Finite Element Method

Every Boeing part undergoes FEM simulation before manufacturing. Coronary stents are FEM-tested under various blood pressures. The iPhone chassis is FEM-optimized for minimum weight. FEM is the language of engineering design.

• **Structural analysis in ANSYS/ABAQUS:** FEM for stresses and deformations in 3D parts; stiffness matrices with millions of DOFs
• **FEniCS in research:** open-source; Python API; used in academic fluid dynamics and biomechanics
• **Electromagnetic simulation:** Maxwell's equations via FEM; ANSYS HFSS, OpenEMS for antennas and ICs

Предварительные знания

• Finite Difference Methods for PDEs

Weak Formulation

The finite element method (FEM) starts from the **weak (variational) formulation** of a PDE. Instead of requiring the equation at every point (strong form), we require it to hold 'on average' against a set of test functions. This allows irregular meshes, discontinuous coefficients, and natural boundary conditions.

**Weak formulation of Poisson's equation:** Strong form: −∇·(k∇u) = f, u|_{∂Ω} = 0 Multiply by test function v and integrate over Ω: ∫_Ω k∇u·∇v dΩ = ∫_Ω f·v dΩ for all v ∈ H₀¹(Ω) (integration by parts removes the second derivative!) **Advantages:** - Requires less smoothness (H¹ instead of H²) - Naturally includes Neumann boundary conditions - Handles discontinuous coefficients k(x)

The key insight: integration by parts shifts the derivative from the unknown u onto the test function v. This symmetrizes the problem (stiffness matrix is symmetric for self-adjoint operators) and reduces smoothness requirements.

Galerkin Method and Matrix Assembly

The Galerkin method approximates u as a finite sum of basis functions φᵢ: u_h = Σᵢ uᵢ·φᵢ. Substituting into the weak form with test functions v = φⱼ yields the system Ku = f, where Kᵢⱼ = ∫ k∇φᵢ·∇φⱼ dΩ is the stiffness matrix.

**Stiffness matrix and load vector:** Kᵢⱼ = ∫_Ω k·∇φᵢ·∇φⱼ dΩ ← stiffness matrix fᵢ = ∫_Ω f·φᵢ dΩ ← load vector **Local-to-global assembly:** 1. For each element Ωₑ: compute K_local and f_local 2. Scatter into global K and f using the DOF table **Basis functions φᵢ:** - P1 (linear): φᵢ = 1 at node i, 0 at others - P2 (quadratic): parabolic interpolation - Higher order: hp-FEM

FEniCS and Applications

FEniCS/FEniCSx is an open-source FEM platform that allows formulating PDEs mathematically and automatically generating optimized code for matrix assembly and solving. Used in aerospace engineering, biomechanics, and electromagnetics.

Key Ideas

• **Weak form:** integration by parts reduces smoothness requirements; natural boundary conditions included automatically
• **Galerkin method:** u = Σ uᵢφᵢ → system Ku = f; Kᵢⱼ = ∫k∇φᵢ·∇φⱼ
• **Sparsity of K:** Kᵢⱼ ≠ 0 only for neighboring nodes; O(n) nonzeros for P1
• **FEniCS:** Python API for automatic assembly and solving; solve(a == L, u, bc)

Related Topics

FEM combines PDEs, linear algebra, and numerical methods:

• Finite Difference Methods for PDEs — FD is a special case of FEM on regular grids; the same sparse linear systems arise
• Sparse Matrices — The stiffness matrix K is sparse; CSR/CSC storage and iterative solvers are mandatory

Вопросы для размышления

• Why does P2 FEM (quadratic elements) give O(h⁴) accuracy instead of O(h²) for P1?
• How is a Neumann boundary condition (specified flux) naturally incorporated into the weak formulation?
• What is hp-adaptivity in FEM and when is it preferred over uniform mesh refinement?

Связанные уроки

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