用PDE-Toolbox求解三类典型方程求解抛物型方程.docx

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1、实验二用PDEToolbox求解三类典型方程一求解抛物型方程一、实验目的本次实验在掌握PDEToolbox的功能和PDE图形用户界面的根底上,求解抛物型方程。二、实验内容1、解抛物型方程的一个例子热传导方程：金属板的导热问题考虑一个带有矩形孔的金属板上的热传导问题。板的左边保持在100C,板的右边热量从板向环境空气定常流动，其他边及内孔边界保持绝缘。初始=f0时板的温度为oc,于是概括为如下定解问题：Q的外边界定点坐标为(-0.5,-0.8),(0.5,-0.8),(0.5,0.8),(-0.5,0.8),内边界顶点坐标为(-00.5,-0.4),(00.5,-0.4),(00.5,0.4),

2、(-00.5,0.4)o使用GUI求解这一问题。在PDEToolbox窗口的工具栏中选择GenericScalar.第一步：区域设置单击工具01.在窗口拖拉出一个矩形，双击矩形区域，在对话框中输入1.eft为-05,BOttoin为-0.8,Width为1.Height为1.6,单击OK按钮，显示矩形区域Rlo用同样方法做内孔R2,只要设置1.eft为-00.5,Bottom为-0.4,Width为0.1,Height为0.8即可。然后在栏中键入RI-R2。第二步：设置边界条件单击工具叵虱，图形边界变红，逐段双击边界，翻开BOinIdaryCondition对话框，输入边界条件。对于同一类型的

3、边界，可按shift键，将多个边界同时选择，统一设置边界条件。此题在左边界上选择DiriChlet条件，输入h为1.r为100；右边界上，选择NeUmann条件，输入g为q为0;其他边界上，选择NeUmann条件,输入g为0,q为0。第三步：设置方程类型单击PDE菜单中PDESPeCifiCation选项，翻开PDESPeCifiCatiOll对话框，选择方程类型。此题单击ParabOliC(抛物型)，输入d为l,c为1.a为0,f为0,然后单击OK按钮。第四步：网格剖分单击工具4,或者单击MeSh菜单中InitiaIiZeMeSh选项，可进行初始网格剖分，如果需要网格加密，再单击二叁，或者单

4、击Mesh菜单中RefineMesh选项。第五步：初值和误差设置单击Solve菜单中ParameterS选项，翻开SolveParameters.对话框，输入Time为0：5,u(t)为0,Relativetolerance0.01,Absolutetolerance0.001,然后单击OK按钮。第六步：解的图形单击PIOt菜单中Parameters.选项，出现PlOtSeleCtion对话框,从中可以选择Color,Contour,Arrows,Deformedmesh,Height(3-Dplot),还可以设置等值线的数目等。本例中选择Color,Contour和Arrows三项，然后单击

5、Plot按钮，窗口显示出时的解的彩色图形。第七步：数值解的输出单击SOlVe菜单中EXPortSoliItiOil选项，在翻开的EXPort对话框中输入U,单击OK按钮确定。再在MAT1.AB命令窗口键入u,按回车键确定，即显示按节点编号排列的数值解UU=079.629195.377498.038098.570698.6395079.629295.377498.038098.570598.6395100.0000100.0000100.0000100.0000100.0000100.0000083.357796.385898.586999.027599.0845083.365396.38749

6、8.587599.027999.0849089.818897.811199.161399.431599.4664089.811897.809699.160799.431099.4660100.0000100.0000100.0000100.0000100.0000100.0000079.332495.313998.013998.554498.6243078.559295.146297.948698.509698.5822077.504594.917197.859298.448298.5244076.590594.720097.783398.396798.4760076.247094.64329

7、7.751698.374098.4545076.572894.716797.782598.396398.4757077.499694.916097.858798.447898.5241078.557595.145897.948598.509598.5821O82.123196.082598.440898.912898.9739O84.960796.739198.728899.127199.1786O88.503697.527199.051599.356599.3960O92.353498.364499.379899.583099.6093O96.194999.189099.694899.795

8、999.8090O80.325695.716398.316798.837298.9046O78.183495.245098.127898.705098.7796O77.157895.020298.038498.642798.7209O77.155295.019698.038198.642598.7207O78.180195.244198.127398.704698.7792O80.324495.715998.316598.837198.9044100.0000100.0000100.0000100.0000100.0000100.0000100.0000100.0000100.0000100.

9、0000100.0000100.0000100.0000100.0000100.0000100.0000100.0000100.0000100.0000100.0000100.0000100.0000100.0000100.0000100.0000100.0000100.0000100.0000100.0000100.0000100.0000100.0000100.0000100.0000100.0000100.0000100.0000100.0000100.0000100.0000100.0000100.0000100.0000100.0000100.0000100.0000100.0000

10、100.0000100.0000100.0000100.0000100.0000100.0000100.0000O96.194699.188999.694799.795999.8090O92.353298.364399.379799.582999.6092O88.510197.528299.051699.356599.3959O84.949196.736998.728399.126999.1784O82.118996.081798.440598.912798.9738O80.301895.620698.208698.726698.7936O92.781198.458999.418099.609

11、899.6347O94.801098.899199.591299.729699.7476O95.720199.098699.669299.783299.7980O92.784298.459699.418299.610099.6348O96.123199.179199.695299.798499.8118O77.913995.144398.055698.638598.7139O96.236699.203499.704599.804799.8176O77.828795.123898.046198.631198.7068O81.275295.891498.360798.855098.9190O93.

12、190298.544999.449599.630499.6539O93.123698.530699.443999.626799.6503O81.315595.901198.365398.858598.9224O97.027799.375099.771499.850699.8609O76.689794.877297.950398.565798.6452O86.442797.071398.866899.226299.2727O86.820097.155098.900999.250399.2955O78.450695.222498.056298.623598.6969O96.920299.34619

13、9.755899.837899.8484O96.958299.354299.758999.839899.8503O78.430095.217198.053498.621298.6947O77.254295.027098.030098.631398.7090O95.976399.151799.687999.795199.8090O77.304295.038498.034998.634998.7125O95.922999.140299.683599.792199.8062O83.671896.446598.604799.036799.0926O90.245097.907199.201499.460499.4940O90.022897.858899.182599.447399.4816O83.502696.407698.587799.024199.0806O97.579399.490099.812799.877299.8855O76.857094.884997.931098.540998.6198O76.715894.84