||||
# Find all .jl files in vendor subdirectories
|
||||
for package_dir in vendor_dir.iterdir():
|
||||
if not package_dir.is_dir():
|
||||
continue
|
||||
|
||||
src_dir = package_dir / "src"
|
||||
if not src_dir.exists():
|
||||
continue
|
||||
|
||||
for jl_file in src_dir.rglob("*.jl"):
|
||||
file_count += 1
|
||||
if fix_element_types(jl_file):
|
||||
print(f"✓ Fixed: {jl_file.relative_to(vendor_dir)}")
|
||||
fixed_count += 1
|
||||
|
||||
print("=" * 60)
|
||||
print(f"Processed {file_count} files")
|
||||
print(f"Fixed {fixed_count} files")
|
||||
|
||||
if fixed_count > 0:
|
||||
print("\n⚠️ IMPORTANT: Review changes before committing!")
|
||||
print("Run: git diff vendor/")
|
||||
|
||||
|
||||
if __name__ == "__main__":
|
||||
main()
|
||||
Executable
+711
@@ -0,0 +1,711 @@
|
||||
#!/usr/bin/env julia
|
||||
|
||||
# ==============================================================================
|
||||
# LAGRANGE BASIS FUNCTION GENERATION SCRIPT
|
||||
# ==============================================================================
|
||||
#
|
||||
# This script generates pre-computed Lagrange basis functions for all standard
|
||||
# finite element types and writes them to src/basis/lagrange_generated.jl
|
||||
#
|
||||
# USAGE:
|
||||
# cd /path/to/JuliaFEM.jl
|
||||
# julia --project=. scripts/generate_lagrange_basis.jl
|
||||
#
|
||||
# OUTPUT:
|
||||
# src/basis/lagrange_generated.jl (commit this to git!)
|
||||
#
|
||||
# WHEN TO RUN:
|
||||
# - Adding new element types
|
||||
# - Fixing bugs in generation logic
|
||||
# - Changing polynomial ansatz strategy
|
||||
# - After modifying lagrange_generator.jl
|
||||
#
|
||||
# THEORY:
|
||||
# See docs/theory/lagrange_basis_functions.md for full mathematical details
|
||||
#
|
||||
# ==============================================================================
|
||||
|
||||
using Pkg
|
||||
Pkg.activate(".")
|
||||
|
||||
using Dates # For timestamp in generated file header
|
||||
|
||||
println("="^80)
|
||||
println("LAGRANGE BASIS FUNCTION GENERATOR")
|
||||
println("="^80)
|
||||
println()
|
||||
|
||||
# Load the symbolic generation engine
|
||||
println("Loading symbolic generator...")
|
||||
include("../src/basis/lagrange_generator.jl")
|
||||
println("✓ Generator loaded")
|
||||
println()
|
||||
|
||||
# ==============================================================================
|
||||
# ELEMENT CATALOG
|
||||
# ==============================================================================
|
||||
#
|
||||
# For each standard Lagrange element, define:
|
||||
# 1. Name (e.g., "Seg2")
|
||||
# 2. Description
|
||||
# 3. Node coordinates in reference element
|
||||
# 4. Polynomial ansatz (monomial basis)
|
||||
#
|
||||
# Reference element conventions:
|
||||
# - 1D (Seg): u ∈ [-1, 1]
|
||||
# - 2D (Tri): (u, v) where u,v ≥ 0, u+v ≤ 1
|
||||
# - 2D (Quad): (u, v) ∈ [-1, 1]²
|
||||
# - 3D (Tet): (u, v, w) where u,v,w ≥ 0, u+v+w ≤ 1
|
||||
# - 3D (Hex): (u, v, w) ∈ [-1, 1]³
|
||||
#
|
||||
# NOTE: Variable names are u, v, w (not ξ, η, ζ)
|
||||
#
|
||||
# ==============================================================================
|
