{"id":1072825,"date":"2025-01-08T11:19:44","date_gmt":"2025-01-08T03:19:44","guid":{"rendered":"https:\/\/docs.pingcode.com\/ask\/ask-ask\/1072825.html"},"modified":"2025-01-08T11:19:46","modified_gmt":"2025-01-08T03:19:46","slug":"python%e5%a6%82%e4%bd%95%e6%b1%82%e8%a7%a3%e5%81%8f%e5%be%ae%e5%88%86%e6%96%b9%e7%a8%8b-2","status":"publish","type":"post","link":"https:\/\/docs.pingcode.com\/ask\/1072825.html","title":{"rendered":"python\u5982\u4f55\u6c42\u89e3\u504f\u5fae\u5206\u65b9\u7a0b"},"content":{"rendered":"

\"python\u5982\u4f55\u6c42\u89e3\u504f\u5fae\u5206\u65b9\u7a0b\"<\/p>\n

Python\u6c42\u89e3\u504f\u5fae\u5206\u65b9\u7a0b\u7684\u5e38\u7528\u65b9\u6cd5\u6709\uff1a\u6709\u9650\u5dee\u5206\u6cd5\u3001\u6709\u9650\u5143\u6cd5\u3001\u8c31\u65b9\u6cd5\u3001\u4ee5\u53ca\u4f7f\u7528\u4e13\u95e8\u7684\u6570\u503c\u8ba1\u7b97\u5e93\u548c\u8f6f\u4ef6\uff0c\u5982SymPy\u3001SciPy\u3001FEniCS\u7b49\u3002\u4e0b\u9762\u5c06\u8be6\u7ec6\u4ecb\u7ecd\u5176\u4e2d\u7684\u4e00\u79cd\u65b9\u6cd5\uff0c\u5e76\u7b80\u8981\u4ecb\u7ecd\u5176\u4ed6\u65b9\u6cd5\u7684\u57fa\u672c\u539f\u7406\u548c\u5e94\u7528\u573a\u666f\u3002<\/strong> <\/p>\n<\/p>\n

\u4e00\u3001\u6709\u9650\u5dee\u5206\u6cd5<\/h3>\n<\/p>\n

\u6709\u9650\u5dee\u5206\u6cd5\u662f\u6c42\u89e3\u504f\u5fae\u5206\u65b9\u7a0b\u7684\u4e00\u79cd\u5e38\u7528\u6570\u503c\u65b9\u6cd5\u3002\u5b83\u901a\u8fc7\u5c06\u8fde\u7eed\u7684\u504f\u5fae\u5206\u65b9\u7a0b\u79bb\u6563\u5316\u4e3a\u4e00\u7ec4\u4ee3\u6570\u65b9\u7a0b\uff0c\u4ece\u800c\u5728\u79bb\u6563\u7f51\u683c\u4e0a\u8fd1\u4f3c\u6c42\u89e3\u504f\u5fae\u5206\u65b9\u7a0b\u3002<\/p>\n<\/p>\n

1.1 \u57fa\u672c\u539f\u7406<\/h4>\n<\/p>\n

\u6709\u9650\u5dee\u5206\u6cd5\u7684\u57fa\u672c\u601d\u60f3\u662f\u7528\u5dee\u5206\u4ee3\u66ff\u5bfc\u6570\uff0c\u5c06\u5fae\u5206\u65b9\u7a0b\u8f6c\u6362\u6210\u5dee\u5206\u65b9\u7a0b\u3002\u4f8b\u5982\uff0c\u5bf9\u4e8e\u4e00\u7ef4\u70ed\u4f20\u5bfc\u65b9\u7a0b\uff1a<\/p>\n<\/p>\n

$$ \\frac{\\partial u}{\\partial t} = \\alpha \\frac{\\partial^2 u}{\\partial x^2} $$<\/p>\n<\/p>\n

\u6211\u4eec\u53ef\u4ee5\u5c06\u5176\u79bb\u6563\u5316\u4e3a\uff1a<\/p>\n<\/p>\n

$$ \\frac{u_i^{n+1} – u_i^n}{\\Delta t} = \\alpha \\frac{u_{i+1}^n – 2u_i^n + u_{i-1}^n}{\\Delta x^2} $$<\/p>\n<\/p>\n

\u5176\u4e2d\uff0c$u_i^n$\u8868\u793a\u5728\u65f6\u95f4\u6b65$n$\u3001\u7a7a\u95f4\u4f4d\u7f6e$i$\u5904\u7684\u6e29\u5ea6\uff0c$\\Delta t$\u548c$\\Delta x$\u5206\u522b\u662f\u65f6\u95f4\u6b65\u957f\u548c\u7a7a\u95f4\u6b65\u957f\u3002<\/p>\n<\/p>\n

1.2 \u5b9e\u73b0\u6b65\u9aa4<\/h4>\n<\/p>\n
    \n
