有限元方法问答 07：建模实践

上一篇：有限元方法问答 06：轴对称单元 · 系列总目录 · 下一篇：有限元方法问答 08：等参单元

L8 Practical — 建模实践

Q50 🔴 长宽比(Aspect Ratio)

EN: The Aspect Ratio is defined as:

$\text{Aspect Ratio} = \frac{\text{longest dimension}}{\text{shortest dimension}}$

As the aspect ratio increases, solution accuracy decreases. The effect is most severe in regions of high stress gradient. In regions where the stress gradient is nearly zero, a larger aspect ratio may be tolerable. The ideal aspect ratio is close to 1.

中文： 长宽比 = 最长尺寸/最短尺寸。长宽比越大，精度越低。应力梯度越大的区域对长宽比越敏感。理想长宽比接近1。应避免过大长宽比、过度扭曲、角度过大或过小的单元。

Q51 ⭐ 需要在哪些位置布置节点？

EN: Nodes (and mesh lines) must be placed at:

Geometric discontinuities — abrupt changes in geometry, thickness changes, corners
Loading points — concentrated forces, distributed load start/end
Material interfaces — changes in material properties
Boundary condition changes — transitions between different support types
Stress concentration regions — holes, notches, sharp corners, fillets

中文： 节点必须布置在：几何突变处、载荷作用点、材料界面、边界条件变化处、应力集中区（孔、槽、尖角）。在变化剧烈处应加密网格。

Q52 🔴 不同单元连接时的自由度匹配

EN: When connecting different element types (e.g., plane elements with beam elements), the DOF compatibility must be carefully considered. Plane elements have only translational DOFs ( $u, v$ ), while beam elements also have rotational DOFs ( $\theta$ ). A direct connection means the plane element cannot transmit the moment associated with the beam’s rotational DOF. Solutions include:

Constraint equations linking DOFs
Transition elements
Avoiding direct connections (use consistent element types throughout)

中文： 平面单元只有平动DOF( $u,v$ )，梁单元还有转动DOF( $\theta$ )。直接连接时，平面单元不能传递梁的弯矩。解决方案：约束方程、过渡单元、或使用一致的元素类型。这是建模中的常见错误来源。

Q53 ⭐ 对称性的使用条件和方法

EN: Symmetry can be exploited to reduce model size when ALL of the following are symmetric:

Geometry
Material properties
Boundary conditions
Loading

Symmetry BCs: On the symmetry plane, the normal displacement component is zero; tangential displacement is free.

Reduction: One symmetry line → half model; two orthogonal symmetry lines → quarter model.

⚠️ CAUTION: Do NOT use symmetry blindly for vibration and buckling analysis. Symmetric structures can have anti-symmetric mode shapes that would be completely missed by a symmetric model.

中文： 必须四者（几何、材料、BC、载荷）都对称才能利用对称性。对称面BC：法向位移=0。一对称线减半，两正交对称线减至四分之一。⚠️ 振动/屈曲分析不能盲目用对称，会遗漏反对称模态！

Q54 🔴 静力凝聚(Static Condensation)

EN: Static condensation is a technique to eliminate internal DOFs without losing accuracy, reducing the system size. Given partitioned equations:

$\begin{bmatrix} \mathbf{K}_{aa} & \mathbf{K}_{ab} \\ \mathbf{K}_{ba} & \mathbf{K}_{bb} \end{bmatrix} \begin{Bmatrix} \mathbf{U}_a \\ \mathbf{U}_b \end{Bmatrix} = \begin{Bmatrix} \mathbf{F}_a \\ \mathbf{F}_b \end{Bmatrix}$

Where $a$ = external (retained) DOFs, $b$ = internal (condensed) DOFs:

$\mathbf{U}_b = \mathbf{K}_{bb}^{-1}(\mathbf{F}_b - \mathbf{K}_{ba}\mathbf{U}_a)$

Substituting back gives the condensed equation:

$\boxed{(\mathbf{K}_{aa} - \mathbf{K}_{ab}\mathbf{K}_{bb}^{-1}\mathbf{K}_{ba})\mathbf{U}_a = \mathbf{F}_a - \mathbf{K}_{ab}\mathbf{K}_{bb}^{-1}\mathbf{F}_b}$

Common use: condensing interior nodes of subdivided quadrilaterals, creating superelements.

