用PDEToolbox求解三類典型方程求解橢圓型方程

1、實(shí)驗(yàn)二 用PDE Toolbox求解三類典型方程-求解橢圓型方程一、實(shí)驗(yàn)?zāi)康谋敬螌?shí)驗(yàn)在掌握PDE Toolbox的功能和PDE圖形用戶界面的基礎(chǔ)上，求解 橢圓型方程。二、實(shí)驗(yàn)內(nèi)容1、解橢圓型方程的一個(gè)例子單位圓上的Poisson方程邊值問(wèn)題：），u = 1,； ： - x, y x2 y2 ：1u 士尸0這一問(wèn)題的精確解為1 - x - yu x, y =-4若使用圖形用戶界面（GUI）,則首先在MATLAB勺工作窗口中鍵入pdetool,按 回車鍵確定，于是出現(xiàn) PDE Toolbox窗口，如果需要坐標(biāo)網(wǎng)格，單擊 Options 菜單下的Grid選項(xiàng)即可。下面分步進(jìn)行操作。第一步：畫(huà)區(qū)域圖

2、單擊工具L田L(fēng)大致在（0, 0）位置單擊鼠標(biāo)右鍵同時(shí) 拖拉鼠標(biāo)到適當(dāng)位置松開(kāi)，繪制圓。為了保證所繪制的圓是標(biāo)準(zhǔn)的單位圓，在 所繪圖上雙擊，打開(kāi) Object Dialog 對(duì)話框，精確地輸入圓心坐標(biāo) X-center為 0,Y-center為0及半徑Radius為1,然后單擊OK按鈕，這樣單位圓已畫(huà)好。Obiect Dialog兇史J,圖形邊界變紅，逐段雙擊邊界，打第二步：設(shè)置邊界條件單擊工具.開(kāi)Boundary Condition對(duì)話框，輸入邊界條件。對(duì)于同一類型的邊界，可按shift鍵，將多個(gè)邊界同時(shí)選擇，統(tǒng)一設(shè)置邊界條件。本題選擇Dirichlet條件，輸入 h為1,r為0,然后單擊。儂

3、鈕。也可以單擊Boundary菜單中Specify Boundary Conditions選項(xiàng)，打開(kāi) Boundary Condition對(duì)話框，輸入邊界條件。第三步：設(shè)置方程 單擊 PDE菜單中 PDE Specification選項(xiàng)，打開(kāi) PDE Specification對(duì)話框，選擇方程類型。本題單擊 Elliptic,輸入c為1, a為0, f為1,然后單擊OKR鈕第四步：網(wǎng)格剖分 單擊工具與，或者單擊Mesh菜單中Initialize Mesh選項(xiàng)， 可進(jìn)行初始網(wǎng)格剖分，這時(shí)在 PDE Toolbox窗口下方的狀態(tài)欄內(nèi)顯示出初始網(wǎng) 格的節(jié)點(diǎn)數(shù)和三角形單元數(shù)。本題節(jié)點(diǎn)數(shù)為144個(gè)，三角

4、形單元數(shù)為254個(gè)。如果需要網(wǎng)格加密，再單擊工匈,或者單擊Mesh菜單中Refine Mesh選項(xiàng)，這 時(shí)節(jié)點(diǎn)數(shù)變?yōu)?41個(gè)，三角形單元數(shù)為1016個(gè)。如此還可繼續(xù)加密。第五步：解方程 單擊工具 -=-,或者單擊Solve菜單中Solve PDE選項(xiàng)，可顯示方程色彩解。如果單擊 Plot菜單中Parameters選項(xiàng)，出現(xiàn)Plot Selection寸 話框，從中可以選擇 Color, Contour, Arrows, Deformed mesh, Height(3-D plot) 還可以設(shè)置等值線的數(shù)目等。本例中選擇 Height(3-D plot),Color, Contour和 Show

