{"id":1029787,"date":"2024-12-31T11:11:28","date_gmt":"2024-12-31T03:11:28","guid":{"rendered":"https:\/\/docs.pingcode.com\/ask\/ask-ask\/1029787.html"},"modified":"2024-12-31T11:11:30","modified_gmt":"2024-12-31T03:11:30","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","status":"publish","type":"post","link":"https:\/\/docs.pingcode.com\/ask\/1029787.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\u65b9\u6cd5\u6709\uff1a\u4f7f\u7528\u6570\u503c\u65b9\u6cd5\uff08\u5982\u6709\u9650\u5dee\u5206\u6cd5\u3001\u6709\u9650\u5143\u6cd5\uff09\u3001\u4f7f\u7528\u7b26\u53f7\u8ba1\u7b97\u5e93\uff08\u5982SymPy\uff09\u3001\u4f7f\u7528\u4e13\u95e8\u7684PDE\u6c42\u89e3\u5e93\uff08\u5982FiPy\uff09<\/strong>\u3002\u672c\u6587\u5c06\u7740\u91cd\u4ecb\u7ecd\u5982\u4f55\u4f7f\u7528\u8fd9\u4e9b\u65b9\u6cd5\u6765\u6c42\u89e3\u504f\u5fae\u5206\u65b9\u7a0b\uff0c\u5e76\u8be6\u7ec6\u8bb2\u89e3\u5176\u4e2d\u7684\u4e00\u79cd\u65b9\u6cd5\u2014\u2014\u4f7f\u7528FiPy\u5e93\u3002<\/p>\n<\/p>\n

\u4e00\u3001\u6570\u503c\u65b9\u6cd5<\/h3>\n<\/p>\n

\u6570\u503c\u65b9\u6cd5\u662f\u6c42\u89e3\u504f\u5fae\u5206\u65b9\u7a0b\u7684\u4e00\u79cd\u5e38\u7528\u65b9\u6cd5\u3002\u6570\u503c\u65b9\u6cd5\u5305\u62ec\u6709\u9650\u5dee\u5206\u6cd5\u3001\u6709\u9650\u5143\u6cd5\u3001\u6709\u9650\u4f53\u79ef\u6cd5\u7b49\u3002\u6570\u503c\u65b9\u6cd5\u7684\u57fa\u672c\u601d\u60f3\u662f\u5c06\u8fde\u7eed\u7684\u504f\u5fae\u5206\u65b9\u7a0b\u79bb\u6563\u5316\u4e3a\u4ee3\u6570\u65b9\u7a0b\uff0c\u7136\u540e\u901a\u8fc7\u6570\u503c\u6c42\u89e3\u4ee3\u6570\u65b9\u7a0b\u6765\u83b7\u5f97\u504f\u5fae\u5206\u65b9\u7a0b\u7684\u8fd1\u4f3c\u89e3\u3002<\/p>\n<\/p>\n

1\u3001\u6709\u9650\u5dee\u5206\u6cd5<\/h4>\n<\/p>\n

\u6709\u9650\u5dee\u5206\u6cd5\u662f\u4e00\u79cd\u5e38\u7528\u7684\u6570\u503c\u65b9\u6cd5\u3002\u5b83\u7684\u57fa\u672c\u601d\u60f3\u662f\u7528\u5dee\u5206\u6765\u8fd1\u4f3c\u5fae\u5206\uff0c\u5c06\u504f\u5fae\u5206\u65b9\u7a0b\u79bb\u6563\u5316\u4e3a\u4ee3\u6570\u65b9\u7a0b\u3002\u4ee5\u4e0b\u662f\u4e00\u4e2a\u4e00\u7ef4\u70ed\u4f20\u5bfc\u65b9\u7a0b\u7684\u6709\u9650\u5dee\u5206\u6cd5\u6c42\u89e3\u793a\u4f8b\uff1a<\/p>\n<\/p>\n

import numpy as np<\/p>\n
import matplotlib.pyplot as plt<\/p>\n