中文： 静力凝聚通过消去内部自由度来减小方程规模，但不损失精度。常用于消去四边形内部节点（4个三角形→5节点→凝聚为4节点）。凝聚后的刚度矩阵 $\mathbf{K}_{aa}-\mathbf{K}_{ab}\mathbf{K}_{bb}^{-1}\mathbf{K}_{ba}$ 称为等效刚度矩阵。

Q55 🟡 子结构(Substructure)思想

EN: A substructure (superelement) treats a portion of a structure as a single component. The interior DOFs are condensed to boundary DOFs via static condensation. Advantages: handles large problems, different teams can model different components, partial redesign requires only partial reanalysis, efficient for local nonlinearities. Disadvantages: increased file management, approximate for dynamic problems.

中文： 子结构 = 将复杂结构的一部分视为一个"超级单元"，内部DOF凝聚到边界DOF。优点：适合大型结构、可分工建模、局部修改只需重新分析局部。缺点：文件管理复杂、动力问题有近似性。

Q56 ⭐ 有限元解的三大误差来源

EN: The three major errors in FEM:

Discretization Error（离散误差）: Error from replacing the continuous structure with a finite number of elements. Decreases as the mesh is refined.
Modeling Error（建模误差）: Error from idealizing the physical problem — geometric simplifications, material model assumptions (e.g., linear elastic vs. plastic), boundary condition idealizations.
Numerical Error（数值误差）: Error from finite-precision computation — round-off error, numerical integration error (insufficient Gauss points), error in solving linear systems.

中文： (1)离散误差——用有限元离散化替代连续体，网格越粗误差越大；(2)建模误差——物理问题理想化带来的误差（几何简化、材料假设、边界条件理想化）；(3)数值误差——计算精度有限导致的舍入误差、数值积分误差、方程求解误差。

Q57 ⭐ 网格收敛性(Mesh Convergence)

EN: Mesh convergence is a measure of the number of elements required in a model to ensure that the results of an analysis are not affected by changing the size of the mesh. The procedure: (1) create an initial mesh; (2) refine the mesh; (3) compare key results (displacements, stresses); (4) if results change significantly, refine further; (5) when further refinement produces negligible change → convergence achieved.

中文： 网格收敛性指：逐步加密网格后，关键结果趋于稳定，不再随网格尺寸减小而明显变化。标准做法：加密→比较→再加密→直至结果稳定。位移收敛通常比应力收敛更快。

EN: Three types of mesh refinement:

h-refinement: Reduce element size $h$ — increase the number of elements. Most common approach.
p-refinement: Increase the polynomial order $p$ of shape functions on the same mesh (e.g., CST → LST). Improves accuracy without changing the mesh.
r-refinement: Reposition nodes while keeping the same number of elements and DOFs — concentrate nodes in high-gradient regions.
hp-refinement: Combination of h- and p-refinement — produces the best convergence rate.

中文： h-细化（缩小单元尺寸）、p-细化（提高形函数阶次）、r-细化（重布节点位置）、hp-细化（组合法，收敛最快）。FEM收敛性分析中h-细化最常用。

Q59 🔴 FEM应力/应变的精度特点

EN: In displacement-based FEM:

Displacement has the highest accuracy and is continuous across element boundaries.
Strain and stress have lower accuracy (derived from displacement derivatives) and are generally discontinuous across element boundaries.
Stresses/strains are most accurate at Gauss integration points (superconvergent points).
Nodal stresses are obtained by extrapolation from Gauss points or by averaging (stress smoothing) from adjacent elements.
Simple averaging: $\boldsymbol{\sigma}_P = \frac{1}{n}\sum \boldsymbol{\sigma}_P^{(i)}$ or area-weighted averaging.

中文： 位移精度 > 应力/应变精度（因为应力/应变来自位移的导数）。位移在单元间连续，应力/应变不连续。Gauss积分点处应力最准确（超收敛点）。节点应力通过外推或平均得到（应力平滑）。

Q60 🟡 直接法 vs 迭代法求解

EN: Direct methods (Gauss elimination): Solution time $\propto N \cdot B^2$ ( $N$ = matrix dimension, $B$ = bandwidth). Good for small-to-medium problems, slender structures (small bandwidth), multiple load cases. Iterative methods: Reduced storage, good for large problems, bulky structures (faster convergence), but unknown solution time and need resolving for different load cases.

中文： 直接法（Gauss消元）：求解时间正比于 $N\times B^2$ ，适合中小型、细长结构、多载荷工况；迭代法：存储量小，适合大型、块状结构，但求解时间不可预测，不同载荷需重新求解。