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和其他流体力学中的数学方法类似，SPH算法同样涉及到“光滑核”的概念，可以这样理解这个概念，粒子的属性都会“扩散”到周围，并且随着距离的增加影响逐渐变小，这种随着距离而衰减的函数被称为“光滑核”函数，最大影响半径为“光滑核半径”。

反过来不难理解，尽管我们将流体视为一个个分散的粒子，但流体毕竟是连续充满整个空间的，流体中每个位置参与运算的值都是由周围一组粒子累加起来的。

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683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"})])])],-1)]))),T[24]||(T[24]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"h")])],-1))]),T[34]||(T[34]=m("是光滑核半径。函数")),Q("mjx-container",R,[(t(),a("svg",O,T[25]||(T[25]=[Q("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[Q("g",{"data-mml-node":"math"},[Q("g",{"data-mml-node":"mi"},[Q("path",{"data-c":"1D44A",d:"M436 683Q450 683 486 682T553 680Q604 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683Z"})])])],-1)]))),T[26]||(T[26]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"W")])],-1))]),T[35]||(T[35]=m("就是光滑核函数。")),T[36]||(T[36]=Q("br",null,null,-1)),T[37]||(T[37]=m(" 光滑核函数两个重要属性，首先一定是偶函数，也就是")),Q("mjx-container",A,[(t(),a("svg",C,T[27]||(T[27]=[l('',1)]))),T[28]||(T[28]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"W"),Q("mo",{stretchy:"false"},"("),Q("mo",null,"−"),Q("mi",null,"r"),Q("mo",{stretchy:"false"},")"),Q("mo",null,"="),Q("mi",null,"W"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"r"),Q("mo",{stretchy:"false"},")")])],-1))]),T[38]||(T[38]=m("，第二，是“规整函数”，也就是"))]),Q("mjx-container",k,[(t(),a("svg",X,T[39]||(T[39]=[l('',1)]))),T[40]||(T[40]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mstyle",{displaystyle:"true",scriptlevel:"0"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mo",{"data-mjx-texclass":"OP"},"∫"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mi",null,"W"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"r"),Q("mo",{stretchy:"false"},")"),Q("mi",null,"d"),Q("mi",null,"r")])]),Q("mo",null,"="),Q("mn",null,"1")])])],-1))]),T[143]||(T[143]=Q("hr",null,null,-1)),T[144]||(T[144]=Q("h4",{id:"sph推导过程",tabindex:"-1"},[Q("a",{class:"header-anchor",href:"#sph推导过程"},[Q("span",null,[Q("strong",null,"SPH推导过程")])])],-1)),Q("p",null,[T[51]||(T[51]=m("我们假设流体中一个位置为")),Q("mjx-container",F,[(t(),a("svg",E,T[41]||(T[41]=[l('',1)]))),T[42]||(T[42]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("msub",null,[Q("mi",null,"r"),Q("mi",null,"i")]),Q("mo",{stretchy:"false"},"→")])])])],-1))]),T[52]||(T[52]=m("的点，此处的密度为")),Q("mjx-container",B,[(t(),a("svg",S,T[43]||(T[43]=[l('',1)]))),T[44]||(T[44]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"ρ"),Q("mo",{stretchy:"false"},"("),Q("msub",null,[Q("mi",null,"r"),Q("mi",null,"i")]),Q("mo",{stretchy:"false"},")")])],-1))]),T[53]||(T[53]=m("、压力为")),Q("mjx-container",J,[(t(),a("svg",G,T[45]||(T[45]=[l('',1)]))),T[46]||(T[46]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"p"),Q("mo",{stretchy:"false"},"("),Q("msub",null,[Q("mi",null,"r"),Q("mi",null,"i")]),Q("mo",{stretchy:"false"},")")])],-1))]),T[54]||(T[54]=m("、速度为")),Q("mjx-container",P,[(t(),a("svg",Y,T[47]||(T[47]=[l('',1)]))),T[48]||(T[48]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"u"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{stretchy:"false"},"("),Q("msub",null,[Q("mi",null,"r"),Q("mi",null,"i")]),Q("mo",{stretchy:"false"},")")])],-1))]),T[55]||(T[55]=m("，那么我们可以根据上一篇的公式2.8，可以推导出此处的加速度")),Q("mjx-container",N,[(t(),a("svg",q,T[49]||(T[49]=[l('',1)]))),T[50]||(T[50]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{stretchy:"false"},"("),Q("msub",null,[Q("mi",null,"r"),Q("mi",null,"i")]),Q("mo",{stretchy:"false"},")")])],-1))]),T[56]||(T[56]=m("为"))]),Q("mjx-container",W,[(t(),a("svg",K,T[57]||(T[57]=[l('',1)]))),T[58]||(T[58]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mtable",{displaystyle:"true"},[Q("mlabeledtr",null,[Q("mtd",null,[Q("mtext",null,"(3.2)")]),Q("mtd",null,[Q("mstyle",{displaystyle:"true",scriptlevel:"0"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"a"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{stretchy:"false"},"("),Q("msub",null,[Q("mi",null,"r"),Q("mi",null,"i")]),Q("mo",{stretchy:"false"},")"),Q("mo",null,"="),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"g"