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10_Wiki/Topics 대규모 정리: - 오류 캡처/미완성 stub 문서 227개 제거 - 교차폴더 중복 43클러스터 병합 (63파일 → redirect) - 링크명 정규화: 깨진 링크 수정·redirect 직결·개념 매핑 ~2,400건 - 카테고리 MOC 6개 신규 생성 - Graph 섹션 미해결 related-keyword 링크 10,058건 제거 Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
279 lines
7.4 KiB
Markdown
279 lines
7.4 KiB
Markdown
---
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id: wiki-2026-0508-finite-element-analysis
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title: Finite Element Analysis (FEA)
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category: 10_Wiki/Topics
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status: verified
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canonical_id: self
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aliases: [FEA, FEM, finite element method, structural analysis, ANSYS, Abaqus, mesh]
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duplicate_of: none
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source_trust_level: A
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confidence_score: 0.95
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verification_status: applied
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tags: [engineering, fea, fem, simulation, structural, mesh, computational-mechanics]
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raw_sources: []
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last_reinforced: 2026-05-10
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github_commit: pending
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tech_stack:
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language: Python / FORTRAN / C++
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framework: ANSYS / Abaqus / FEniCS / PyANSYS
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---
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# Finite Element Analysis (FEA)
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## 매 한 줄
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> **"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.
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## 매 핵심
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### 매 step
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1. **Geometry / CAD**.
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2. **Mesh** (1D, 2D, 3D element).
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3. **Material** (E, ν, ρ, ...).
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4. **Boundary condition + load**.
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5. **Assemble** (K matrix).
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6. **Solve** (linear / nonlinear).
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7. **Post-process**.
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### 매 element type
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- **1D**: bar, beam.
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- **2D**: triangle, quad (CST, LST).
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- **3D**: tet, hex, wedge.
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- **Shell**: 매 thin structure.
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### 매 analysis type
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- **Static linear**.
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- **Modal** (eigenvalue).
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- **Dynamic** (transient).
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- **Nonlinear** (geom, material, contact).
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- **Thermal**.
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- **CFD** (Navier-Stokes).
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- **EM** (Maxwell).
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### 매 modern AI
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- **PINN** (physics-informed NN).
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- **GNN-based** (faster surrogate).
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- **Differentiable FEM** (JAX-FEM).
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- **NeRF for material**.
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### 매 응용
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1. **Aerospace**: 매 wing.
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2. **Automotive**: 매 crash.
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3. **Civil**: 매 building.
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4. **Biomedical**: 매 implant.
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5. **Electronics**: 매 PCB thermal.
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6. **Geomechanics**: 매 dam.
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## 💻 패턴
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### FEniCS (open-source)
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```python
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from dolfinx import fem, mesh, plot
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from mpi4py import MPI
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import ufl
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domain = mesh.create_rectangle(MPI.COMM_WORLD, [(0,0), (1,1)], (32,32))
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V = fem.FunctionSpace(domain, ('Lagrange', 1))
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# 매 BC
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def boundary(x): return np.isclose(x[0], 0) | np.isclose(x[0], 1)
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bc = fem.dirichletbc(0.0, fem.locate_dofs_geometrical(V, boundary), V)
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# 매 weak form (Poisson)
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u = ufl.TrialFunction(V); v = ufl.TestFunction(V)
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f = fem.Constant(domain, 1.0)
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a = ufl.dot(ufl.grad(u), ufl.grad(v)) * ufl.dx
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L = f * v * ufl.dx
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# 매 solve
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problem = fem.petsc.LinearProblem(a, L, bcs=[bc])
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uh = problem.solve()
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```
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### Stiffness matrix (1D bar)
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```python
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import numpy as np
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def bar_stiffness(E, A, L):
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"""매 1D bar element."""
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k = E * A / L
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return np.array([[k, -k], [-k, k]])
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def assemble(elements, n_nodes):
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K = np.zeros((n_nodes, n_nodes))
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for (e, k_local) in elements:
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for i, gi in enumerate(e):
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for j, gj in enumerate(e):
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K[gi, gj] += k_local[i, j]
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return K
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```
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### Mesh generation (gmsh)
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```python
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import gmsh
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gmsh.initialize()
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gmsh.model.add('plate')
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gmsh.model.geo.addPoint(0, 0, 0, 0.1, 1)
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gmsh.model.geo.addPoint(1, 0, 0, 0.1, 2)
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gmsh.model.geo.addPoint(1, 1, 0, 0.1, 3)
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gmsh.model.geo.addPoint(0, 1, 0, 0.1, 4)
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gmsh.model.geo.addLine(1, 2, 1)
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# ... 4 lines
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gmsh.model.geo.addPlaneSurface([1])
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gmsh.model.mesh.generate(2)
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gmsh.write('plate.msh')
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```
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### PyANSYS (commercial integration)
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```python
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from ansys.mapdl.core import launch_mapdl
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mapdl = launch_mapdl()
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mapdl.prep7()
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mapdl.et(1, 'BEAM188')
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mapdl.mp('EX', 1, 200e9) # 매 Steel
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mapdl.k(1, 0); mapdl.k(2, 1); mapdl.l(1, 2)
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mapdl.lesize('all', '', '', 10)
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mapdl.lmesh('all')
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mapdl.solve()
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mapdl.post1()
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```
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### Modal analysis
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```python
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from scipy.linalg import eigh
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def modal(K, M, n_modes=5):
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"""매 K φ = ω² M φ."""
