This interactive simulation demonstrates 2D Finite Element Analysis (FEA) using Linear Triangle (T3 / Constant Strain Triangle) elements in plane stress. A rectangular mesh is built from nodes and triangular elements; the global stiffness matrix is assembled from each element’s B (strain–displacement) and D (constitutive) matrices. Fixed nodes (left edge by default) are enforced with the penalty method; the governing equation K d = F_ext + F_p is solved with the Conjugate Gradient method. Displacements are exaggerated for visualization; triangles are colored by Von Mises stress (blue = low, red = high).

Adjust Young’s Modulus E and Poisson’s ratio ν in the sidebar. Use Apply force (drag) to drag a node and set force; use Toggle fix node to fix or free a node. Use Mesh Density to change grid size; Interpolation for None/Linear/Shape Functions; Principal Stresses, Nodal Influence, Residuals for extra visuals. The Math Monitor shows K, d, F_ext, F_p; click K for assembly animation. Reset restores mesh and left-edge fix. The gray outline is the original mesh; the colored mesh is the deformed configuration (displacement scaled for visibility).

 

Math behind the simulation

1. Goal

Solve for nodal displacements d given applied forces and boundary conditions:

where K is the global stiffness matrix, F_ext is external force, and F_p is plastic force.

2. Plane stress and D matrix

For plane stress, stress and strain are related by the constitutive (D) matrix. With Young’s modulus E and Poisson’s ratio ν:

3. Constant Strain Triangle (CST)

Each triangle has linear displacement; strain is constant in the element. The B (strain–displacement) matrix depends on the node coordinates and area:

where u_e is the 6×1 vector of nodal displacements (u, v for each of 3 nodes).

4. Element stiffness and assembly

Local stiffness is integrated over the element (area A, thickness t = 1 here):

k_e is assembled into the global K by mapping local DOFs to global node indices. Click K in the equation box to animate this assembly.

5. Boundary conditions and solver

Fixed degrees of freedom are enforced with the penalty method: a large value is added to the diagonal of K for that DOF and the force is set to zero. The system is solved with the Conjugate Gradient method (iterative; scales with mesh size).

6. Von Mises stress and plasticity

For each element, strain ε = B u_e and stress σ = D ε. The Von Mises equivalent stress (plane stress) is:

Colors map low (blue) to high (red) stress. With plasticity enabled, excess stress accumulates permanent strain and F_p is added to the right-hand side.

Simulation

The interactive simulator is below. Use the controls to explore the concepts described above.

 

Usage

1. Apply force: Set mode to “Apply force (drag)”, then click and drag a node. The drag vector sets the force at that node; the mesh deforms and stress updates.
2. Fix / free nodes: Set mode to “Toggle fix node”, then click a node to fix or free it in X and Y. By default the left edge (x = 0) is fixed.
3. Material: Change Young’s Modulus E and Poisson’s ratio ν. Use Mesh Density (4–30) to change grid size (e.g. 10×5).
4. Plasticity: Enable and set Yield strength; drag hard to see permanent deformation. Clear Plasticity resets.
5. Interpolation: None (flat), Linear (Gouraud), or Shape Functions (with sampling markers).
6. Math Monitor: Click K in the equation to run assembly animation. Hover the K grid for K[i,j]; hover a node (with Nodal Influence on) to see its DOF rows.
7. Reset: Restores mesh and left-edge fix; keeps current Mesh Density; clears applied forces.

Parameters

Control	Range / options	Effect
Young’s Modulus E	e.g. 1e6	Material stiffness. Higher E ⇒ smaller displacements for the same force.
Poisson’s ratio ν	0 < ν < 0.5	Lateral strain / axial strain. Larger ν ⇒ more "bulge" perpendicular to load.
Mesh Density	4–30	Grid size (e.g. 10 → 10×5). Finer mesh ⇒ more DOFs and a larger K matrix.
Plasticity / Yield strength	on / off + value	Stress above yield accumulates permanent strain; Fp appears in the equation monitor.
Interpolation	None / Linear / Shape Functions	Flat shading, Gouraud smoothing, or shape-function sampling markers.

Visualization

• Gray outline: Undeformed mesh.
• Colored triangles: Deformed mesh; color = Von Mises stress (blue low → red high). Interpolation controls smooth vs. flat shading.
• Nodes: Blue = unloaded; gray = fixed; red = has applied force.
• Principal Stresses: Green/red crosses on elements (tension/compression).
• Nodal Influence: Hover a node to highlight connected elements and its global DOF rows.
• Residuals: Red arrows showing F − K d at each node.
• Math Monitor: Governing equation, dimensions, sparsity %, and K×d=F_ext+F_p canvases (K sparsity grid and d, F_ext, F_p vector ribbons).

Limitations

• Linear, small-strain elasticity. The solver assumes infinitesimal strains and a fixed stiffness matrix; large rotations and geometric (large-deformation) nonlinearity are not modeled. Displacements are exaggerated purely for visualization.
• Constant Strain Triangles only. T3 elements give a piecewise-constant stress field, so stresses are discontinuous between elements and converge slowly — this demo trades accuracy for transparency.
• 2D plane stress. Only thin-plate, in-plane loading is represented; no plane strain, no out-of-plane bending, no 3D effects.
• Penalty boundary conditions. Fixed DOFs use a large diagonal penalty rather than exact constraint elimination, which can mildly perturb the solution and conditioning.
• Simplified plasticity. The plastic-force update is a basic excess-stress accumulation, not a full return-mapping / hardening model, so it is illustrative rather than quantitatively accurate.
• Iterative CG solver. Conjugate Gradient convergence depends on conditioning; very fine meshes or extreme material ratios can slow or degrade the solve.