  1. \u5b9a\u4e49\u7f51\u683c\u548c\u521d\u59cb\u6761\u4ef6<\/strong>\uff1a\u786e\u5b9a\u65f6\u95f4\u548c\u7a7a\u95f4\u7684\u79bb\u6563\u70b9\uff0c\u5e76\u8bbe\u7f6e\u521d\u59cb\u6761\u4ef6\u3002<\/li>\n
  2. \u79bb\u6563\u5316\u65b9\u7a0b<\/strong>\uff1a\u5c06\u504f\u5fae\u5206\u65b9\u7a0b\u8f6c\u6362\u4e3a\u5dee\u5206\u65b9\u7a0b\u3002<\/li>\n
  3. \u8fed\u4ee3\u6c42\u89e3<\/strong>\uff1a\u6839\u636e\u79bb\u6563\u5316\u7684\u5dee\u5206\u65b9\u7a0b\u8fdb\u884c\u8fed\u4ee3\u8ba1\u7b97\u3002<\/li>\n<\/ol>\n

    1.3 \u793a\u4f8b\u4ee3\u7801<\/h4>\n<\/p>\n

    \u4ee5\u4e0b\u662f\u7528Python\u5b9e\u73b0\u4e00\u7ef4\u70ed\u4f20\u5bfc\u65b9\u7a0b\u7684\u6709\u9650\u5dee\u5206\u6cd5\u793a\u4f8b\u4ee3\u7801\uff1a<\/p>\n<\/p>\n

    import numpy as np<\/p>\n
    import matplotlib.pyplot as plt<\/p>\n
    \u5b9a\u4e49\u53c2\u6570<\/strong><\/h2>\n
    alpha = 0.01  # \u70ed\u6269\u6563\u7cfb\u6570<\/p>\n
    L = 10.0      # \u7a7a\u95f4\u957f\u5ea6<\/p>\n
    T = 2.0       # \u65f6\u95f4\u603b\u957f\u5ea6<\/p>\n
    nx = 100      # \u7a7a\u95f4\u79bb\u6563\u70b9\u6570<\/p>\n
    nt = 500      # \u65f6\u95f4\u6b65\u6570<\/p>\n
    dx = L \/ (nx - 1)<\/p>\n
    dt = T \/ nt<\/p>\n
    \u521d\u59cb\u5316\u6e29\u5ea6\u5206\u5e03<\/strong><\/h2>\n
    u = np.zeros(nx)<\/p>\n
    u_new = np.zeros(nx)<\/p>\n
    \u521d\u59cb\u6761\u4ef6\uff1a\u4e2d\u592e\u4f4d\u7f6e\u6e29\u5ea6\u4e3a1\uff0c\u5176\u4f59\u4f4d\u7f6e\u6e29\u5ea6\u4e3a0<\/strong><\/h2>\n
    u[int(nx \/ 2)] = 1<\/p>\n
    \u8fed\u4ee3\u6c42\u89e3<\/strong><\/h2>\n
    for n in range(nt):<\/p>\n
        for i in range(1, nx - 1):<\/p>\n
            u_new[i] = u[i] + alpha * dt \/ dx2 * (u[i + 1] - 2 * u[i] + u[i - 1])<\/p>\n
        u[:] = u_new[:]<\/p>\n
    \u7ed8\u5236\u7ed3\u679c<\/strong><\/h2>\n
    plt.plot(np.linspace(0, L, nx), u)<\/p>\n
    plt.xlabel('\u7a7a\u95f4\u4f4d\u7f6e')<\/p>\n
    plt.ylabel('\u6e29\u5ea6')<\/p>\n
    plt.title('\u4e00\u7ef4\u70ed\u4f20\u5bfc\u65b9\u7a0b\u7684\u6709\u9650\u5dee\u5206\u89e3')<\/p>\n
    plt.show()<\/p>\n
    <\/code><\/pre>\n<\/p>\n
    \u4e8c\u3001\u6709\u9650\u5143\u6cd5<\/h3>\n<\/p>\n
    \u6709\u9650\u5143\u6cd5\u662f\u4e00\u79cd\u7528\u4e8e\u6c42\u89e3\u504f\u5fae\u5206\u65b9\u7a0b\u7684\u6570\u503c\u6280\u672f\uff0c\u7279\u522b\u9002\u7528\u4e8e\u590d\u6742\u51e0\u4f55\u5f62\u72b6\u548c\u8fb9\u754c\u6761\u4ef6\u7684\u95ee\u9898\u3002\u5b83\u5c06\u8fde\u7eed\u57df\u5206\u5272\u6210\u6709\u9650\u4e2a\u5b50\u57df\uff08\u5355\u5143\uff09\uff0c\u5e76\u5728\u6bcf\u4e2a\u5b50\u57df\u4e0a\u6784\u9020\u8fd1\u4f3c\u89e3\u3002<\/p>\n<\/p>\n
    2.1 \u57fa\u672c\u539f\u7406<\/h4>\n<\/p>\n