\u53c2\u6570\u8bbe\u7f6e<\/strong><\/h2>\n
L = 1.0    # \u6746\u957f<\/p>\n
T = 0.5    # \u603b\u65f6\u95f4<\/p>\n
Nx = 10    # \u7a7a\u95f4\u79bb\u6563\u70b9\u6570<\/p>\n
Nt = 100   # \u65f6\u95f4\u79bb\u6563\u70b9\u6570<\/p>\n
alpha = 0.01  # \u70ed\u6269\u6563\u7cfb\u6570<\/p>\n
dx = L \/ (Nx - 1)<\/p>\n
dt = T \/ Nt<\/p>\n
x = np.linspace(0, L, Nx)<\/p>\n
u = np.zeros(Nx)<\/p>\n
u_new = np.zeros(Nx)<\/p>\n
\u521d\u59cb\u6761\u4ef6<\/strong><\/h2>\n
u[int(Nx\/2)] = 1<\/p>\n
\u65f6\u95f4\u6b65\u8fdb<\/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
\u7ed3\u679c\u7ed8\u56fe<\/strong><\/h2>\n
plt.plot(x, u)<\/p>\n
plt.xlabel('x')<\/p>\n
plt.ylabel('u')<\/p>\n
plt.title('Heat Conduction')<\/p>\n
plt.show()<\/p>\n
<\/code><\/pre>\n<\/p>\n
2\u3001\u6709\u9650\u5143\u6cd5<\/h4>\n<\/p>\n
\u6709\u9650\u5143\u6cd5\u662f\u4e00\u79cd\u57fa\u4e8e\u53d8\u5206\u539f\u7406\u7684\u6570\u503c\u65b9\u6cd5\u3002\u5b83\u5c06\u504f\u5fae\u5206\u65b9\u7a0b\u8f6c\u5316\u4e3a\u53d8\u5206\u95ee\u9898\uff0c\u901a\u8fc7\u79bb\u6563\u5316\u6c42\u89e3\u53d8\u5206\u95ee\u9898\u6765\u83b7\u5f97\u504f\u5fae\u5206\u65b9\u7a0b\u7684\u8fd1\u4f3c\u89e3\u3002\u4e0b\u9762\u662f\u4e00\u4e2a\u4f7f\u7528FEniCS\u5e93\u6c42\u89e3\u4e00\u7ef4\u6cca\u677e\u65b9\u7a0b\u7684\u793a\u4f8b\uff1a<\/p>\n<\/p>\n
from fenics import *<\/p>\n
\u521b\u5efa\u7f51\u683c\u548c\u51fd\u6570\u7a7a\u95f4<\/strong><\/h2>\n
mesh = UnitIntervalMesh(10)<\/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]', 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
\u7ed3\u679c\u7ed8\u56fe<\/strong><\/h2>\n
import matplotlib.pyplot as plt<\/p>\n
plot(u)<\/p>\n
plt.show()<\/p>\n
<\/code><\/pre>\n<\/p>\n
\u4e8c\u3001\u7b26\u53f7\u8ba1\u7b97\u5e93<\/h3>\n<\/p>\n
\u7b26\u53f7\u8ba1\u7b97\u5e93\uff08\u5982SymPy\uff09\u53ef\u4ee5\u7528\u4e8e\u89e3\u6790\u6c42\u89e3\u7b80\u5355\u7684\u504f\u5fae\u5206\u65b9\u7a0b\u3002SymPy\u63d0\u4f9b\u4e86\u4e00\u4e2adsolve<\/code>\u51fd\u6570\uff0c\u53ef\u4ee5\u7528\u4e8e\u6c42\u89e3\u504f\u5fae\u5206\u65b9\u7a0b\u3002\u4ee5\u4e0b\u662f\u4e00\u4e2a\u4f7f\u7528SymPy\u6c42\u89e3\u4e00\u7ef4\u6ce2\u52a8\u65b9\u7a0b\u7684\u793a\u4f8b\uff1a<\/p>\n<\/p>\n
