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docs(user): Add linear elasticity quickstart tutorial
- Complete cantilever beam example from mesh to visualization - Gmsh mesh creation with physical groups for BCs - Material definition (Young's modulus and Poisson's ratio) - Dirichlet (fixed) and Neumann (pressure) boundary conditions - ElasticityPhysics problem setup and solve!() call - Results visualization with stress and displacement - 361 lines: Step-by-step user tutorial for beginners
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Jukka Aho
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---
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title: "Quick Start: Linear Elasticity"
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date: 2025-11-10
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author: "JuliaFEM Team"
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status: "Draft"
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tags: ["user-guide", "elasticity", "quickstart", "tutorial"]
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---
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## Overview
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This guide shows how to solve a simple linear elasticity problem using JuliaFEM's GPU-accelerated solver.
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**What you'll learn:**
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- How to create a mesh with Gmsh
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- How to define materials and boundary conditions
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- How to solve and visualize results
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**Example problem:** Cantilever beam with fixed end and pressure load.
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## Step 1: Create the Mesh
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First, create a 3D mesh using Gmsh:
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```julia
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using Gmsh
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# Initialize Gmsh
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gmsh.initialize()
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gmsh.model.add("cantilever")
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# Geometry: 10m × 1m × 1m beam
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L, W, H = 10.0, 1.0, 1.0
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box = gmsh.model.occ.addBox(0, 0, 0, L, W, H)
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gmsh.model.occ.synchronize()
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# Define physical groups for boundary conditions
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surfaces = gmsh.model.getBoundary([(3, box)], false, false, true)
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for surf in surfaces
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_, surf_id = surf
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# Get surface center to identify it
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com = gmsh.model.occ.getCenterOfMass(2, surf_id)
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if abs(com[1]) < 1e-6 # X = 0 (fixed end)
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gmsh.model.addPhysicalGroup(2, [surf_id], -1, "FixedEnd")
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elseif abs(com[3] - H) < 1e-6 # Z = H (top surface for pressure)
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gmsh.model.addPhysicalGroup(2, [surf_id], -1, "PressureSurface")
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end
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end
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# Add volume physical group
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gmsh.model.addPhysicalGroup(3, [box], -1, "Volume")
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# Generate 3D tetrahedral mesh
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gmsh.option.setNumber("Mesh.MeshSizeMax", 0.5)
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gmsh.model.mesh.generate(3)
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# Save mesh
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gmsh.write("cantilever_beam.msh")
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gmsh.finalize()
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```
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**Key concepts:**
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- Physical groups label surfaces/volumes for boundary conditions
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- `"FixedEnd"` - nodes that will be constrained (Dirichlet BC)
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- `"PressureSurface"` - nodes where pressure is applied (Neumann BC)
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- `"Volume"` - elements for assembly
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## Step 2: Load the Mesh
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```julia
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include("src/gpu_elasticity.jl")
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using .GPUElasticity
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# Read the mesh file
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mesh = read_gmsh_mesh("cantilever_beam.msh")
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println("Mesh info:")
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println(" Nodes: $(size(mesh.nodes, 2))")
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println(" Elements: $(size(mesh.elements, 2))")
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```
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**What you get:**
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- `mesh.nodes` - 3×n_nodes matrix of coordinates
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- `mesh.elements` - 4×n_elements matrix of connectivity (Tet4)
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- `mesh.physical_groups` - Dictionary mapping names to node/element IDs
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## Step 3: Extract Boundary Condition Nodes
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```julia
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using .GPUElasticity.GmshReader: get_surface_nodes
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# Get nodes for each boundary condition
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fixed_nodes = get_surface_nodes(mesh, "FixedEnd")
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pressure_nodes = get_surface_nodes(mesh, "PressureSurface")
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println("Boundary conditions:")
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println(" Fixed nodes: $(length(fixed_nodes))")
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println(" Pressure nodes: $(length(pressure_nodes))")
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```
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**Boundary condition types:**
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1. **Dirichlet (fixed_nodes):** Zero displacement constraint
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- Nodes cannot move (u = 0)
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- Models supports, clamps, symmetry
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2. **Neumann (pressure_nodes):** Applied force/pressure
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- External load on surface
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- Models traction, pressure, point forces
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## Step 4: Define Material
