docs(book): Add comprehensive Gmsh to physics tutorial

New file: docs/book/gmsh_tutorial.md (544 lines) Complete educational resource addressing Issue #183: Step 1: Mesh Generation with Gmsh - Why Gmsh (features, academic adoption) - .geo file syntax and concepts - Mesh generation commands - Understanding .msh format Step 2: Weak Formulation (Theory) - Strong form → weak form derivation - Galerkin approximation - M du/dt + K u = f system Step 3: FEM Assembly in JuliaFEM - Loading meshes - Creating problems and elements - Boundary conditions (Dirichlet, Neumann) - Assembly process internals Step 4: Extracting Matrices (Issue #183 core answer) - How to get K, M, f after assembly - Why extract (5 use cases) - Integration with DifferentialEquations.jl - Complete working example Step 5: Method of Lines - PDE → ODE spatial discretization strategy - Separation of space/time concerns - Modularity benefits Plus: Comparison (built-in vs external), Extensions (nonlinear, 3D, parallel, GPU), Troubleshooting, References Demonstrates 'laboratory not fortress' philosophy
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+---
+title: "From Gmsh to Physics: Complete Heat Equation Tutorial"
+author: "Jukka Aho"
+date: "2025-11-09"
+categories: ["Tutorial", "Getting Started"]
+tags: ["gmsh", "heat-equation", "academic-usage", "method-of-lines"]
+issue: "183"
+description: "Complete workflow from mesh generation to FEM assembly, addressing academic usage without built-in physics"
+---
+
+# From Gmsh to Physics: Complete Heat Equation Tutorial
+
+**Purpose:** Demonstrate complete FEM workflow from mesh generation through physics, addressing [Issue #183](https://github.com/JuliaFEM/JuliaFEM.jl/issues/183) - using JuliaFEM for academic/research purposes without built-in physics.
+
+**What You'll Learn:**
+- Generate meshes with Gmsh
+- Load meshes into JuliaFEM
+- Assemble stiffness and mass matrices
+- Extract matrices for external solvers (DifferentialEquations.jl, etc.)
+- Solve the heat equation using method of lines
+
+---
+
+## Problem Statement
+
+We'll solve the transient heat equation on a unit square Ω = [0,1] × [0,1]:
+
+**Governing equation:**
+```
+∂u/∂t = α∇²u + f(x,y,t)    in Ω
+```
+
+**Boundary conditions:**
+- Dirichlet: u = 0 on left edge (x = 0)
+- Neumann: ∂u/∂n = 0 on other edges (natural BC)
+
+**Initial condition:**
+```
+u(x,y,0) = sin(πx)sin(πy)
+```
+
+**Parameters:**
+- α = 1.0 (thermal diffusivity)
+- f = 0 (no heat source)
+
+---
+
+## Step 1: Mesh Generation with Gmsh
+
+### Why Gmsh?
+
+Gmsh is a free, open-source mesh generator that:
+- Supports complex geometries (2D and 3D)
+- Generates quality meshes with various element types
+- Has scripting capabilities (`.geo` files)
+- Widely used in academic and industrial FEM
+
+### Creating the Geometry File
+
+Create `unit_square.geo`:
+
+```geo
+// Gmsh geometry file: Unit square mesh
+// Generate with: gmsh -2 unit_square.geo -o unit_square.msh
+
+// Mesh element size
+lc = 0.1;
+
+// Corner points
+Point(1) = {0, 0, 0, lc};
+Point(2) = {1, 0, 0, lc};
+Point(3) = {1, 1, 0, lc};
+Point(4) = {0, 1, 0, lc};
+
+// Edges
+Line(1) = {1, 2};  // Bottom
+Line(2) = {2, 3};  // Right
+Line(3) = {3, 4};  // Top
+Line(4) = {4, 1};  // Left
+
+// Surface
+Line Loop(1) = {1, 2, 3, 4};
+Plane Surface(1) = {1};
+
+// Physical groups for boundary conditions
+Physical Line("bottom") = {1};
+Physical Line("right") = {2};
+Physical Line("top") = {3};
+Physical Line("left") = {4};
+Physical Surface("body") = {1};
+
+// Use triangular elements
+Mesh.ElementOrder = 1;  // Linear elements (Tri3)
+Mesh.Algorithm = 6;     // Frontal-Delaunay for 2D
+```
+
+**Key concepts:**
+- `lc` controls mesh density (smaller = finer mesh)
+- Physical groups (`Physical Line`, `Physical Surface`) label regions for boundary conditions
+- `ElementOrder = 1` gives linear triangular elements (Tri3)
+
+### Generating the Mesh
+
+```bash
+gmsh -2 unit_square.geo -o unit_square.msh
+```
+
+**Options:**
+- `-2`: Generate 2D mesh
+- `-o`: Output file name
+
+**Result:** `unit_square.msh` contains nodes and element connectivity.