||||
|
||||
elements = []
|
||||
|
||||
# ------------------------------------------------------------------------------
|
||||
# 1D ELEMENTS
|
||||
# ------------------------------------------------------------------------------
|
||||
|
||||
# Seg2: 2-node linear segment
|
||||
push!(elements, (
|
||||
name = "Seg2",
|
||||
description = "2-node linear segment element",
|
||||
coordinates = [
|
||||
[-1.0], # Node 1
|
||||
[ 1.0] # Node 2
|
||||
],
|
||||
ansatz = [
|
||||
:(1), # 1
|
||||
:(u) # u
|
||||
]
|
||||
))
|
||||
|
||||
# Seg3: 3-node quadratic segment
|
||||
push!(elements, (
|
||||
name = "Seg3",
|
||||
description = "3-node quadratic segment element",
|
||||
coordinates = [
|
||||
[-1.0], # Node 1
|
||||
[ 1.0], # Node 2
|
||||
[ 0.0] # Node 3 (midpoint)
|
||||
],
|
||||
ansatz = [
|
||||
:(1), # 1
|
||||
:(u), # u
|
||||
:(u^2) # u²
|
||||
]
|
||||
))
|
||||
|
||||
# ------------------------------------------------------------------------------
|
||||
# 2D TRIANGULAR ELEMENTS
|
||||
# ------------------------------------------------------------------------------
|
||||
|
||||
# Tri3: 3-node linear triangle
|
||||
push!(elements, (
|
||||
name = "Tri3",
|
||||
description = "3-node linear triangular element",
|
||||
coordinates = [
|
||||
[0.0, 0.0], # Node 1
|
||||
[1.0, 0.0], # Node 2
|
||||
[0.0, 1.0] # Node 3
|
||||
],
|
||||
ansatz = [
|
||||
:(1), # 1
|
||||
:(u), # u
|
||||
:(v) # v
|
||||
]
|
||||
))
|
||||
|
||||
# Tri6: 6-node quadratic triangle
|
||||
push!(elements, (
|
||||
name = "Tri6",
|
||||
description = "6-node quadratic triangular element",
|
||||
coordinates = [
|
||||
[0.0, 0.0], # Node 1
|
||||
[1.0, 0.0], # Node 2
|
||||
[0.0, 1.0], # Node 3
|
||||
[0.5, 0.0], # Node 4 (edge 1-2)
|
||||
[0.5, 0.5], # Node 5 (edge 2-3)
|
||||
[0.0, 0.5] # Node 6 (edge 3-1)
|
||||
],
|
||||
ansatz = [
|
||||
:(1), # 1
|
||||
:(u), # u
|
||||
:(v), # v
|
||||
:(u^2), # u²
|
||||
:(u*v), # uv
|
||||
:(v^2) # v²
|
||||
]
|
||||
))
|
||||
|
||||
# Tri6: 6-node quadratic triangle
|
||||
push!(elements, (
|
||||
name="Tri6",
|
||||
description="6-node quadratic triangular element",
|
||||
coordinates=[
|
||||
[0.0, 0.0], # Node 1
|
||||
[1.0, 0.0], # Node 2
|
||||
[0.0, 1.0], # Node 3
|
||||
[0.5, 0.0], # Node 4 (edge 1-2)
|
||||
[0.5, 0.5], # Node 5 (edge 2-3)
|
||||
[0.0, 0.5] # Node 6 (edge 3-1)
|
||||
],
|
||||
ansatz=[
|
||||
:(1), # 1
|
||||
:( u), # ξ
|
||||
:( v), # η
|
||||
:(u^2), # ξ²
|
||||
:(ξ * η), # ξη
|
||||
:(v^2) # η²
|
||||
]
|
||||
))
|
||||
|
||||
# ------------------------------------------------------------------------------
|
||||
# 2D QUADRILATERAL ELEMENTS
|
||||
# ------------------------------------------------------------------------------
|
||||
|
||||
# Quad4: 4-node bilinear quadrilateral
|
||||
# Quad4: 4-node bilinear quadrilateral
|
||||
push!(elements, (
|
||||
name = "Quad4",
|
||||
description = "4-node bilinear quadrilateral element",
|
||||
coordinates = [
|
||||
[-1.0, -1.0], # Node 1
|
||||
[ 1.0, -1.0], # Node 2
|
||||
[ 1.0, 1.0], # Node 3
|
||||
[-1.0, 1.0] # Node 4
|
