有限元方法问答 09：数值积分

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L10 — Numerical Integration（数值积分）

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Q67 ⭐ Gauss积分精度公式

EN: With $n$ Gauss integration points, a polynomial of degree $2n-1$ is integrated exactly:

$$\boxed{\int_{-1}^{1} f(x)\,dx \approx \sum_{i=1}^{n} W_i f(x_i) \quad \text{exact for } f(x) \text{ of degree } \leq 2n-1}$$

Examples:

• $n=1$ (1 point): exact for linear ($p=1$)
• $n=2$ (2 points): exact for cubic ($p=3$)
• $n=3$ (3 points): exact for quintic ($p=5$)

中文： $n$ 点 Gauss 积分精确到 $2n-1$ 次多项式。1点精确1次，2点精确3次，3点精确5次。这是Gauss积分比Newton-Cotes更高效的根本原因。

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Q68 🔴 Newton-Cotes积分精度

EN: Newton-Cotes formulas use equally spaced sampling points. With $n$ points, they are exact for polynomials of degree $n-1$ (or $n$ for even $n$ in some variants). This is much less efficient than Gauss quadrature for the same number of points.

Comparison: With 3 points — Gauss exact to degree 5, Newton-Cotes exact to degree 2 (Simpson’s rule) or 3. FEM programs generally prefer Gauss integration for efficiency.

中文： Newton-Cotes用等距积分点，$n$点精确 $n-1$ 次（或n次），效率远低于Gauss。3点Gauss精确5次，3点Newton-Cotes仅精确2-3次。有限元优先使用Gauss积分。

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Q69 🔴 Gauss vs Newton-Cotes 对比

EN:

Feature	Gauss	Newton-Cotes
Sampling points	Non-equally spaced	Equally spaced
Boundary points sampled?	No	Yes
Accuracy ($n$ points)	$2n-1$ degree	$n-1$ (or $n$) degree
Efficiency	Higher	Lower
Usage in FEM	Preferred	Rare (plasticity starting from surfaces)

Gauss is more efficient because it optimizes both point locations AND weights, while Newton-Cotes fixes point locations and optimizes only weights. However, in some applications like plasticity where yielding starts at element surfaces, Newton-Cotes may be used because it samples boundary points.

中文： Gauss用非等距点（内部点+优化权重），精度远高于等距的Newton-Cotes。有限元优先用Gauss。但Newton-Cotes采样边界点，在塑性（从表面开始屈服）中有应用。

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Q70 ⭐ 4节点和8节点等参单元的完全积分

EN: Recommended full Gauss integration orders:

Element	Shape function order $p$	Full integration	Gauss points
4-node 2D quadrilateral	$p=2$	$2 \times 2$	4 points
8-node 2D quadrilateral	$p=3$	$3 \times 3$	9 points
8-node 3D brick	$p=2$	$2 \times 2 \times 2$	8 points
20-node 3D brick	$p=3$	$3 \times 3 \times 3$	27 points

The requirement comes from: $2n-1 \geq 2(p-1)$, where $p$ is the shape function polynomial order and $p-1$ is the approximate B-matrix polynomial order (after differentiation).

中文： 4节点四边形：2×2 Gauss（4点）；8节点四边形：3×3 Gauss（9点）。完全积分确保被积函数 $\mathbf{B}^T\mathbf{D}\mathbf{B}|\mathbf{J}|$ 中的所有多项式项被精确积分。

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Q71 🔴 减缩积分(Reduced Integration)

EN: Reduced integration uses fewer Gauss points than the full integration scheme:

Element	Full	Reduced
4-node quad	$2\times2$	$1\times1$ (1 point)
8-node quad	$3\times3$	$2\times2$ (4 points)

Advantages: (1) Lower computational cost; (2) Can alleviate shear locking and volumetric locking in certain problems.

Disadvantages/Risks: May introduce spurious zero-energy modes — deformation patterns that produce zero strain at all integration points, leading to hourglass modes. The element has no stiffness against such deformation.