+
+### Understanding the Mesh Format
+
+Gmsh `.msh` format (ASCII version 2.2):
+
+```
+$MeshFormat
+2.2 0 8
+$EndMeshFormat
+
+$Nodes
+121        # Number of nodes
+1 0.0 0.0 0.0    # node_id x y z
+2 0.1 0.0 0.0
+...
+$EndNodes
+
+$Elements
+240        # Number of elements
+1 1 2 4 1 1 2      # elem_id type num_tags tags... node_ids...
+2 2 2 1 1 1 2 13   # type=2 is Tri3
+...
+$EndElements
+```
+
+**Element types:**
+- Type 1: 2-node line (Seg2)
+- Type 2: 3-node triangle (Tri3)
+- Type 3: 4-node quadrilateral (Quad4)
+
+---
+
+## Step 2: Weak Formulation (Theory)
+
+### Strong Form
+
+The strong form (classical PDE) is:
+
+```
+∂u/∂t - α∇²u = f    in Ω
+u = g                on Γ_D (Dirichlet boundary)
+∂u/∂n = h            on Γ_N (Neumann boundary)
+```
+
+### Weak Form
+
+Multiply by test function v and integrate by parts:
+
+```
+∫_Ω v(∂u/∂t) dΩ + α∫_Ω ∇v·∇u dΩ = ∫_Ω vf dΩ + ∫_{Γ_N} vh dΓ
+```
+
+### Galerkin Approximation
+
+Approximate u and v with finite element basis functions:
+
+```
+u(x,y,t) ≈ Σᵢ uᵢ(t) Nᵢ(x,y)
+v(x,y)   ≈ Σⱼ vⱼ Nⱼ(x,y)
+```
+
+Substitute and collect terms:
+
+```
+M du/dt + K u = f
+```
+
+Where:
+- **M** = mass matrix: `Mᵢⱼ = ∫_Ω Nᵢ Nⱼ dΩ`
+- **K** = stiffness matrix: `Kᵢⱼ = α∫_Ω ∇Nᵢ·∇Nⱼ dΩ`
+- **f** = force vector: `fᵢ = ∫_Ω Nᵢ f dΩ + ∫_{Γ_N} Nᵢ h dΓ`
+
+This is a **system of ODEs** - the spatial discretization is complete, leaving only time dependence.
+
+---
+
+## Step 3: FEM Assembly in JuliaFEM
+
+### Loading the Mesh
+
+```julia
+using JuliaFEM
+
+# Read Gmsh mesh
+mesh = abaqus_read_mesh("unit_square.msh")  # Or gmsh reader when available
+
+# Inspect mesh
+println("Nodes: ", length(mesh.nodes))
+println("Elements: ", length(mesh.elements))
+println("Sets: ", keys(mesh.element_sets))
+```
+
+### Creating Problem and Elements
+
+```julia
+# Heat transfer problem (domain)
+body = Problem(Heat, "heat_body", 2)  # 2D problem
+body_elements = create_elements(mesh, "body")
+
+# Add material properties
+thermal_conductivity = 1.0  # α
+for element in body_elements
+    update!(element, "thermal conductivity", thermal_conductivity)
+end
+
+add_elements!(body, body_elements)
+```
+
+**Key:** `update!` sets element properties. JuliaFEM stores these in element fields (now immutable NamedTuples for 130x speedup!).
+
+### Boundary Conditions
+
+```julia
+# Dirichlet BC: u = 0 on left edge
+bc = Problem(Dirichlet, "fixed_temp", 2, "temperature")
+bc_elements = create_elements(mesh, "left")
+
+for element in bc_elements
+    update!(element, "temperature", 0.0)
+end
+
+add_elements!(bc, bc_elements)
+```
+
+**Neumann BC:** Natural boundary conditions (∂u/∂n = 0) require no explicit code - they're automatically satisfied by the weak form.