||||
],
|
||||
ansatz = [
|
||||
:(1), # 1
|
||||
:(u), # u
|
||||
:(v), # v
|
||||
:(u*v) # uv
|
||||
]
|
||||
))
|
||||
|
||||
# Quad8: 8-node serendipity quadrilateral (quadratic edges, no center node)
|
||||
push!(elements, (
|
||||
name = "Quad8",
|
||||
description = "8-node serendipity quadrilateral element",
|
||||
coordinates = [
|
||||
[-1.0, -1.0], # Node 1
|
||||
[ 1.0, -1.0], # Node 2
|
||||
[ 1.0, 1.0], # Node 3
|
||||
[-1.0, 1.0], # Node 4
|
||||
[ 0.0, -1.0], # Node 5 (edge 1-2)
|
||||
[ 1.0, 0.0], # Node 6 (edge 2-3)
|
||||
[ 0.0, 1.0], # Node 7 (edge 3-4)
|
||||
[-1.0, 0.0] # Node 8 (edge 4-1)
|
||||
],
|
||||
ansatz = [
|
||||
:(1), # 1
|
||||
:(u), # u
|
||||
:(v), # v
|
||||
:(u^2), # u²
|
||||
:(u*v), # uv
|
||||
:(v^2), # v²
|
||||
:(u^2*v), # u²v
|
||||
:(u*v^2) # uv²
|
||||
]
|
||||
))
|
||||
|
||||
# Quad9: 9-node biquadratic quadrilateral (complete quadratic)
|
||||
push!(elements, (
|
||||
name = "Quad9",
|
||||
description = "9-node biquadratic quadrilateral element",
|
||||
coordinates = [
|
||||
[-1.0, -1.0], # Node 1
|
||||
[ 1.0, -1.0], # Node 2
|
||||
[ 1.0, 1.0], # Node 3
|
||||
[-1.0, 1.0], # Node 4
|
||||
[ 0.0, -1.0], # Node 5 (edge 1-2)
|
||||
[ 1.0, 0.0], # Node 6 (edge 2-3)
|
||||
[ 0.0, 1.0], # Node 7 (edge 3-4)
|
||||
[-1.0, 0.0], # Node 8 (edge 4-1)
|
||||
[ 0.0, 0.0] # Node 9 (center)
|
||||
],
|
||||
ansatz = [
|
||||
:(1), # 1
|
||||
:(u), # u
|
||||
:(v), # v
|
||||
:(u^2), # u²
|
||||
:(u*v), # uv
|
||||
:(v^2), # v²
|
||||
:(u^2*v), # u²v
|
||||
:(u*v^2), # uv²
|
||||
:(u^2*v^2) # u²v²
|
||||
]
|
||||
))
|
||||
|
||||
# Quad8: 8-node serendipity quadrilateral (quadratic edges, no center node)
|
||||
push!(elements, (
|
||||
name="Quad8",
|
||||
description="8-node serendipity quadrilateral element",
|
||||
coordinates=[
|
||||
[-1.0, -1.0], # Node 1
|
||||
[1.0, -1.0], # Node 2
|
||||
[1.0, 1.0], # Node 3
|
||||
[-1.0, 1.0], # Node 4
|
||||
[0.0, -1.0], # Node 5 (edge 1-2)
|
||||
[1.0, 0.0], # Node 6 (edge 2-3)
|
||||
[0.0, 1.0], # Node 7 (edge 3-4)
|
||||
[-1.0, 0.0] # Node 8 (edge 4-1)
|
||||
],
|
||||
ansatz=[
|
||||
:(1), # 1
|
||||
:( u), # ξ
|
||||
:( v), # η
|
||||
:(u^2), # ξ²
|
||||
:(ξ * η), # ξη
|
||||
:(v^2), # η²
|
||||
:(u^2 * η), # ξ²η
|
||||
:(ξ * v^2) # ξη²
|
||||
]
|
||||
))
|
||||
|
||||
# Quad9: 9-node biquadratic quadrilateral (complete quadratic)
|
||||
push!(elements, (
|
||||
name="Quad9",
|
||||
description="9-node biquadratic quadrilateral element",
|
||||
coordinates=[
|
||||
[-1.0, -1.0], # Node 1
|
||||
[1.0, -1.0], # Node 2
|
||||
[1.0, 1.0], # Node 3
|
||||
[-1.0, 1.0], # Node 4
|
||||
[0.0, -1.0], # Node 5 (edge 1-2)
|
||||
[1.0, 0.0], # Node 6 (edge 2-3)
|
||||
[0.0, 1.0], # Node 7 (edge 3-4)
|
||||
[-1.0, 0.0], # Node 8 (edge 4-1)
|
||||
[0.0, 0.0] # Node 9 (center)
|
||||
],
|
||||
ansatz=[
|
||||
:(1), # 1
|
||||
:( u), # ξ
|
||||
:( v), # η
|
||||
:(u^2), # ξ²
|
||||
:(ξ * η), # ξη
|
||||
:(v^2), # η²
|
||||
:(u^2 * η), # ξ²η
|
||||
:(ξ * v^2), # ξη²
|
||||
:(u^2 * v^2) # ξ²η²
|
||||
]
|
||||
))
|
||||
|
||||
# ------------------------------------------------------------------------------
|
