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docs: Reorganize documentation into three-tier structure
**Three Manuals for Three Audiences:** 1. **User Manual** (docs/user/) - "Just Get It Done" - For end users, engineers, students - Simple, practical, step-by-step - Quick start, tutorials, examples, troubleshooting - Philosophy: Show me how to solve my problem 2. **Contributor Manual** (docs/contributor/) - "Show Me the Code" - For developers, contributors, advanced users - Technical, detailed, design rationale - Testing, architecture, performance, CI/CD - Philosophy: Explain HOW and WHY 3. **The JuliaFEM Book** (docs/book/) - "Let Me Show You How I Think" - For researchers, theory nerds, and Jukka - Comprehensive, educational, opinionated, personal - Math foundations, design philosophy, history, research - Philosophy: Mix theory, code, and personal experience **Reorganization:** - Moved: TESTING_PHILOSOPHY.md → contributor/testing_philosophy.md - Moved: STATUS.md → contributor/status.md - Moved: TEST_FIXES_NEEDED.md → contributor/test_fixes_needed.md - Moved: lagrange_basis_functions.md → book/lagrange_basis_functions.md - Moved: benchmarks/ → book/benchmarks/ - Created: docs/README.md (main index explaining structure) - Created: README.md in each section explaining audience and contents - Updated: All references in scripts and source files **Naming:** All docs now lowercase (testing_philosophy not TESTING_PHILOSOPHY) **Benefits:** - Clear separation of concerns - Users don't get overwhelmed with implementation details - Contributors get technical depth - Book preserves deep theory and personal insights - Each manual optimized for its audience **Next:** Populate each section with appropriate content
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# JuliaFEM Documentation
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Welcome! JuliaFEM documentation is organized into **three manuals** for three different audiences:
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---
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## 📘 [User Manual](user/) - "Just Get It Done"
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**For:** End users, engineers, students who want to run simulations.
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**Style:** Simple, practical, step-by-step.
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**Contents:**
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- Quick start and installation
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- Tutorials and examples
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- API reference
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- Troubleshooting
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**Philosophy:** Show me how to solve my problem, skip the lectures.
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👉 **[Start Here](user/README.md)** if you want to run simulations.
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---
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## 🔧 [Contributor Manual](contributor/) - "Show Me the Code"
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**For:** Developers, contributors, advanced users who want to extend JuliaFEM.
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**Style:** Technical, detailed, design rationale.
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**Contents:**
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- Testing philosophy
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- Code style and architecture
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- Performance guidelines
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- How to add elements
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- CI/CD and git workflow
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**Philosophy:** Explain HOW the code works and WHY we made these choices.
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👉 **[Start Here](contributor/README.md)** if you want to contribute code.
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---
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## 📖 [The JuliaFEM Book](book/) - "Let Me Show You How I Think"
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**For:** Advanced researchers, theory nerds, those who want to understand deeply. And Jukka.
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**Style:** Comprehensive, educational, opinionated, personal.
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**Contents:**
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- Mathematical foundations (Lagrange basis, contact mechanics, etc.)
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- Design philosophy and technical vision
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- Strategic mistakes and lessons learned (2015-2019)
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- Research directions (nodal assembly, matrix-free, etc.)
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- Personal reflections on the journey
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**Philosophy:** Mix theory, software design, and personal experience. Teach FEM through implementation.
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👉 **[Start Here](book/README.md)** if you love deep dives and want to understand the "why" behind everything.
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---
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## Quick Navigation
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### I want to...
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- **Solve a heat transfer problem** → [User Manual](user/)
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- **Add a new element type** → [Contributor Manual](contributor/)
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- **Understand Lagrange basis functions** → [Book: Lagrange Basis](book/lagrange_basis_functions.md)
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- **Learn about testing** → [Contributor: Testing Philosophy](contributor/testing_philosophy.md)
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- **See benchmark results** → [Book: Benchmarks](book/benchmarks/)
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- **Understand the design philosophy** → [Book: Philosophy](book/)
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- **Report a bug** → GitHub Issues
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- **Ask a question** → GitHub Discussions
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---
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## Documentation Philosophy
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### Why Three Manuals?
