---
id: wiki-2026-0508-finite-element-analysis
title: Finite Element Analysis (FEA)
category: 10_Wiki/Topics
status: verified
canonical_id: self
aliases: [FEA, FEM, finite element method, structural analysis, ANSYS, Abaqus, mesh]
duplicate_of: none
source_trust_level: A
confidence_score: 0.95
verification_status: applied
tags: [engineering, fea, fem, simulation, structural, mesh, computational-mechanics]
raw_sources: []
last_reinforced: 2026-05-10
github_commit: pending
tech_stack:
  language: Python / FORTRAN / C++
  framework: ANSYS / Abaqus / FEniCS / PyANSYS
---

# Finite Element Analysis (FEA)

## 매 한 줄
> **"매 PDE 의 의 의 mesh 의 element 의 의 discretize 의 solve"**. 매 structural, thermal, fluid, EM. 매 famous: ANSYS, Abaqus, NASTRAN. 매 modern: 매 FEniCS (open), 매 ML-augmented (PINN, GNN), 매 cloud HPC.

## 매 핵심

### 매 step
1. **Geometry / CAD**.
2. **Mesh** (1D, 2D, 3D element).
3. **Material** (E, ν, ρ, ...).
4. **Boundary condition + load**.
5. **Assemble** (K matrix).
6. **Solve** (linear / nonlinear).
7. **Post-process**.

### 매 element type
- **1D**: bar, beam.
- **2D**: triangle, quad (CST, LST).
- **3D**: tet, hex, wedge.
- **Shell**: 매 thin structure.

### 매 analysis type
- **Static linear**.
- **Modal** (eigenvalue).
- **Dynamic** (transient).
- **Nonlinear** (geom, material, contact).
- **Thermal**.
- **CFD** (Navier-Stokes).
- **EM** (Maxwell).

### 매 modern AI
- **PINN** (physics-informed NN).
- **GNN-based** (faster surrogate).
- **Differentiable FEM** (JAX-FEM).
- **NeRF for material**.

### 매 응용
1. **Aerospace**: 매 wing.
2. **Automotive**: 매 crash.
3. **Civil**: 매 building.
4. **Biomedical**: 매 implant.
5. **Electronics**: 매 PCB thermal.
6. **Geomechanics**: 매 dam.

## 💻 패턴

### FEniCS (open-source)
```python
from dolfinx import fem, mesh, plot
from mpi4py import MPI
import ufl

domain = mesh.create_rectangle(MPI.COMM_WORLD, [(0,0), (1,1)], (32,32))
V = fem.FunctionSpace(domain, ('Lagrange', 1))

# 매 BC
def boundary(x): return np.isclose(x[0], 0) | np.isclose(x[0], 1)
bc = fem.dirichletbc(0.0, fem.locate_dofs_geometrical(V, boundary), V)

# 매 weak form (Poisson)
u = ufl.TrialFunction(V); v = ufl.TestFunction(V)
f = fem.Constant(domain, 1.0)
a = ufl.dot(ufl.grad(u), ufl.grad(v)) * ufl.dx
L = f * v * ufl.dx

# 매 solve
problem = fem.petsc.LinearProblem(a, L, bcs=[bc])
uh = problem.solve()
```

### Stiffness matrix (1D bar)
```python
import numpy as np

def bar_stiffness(E, A, L):
    """매 1D bar element."""
    k = E * A / L
    return np.array([[k, -k], [-k, k]])

def assemble(elements, n_nodes):
    K = np.zeros((n_nodes, n_nodes))
    for (e, k_local) in elements:
        for i, gi in enumerate(e):
            for j, gj in enumerate(e):
                K[gi, gj] += k_local[i, j]
    return K
```

### Mesh generation (gmsh)
```python
import gmsh
gmsh.initialize()
gmsh.model.add('plate')
gmsh.model.geo.addPoint(0, 0, 0, 0.1, 1)
gmsh.model.geo.addPoint(1, 0, 0, 0.1, 2)
gmsh.model.geo.addPoint(1, 1, 0, 0.1, 3)
gmsh.model.geo.addPoint(0, 1, 0, 0.1, 4)
gmsh.model.geo.addLine(1, 2, 1)
# ... 4 lines
gmsh.model.geo.addPlaneSurface([1])
gmsh.model.mesh.generate(2)
gmsh.write('plate.msh')
```

### PyANSYS (commercial integration)
```python
from ansys.mapdl.core import launch_mapdl
mapdl = launch_mapdl()
mapdl.prep7()
mapdl.et(1, 'BEAM188')
mapdl.mp('EX', 1, 200e9)  # 매 Steel
mapdl.k(1, 0); mapdl.k(2, 1); mapdl.l(1, 2)
mapdl.lesize('all', '', '', 10)
mapdl.lmesh('all')
mapdl.solve()
mapdl.post1()
```