||||
# 3D TETRAHEDRAL ELEMENTS
|
||||
# ------------------------------------------------------------------------------
|
||||
|
||||
# Tet4: 4-node linear tetrahedron
|
||||
# Tet4: 4-node linear tetrahedron
|
||||
push!(elements, (
|
||||
name = "Tet4",
|
||||
description = "4-node linear tetrahedral element",
|
||||
coordinates = [
|
||||
[0.0, 0.0, 0.0], # Node 1
|
||||
[1.0, 0.0, 0.0], # Node 2
|
||||
[0.0, 1.0, 0.0], # Node 3
|
||||
[0.0, 0.0, 1.0] # Node 4
|
||||
],
|
||||
ansatz = [
|
||||
:(1), # 1
|
||||
:(u), # u
|
||||
:(v), # v
|
||||
:(w) # w
|
||||
]
|
||||
))
|
||||
|
||||
# Tet10: 10-node quadratic tetrahedron
|
||||
push!(elements, (
|
||||
name = "Tet10",
|
||||
description = "10-node quadratic tetrahedral element",
|
||||
coordinates = [
|
||||
[0.0, 0.0, 0.0], # Node 1
|
||||
[1.0, 0.0, 0.0], # Node 2
|
||||
[0.0, 1.0, 0.0], # Node 3
|
||||
[0.0, 0.0, 1.0], # Node 4
|
||||
[0.5, 0.0, 0.0], # Node 5 (edge 1-2)
|
||||
[0.5, 0.5, 0.0], # Node 6 (edge 2-3)
|
||||
[0.0, 0.5, 0.0], # Node 7 (edge 3-1)
|
||||
[0.0, 0.0, 0.5], # Node 8 (edge 1-4)
|
||||
[0.5, 0.0, 0.5], # Node 9 (edge 2-4)
|
||||
[0.0, 0.5, 0.5] # Node 10 (edge 3-4)
|
||||
],
|
||||
ansatz = [
|
||||
:(1), # 1
|
||||
:(u), # u
|
||||
:(v), # v
|
||||
:(w), # w
|
||||
:(u^2), # u²
|
||||
:(u*v), # uv
|
||||
:(u*w), # uw
|
||||
:(v^2), # v²
|
||||
:(v*w), # vw
|
||||
:(w^2) # w²
|
||||
]
|
||||
))
|
||||
|
||||
# Tet10: 10-node quadratic tetrahedron
|
||||
push!(elements, (
|
||||
name="Tet10",
|
||||
description="10-node quadratic tetrahedral element",
|
||||
coordinates=[
|
||||
[0.0, 0.0, 0.0], # Node 1
|
||||
[1.0, 0.0, 0.0], # Node 2
|
||||
[0.0, 1.0, 0.0], # Node 3
|
||||
[0.0, 0.0, 1.0], # Node 4
|
||||
[0.5, 0.0, 0.0], # Node 5 (edge 1-2)
|
||||
[0.5, 0.5, 0.0], # Node 6 (edge 2-3)
|
||||
[0.0, 0.5, 0.0], # Node 7 (edge 3-1)
|
||||
[0.0, 0.0, 0.5], # Node 8 (edge 1-4)
|
||||
[0.5, 0.0, 0.5], # Node 9 (edge 2-4)
|
||||
[0.0, 0.5, 0.5] # Node 10 (edge 3-4)
|
||||
],
|
||||
ansatz=[
|
||||
:(1), # 1
|
||||
:( u), # ξ
|
||||
:( v), # η
|
||||
:( w), # ζ
|
||||
:(u^2), # ξ²
|
||||
:(ξ * η), # ξη
|
||||
:(ξ * ζ), # ξζ
|
||||
:(v^2), # η²
|
||||
:(η * ζ), # ηζ
|
||||
:(w^2) # ζ²
|
||||
]
|
||||
))
|
||||
|
||||
# ------------------------------------------------------------------------------
|
||||
# 3D HEXAHEDRAL ELEMENTS
|
||||
# ------------------------------------------------------------------------------
|
||||
|
||||
# Hex8: 8-node trilinear hexahedron (brick)
|
||||
push!(elements, (
|
||||
name="Hex8",
|
||||
description="8-node trilinear hexahedral element",
|
||||
coordinates=[
|
||||
[-1.0, -1.0, -1.0], # Node 1
|
||||
[1.0, -1.0, -1.0], # Node 2
|
||||
[1.0, 1.0, -1.0], # Node 3
|
||||
[-1.0, 1.0, -1.0], # Node 4
|
||||
[-1.0, -1.0, 1.0], # Node 5
|
||||
[1.0, -1.0, 1.0], # Node 6
|
||||
[1.0, 1.0, 1.0], # Node 7
|
||||
[-1.0, 1.0, 1.0] # Node 8
|
||||
],
|
||||
ansatz=[
|
||||
:(1), # 1
|
||||
:( u), # ξ
|
||||
:( v), # η
|
||||
:( w), # ζ
|
||||
:(ξ * η), # ξη
|
||||
:(ξ * ζ), # ξζ
|
||||
:(η * ζ), # ηζ
|
||||
:(ξ * η * ζ) # ξηζ
|
||||
]
|
||||
))
|
||||
|
||||
