layout: true class: typo, typo-selection --- count: false class: nord-dark, middle, center # **Discrete Exterior Calculus** @luk036 👨‍💻 · 2026 📅 --- ### 📐 Discrete Exterior Calculus (DEC) ✨ .mermaid[ <pre> graph LR A[Continuous Exterior Calculus] -->|Discretize| B[DEC] B --> C[Simplicial Meshes] B --> D[Differential Forms] B --> E[Structure Preservation] style A fill:#f9f,stroke:#333 style B fill:#bbf,stroke:#333 style C fill:#f96,stroke:#333 style D fill:#6f9,stroke:#333 style E fill:#9cf,stroke:#333 </pre> ] * **What is DEC?** 🤔💡 * A framework for computational modeling. ⚙️ * Based on differential forms and simplicial meshes. 📊 * Extends continuous exterior calculus to a discrete setting. 🔄 * Goal: Preserve geometric and physical structure in numerical simulations. 🎯 --- ### 🌌 Exterior Calculus Basics 🔍 .mermaid[ <pre> pie title Differential Forms "0-form (Scalar)" : 30 "1-form (Line Integral)" : 30 "2-form (Surface Integral)" : 20 "3-form (Volume Integral)" : 20 </pre> ] * Exterior calculus deals with **differential forms (k-forms)**. 📜 * They are related to quantities integrated over geometric objects. 📏 * **0-form:** Like a scalar function (e.g., elevation at a point). 📍 * **1-form:** Like a quantity integrated along a curve (e.g., elevation *gain* along a path). 🛤️ * Closely related to vector calculus operations. 🔄 * Exterior derivative of a 1-form looks like the curl of a vector field. 🌪️ --- ### 🤖 DEC on Simplicial Complexes 🏗️ .mermaid[ <pre> graph LR V[Vertex - 0-form] -->|d0| E[Edge - 1-form] E -->|d1| F[Face - 2-form] F -->|d2| V3[Volume - 3-form] style V fill:#f9f,stroke:#333 style E fill:#6f9,stroke:#333 style F fill:#f96,stroke:#333 style V3 fill:#9cf,stroke:#333 </pre> ] * DEC operates on **simplicial complexes** (meshes) and their associated **dual meshes**. 🔲 * A **discrete k-form** represents the integral of a continuous k-form over a k-dimensional mesh object (primal or dual). 📊 * Discrete **0-form**: Scalar value at a **vertex**. ⚫ * Discrete **1-form**: Value associated with an **edge** (integral along an edge). ➖ * Discrete **2-form** (in 2D): Value associated with a **triangle**. 🔺 * Discrete variables require specifying an **orientation** for each mesh element, just like elevation gain needs direction (+151m vs -151m). 🔄 * Discrete differential forms are stored with both a number and the specified orientation. 🔢 --- ### ✨ The Core Trio: d, ★, ∧ * **Exterior Derivative (d)**: Generalizes differentiation. Maps k-forms to (k+1)-forms. 🔄 * `d0` maps values on vertices (0-forms) to values on edges (1-forms). ↗️ * Requires taking orientation into account. 🔄 * In continuous calculus, `d` on 1-forms looks like curl. 🌪️ * A defining property: `d` applied twice is zero (`dp+1dp = 0`). ❌ * **Hodge Star Operator (★)**: 💫🌟 Relates k-forms to (N-k)-forms in N dimensions. Incorporates metric information. Often transfers forms between primal and dual complexes. A Delaunay Hodge star exists. ⭐ * **Wedge Product (∧)**: 🌱✖️ Multiplies forms. The discrete definition involves integration. For instance, a 1-form wedge a 0-form results in a 1-form. ➗ --- ### 👍 Advantages for Computation 🚀 * **Structure Preservation** 🏛️🔒: DEC discretizations aim to preserve the physical and mathematical properties (structure) of continuous systems. * Proven successful at preserving the **homological structure** of a surface. 🧬 * Can lead to "structure-preserving algorithms". ✅ * Helps preserve **inviscid invariants** over extended periods. ⏳ * **Geometric Flexibility** 🌐🌀: Works over **surface simplicial meshes** and on **arbitrary curved surfaces/manifolds**. Tested on flows past cylinders on spherical surfaces, double periodic shear layers on a plane, and harmonic waves on rotating spheres. 🌍 * Leads to numerically stable and accurate results for various problems. 📊 --- ### 🖼️ Shaping the Digital World 💻 .mermaid[ <pre> graph TD DEC --> A[Poisson Equations] DEC --> B[Mesh Smoothing] DEC --> C[Surface Parameterization] DEC --> D[Vector Fields] DEC --> E[Shortest Paths] DEC --> F[Elasticity] style DEC fill:#bbf,stroke:#333 style A fill:#f9f,stroke:#333 style B fill:#6f9,stroke:#333 style C fill:#f96,stroke:#333 style D fill:#9cf,stroke:#333 style E fill:#c9f,stroke:#333 style F fill:#fc9,stroke:#333 </pre> ] * A flexible and efficient framework for digital geometry processing. 