    \u6709\u9650\u5143\u6cd5\u7684\u6838\u5fc3\u601d\u60f3\u662f\u5c06\u504f\u5fae\u5206\u65b9\u7a0b\u7684\u89e3\u8868\u793a\u4e3a\u4e00\u7ec4\u57fa\u51fd\u6570\u7684\u7ebf\u6027\u7ec4\u5408\uff0c\u7136\u540e\u901a\u8fc7\u5f31\u5f62\u5f0f\u548c\u52a0\u6743\u6b8b\u91cf\u6cd5\uff0c\u5c06\u5fae\u5206\u65b9\u7a0b\u8f6c\u5316\u4e3a\u4ee3\u6570\u65b9\u7a0b\u7ec4\u3002<\/p>\n<\/p>\n
    2.2 \u5b9e\u73b0\u6b65\u9aa4<\/h4>\n<\/p>\n
      \n
    1. \u7f51\u683c\u5212\u5206<\/strong>\uff1a\u5c06\u6c42\u89e3\u57df\u5212\u5206\u4e3a\u6709\u9650\u4e2a\u5355\u5143\u3002<\/li>\n
    2. \u9009\u62e9\u57fa\u51fd\u6570<\/strong>\uff1a\u9009\u62e9\u9002\u5f53\u7684\u57fa\u51fd\u6570\uff0c\u7528\u4e8e\u8868\u793a\u6bcf\u4e2a\u5355\u5143\u4e0a\u7684\u8fd1\u4f3c\u89e3\u3002<\/li>\n
    3. \u6784\u9020\u5f31\u5f62\u5f0f<\/strong>\uff1a\u5c06\u504f\u5fae\u5206\u65b9\u7a0b\u8f6c\u5316\u4e3a\u5f31\u5f62\u5f0f\uff0c\u5e76\u901a\u8fc7\u52a0\u6743\u6b8b\u91cf\u6cd5\u6784\u9020\u4ee3\u6570\u65b9\u7a0b\u7ec4\u3002<\/li>\n
    4. \u6c42\u89e3\u4ee3\u6570\u65b9\u7a0b\u7ec4<\/strong>\uff1a\u5229\u7528\u6570\u503c\u65b9\u6cd5\u6c42\u89e3\u4ee3\u6570\u65b9\u7a0b\u7ec4\uff0c\u5f97\u5230\u8fd1\u4f3c\u89e3\u3002<\/li>\n<\/ol>\n
      2.3 \u793a\u4f8b\u4ee3\u7801<\/h4>\n<\/p>\n
      \u4ee5\u4e0b\u662f\u4f7f\u7528FEniCS\u6c42\u89e3\u4e8c\u7ef4\u6cca\u677e\u65b9\u7a0b\u7684\u793a\u4f8b\u4ee3\u7801\uff1a<\/p>\n<\/p>\n
      from dolfin import *<\/p>\n
      \u521b\u5efa\u7f51\u683c\u548c\u51fd\u6570\u7a7a\u95f4<\/strong><\/h2>\n
      mesh = UnitSquareMesh(32, 32)<\/p>\n
      V = FunctionSpace(mesh, 'P', 1)<\/p>\n
      \u5b9a\u4e49\u8fb9\u754c\u6761\u4ef6<\/strong><\/h2>\n
      u_D = Expression('1 + x[0]*x[0] + 2*x[1]*x[1]', degree=2)<\/p>\n
      bc = DirichletBC(V, u_D, 'on_boundary')<\/p>\n
      \u5b9a\u4e49\u53d8\u5206\u95ee\u9898<\/strong><\/h2>\n
      u = TrialFunction(V)<\/p>\n
      v = TestFunction(V)<\/p>\n
      f = Constant(-6.0)<\/p>\n
      a = dot(grad(u), grad(v))*dx<\/p>\n
      L = f*v*dx<\/p>\n
      \u6c42\u89e3<\/strong><\/h2>\n
      u = Function(V)<\/p>\n
      solve(a == L, u, bc)<\/p>\n
      \u7ed8\u5236\u7ed3\u679c<\/strong><\/h2>\n
      import matplotlib.pyplot as plt<\/p>\n
      plot(u)<\/p>\n
      plt.show()<\/p>\n
      <\/code><\/pre>\n<\/p>\n
      \u4e09\u3001\u8c31\u65b9\u6cd5<\/h3>\n<\/p>\n
      \u8c31\u65b9\u6cd5\u662f\u4e00\u79cd\u9ad8\u7cbe\u5ea6\u7684\u6570\u503c\u65b9\u6cd5\uff0c\u7279\u522b\u9002\u7528\u4e8e\u6c42\u89e3\u5149\u6ed1\u57df\u4e0a\u7684\u504f\u5fae\u5206\u65b9\u7a0b\u3002\u5b83\u901a\u8fc7\u5085\u91cc\u53f6\u7ea7\u6570\u6216Chebyshev\u591a\u9879\u5f0f\u7b49\u5168\u5c40\u57fa\u51fd\u6570\u6765\u8868\u793a\u89e3\u3002<\/p>\n<\/p>\n
      3.1 \u57fa\u672c\u539f\u7406<\/h4>\n<\/p>\n
