# 在 COMSOL Multiphysics® 中曲线拟合解数据

2021年 7月 27日

﻿在 COMSOL Multiphysics® 软件中求解一个模型后，我们可能希望将求解数据拟合到在仿真域中定义的一组函数中。在之前的博文中，我们解释了如何将离散试验数据拟合到曲线上。今天，我们将考虑对连续解数据进行拟合。之后，我们将介绍正交性的概念，并解释如何将解数据拟合到一组正交函数中，从而简化为简单方便的后处理操作。

### 最小二乘拟合概述

u(\mathbf{r}) \approx \sum_{i=1}^N c_i f_i(\mathbf{r})

\int_\Omega \left(u(\mathbf{r})-\sum_{i=1}^N c_if_i(\mathbf{r})\right)^2\textrm{d}\Omega

\frac{\partial}{\partial c_j}\left[\int_\Omega \left(u(\mathbf{r})-\sum_{i=1}^N c_if_i(\mathbf{r})\right)^2\textrm{d}\Omega\right]=0

2\int_\Omega \left(u(\mathbf{r})-\sum_{i=1}^N c_if_i(\mathbf{r})\right)\left(-f_j(\mathbf{r})\right)\textrm{d}\Omega=0

\sum_{i=1}^N c_i \int_\Omega f_i(\mathbf{r}) f_j(\mathbf{r}) \textrm{d}\Omega = \int_\Omega u(\mathbf{r})f_j(\mathbf{r})\textrm{d}\Omega

\begin{aligned}
&\begin{bmatrix}
A_{11} & A_{12} & A_{13}\\
A_{21} & A_{22} & A_{23}\\
A_{31} & A_{32} & A_{33}
\end{bmatrix}
\begin{bmatrix}
c_1\\
c_2\\
c_3
\end{bmatrix}
=
\begin{bmatrix}
B_1\\B_2\\B_3\\
\end{bmatrix}\\
&A_{ij} \equiv \int_\Omega f_i(\mathbf{r})f_j(\mathbf{r})\textrm{d}\Omega
\qquad B_{j} \equiv \int_\Omega u(\mathbf{r})f_j(\mathbf{r})\textrm{d}\Omega
\end{aligned}

### 正交函数简介

\langle f_i,f_j\rangle \equiv \int_\Omega f_i(\mathbf{r})f_j(\mathbf{r})w(\mathbf{r})\textrm{d}\mathbf{r}

\langle f_i, f_j \rangle \equiv \int_{-\pi}^{\pi}f_i(x)f_j(x)\textrm{d}x

\begin{aligned}
\int_{-\pi}^\pi (1)^2 \textrm{d}x &= 2\pi \qquad \int_{-\pi}^\pi (\sin x)^2 \textrm{d}x = \pi \qquad \int_{-\pi}^\pi (\cos x)^2 \textrm{d}x = \pi \\
\int_{-\pi}^\pi (1)(\sin x) \textrm{d}x &= 0 \qquad \int_{-\pi}^\pi (1)(\cos x) \textrm{d}x = 0 \qquad\int_{-\pi}^\pi (\sin x)(\cos x) \textrm{d}x = 0\\
\end{aligned}

### 正交函数的最小二乘拟合

\int_\Omega \sum_{i=1}^N c_i f_i(\mathbf{r}) f_j(\mathbf{r}) \textrm{d}\Omega = \int_\Omega u(\mathbf{r})f_j(\mathbf{r})\textrm{d}\Omega

\langle f_i, f_j \rangle \equiv \int_\Omega f_i(\mathbf{r})f_j(\mathbf{r})\textrm{d}\Omega

c_i = \frac{\int_\Omega u(\mathbf{r})f_i(\mathbf{r})\textrm{d}\Omega}{\int_\Omega f_i(\mathbf{r}
)^2\textrm{d}\Omega}

c_i = \int_\Omega u(\mathbf{r})f_i(\mathbf{r})\textrm{d}\Omega

### 泽尼克多项式

Z_n^m(\rho,\theta) = N_n^m R_n^{|m|}(\rho)M(m\theta)

• ρ 是径向坐标(0\leq\rho\leq 1)
• θ 是方位角 (0\leq \theta < 2\pi)
• N_n^m 是归一化项
• R_n^{|m|}(\rho) 是径向项
• M(m\theta) 是子午项或方位项
• n 是径向指数 (n \in \left\\{0,1,2,\dots\right})
• m 是子午线或方位角指数（对于给定的 n，m \in \left
{-n, -n+2, -n+4, \dots , n-2, n\right}
)

Z_0^0 1 平移
Z_1^{-1} 2\rho \sin(\theta) 垂直倾斜
Z_1^1 2\rho \cos(\theta) 水平倾斜
Z_2^{-2} \sqrt{6}\rho^2 \sin(2\theta) 斜散像差
Z_2^0 \sqrt{3}\left(2\rho^2-1\right) 散焦
Z_2^2 \sqrt{6}\rho^2 \cos(2\theta) 散光
Z_3^{-3} \sqrt{8}\rho^3 \sin(3\theta) 斜向三叶草像差
Z_3^{-1} \sqrt{8}\left(3\rho^3-2\rho\right)\sin(\theta) 垂直慧差
Z_3^1 \sqrt{8}\left(3\rho^3-2\rho\right)\cos(\theta) 水平慧差
Z_3^3 \sqrt{8}\rho^3 \cos(3\theta) 水平三叶草像差
Z_4^{-4} \sqrt{10}\rho^4\sin(4\theta) 斜四叶草像差
Z_4^{-2} \sqrt{10}\left(4\rho^4-3\rho^2\right)\sin(2\theta) 斜次阶散光像差
Z_4^0 \sqrt{5}\left(6\rho^4-6\rho^2+1\right) 球差
Z_4^2 \sqrt{10}\left(4\rho^4-3\rho^2\right)\cos(2\theta) 阶散光像差
Z_4^4 \sqrt{10}\rho^4\cos(4\theta) 水平四叶草像差

\int_0^{2\pi}\int_0^1 Z_n^m(\rho,\theta)Z_p^q(\rho,\theta)\rho\textrm{d}\rho\textrm{d}\theta=\pi\delta_{n,p}\delta_{m,q}

### 参考文献

1. ISO 24157:2008: Ophthalmic optics and instruments — Reporting aberrations of the human eye, International Organization for Standardization, Geneva, Switzerland. Amendment 1, ibid., 2019.
2. ANSI Z80.28-2017: American National Standard for Ophthalmics — Methods of Reporting Optical Aberrations of Eyes. American National Standards Institute, Alexandria, VA.

#### 评论 (2)

2021-10-12

##### 洋洋 张
2021-10-20 COMSOL 员工