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```julia
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# Create material (steel)
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material = ElasticMaterial(
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210e9, # E - Young's modulus [Pa]
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0.3 # ν - Poisson's ratio [-]
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)
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```
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**Common materials:**
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| Material | E (GPa) | ν |
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|----------|---------|---|
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| Steel | 200-210 | 0.27-0.30 |
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| Aluminum | 69 | 0.33 |
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| Concrete | 30-40 | 0.15-0.20 |
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| Rubber | 0.01-0.1 | 0.48-0.50 |
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## Step 5: Create Physics Problem
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```julia
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# Define the complete problem
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physics = ElasticityPhysics(
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mesh, # Mesh with geometry
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material, # Material properties
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fixed_nodes, # Dirichlet BC nodes
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pressure_nodes, # Neumann BC nodes
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10e6 # Pressure magnitude [Pa] = 10 MPa
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)
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```
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**What ElasticityPhysics contains:**
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- Mesh (nodes, elements, connectivity)
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- Material (E, ν for isotropic linear elasticity)
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- Fixed nodes (where displacement = 0)
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- Pressure nodes (where external load is applied)
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- Pressure value (load magnitude)
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## Step 6: Solve
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```julia
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# Solve on GPU with iterative solver
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result = solve_elasticity_gpu(physics, tol=1e-6, max_iter=1000)
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println("\nSolution converged!")
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println(" CG iterations: $(result.iterations)")
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println(" Final residual: $(result.residual)")
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```
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**Solver parameters:**
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- `tol` - Convergence tolerance (default: 1e-6)
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- `max_iter` - Maximum CG iterations (default: 1000)
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**What you get:**
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- `result.u` - Displacement field (3n_nodes vector)
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- `result.iterations` - Number of CG iterations
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- `result.residual` - Final residual norm
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## Step 7: Post-Process Results
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```julia
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# Extract displacement components
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n_nodes = size(mesh.nodes, 2)
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u_x = result.u[1:3:end]
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u_y = result.u[2:3:end]
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u_z = result.u[3:3:end]
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# Find maximum displacement
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u_magnitude = sqrt.(u_x.^2 + u_y.^2 + u_z.^2)
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max_disp = maximum(u_magnitude)
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max_node = argmax(u_magnitude)
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println("\nResults:")
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println(" Max displacement: $(max_disp * 1000) mm")
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println(" At node: $(max_node)")
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println(" Location: $(mesh.nodes[:, max_node])")
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# Compute stresses (requires element loop - TODO)
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```
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## Complete Example
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Here's the full script combining all steps:
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```julia
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using Gmsh
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include("src/gpu_elasticity.jl")
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using .GPUElasticity
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using .GPUElasticity.GmshReader: get_surface_nodes
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# 1. Generate mesh
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gmsh.initialize()
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gmsh.model.add("cantilever")
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L, W, H = 10.0, 1.0, 1.0
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box = gmsh.model.occ.addBox(0, 0, 0, L, W, H)
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gmsh.model.occ.synchronize()
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# Label surfaces
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surfaces = gmsh.model.getBoundary([(3, box)], false, false, true)
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for surf in surfaces
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_, surf_id = surf
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com = gmsh.model.occ.getCenterOfMass(2, surf_id)
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if abs(com[1]) < 1e-6
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gmsh.model.addPhysicalGroup(2, [surf_id], -1, "FixedEnd")
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elseif abs(com[3] - H) < 1e-6
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gmsh.model.addPhysicalGroup(2, [surf_id], -1, "PressureSurface")
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end
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end
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gmsh.model.addPhysicalGroup(3, [box], -1, "Volume")
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# Generate and save
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gmsh.option.setNumber("Mesh.MeshSizeMax", 0.5)
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gmsh.model.mesh.generate(3)
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gmsh.write("cantilever.msh")
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gmsh.finalize()
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# 2. Load mesh
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mesh = read_gmsh_mesh("cantilever.msh")