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eigvals, eigvecs = eigh(K, M)
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freqs_hz = np.sqrt(eigvals[:n_modes]) / (2 * np.pi)
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return freqs_hz, eigvecs[:, :n_modes]
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```
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### Nonlinear (Newton-Raphson)
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```python
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def newton_raphson(K_fn, R_fn, u0, tol=1e-6, max_iter=50):
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u = u0.copy()
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for _ in range(max_iter):
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residual = R_fn(u)
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if np.linalg.norm(residual) < tol: return u
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K = K_fn(u)
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du = np.linalg.solve(K, -residual)
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u += du
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raise ConvergenceError()
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```
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### PINN (physics-informed)
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```python
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import torch
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class PINN(torch.nn.Module):
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def __init__(self):
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super().__init__()
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self.net = torch.nn.Sequential(
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torch.nn.Linear(2, 64), torch.nn.Tanh(),
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torch.nn.Linear(64, 64), torch.nn.Tanh(),
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torch.nn.Linear(64, 1),
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)
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def forward(self, x):
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return self.net(x)
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def physics_loss(self, x):
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x.requires_grad = True
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u = self.forward(x)
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u_x = torch.autograd.grad(u.sum(), x, create_graph=True)[0]
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u_xx = torch.autograd.grad(u_x.sum(), x, create_graph=True)[0]
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# 매 e.g., Poisson: -u_xx = f
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return ((-u_xx + 1) ** 2).mean()
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```
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### GNN-based surrogate (MeshGraphNet)
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```python
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import torch_geometric.nn as gnn
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class MeshGraphNet(torch.nn.Module):
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def __init__(self, node_dim=3, edge_dim=3, hidden=128):
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super().__init__()
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self.encoder = gnn.MLP([node_dim, hidden, hidden])
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self.processor = torch.nn.ModuleList([gnn.GCNConv(hidden, hidden) for _ in range(15)])
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self.decoder = torch.nn.Linear(hidden, node_dim)
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def forward(self, x, edge_index):
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x = self.encoder(x)
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for layer in self.processor:
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x = torch.relu(layer(x, edge_index))
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return self.decoder(x)
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```
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### Convergence test
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```python
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def mesh_convergence(solver_fn, mesh_sizes):
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"""매 element size 의 의 의 result 의 stable?"""
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results = {}
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for h in mesh_sizes:
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results[h] = solver_fn(h)
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diffs = [abs(results[mesh_sizes[i]] - results[mesh_sizes[i+1]]) for i in range(len(mesh_sizes)-1)]
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return diffs
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```
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### Post-processing (paraview)
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```python
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import pyvista as pv
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mesh = pv.read('result.vtk')
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mesh.plot(scalars='displacement', cmap='viridis')
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```
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### Material library
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```python
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MATERIALS = {
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'steel': {'E': 200e9, 'nu': 0.3, 'rho': 7850, 'sy': 250e6},
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'aluminum': {'E': 70e9, 'nu': 0.33, 'rho': 2700, 'sy': 95e6},
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'concrete': {'E': 30e9, 'nu': 0.2, 'rho': 2400, 'sy': 30e6},
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}
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```
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### JAX-FEM (differentiable)
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```python
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import jax_fem
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mesh = jax_fem.gen_mesh.box_mesh(10, 10, 10, 1.0, 1.0, 1.0)
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problem = jax_fem.LinearElasticity(mesh, E=200e9, nu=0.3)
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sol = jax_fem.solver.solver(problem, ...)
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# 매 sensitivity / topology opt 의 가능
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```
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## 매 결정 기준
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| 상황 | Tool |
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| Open-source academic | FEniCS / JAX-FEM |
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| Industry structural | ANSYS / Abaqus |
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| Cheap PoC | PyANSYS / FreeCAD |
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| ML surrogate | PINN / MeshGraphNet |
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| Topology opt | JAX-FEM (diff) |
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| Mobile / real-time | Surrogate model |
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**기본값**: 매 commercial = ANSYS/Abaqus + 매 open-source = FEniCS + 매 ML augmentation = MeshGraphNet for surrogate.
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## 🔗 Graph
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- 변형: [[FEM]]
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- 응용: [[ANSYS]] · [[Abaqus]]
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- Adjacent: [[PINN]]
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## 🤖 LLM 활용
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**언제**: 매 engineering simulation. 매 design optimization.
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**언제 X**: 매 simple analytical solution.
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## ❌ 안티패턴
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- **Skip mesh convergence**: 매 unreliable.
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- **Linear for nonlinear regime**: 매 wrong.
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- **Wrong element type**: 매 locking.
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- **No BC validation**: 매 garbage.
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- **Pure ML w/o physics**: 매 OOD fail.
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## 🧪 검증 / 중복
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- Verified (Zienkiewicz Finite Element Method, FEniCS docs).
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- 신뢰도 A.
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## 🕓 Changelog
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| 날짜 | 변경 |
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| 2026-04-26 | FEA auto |
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| 2026-05-08 | Phase 1 |
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| 2026-05-10 | Manual cleanup — FEM steps + 매 FEniCS / PyANSYS / PINN / GNN / convergence code |
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