🛠️ * Applications include: * Solving the **scalar Poisson equation** (`∆u = f`) on meshes. Involves building a sparse matrix for the Laplacian. 🔢 * Implicit **mean curvature flow** for mesh smoothing. 🌀 * Surface **parameterization** (e.g., conformal mapping). Can compute via the first non-trivial eigenvector of conformal energy. 🗺️ * **Vector field** decomposition (Helmholtz-Hodge) and design (using discrete connections and parallel transport). 🌪️ * Computing **shortest paths** or geodesic distance (by solving Poisson or integrating heat flow). 🛣️ * Improving **mesh quality**. 🔧 * Simulating mechanical phenomena like **elastic bodies**. 🏋️ --- ### 🌊 Simulating Flows 💧🌀 .mermaid[ <pre> graph TB NS[Navier-Stokes] --> DEC DEC --> Cylinder[Flow Past Cylinders] DEC --> Airfoil[Flow Past Airfoils] DEC --> Vortices[Taylor-Green Vortices] DEC --> Shear[Shear Layers] DEC --> Sphere[Rotating Spheres] style NS fill:#bbf,stroke:#333 style DEC fill:#9cf,stroke:#333 </pre> ] * Used for discretizing the **incompressible Navier–Stokes equations** over surface simplicial meshes. 📜 * Uses a **primitive variable DEC formulation** with velocity as a one-form (often dual) and pressure as a zero-form (often dual). This choice means velocity integrals along dual edges correspond to mass flux across primal edges. ⚡ * Tested on various benchmarks: * Flow past stationary and rotating **circular cylinders** (flat and on spherical surfaces). Results like drag/lift coefficients and Strouhal number show good agreement with literature. 🌀 * Flow past **airfoils**. ✈️ * **Taylor–Green vortices**. Used for error convergence studies. 🔄 * Double periodic **shear layer**. Used to show kinetic energy conservation and convergence. ⚡ * Motion of harmonic waves and flows on **rotating spheres**. Can preserve stationary states and inviscid invariants. 🌍 --- ### 🔬 Beyond Geometry & Fluids 🧪🌐 .mermaid[ <pre> pie title DEC Applications "Geometry Processing" : 35 "Fluid Dynamics" : 25 "Electromagnetics" : 15 "Elasticity" : 15 "Computer Vision" : 10 </pre> ] * Applicable to **variational problems** in computer vision and graphics. 👁️🎨 * Connections to **geometric mechanics** and discrete connections. ⚙️ * Mentions of related use in: * Computer Vision (e.g., template matching). 👁️ * Computational Modeling. 🧮 * Darcy flow. 💧 * Elasticity. 🏋️ * Electromagnetic problems. ⚡ * **Active Research Area** 🤔🔍: Developing a complete discrete calculus where *all* continuous operators (d, ∧, ★, Lie derivative (ℒ_X), interior product (iα), etc.) work well together in the discrete setting. 🏗️ --- ### 💻 Putting it into Practice 🖥️🔧 .mermaid[ <pre> graph LR Theory[DEC Theory] -->|Implementation| Code Code --> Cpp[C++ Framework] Code --> Python[PyDEC] Code --> Convergence[Convergence Analysis] style Theory fill:#bbf,stroke:#333 style Code fill:#9cf,stroke:#333 style Cpp fill:#f9f,stroke:#333 style Python fill:#6f9,stroke:#333 style Convergence fill:#f96,stroke:#333 </pre> ] * Discrete operators (like the discrete Laplacian or exterior derivative matrices) are often implemented as **sparse matrices**. 🔢 * Implementation examples and libraries exist: * A supplementary C++ framework is available. 📦 * **PyDEC** is a Python library for DEC on simplicial and cubical complexes of any dimension, implementing discrete exterior derivative (as coboundary), Delaunay Hodge star, and Whitney forms. 🐍 * **Convergence**: * Rigorous convergence analysis for the scalar Poisson problem on arbitrary domains has been established. ✅ * Numerical tests show convergence. For Taylor-Green vortices, the velocity *flux* error shows second-order convergence on structured meshes and better than first-order on unstructured meshes. Interpolated velocity error is slightly better than first-order. 📈 * The kinetic energy relative error converges in a second-order fashion with mesh size for the double periodic shear layer case. ⚡ --- count: false class: nord-dark, middle, center # Q&A 🎤