      \u8c31\u65b9\u6cd5\u7684\u57fa\u672c\u601d\u60f3\u662f\u5c06\u504f\u5fae\u5206\u65b9\u7a0b\u7684\u89e3\u8868\u793a\u4e3a\u4e00\u7ec4\u5168\u5c40\u57fa\u51fd\u6570\uff08\u5982\u5085\u91cc\u53f6\u7ea7\u6570\u6216Chebyshev\u591a\u9879\u5f0f\uff09\u7684\u7ebf\u6027\u7ec4\u5408\uff0c\u7136\u540e\u901a\u8fc7\u52a0\u6743\u6b8b\u91cf\u6cd5\u6216\u4f3d\u8fbd\u91d1\u6cd5\uff0c\u5c06\u5fae\u5206\u65b9\u7a0b\u8f6c\u5316\u4e3a\u4ee3\u6570\u65b9\u7a0b\u7ec4\u3002<\/p>\n<\/p>\n
      3.2 \u5b9e\u73b0\u6b65\u9aa4<\/h4>\n<\/p>\n
        \n
      1. \u9009\u62e9\u57fa\u51fd\u6570<\/strong>\uff1a\u9009\u62e9\u9002\u5f53\u7684\u5168\u5c40\u57fa\u51fd\u6570\uff0c\u7528\u4e8e\u8868\u793a\u89e3\u3002<\/li>\n
      2. \u6784\u9020\u5f31\u5f62\u5f0f<\/strong>\uff1a\u5c06\u504f\u5fae\u5206\u65b9\u7a0b\u8f6c\u5316\u4e3a\u5f31\u5f62\u5f0f\uff0c\u5e76\u901a\u8fc7\u52a0\u6743\u6b8b\u91cf\u6cd5\u6216\u4f3d\u8fbd\u91d1\u6cd5\u6784\u9020\u4ee3\u6570\u65b9\u7a0b\u7ec4\u3002<\/li>\n
      3. \u6c42\u89e3\u4ee3\u6570\u65b9\u7a0b\u7ec4<\/strong>\uff1a\u5229\u7528\u6570\u503c\u65b9\u6cd5\u6c42\u89e3\u4ee3\u6570\u65b9\u7a0b\u7ec4\uff0c\u5f97\u5230\u8fd1\u4f3c\u89e3\u3002<\/li>\n<\/ol>\n
        3.3 \u793a\u4f8b\u4ee3\u7801<\/h4>\n<\/p>\n
        \u4ee5\u4e0b\u662f\u4f7f\u7528SciPy\u6c42\u89e3\u4e00\u7ef4\u70ed\u4f20\u5bfc\u65b9\u7a0b\u7684\u8c31\u65b9\u6cd5\u793a\u4f8b\u4ee3\u7801\uff1a<\/p>\n<\/p>\n
        import numpy as np<\/p>\n
        from scipy.fftpack import fft, ifft<\/p>\n
        \u5b9a\u4e49\u53c2\u6570<\/strong><\/h2>\n
        L = 2*np.pi  # \u7a7a\u95f4\u957f\u5ea6<\/p>\n
        T = 2.0      # \u65f6\u95f4\u603b\u957f\u5ea6<\/p>\n
        nx = 128     # \u7a7a\u95f4\u79bb\u6563\u70b9\u6570<\/p>\n
        nt = 500     # \u65f6\u95f4\u6b65\u6570<\/p>\n
        dx = L \/ nx<\/p>\n
        dt = T \/ nt<\/p>\n
        \u521d\u59cb\u5316\u6e29\u5ea6\u5206\u5e03<\/strong><\/h2>\n
        x = np.linspace(0, L, nx, endpoint=False)<\/p>\n
        u = np.sin(x)<\/p>\n
        \u8fed\u4ee3\u6c42\u89e3<\/strong><\/h2>\n
        for n in range(nt):<\/p>\n
            u_hat = fft(u)<\/p>\n
            k = np.fft.fftfreq(nx, d=dx)<\/p>\n
            u_hat_new = u_hat * np.exp(-k2 * dt)<\/p>\n
            u = np.real(ifft(u_hat_new))<\/p>\n
        \u7ed8\u5236\u7ed3\u679c<\/strong><\/h2>\n
        import matplotlib.pyplot as plt<\/p>\n
        plt.plot(x, u)<\/p>\n
        plt.xlabel('\u7a7a\u95f4\u4f4d\u7f6e')<\/p>\n
        plt.ylabel('\u6e29\u5ea6')<\/p>\n
        plt.title('\u4e00\u7ef4\u70ed\u4f20\u5bfc\u65b9\u7a0b\u7684\u8c31\u65b9\u6cd5\u89e3')<\/p>\n
        plt.show()<\/p>\n
        <\/code><\/pre>\n<\/p>\n
