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L2 — Preknowledge（预备知识）

Q7 🟡 线弹性问题的三类基本方程

EN: The three sets of governing equations for linear elasticity are:

Type Equation Meaning
Kinematic (Geometric) ε=∇u\boldsymbol{\varepsilon} = \nabla\mathbf{u} Displacement → Strain (6 equations)
Constitutive σ=Dε\boldsymbol{\sigma} = \mathbf{D}\boldsymbol{\varepsilon} Strain → Stress (6 equations)
Equilibrium ∇Tσ+f=0\nabla^T\boldsymbol{\sigma} + \mathbf{f} = \mathbf{0} Force balance (3 equations)

Total: 15 equations for 15 unknowns (6 stresses + 6 strains + 3 displacements).

中文： 三类方程——(1)几何方程 ε=∇u\boldsymbol{\varepsilon}=\nabla\mathbf{u}：位移求导得应变；(2)本构方程 σ=Dε\boldsymbol{\sigma}=\mathbf{D}\boldsymbol{\varepsilon}：应变映射为应力（Hooke定律）；(3)平衡方程 ∇Tσ+f=0\nabla^T\boldsymbol{\sigma}+\mathbf{f}=0：力平衡。共15个方程，15个未知量。

Q8 🟡 最小势能原理

EN: The Principle of Minimum Potential Energy states that among all kinematically admissible displacement fields, the one that satisfies equilibrium minimizes the total potential energy:

Πp=U+W=12∫VεTσ dV−∫VuTf dV−∫SuTT dS\Pi_p = U + W = \frac{1}{2}\int_V \boldsymbol{\varepsilon}^T\boldsymbol{\sigma}\,dV - \int_V \mathbf{u}^T\mathbf{f}\,dV - \int_S \mathbf{u}^T\mathbf{T}\,dS

The equilibrium condition is δΠp=0\delta\Pi_p = 0. Substituting u=Nd\mathbf{u} = \mathbf{Nd} and ε=Bd\boldsymbol{\varepsilon} = \mathbf{Bd} yields the FEM equation Kd=F\mathbf{Kd} = \mathbf{F}.

中文： 在所有运动学容许的位移场中，使总势能（应变能+外力势能）取最小值的那个满足平衡条件。δΠp=0\delta\Pi_p=0 → 代入 u=Nd\mathbf{u}=\mathbf{Nd}, ε=Bd\boldsymbol{\varepsilon}=\mathbf{Bd} → 得FEM方程 Kd=F\mathbf{Kd}=\mathbf{F}。

Q9 ⭐ 单元刚度矩阵的一般公式

EN: The element stiffness matrix in its most general form is:

ke=∫VeBTDB dV\boxed{\mathbf{k}_e = \int_{V_e} \mathbf{B}^T \mathbf{D} \mathbf{B} \, dV}

where B\mathbf{B} is the strain-displacement matrix (ε=Bd\boldsymbol{\varepsilon} = \mathbf{Bd}), D\mathbf{D} is the constitutive (material) matrix (σ=Dε\boldsymbol{\sigma} = \mathbf{D}\boldsymbol{\varepsilon}), and the integration is over the element volume.

中文： ke=∫BTDB dV\mathbf{k}_e=\int\mathbf{B}^T\mathbf{D}\mathbf{B}\,dV 是单元刚度矩阵的通式。B是应变-位移矩阵，D是材料本构矩阵。这是FEM最重要的公式之一，所有单元刚度矩阵都由它推导。

Q10 ⭐ 位移插值关系

EN: In FEM, the displacement field within an element is interpolated from nodal displacements using shape functions:

u=Nd\mathbf{u} = \mathbf{N}\mathbf{d}

The strain is then: ε=Bd\boldsymbol{\varepsilon} = \mathbf{B}\mathbf{d}, where B\mathbf{B} is the derivative of N\mathbf{N}.

The stress is: σ=DBd\boldsymbol{\sigma} = \mathbf{D}\mathbf{B}\mathbf{d}.

中文： 三大基本关系——位移插值 u=Nd\mathbf{u}=\mathbf{Nd}（N是形函数矩阵）；应变 ε=Bd\boldsymbol{\varepsilon}=\mathbf{Bd}（B是N的导数）；应力 σ=DBd\boldsymbol{\sigma}=\mathbf{D}\mathbf{B}\mathbf{d}。这是FEM推导的三步曲。

Q11 ⭐ 刚度矩阵奇异性

EN: Before applying sufficient boundary conditions, the global stiffness matrix K\mathbf{K} is singular (determinant = 0, cannot be inverted). This is because the structure can undergo rigid-body motion — translating or rotating without developing any strain or stress. Sufficient constraints must be applied to eliminate all rigid-body modes before solving for displacements.

中文： 未施加足够约束前，K是奇异的（不可逆），因为结构可发生刚体位移（平动/转动不产生应变）。必须施加足够的位移约束消除所有刚体模态后，才能求解位移。

Q12 🔴 位移型FEM给出精确解的"下界"

EN: In displacement-based FEM, the model is stiffer than the actual structure because the assumed displacement field (shape functions) constrains the deformation. As a result, FEM displacements are smaller in magnitude than the exact solution — the FEM solution provides a lower bound and converges to the exact solution from below as the mesh is refined. This is known as the “stiffening effect” of displacement-based FEA. Note: this applies only to displacement-based formulations, not mixed or hybrid methods.

中文： 位移型FEM偏硬（stiffening effect），因为假定的形函数约束了变形模式，使结构比实际更"刚"。因此FEM位移解小于精确解（下界解），随网格加密从下方收敛。注意这只适用于位移型FEM，混合法和杂交法不适用。

Q13 🟢 对称矩阵与正交矩阵

EN: A symmetric matrix satisfies AT=A\mathbf{A}^T = \mathbf{A} (e.g., K\mathbf{K}, D\mathbf{D}, M\mathbf{M} in FEM). An orthogonal matrix satisfies ATA=I\mathbf{A}^T\mathbf{A} = \mathbf{I}, meaning A−1=AT\mathbf{A}^{-1} = \mathbf{A}^T (e.g., rotation/coordinate transformation matrices).

中文： 对称矩阵：AT=A\mathbf{A}^T=\mathbf{A}（如K,D,M都是对称的）；正交矩阵：ATA=I\mathbf{A}^T\mathbf{A}=\mathbf{I}（如坐标变换矩阵）。利用对称性可节省一半以上的存储和计算量。

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