{"id":473,"date":"2014-07-31T15:14:56","date_gmt":"2014-07-31T07:14:56","guid":{"rendered":"http:\/\/blog.xuhao1.me\/?p=473"},"modified":"2014-08-03T06:29:21","modified_gmt":"2014-08-02T22:29:21","slug":"fea%e4%bc%98%e5%8c%96%e9%a3%9e%e6%9c%ba%e7%bb%93%e6%9e%84%e8%ae%be%e8%ae%a1","status":"publish","type":"post","link":"http:\/\/blog.xuhao1.me\/?p=473","title":{"rendered":"\u4e0a\u5e1d\u8bf4\uff1a\u8981\u6709\u5149   &#8211;PDE \u6742\u8bba"},"content":{"rendered":"<p>\u4e0a\u5e1d\u8bf4\u8981\u6709\u5149 \u4e8e\u662f\u6709\u4e86\u9ea6\u514b\u65af\u97e6\u65b9\u7a0b\u3002 \u4e8e\u662f\u5b87\u5b99\u5f00\u59cb\u6c42\u89e3PDE\u3002 \u5199\u8fd9\u4e2a\u6559\u7a0b\u662fPDE\u8fd8\u662f\u5f88\u6709\u4e0a\u5e1d\u7684\u611f\u89c9\u7684 <!--more--><\/p>\n<p><a href=\"#Build_Func\">\u65b9\u7a0b\u7684\u5efa\u7acb<\/a><\/p>\n<p><a href=\"#Solve_Func\">\u65b9\u7a0b\u7684\u89e3\u6cd5<\/a><\/p>\n<p>\uff08\u4f9d\u8d56\u4e8e\u65f6\u95f4\u7684\u77ac\u6001\u95ee\u9898\uff09 \u5bf9\u5e38\u5fae\u5206\u65b9\u7a0b\u76f4\u63a5\u8fdb\u884c\u6570\u503c\u79ef\u5206\u7684\u65f6\u95f4\u9010\u6b65\u79ef\u5206\u6cd5 \u65f6\u95f4\u6b65\u957f\u53ea\u53d7\u6c42\u89e3\u7cbe\u5ea6\u9650\u5236\u7684\u9690\u5f0f\u7b97\u6cd5\uff08Newmark\u6cd5\uff09<\/p>\n<p>\u6c42\u89e3\u65f6\u95f4\u6b65\u957f\u53d7\u7b97\u6cd5\u7a33\u5b9a\u9650\u5236\u7684\u663e\u793a\u7b97\u6cd5\uff08\u4e2d\u5fc3\u5dee\u5206\u6cd5\uff09<\/p>\n<h2><a name=\"Build_Func\"><\/a>\u65b9\u7a0b\u7684\u5efa\u7acb<\/h2>\n<h3>\u5fae\u5206\u65b9\u7a0b\u7684\u7b49\u6548\u79ef\u5206\u5f62\u5f0f\u548c\u52a0\u6743\u4f59\u91cf\u6cd5<\/h3>\n<p>$$!A(u)=\\left(\\begin{array}{c}A_{1}(u)\\\\ A_{2}(u)\\\\ A_{3}(u)\\\\ A_{4}(u)\\\\ . \\end{array}\\right )=0$$ $$!B(u)=\\left(\\begin{array}{c}B_{1}(u)\\\\ B_{2}(u)\\\\ B_{3}(u)\\\\ B_{4}(u)\\\\ . \\end{array}\\right )=0$$ \u6bd4\u5982\u5bf9\u4e8e\u70ed\u4f20\u5bfc\u65b9\u7a0b\u6709 $$!A(\\phi)=\\frac{\\partial}{\\partial x}(k\\frac{\\partial\\phi}{\\partial x})+\\frac{\\partial}{\\partial y}(k\\frac{\\partial\\phi}{\\partial y})+Q=0$$ &#8211;Equ .0 \u5728\u03a9\u4e0a \u4ee5\u53ca $$!B(\\phi)=\\begin{cases}\\phi-\\overline{\\phi}\\\\k\\frac{\\partial\\phi}{\\partial n}-\\overline{q}\\end{cases}=0$$\u5728\u0393\u4e0a\u3002 &#8211;Equ .1 \u5176\u4e2d$$\\phi$$\u4e3a\u6e29\u5ea6\uff0ck\u4e3a\u70ed\u4f20\u5bfc\u7cfb\u6570$$\\overline{\\phi},\\overline{q}$$\u5206\u522b\u662f\u8fb9\u754c\u4e0a\u6e29\u5ea6\u548c\u70ed\u6d41\u7684\u7ed9\u5b9a\u503c \u7531\u4e8e\u65b9\u7a0b\u7ec4\u5728\u57df\u03a9\u4e0a\u6bcf\u4e00\u70b9\u90fd\u5fc5\u987b\u4e3a\u96f6\u3002\u6240\u4ee5\u6709 $$!\\int_\\Omega v^{T}A(u)d\\Omega\\equiv\\int_{\\Omega}(v_{1}A_{1}(u)+v_{2}A_{2}(u)+&#8230;)d\\Omega\\equiv 0$$ &#8211;Equ .2 \u5176\u4e2d\uff0c$$!v=\\left(\\begin{array}{c}v_{1}\\\\ v_{2}\\\\ v_{3}\\\\ v_{4}\\\\ .\\\\.\\\\. \\end{array}\\right )$$ &#8211;Equ .3 \u4e3a\u51fd\u6570\u5411\u91cf\uff0c\u662f\u4e00\u7ec4\u548c\u5fae\u5206\u65b9\u7a0b\u6570\u91cf\u76f8\u7b49\u7684<span style=\"color: #ff0000;\">\u4efb\u610f\u51fd\u6570\u3002<\/span> \u5176\u4e2d\u0395qu.3\u548cEqu.0\u662f\u5b8c\u5168\u7b49\u6548\u7684\u79ef\u5206\u5f62\u5f0f\u3002\u53ef\u4ee5\u65ad\u8a00\uff1a \u82e5Equ.3\u5bf9\u4e8e\u4efb\u4f55\u7684v\u90fd\u53ef\u4ee5\u6210\u7acb\uff0c\u5219Equ.0\u5728\u57df\u5185\u4efb\u4f55\u4e00\u5b9a\u5f97\u5230\u6ee1\u8db3\u3002 \u540c\u6837\u7684\uff0cB(u)\u4e5f\u53ef\u4ee5\u5199\u6210\u5982\u4e0b\u7684\u79ef\u5206\u5f62\u5f0f $$!\\int_{\\text{\u03a9}}\\overline{v}^{T}B(u)d\\Omega\\equiv\\int_{\\Omega}(\\overline{v}_{1}B_{1}(u)+\\overline{v}_{2}B_{2}(u)+&#8230;)d\\Omega\\equiv 0$$ &#8211;Equ .4 \u4e5f\u5c31\u662f\u8bf4\uff0c\u79ef\u5206\u5f62\u5f0f $$!\\int_{\\Omega}v^{T}A(u)d\\Omega+\\int_\\Omega \\overline{v^{T}}B(u)d\\Gamma\\equiv 0$$<\/p>\n<h2>MATHEMATICA<\/h2>\n<p>\u7136\u540e\u5c31\u662f\u5927\u90e8\u5206\u4e1c\u897f\u90fd\u53ef\u4ee5\u5728MMA\u7684Documation\u76f4\u63a5\u641c\u7d22FEA\u5f97\u5230\uff0c\u535a\u4e3b\u8fd9\u8fb9\u4e3b\u8981\u662f\u6839\u636e\u81ea\u5df1\u7684\u4e00\u4e9b\u4e86\u89e3\u8fdb\u884c\u6269\u5145\uff0c\u987a\u4fbf\u52a0\u5165\u4e00\u4e9b\u5927\u5b66\u4e8c\u5e74\u7ea7\u4e09\u5e74\u7ea7\u7269\u7406\u4e13\u4e1a\u6240\u6d89\u53ca\u7684\u4e1c\u897f\uff08\u5305\u62ec\u56db\u5927\u529b\u5b66\u7684\u4e00\u4e9b\u7ecf\u5178\u95ee\u9898\u7684\uff0c\u4ee5\u53ca\u6d41\u4f53\u529b\u5b66\u76f8\u5173\u7684\u5185\u5bb9\uff09\uff0c\u6765\u4f53\u4f1a\u4e0b\u4e00\u4e2a\u6570\u503c\u89e3PDE\u5230\u5e95\u53ef\u4ee5\u5e72\u4ec0\u4e48 \u53e6\u5916\u662f\u8fd9\u91cc\u9762\u7684\u4ee3\u7801\u5b9e\u4f8b\u5c3d\u53ef\u80fd\u548cMMA\u4fdd\u6301\u89c6\u89c9\u4e0a\u7684\u4e00\u81f4\uff0c\u6bd4\u5982$$\\in$$\u5728blog\u91cc\u9762\u5c31\u662f\u4ee5Latex\u8f93\u5165\u7684\u3002\u6240\u6709\u4ee3\u7801\u8bf7\u89c1\u6211\u7684<a href=\"http:\/\/xuhao1.github.io\/mma\/\">Github.io<\/a><\/p>\n<h2>\u6982\u62ec<\/h2>\n<p>\u73b0\u8c08\u8bba\u4e0bPDE \u5427\uff0c\u4e5f\u5c31\u662f\u6570\u7406\u65b9\u7a0b\u5904\u7406\u7684\u95ee\u9898\uff0c\u672c\u6765\uff0c\u4f5c\u4e3a\u4e00\u4e2a\u6570\u7406\u65b9\u7a0b\u521a\u53ca\u683c\u7684\u5b66\u6e23\uff0c\u5e94\u8be5\u9762\u58c1\uff0c\u4e0d\u8fc7\u9274\u4e8e\u4e00\u76f4\u5bf9\u4e8e\u8ba1\u7b97\u6bd4\u8f83\u611f\u5174\u8da3\uff0c\u5728\u8fd9\u91cc\u5984\u8c08\u4e00\u4e0b\u6570\u503c\u8ba1\u7b97PDE\u3002 \u00a0 \u5982\u6211\u4eec\u6240\u77e5\uff0c\u6570\u503c\u8ba1\u7b97PDE\u7684\u65b9\u6cd5\u4f17\u591a\uff0c\u5e38\u7528\u4e8e\u5de5\u7a0b\u9886\u57df\u7684\u5305\u62ec\uff1a \u5728CFD\uff08\u8ba1\u7b97\u6d41\u4f53\u529b\u5b66\uff09\u5e38\u7528\u7684\u662f\u6709\u9650\u5dee\u5206\u6cd5 \u6216\u8005 \u6709\u9650\u4f53\u79ef\u6cd5 \u5728\u7ed3\u6784\u529b\u5b66\u5de5\u7a0b\u9886\u57df\u5e38\u89c1\u7684\u662f\u6709\u9650\u5143\u65b9\u6cd5\u3002 \u8bf7\u53c2\u7167\u6b64\u7bc7blog \u5173\u4e8eFEA\u7684\u4e00\u4e9b\u8ba8\u8bba\uff0c\u4ee5\u540e\u4f1a\u628aFEA\u7684\u4ee3\u7801\u5b9e\u73b0\u81ea\u5df1\u8fc7\u4e00\u904d\uff0c\u7528GPU\u52a0\u901f\u7684\u65b9\u5f0f\u3002\u8fd9\u662f\u4e00\u79cd\u81ea\u6211\u78e8\u783a\u3002 http:\/\/blog.stlover.org\/?p=239 \u5176\u4ed6\u5bf9\u4e8e\u6bd4\u8f83\u5bf9\u79f0\u7684\u8fb9\u754c\u6761\u4ef6\u8fd8\u6709\u751a\u591a\u65b9\u6cd5\uff0c\u8bf8\u5982\u4e00\u5806\u8c31\u65b9\u6cd5\uff08\u57fa\u4e8e\u5085\u7acb\u53f6\u53d8\u6362\uff09\u3002 \u5148\u629b\u53bbFEA\uff08\u6709\u9650\u5143\uff09\u4e4b\u7c7b\u7684\u683c\u5f0f\u4e0d\u8c08\uff0c\u6211\u4eec\u5148\u8bf4\u4e0b\u6211\u4eec\u8981\u5173\u6ce8\u7684\u662f\u4ec0\u4e48<\/p>\n<h2>PDE<\/h2>\n<p>\u8ba1\u7b97\u4e00\u4e2aPDE\uff0c\u8981\u8003\u8651\u7684\u95ee\u9898\u6bd4\u8f83\u590d\u6742\uff0c\u6d69\u6d69\u4e5f\u662f\u521d\u5165\u95e8\u5f84 \u9996\u5148\u662f\u8ba1\u7b97\u7684\u7269\u7406\u5bf9\u8c61\u7684\u5c3a\u5ea6\u548c\u8981\u6c42\u7684\u7cbe\u5ea6\uff0c\u5bf9\u4e0d\u540c\u7684\u7cbe\u5ea6\u548c\u5c3a\u5ea6\u6709\u4e0d\u540c\u7684\u65b9\u6cd5\uff0c\u4e3e\u4e2a\u4f8b\u5b50\uff0c\u5728\u7b49\u79bb\u5b50\u4f53\u7684\u8ba1\u7b97\u4e2d\uff0c\u5c31\u6709\u5c06\u6d41\u4f53\u6a21\u62df\u4e3a\u4f8b\u5b50\uff0c\u4ec5\u4ec5\u5c06\u7535\u78c1\u573a\u4f5c\u4e3a\u6709\u9650\u5143\u5904\u7406\u7684\u3002 \u518d\u6709\u7684\u95ee\u9898\u662f\u6211\u4eec\u8981\u5904\u7406\u7684PDE\u7684\u4e00\u4e9b\u7279\u6027\uff0c\u6bd4\u5982\u662f\u692d\u5706\uff0c\u629b\u7269\u8fd8\u662f\u53cc\u66f2\uff0c\u5f88\u91cd\u8981\u7684\u4e00\u70b9\u662f\u662f\u5426\u65f6\u95f4\u76f8\u5173\u3002 \u6bd4\u5982\uff0c\u5982\u679c\u53bb\u6c42\u89e3\u4e00\u4e2a\u7535\u78c1\u573a\u7684\u7a33\u6001\uff0c\u6216\u8005\u65e0\u7c98\u6d41\u4f53\uff0c\u6216\u8005\u5728\u5916\u529b\u4e0b\u7684\u5e73\u677f\u5f2f\u66f2\uff0c\u6211\u4eec\u53ef\u4ee5\u8ba4\u4e3a\u7269\u4f53\u4f1a\u6210\u4e3a\u4e00\u4e2a\u7a33\u5b9a\u72b6\u6001\uff0c\u8fd9\u65f6\u5019\u6211\u4eec\u4e0d\u9700\u8981\u8003\u8651\u65f6\u95f4\u76f8\u5173\u7684\u56e0\u7d20\u3002 \u7136\u800c\u5728\u5f88\u591a\u95ee\u9898\uff0c\u8bf8\u5982\uff0c\u58f0\u573a\u8ba1\u7b97\uff0c\u57fa\u4e8eN\uff0dS\u65b9\u7a0b\u7684\u5361\u95e8\u6da1\u8857\uff0c\u8fd9\u4e9b\u90fd\u662f\u975e\u7a33\u6001\u7684\u95ee\u9898\uff0c\u6211\u4eec\u5219\u5fc5\u987b\u52a0\u5165\u65f6\u95f4\u8fd9\u4e2a\u81ea\u53d8\u91cf\u3002 \u5f53\u7136\uff0c\u8fd9\u5c31\u6709\u53e6\u4e00\u4e2a\u95ee\u9898\u4e86\uff0c\u65b9\u7a0b\u662f\u5426\u662f\u7ebf\u6027\u6216\u8005\u975e\u7ebf\u6027\u7684\u3002\u5ef6\u4f38\u4e0b\u6765\u5c31\u592a\u591a\u4e86\u3002 \u5176\u6b21\u7684\u95ee\u9898\u662f\uff0c\u79bb\u6563\u5316\uff0c\u6211\u4eec\u8981\u6c42\u4e00\u4e2a\u9886\u57df\u4e0a\u7684\u89e3\uff0c\u4e0d\u5f97\u4e0d\u8003\u8651\u5c06\u8fd9\u4e2a\u89e3\u53d8\u6210\u4e00\u5806\u7684\u91c7\u6837\u70b9\uff0c\u51e0\u4e4e\u6240\u6709\u7684\u6570\u503cPDE\u8ba1\u7b97\u90fd\u662f\u5148\u8bb2\u6570\u636e\u79bb\u6563\u5316\uff0c\u7136\u540e\u518d\u8fdb\u884c\u5904\u7406\uff0c\u8fd9\u5c31\u662f\u6d89\u53ca\u5230\u53e6\u4e00\u4e2a\u590d\u6742\u7684\u95ee\u9898\uff1a<\/p>\n<blockquote><p>\u7f51\u683c\u5212\u5206\u3002<\/p><\/blockquote>\n<p>\u6bd4\u5982\u6211\u4eec\u8981\u8ba1\u7b97\u4e00\u4e2a\u673a\u7ffc\u7684\u5347\u529b\uff0c\u663e\u800c\u6613\u89c1\u7684\u662f\u8ddd\u79bb\u673a\u7ffc\u6bd4\u8f83\u8fdb\u7684\u5730\u65b9\u9700\u8981\u8ba1\u7b97\u7cbe\u7ec6\u4e00\u4e9b\uff0c\u8ddd\u79bb\u673a\u7ffc\u6bd4\u8f83\u8fdc\u7684\u5730\u65b9\uff0c\u9700\u8981\u8ba1\u7b97\u7684\u758f\u677e\u4e00\u4e9b\uff0c\u8fd9\u4e2d\u95f4\u5c31\u9700\u8981\u5f88\u591a\u7f51\u683c\u5212\u5206\u7684\u95ee\u9898\u3002 \u518d\u6709\u7684\u95ee\u9898\u662f\u9009\u62e9\u5408\u9002\u7684\u201c\u683c\u5f0f\u201d\uff0c\u5c31\u662f\u5982\u4f55\u8868\u793a\u8fd9\u4e9b\u7269\u4f53\uff0c\u8fd9\u5c31\u662f\u6240\u8c13\u7684\u6709\u9650\u5143\uff0c\u6709\u9650\u5dee\u5206\uff0c\u6709\u9650\u4f53\u79ef\u7684\u6765\u5386\u4e86\u3002 \u4e0d\u8fc7\u5728\u672c\u6587\u4e2d\uff0c\u6211\u4eec\u6682\u65f6\u4e0d\u8003\u8651\u8fd9\u4e9b\u95ee\u9898\uff0c\u4ec5\u4ec5\u4f7f\u7528Mathematica\u63d0\u4f9b\u7684\u6709\u9650\u5143\u65b9\u6cd5\u3002\u672a\u6765\u65f6\u95f4\u5145\u88d5\u6d69\u6d69\u53ef\u80fd\u4f1a\u5199\u4e00\u4e2a\u7b80\u5355\u7684CUDA\u7684N\uff0dS\u65b9\u7a0b\u89e3\u7b97\u5668\uff0c\u518d\u8bb2\u4e0d\u8fdf\u3002<\/p>\n<h2>\u5165\u624b<\/h2>\n<p>\u9996\u5148\uff0c\u4f60\u9700\u8981\u51c6\u5907\u4e00\u4e9b\u4e1c\u897f\uff0c\u4e3b\u8981\u5305\u62ec\uff1a 1.\u8bf7\u5b66\u4f1a\u6b63\u786e\u62fc\u5199PDE\u548c\u77e5\u9053\u4ec0\u4e48\u662fPDE 2.\u8bf7\u81f3\u5c11\u83b7\u5f97\u6570\u7406\u65b9\u7a0b\u53ca\u683c\u7684\u6210\u7ee9\uff08\u5443\u3002\u3002\u6d69\u6d69\u5c31\u662f\u4e0d\u4f5c\u4e0d\u6b7b\u7684\u521a\u53ca\u683c\uff09 3.\u8bf7\u83b7\u5f97\u4e00\u53f0\u7535\u8111 4.\u5b89\u88c5Mathematica 10\uff0c\u6ce8\u610fMathematica\u5b98\u7f51\u4f1a\u81ea\u52a8\u8df3\u8f6c\u5230\u4e2d\u6587\uff0c\u4e2d\u6587\u53ea\u67099\uff0c\u8bf7\u5207\u6362\u5230\u82f1\u6587\u4e0b\u8f7d10\u7248 5.\u5c06\u672cblog\u6dfb\u52a0\u5230\u4f60\u7684\u6536\u85cf\u5939\u3002 6.\u6240\u6709\u7684\u6e90\u4ee3\u7801\u90fd\u53ef\u4ee5\u5728\u6211\u7684<a href=\"http:\/\/xuhao1.github.io\/mma\/\">Github.io<\/a>\u4e0a\u627e\u5230 7.\u4e14\u884c\u4e14\u73cd\u60dc\uff0c\u5378\u8f7dMATLAB\u5165\u6211MMA\u5927\u6cd5\u597d<\/p>\n<h2>\u5165\u95e8<\/h2>\n<p>\u6211\u4eec\u5148\u4ece\u6700\u7b80\u5355\u7684\u5f00\u59cb\uff0c\u8ba1\u7b97\u4e00\u4e2a\u7a33\u5b9a\u72b6\u51b5\u4e0b\u7684\u70ed\u5206\u5e03 \u9996\u5148\uff0c\u6211\u4eec\u77e5\u9053\uff0c\u5bf9\u4e8e\u4e00\u4e2a\u5e73\u9762\u6e29\u5ea6\u573a\u800c\u8a00\uff0c\u5176\u7a33\u5b9a\u72b6\u51b5\u4e0b\u65b9\u7a0b\u53ef\u4ee5\u5199\u505a $$!