# Hex20: 20-node serendipity hexahedron (quadratic edges, no face/center nodes)
|
||||
push!(elements, (
|
||||
name="Hex20",
|
||||
description="20-node serendipity hexahedral element",
|
||||
coordinates=[
|
||||
# Corner nodes
|
||||
[-1.0, -1.0, -1.0], # 1
|
||||
[1.0, -1.0, -1.0], # 2
|
||||
[1.0, 1.0, -1.0], # 3
|
||||
[-1.0, 1.0, -1.0], # 4
|
||||
[-1.0, -1.0, 1.0], # 5
|
||||
[1.0, -1.0, 1.0], # 6
|
||||
[1.0, 1.0, 1.0], # 7
|
||||
[-1.0, 1.0, 1.0], # 8
|
||||
# Edge nodes (bottom face)
|
||||
[0.0, -1.0, -1.0], # 9
|
||||
[1.0, 0.0, -1.0], # 10
|
||||
[0.0, 1.0, -1.0], # 11
|
||||
[-1.0, 0.0, -1.0], # 12
|
||||
# Edge nodes (top face)
|
||||
[0.0, -1.0, 1.0], # 13
|
||||
[1.0, 0.0, 1.0], # 14
|
||||
[0.0, 1.0, 1.0], # 15
|
||||
[-1.0, 0.0, 1.0], # 16
|
||||
# Edge nodes (vertical)
|
||||
[-1.0, -1.0, 0.0], # 17
|
||||
[1.0, -1.0, 0.0], # 18
|
||||
[1.0, 1.0, 0.0], # 19
|
||||
[-1.0, 1.0, 0.0] # 20
|
||||
],
|
||||
ansatz=[
|
||||
:(1), # 1
|
||||
:( u), :( v), :( w), # linear
|
||||
:(u^2), :(v^2), :(w^2), # pure quadratic
|
||||
:(ξ * η), :(ξ * ζ), :(η * ζ), # bilinear
|
||||
:(u^2 * η), :(u^2 * ζ), :(v^2 * ξ), :(v^2 * ζ), :(w^2 * ξ), :(w^2 * η), # quadratic-linear
|
||||
:(ξ * η * ζ), # trilinear
|
||||
:(u^2 * η * ζ), :(ξ * v^2 * ζ), :(ξ * η * w^2) # quadratic-bilinear
|
||||
]
|
||||
))
|
||||
|
||||
# Hex27: 27-node triquadratic hexahedron (complete quadratic)
|
||||
push!(elements, (
|
||||
name="Hex27",
|
||||
description="27-node triquadratic hexahedral element",
|
||||
coordinates=[
|
||||
# Corner nodes
|
||||
[-1.0, -1.0, -1.0], # 1
|
||||
[1.0, -1.0, -1.0], # 2
|
||||
[1.0, 1.0, -1.0], # 3
|
||||
[-1.0, 1.0, -1.0], # 4
|
||||
[-1.0, -1.0, 1.0], # 5
|
||||
[1.0, -1.0, 1.0], # 6
|
||||
[1.0, 1.0, 1.0], # 7
|
||||
[-1.0, 1.0, 1.0], # 8
|
||||
# Edge nodes (bottom face)
|
||||
[0.0, -1.0, -1.0], # 9
|
||||
[1.0, 0.0, -1.0], # 10
|
||||
[0.0, 1.0, -1.0], # 11
|
||||
[-1.0, 0.0, -1.0], # 12
|
||||
# Edge nodes (top face)
|
||||
[0.0, -1.0, 1.0], # 13
|
||||
[1.0, 0.0, 1.0], # 14
|
||||
[0.0, 1.0, 1.0], # 15
|
||||
[-1.0, 0.0, 1.0], # 16
|
||||
# Edge nodes (vertical)
|
||||
[-1.0, -1.0, 0.0], # 17
|
||||
[1.0, -1.0, 0.0], # 18
|
||||
[1.0, 1.0, 0.0], # 19
|
||||
[-1.0, 1.0, 0.0], # 20
|
||||
# Face center nodes
|
||||
[0.0, 0.0, -1.0], # 21 (bottom)
|
||||
[0.0, 0.0, 1.0], # 22 (top)
|
||||
[0.0, -1.0, 0.0], # 23 (front)
|
||||
[1.0, 0.0, 0.0], # 24 (right)
|
||||
[0.0, 1.0, 0.0], # 25 (back)
|
||||
[-1.0, 0.0, 0.0], # 26 (left)
|
||||
# Volume center node
|
||||
[0.0, 0.0, 0.0] # 27 (center)
|
||||
],
|
||||
ansatz=[
|
||||
:(1), # 1
|
||||
:( u), :( v), :( w), # linear (3)
|
||||
:(u^2), :(v^2), :(w^2), # pure quadratic (3)
|
||||
:(ξ * η), :(ξ * ζ), :(η * ζ), # bilinear (3)
|
||||
:(u^2 * η), :(u^2 * ζ), :(v^2 * ξ), :(v^2 * ζ), :(w^2 * ξ), :(w^2 * η), # quad-linear (6)
|
||||
:(ξ * η * ζ), # trilinear (1)
|
||||
:(u^2 * η * ζ), :(ξ * v^2 * ζ), :(ξ * η * w^2), # quad-bilinear (3)
|
||||
:(u^2 * v^2), :(u^2 * w^2), :(v^2 * w^2), # biquadratic (3)
|
||||
:(u^2 * v^2 * ζ), :(u^2 * η * w^2), :(ξ * v^2 * w^2), # biquad-linear (3)
|
||||
:(u^2 * v^2 * w^2) # triquadratic (1)