from sympy import symbols, Function, dsolve, Eq<\/p>\n
\u5b9a\u4e49\u7b26\u53f7<\/strong><\/h2>\n
x, t = symbols('x t')<\/p>\n
u = Function('u')(x, t)<\/p>\n
\u5b9a\u4e49\u504f\u5fae\u5206\u65b9\u7a0b<\/strong><\/h2>\n
wave_eq = Eq(u.diff(t, t), u.diff(x, x))<\/p>\n
\u6c42\u89e3\u504f\u5fae\u5206\u65b9\u7a0b<\/strong><\/h2>\n
sol = dsolve(wave_eq)<\/p>\n
print(sol)<\/p>\n
<\/code><\/pre>\n<\/p>\n
\u4e09\u3001\u4e13\u95e8\u7684PDE\u6c42\u89e3\u5e93<\/h3>\n<\/p>\n
\u4e13\u95e8\u7684PDE\u6c42\u89e3\u5e93\uff08\u5982FiPy\uff09\u63d0\u4f9b\u4e86\u9ad8\u6548\u7684\u6570\u503c\u6c42\u89e3\u504f\u5fae\u5206\u65b9\u7a0b\u7684\u65b9\u6cd5\u3002FiPy\u662f\u4e00\u4e2a\u57fa\u4e8e\u6709\u9650\u4f53\u79ef\u6cd5\u7684Python\u5e93\uff0c\u9002\u7528\u4e8e\u6c42\u89e3\u5404\u79cd\u504f\u5fae\u5206\u65b9\u7a0b\u3002\u4ee5\u4e0b\u662f\u4e00\u4e2a\u4f7f\u7528FiPy\u6c42\u89e3\u6269\u6563\u65b9\u7a0b\u7684\u793a\u4f8b\uff1a<\/p>\n<\/p>\n
from fipy import CellVariable, Grid1D, TransientTerm, DiffusionTerm<\/p>\n
import matplotlib.pyplot as plt<\/p>\n
\u521b\u5efa\u7f51\u683c<\/strong><\/h2>\n
nx = 50<\/p>\n
dx = 1.0<\/p>\n
mesh = Grid1D(nx=nx, dx=dx)<\/p>\n
\u521b\u5efa\u53d8\u91cf<\/strong><\/h2>\n
phi = CellVariable(name='solution variable', mesh=mesh, value=0.0)<\/p>\n
\u8bbe\u7f6e\u521d\u59cb\u6761\u4ef6<\/strong><\/h2>\n
phi.setValue(1.0, where=mesh.cellCenters[0] < dx)<\/p>\n
\u5b9a\u4e49\u65b9\u7a0b<\/strong><\/h2>\n
eq = TransientTerm() == DiffusionTerm(coeff=1.0)<\/p>\n
\u65f6\u95f4\u6b65\u8fdb<\/strong><\/h2>\n
timeStepDuration = 0.9 * dx2 \/ 2<\/p>\n
steps = 100<\/p>\n
\u7ed3\u679c\u5b58\u50a8<\/strong><\/h2>\n
result = []<\/p>\n
\u6c42\u89e3<\/strong><\/h2>\n
for step in range(steps):<\/p>\n
    eq.solve(var=phi, dt=timeStepDuration)<\/p>\n
    result.append(phi.copy())<\/p>\n
\u7ed3\u679c\u7ed8\u56fe<\/strong><\/h2>\n
for i in range(0, steps, int(steps\/10)):<\/p>\n
    plt.plot(result[i], label=f't={i*timeStepDuration:.2f}')<\/p>\n
plt.xlabel('x')<\/p>\n
plt.ylabel('phi')<\/p>\n
plt.legend()<\/p>\n
plt.show()<\/p>\n
<\/code><\/pre>\n<\/p>\n
\u56db\u3001\u603b\u7ed3<\/h3>\n<\/p>\n