        \u56db\u3001\u6570\u503c\u8ba1\u7b97\u5e93\u548c\u8f6f\u4ef6<\/h3>\n<\/p>\n
        \u9664\u4e86\u624b\u52a8\u5b9e\u73b0\u6709\u9650\u5dee\u5206\u6cd5\u3001\u6709\u9650\u5143\u6cd5\u548c\u8c31\u65b9\u6cd5\u5916\uff0cPython\u8fd8\u63d0\u4f9b\u4e86\u8bb8\u591a\u5f3a\u5927\u7684\u6570\u503c\u8ba1\u7b97\u5e93\u548c\u8f6f\u4ef6\uff0c\u53ef\u4ee5\u7528\u4e8e\u6c42\u89e3\u504f\u5fae\u5206\u65b9\u7a0b\u3002<\/p>\n<\/p>\n
        4.1 SymPy<\/h4>\n<\/p>\n
        SymPy\u662f\u4e00\u4e2a\u7528\u4e8e\u7b26\u53f7\u6570\u5b66\u8ba1\u7b97\u7684Python\u5e93\uff0c\u53ef\u4ee5\u7528\u4e8e\u89e3\u6790\u6c42\u89e3\u504f\u5fae\u5206\u65b9\u7a0b\u3002\u4ee5\u4e0b\u662f\u4e00\u4e2a\u4f7f\u7528SymPy\u6c42\u89e3\u4e00\u7ef4\u70ed\u4f20\u5bfc\u65b9\u7a0b\u7684\u793a\u4f8b\uff1a<\/p>\n<\/p>\n
        import sympy as sp<\/p>\n
        \u5b9a\u4e49\u7b26\u53f7<\/strong><\/h2>\n
        x, t, alpha = sp.symbols('x t alpha')<\/p>\n
        u = sp.Function('u')(x, t)<\/p>\n
        \u5b9a\u4e49\u504f\u5fae\u5206\u65b9\u7a0b<\/strong><\/h2>\n
        pde = sp.Eq(u.diff(t), alpha * u.diff(x, x))<\/p>\n
        \u6c42\u89e3\u504f\u5fae\u5206\u65b9\u7a0b<\/strong><\/h2>\n
        sol = sp.dsolve(pde)<\/p>\n
        print(sol)<\/p>\n
        <\/code><\/pre>\n<\/p>\n
        4.2 SciPy<\/h4>\n<\/p>\n
        SciPy\u662f\u4e00\u4e2a\u7528\u4e8e\u79d1\u5b66\u8ba1\u7b97\u7684Python\u5e93\uff0c\u63d0\u4f9b\u4e86\u8bb8\u591a\u6570\u503c\u65b9\u6cd5\u548c\u5de5\u5177\u3002\u4ee5\u4e0b\u662f\u4f7f\u7528SciPy\u6c42\u89e3\u4e00\u7ef4\u6ce2\u52a8\u65b9\u7a0b\u7684\u793a\u4f8b\uff1a<\/p>\n<\/p>\n
        import numpy as np<\/p>\n
        from scipy.integrate import solve_ivp<\/p>\n
        \u5b9a\u4e49\u53c2\u6570<\/strong><\/h2>\n
        L = 1.0  # \u7a7a\u95f4\u957f\u5ea6<\/p>\n
        T = 1.0  # \u65f6\u95f4\u603b\u957f\u5ea6<\/p>\n
        nx = 100  # \u7a7a\u95f4\u79bb\u6563\u70b9\u6570<\/p>\n
        nt = 1000  # \u65f6\u95f4\u6b65\u6570<\/p>\n
        dx = L \/ (nx - 1)<\/p>\n
        dt = T \/ nt<\/p>\n
        \u5b9a\u4e49\u521d\u59cb\u6761\u4ef6<\/strong><\/h2>\n
        x = np.linspace(0, L, nx)<\/p>\n
        u0 = np.sin(np.pi * x)<\/p>\n
        v0 = np.zeros(nx)<\/p>\n
        \u5b9a\u4e49\u6ce2\u52a8\u65b9\u7a0b<\/strong><\/h2>\n
        def wave_eq(t, y):<\/p>\n
            u = y[:nx]<\/p>\n
            v = y[nx:]<\/p>\n
            dudt = v<\/p>\n
            dvdt = np.zeros(nx)<\/p>\n
            dvdt[1:-1] = (u[2:] - 2 * u[1:-1] + u[:-2]) \/ dx2<\/p>\n
            return np.concatenate([dudt, dvdt])<\/p>\n
        \u6c42\u89e3\u6ce2\u52a8\u65b9\u7a0b<\/strong><\/h2>\n
        sol = solve_ivp(wave_eq, (0, T), np.concatenate([u0, v0]), t_eval=np.linspace(0, T, nt))<\/p>\n