5、 mesh四項(xiàng)，然后單擊Plot按鈕，方程的圖形解如圖所示。 PDE Toolbox - UrititlledFito Ecfl. 例Afis Draw Boundary PDE Mesh Solve Plot Mndow Helpz口| 田|o| 到a=gfcrtar元 一露 4”丫二-0/157Set lonmiJaColor: uInfo* 宅 eH b nsw pidl s 0r change mode ia a ter PDE mesh, or baindariesul StlecliiuH-lol X|Plcrttpei:Property:User errtry:Plot styl

6、e:p 8cr|ud 11rierpoMed shad.FContour Arrows1卡前J 11r Delorme mesh| -srati&J)d 17(眈 口啕hd 1Q怖ijt舊1ir可站吧Contour plot levels-p Plot sofutkxi automatically尼新縉咚3kColDrmap:|coolPlotDoneCancell第六步：與精確解作比較 單擊Plot菜單中Parameters選項(xiàng)，打開(kāi) PlotSelection對(duì)話框，在 Height(3-D plot)行 Property 的下拉框中選 user entry,且 在該行的user entr

7、y輸入框中鍵入u-(1-x.A2-y.A2)/4,單擊Plot按鈕就可以看到0解的絕對(duì)誤差圖形，可見(jiàn)在邊界處誤差為 0J第七步：輸出網(wǎng)格節(jié)點(diǎn)的編號(hào)、單元編號(hào)以及節(jié)點(diǎn)坐標(biāo)單擊Mesh菜單中Show Node Labels選項(xiàng)，冉單擊工具 仝或亞冬，即可顯示節(jié)點(diǎn)編號(hào)。若要輸 出節(jié)點(diǎn)坐標(biāo)，只需單擊 Mesh菜單中Export Mesh選項(xiàng)，這時(shí)打開(kāi)的Expor對(duì) 話框中的默認(rèn)值為 p e t,這里p e t分別表示points（點(diǎn)）、edges邊）、triangles。 角形），單擊OK按鈕。然后在MATLA熱令窗口鍵入p,按回車鍵確定，即可顯 示出節(jié)點(diǎn)按編號(hào)排列的坐標(biāo)（二維數(shù)組）；鍵入e,按回車鍵

8、，則顯示邊界線段數(shù)據(jù)矩陣（7維數(shù)組）；鍵入t,按回車鍵，則顯示三角形單元數(shù)據(jù)矩陣（4維 數(shù)組）。 pColumns 1 through 6-1.00000.00001.0000-0.0000-1.000000.00001.0000-0.7071-0.70710.7071-0.7071Columns 7 through 120.7071-0.7071-0.98080.70710.7071-0.1951-0.9239-0.3827-0.8315-0.5556-0.5556-0.8315Columns 13 through 180.38270.55560.8315-0.9239-0.8315-0.5

9、556-0.3827-0.19510.1951-0.9239-0.9808-0.9808Columns 19 through 240.92390.83150.55560.38270.55560.83150.92390.98080.9808-0.3827-0.19510.1951Columns 25 through 30-0.3827-0.5556-0.83150.92390.83150.55560.38270.1951-0.19510.92390.98080.9808Columns 31 through 36-0.49190.43440.41740.19260.2283-0.3200-0.92

10、39-0.98080.00960.38270.19510.0275Columns 37 through 42-0.40300.19420.64790.4847-0.5961-0.0626-0.1269-0.0494-0.5371-0.49270.5377-0.2817Columns 43 through 480.6544-0.2005-0.07410.3774-0.70180.77300.2906-0.72480.61810.55850.1984-0.4655Columns 49 through 540.1221-0.2130-0.4589-0.32430.2570-0.5475-0.3284

11、0.16350.3449-0.05130.3109-0.0420Columns 55 through 600.18480.05150.79770.6991-0.72340.0628-0.7174-0.61750.7301-0.07500.5183-0.2667Columns 61 through 66-0.70450.3744-0.8352-0.37470.72840.24180.50740.3780-0.31390.5346-0.71210.6522Columns 67 through 72-0.2495-0.1154-0.3813-0.8316-0.2527-0.37330.6894-0.

