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GAMES101-22：动画2<a href="https://github.com/KXAND/BlogSource/edit/source/_posts/GAMES101/GAMES101-20.md" class="post-edit-link" title="编辑" rel="noopener" target="_blank"><i class="fa fa-pen-nib"></i></a>
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<div class="post-body" itemprop="articleBody"><h2 id="前言"><a class="markdownIt-Anchor" href="#前言"></a> 前言</h2>
<p><a href="https://www.bilibili.com/video/BV1X7411F744/?p=22">GAMES101-22</a>:单粒子的运动的模拟方法（欧拉与其各种改进）、刚体模拟、流体模拟、网格法与质点法处理大量物体的移动问题 <span id="more"></span></p>
<h2 id="单粒子模拟"><a class="markdownIt-Anchor" href="#单粒子模拟"></a> 单粒子模拟</h2>
<p>考虑一个粒子在速度场中速度随着时间的变化的模拟。即算常微分方程 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>x</mi><mo>˙</mo></mover><mo>=</mo><mi>v</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\dot{x} = v(x,t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66786em;vertical-align:0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.66786em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">x</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.11111000000000001em;"><span class="mord">˙</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span></p>
<p>简单而基础的方法：（前向）<strong>欧拉方法</strong></p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>x</mi><mrow><mi>t</mi><mo>+</mo><mi mathvariant="normal">Δ</mi><mi>t</mi></mrow></msup><mo>=</mo><msup><mi>x</mi><mi>t</mi></msup><mo>+</mo><mi mathvariant="normal">Δ</mi><mi>t</mi><msup><mover accent="true"><mi>x</mi><mo>˙</mo></mover><mi>t</mi></msup><mspace linebreak="newline"></mspace><msup><mover accent="true"><mi>x</mi><mo>˙</mo></mover><mrow><mi>t</mi><mo>+</mo><mi mathvariant="normal">Δ</mi><mi>t</mi></mrow></msup><mo>=</mo><msup><mover accent="true"><mi>x</mi><mo>˙</mo></mover><mi>t</mi></msup><mo>+</mo><mi mathvariant="normal">Δ</mi><mi>t</mi><msup><mover accent="true"><mi>x</mi><mo>¨</mo></mover><mi>t</mi></msup></mrow><annotation encoding="application/x-tex">x^{t+\Delta t} = x^t + \Delta t \dot{x}^t \\
\dot{x}^{t+\Delta t} = \dot{x}^t + \Delta t \ddot{x}^t
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.891331em;vertical-align:0em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.891331em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">+</span><span class="mord mtight">Δ</span><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.926886em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.843556em;vertical-align:0em;"></span><span class="mord">Δ</span><span class="mord mathnormal">t</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.66786em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">x</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.11111000000000001em;"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.891331em;vertical-align:0em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.66786em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">x</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.11111000000000001em;"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.891331em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">+</span><span class="mord mtight">Δ</span><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.926886em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.66786em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">x</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.11111000000000001em;"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.843556em;vertical-align:0em;"></span><span class="mord">Δ</span><span class="mord mathnormal">t</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.66786em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">x</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.22222em;"><span class="mord">¨</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span></span></span></span></span></p>
<p>其中右上角只是标记，不是指数；而一个点表示一次求导（即速度）。这种方式用上一帧的数据计算下一帧的数据。其问题在于：</p>