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Different readers have different needs:
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1. **Users** don't care about implementation details - they just want working code.
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2. **Contributors** need technical depth but not necessarily all the theory.
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3. **Researchers** (and Jukka) want to understand everything from first principles.
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Mixing these audiences in one manual makes it too complex for users and too shallow for researchers.
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### Design Principles
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- **User Manual:** Optimize for time-to-first-result
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- **Contributor Manual:** Optimize for correctness and maintainability
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- **Book:** Optimize for understanding and education
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### Cross-References
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Manuals link to each other when appropriate:
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- User manual links to theory when deeper understanding helps
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- Contributor manual links to book for design rationale
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- Book links to code examples and practical guides
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---
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## Contributing to Documentation
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Documentation improvements are always welcome!
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- **User docs:** Fix errors, add examples, improve clarity
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- **Contributor docs:** Update for new features, clarify architecture
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- **Book:** Add theory, share insights, document research
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See [Contributor Manual](contributor/) for guidelines.
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---
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**License:** MIT (same as code)
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**Questions?** Open an issue or discussion on GitHub
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# The JuliaFEM Book
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**Audience:** Advanced researchers, theory nerds, those who want to understand the "why" and "how" at a deep level. And Jukka.
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This is the **JuliaFEM Bible** - a comprehensive manual mixing theory, philosophy, software design, and personal experience. It's educational, opinionated, and unapologetically deep.
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## What's Here
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- **Mathematical Foundations:** Lagrange basis functions, weak forms, contact mechanics
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- **Design Philosophy:** Why JuliaFEM exists, what problems it solves (and doesn't)
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- **Technical Vision:** Strategic mistakes from 2015-2019, lessons learned
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- **Research Directions:** Experimental ideas (nodal assembly, matrix-free, etc.)
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- **Personal Notes:** The journey, the failures, the "aha!" moments
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- **Theory + Code:** How mathematics becomes software
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## What's NOT Here
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- "How do I install?" (see `docs/user/`)
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- "How do I add a feature?" (see `docs/contributor/`)
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- Short answers (everything here is DEEP)
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## Philosophy
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**"Let me show you how I think about FEM."**
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This is:
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- **Educational:** Teach FEM through implementation
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- **Personal:** Written in Jukka's voice, reflecting 8+ years of experience
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- **Opinionated:** Strong views on what works and what doesn't
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- **Comprehensive:** From first principles to cutting-edge research
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- **Honest:** Documents failures as much as successes
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We assume you:
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- Love mathematics AND programming
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- Want to understand WHY, not just HOW
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- Have time to read deeply
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- Are curious about unconventional approaches
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- Might be me, 5 years from now, trying to remember why I did this
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## Structure
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### Part I: Foundations
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- Finite Element Method (brief review)
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- Lagrange Basis Functions (deep dive)
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- Assembly and Solving
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- Contact Mechanics
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### Part II: Software Design
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- Type Stability and Performance
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- Zero-Allocation Design
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- Immutability and Composition
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- Field System Architecture
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### Part III: History and Vision
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- Strategic Mistakes (2015-2019)
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- Why JuliaFEM is Different
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- Contact Mechanics Focus
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- Laboratory Philosophy
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### Part IV: Research
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- Nodal Assembly (experimental)
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- Matrix-Free Methods
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- Automatic Differentiation
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- GPU Acceleration
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### Part V: The Journey
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- Personal Reflections
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- Lessons Learned
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- Future Directions
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- Open Questions
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## Reading Guide
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- **For Theory:** Start with Part I
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- **For Design Rationale:** Start with Part II
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- **For History:** Start with Part III
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- **For Research Ideas:** Start with Part IV
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- **For Philosophy:** Read Part V first, then everything else
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---
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**Start here:** [Mathematical Foundations](foundations.md) | [Strategic Mistakes](strategic_mistakes.md) | [Why JuliaFEM?](philosophy.md)
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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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|
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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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|
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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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|
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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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|
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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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@@ -0,0 +1,53 @@
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# JuliaFEM Contributor Manual
|
||||
|
||||
**Audience:** Developers, contributors, advanced users who want to extend or modify JuliaFEM.
|
||||
|
||||
This manual is **technical and detailed** - it explains HOW the code works and WHY we made certain design choices.