        \u7ed8\u5236\u7ed3\u679c<\/strong><\/h2>\n
        import matplotlib.pyplot as plt<\/p>\n
        for i in range(0, nt, nt \/\/ 10):<\/p>\n
            plt.plot(x, sol.y[:nx, i], label=f't={sol.t[i]:.2f}')<\/p>\n
        plt.xlabel('\u7a7a\u95f4\u4f4d\u7f6e')<\/p>\n
        plt.ylabel('\u4f4d\u79fb')<\/p>\n
        plt.title('\u4e00\u7ef4\u6ce2\u52a8\u65b9\u7a0b\u7684\u6570\u503c\u89e3')<\/p>\n
        plt.legend()<\/p>\n
        plt.show()<\/p>\n
        <\/code><\/pre>\n<\/p>\n
        4.3 FEniCS<\/h4>\n<\/p>\n
        FEniCS\u662f\u4e00\u4e2a\u7528\u4e8e\u6709\u9650\u5143\u65b9\u6cd5\u6c42\u89e3\u504f\u5fae\u5206\u65b9\u7a0b\u7684\u5f00\u6e90\u8f6f\u4ef6\u3002\u4ee5\u4e0b\u662f\u4f7f\u7528FEniCS\u6c42\u89e3\u4e8c\u7ef4\u6cca\u677e\u65b9\u7a0b\u7684\u793a\u4f8b\uff1a<\/p>\n<\/p>\n
        from dolfin import *<\/p>\n
        \u521b\u5efa\u7f51\u683c\u548c\u51fd\u6570\u7a7a\u95f4<\/strong><\/h2>\n
        mesh = UnitSquareMesh(32, 32)<\/p>\n
        V = FunctionSpace(mesh, 'P', 1)<\/p>\n
        \u5b9a\u4e49\u8fb9\u754c\u6761\u4ef6<\/strong><\/h2>\n
        u_D = Expression('1 + x[0]*x[0] + 2*x[1]*x[1]', degree=2)<\/p>\n
        bc = DirichletBC(V, u_D, 'on_boundary')<\/p>\n
        \u5b9a\u4e49\u53d8\u5206\u95ee\u9898<\/strong><\/h2>\n
        u = TrialFunction(V)<\/p>\n
        v = TestFunction(V)<\/p>\n
        f = Constant(-6.0)<\/p>\n
        a = dot(grad(u), grad(v))*dx<\/p>\n
        L = f*v*dx<\/p>\n
        \u6c42\u89e3<\/strong><\/h2>\n
        u = Function(V)<\/p>\n
        solve(a == L, u, bc)<\/p>\n
        \u7ed8\u5236\u7ed3\u679c<\/strong><\/h2>\n
        import matplotlib.pyplot as plt<\/p>\n
        plot(u)<\/p>\n
        plt.show()<\/p>\n
        <\/code><\/pre>\n<\/p>\n
        \u4e94\u3001\u603b\u7ed3<\/h3>\n<\/p>\n
        Python\u6c42\u89e3\u504f\u5fae\u5206\u65b9\u7a0b\u7684\u65b9\u6cd5\u591a\u79cd\u591a\u6837\uff0c\u5305\u62ec\u6709\u9650\u5dee\u5206\u6cd5\u3001\u6709\u9650\u5143\u6cd5\u3001\u8c31\u65b9\u6cd5\u4ee5\u53ca\u4f7f\u7528\u4e13\u95e8\u7684\u6570\u503c\u8ba1\u7b97\u5e93\u548c\u8f6f\u4ef6\u3002\u6bcf\u79cd\u65b9\u6cd5\u90fd\u6709\u5176\u4f18\u7f3a\u70b9\u548c\u9002\u7528\u573a\u666f\uff0c\u9009\u62e9\u5408\u9002\u7684\u65b9\u6cd5\u5bf9\u4e8e\u9ad8\u6548\u51c6\u786e\u5730\u6c42\u89e3\u504f\u5fae\u5206\u65b9\u7a0b\u81f3\u5173\u91cd\u8981\u3002<\/p>\n<\/p>\n
          \n
        1. \u6709\u9650\u5dee\u5206\u6cd5<\/strong>\uff1a\u9002\u7528\u4e8e\u7b80\u5355\u51e0\u4f55\u5f62\u72b6\u548c\u8fb9\u754c\u6761\u4ef6\u7684\u504f\u5fae\u5206\u65b9\u7a0b\uff0c\u6613\u4e8e\u7406\u89e3\u548c\u5b9e\u73b0\u3002<\/li>\n