<ul>
<li>不准确：步长（即两帧之间的间隔 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">\Delta t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord">Δ</span><span class="mord mathnormal">t</span></span></span></span>）越大，结果越不准；</li>
<li>不稳定：对于圆周运动的情况，会变成螺旋形飞出去；</li>
</ul>
<p>不准确在 CG 中往往可以理解容忍，但是不稳定、无论如何都得不到准确的结果是难以接受的。这种不稳定现象是<strong>欧拉方法试图用数值方法解微分方程</strong>的必然结果。</p>
<p>下面是一些解决了不稳定性的方法。</p>
<h3 id="中点法"><a class="markdownIt-Anchor" href="#中点法"></a> 中点法</h3>
<ul>
<li>使用欧拉方法计算出在 o 点的粒子 p 下一步长所在的位置 a；</li>
<li>取 oa 中点 b 的速度场方向；</li>
<li>对（依然在 o 点的 p）应用 b 点的速度计算欧拉方法，得到实际到达的位置 c。</li>
</ul>
<p>这样，中点法取了一个中间、更能代表平均运动方向的速度。</p>
<p>如果我们将中点法的公式推出来，那么，我们会发现中点法更准确的原因在于它引入了二阶导，而不是普通欧拉的一阶导。</p>
<h3 id="自适应步长"><a class="markdownIt-Anchor" href="#自适应步长"></a> 自适应步长</h3>
<ul>
<li>使用欧拉方法计算出在 o 点的粒子 p 下一步长所在的位置 a；</li>
<li>使用欧拉方法计算出在 oa 中点的粒子下一步长所在的位置 b；</li>
<li>如果 a、b 距离甚远，那么说明步长过大，缩小步长重新计算；</li>
<li>如果 a、b 距离不大，那么说明步长已经缩得够小，取一个作为结果；</li>
</ul>
<p>这种办法步长可变，所以能模拟得很好。</p>
<h3 id="隐式后向的欧拉方法"><a class="markdownIt-Anchor" href="#隐式后向的欧拉方法"></a> 隐式/后向的欧拉方法</h3>
<p>使用下一帧的速度而不是上一帧的速度。</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>x</mi><mrow><mi>t</mi><mo>+</mo><mi mathvariant="normal">Δ</mi><mi>t</mi></mrow></msup><mo>=</mo><msup><mi>x</mi><mi>t</mi></msup><mo>+</mo><mi mathvariant="normal">Δ</mi><mi>t</mi><msup><mover accent="true"><mi>x</mi><mo>˙</mo></mover><mrow><mi>t</mi><mo>+</mo><mi mathvariant="normal">Δ</mi><mi>t</mi></mrow></msup><mspace linebreak="newline"></mspace><msup><mover accent="true"><mi>x</mi><mo>˙</mo></mover><mrow><mi>t</mi><mo>+</mo><mi mathvariant="normal">Δ</mi><mi>t</mi></mrow></msup><mo>=</mo><msup><mover accent="true"><mi>x</mi><mo>˙</mo></mover><mi>t</mi></msup><mo>+</mo><mi mathvariant="normal">Δ</mi><mi>t</mi><msup><mover accent="true"><mi>x</mi><mo>¨</mo></mover><mrow><mi>t</mi><mo>+</mo><mi mathvariant="normal">Δ</mi><mi>t</mi></mrow></msup></mrow><annotation encoding="application/x-tex">x^{t+\Delta t} = x^t + \Delta t \dot{x}^{t+\Delta t} \\
\dot{x}^{t+\Delta t} = \dot{x}^t + \Delta t \ddot{x}^{t+\Delta t}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.891331em;vertical-align:0em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.891331em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">+</span><span class="mord mtight">Δ</span><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.926886em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.891331em;vertical-align:0em;"></span><span class="mord">Δ</span><span class="mord mathnormal">t</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.66786em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">x</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.11111000000000001em;"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.891331em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">+</span><span class="mord mtight">Δ</span><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.891331em;vertical-align:0em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.66786em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">x</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.11111000000000001em;"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.891331em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">+</span><span class="mord mtight">Δ</span><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.926886em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.66786em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">x</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.11111000000000001em;"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.891331em;vertical-align:0em;"></span><span class="mord">Δ</span><span class="mord mathnormal">t</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.66786em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">x</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.22222em;"><span class="mord">¨</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.891331em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">+</span><span class="mord mtight">Δ</span><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span></span></span></span></span></span></p>