\\Delta u(x,y) = 0$$ \u7136\u540e\u6211\u4eec\u53ef\u4ee5\u7ed9\u5b9a\u7b2c\u67d0\u7c7b\u8fb9\u754c\u6761\u4ef6\uff08\u535a\u4e3b\u7684\u5b66\u6e23\u672c\u8d28\u66b4\u9732\u65e0\u7591\uff09 $$! U(x,y)=\\Gamma(x,y)\uff0c (x,y)\\in \\partial \\Omega $$ \u8fd9\u6837\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u7b97\u51fa\u6765\u533a\u57df$$\\Omega$$\u7684\u6e29\u5ea6\u5206\u5e03. \u600e\u4e48\u505a\u5462\uff0c\u9996\u5148\u662f\u8868\u793a\u5728Mathematica\u91cc\u9762\u5566<\/p>\n<pre><code lang=\"mathematica\">\r\nEqu = {\r\n  Laplacian[u[x, y], {x, y}] == 0,        \r\n  u[x, 0] == 1,\r\n  u[x, 1] == 0,\r\n  u[1, y] == 0,\r\n  u[0, y] == 0\r\n  }<\/code><\/pre>\n<p>\u8fd9\u5c31\u7ed9\u5b9a\u4e86\u4e00\u4e2a\u5b8c\u6574\u7684\u65b9\u7a0b\uff0c\u6ce8\u610f\uff0c [code lang=&#8221;mathematica&#8221;]Laplacian[u,{x1,x2,x3}][\/code]\u5728MMA\u91cc\u8868\u793a $$!\\Delta u_{x1,x2,x3}$$ \u8bb0\u5f97\u63d0\u524d\u8981\u8f7d\u5165FEM\u5305 [code lang=&#8221;mathematica&#8221;]Needs[&#8220;NDSolve`FEM`&#8221;][\/code] \u7136\u540e\u6211\u4eec\u5c06\u8fd9\u4e2a\u65b9\u7a0b\u62ff\u53bb\u8ba1\u7b97 [code lang=&#8221;mathematica&#8221;]uif = NDSolveValue[Equ, u, {x, 0, 1}, {y, 0, 1}][\/code] \u53ef\u4ee5\u5f97\u5230\u4e00\u4e2a\u63d2\u503c\u540e\u7684uif\u5bf9\u8c61 \u753b\u56fe [code lang=&#8221;Mathematica&#8221;]Plot3D[uif[x, y], {x, 0, 1}, {y, 0, 1}][\/code] \u540e\u5f97\u5230 \u4e0a\u9762\u662f\u5d4c\u5165\u7f51\u9875\u7684Mathematica\u4ee3\u7801\uff0c\u5982\u679c\u4f60\u7684\u7535\u8111\u914d\u7f6e\u6b63\u786e\u5e94\u8be5\u53ef\u4ee5\u6b23\u8d4f\u5230\u4e00\u4e2a\u52a8\u6001\u7684\u53ef\u4ee5\u62ff\u9f20\u6807\u62c9\u6765\u62c9\u53bb\u7684\u70ed\u6d41\u5206\u5e03\u56fe\uff0c\u6258\u7ba1\u4e8eGithub\uff0c\u8bf7\u70b9\u51fb\u4fe1\u4efb\u63d2\u4ef6\uff0c\u6d69\u6d69\u4e0d\u4fdd\u8bc1IE 360\u6d4f\u89c8\u5668\u53ef\u4ee5\u987a\u7545\u8fd0\u884c\u3002\u6ca1\u6709\u7684\u8bdd\u8bf7\u81ea\u884c\u5b89\u88c5MMA 10\u4ee5\u53caChrome\u6d4f\u89c8\u5668\u3002 \u5982\u6b64\u4e00\u6765\uff0c\u6211\u4eec\u5c31\u4ef0\u4ed7\u6700\u7b80\u5355\u7684Mathematica\u8ba1\u7b97\u51fa\u6765\u4e86\u4e00\u4e2a\u7b80\u5355\u7684\u70ed\u6d41\u5206\u5e03\u3002 \u53ef\u80fd\u4f60\u4f1a\u95ee\u6211\uff0c\u4f60\u5728\u9017\u6211\u5462\uff0c\u8bf4\u597d\u7684\u7f51\u683c\u5212\u5206\u5462\uff1f\u5176\u5b9e\u4e0a\u9762\u7684\u8fd9\u4e2a\u65b9\u6cd5\u662fMMA\u5728\u8001\u7248\u672c\u5c31\u5185\u7f6e\u7684PDE\u6c42\u89e3\u5668\uff0c\u4ec5\u4ec5\u53ef\u4ee5\u6c42\u89e3\u4e00\u4e9b\u975e\u590d\u6742\u60c5\u51b5\uff0c\u5982\u679c\u7a0d\u5fae\u590d\u6742\u5c31\u8dd1\u4e0d\u52a8\u4e86\uff0c\u4e0d\u8fc7\u8fd9\u4e0d\u59a8\u788d\u6211\u4eec\u62ff\u4e4b\u5f00\u5200\u3002<\/p>\n<h2>\u533a\u57df\u4e0e\u8fb9\u754c<\/h2>\n<p>\u6b64\u90e8\u5206\u9700\u8981\u9996\u5148\u5bf9\u4e8e\u5e03\u5c14\u8fd0\u7b97\u6709\u4e00\u5b9a\u4e86\u89e3\u3002 \u6309\u7167\u725b\u8001\u7237\u5b50\u7684\u8bc1\u660e\uff0c\u4e0a\u5e1d\u505a\u7684\u66f4\u591a\u7684\u5728\u4e8e\u7b2c\u4e00\u63a8\u52a8\uff0c\u6240\u8c13\u7b2c\u4e00\u63a8\u52a8\u561b\uff0c\u5c31\u662f\u8fb9\u754c\u6761\u4ef6\uff0c\u53ef\u89c1\u6709\u65f6\u5019\u8fb9\u754c\u6bd4\u4ec0\u4e48\u90fd\u91cd\u8981\u3002 Mathematica\u7684PDE\u529f\u80fd\u5728\u524d\u51e0\u7248\u90fd\u662f\u5e9f\u7684\uff0c\u4e3b\u8981\u539f\u56e0\u5c31\u5728\u4e8e\u524d\u51e0\u7248\u6ca1\u6709\u63d0\u4f9b\u6f02\u4eae\u7684\u533a\u57df\u5b9a\u4e49\u4e0e\u8fb9\u754c\u5b9a\u4e49\uff0c\u800c\u8fd9\u4e00\u7248\u7ed9\u51fa\u4e86\u4e00\u4e2a\u4f18\u96c5\u7684\u5b9e\u73b0\u3002\u6211\u4eec\u5148\u4ece\u89e3\u6790\u7684\u90e8\u5206\u8bb2\u8d77\u3002 \u9996\u5148\u6211\u4eec\u5b9a\u4e49\u4e00\u4e2a\u533a\u57df\uff0c\u53ef\u4ee5\u7528\u4e00\u79cd\u5f88\u7b80\u5355\u7684\u65b9\u6cd5\u5c31\u662f\u8868\u8fbe\u5f0f\uff0c\u5728\u5f0f\u4e3a\u771f\u7684\u65f6\u5019\u4e3a\u9886\u57df\u3002MMA 10 \u5c31\u771f\u7684\u7ed9\u6211\u4eec\u63d0\u4f9b\u4e86\u8fd9\u79cd\u65b9\u6cd5\uff0c\u611f\u8c22\u5927\u6c11\u79d1Wolfram [code lang=&#8221;mathematica&#8221;]ImplicitRegion[! (x^2 + y^2 &lt; 5), {{x, -20, 20}, {y, -20, 20}}];[\/code] \u5b66\u8fc7c\u8bed\u8a00\u7684\u5e94\u8be5\u80fd\u770b\u51fa\u8fd9\u4e2a\u611f\u53f9\u53f7\u7684\u610f\u601d \u8fd9\u4e2a\u5f0f\u5b50\u7684\u610f\u601d\u662f\uff0c\u5728$$x\\in(\uff0d20,20)$$\u7684\u65f6\u5019\uff0c\u5982\u679c\u5f0f\u5b57$$ !(x^2 + y^2 &lt; 5)$$\u4e3a\u771f\uff0c\u5219\u662f\u6211\u4eec\u7684\u9886\u57df\uff0c\u4e5f\u5c31\u662f\u4e00\u4e2a\u65b9\u5757\u6316\u51fa\u4e86\u4e00\u4e2a\u5706 \u753b\u51fa\u56fe\u662f [code lang=&#8221;mathematica&#8221;]EquArea = ImplicitRegion[! (x^2 + y^2 &lt; 5), {{x, -20, 20}, {y, -20, 20}}]; RegionPlot[EquArea, AspectRatio -&gt; Automatic][\/code] \u5982\u56fe\u7684\u6837\u5b50RegionPlot\u51fd\u6570\u5373\u662f\u753b\u51fa\u4e00\u4e2a\u533a\u57df <a href=\"http:\/\/blog.stlover.org\/wp-content\/uploads\/2014\/07\/pdetutorial.jpg\"><img loading=\"lazy\" class=\"alignnone size-full wp-image-368\" src=\"http:\/\/blog.stlover.org\/wp-content\/uploads\/2014\/07\/pdetutorial.jpg\" alt=\"pdetutorial\" width=\"569\" height=\"565\" srcset=\"http:\/\/blog.xuhao1.me\/wp-content\/uploads\/2014\/07\/pdetutorial.jpg 569w, http:\/\/blog.xuhao1.me\/wp-content\/uploads\/2014\/07\/pdetutorial-150x150.jpg 150w, http:\/\/blog.xuhao1.me\/wp-content\/uploads\/2014\/07\/pdetutorial-300x297.jpg 300w\" sizes=\"(max-width: 569px) 100vw, 569px\" \/><\/a> \u7136\u540e\u6211\u4eec\u9700\u8981\u5b9a\u4e49\u4e00\u4e0b\u8fb9\u754c\u7684\u60c5\u51b5\u3002MMA 10\u540c\u6837\u7ed9\u4e86\u4e00\u4e2a\u6f02\u4eae\u7684\u65b9\u5f0f\u7ed9\u6211\u4eec\uff08\u5f53\u7136\u4e0d\u4ec5\u4ec5\u9650\u4e8e\u6b64\uff09 [code]DirichletCondition[beqn,pred];[\/code] \u8fd9\u4e2a\u5f0f\u5b57\u7684\u610f\u601d\u5462\u5c31\u662f\u5728\u540e\u9762\u4e00\u4e2a\u5f0f\u5b50\u6210\u7acb\u7684\u65f6\u5019pred\u6210\u7acb\u7684\u65f6\u5019\uff0c\u8fb9\u754c\u6761\u4ef6\u4f7f\u5f97beqn\u6210\u7acb\uff0c\u5c31\u662f\u540e\u9762\u7ea6\u5b9a\u4e86\u8fb9\u754c\u7684\u5b9a\u4e49\uff0c\u524d\u9762\u7ea6\u5b9a\u4e86\u8fb9\u754c\u7684\u6761\u4ef6 \u6bd4\u5982 [code]DirichletCondition[u[x, y] == a, x^2 + y^2 == 5];[\/code] \u5c31\u8868\u793a\uff0c x^2 + y^2 == 5 \u6210\u7acb\u7684\u60c5\u51b5\u4e0b\uff0c\u8fb9\u754c\u6ee1\u8db3u[x, y] == a\u3002 \u6211\u4eec\u73b0\u5728\u505a\u4e00\u4e2a\u7b80\u5355\u7684\u95ee\u9898\uff0c\u4e00\u4e2a\u4e8c\u7ef4\u7684\u76d2\u5b50\u63a5\u5730\u91cc\u9762\u88c5\u4e86\u4e00\u4e2a\u4e8c\u7ef4\u7684\u7b49\u52bf\u4f53\uff0c\u6c42\u7535\u573a\u5206\u5e03\u3002 \u5982\u679c\u7528\u7535\u52a8\u529b\u5b66\u7684\u65b9\u6cd5\uff0c\u6211\u4eec\u53ef\u4ee5\u5199\u51fa\u8fd9\u4e2a\u7a7a\u95f4\u7684\u52d2\u8ba9\u5fb7\u591a\u9879\u5f0f\uff0c\u7136\u540e\u4f7f\u5f97\u6ee1\u8db3\u8fb9\u754c\u6761\u4ef6\uff0c\u7136\u540e\u5f97\u5230\u89e3\u3002\u975e\u5e38\u7e41\u7410\u3002\u5728\u8fd9\u91cc\u9762\u6211\u4eec\u5b9a\u4e49\u4e86\u4ee5\u4e0b\u65b9\u7a0b [code lang=&#8221;mathematica&#8221;]Equ0 = Laplacian[u[x, y], {x, y}] == 0; a = 10; bdn1 = DirichletCondition[u[x, y] == a, x^2 + y^2 == 5]; bdn2 = DirichletCondition[u[x, y] == 0, x == 20 || x == -20 || y == 20 || y == -20]; Equs = {Equ0, bdn1, bdn2};[\/code] Equ0\u8868\u793a\u4e86\u9759\u7535\u65b9\u7a0b bdn1\u662f\u4e8c\u7ef4\u7b49\u52bf\u4f53\uff08\u8fd9\u91cc\u662f\u4e00\u4e2a\u5706\u67f1\uff09\u7684\u7535\u52bf\uff0c\u6211\u4eec\u5b9a\u4e49\u4e3aa bd2\u5219\u8868\u793a\u4e86\u76d2\u5b50\u63a5\u5730\uff0c\u6ce8\u610f\u6211\u4eec\u8fd9\u91cc\u7684\u5e03\u5c14\u8fd0\u7b97\u7b26\u548c\u7c7bc\u8bed\u8a00(c,c++,c#,js etc.)\u7c7b\u4f3c\uff0c\u4f46\u662f a \u6216 b \u53ea\u80fd\u7528 a||b\u8868\u793a\uff0c\u8fd9\u662f\u56e0\u4e3a\u5728\u7c7bc\u8bed\u8a00\u4e2d\uff0c\u5f88\u5e38\u89c1\u7684\u662f\u5355\u7ad6\u7ebf\u8868\u793a<a href=\"http:\/\/zh.wikipedia.org\/wiki\/\u4f4d\u64cd\u4f5c\">\u4f4d\u8fd0\u7b97<\/a>\uff0c\u800cbool\u662f\u4e00\u4e2abit\uff0c\u5176\u4f4d\u8fd0\u7b97\u5f53\u7136\u548c\u5e03\u5c14\u8fd0\u7b97\u76f8\u7b49\uff0c\u4f46\u662fmma\u53ef\u4e0d\u8ba4\u4f60\u8fd9\u4e00\u5957 \u8fd9\u6837\uff0c\u6211\u4eec\u6709\u4e86\u4e00\u5806\u65b9\u7a0b\uff0c\u8fb9\u754c\u8fd8\u6709\u4e00\u4e2a\u8ba1\u7b97\u533a\u57df\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u76f4\u63a5\u62ff\u53bb\u7b97\u4e86 [code]uif = NDSolveValue[Equs, u, {x, y} $$\\in$$ EquArea];[\/code] \u8fd9\u91cc{x, y} $$\\in$$ EquArea\u662f\u4ec0\u4e48\u610f\u601d\u5404\u4f4d\u7406\u5de5\u751f\u4e00\u770b\u5c31\u80fd\u7406\u89e3\uff0c\u8fd9\u5c31\u662f\u5728\u4e00\u4e2a\u533a\u57df\u5185\uff0c\u5bf9Equs\u8fdb\u884c\u6c42\u89e3\u3002 \u6ce8\u610f\u7684\u662f\u7b26\u53f7$$\\in$$\u5728MMA\u9700\u8981\u6441\u4e00\u4e0bESC\u5728\u8f93\u5165el\u518d\u6441\u4e00\u4e0bESC\u8f93\u5165\uff0c\u8fd9\u79cd\u4e24\u4e0bESC\u89c1\u8f93\u5165\u5b57\u7b26\u662fMMA\u72ec\u6709\u7684\uff0c\u5f53\u7136\u4f60\u4e5f\u53ef\u4ee5\u7528Math\u8f93\u5165\u52a9\u624b\uff0c\u867d\u7136\u6162\u4e86\u70b9\uff0c\u4f46\u662f\u6bd4\u4f60\u89e3\u6570\u7406\u65b9\u7a0b\u4e00\u9053\u9898\u4e00\u5c0f\u65f6\u80fd\u597d\u70b9\u3002 \u6211\u4eec\u53ef\u4ee5\u6ce8\u610f\u5230\u4e00\u4e2a\u4e8b\u60c5\u5927\u62ec\u53f7{}\u5728MMA\u91cc\u9762\u5f88\u5168\u80fd\uff0c\u5728Equs\u7684\u5b9a\u4e49\u91cc\u9762\u4ed6\u8868\u8fbe\u4e86\u4e00\u4e2a\u65b9\u7a0b\u6570\u7ec4\u7684\u610f\u601d\uff0c\u5728{x,y}\u5219\u662f\u4e00\u4e2a\u5750\u6807\uff0c\u719f\u6089\u7684\u4eba\u77e5\u9053\u5176\u5b9e\u8fd9\u8d27\u8fd8\u662f\u6570\u7ec4\uff0c\u8fd9\u5c31\u662f\u9762\u5bf9\u7b26\u53f7\u7684MMA\u7684\u9b45\u529b\u6240\u5728\uff0c\u4f60\u53ef\u4ee5\u5728\u6570\u7ec4\u91cc\u9762\u5b58\u4efb\u4f55\u4e1c\u897f\uff0c\u4f60\u7684\u65b9\u7a0b\uff0c\u4f60\u7684\u51fd\u6570\uff0c\u4ee5\u81f3\u4e8e\u5973\u795e\u540c\u5b66\u7684\u7167\u7247\uff0c\u6216\u8005\u535a\u4e3b\u548c\u67d0\u540c\u5b66\u534a\u5e74\u524d\u5435\u67b6\u90a3\u5929\u665a\u4e0a\u4e91\u540e\u7684\u5317\u6597\u4e03\u661f\u7684\u4f4d\u7f6e\uff08\u4ee5\u540e\u518d\u8bb2MMA\u7684\u6570\u636e\u5e93\uff09\u3002 \u8fd9\u4e2a\u89e3\u51fa\u6765\u7684uif\u5b9e\u9645\u4e0a\u662f\u4e00\u4e2a\u5dee\u503c\u51fd\u6570\uff0c\u6211\u4eec\u53ef\u4ee5\u5f88\u5bb9\u6613\u753b\u51fa\u6765\u4ed6\u7684\u7535\u52bf\u9ad8\u5ea6\u548c\u7535\u573a\u5411\u91cf [code lang=&#8221;mathematica&#8221;]p1 = ContourPlot[uif[x, y], {x, y} $$\\in$$ EquArea, ColorFunction -&gt; &#8220;Temperature&#8221;, AspectRatio -&gt; Automatic, PlotLegends -&gt; Automatic]; vetorField = -Grad[uif[x, y], {x, y}] p2 = StreamPlot[ vetorField, {x, -20, 20}, {y, -20, 20}]; Show[p1, p2][\/code] \u6bcf\u4e00\u4e2a\u53c2\u6570\u662f\u505a\u4ec0\u4e48\u7684\u5efa\u8bae\u641c\u7d22\u4e0b\u6587\u6863\uff0c\u9700\u8981\u77e5\u9053MMA\u662f\u7528\u7bad\u5934\u7ed9\u53ef\u9009\u53c2\u6570\u8d4b\u503c\u7684 \u8fd9\u91cc\u7528\u4e86\u70b9MMA\u7684\u77e2\u91cf\u8fd0\u7b97[code lang=&#8221;mathematica&#8221;]-Grad[u[x,y],{x,y}][\/code] \u8868\u793a $$\\Delta_{x,y}u(x,y)$$ \u7136\u540e\u7528stream\u753b\u51fa\u6765\uff0c\u6ce8\u610f\u56e0\u4e3amma\u7684\u4e00\u70b9bug\u76f4\u63a5\u628avectorField\u90a3\u5757\u6362\u6210grad\u662f\u4f1a\u6302\u7684\u3002 \u5219\u51fa\u73b0\u4e86\u4e0b\u56fe\u7684\u6837\u5b50 <a href=\"http:\/\/blog.stlover.org\/wp-content\/uploads\/2014\/07\/pdetutorial2.jpg\"><img loading=\"lazy\" class=\"alignnone size-full wp-image-371\" src=\"http:\/\/blog.stlover.org\/wp-content\/uploads\/2014\/07\/pdetutorial2.jpg\" alt=\"pdetutorial2\" width=\"360\" height=\"356\" srcset=\"http:\/\/blog.xuhao1.me\/wp-content\/uploads\/2014\/07\/pdetutorial2.jpg 360w, http:\/\/blog.xuhao1.me\/wp-content\/uploads\/2014\/07\/pdetutorial2-300x296.jpg 300w\" sizes=\"(max-width: 360px) 100vw, 