|
||||
|
||||
## What's Here
|
||||
|
||||
- **Testing Philosophy:** How and why we test
|
||||
- **Code Style:** Conventions and best practices
|
||||
- **Architecture:** Module structure, data flow, key abstractions
|
||||
- **Performance:** Zero-allocation design, profiling, benchmarking
|
||||
- **Adding Elements:** How to implement new element types
|
||||
- **CI/CD:** Continuous integration, releases, versioning
|
||||
- **Git Workflow:** Branching, commits, pull requests
|
||||
|
||||
## What's NOT Here
|
||||
|
||||
- User tutorials (see `docs/user/` for that)
|
||||
- Deep mathematical theory (see `docs/book/` for that)
|
||||
- "How do I solve problem X?" (that's user docs)
|
||||
|
||||
## Philosophy
|
||||
|
||||
**"Show me the code AND tell me why."**
|
||||
|
||||
We assume you:
|
||||
- Know Julia reasonably well
|
||||
- Understand FEM basics
|
||||
- Want to add features or fix bugs
|
||||
- Care about performance and correctness
|
||||
- Need to understand design rationale
|
||||
|
||||
## Before Contributing
|
||||
|
||||
1. Read [Testing Philosophy](testing_philosophy.md)
|
||||
2. Understand [Architecture](architecture.md)
|
||||
3. Follow [Code Style](code_style.md)
|
||||
4. Check [Performance Guidelines](performance.md)
|
||||
5. Review [Git Workflow](git_workflow.md)
|
||||
|
||||
## Key Principles
|
||||
|
||||
- **Type stability:** No `Any`, no `Dict` without types
|
||||
- **Zero allocations:** Hot paths should allocate nothing
|
||||
- **Immutability:** Prefer `struct` over `mutable struct`
|
||||
- **Composition:** Use tuples and free functions, not OOP hierarchies
|
||||
- **Explicit:** No magic, user knows what happens
|
||||
- **Test first:** Write tests before fixing bugs
|
||||
|
||||
---
|
||||
|
||||
**Start here:** [Testing Philosophy](testing_philosophy.md) | [Architecture Overview](architecture.md)
|
||||
@@ -0,0 +1,40 @@
|
||||
# JuliaFEM User Manual
|
||||
|
||||
**Audience:** End users, engineers, students who want to run simulations and get results.
|
||||
|
||||
This manual is designed to be **simple and practical** - get you from zero to running simulations as quickly as possible.
|
||||
|
||||
## What's Here
|
||||
|
||||
- **Quick Start:** Installation and first simulation in 5 minutes
|
||||
- **Tutorials:** Step-by-step guides for common problems
|
||||
- **Examples:** Pre-built simulations you can run and modify
|
||||
- **API Reference:** Function documentation (what does this do?)
|
||||
- **Troubleshooting:** Common errors and how to fix them
|
||||
|
||||
## What's NOT Here
|
||||
|
||||
- Deep theory (see `docs/book/` for that)
|
||||
- How to contribute code (see `docs/contributor/` for that)
|
||||
- Internal architecture details
|
||||
|
||||
## Philosophy
|
||||
|
||||
**"Just show me how to solve my problem."**
|
||||
|
||||
We assume you:
|
||||
- Have a problem to solve (heat transfer, elasticity, contact)
|
||||
- Want working code, not lectures
|
||||
- Will read theory when YOU need it, not when WE think you should
|
||||
|
||||
## Getting Help
|
||||
|
||||
1. Start with the Quick Start
|
||||
2. Find an example similar to your problem
|
||||
3. Modify it to fit your needs
|
||||
4. If stuck, check Troubleshooting
|
||||
5. Still stuck? Ask on GitHub Discussions
|
||||
|
||||
---
|
||||
|
||||
**Next:** Start with [Quick Start](quickstart.md) or browse [Examples](../examples/)
|
||||
Reference in New Issue
Block a user