<p>在实际情况中速度和加速度之间的关系往往并不是那么好直接解出。往往可能需要求根公式等导致速度会比显示慢得多————当然，也变得稳定得多。</p>
<h3 id="龙格-库塔方法"><a class="markdownIt-Anchor" href="#龙格-库塔方法"></a> 龙格-库塔方法</h3>
<p>通过中点法我们已经知道阶数越高效果越好。而龙格-库塔方法是一系列解常微分方程的很好的方法（同样基于欧拉），其中四阶的龙格库塔方法（RK4）最常用，它形如</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>x</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>x</mi><mi>n</mi></msub><mo>+</mo><mfrac><mn>1</mn><mn>6</mn></mfrac><mi>h</mi><mo stretchy="false">(</mo><msub><mi>k</mi><mn>1</mn></msub><mo>+</mo><mn>2</mn><msub><mi>k</mi><mn>2</mn></msub><mo>+</mo><mn>2</mn><msub><mi>k</mi><mn>3</mn></msub><mo>+</mo><mn>4</mn><msub><mi>k</mi><mn>4</mn></msub><mo stretchy="false">)</mo><mspace linebreak="newline"></mspace><msub><mi>t</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>t</mi><mi>n</mi></msub><mo>+</mo><mi>h</mi></mrow><annotation encoding="application/x-tex">x_{n+1} = x_n + \frac{1}{6}h(k_1 + 2k_2 + 2k_3 + 4k_4) \\
t_{n+1} = t_n + h
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.638891em;vertical-align:-0.208331em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:2.00744em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">6</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">4</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.8234109999999999em;vertical-align:-0.208331em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.76508em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal">h</span></span></span></span></span></p>
<p>其中四个 k 是四个关于 x、t 的函数，懒得记了，用到再说。</p>
<blockquote>
<p>对推导感兴趣的话请学习数值分析。</p>
</blockquote>
<h3 id="不基于物理的方法position-based-verlet-intergration"><a class="markdownIt-Anchor" href="#不基于物理的方法position-based-verlet-intergration"></a> 不基于物理的方法（Position-based / Verlet Intergration）</h3>
<p>不基于物理的方法难以保证能量守恒，但是又快又简单。</p>
<p>参见作业 8.</p>
<h3 id="衡量稳定性"><a class="markdownIt-Anchor" href="#衡量稳定性"></a> 衡量稳定性</h3>
<p>通常我们不关心单步的误差（局部误差）和整体误差的值是多少，但是会关心它们和步长之间的关系。</p>
<p>以隐式欧拉为例，步长为 h，整体误差是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(h)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">h</span><span class="mclose">)</span></span></span></span>，局部误差是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>h</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(h^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">h</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>，即如果步长减少一半，那么可以期望整体误差变为原来的一半，而局部误差可以期望变为原来的四分之一。</p>
<h2 id="刚体模拟"><a class="markdownIt-Anchor" href="#刚体模拟"></a> 刚体模拟</h2>
<p>刚体即不可变形的物体，刚体的模拟本质上就是一个大单体粒子的模拟。当然我们还需要考虑一些其他的属性，例如：</p>
<ol>
<li>角度、角速度（一阶导，F/M）、角加速度（二阶导）</li>
<li>位置、速度、加速度</li>
</ol>
<h2 id="流体模拟"><a class="markdownIt-Anchor" href="#流体模拟"></a> 流体模拟</h2>
<p>我们通过模拟小的刚体球（粒子）近似水体；并且认为水的密度是不变的，如果某处的密度改变了，那么就通过修正粒子的位置来进行修正。</p>
<blockquote>
<p>物理模拟和渲染是有前后的两步，所以近似为小球不影响渲染。</p>
</blockquote>
<p>每个粒子都可以对空间中的任意一点的密度产生影响。因此，为了达到“改变粒子的位置使得此处的密度降低”的目的，我们就需要知道密度对于所有粒子的位置的导数梯度。然后应用梯度下降法即可。（即改变对此处密度影响最大的粒子的位置）</p>
<blockquote>
<p>以防万一：梯度是导数下降最快的方向。</p>
</blockquote>
<p>这种模拟是不基于物理的，例如我们直接改了粒子的位置。当然为了避免水的运动停不下来，我们也可以加入能量衰减的公式。</p>
<h2 id="大量物体的模拟方法视角质点法与网格法"><a class="markdownIt-Anchor" href="#大量物体的模拟方法视角质点法与网格法"></a> 大量物体的模拟方法（视角）：质点法与网格法</h2>
<ul>
<li>质点法：又称拉格朗日法，以质点为主体，研究观察质点的位置变化情况；</li>
<li>网格法：又称欧拉法，以空间网格为主体，研究观察一个空间中的固定位置的密度等信息（类似盯着那种大颗粒 LED 滚动屏的单个珠子看）。</li>
<li>材质点法：混合网格质点的方法，例如用粒子记录材质信息，然后用网格计算粒子的变迁，最后写回粒子里。</li>
</ul>
<h2 id="尾声"><a class="markdownIt-Anchor" href="#尾声"></a> 尾声</h2>
<p>见笔记1。</p>
<h2 id="跳转"><a class="markdownIt-Anchor" href="#跳转"></a> 跳转</h2>
<p>Home:<a href="GAMES10101.html">GAMES101-1：课程总览与笔记导航</a></p>
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<p>Next：完结撒花！</p>
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