360px\" \/><\/a> \u7136\u540e\u6211\u4eec\u518d\u7814\u7a76\u4e00\u4e2a\u95ee\u9898\uff0c\u4f5c\u4e3a\u5bf9\u4e8e\u67d0\u51ac\u971c\u7684\u5927\u4e00\u7684\u7535\u78c1\u5b66\u5927\u4f5c\u4e1a\u5632\u8bbd\u3002 \u67d0\u51ac\u971c\u540c\u5b66\u5728\u5927\u4e00\u7684\u7535\u78c1\u5b66\u514d\u4fee\u8003\u8bd5\u5f97\u4e8698\u5427\u5206\u4ee5\u540e\uff0c\u89c9\u5f97\u5c11\u4e86\u4e24\u5206\u4e0d\u5f97\u4e0d\u62b1\u61be\u7ec8\u8eab\uff0c\u4e8e\u662f\u4e00\u8fb9\u5728\u590d\u4e60\u7535\u52a8\u529b\u5b66\u514d\u4fee\u8003\u8bd5\uff0c\u4e00\u8fb9\u8ff7\u4e0a\u4e86\u7535\u78c1\u5b66\u5927\u4f5c\u4e1a\uff0c\u51ac\u971c\u540c\u5b66\u7814\u7a76\u8fd9\u4e48\u4e00\u4e2a\u95ee\u9898<\/p>\n<blockquote><p>\u4e00\u4e2a\u4ee5\u63a5\u8fd1\u5149\u901f\u8fd0\u52a8\u7684\u7403\u4f53\u7535\u5bb9\u5668\u662f\u5426\u4e0e\u4e00\u4e2a\u62c9\u666e\u62c9\u65af\u53d8\u6362\u5750\u6807\u540e\u5f62\u6210\u7684\u692d\u7403\u4f53\u7535\u5bb9\u5668\u7b49\u4ef7<\/p><\/blockquote>\n<p>\u6211\u4eec\u5148\u4e0d\u7528\u5173\u5fc3\u4ec0\u4e48\u662f\u4ed6\u55b5\u7684\u5149\u901f\u60c5\u51b5\u4e0b\u7684\u7403\u4f53\u7535\u5bb9\u5668\uff0c\u6211\u4eec\u53ef\u4ee5\u8bd5\u8bd5\u7b97\u4e00\u4e0b\u692d\u7403\u4f53\u7535\u5bb9\u5668\u7684\u7535\u5bb9\u3002\u5728\u5927\u4e00\u90a3\u4e2aMMA\u8981\u505a\u77e2\u91cf\u8fd0\u7b97\u9700\u8981\u52a0\u8f7d\u5305\u7684\u5e74\u5934\uff0c\u8fd9\u8d27\u51fa\u4e86\u4ece\u56fe\u4e66\u9986\u627e\u5230\u4e86\u4e00\u672c\u5341\u6570\u5e74\u65e0\u4eba\u95ee\u6d25\u7684\u7535\u78c1\u4e66\u6765\u89e3\u6790\u89e3\u4ee5\u5916\u3002\u6211\u4eec\u7528\u4e86\u4e00\u5957\u8499\u7279\u5361\u6d1b\u65b9\u6cd5\u6765\u8ba1\u7b97\u8fd9\u4e2a\u95ee\u9898\uff1a<\/p>\n<blockquote><p>\u7ed9\u5b9a\u4e00\u5806\u7535\u5b50\uff0c\u7ea6\u675f\u4f7f\u5f97\u4ed6\u4eec\u53ea\u80fd\u5728\u692d\u7403\u4e0a\u8fd0\u884c\uff0c\u6bcf\u4e2a\u7535\u5b50\u8fd0\u52a8\u8d77\u6765\u53ea\u53d7\u9759\u7535\u529b\uff0c\u4e14\u4f1a\u56e0\u4e3a\u6469\u64e6\u9020\u6210\u52a8\u91cf\u635f\u5931\uff0c\u8ba9\u8fd9\u4e9b\u767e\u4e07\u4e2a\u7535\u5b50\u8dd1\u4e00\u8dd1\uff0c\u6211\u4eec\u5c31\u80fd\u5f97\u5230\u4e00\u4e2a\u692d\u7403\u4e0a\u7684\u7535\u8377\u5206\u5e03\uff0c\u4e5f\u5c31\u662f\u5176\u7535\u5bb9\u7b49\u5c5e\u6027<\/p><\/blockquote>\n<p>\u5927\u4e8c\u5b66\u8fc7\u8ba1\u7b97\u7269\u7406\u4ee5\u540e\u77e5\u9053\u8fd9\u53eb\u8499\u7279\u5361\u6d1b\u3002 \u4e0d\u8fc7\u6211\u4eec\u53ef\u4ee5\u5728\u4eca\u5929\u7684MMA\u7528\u4e00\u79cd\u66f4\u52a0\u4f18\u96c5\u7684\u65b9\u5f0f\u89e3\u51b3 \u601d\u8def\u5982\u4e0b\uff0c \u7ed9\u5b9a\u4e49\u4e00\u4e2a\u5f88\u5927\u7684\u4e09\u7ef4\u7a7a\u95f4\uff0c\u7ed9\u4e00\u4e2a\u692d\u7403\u7535\u5bb9\u5668\u53c2\u7167\u4e0a\u9762\uff0c\u53ea\u8981\u7ed9\u5b9a\u7535\u52bf\uff0c\u5c31\u53ef\u4ee5\u5f97\u5230\u4e00\u4e2a\u7535\u573a\u5206\u5e03\uff0c\u6839\u636e\u7535\u52a8\u529b\u5b66\u7684\u8fb9\u754c\u6761\u4ef6\u5173\u7cfb\uff0c\u5f88\u5bb9\u6613\u5f97\u5230\u692d\u7403\u7535\u5bb9\u5668\u6bcf\u4e00\u70b9\u7684\u7535\u8377\u5bc6\u5ea6\uff0c\u79ef\u5206\u5f97\u5230\u7535\u5bb9\u3002<\/p>\n<h2>\u7f51\u683c\u5212\u5206<\/h2>\n<p>\u4e0a\u9762\u8fd9\u4e2a\u95ee\u9898\u6309\u7167\u6211\u4eec\u4e4b\u524d\u7684\u601d\u8def\uff0c\u4f3c\u4e4e\u5f88\u662f\u7b80\u5355 [code lang=&#8221;mathematica&#8221;]Equ0 = Laplacian[u[x, y, z], {x, y, z}] == 0; a = 10; bdn1 = DirichletCondition[u[x, y, z] == a, x^2 + 3 y^2 + z^2 == 5]; bdn2 = DirichletCondition[u[x, y, z] == 0, x == 20 || x == -20 || y == 20 || y == -20 || x == 20 || x == -20]; Equs = {Equ0, bdn1, bdn2}; EquArea = ImplicitRegion[! ($$x^2 + 3 y^2 + z^2$$ &lt; 5), {{x, -20, 20}, {y, -20, 20}, {z, -20, 20}}]; RegionPlot3D[EquArea, AspectRatio -&gt; Automatic, PlotStyle -&gt; Directive[Yellow, Opacity[0.5]]] uif = NDSolveValue[Equs, u, {x, y, z} $$\\in$$ EquArea];[\/code] \u4f46\u662f\u89e3\u8fd9\u4e2a\u65b9\u7a0b\u4f60\u4f1a\u53d1\u73b0 \u8fd9\u4e2a\u56fe\u4e0d\u600e\u4e48\u597d\u770b\u561b \u7136\u540e<\/p>\n<blockquote><p>NDSolveValue::bcnop: No places were found on the boundary where x^2+3 y^2+z^2==5 was True, so DirichletCondition[u==10,x^2+3 y^2+z^2==5] will effectively be ignored. &gt;&gt;<\/p><\/blockquote>\n<p>MMA\u4f3c\u4e4e\u50bb\u4e86\uff0c\u8fd9\u4e5f\u5c31\u610f\u5473\u7740\uff0c\u5bf9\u4e8e\u8fd9\u4e2a\u590d\u6742\u76843D\u95ee\u9898\uff0cMMA\u4e27\u5931\u4e86\u5212\u5206\u7f51\u683c\u7684\u80fd\u529b\u3002 \u5176\u5b9e\u4e4b\u524d\u6211\u4eec\u505a\u7684\u867d\u7136\u6ca1\u6709\u663e\u5f0f\u7684\u5236\u5b9a\u7f51\u683c\uff0c\u5176\u5b9eMMA\u90fd\u66ff\u6211\u4eec\u505a\u4e86\u8fd9\u90e8\u5206\u7684\u5de5\u4f5c\uff0c\u73b0\u5728\u6211\u4eec\u8fd8\u662f\u5f97\u4e00\u6b65\u6b65\u6765\u5427\u3002 \u56de\u5230\u90a3\u4e2a2D\u5e73\u9762\u7684\u7535\u5bb9\u95ee\u9898 \u6211\u4eec\u53ef\u4ee5\u753b\u51fauif\u89e3\u7684\u7f51\u683c [code lang=&#8221;mathematica&#8221;]uif[&#8220;ElementMesh&#8221;][&#8220;Wireframe&#8221;][\/code] \u90a3\u4e48\u5982\u56fe <a href=\"http:\/\/blog.stlover.org\/wp-content\/uploads\/2014\/07\/pdetutorial1.jpg\"><img loading=\"lazy\" class=\"alignnone size-full wp-image-401\" src=\"http:\/\/blog.stlover.org\/wp-content\/uploads\/2014\/07\/pdetutorial1.jpg\" alt=\"pdetutorial\" width=\"574\" height=\"573\" srcset=\"http:\/\/blog.xuhao1.me\/wp-content\/uploads\/2014\/07\/pdetutorial1.jpg 574w, http:\/\/blog.xuhao1.me\/wp-content\/uploads\/2014\/07\/pdetutorial1-150x150.jpg 150w, http:\/\/blog.xuhao1.me\/wp-content\/uploads\/2014\/07\/pdetutorial1-300x300.jpg 300w\" sizes=\"(max-width: 574px) 100vw, 574px\" \/><\/a> \u6211\u4eec\u6240\u6709\u7684\u8ba1\u7b97\uff0c\u90fd\u662f\u5728\u8fd9\u4e9b\u7f51\u683c\u70b9\u4e0a\u5b8c\u6210\u7684\u3002\u3002\u662f\u4e0d\u662f\u5341\u5206\u7684\u7c97\u7cd9\uff0c\u73b0\u5728\u6211\u4eec\u5c31\u6765\u7814\u7a76\u5982\u4f55\u83b7\u5f97\u4e00\u4e2a\u5408\u9002\u7684\uff0c\u597d\u770b\u7684\u7f51\u683c\u70b9\u3002 \u9996\u5148\u6211\u4eec\u671f\u5f85\uff0c\u5728\u4e2d\u5fc3\u7684\u5706\u9644\u8fd1\u7684\u7f51\u683c\u53ef\u4ee5\u66f4\u52a0\u7684\u7a20\u5bc6\uff0c\u5728\u8fb9\u754c\u5219\u758f\u677e\u4e00\u4e9b\u3002 \u5176\u6b21\u662f\u8981\u591f\u597d\u770b [code lang=&#8221;mathematica&#8221;]EquArea = ImplicitRegion[ ! ((x)^2 + (y)^2 &lt; 5), {{x, -20, 20}, {y, -20, 20}}]; bmesh = ToBoundaryMesh[EquArea, &#8220;BoundaryMeshGenerator&#8221; -&gt; {&#8220;RegionPlot&#8221;, MaxRecursion -&gt; 10}, &#8220;MaxBoundaryCellMeasure&#8221; -&gt; .5]; mesh = ToElementMesh[bmesh] mesh[&#8220;Wireframe&#8221;][\/code] \u8fd9\u4e2a\u5c31\u662f\u4f7f\u7528\u4e00\u4e2a\u533a\u57df\u6765\u751f\u6210\u4e00\u4e2a\u7f51\u683c\uff0c\u6211\u4eec\u5148\u751f\u6210\u8fb9\u754c\u7f51\u7edcbmesh\uff0c\u518d\u4f7f\u7528\u8fb9\u754c\u7f51\u7edc\u751f\u6210mesh\u3002 \u753b\u51fa\u6765\u5982\u56fe <a href=\"http:\/\/blog.stlover.org\/wp-content\/uploads\/2014\/07\/mesh0.jpg\"><img loading=\"lazy\" class=\"alignnone size-full wp-image-402\" src=\"http:\/\/blog.stlover.org\/wp-content\/uploads\/2014\/07\/mesh0.jpg\" alt=\"mesh0\" width=\"604\" height=\"603\" srcset=\"http:\/\/blog.xuhao1.me\/wp-content\/uploads\/2014\/07\/mesh0.jpg 604w, http:\/\/blog.xuhao1.me\/wp-content\/uploads\/2014\/07\/mesh0-150x150.jpg 150w, http:\/\/blog.xuhao1.me\/wp-content\/uploads\/2014\/07\/mesh0-300x300.jpg 300w\" sizes=\"(max-width: 604px) 100vw, 604px\" \/><\/a> \u5728\u5916\u8fb9\u6846\u6709\u70b9\u4e0d\u597d\u770b\uff0c\u4e0d\u8fc7\u53ef\u4ee5\u8bd5\u8bd5\u561b [code lang=&#8221;mathematica&#8221;]Equ0 = Laplacian[u[x, y], {x, y}] == 0; a = 10; bdn1 = DirichletCondition[u[x, y] == a, x^2 + y^2 == 5]; bdn2 = DirichletCondition[u[x, y] == 0, x == 20 || x == -20 || y == 20 || y == -20]; Equs = {Equ0, bdn1, bdn2}; uif = NDSolveValue[Equs, u, {x, y} \\[Element] mesh]; p1 = ContourPlot[uif[x, y], {x, y} \\[Element] EquArea, ColorFunction -&gt; &#8220;Temperature&#8221;, AspectRatio -&gt; Automatic, PlotLegends -&gt; Automatic]; vetorField = -Grad[uif[x, y], {x, y}] p2 = StreamPlot[ vetorField, {x, -20, 20}, {y, -20, 20}]; Show[p1, p2][\/code] \u7ed3\u679c\u5982\u56fe <a href=\"http:\/\/blog.stlover.org\/wp-content\/uploads\/2014\/07\/meshpde1.jpg\"><img loading=\"lazy\" class=\"alignnone size-full wp-image-403\" src=\"http:\/\/blog.stlover.org\/wp-content\/uploads\/2014\/07\/meshpde1.jpg\" alt=\"meshpde1\" width=\"731\" height=\"727\" srcset=\"http:\/\/blog.xuhao1.me\/wp-content\/uploads\/2014\/07\/meshpde1.jpg 731w, http:\/\/blog.xuhao1.me\/wp-content\/uploads\/2014\/07\/meshpde1-150x150.jpg 150w, http:\/\/blog.xuhao1.me\/wp-content\/uploads\/2014\/07\/meshpde1-300x298.jpg 300w\" sizes=\"(max-width: 731px) 100vw, 731px\" \/><\/a> \u597d\u50cf\u662f\u8981\u597d\u770b\u4e00\u4e9b\u5566\uff5e\uff5e \u4e0d\u8fc7\u6211\u4eec\u5bf9\u4e8e\u5916\u8fb9\u754c\u7684\u8fd9\u4e2a\u803f\u803f\u4e8e\u6000\uff0c\u56e0\u4e3a\u5982\u679cMaxBoundaryCellMeasure\u518d\u53d6\u5f97\u7ec6\u81f4\uff0c\u8ba1\u7b97\u91cf\u4f1a\u589e\u5927\uff0c\u4f46\u662f\u5916\u8fb9\u754c\u4e0d\u9700\u8981\u90a3\u4e48\u7ec6\u81f4\uff0c\u7279\u522b\u5bf9\u4e8e\u672a\u6765\u7684\u6d41\u4f53\u662f\u5f88\u660e\u663e\u4e86\uff0c\u4f46\u662f\u8981\u5982\u4f55\u4f18\u5316\u5462 \u8fd9\u65f6\u5019\u6211\u7ffb\u9605\u4e86\u6587\u6863\uff0c\u53d1\u73b0\u6211\u4eec\u5f97\u91c7\u7528\u4e00\u79cd\u4e0d\u662f\u90a3\u4e48\u4f18\u7f8e\u7684\u65b9\u5f0f\u4e86&#8230; \u8fd9\u91cc\u6211\u4eec\u4ee5\u540e\u518d\u8f6c\u95e8\u8bb2 \u6211\u4eec\u53ef\u4ee5\u8bd5\u8bd5\u7528\u8fd9\u4e2a\u65b9\u6cd5\u5904\u7406\u692d\u5706<\/p>\n<h2>\u5fb7\u83b1\u5c3c\u4e09\u89d2<\/h2>\n<p>\u9884\u5904\u7406\u5de5\u4f5c\uff0c\u4e5f\u5c31\u662fDelaunay\u4e09\u89d2\u5228\u5206\u4e86\u3002<\/p>\n<p>\u6240\u8c13\u5fb7\u83b1\u5c3c\u4e09\u89d2\u5228\u5206\uff0c\u662f\u5bf9\u5e73\u9762\u533a\u57df\u6839\u636e\u70b9\u96c6\u5408\u7684\u4e00\u4e2a\u4f18\u96c5\u5228\u5206\uff0c\u4f7f\u5f97\u5e73\u9762\u5212\u5206\u66f4\u52a0\u201c\u79d1\u5b66\u5408\u7406\u201d\uff0c\u6b64\u5904\u7684\u201c\u79d1\u5b66\u5408\u7406\u201d\u6211\u4eec\u65e0\u59a8\u8ba4\u4e3a\u6307\u7684\u662f\u5728\u6211\u4eec\u505a\u66f2\u9762\u79ef\u5206\u6216\u8005\u7ebf\u79ef\u5206\u7684\u65f6\u5019\u6709\u66f4\u5c0f\u8bef\u5dee\uff0c\u56e0\u4e3a\u6211\u4eec\u8981\u7b97\u4e1c\u897f\u5fc5\u987b\u9996\u5148\u5206\u5316\u4e4b\u3002\u6240\u4ee5\u5fb7\u83b1\u5c3c\u4e09\u89d2\u5228\u5206\u7684\u5e94\u7528\u4e0d\u53ef\u8c13\u4e0d\u5c11\u3002<\/p>\n<p>\u9996\u5148\u901a\u8fc7\u4e00\u4e9b\u7b80\u5355\u7684\u4f8b\u5b50\u4e86\u89e3\u4e0b\u4ec0\u4e48\u662f\u5fb7\u83b1\u5c3c\u4e09\u89d2\u5228\u5206\uff0c\u4e8e\u662f\u6709\u8bf7Mathematica<\/p>\n<ol class=\"linenums\">\n<li class=\"L0\"><span class=\"pun\">(*<\/span><span class=\"kwd\">this<\/span><span class=\"pln\">\u00a0is\u00a0mathematica\u00a0code<\/span><span class=\"pun\">*)<\/span><\/li>\n<li class=\"L1\"><span class=\"typ\">Needs<\/span><span class=\"pun\">[<\/span><span class=\"str\">&#8220;ComputationalGeometry`&#8221;<\/span><span class=\"pun\">](*\u5f15\u5165\u8ba1\u7b97\u5305*)<\/span><\/li>\n<li class=\"L2\"><span class=\"pun\">(*\u4ea7\u751f\u4e00\u4e2a\u968f\u673a\u70b9\u5217\uff0c\u4ee5\u53ca\u4e0e\u4e4b\u76f8\u5e94\u7684\u70b9\u5750\u6807\u89c4\u5219*)<\/span><\/li>\n<li class=\"L3\"><span class=\"pln\">points\u00a0<\/span><span class=\"pun\">=<\/span><span class=\"pln\">\u00a0<\/span><span class=\"typ\">Table<\/span><span class=\"pun\">[<\/span><\/li>\n<li class=\"L4\"><span class=\"pln\">\u00a0\u00a0\u00a0<\/span><span class=\"pun\">{<\/span><span class=\"typ\">RandomReal<\/span><span class=\"pun\">[<\/span><span class=\"lit\">100<\/span><span class=\"pun\">],<\/span><span