Python\u63d0\u4f9b\u4e86\u591a\u79cd\u65b9\u6cd5\u6765\u6c42\u89e3\u504f\u5fae\u5206\u65b9\u7a0b\uff0c\u5305\u62ec\u6570\u503c\u65b9\u6cd5\u3001\u7b26\u53f7\u8ba1\u7b97\u5e93\u548c\u4e13\u95e8\u7684PDE\u6c42\u89e3\u5e93\u3002\u6570\u503c\u65b9\u6cd5\u5982\u6709\u9650\u5dee\u5206\u6cd5\u548c\u6709\u9650\u5143\u6cd5\u9002\u7528\u4e8e\u6c42\u89e3\u590d\u6742\u7684\u504f\u5fae\u5206\u65b9\u7a0b\uff0c\u4f46\u9700\u8981\u7f16\u5199\u8f83\u591a\u7684\u4ee3\u7801\u3002\u7b26\u53f7\u8ba1\u7b97\u5e93\u5982SymPy\u9002\u7528\u4e8e\u89e3\u6790\u6c42\u89e3\u7b80\u5355\u7684\u504f\u5fae\u5206\u65b9\u7a0b\uff0c\u4f46\u5bf9\u4e8e\u590d\u6742\u7684\u65b9\u7a0b\u53ef\u80fd\u65e0\u6cd5\u6c42\u89e3\u3002\u4e13\u95e8\u7684PDE\u6c42\u89e3\u5e93\u5982FiPy\u63d0\u4f9b\u4e86\u9ad8\u6548\u7684\u6570\u503c\u6c42\u89e3\u65b9\u6cd5\uff0c\u9002\u7528\u4e8e\u6c42\u89e3\u5404\u79cd\u504f\u5fae\u5206\u65b9\u7a0b\u3002\u9009\u62e9\u5408\u9002\u7684\u65b9\u6cd5\u548c\u5de5\u5177\u53ef\u4ee5\u5e2e\u52a9\u6211\u4eec\u66f4\u9ad8\u6548\u5730\u6c42\u89e3\u504f\u5fae\u5206\u65b9\u7a0b\u3002<\/p>\n<\/p>\n
\u76f8\u5173\u95ee\u7b54FAQs\uff1a<\/strong><\/h2>\n
 \u5982\u4f55\u5728Python\u4e2d\u9009\u62e9\u5408\u9002\u7684\u5e93\u6765\u6c42\u89e3\u504f\u5fae\u5206\u65b9\u7a0b\uff1f<\/strong>
\u5728Python\u4e2d\uff0c\u5e38\u7528\u7684\u5e93\u6709SciPy\u3001NumPy\u548cSymPy\u3002SciPy\u63d0\u4f9b\u4e86\u6570\u503c\u6c42\u89e3\u7684\u529f\u80fd\uff0c\u9002\u5408\u5904\u7406\u590d\u6742\u7684\u8fb9\u503c\u548c\u521d\u503c\u95ee\u9898\uff1bNumPy\u5219\u7528\u4e8e\u6570\u7ec4\u64cd\u4f5c\u548c\u57fa\u7840\u6570\u5b66\u8ba1\u7b97\uff1bSymPy\u9002\u5408\u7b26\u53f7\u8ba1\u7b97\uff0c\u53ef\u4ee5\u6c42\u89e3\u89e3\u6790\u89e3\u3002\u6839\u636e\u4f60\u7684\u9700\u6c42\uff0c\u53ef\u4ee5\u9009\u62e9\u5408\u9002\u7684\u5e93\u8fdb\u884c\u6c42\u89e3\u3002<\/p>\n
\u504f\u5fae\u5206\u65b9\u7a0b\u6c42\u89e3\u7684\u6b65\u9aa4\u662f\u4ec0\u4e48\uff1f<\/strong>
\u6c42\u89e3\u504f\u5fae\u5206\u65b9\u7a0b\u901a\u5e38\u5305\u62ec\u51e0\u4e2a\u5173\u952e\u6b65\u9aa4\uff1a\u9996\u5148\uff0c\u9700\u8981\u5c06\u65b9\u7a0b\u8f6c\u5316\u4e3a\u9002\u5408\u8ba1\u7b97\u7684\u5f62\u5f0f\uff1b\u63a5\u7740\uff0c\u9009\u62e9\u5408\u9002\u7684\u8fb9\u754c\u6761\u4ef6\u548c\u521d\u59cb\u6761\u4ef6\uff1b\u7136\u540e\uff0c\u4f7f\u7528\u6570\u503c\u65b9\u6cd5\uff08\u5982\u6709\u9650\u5dee\u5206\u6cd5\u6216\u6709\u9650\u5143\u6cd5\uff09\u8fdb\u884c\u6c42\u89e3\uff0c\u6700\u540e\u5bf9\u7ed3\u679c\u8fdb\u884c\u53ef\u89c6\u5316\u5206\u6790\uff0c\u4ee5\u9a8c\u8bc1\u89e3\u7684\u5408\u7406\u6027\u3002<\/p>\n