12、08560.7630-0.54720.87960.4201Columns 74 through 78-0.53240.2505-0.0174-0.0988-0.20730.2338Columns 79 through 84-0.5385-0.33880.80750.3474 0.1391 -0.1332Columns 85 through 900.21190.5371-0.2152-0.8010-0.18400.5204Columns 91 through 960.75800.5477-0.72710.2617-0.60370.3539Columns 97 through 102-0.0858

13、0.06860.6451-0.82740.83540.52200.30460.59350.38070.10410.1146-0.51400.1139-0.0031-0.49880.5561-0.56130.66170.3265-0.8205-0.38090.38350.0799-0.7221-0.6318-0.80810.7465-0.5469-0.2201-0.4030-0.22730.24140.53650.80710.83390.6535Columns 103 through 1080.18390.04100.5112-0.0243-0.17970.4034Columns 109 thr

14、ough 114-0.1699-0.2602-0.65920.0843-0.2272-0.2103Columns 115 through 120-0.13660.1539-0.5424-0.09980.1487-0.4250Columns 121 through 126-0.0254-0.26420.1051-0.37780.3913-0.4530Columns 127 through 132-0.3957-0.07600.01700.7755-0.68320.67800.6019-0.6233-0.2693-0.32590.0830-0.54520.43380.2437-0.37420.07

15、32-0.40280.3017-0.5470-0.86500.0476-0.6711-0.08220.4134-0.11810.83710.07400.3926-0.2762-0.8670-0.1437-0.46180.40730.66200.0524-0.1589Columns 133 through 1380.2771-0.6205-0.40690.23600.2346-0.2067Columns 139 through 1440.51160.41340.6636-0.44620.4958-0.19450.1892-0.19650.59090.4454-0.37840.27770.4932

16、-0.13620.4255-0.0428-0.59760.5949 ee =Columns 1 through 61.00009.000010.000011.00005.000012.00009.000010.000011.00005.000012.000013.000000.12500.25000.37500.50000.62500.12500.25000.37500.50000.62500.75001.00001.00001.00001.00001.00001.00001.00001.00001.00001.00001.00001.000000000Columns 7 through 12

17、13.000014.00000.75000.87501.00001.0000014.00002.00000.87501.00001.00001.000002.000015.000000.12502.00001.0000015.000016.00000.12500.25002.00001.0000016.000017.00000.25000.37502.00001.0000017.00006.00000.37500.50002.00001.00000Columns 13 through 186.000018.00000.50000.62502.00001.0000018.000019.00000

18、.62500.75002.00001.0000019.000020.00000.75000.87502.00001.0000020.00003.00000.87501.00002.00001.000003.000021.000000.12503.00001.0000021.000022.00000.12500.25003.00001.00000Columns 19 through 2422.000023.000023.00007.00000.25000.37500.37500.50003.00003.00001.00001.0000007.000024.000024.000025.00000.

19、50000.62500.62500.75003.00003.00001.00001.00000025.000026.000026.00004.00000.75000.87500.87501.00003.00003.00001.00001.000000674.000027.000028.000029.00008.000030.000027.000028.000029.00008.000030.000031.000000.12500.25000.37500.50000.62500.12500.25000.37500.50000.62500.75004.00004.00004.00004.00004

20、.00004.00001.00001.00001.00001.00001.00001.000000000Columns 25 through 300Columns 31 through 3231.000032.00000.75000.87504.00001.0000032.00001.00000.87501.00004.00001.00000 t3214202629172310011661234867285328997819884929912794891111111111Columns 12 through 2219945121397262691051213142151718Columns 1

21、 through 1189119189615630931151685183411231654314411995118118907012612692671111111111Columns 23 through 339636022992465254819212123725252627301256091691026510110168561111111111Columns 34 through 4475681047721106163010779332733352236173134341161001151129113962931341141111111111Columns 45 through 5510

22、1201978710510828808711382036243537293440641241251131021381371271311221111111111Columns 56 through 667486768359638790648236423537414038473938132142133143838713010811112011111111111Columns 67 through 785266449388351059245139110711157481421418843136121711377651112804910975501207339135704790641194734913