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"−"),Q("mfrac",null,[Q("mrow",null,[Q("mi",{mathvariant:"normal"},"∇"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mi",null,"p"),Q("mo",{stretchy:"false"},"("),Q("msub",null,[Q("mi",null,"r"),Q("mi",null,"i")]),Q("mo",{stretchy:"false"},")")])]),Q("mrow",null,[Q("mi",null,"ρ"),Q("mo",{stretchy:"false"},"("),Q("msub",null,[Q("mi",null,"r"),Q("mi",null,"i")]),Q("mo",{stretchy:"false"},")")])]),Q("mo",null,"+"),Q("mfrac",null,[Q("mrow",null,[Q("mi",null,"μ"),Q("msup",null,[Q("mi",{mathvariant:"normal"},"∇"),Q("mn",null,"2")]),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("mi",null,"u"),Q("mo",{stretchy:"false"},"→")])]),Q("mo",{stretchy:"false"},"("),Q("msub",null,[Q("mi",null,"r"),Q("mi",null,"i")]),Q("mo",{stretchy:"false"},")")]),Q("mrow",null,[Q("mi",null,"ρ"),Q("mo",{stretchy:"false"},"("),Q("msub",null,[Q("mi",null,"r"),Q("mi",null,"i")]),Q("mo",{stretchy:"false"},")")])])])])])])])])],-1))]),T[145]||(T[145]=Q("p",null,"对于SPH算法来说，基本流程就是这样，根据光滑核函数逐个推出流体中某点的密度，压力，速度相关的累加函数，进而推导出此处的加速度，从而模拟流体的运动趋势，下面我们逐个来分析",-1)),T[146]||(T[146]=Q("hr",null,null,-1)),T[147]||(T[147]=Q("h4",{id:"密度",tabindex:"-1"},[Q("a",{class:"header-anchor",href:"#密度"},[Q("span",null,[Q("strong",null,"密度")])])],-1)),Q("p",null,[T[63]||(T[63]=m("根据公式3.1，用密度")),Q("mjx-container",I,[(t(),a("svg",z,T[59]||(T[59]=[Q("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[Q("g",{"data-mml-node":"math"},[Q("g",{"data-mml-node":"mi"},[Q("path",{"data-c":"1D70C",d:"M58 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716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"})])])],-1)]))),T[62]||(T[62]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"A")])],-1))]),T[65]||(T[65]=m("，可以得到"))]),Q("mjx-container",_,[(t(),a("svg",Q1,T[66]||(T[66]=[l('',1)]))),T[67]||(T[67]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mtable",{displaystyle:"true"},[Q("mlabeledtr",null,[Q("mtd",null,[Q("mtext",null,"(3.3)")]),Q("mtd",null,[Q("mstyle",{displaystyle:"true",scriptlevel:"0"},[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mi",null,"ρ"),Q("mo",{stretchy:"false"},"("),Q("msub",null,[Q("mi",null,"r"),Q("mi",null,"i")]),Q("mo",{stretchy:"false"},")"),Q("mo",null,"="),Q("munder",null,[Q("mo",{"data-mjx-texclass":"OP"},"∑"),Q("mi",null,"j")]),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("msub",null,[Q("mi",null,"ρ"),Q("mi",null,"j")]),Q("mfrac",null,[Q("msub",null,[Q("mi",null,"m"),Q("mi",null,"j")]),Q("msub",null,[Q("mi",null,"ρ"),Q("mi",null,"j")])]),Q("mi",null,"W"),Q("mo",{stretchy:"false"},"("),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("msub",null,[Q("mi",null,"r"),Q("mi",null,"i")]),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"−"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("msub",null,[Q("mi",null,"r"),Q("mi",null,"j")]),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,","),Q("mi",null,"h"),Q("mo",{stretchy:"false"},")")]),Q("mo",null,"="),Q("munder",null,[Q("mo",{"data-mjx-texclass":"OP"},"∑"),Q("mi",null,"j")]),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("msub",null,[Q("mi",null,"m"),Q("mi",null,"j")]),Q("mi",null,"W"),Q("mo",{stretchy:"false"},"("),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("msub",null,[Q("mi",null,"r"),Q("mi",null,"i")]),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,"−"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mover",null,[Q("msub",null,[Q("mi",null,"r"),Q("mi",null,"j")]),Q("mo",{stretchy:"false"},"→")])]),Q("mo",null,","),Q("mi",null,"h"),Q("mo",{stretchy:"false"},")")])])])])])])])],-1))]),T[148]||(T[148]=Q("p",null,"计算使用的光滑核函数称为Poly6函数，具体形式为:",-1)),Q("mjx-container",T1,[(t(),a("svg",a1,T[68]||(T[68]=[l('',1)]))),T[69]||(T[69]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mtable",{displaystyle:"true"},[Q("mlabeledtr",null,[Q("mtd",null,[Q("mtext",null,"(3.4)")]),Q("mtd",null,[Q("msub",null,[Q("mi",null,"W"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mi",null,"P"),Q("mi",null,"o"),Q("mi",null,"l"),Q("mi",null,"y"),Q("mn",null,"6")])]),Q("mtext",null," 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