Finite element method — nodes, elements, stiffness matrix

Why FEM Exists

Real spacecraft structures have complex geometries: curved panels, ribs, cutouts for equipment. Solving the partial differential equations of elasticity for these shapes analytically is impossible for anything beyond textbook examples.

FEM's philosophy: discretization. Replace the continuous structure (infinite degrees of freedom) with a finite collection of simple shapes whose behavior we can calculate exactly.

The Three Pillars

1. Nodes

Why nodes? We can't track displacement at every point—that's infinite unknowns. By tracking only at nodes and interpolating between them, we reduce the problem to a finite set of equations.

Physical meaning: Each node represents a physical location. Applied forces/moments go on nodes. Boundary conditions (fixed supports) constrain node displacements.

2. Elements

Common spacecraft elements:

• Beam elements: 2 nodes, models truss members, longerons
• Shell elements: 3-4 nodes, models thin panels, skin
• Solid elements: 4-8 nodes, models thick blocks, joints

Shape Functions — The Interpolation Magic

For a2-node beam element with nodes1 and 2, the displacement $u(x)$ at any point $x$ along the element is:

$u(x) = N_1(x) u_1 + N_2(x) u_2$

where shape functions $N_1, N_2$ are chosen so:

• $N_1 = 1$ at node 1, $N_1 = 0$ at node 2
• $N_2 = 0$ at node 1, $N_2 = 1$ at node 2
• $N_1 + N_2 = 1$ everywhere (partition of unity)

For linear interpolation over length $L$:

$N_1(x) = 1 - \frac{x}{L}, \quad N_2(x) = \frac{x}{L}$

Why this works: At the nodes, we get exact values. Between nodes, we get a smooth blend. This turns a continuous field into a function of just the node values.

3. Stiffness Matrix

Derivation from First Principles (1D Bar Element)

Consider a uniform bar, length $L$, cross-section $A$, Young's modulus $E$. Two nodes, each with 1 DOF (axial displacement).

Step 1: Displacement field using shape functions

$u(x) = N_1(x) d_1 + N_2(x) d_2 = \left(1 - \frac{x}{L}\right) d_1 + \frac{x}{L} d_2$

Step 2: Strain-displacement relation

$\epsilon(x) = \frac{du}{dx} = -\frac{1}{L} d_1 + \frac{1}{L} d_2 = \frac{1}{L}(d_2 - d_1)$

Why this step? Strain is the gradient of displacement. Our linear shape functions produce constant strain in the element (acceptable for small elements).

Step 3: Stress from Hooke's law

$\sigma = E\epsilon = \frac{E}{L}(d_2 - d_1)$

Step 4: Internal force

$F_{internal} = \sigma \cdot A = \frac{EA}{L}(d_2 - d_1)$

Step 5: Equilibrium at nodes

Node 1 experiences tension from element: $F_1 = -F_{internal} = -\frac{EA}{L}(d_2 - d_1) = \frac{EA}{L}(d_1 - d_2)$

Node 2 experiences compression: $F_2 = +F_{internal} = \frac{EA}{L}(d_2 - d_1)$

Why opposite signs? Newton's third law—element pulls on node 1 as node 1 pulls on element.

Step 6: Matrix form

$\begin{bmatrix} F_1 \\ F_2 \end{bmatrix} = \frac{EA}{L} \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix} \begin{bmatrix} d_1 \\ d_2 \end{bmatrix}$

The element stiffness matrix is:

Global Assembly

Each element contributes to nodes it connects. For a structure with $N$ nodes and multiple elements:

1. Create global stiffness matrix $\mathbf{K}_{global}$ (size $N \times N$ for 1D, $6N \times 6N$ for 3D)
2. For each element, map element DOFs to global DOFs
3. Add $\mathbf{K}^e$ entries to corresponding global positions

Example: Element connecting global nodes 2 and 5. Its $K^e_{11}$ adds to $K_{global,22}$, $K^e_{12}$ adds to $K_{global,25}$, etc.

The FEM Workflow

1. Preprocessing: Define geometry, create mesh (place nodes, define elements), assign material properties, apply loads/BCs
2. Element calculations: Compute $\mathbf{K}^e$ for each element (use analytical formulas or numerical integration for complex shapes)
3. Assembly: Build global $\mathbf{K}$
4. Boundary conditions: Eliminate rows/columns for fixed DOFs (or use penalty method)
5. Solve: $\mathbf{K}\mathbf{d} = \mathbf{F}$ (typically sparse matrix solver, since most nodes don't connect to most other nodes)
6. Postprocessing: Calculate strains, stresses from $\mathbf{d}$; check Von Mises stress, plot deformed shape

Convergence & Mesh Refinement

FEM is an approximation. Accuracy improves with:

• More elements (h-refinement): Smaller elements → better capture of gradients
• Higher-order shape functions (p-refinement): Quadratic/cubic interpolation instead of linear

Rule of thumb: Place more elements where stress gradients are high (near holes, filets, joints).

Energy Interpretation

The stiffness matrix emerges from minimizing potential energy. For a structure:

$\Pi = U_{strain} - W_{external}$

where strain energy:

$U = \frac{1}{2} \mathbf{d}^T \mathbf{K} \mathbf{d}$

At equilibrium, $\frac{\partial \Pi}{\partial d_i} = 0$ for all $i$ → $\mathbf{K}\mathbf{d} = \mathbf{F}$.

Why this matters: FEM guarantees we find the displacement field with lowest total energy consistent with BCs. This is nature's actual behavior (minimum energy principle).

Connections

• 3.6.1-Stress-and-strain-in-structural-members — FEM solves for these quantities
• 3.6.5-Buckling-of-columns-and-shells — Eigenvalue problem: $(\mathbf{K} - \lambda \mathbf{K}_G)\boldsymbol{\phi} = 0$
• 3.6.12-Modal-analysis-and-natural-frequencies — $(\mathbf{K} - \omega^2 \mathbf{M})\boldsymbol{\phi} = 0$, same $\mathbf{K}$ + mass matrix
• 2.4.8-Numerical-methods-for-differential-equations — FEM is the spatial discretization for PDEs
• 3.7.4-Thermal-analysis-of-spacecraft-components — Same FEM framework, replace stiffness with conductance

#flashcards/physics

Question: What is the fundamental idea behind the finite element method? :: Replace a continuous structure (infinite unknowns) with a finite mesh of simple elements connected at nodes (finite unknowns), where displacement is interpolated within each element using shape functions.

Concept Map

Hinglish (regional understanding)