|
||||
]
|
||||
))
|
||||
|
||||
# ------------------------------------------------------------------------------
|
||||
# 3D PYRAMID ELEMENTS
|
||||
# ------------------------------------------------------------------------------
|
||||
|
||||
# Pyr5: 5-node linear pyramid
|
||||
push!(elements, (
|
||||
name="Pyr5",
|
||||
description="5-node linear pyramid element",
|
||||
coordinates=[
|
||||
[-1.0, -1.0, 0.0], # Node 1 (base)
|
||||
[1.0, -1.0, 0.0], # Node 2 (base)
|
||||
[1.0, 1.0, 0.0], # Node 3 (base)
|
||||
[-1.0, 1.0, 0.0], # Node 4 (base)
|
||||
[0.0, 0.0, 1.0] # Node 5 (apex)
|
||||
],
|
||||
ansatz=[
|
||||
:(1), # 1
|
||||
:( u), # ξ
|
||||
:( v), # η
|
||||
:( w), # ζ
|
||||
:(ξ * η) # ξη (needed for symmetry)
|
||||
]
|
||||
))
|
||||
|
||||
# ------------------------------------------------------------------------------
|
||||
# 3D WEDGE ELEMENTS (Prism/Pentahedron)
|
||||
# ------------------------------------------------------------------------------
|
||||
|
||||
# Wedge6: 6-node linear wedge (triangular prism)
|
||||
push!(elements, (
|
||||
name="Wedge6",
|
||||
description="6-node linear wedge element (triangular prism)",
|
||||
coordinates=[
|
||||
[0.0, 0.0, -1.0], # Node 1 (bottom triangle)
|
||||
[1.0, 0.0, -1.0], # Node 2 (bottom triangle)
|
||||
[0.0, 1.0, -1.0], # Node 3 (bottom triangle)
|
||||
[0.0, 0.0, 1.0], # Node 4 (top triangle)
|
||||
[1.0, 0.0, 1.0], # Node 5 (top triangle)
|
||||
[0.0, 1.0, 1.0] # Node 6 (top triangle)
|
||||
],
|
||||
ansatz=[
|
||||
:(1), # 1
|
||||
:( u), # ξ
|
||||
:( v), # η
|
||||
:( w), # ζ
|
||||
:(ξ * ζ), # ξζ
|
||||
:(η * ζ) # ηζ
|
||||
]
|
||||
))
|
||||
|
||||
# Wedge15: 15-node quadratic wedge
|
||||
push!(elements, (
|
||||
name="Wedge15",
|
||||
description="15-node quadratic wedge element",
|
||||
coordinates=[
|
||||
# Bottom triangle
|
||||
[0.0, 0.0, -1.0], # 1
|
||||
[1.0, 0.0, -1.0], # 2
|
||||
[0.0, 1.0, -1.0], # 3
|
||||
# Top triangle
|
||||
[0.0, 0.0, 1.0], # 4
|
||||
[1.0, 0.0, 1.0], # 5
|
||||
[0.0, 1.0, 1.0], # 6
|
||||
# Mid-edge nodes (bottom triangle)
|
||||
[0.5, 0.0, -1.0], # 7
|
||||
[0.5, 0.5, -1.0], # 8
|
||||
[0.0, 0.5, -1.0], # 9
|
||||
# Mid-edge nodes (top triangle)
|
||||
[0.5, 0.0, 1.0], # 10
|
||||
[0.5, 0.5, 1.0], # 11
|
||||
[0.0, 0.5, 1.0], # 12
|
||||
# Mid-edge nodes (vertical)
|
||||
[0.0, 0.0, 0.0], # 13
|
||||
[1.0, 0.0, 0.0], # 14
|
||||
[0.0, 1.0, 0.0] # 15
|
||||
],
|
||||
ansatz=[
|
||||
:(1), # 1
|
||||
:( u), :( v), :( w), # linear (3)
|
||||
:(u^2), :(v^2), :(w^2), # pure quadratic (3)
|
||||
:(ξ * η), # triangle bilinear (1)
|
||||
:(ξ * ζ), :(η * ζ), # prism bilinear (2)
|
||||
:(u^2 * ζ), :(v^2 * ζ), :(ξ * η * ζ), # mixed (3)
|
||||
:(ξ * w^2), :(η * w^2) # mixed (2)
|
||||
]
|
||||
))
|
||||
|
||||
println("Element catalog loaded: $(length(elements)) element types")
|
||||
println()
|
||||
|
||||
# ==============================================================================
|
||||
# GENERATION LOOP
|
||||
# ==============================================================================
|
||||
|
||||
println("Generating basis functions...")
