feat: add document conversion templates
Add md-to-pdf and md-to-docx templates for mytoolkit convert command. Migrated from workspace/templates.
This commit is contained in:
Zhengshou Lai
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---
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title: 计算力学导论
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author: 赖正首
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lang: zh
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date: 2025 年
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---
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# 引言
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计算力学是利用数值方法求解力学问题的学科。本章将介绍有限元方法的基本概念。
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## 历史背景
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有限元方法起源于 20 世纪 50 年代的航空工业。Turner 等人于 1956 年首次发表了关于有限元方法的系统性论文。
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## 本章概述
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本书将按以下结构组织：
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- **第一章** 介绍有限元方法的基本原理
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- **第二章** 讨论变分原理与弱形式
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- **第三章** 探讨高阶有限元与谱方法
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---
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# 有限元方法基础
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## 强形式与弱形式
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考虑一维泊松方程：
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$$-u''(x) = f(x), \quad x \in (0, 1)$$
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边界条件为 $u(0) = u(1) = 0$。
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以下定理框使用 Typst 语法；`mytoolkit` 会在转 Typst 前自动包一层 raw，避免行首 `#` 被 Pandoc 转义。
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#theorem(title: "解的存在唯一性")[
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设 $f \in L^2(0,1)$，则上述边值问题存在唯一弱解 $u \in H_0^1(0,1)$。
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]
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#proof[
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根据 Lax-Milgram 定理，双线性形式 $a(u, v) = \int_0^1 u' v' \, \mathrm{d}x$ 在 $H_0^1(0,1)$ 上是连续且强制的，线性泛函 $L(v) = \int_0^1 f v \, \mathrm{d}x$ 是连续的。因此存在唯一解。
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]
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## 伽辽金方法
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#example(title: "一维泊松方程")[
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取试探空间 $V_h = \mathrm{span}\{\phi_1, \phi_2, \dots, \phi_n\}$，其中 $\phi_i$ 为分段线性帽函数。则近似解可写为：
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$$u_h(x) = \sum_{j=1}^n u_j \phi_j(x)$$
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代入弱形式并取测试函数 $v = \phi_i$，得到线性方程组 $K u = F$。
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]
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---
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# 变分原理
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## 极小位能原理
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#lemma(title: "能量泛函的凸性")[
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定义能量泛函 $J(v) = \frac{1}{2} a(v, v) - L(v)$，则 $J$ 是严格凸的。
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]
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## 拉格朗日乘子法
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对于有约束的优化问题，引入拉格朗日乘子：
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$$\mathcal{L}(u, \lambda) = J(u) + \lambda^T (B u - g)$$
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---
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# 高级主题
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## 代码示例
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以下是用 Python 实现的简单有限元求解器：
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```python
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import numpy as np
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from scipy.sparse import csr_matrix
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from scipy.sparse.linalg import spsolve
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def assemble_stiffness(n):
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"""组装一维刚度矩阵"""
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h = 1.0 / (n + 1)
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data = np.array([1/h] * (n-1) + [2/h] * n + [1/h] * (n-1))
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rows = np.array(list(range(n-1)) + list(range(n)) + list(range(1, n)))
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cols = np.array(list(range(1, n)) + list(range(n)) + list(range(n-1)))
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return csr_matrix((data, (rows, cols)), shape=(n, n))
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def solve_poisson_1d(f, n):
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"""求解一维泊松方程"""
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K = assemble_stiffness(n)
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h = 1.0 / (n + 1)
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F = h * f(np.linspace(h, 1-h, n))
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return spsolve(K, F)
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```
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## 练习
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#exercise(title: "3.1")[
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证明一维线性有限元方法的收敛阶为 $O(h^2)$，即：
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$$|u - u_h|_1 \leq C h |u|_2$$
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其中 $C$ 为与 $h$ 无关的常数。
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]
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#exercise(title: "3.2")[
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编写程序计算二维正方形区域上的泊松方程，并与解析解比较收敛阶。
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]
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@@ -0,0 +1,194 @@
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// 教材模板 —— 定理/代码/目录 + 封面页 + 三线表
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#import "@preview/bookly:3.1.0": *
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#import "@preview/great-theorems:0.1.2": *
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#import "@preview/codly:1.3.0": *
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#import "@preview/showybox:2.0.4": showybox
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#let primary = rgb("#1F4E79")
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#let secondary = rgb("#2C3E50")
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#let accent = rgb("#3467A8")
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#let light = rgb("#96A0AA")
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#let rule-blue = accent.lighten(46%)
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// Filled mathblocks: inset so fill does not hug text; radius slightly larger than codly (4pt)
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#let _mathblock-inset = (x: 14pt, y: 12pt)
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#let _mathblock-radius = 5pt
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// Theorem-like environments (great-theorems)
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#let theorem = mathblock(
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blocktitle: "Theorem",
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fill: rgb("#E8F0FE"),
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inset: _mathblock-inset,
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radius: _mathblock-radius,
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)
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#let lemma = mathblock(
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blocktitle: "Lemma",
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fill: rgb("#F3E8FF"),
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inset: _mathblock-inset,
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radius: _mathblock-radius,
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)
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#let proof = proofblock(
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inset: _mathblock-inset,
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radius: _mathblock-radius,
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)
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#let example = mathblock(
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blocktitle: "Example",
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fill: rgb("#F0FDF4"),
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inset: _mathblock-inset,
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radius: _mathblock-radius,
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)
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// Uncounted block: titles must use `title: ...`, not a positional string (see great-theorems README).