23、1Columns 78 through 8831326660869827621086969584281364828414746234310612910085143919965Columns 89 through 9967185612972739810593794592199640483955586144568413011711110114013493Columns 100 through 110786292684898100779170971091139463604614594952541309113897128115123117Columns 111 through 121110891031

24、047412271883542495533715153104507660135107104121103124115133112771111111111113159831281955511913964117184631271Columns 122 through 132418040107751063534312162535673535760755883781147913110914181124821231111111111Columns 133 through 14384056638484114116596285851196112575110614211114635387122130119911

25、6617610513311Columns 144 through 15444721322356363134906654706977677866795489108909113892139931341181111111111Columns 155 through 165951081458547824140466464576759705868616911195125961269712998144991111111111Columns 166 through 176485826246150515274102686598659975767410365100101101102102116103104104

26、1441111111111Columns 177 through 187574555441105433533739819673897290758083721411061071071371081091091211351111111111Columns 188 through 198554213236414034334910395778674787980109110761111421421131131141141151151161111111111Columns 199 through 20954641254551204352727294909489955082104108469910513973

27、11113310311613887124163100117117118118119119136136121137Columns 210 through 220404152758720571559291141131211201228196859784122123123124124125125126126127128135881331461051381611021441Columns 221 through 231471289783128143113858829812912911483810012913013011493480107131131115136748613213211Columns 2

28、32 through 2425079116931331344449107731341357250110881351368271120110136137377712191137138Columns 243 through 253784543884286775137479210688105861061121321081281391391401401411411421421431431111111111Column 254431401441第八步：輸出解的數(shù)值 單擊Solve菜單中Export Solution選項(xiàng)，在打開(kāi)的Export對(duì)話框中輸入u,單擊OK鈕確定。冉在MATLA晞令窗口鍵入u,按

29、回 車鍵確定，即顯示按節(jié)點(diǎn)編號(hào)排列的解的數(shù)值J ExportCancelOK uu =000000000000000000000000000000000.24980.18010.18990.18090.18500.17700.15800.15080.15190.14400.15090.10850.10000.10700.11660.09890.22250.21900.21970.21960.22200.12230.11980.08750.09860.11920.11840.08990.11380.08750.11890.09100.08240.06070.05600.05440.06000.0

30、6130.23040.17890.17680.22370.23640.22400.15880.14770.14720.21640.08260.16930.17120.07830.07830.16930.17060.18660.08020.08350.08940.08400.08660.07530.07450.07000.07710.07440.07790.07430.06130.07110.24130.24170.14400.13280.15120.15760.24100.22010.13000.20130.19450.19210.24280.23840.13100.06230.06090.2

31、0660.21420.19390.19560.20790.05540.06040.06020.13150.13490.13520.19570.20190.21660.13960.19760.19110.20440.14310.13450.14540.13010.18860.15570.1159000000000000000000000000000000000.04650.04670.12200.13700.13740.04780.04820.11940.04660.04660.12260.13710.04840.04850.12660.04380.04350.11880.10300.03210

32、.03190.08550.02970.02930.07880.02860.02870.07750.03180.03150.08440.03230.03230.09020.17040.15280.16940.20220.15040.20470.22390.22360.24570.23010.23160.20880.22320.17650.15350.12710.16670.15230.12010.16530.15130.11960.19980.22160.22110.04470.04340.11510.09210.08860.17480.16200.14580.17930.16400.14660

33、.04200.04110.11700.08510.10070.04120.04190.11770.10030.08510.17710.15830.17550.16290.14650.19040.17060.20470.04240.04240.09580.10210.07210.04430.04420.10210.10510.07420.04800.04720.09970.09220.07620.12700.04420.04520.09350.08820.07140.11800.04650.04570.09900.08940.07510.11910.03960.04020.10060.08530

34、.03920.03980.09900.08480.03690.03700.08630.08550.06450.04080.04070.09860.09950.07090.03910.03950.08820.09870.06600.04130.04110.09370.09700.07050.03900.03940.08810.09850.06600.03250.03220.09160.07370.06970.03760.03760.09340.07870.07620.24760.23220.23480.23460.24840.23240.23790.23530.24430.16910.12680.13000.16730.11270.15890.11760.11