|
||||
println("─"^80)
|
||||
|
||||
output = IOBuffer()
|
||||
|
||||
# File header
|
||||
println(output, "# This file is a part of JuliaFEM.")
|
||||
println(output, "# License is MIT: see https://github.com/JuliaFEM/JuliaFEM.jl/blob/master/LICENSE")
|
||||
println(output)
|
||||
println(output, "# ============================================================================")
|
||||
println(output, "# AUTO-GENERATED LAGRANGE BASIS FUNCTIONS")
|
||||
println(output, "# ============================================================================")
|
||||
println(output, "#")
|
||||
println(output, "# WARNING: DO NOT EDIT THIS FILE MANUALLY!")
|
||||
println(output, "#")
|
||||
println(output, "# This file was automatically generated by:")
|
||||
println(output, "# scripts/generate_lagrange_basis.jl")
|
||||
println(output, "#")
|
||||
println(output, "# To regenerate (e.g., after adding new element types):")
|
||||
println(output, "# cd /path/to/JuliaFEM.jl")
|
||||
println(output, "# julia --project=. scripts/generate_lagrange_basis.jl")
|
||||
println(output, "#")
|
||||
println(output, "# Theory:")
|
||||
println(output, "# See docs/theory/lagrange_basis_functions.md")
|
||||
println(output, "#")
|
||||
println(output, "# Generator:")
|
||||
println(output, "# src/basis/lagrange_generator.jl (symbolic engine)")
|
||||
println(output, "#")
|
||||
println(output, "# Generated: $(Dates.format(now(), "yyyy-mm-dd HH:MM:SS"))")
|
||||
println(output, "# ============================================================================")
|
||||
println(output)
|
||||
|
||||
for (i, elem) in enumerate(elements)
|
||||
println("[$i/$(length(elements))] Generating $(elem.name)...")
|
||||
|
||||
try
|
||||
# Convert coordinates to proper format (tuples, not vectors)
|
||||
coords = [tuple(coord...) for coord in elem.coordinates]
|
||||
|
||||
# Build polynomial ansatz as single expression: term1 + term2 + ...
|
||||
# The ansatz is a vector of terms [:1, :ξ, :η, ...] → combine with +
|
||||
if length(elem.ansatz) == 1
|
||||
ansatz_expr = elem.ansatz[1]
|
||||
else
|
||||
ansatz_expr = Expr(:call, :+, elem.ansatz...)
|
||||
end
|
||||
|
||||
# Generate basis code using symbolic engine
|
||||
# Returns an Expr (quoted code)
|
||||
basis_code_expr = create_basis(
|
||||
Symbol(elem.name),
|
||||
elem.description,
|
||||
coords,
|
||||
ansatz_expr
|
||||
)
|
||||
|
||||
# Convert Expr to readable Julia code string
|
||||
basis_code_str = string(basis_code_expr)
|
||||
|
||||
# Pretty-print: The generated code is in a quote block, extract inner code
|
||||
# Handle both `quote ... end` and `begin ... end` forms
|
||||
basis_code_str = replace(basis_code_str, r"^(quote|begin)\s+" => "")
|
||||
basis_code_str = replace(basis_code_str, r"\s*end$" => "")
|
||||
|
||||
# Write to output with nice formatting
|
||||
println(output, "# " * "─"^78)
|
||||
println(output, "# $(elem.name): $(elem.description)")
|
||||
println(output, "# " * "─"^78)
|
||||
println(output)
|
||||
println(output, basis_code_str)
|
||||
println(output)
|
||||
|
||||
catch e
|
||||
println(" ⚠ Error generating $(elem.name):")
|
||||
println(" $e")
|
||||
if isa(e, ErrorException)
|
||||
# Print stacktrace for debugging
|
||||
for (exc, bt) in Base.catch_stack()
|
||||
showerror(stdout, exc, bt)
|
||||
println()
|
||||
end
|
||||
end
|
||||
println(" Skipping...")
|
||||
end
|
||||
end
|
||||
|
||||
println("─"^80)
|
||||
println()
|
||||
|
||||
# ==============================================================================
|
||||
# WRITE OUTPUT FILE
|
||||
# ==============================================================================
|
||||
|
||||
output_path = joinpath(@__DIR__, "..", "src", "basis", "lagrange_generated.jl")
|
||||
println("Writing to: $output_path")
|
||||
|
||||
output_content = String(take!(output))
|
||||
write(output_path, output_content)
|
||||
|
||||
println("✓ Generation complete!")
