瞬态诺伊曼值 

将冷却液注入管道的轴对称切面，计算输送管的冷却效果..

指定求解域.

In[1]:=

X \[CapitalOmega] = ImplicitRegion[ y >= 0 && (x - 3/2)^2 + y^2 >= 1/25 && x^2 + y^2 >= 1 && x^2 + y^2 <= 4 && y <= x Tan[\[Pi]/8], {x, y}];

指定依据时间和平稳狄利克雷条件的诺伊曼值.

In[2]:=

X Subscript[\[CapitalGamma], N][t_] = NeumannValue[ If[t <= 10., -100. t, -1000.], (x - 3/2)^2 + y^2 == 1/25];

In[3]:=

X Subscript[\[CapitalGamma], D][ t_] = {DirichletCondition[u[t, x, y] == 200., x^2 + y^2 == 1], DirichletCondition[u[t, x, y] == 15., x^2 + y^2 == 4]};

求解与边界条件为 对应的稳态问题，从而找出初始条件. 用 "Continuation" 网格生成器可更好求解定义域 的角和边缘.

In[4]:=

X uinit = NDSolveValue[{10 \!\( \*SubsuperscriptBox[\(\[Del]\), \({x, y}\), \(2\)]\(u[0, x, y]\)\) == 0, Subscript[\[CapitalGamma], D][0]}, u[0, x, y], {x, y} \[Element] \[CapitalOmega], Method -> {"PDEDiscretization" -> {"FiniteElement", "MeshOptions" -> {"BoundaryMeshGenerator" -> "Continuation"}}}];

求解以稳态解为初始条件，具有在外边界的温度集和内边界的时间相关的流量的热方程. 为确保最佳的初始化，对稳态解使用相同空间网格.

In[5]:=

X uif = NDSolveValue[{D[u[t, x, y], t] - 10 \!\( \*SubsuperscriptBox[\(\[Del]\), \({x, y}\), \(2\)]\(u[t, x, y]\)\) == Subscript[\[CapitalGamma], N][t], Subscript[\[CapitalGamma], D][0], u[0, x, y] == uinit}, u, {t, 0, 25}, {x, y} \[Element] Head[uinit]["ElementMesh"]];

绘制解.

显示完整的 Wolfram 语言输入隐藏输入

In[6]:=

X framesNTV = Table[ContourPlot[ uif[t, x, y], {x, y} \[Element] uif["ElementMesh"], Mesh -> None, ColorFunction -> "TemperatureMap", Contours -> 21, PlotRange -> All, AspectRatio -> Automatic, Frame -> False], {t, 1/2, 15, 1/2}];

In[7]:=

X Manipulate[framesNTV[[i]], {{i, 9, "time"}, 1, Length[framesNTV], 1}, SaveDefinitions -> True]

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