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# 3. Define boundary conditions
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fixed_nodes = get_surface_nodes(mesh, "FixedEnd")
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pressure_nodes = get_surface_nodes(mesh, "PressureSurface")
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# 4. Define material (steel)
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material = ElasticMaterial(210e9, 0.3)
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# 5. Create physics
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physics = ElasticityPhysics(
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mesh,
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material,
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fixed_nodes,
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pressure_nodes,
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10e6 # 10 MPa pressure
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)
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# 6. Solve
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result = solve_elasticity_gpu(physics)
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# 7. Results
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n_nodes = size(mesh.nodes, 2)
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u_mag = sqrt.(
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result.u[1:3:end].^2 +
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result.u[2:3:end].^2 +
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result.u[3:3:end].^2
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)
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println("Max displacement: $(maximum(u_mag) * 1000) mm")
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```
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## Current Limitations (Linear Elasticity)
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The current implementation (`gpu_elasticity.jl`) supports:
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✅ **Working:**
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- Linear elastic material (Hooke's law)
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- Isotropic materials (E, ν constant)
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- Small strain assumption
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- Dirichlet BC (fixed displacement)
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- Neumann BC (pressure on surfaces)
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- Matrix-free CG solver
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- GPU acceleration
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❌ **Not yet implemented:**
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- Nonlinear materials (plasticity, hyperelasticity)
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- Large deformations (geometric nonlinearity)
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- Material state variables (plastic strain, damage)
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- Contact mechanics
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- Dynamic analysis (time integration)
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- Point forces (only surface pressure)
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## Next Steps
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**To extend to nonlinear elasticity**, we need:
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1. **Material state at integration points**
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- Store plastic strain εₚ, hardening α, etc.
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- Update state during Newton iterations
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2. **Newton-Raphson solver**
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- Replace CG with Newton loop
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- Compute tangent stiffness and residual
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- Line search for globalization
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3. **Stress update algorithms**
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- Radial return for plasticity
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- Hyperelastic stress from strain energy
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- State management (old vs new state)
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4. **Boundary condition updates**
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- Prescribed displacement (not just zero)
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- Follower forces (load direction changes)
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- Contact constraints
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See `docs/src/book/` for design documents on these extensions.
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## Troubleshooting
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### Gmsh not found
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```julia
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using Pkg
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Pkg.add("Gmsh")
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```
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### No CUDA device
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CPU-only version coming soon. For now, requires NVIDIA GPU with CUDA.
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### CG doesn't converge
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- Increase `max_iter` parameter
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- Check boundary conditions (mesh must be constrained)
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- Add preconditioner (future work)
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### Out of GPU memory
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- Reduce mesh size (fewer elements)
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- Use coarser mesh (`Mesh.MeshSizeMax` larger)
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- Future: Distributed multi-GPU solver
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## Where to Learn More
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- **Theory:** `docs/src/book/elasticity_theory.md`
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- **Implementation:** `src/gpu_elasticity.jl` (477 lines, well-commented)
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- **Test:** `test/test_gpu_elasticity.jl`
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- **Demo:** `demos/cantilever_beam_demo.jl`
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- **Design:** `docs/src/book/design/gpu_elasticity_implementation.md`
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## Summary
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**Workflow:**
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1. Generate mesh with Gmsh (label surfaces for BCs)
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2. Load mesh into JuliaFEM
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3. Extract boundary condition nodes
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4. Define material properties
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5. Create `ElasticityPhysics` struct
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6. Solve with `solve_elasticity_gpu()`
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7. Post-process displacement field
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**Current status:** Linear elasticity works. Nonlinear extensions in progress.
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