中文： 减缩积分 = 用比完全积分更少的Gauss点。优点：计算量小、可缓解剪切锁死和体积锁死。缺点：可能引入虚假零能模式（沙漏模式）——单元变形但积分点处应变为零，刚度无法抵抗该变形。

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Q72 ⭐ 沙漏模式(Hourglass Mode)

EN: Hourglass mode (also called zero-energy mode, spurious mode, or kinematic mode) occurs when an element deforms in such a way that all integration points register zero strain and stress, yet the element has clearly deformed. The element therefore has no stiffness against this deformation mode.

This most commonly occurs with reduced integration schemes (e.g., $1\times1$ Gauss for a 4-node quadrilateral). With only one integration point at the element center, deformation patterns that preserve the center’s shape produce zero strain energy at that point.

Solutions: (1) Use full integration; (2) Use hourglass control algorithms; (3) Change element type; (4) Improve mesh quality.

中文： 沙漏模式 = 单元变形了但积分点处应变为零，单元对该变形模式无刚度。最常见于4节点单元的1点减缩积分（只有中心一个积分点）。解决方法：用完全积分、沙漏控制、换单元类型、改进网格质量。

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Q73 ⭐ 位移 vs 应力精度

EN: In displacement-based FEM:

• Displacements are most accurate, continuous across elements → primary unknowns
• Strains are less accurate (from displacement derivatives), discontinuous across element boundaries
• Stresses ($\boldsymbol{\sigma} = \mathbf{D}\boldsymbol{\varepsilon}$) are similarly less accurate and discontinuous
• Gauss points give the most accurate stress/strain values (superconvergent points)
• Nodal stresses are typically obtained by extrapolation from Gauss points or averaging across adjacent elements

中文： 精度排序：位移 > 应变 > 应力。位移连续，应力/应变在单元间不连续。Gauss点应力和应变最准（超收敛点）。节点应力需通过外推或平均得到。

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Q74 🔴 Gauss点处应力最准确的原因

EN: Gauss integration points are “superconvergent” points — the error in stress/strain at these points converges at a higher rate than elsewhere in the element. For a 4-node quadrilateral, the $2\times2$ Gauss points give stresses that are one order more accurate than at nodes. This is why commercial FEA software typically reports stresses at integration points and then extrapolates to nodes.

中文： Gauss点处的应力具有"超收敛性"——误差递减速率比其他位置快一阶。因此商业FEM软件通常在积分点计算应力，再外推到节点。这也是节点应力需要平均处理的根本原因。

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Q75 🟡 应力平滑

EN: Since stresses are discontinuous across element boundaries, post-processing often involves stress smoothing to obtain continuous nodal stress fields:

• Simple averaging: $\bar{\boldsymbol{\sigma}}_P = \frac{1}{m}\sum_{e=1}^{m} \boldsymbol{\sigma}_P^{(e)}$ (average of all elements sharing node $P$)
• Area-weighted averaging: $\bar{\boldsymbol{\sigma}}_P = \frac{\sum A_e \boldsymbol{\sigma}_P^{(e)}}{\sum A_e}$ (weighted by element areas)

This produces visually smoother stress contours for engineering interpretation.

中文： 应力平滑——对共享节点的各单元在节点处的应力做平均（简均或面积加权平均），得到连续的节点应力分布，用于绘制光滑的应力云图。

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Q76 🟡 2点Gauss积分点位置和权重

EN: For 1D Gauss quadrature with $n=2$:

• Integration points: $x_1 = -1/\sqrt{3} \approx -0.57735$, $x_2 = +1/\sqrt{3} \approx +0.57735$
• Weights: $W_1 = W_2 = 1.0$

For $n=1$: $x_1 = 0$, $W_1 = 2$.

For $n=3$: $x = \{-0.7746, 0, +0.7746\}$, $W = \{5/9, 8/9, 5/9\}$.

In 2D, the points are the tensor product: $(s_i, t_j)$ with weights $W_i W_j$.

中文： 2点Gauss：$x=\pm0.57735$，权重各1.0。3点：$x=\{0,\pm0.7746\}$，权重 $\{8/9,5/9,5/9\}$。二维是张量积：点 $(s_i,t_j)$，权重 $W_iW_j$。

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