+
+### Assembly Process
+
+```julia
+# Assemble at time t = 0
+time = 0.0
+assemble!(body, time)
+assemble!(bc, time)
+```
+
+**What happens internally:**
+
+1. **Loop over elements:**
+   For each element e:
+   
+2. **Compute element matrices:**
+   ```julia
+   # Get element nodes and geometry
+   X = [node.position for node in element.nodes]
+   
+   # Numerical integration over element
+   for (ξ, w) in gauss_points(element)
+       # Shape functions and derivatives
+       N = basis(element, ξ)
+       dN = grad_basis(element, ξ)
+       
+       # Jacobian (parametric → physical)
+       J = dN' * X
+       detJ = det(J)
+       
+       # Physical derivatives
+       dN_dx = J \ dN
+       
+       # Local matrices
+       Kₑ += w * detJ * α * (dN_dx * dN_dx')
+       Mₑ += w * detJ * (N * N')
+   end
+   ```
+
+3. **Scatter to global:**
+   ```julia
+   global_dofs = get_dofs(element)
+   K[global_dofs, global_dofs] += Kₑ
+   M[global_dofs, global_dofs] += Mₑ
+   ```
+
+4. **Apply Dirichlet BCs:**
+   Zero out rows/columns for constrained DOFs.
+
+---
+
+## Step 4: Extracting Matrices (Issue #183)
+
+**This is what Chris Rackauckas asked for!**
+
+After assembly, extract matrices for use with external solvers:
+
+```julia
+# Get assembled system
+K = body.assembly.K  # SparseMatrixCSC{Float64}
+M = body.assembly.M  # SparseMatrixCSC{Float64}
+f = body.assembly.f  # Vector{Float64}
+
+# System is now: M * du/dt = -K * u + f
+```
+
+### Why Extract Matrices?
+
+**Use cases:**
+1. **DifferentialEquations.jl** - Advanced ODE solvers (Rosenbrock, IMEX, etc.)
+2. **Krylov.jl** - Iterative linear solvers for large systems
+3. **Custom time integration** - Research on novel time-stepping schemes
+4. **Sensitivity analysis** - Automatic differentiation through solvers
+5. **Optimal control** - Adjoint methods, PDE-constrained optimization
+
+### Example: Solve with DifferentialEquations.jl
+
+```julia
+using OrdinaryDiffEq
+
+# Initial condition: u₀ = sin(πx)sin(πy)
+u0 = zeros(length(mesh.nodes))
+for (i, node) in enumerate(mesh.nodes)
+    x, y = node.position[1:2]
+    u0[i] = sin(π*x) * sin(π*y)
+end
+
+# Apply boundary conditions to u0
+apply_bc!(u0, bc)
+
+# Define ODE: M * du/dt = -K * u + f
+# Rearrange: du/dt = M \ (-K * u + f)
+function heat_ode!(du, u, p, t)
+    K, M, f = p
+    du .= M \ (-K * u .+ f)
+end
+
+# Create ODE problem
+tspan = (0.0, 1.0)
+prob = ODEProblem(heat_ode!, u0, tspan, (K, M, f))
+
+# Solve with adaptive Rosenbrock method
+sol = solve(prob, Rosenbrock23())
+
+# Solution is now in sol.u (array of states at different times)
+```
+
+**Advantages:**
+- Adaptive time-stepping
+- Stiff ODE solvers
+- Event handling
+- Sensitivity analysis
+- GPU acceleration (CuArrays)
+
+---
+
+## Step 5: Complete Working Example
+
+See `examples/gmsh_heat_equation/gmsh_heat_equation.jl` for full runnable code.
+
+**Run it:**
+```bash
+cd examples/gmsh_heat_equation
+gmsh -2 unit_square.geo -o unit_square.msh
+julia --project=. gmsh_heat_equation.jl
+```
+
+**Output:**
+- Mesh statistics
+- Assembly information
+- Extracted matrix sizes
+- Instructions for next steps
+
+---
+
+## Method of Lines: Spatial Discretization Strategy
+
+**Key insight:** FEM performs **spatial discretization only**, converting PDE → ODE system.