### Modal analysis
```python
from scipy.linalg import eigh
def modal(K, M, n_modes=5):
    """매 K φ = ω² M φ."""
    eigvals, eigvecs = eigh(K, M)
    freqs_hz = np.sqrt(eigvals[:n_modes]) / (2 * np.pi)
    return freqs_hz, eigvecs[:, :n_modes]
```

### Nonlinear (Newton-Raphson)
```python
def newton_raphson(K_fn, R_fn, u0, tol=1e-6, max_iter=50):
    u = u0.copy()
    for _ in range(max_iter):
        residual = R_fn(u)
        if np.linalg.norm(residual) < tol: return u
        K = K_fn(u)
        du = np.linalg.solve(K, -residual)
        u += du
    raise ConvergenceError()
```

### PINN (physics-informed)
```python
import torch
class PINN(torch.nn.Module):
    def __init__(self):
        super().__init__()
        self.net = torch.nn.Sequential(
            torch.nn.Linear(2, 64), torch.nn.Tanh(),
            torch.nn.Linear(64, 64), torch.nn.Tanh(),
            torch.nn.Linear(64, 1),
        )
    
    def forward(self, x):
        return self.net(x)
    
    def physics_loss(self, x):
        x.requires_grad = True
        u = self.forward(x)
        u_x = torch.autograd.grad(u.sum(), x, create_graph=True)[0]
        u_xx = torch.autograd.grad(u_x.sum(), x, create_graph=True)[0]
        # 매 e.g., Poisson: -u_xx = f
        return ((-u_xx + 1) ** 2).mean()
```

### GNN-based surrogate (MeshGraphNet)
```python
import torch_geometric.nn as gnn

class MeshGraphNet(torch.nn.Module):
    def __init__(self, node_dim=3, edge_dim=3, hidden=128):
        super().__init__()
        self.encoder = gnn.MLP([node_dim, hidden, hidden])
        self.processor = torch.nn.ModuleList([gnn.GCNConv(hidden, hidden) for _ in range(15)])
        self.decoder = torch.nn.Linear(hidden, node_dim)
    
    def forward(self, x, edge_index):
        x = self.encoder(x)
        for layer in self.processor:
            x = torch.relu(layer(x, edge_index))
        return self.decoder(x)
```

### Convergence test
```python
def mesh_convergence(solver_fn, mesh_sizes):
    """매 element size 의 의 의 result 의 stable?"""
    results = {}
    for h in mesh_sizes:
        results[h] = solver_fn(h)
    diffs = [abs(results[mesh_sizes[i]] - results[mesh_sizes[i+1]]) for i in range(len(mesh_sizes)-1)]
    return diffs
```

### Post-processing (paraview)
```python
import pyvista as pv
mesh = pv.read('result.vtk')
mesh.plot(scalars='displacement', cmap='viridis')
```

### Material library
```python
MATERIALS = {
    'steel': {'E': 200e9, 'nu': 0.3, 'rho': 7850, 'sy': 250e6},
    'aluminum': {'E': 70e9, 'nu': 0.33, 'rho': 2700, 'sy': 95e6},
    'concrete': {'E': 30e9, 'nu': 0.2, 'rho': 2400, 'sy': 30e6},
}
```

### JAX-FEM (differentiable)
```python
import jax_fem
mesh = jax_fem.gen_mesh.box_mesh(10, 10, 10, 1.0, 1.0, 1.0)
problem = jax_fem.LinearElasticity(mesh, E=200e9, nu=0.3)
sol = jax_fem.solver.solver(problem, ...)
# 매 sensitivity / topology opt 의 가능
```

## 매 결정 기준
| 상황 | Tool |
|---|---|
| Open-source academic | FEniCS / JAX-FEM |
| Industry structural | ANSYS / Abaqus |
| Cheap PoC | PyANSYS / FreeCAD |
| ML surrogate | PINN / MeshGraphNet |
| Topology opt | JAX-FEM (diff) |
| Mobile / real-time | Surrogate model |

**기본값**: 매 commercial = ANSYS/Abaqus + 매 open-source = FEniCS + 매 ML augmentation = MeshGraphNet for surrogate.

## 🔗 Graph
- 변형: [[FEM]]
- 응용: [[ANSYS]] · [[Abaqus]]
- Adjacent: [[PINN]]

## 🤖 LLM 활용
**언제**: 매 engineering simulation. 매 design optimization.
**언제 X**: 매 simple analytical solution.

## ❌ 안티패턴
- **Skip mesh convergence**: 매 unreliable.
- **Linear for nonlinear regime**: 매 wrong.
- **Wrong element type**: 매 locking.
- **No BC validation**: 매 garbage.
- **Pure ML w/o physics**: 매 OOD fail.

## 🧪 검증 / 중복
- Verified (Zienkiewicz Finite Element Method, FEniCS docs).
- 신뢰도 A.

## 🕓 Changelog
| 날짜 | 변경 |
|---|---|
| 2026-04-26 | FEA auto |
| 2026-05-08 | Phase 1 |
| 2026-05-10 | Manual cleanup — FEM steps + 매 FEniCS / PyANSYS / PINN / GNN / convergence code |