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#let exercise = mathblock(
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blocktitle: "Exercise",
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prefix: [],
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titlix: title => [*习题*~#title~。#h(0.25em)],
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fill: rgb("#FEF3C7"),
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inset: (x: 14pt, y: 13pt),
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radius: _mathblock-radius,
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)
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#let cover-page(title, subtitle, date) = {
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page(margin: (top: 3cm, bottom: 3cm, left: 3cm, right: 3cm))[
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#align(center + horizon)[
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#v(2cm)
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#text(size: 28pt, weight: "bold", fill: primary, title)
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#if subtitle != none {
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v(0.8cm)
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text(size: 16pt, fill: accent, subtitle)
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}
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#if date != none {
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v(2cm)
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text(size: 12pt, fill: light, date)
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}
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]
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]
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}
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#let three-line-table(..args) = {
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let kwargs = args.named()
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let children = args.pos()
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let new_children = ()
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let header_seen = false
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for child in children {
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if child.func() == table.header {
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new_children.push(table.hline(stroke: 1.25pt + rule-blue))
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new_children.push(child)
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header_seen = true
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} else if header_seen and child.func() == table.hline {
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new_children.push(table.hline(stroke: 0.75pt + rule-blue))
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header_seen = false
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} else {
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new_children.push(child)
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}
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}
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new_children.push(table.hline(stroke: 1.25pt + rule-blue))
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kwargs.stroke = none
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kwargs.inset = (x: 8pt, y: 5pt)
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table(..new_children, ..kwargs)
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}
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#let textbook(
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title: none,
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subtitle: none,
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date: none,
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body,
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) = {
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if title != none {
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cover-page(title, subtitle, date)
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pagebreak()
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}
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set text(
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font: (
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"Cambria",
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"PingFang SC",
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),
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size: 11pt,
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lang: "zh",
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)
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show raw: set text(font: (
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"Courier New",
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"Menlo",
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"Monaco",
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"SimSun",
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))
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// Long-form reading: most generous body rhythm in this family
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set par(
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justify: true,
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leading: 0.86em,
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spacing: 1.2em,
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first-line-indent: 2em,
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)
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show: codly-init
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codly(
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stroke: 0.5pt + rule-blue,
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fill: rgb("#F8F9FA"),
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zebra-fill: none,
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lang-stroke: none,
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lang-fill: light.lighten(60%),
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radius: 4pt,
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)
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show: great-theorems-init
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set heading(numbering: "1.1")
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// Same 11pt body as default: match list rhythm to default template
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set list(indent: 1.2em, body-indent: 0.45em, spacing: 0.62em)
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set enum(indent: 1.2em, body-indent: 0.45em, spacing: 0.62em)
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// Vertical rhythm matches manual (long-form); font sizes differ
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show heading.where(level: 1): it => {
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v(1.45em)
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text(size: 18pt, weight: "bold", fill: primary, it)
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v(0.55em)
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}
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show heading.where(level: 2): it => {
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v(1.1em)
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text(size: 14pt, weight: "bold", fill: accent, it)
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v(0.5em)
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}
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show heading.where(level: 3): it => {
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v(0.8em)
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text(size: 12pt, weight: "bold", fill: secondary, it)
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v(0.4em)
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}
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set page(
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paper: "a4",
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margin: (top: 2.5cm, bottom: 2.5cm, left: 2.5cm, right: 2.5cm),
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header: context {
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let h = query(selector(heading.where(level: 1))).find(it => it.location().page() == here().page())
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if h != none {
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align(right)[
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#text(size: 9pt, fill: light)[
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#h.body
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]
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]
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}
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},
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footer: context {
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align(center)[
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#text(size: 9pt, fill: light)[
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— #counter(page).display("1") —
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]
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]
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},
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)
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show link: it => text(fill: accent, it)
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outline(title: "目录", depth: 2)
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pagebreak()
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body
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}
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