        2. \u6709\u9650\u5143\u6cd5<\/strong>\uff1a\u9002\u7528\u4e8e\u590d\u6742\u51e0\u4f55\u5f62\u72b6\u548c\u8fb9\u754c\u6761\u4ef6\u7684\u504f\u5fae\u5206\u65b9\u7a0b\uff0c\u5177\u6709\u8f83\u9ad8\u7684\u7075\u6d3b\u6027\u548c\u7cbe\u5ea6\u3002<\/li>\n
        3. \u8c31\u65b9\u6cd5<\/strong>\uff1a\u9002\u7528\u4e8e\u5149\u6ed1\u57df\u4e0a\u7684\u504f\u5fae\u5206\u65b9\u7a0b\uff0c\u5177\u6709\u9ad8\u7cbe\u5ea6\uff0c\u4f46\u5bf9\u95ee\u9898\u7684\u5149\u6ed1\u6027\u8981\u6c42\u8f83\u9ad8\u3002<\/li>\n
        4. \u6570\u503c\u8ba1\u7b97\u5e93\u548c\u8f6f\u4ef6<\/strong>\uff1a\u5982SymPy\u3001SciPy\u3001FEniCS\u7b49\uff0c\u63d0\u4f9b\u4e86\u5f3a\u5927\u7684\u5de5\u5177\u548c\u51fd\u6570\u5e93\uff0c\u7b80\u5316\u4e86\u6c42\u89e3\u504f\u5fae\u5206\u65b9\u7a0b\u7684\u8fc7\u7a0b\u3002<\/li>\n<\/ol>\n
          \u901a\u8fc7\u4ee5\u4e0a\u4ecb\u7ecd\uff0c\u76f8\u4fe1\u8bfb\u8005\u5df2\u7ecf\u5bf9Python\u6c42\u89e3\u504f\u5fae\u5206\u65b9\u7a0b\u7684\u65b9\u6cd5\u6709\u4e86\u8f83\u4e3a\u5168\u9762\u7684\u4e86\u89e3\u3002\u9009\u62e9\u5408\u9002\u7684\u65b9\u6cd5\u548c\u5de5\u5177\uff0c\u53ef\u4ee5\u5e2e\u52a9\u6211\u4eec\u9ad8\u6548\u51c6\u786e\u5730\u89e3\u51b3\u5b9e\u9645\u95ee\u9898\u3002<\/p>\n<\/p>\n
          \u76f8\u5173\u95ee\u7b54FAQs\uff1a<\/strong><\/h2>\n
           \u5982\u4f55\u7528Python\u6c42\u89e3\u504f\u5fae\u5206\u65b9\u7a0b\u7684\u57fa\u672c\u6b65\u9aa4\u662f\u4ec0\u4e48\uff1f<\/strong>
          \u5728Python\u4e2d\u6c42\u89e3\u504f\u5fae\u5206\u65b9\u7a0b\u901a\u5e38\u6d89\u53ca\u51e0\u4e2a\u6b65\u9aa4\u3002\u9996\u5148\uff0c\u9009\u62e9\u5408\u9002\u7684\u6570\u503c\u65b9\u6cd5\uff0c\u6bd4\u5982\u6709\u9650\u5dee\u5206\u6cd5\u3001\u6709\u9650\u5143\u6cd5\u6216\u8c31\u65b9\u6cd5\u7b49\u3002\u63a5\u7740\uff0c\u4f7f\u7528Python\u7684\u79d1\u5b66\u8ba1\u7b97\u5e93\uff0c\u5982NumPy\u548cSciPy\uff0c\u6765\u5b9e\u73b0\u8fd9\u4e9b\u65b9\u6cd5\u3002\u6700\u540e\uff0c\u5229\u7528Matplotlib\u7b49\u5e93\u53ef\u89c6\u5316\u7ed3\u679c\uff0c\u5e2e\u52a9\u7406\u89e3\u89e3\u7684\u884c\u4e3a\u3002<\/p>\n
          \u6709\u54ea\u4e9bPython\u5e93\u53ef\u4ee5\u5e2e\u52a9\u6c42\u89e3\u504f\u5fae\u5206\u65b9\u7a0b\uff1f<\/strong>
          Python\u4e2d\u6709\u591a\u4e2a\u5e93\u53ef\u4ee5\u6709\u6548\u6c42\u89e3\u504f\u5fae\u5206\u65b9\u7a0b\u3002\u4f8b\u5982\uff0cSciPy\u63d0\u4f9b\u4e86\u4e00\u4e9b\u57fa\u7840\u7684\u6570\u503c\u6c42\u89e3\u5de5\u5177\uff0cFEniCS\u548cFiPy\u4e13\u6ce8\u4e8e\u6709\u9650\u5143\u548c\u6709\u9650\u4f53\u79ef\u65b9\u6cd5\uff0c\u5206\u522b\u9002\u7528\u4e8e\u4e0d\u540c\u7c7b\u578b\u7684\u504f\u5fae\u5206\u65b9\u7a0b\u3002\u5bf9\u4e8e\u66f4\u590d\u6742\u7684\u9700\u6c42\uff0c\u53ef\u4ee5\u8003\u8651\u4f7f\u7528SymPy\u8fdb\u884c\u7b26\u53f7\u8ba1\u7b97\uff0c\u6216\u8005\u4f7f\u7528TensorFlow\u548cPyTorch\u8fdb\u884c\u6df1\u5ea6\u5b66\u4e60\u65b9\u6cd5\u7684\u6c42\u89e3\u3002<\/p>\n
