Numerical Integration¶
Newton-Cotes Formulas (Quadrature)¶
Lagrangian interpolation
\[ l_j(x) = \prod_{k=0, k \neq j}^n \frac{x - x_k}{x_j - x_k} \]
Newton-Cotes formulas
\[ \int_a^b f(x) \, \mathrm{d} x \approx \sum_{j=0}^n C_j f(x_j), \quad C_j = \int_a^b l_j(x) \, \mathrm{d} x \]
Gaussian Quadrature¶
\[ 2n - 1 \geq 2(p - 1) \implies n \geq \frac{2p - 1}{2} \]
Reduc¶
Hourglass Mode
Spurious zero-energy modes that can arise in under-integrated elements, leading to non-physical deformations.
Use full integration schemes or apply hourglass control techniques to mitigate this issue.
Accuracy of Stress and Strain
- 应变和应力的精确度比位移低
- 应力和应变在高斯点上的精确度更高
不光滑的应变区域边界 \(\longrightarrow\) 光滑化