|
||||
println()
|
||||
println("Generated $(length(elements)) element types:")
|
||||
for elem in elements
|
||||
println(" - $(elem.name): $(elem.description)")
|
||||
end
|
||||
println()
|
||||
println("Next steps:")
|
||||
println(" 1. Review: src/basis/lagrange_generated.jl")
|
||||
println(" 2. Update module to include this file")
|
||||
println(" 3. Remove __precompile__(false) from main module")
|
||||
println(" 4. Test: julia --project=. -e 'using JuliaFEM'")
|
||||
println(" 5. Run tests: julia --project=. -e 'using Pkg; Pkg.test()'")
|
||||
println(" 6. Commit: git add src/basis/lagrange_generated.jl && git commit")
|
||||
println()
|
||||
println("="^80)
|
||||
+3
-1
@@ -137,7 +137,9 @@ end
|
||||
include("basis/abstract.jl")
|
||||
include("basis/subs.jl") # Symbolic substitution (includes minimal simplify from SymDiff.jl)
|
||||
include("basis/vandermonde.jl")
|
||||
include("basis/create_basis.jl") # Basis generation (includes minimal differentiate from SymDiff.jl)
|
||||
# NOTE: lagrange_generator.jl is NOT included here - it's a tool, not runtime code!
|
||||
# It's only loaded by scripts/generate_lagrange_basis.jl during pre-generation.
|
||||
# The generated code is in lagrange_generated.jl (to be created).
|
||||
include("basis/lagrange_segments.jl")
|
||||
include("basis/lagrange_quadrangles.jl")
|
||||
include("basis/lagrange_triangles.jl")
|
||||
|
||||
@@ -1,7 +1,57 @@
|
||||
# This file is a part of JuliaFEM.
|
||||
# License is MIT: see https://github.com/JuliaFEM/jl/blob/master/LICENSE
|
||||
# License is MIT: see https://github.com/JuliaFEM/JuliaFEM.jl/blob/master/LICENSE
|
||||
|
||||
__precompile__(false)
|
||||
# ==============================================================================
|
||||
# LAGRANGE BASIS FUNCTION GENERATOR
|
||||
# ==============================================================================
|
||||
#
|
||||
# This file contains the symbolic engine for generating Lagrange basis functions.
|
||||
# It is NOT loaded at runtime - it's a TOOL used during development.
|
||||
#
|
||||
# PURPOSE:
|
||||
# Generate pre-computed basis functions and derivatives for all standard
|
||||
# Lagrange finite elements (Seg2, Tri3, Quad4, Tet10, Hex8, etc.)
|
||||
#
|
||||
# THEORY:
|
||||
# Lagrange basis functions satisfy the Kronecker delta property:
|
||||
#
|
||||
# N_i(x_j) = δ_ij = { 1 if i = j
|
||||
# { 0 if i ≠ j
|
||||
#
|
||||
# Given:
|
||||
# - n nodes with coordinates {x₁, x₂, ..., xₙ} in reference element
|
||||
# - Polynomial ansatz {p₁(x), p₂(x), ..., pₙ(x)} (complete to order k)
|
||||
#
|
||||
# We construct: N_i(x) = Σⱼ αᵢⱼ pⱼ(x)
|
||||
#
|
||||
# The Kronecker property gives: V α_i = e_i
|
||||
#
|
||||
# Where Vandermonde matrix: V_kj = pⱼ(x_k)
|
||||
#
|
||||
# Solving these n systems gives all basis functions explicitly.
|
||||
# Then symbolic differentiation provides derivatives.
|
||||
#
|
||||
# USAGE:
|
||||
# This file is loaded by scripts/generate_lagrange_basis.jl which:
|
||||
# 1. Defines all standard element types (coords + polynomial ansatz)
|
||||
# 2. Calls generate_lagrange_basis() for each
|
||||
# 3. Writes clean Julia code to src/basis/lagrange_generated.jl
|
||||
#
|
||||
# WHY GENERATE ONCE?
|
||||
# - Symbolic math is expensive (100+ ms per element type)
|
||||
# - Generated code is constant (mathematics doesn't change!)
|
||||
# - Pre-compilation is much faster
|
||||
# - Generated code is readable and debuggable
|
||||
# - Version control shows what changed
|
||||
#
|
||||
# SEE:
|
||||
# - docs/theory/lagrange_basis_functions.md (mathematical explanation)
|
||||
# - scripts/generate_lagrange_basis.jl (generation script)
|
||||
# - src/basis/lagrange_generated.jl (output - do not edit manually!)
|
||||
#
|
||||
# ==============================================================================
|
||||
|
||||
__precompile__(false) # This is a tool, not runtime code
|
||||
|
||||
# Minimal symbolic differentiation for polynomial basis functions
|
||||
# Adapted from SymDiff.jl by Jukka Aho - zero dependencies!
|
||||
Reference in New Issue
Block a user