class=\"pln\">\u00a0<\/span><span class=\"typ\">RandomReal<\/span><span class=\"pun\">[<\/span><span class=\"lit\">100<\/span><span class=\"pun\">]}<\/span><\/li>\n<li class=\"L5\"><span class=\"pln\">\u00a0\u00a0\u00a0<\/span><span class=\"pun\">,<\/span><span class=\"pln\">\u00a0<\/span><span class=\"pun\">{<\/span><span class=\"pln\">i<\/span><span class=\"pun\">,<\/span><span class=\"pln\">\u00a0<\/span><span class=\"lit\">1<\/span><span class=\"pun\">,<\/span><span class=\"pln\">\u00a0<\/span><span class=\"lit\">1000<\/span><span class=\"pun\">}];<\/span><\/li>\n<li class=\"L6\"><span class=\"pln\">coorrule\u00a0<\/span><span class=\"pun\">=<\/span><span class=\"pln\">\u00a0<\/span><span class=\"typ\">Table<\/span><span class=\"pun\">[<\/span><\/li>\n<li class=\"L7\"><span class=\"pln\">\u00a0\u00a0\u00a0i\u00a0<\/span><span class=\"pun\">-&gt;<\/span><span class=\"pln\">\u00a0points<\/span><span class=\"pun\">[[<\/span><span class=\"pln\">i<\/span><span class=\"pun\">]]<\/span><\/li>\n<li class=\"L8\"><span class=\"pln\">\u00a0\u00a0\u00a0<\/span><span class=\"pun\">,<\/span><span class=\"pln\">\u00a0<\/span><span class=\"pun\">{<\/span><span class=\"pln\">i<\/span><span class=\"pun\">,<\/span><span class=\"pln\">\u00a0<\/span><span class=\"lit\">1<\/span><span class=\"pun\">,<\/span><span class=\"pln\">\u00a0<\/span><span class=\"typ\">Length<\/span><span class=\"pun\">[<\/span><span class=\"pln\">points<\/span><span class=\"pun\">]}<\/span><\/li>\n<li class=\"L9\"><span class=\"pln\">\u00a0\u00a0\u00a0<\/span><span class=\"pun\">];<\/span><\/li>\n<li class=\"L0\"><span class=\"pun\">(*\u8fdb\u884c\u4e09\u89d2\u5228\u5206\uff0c\u4ee5\u53ca\u5c06\u4e09\u89d2\u5228\u5206\u7684\u7ed3\u679c\u8f6c\u5316\u4e3a<\/span><span class=\"pln\">mathematica<\/span><span class=\"pun\">\u56fe\u8bba\u7ec4\u5efa\u9700\u8981\u7684\u5305*)<\/span><\/li>\n<li class=\"L1\"><span class=\"pln\">dela\u00a0<\/span><span class=\"pun\">=<\/span><span class=\"pln\">\u00a0<\/span><span class=\"typ\">DelaunayTriangulation<\/span><span class=\"pun\">[<\/span><span class=\"pln\">points<\/span><span class=\"pun\">];<\/span><\/li>\n<li class=\"L2\"><span class=\"pln\">fir\u00a0<\/span><span class=\"pun\">=<\/span><span class=\"pln\">\u00a0<\/span><span class=\"typ\">Fold<\/span><span class=\"pun\">[<\/span><\/li>\n<li class=\"L3\"><span class=\"pln\">\u00a0\u00a0<\/span><span class=\"pun\">(<\/span><span class=\"typ\">Join<\/span><span class=\"pun\">[#<\/span><span class=\"lit\">1<\/span><span class=\"pun\">,<\/span><span class=\"pln\">\u00a0<\/span><\/li>\n<li class=\"L4\"><span class=\"pln\">\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><span class=\"typ\">Table<\/span><span class=\"pun\">[#<\/span><span class=\"lit\">2<\/span><span class=\"pun\">[[<\/span><span class=\"lit\">1<\/span><span class=\"pun\">]]<\/span><span class=\"pln\">\u00a0<\/span><span class=\"pun\">-&gt;<\/span><span class=\"pln\">\u00a0<\/span><span class=\"pun\">#<\/span><span class=\"lit\">2<\/span><span class=\"pun\">[[<\/span><span class=\"lit\">2<\/span><span class=\"pun\">]][[<\/span><span class=\"pln\">j<\/span><span class=\"pun\">]],<\/span><span class=\"pln\">\u00a0<\/span><span class=\"pun\">{<\/span><span class=\"pln\">j<\/span><span class=\"pun\">,<\/span><span class=\"pln\">\u00a0<\/span><span class=\"lit\">1<\/span><span class=\"pun\">,<\/span><span class=\"pln\">\u00a0<\/span><span class=\"typ\">Length<\/span><span class=\"pun\">[<\/span><span class=\"pln\">\u00a0\u00a0<\/span><span class=\"pun\">#<\/span><span class=\"lit\">2<\/span><span class=\"pun\">[[<\/span><span class=\"lit\">2<\/span><span class=\"pun\">]]<\/span><span class=\"pln\">\u00a0<\/span><span class=\"pun\">]}]])<\/span><span class=\"pln\">\u00a0<\/span><span class=\"pun\">&gt;,<\/span><\/li>\n<li class=\"L5\"><span class=\"pln\">\u00a0\u00a0<\/span><span class=\"pun\">{},<\/span><span class=\"pln\">\u00a0dela<\/span><span class=\"pun\">]<\/span><\/li>\n<li class=\"L6\"><span class=\"pun\">(*\u753b\u51fa\u7ed3\u679c*)<\/span><\/li>\n<li class=\"L7\"><span class=\"typ\">GraphPlot<\/span><span class=\"pun\">[<\/span><span class=\"pln\">fir<\/span><span class=\"pun\">,<\/span><span class=\"pln\">\u00a0<\/span><span class=\"typ\">VertexCoordinateRules<\/span><span class=\"pln\">\u00a0<\/span><span class=\"pun\">-&gt;<\/span><span class=\"pln\">\u00a0coorrule<\/span>, DirectedEdges -&gt; False, MultiedgeStyle -&gt; None<span style=\"color: #666600; line-height: 1.5;\">]<\/span><\/li>\n<\/ol>\n<p>\u7ed3\u679c\u5982\u4e0b\uff1a<\/p>\n<p><a href=\"http:\/\/blog.stlover.org\/wp-content\/uploads\/2013\/08\/delaunay1.jpg\"><img loading=\"lazy\" class=\"alignnone size-full wp-image-122\" src=\"http:\/\/blog.stlover.org\/wp-content\/uploads\/2013\/08\/delaunay1.jpg\" alt=\"delaunay\" width=\"768\" height=\"768\" srcset=\"http:\/\/blog.xuhao1.me\/wp-content\/uploads\/2013\/08\/delaunay1.jpg 768w, http:\/\/blog.xuhao1.me\/wp-content\/uploads\/2013\/08\/delaunay1-150x150.jpg 150w, http:\/\/blog.xuhao1.me\/wp-content\/uploads\/2013\/08\/delaunay1-300x300.jpg 300w\" sizes=\"(max-width: 767px) 89vw, (max-width: 1000px) 54vw, (max-width: 1071px) 543px, 580px\" \/><\/a><\/p>\n<p>&nbsp;<\/p>\n<p>\u54dd\uff5e\u4e00\u4e2a\u6f02\u4eae\u7684\u5228\u5206\u51fa\u73b0\u4e86\u3002\u8fd9\u5c31\u662f\u6211\u4eec\u8981\u6c42\u7684\u7ed3\u679c\u3002\u51fa\u73b0\u5c3d\u91cf\u66f4\u591a\u201c\u79d1\u5b66\u5408\u7406\u7684\u201d\u5228\u5206\u3002<\/p>\n<p>\u90a3\u6211\u4eec\u65e0\u59a8\u50cf\u8fd9\u6837\u4e00\u4e2a\u95ee\u9898\uff0c\u4ec0\u4e48\u6837\u7684\u5228\u5206\u662f\u201c\u79d1\u5b66\u5408\u7406\u201d\u7684\uff1f<\/p>\n<p>1.