\u54ea\u4e9b\u7c7b\u578b\u7684\u504f\u5fae\u5206\u65b9\u7a0b\u53ef\u4ee5\u4f7f\u7528Python\u8fdb\u884c\u6c42\u89e3\uff1f<\/strong>
Python\u80fd\u591f\u5904\u7406\u591a\u79cd\u7c7b\u578b\u7684\u504f\u5fae\u5206\u65b9\u7a0b\uff0c\u5305\u62ec\u4f46\u4e0d\u9650\u4e8e\u70ed\u4f20\u5bfc\u65b9\u7a0b\u3001\u6ce2\u52a8\u65b9\u7a0b\u548c\u62c9\u666e\u62c9\u65af\u65b9\u7a0b\u7b49\u3002\u5177\u4f53\u7684\u6c42\u89e3\u65b9\u6cd5\u53ef\u80fd\u56e0\u65b9\u7a0b\u7684\u6027\u8d28\u800c\u5f02\uff0c\u7528\u6237\u53ef\u4ee5\u6839\u636e\u5177\u4f53\u7684\u65b9\u7a0b\u5f62\u5f0f\u548c\u8fb9\u754c\u6761\u4ef6\u9009\u62e9\u5408\u9002\u7684\u6570\u503c\u65b9\u6cd5\u8fdb\u884c\u6c42\u89e3\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"Python\u6c42\u89e3\u504f\u5fae\u5206\u65b9\u7a0b\u7684\u65b9\u6cd5\u6709\uff1a\u4f7f\u7528\u6570\u503c\u65b9\u6cd5\uff08\u5982\u6709\u9650\u5dee\u5206\u6cd5\u3001\u6709\u9650\u5143\u6cd5\uff09\u3001\u4f7f\u7528\u7b26\u53f7\u8ba1\u7b97\u5e93\uff08\u5982SymPy\uff09\u3001\u4f7f […]","protected":false},"author":3,"featured_media":1029794,"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\/1029787"}],"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=1029787"}],"version-history":[{"count":"1","href":"https:\/\/docs.pingcode.com\/wp-json\/wp\/v2\/posts\/1029787\/revisions"}],"predecessor-version":[{"id":1029796,"href":"https:\/\/docs.pingcode.com\/wp-json\/wp\/v2\/posts\/1029787\/revisions\/1029796"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/docs.pingcode.com\/wp-json\/wp\/v2\/media\/1029794"}],"wp:attachment":[{"href":"https:\/\/docs.pingcode.com\/wp-json\/wp\/v2\/media?parent=1029787"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/docs.pingcode.com\/wp-json\/wp\/v2\/categories?post=1029787"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/docs.pingcode.com\/wp-json\/wp\/v2\/tags?post=1029787"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}