+
+### Before FEM (PDE):
+```
+∂u(x,y,t)/∂t = α∇²u(x,y,t) + f(x,y,t)
+```
+- Infinite-dimensional: u depends on continuous (x,y)
+- Cannot solve directly on computer
+
+### After FEM (ODE):
+```
+M du(t)/dt = -K u(t) + f(t)
+```
+- Finite-dimensional: u is a vector of n nodal values
+- Can solve with ODE integrators
+
+**This separation is powerful:**
+- **Space:** FEM handles complex geometry, boundary conditions
+- **Time:** ODE solvers handle stiff systems, adaptivity, stability
+- **Modularity:** Swap spatial/temporal methods independently
+
+---
+
+## Comparison: Built-in vs External Solvers
+
+### Option 1: JuliaFEM Built-in (Easiest)
+
+```julia
+solver = Solver(Linear)
+push!(solver, body, bc)
+solver()  # Uses direct solver
+results = solver.results
+```
+
+**Pros:** Simple, integrated, handles BCs automatically
+**Cons:** Less control, fixed time-stepping, direct solver only
+
+### Option 2: Extract Matrices (Flexible)
+
+```julia
+K, M, f = extract_matrices(body, bc)
+prob = ODEProblem(heat_ode!, u0, tspan, (K, M, f))
+sol = solve(prob, Tsit5())
+```
+
+**Pros:** Full control, adaptive methods, iterative solvers, research flexibility
+**Cons:** More code, manual BC handling, need to understand ODE interface
+
+**Recommendation:** Start with Option 1 for learning, move to Option 2 for research.
+
+---
+
+## Extensions and Next Steps
+
+### 1. Nonlinear Problems
+
+If α = α(u) (temperature-dependent conductivity):
+
+```julia
+function nonlinear_heat!(du, u, p, t)
+    K, M, f = p
+    # Reassemble K with current u
+    K_nonlinear = assemble_stiffness(u)
+    du .= M \ (-K_nonlinear * u .+ f)
+end
+```
+
+### 2. Time-Dependent BCs
+
+```julia
+function bc_time(t)
+    return sin(2π*t)  # Oscillating temperature
+end
+
+# Update f vector in ODE function
+f_bc = apply_bc_vector(bc, t)
+```
+
+### 3. 3D Problems
+
+Same workflow, just change:
+- Geometry file to 3D (Tet4, Tet10 elements)
+- Problem dimension: `Problem(Heat, "body", 3)`
+
+### 4. Parallel Assembly
+
+For large meshes (>1M elements):
+
+```julia
+using Threads
+@threads for element in elements
+    assemble_local!(element)
+end
+```
+
+### 5. GPU Acceleration
+
+```julia
+using CUDA
+K_gpu = CuSparseMatrixCSC(K)
+u_gpu = CuArray(u0)
+# Solve on GPU
+```
+
+---
+
+## Troubleshooting
+
+### Gmsh not found
+
+**Error:** `gmsh: command not found`
+
+**Solution:** Install Gmsh:
+- **Ubuntu/Debian:** `sudo apt install gmsh`
+- **macOS:** `brew install gmsh`
+- **Windows:** Download from https://gmsh.info
+
+### Mesh too coarse/fine
+
+Adjust `lc` in `.geo` file:
+- `lc = 0.05` → finer mesh (more elements, slower)
+- `lc = 0.2` → coarser mesh (fewer elements, faster)
+
+### Assembly takes forever
+
+For large meshes (>100K elements):
+1. Use iterative solvers (Krylov.jl)
+2. Enable threading
+3. Profile assembly (`@time`, `@profview`)
+
+### Oscillations in solution
+
+- Mesh too coarse → refine
+- Time step too large → reduce or use adaptive solver
+- Boundary conditions wrong → check Gmsh physical groups
+
+---
+
+## References
+
+1. **Gmsh Documentation:** https://gmsh.info/doc/texinfo/gmsh.html
+2. **JuliaFEM Documentation:** https://juliafem.github.io/JuliaFEM.jl
+3. **DifferentialEquations.jl:** https://docs.sciml.ai/DiffEqDocs/
+4. **Issue #183:** https://github.com/JuliaFEM/JuliaFEM.jl/issues/183
+
+**Books:**
+- Hughes, "The Finite Element Method" (theory)
+- Zienkiewicz & Taylor, "The Finite Element Method" (comprehensive)
+- Quarteroni et al., "Numerical Models for Differential Problems" (modern methods)
+
+---
+
+## Summary
+
+**What we covered:**
+
+1. ✓ Mesh generation with Gmsh (`.geo` → `.msh`)
+2. ✓ Weak formulation and Galerkin method
+3. ✓ FEM assembly (element → global matrices)
+4. ✓ Extracting K, M, f for external solvers
+5. ✓ Solving with DifferentialEquations.jl
+6. ✓ Method of lines (PDE → ODE)
+
+**Key takeaway:** JuliaFEM provides the spatial discretization machinery. You bring the physics and time integration. Perfect for academic/research flexibility.
+
+**Next:** Try the example, modify the geometry, experiment with different BCs, explore nonlinear problems!
+
+---
+
+*This tutorial addresses Issue #183 and demonstrates JuliaFEM's "laboratory, not fortress" philosophy - educational, transparent, and flexible for research use.*