\u4f55\u8c13\u6f02\u4eae\uff1f<\/p>\n<p>\u6709\u8fd9\u6837\u4e00\u79cd\u89c2\u70b9\uff0c\u6f02\u4eae\u7684\u4e1c\u897f\u4e00\u822c\u90fd\u662f\u66f4\u6613\u4e8e\u5b58\u5728\u7684\uff0c\u56e0\u4e3a\u4eba\u7c7b\u7684\u76f4\u89c9\u548c\u5bf9\u7f8e\u7684\u5b9a\u4e49\u4e00\u822c\u662f\u533a\u57df\u5408\u7406\u7684\uff0c\u6240\u4ee5\u6b63\u5e38\u5ba1\u7f8e\u4e2d\u7684\u7f8e\u5973\u4e00\u822c\u6709\u66f4\u5f3a\u7684\u751f\u80b2\u80fd\u529b\uff0c\u518d\u6b21\u6211\u4eec\u53ef\u4ee5\u8ba4\u4e3a\u90a3\u4e9b\u8ffd\u9010\u51cf\u80a5\u8fc7\u5ea6\u7684\u5973\u751f\u662f\u5728\u53cd\u4eba\u7c7b\u3002\u540c\u6837\uff0c\u521a\u624d\u6211\u4eec\u4e5f\u770b\u5230\u7684\u662f\u4e00\u4e2a\u6f02\u4eae\u7684\u5228\u5206\uff0c\u4ed6\u662f\u5408\u7406\u7684\u3002\u56e0\u4e3a\u5bf9\u540c\u610f\u7684\u9762\u79ef\u533a\u57df\uff0c\u603b\u6709\u66f4\u5c0f\u7684\u8fb9\u957f\u5c06\u81f3\u56f4\u8d77\u6765\uff0c\u8fd9\u6837\u5728\u5fae\u79ef\u5206\u6570\u503c\u8fd0\u7b97\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u5728\u66f4\u5c0f\u7684\u8fb9\u957f\u4e0a\u4f5c\u8fd1\u4f3c\uff0c\u4e5f\u5c31\u4f1a\u5f97\u5230\u66f4\u5408\u7406\u7684\u7ed3\u679c\u3002\u52a0\u4e4b\u4e8e\u5730\u56fe\u7ed8\u5236\uff0c\u5408\u7406\u7684\u5228\u5206\u80fd\u591f\u4ece\u9ad8\u5ea6\u91c7\u6837\u83b7\u5f97\u6f02\u4eae\u7684\u5730\u56fe\uff0c\u8fd9\u4e00\u70b9\u6211\u4eec\u4f1a\u518d\u540e\u9762\u6f14\u793a\u3002\u90a3\u4e48\u6211\u4eec\u53ef\u4ee5\u4e0b\u4e00\u79cd\u5b9a\u4e49\u6765\u5bfb\u627e\u4e00\u4e2a\u6700\u6f02\u4eae\u7684\u5228\u5206\uff0c\u7136\u540e\u518d\u8ba8\u8bba\u5982\u4f55\u5b9e\u73b0\u5b83\u3002<\/p>\n<p>\u663e\u7136\uff0c\u6211\u4eec\u754f\u60e7\u7684\u662f\u5c0f\u89d2\u5ea6\u89d2\u7684\u51fa\u73b0\uff08\u8fd9\u610f\u5473\u7740\u5c0f\u9762\u79ef\u548c\u957f\u8fb9\u7834\u574f\u4e86\u79ef\u5206\u7684\u8bef\u5dee\uff09\uff0c\u90a3\u4e48\u65e0\u59a8\uff08\u4e8b\u5b9e\u4e0a\u662f\u4e66\u4e0a\u544a\u8bc9\u6211\u7684\uff09\u505a\u8fd9\u6837\u7684\u4e00\u4e2a\u5b9a\u4e49\uff0c\u6700\u5c0f\u89d2\u6700\u5927\uff0c\u6b21\u5c0f\u89d2\uff0c\u6b21\u6b21\u5c0f\u89d2\u3002\u3002\u3002\u90fd\u662f\u6700\u5927\u4e00\u4e2a\u4e09\u89d2\u5228\u5206\u662f\u6700\u5408\u7406\u7684\u3002<\/p>\n<p>\u7528\u6570\u5b66\u5316\u7684\u8bed\u8a00\u8868\u793a\uff0c\u5c31\u662f\u5228\u5206A\u7684\u89d2\u5ea6\u6392\u5e8f$$\\beta=(a0,a.,&#8230;an)$$\u4e3aA\u7684\u89d2\u5ea6\u5411\u91cf\uff0c\u5b9a\u4e49$$\\beta_0&gt;\\beta_1$$\u4e3a\u6309\u7167\u5b57\u5178\u5e8f\u4f9d\u6b21\u6bd4\u8f83\u3002<\/p>\n<p>\u8fd9\u6837\u6211\u4eec\u5c31\u53ef\u4ee5\u5f97\u5230\uff0c\u6bcf\u4e2a\u89d2\u90fd\u662f\u6700\u5927\u60c5\u51b5\u7684\u4e00\u4e2a\u5228\u5206\uff0c\u73b0\u5728\u6709\u4e00\u4e2a\u6570\u5b66\u5bb6\u770b\u8d77\u6765\u6beb\u4e0d\u8d77\u773c\u7684\u95ee\u9898\uff0c\u5982\u4f55\u5b9e\u73b0\uff1f<\/p>\n<h2>COMSOL<\/h2>\n<p>\u6298\u817e\u4e86\u4fe9\u5c0f\u65f6\u7528COMSOL\u8dd1\u51fa\u4e86\u4e00\u70b9\u6d41\u4f53\u3002 \u00a0 \u5176\u5b9e\u8fd9\u73a9\u610f\u7528\u8d77\u6765\u5f88\u65e0\u8111\u7684\u3002\u6bd4\u6211\u9ad8\u4e2d\u65f6\u671f\u7528\u7684Fluent\u7528\u8d77\u6765\u7b80\u5355\u591a\u4e86\uff0c\u5bf9\u4ed8\u4e00\u4e9b\u4e0d\u590d\u6742\u7684\u9700\u6c42\u8fd8\u662f\u53ef\u4ee5\u7684\uff0c\u6bd4\u5982\u7ecf\u5178\u95ee\u9898\uff0c\u5361\u95e8\u6da1\u8857\u3002\u00a0<a style=\"color: #bc360a;\" href=\"http:\/\/blog.stlover.org\/wp-content\/uploads\/2014\/07\/%E5%B1%8F%E5%B9%95%E5%BF%AB%E7%85%A7-2014-07-21-%E4%B8%8A%E5%8D%8812.35.05.png\"><img loading=\"lazy\" class=\"alignnone size-full wp-image-446\" src=\"http:\/\/blog.stlover.org\/wp-content\/uploads\/2014\/07\/%E5%B1%8F%E5%B9%95%E5%BF%AB%E7%85%A7-2014-07-21-%E4%B8%8A%E5%8D%8812.35.05.png\" alt=\"\u5c4f\u5e55\u5feb\u7167 2014-07-21 \u4e0a\u534812.35.05\" width=\"1090\" height=\"680\" \/><\/a><\/p>\n<h2>SOLIDWORKS<\/h2>\n<p>\u56db\u4e2a\u87ba\u4e1d\u7ed9\u5230100N<a href=\"http:\/\/blog.xuhao1.me\/wp-content\/uploads\/2014\/07\/Throttle^Main-\u9759\u5e94\u529b\u5206\u6790-2-\u7ed3\u679c-\u5e94\u529b1.analysis.jpg\"><img loading=\"lazy\" class=\"alignnone size-full wp-image-474\" src=\"http:\/\/blog.xuhao1.me\/wp-content\/uploads\/2014\/07\/Throttle^Main-\u9759\u5e94\u529b\u5206\u6790-2-\u7ed3\u679c-\u5e94\u529b1.analysis.jpg\" alt=\"Throttle^Main-\u9759\u5e94\u529b\u5206\u6790 2-\u7ed3\u679c-\u5e94\u529b1.analysis\" width=\"1565\" height=\"1492\" srcset=\"http:\/\/blog.xuhao1.me\/wp-content\/uploads\/2014\/07\/Throttle^Main-\u9759\u5e94\u529b\u5206\u6790-2-\u7ed3\u679c-\u5e94\u529b1.analysis.jpg 1565w, http:\/\/blog.xuhao1.me\/wp-content\/uploads\/2014\/07\/Throttle^Main-\u9759\u5e94\u529b\u5206\u6790-2-\u7ed3\u679c-\u5e94\u529b1.analysis-300x286.jpg 300w, http:\/\/blog.xuhao1.me\/wp-content\/uploads\/2014\/07\/Throttle^Main-\u9759\u5e94\u529b\u5206\u6790-2-\u7ed3\u679c-\u5e94\u529b1.analysis-1024x976.jpg 1024w\" sizes=\"(max-width: 767px) 89vw, (max-width: 1000px) 54vw, (max-width: 1071px) 543px, 580px\" \/><\/a> \u5728\u63a5\u53e3\u524d\u7684\u5730\u65b9\u662f\u6700\u8106\u5f31\u7684\u5730\u65b9\uff0c\u4e0e\u4e4b\u524d\u7684\u8ba4\u8bc6\u5b8c\u5168\u76f8\u53cd\uff0c\u4e00\u60f3\u4e5f\u662f\u7684\uff0c\u4f4d\u79fb\u57282mm\u78b3\u7ea4\u7ef4\u7684\u60c5\u51b5\u4e0b\u67093mm\uff0c\u57282mm\u94dd\u5408\u91d1\u662f8mm\uff01\u9876\u70b9\u57282mm\u949b\u5408\u91d1\u662f4-6mm \u4e8e\u662f\u505a\u4e86\u4e2a\u52a0\u5f3a\u4ef6\uff0c\u4f18\u5316\u6781\u5176\u660e\u663e\uff0c\u6700\u5927\u5e94\u529b\u4e0b\u964d\u4e09\u4e2a\u6570\u91cf\u7ea7\uff0c\u8c03\u5230\u4e86\u5927\u6c14\u538b\u6c34\u51c6\uff0c\u4f4d\u79fb\u6389\u4e86\u516d\u4e2a\u6570\u91cf\u7ea7\u3002\u3002\u3002<a href=\"http:\/\/blog.xuhao1.me\/wp-content\/uploads\/2014\/07\/Throttle^Main-\u9759\u5e94\u529b\u5206\u6790-3-\u7ed3\u679c-\u4f4d\u79fb1.analysis.jpg\"><img loading=\"lazy\" class=\"alignnone size-full wp-image-475\" 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