          \u5982\u4f55\u9a8c\u8bc1\u7528Python\u6c42\u89e3\u7684\u504f\u5fae\u5206\u65b9\u7a0b\u7684\u7ed3\u679c\u662f\u5426\u6b63\u786e\uff1f<\/strong>
          \u9a8c\u8bc1\u7ed3\u679c\u7684\u51c6\u786e\u6027\u53ef\u4ee5\u901a\u8fc7\u591a\u79cd\u65b9\u5f0f\u8fdb\u884c\u3002\u53ef\u4ee5\u4e0e\u5df2\u77e5\u89e3\u8fdb\u884c\u6bd4\u8f83\uff0c\u7279\u522b\u662f\u5bf9\u4e8e\u7ecf\u5178\u7684\u504f\u5fae\u5206\u65b9\u7a0b\u3002\u540c\u65f6\uff0c\u6267\u884c\u7f51\u683c\u6536\u655b\u6027\u6d4b\u8bd5\uff0c\u4ee5\u67e5\u770b\u89e3\u7684\u7a33\u5b9a\u6027\u548c\u51c6\u786e\u6027\u3002\u6b64\u5916\uff0c\u4f7f\u7528\u4e0d\u540c\u7684\u6570\u503c\u65b9\u6cd5\u6c42\u89e3\u540c\u4e00\u65b9\u7a0b\uff0c\u5e76\u6bd4\u8f83\u7ed3\u679c\u7684\u76f8\u4f3c\u6027\u4e5f\u662f\u4e00\u79cd\u6709\u6548\u7684\u9a8c\u8bc1\u624b\u6bb5\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"Python\u6c42\u89e3\u504f\u5fae\u5206\u65b9\u7a0b\u7684\u5e38\u7528\u65b9\u6cd5\u6709\uff1a\u6709\u9650\u5dee\u5206\u6cd5\u3001\u6709\u9650\u5143\u6cd5\u3001\u8c31\u65b9\u6cd5\u3001\u4ee5\u53ca\u4f7f\u7528\u4e13\u95e8\u7684\u6570\u503c\u8ba1\u7b97\u5e93\u548c\u8f6f\u4ef6\uff0c\u5982Sym […]","protected":false},"author":3,"featured_media":1072829,"comment_status":"closed","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[37],"tags":[],"acf":[],"_links":{"self":[{"href":"https:\/\/docs.pingcode.com\/wp-json\/wp\/v2\/posts\/1072825"}],"collection":[{"href":"https:\/\/docs.pingcode.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/docs.pingcode.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/docs.pingcode.com\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/docs.pingcode.com\/wp-json\/wp\/v2\/comments?post=1072825"}],"version-history":[{"count":"1","href":"https:\/\/docs.pingcode.com\/wp-json\/wp\/v2\/posts\/1072825\/revisions"}],"predecessor-version":[{"id":1072830,"href":"https:\/\/docs.pingcode.com\/wp-json\/wp\/v2\/posts\/1072825\/revisions\/1072830"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/docs.pingcode.com\/wp-json\/wp\/v2\/media\/1072829"}],"wp:attachment":[{"href":"https:\/\/docs.pingcode.com\/wp-json\/wp\/v2\/media?parent=1072825"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/docs.pingcode.com\/wp-json\/wp\/v2\/categories?post=1072825"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/docs.pingcode.com\/wp-json\/wp\/v2\/tags?post=1072825"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}