有限元例子2-简支梁受均布荷载.doc

1)二维承压地下水水流模型算例假设承压含水层区域是一边长为a的正方形，东西边界为定水头边界，水头为，南北边界为隔水边界，区域中心有一抽水井以流量Q抽水，承压含水层的导水系数为T。稳定流定解问题如下： (1) (2) (3)非稳定流定解问题如下： (4) (5) (6) (7)此两个定解问题的解析解由Chan，Mullineux和Reed(1976)给出稳定流解为： (8)非稳定流解为： (9)其中：抽水井的坐标；，整数变量；在计算时，正方形的边长为1200m，计算剖分图见图1，T的单位为，S为无量纲变量，H的单位为m，Q的单位为，确定性模型计算时m，Q=1000.0。图1 地下水水流模型平面有限元剖分Fig.1 FEM mesh of groundwater flow model in horizontal direction表1二维承压稳定流水头解析解和有限元解对比表(T=100.0)节点号152 56 57 58 59 60 127 61解析解100.000 99.555 99.038 98.391 97.554 96.324 95.496 92.387有限元解100.000 99.501 98.976 98.383 97.645 96.567 95.593 92.396绝对误差0.000 0.054 0.062 0.008 0.091 0.243 0.097 0.009表2a二维承压非稳定流水头解析解和有限元解对比表(T=100.0,S=1.0E-4,Time=1.0)节点号 152 56 57 58 59 60 127 61解析解 100.000 99.556 99.039 98.393 97.556 96.326 95.498 92.389有限元解 100.000 99.504 98.981 98.391 97.654 96.577 95.603 92.406绝对误差 0.000 0.052 0.047 0.002 0.098 0.251 0.105 0.017表2b二维承压非稳定流58号节点水头随时间变化计算结果(T=100.0,S=1.0E-4)时间 0.100 0.200 0.400 0.700 1.000 2.000 4.000 7.000 10.000解析解 99.132 98.756 98.484 98.403 98.393 98.391 98.391 98.391 98.391有限元解 99.322 98.912 98.599 98.453 98.406 98.386 98.384 98.384 98.384绝对误差 0.190 0.156 0.115 0.050 0.013 0.005 0.007 0.007 0.0072)三维承压地下水水流模型算例所考虑的承压含水系统有三个水平层，每层厚度为20m，上、下两层为导水性较强的含水层，中间一层导水性较弱，。从平面上来看，含水系统为一正方形，和前述二维承压地下水水流模型算例相同(见图2.6-1)，边长为1200m，东、西部边界BC和AD为两条水头均为的河流，切割三个含水层，南，北部边界AB和CD为不透水边界，在含水层的中心有一抽水井，并且只从上层抽水。含水层剖面结构如图2，含水系统的上、下均为不透水层。定解问题描述如下: (49) (50) (51) (52)其中：为地下水水头L；为贮水系数无量纲；K为渗透系数L/T；(x,y,z)为笛卡尔坐标系的坐标；t为时间；G为不含边界的研究区域；为第一类边界；为第二类边界；为边界；为包括边界的研究区域；为边界单位外法向量；为初始水头和第一类边界上的水头。 (53)其中为抽水井数目,此例中=1；为分层数，此例中=1；为j号井中层的出水量。因为此问题无解析解，和为确定性变量Q=10000.0，。图2 三维水流模型剖面示意图Fig.2 Profile sketch map of 3-D groundwater flow model表3三维承压地下水水流问题有限元计算结果第一水平层位对应部分节点节点号0.1天水头1.0天水头10.天水头稳定水头152 56 57 58 59 60 127 61100.000 99.996 99.982 99.929 99.707 98.741 96.355 83.275100.000 99.687 99.289 98.673 97.585 95.339 92.167 78.395100.000 98.970 97.868 96.577 94.891 92.184 88.903 75.084100.000 98.810 97.557 96.135 94.345 91.572 88.276 74.451第二水平层位对应部分节点0.1天水头1.0天水头10.天水头稳定水头100.000 99.996 99.982 99.930 99.710 98.751 96.433 84.269100.000 99.686 99.289 98.677 97.592 95.347 92.229 79.369100.000 98.970 97.868 96.584 94.900 92.191 88.962 76.055100.000 98.810 97.557 96.142 94.355 91.579 88.335 75.422第三水平层位对应部分节点0.1天水头1.0天水头10.天水头稳定水头100.000 99.996 99.983 99.934 99.749 99.098 98.335 96.529100.000 99.687 99.290 98.685 97.655 95.816 94.510 92.329100.000 98.970 97.868 96.593 94.965 92.665 91.252 89.022100.000 98.810 97.558 96.151 94.420 92.053 90.625 88.389第四水平层位对应部分节点0.1天水头1.0天水头10.天水头稳定水头100.000 99.996 99.983 99.934 99.748 99.104 98.394 97.023100.000 99.686 99.289 98.686 97.659 95.831 94.601 92.891100.000 98.970 97.868 96.599 94.974 92.685 91.346 89.589100.000 98.810 97.558 96.159 94.430 92.073 90.720 88.957例 简支梁受均布荷载 计算简图：图1-(a)所示一简支梁，高3 m，长18 m，承受均布荷载10 N/m2，E=21010Pa ,= 0. 167，取t=1 m,作为平面应力问题。由于对称，只对右边一半进行有限单元法计算，如图1-(b)所示，而在y轴上的各结点处布置水平连杆支座。 图1 计算简图图2 计算剖分图计算结果文件 NODE NO WX WY 1 -0.566702E-08 0.284110E-06 2 -0.113487E-07 0.277090E-06 3 -0.168416E-07 0.265805E-06 4 -0.220905E-07 0.250302E-06 5 -0.270217E-07 0.230758E-06 6 -0.315560E-07 0.207440E-06 7 -0.356141E-07 0.180684E-06 8 -0.391179E-07 0.150885E-06 9 -0.419879E-07 0.118452E-06 10 -0.441365E-07 0.836906E-07 11 -0.455015E-07 0.464555E-07 12 -0.274740E-08 0.284613E-06 13 -0.557603E-08 0.277685E-06 14 -0.834469E-08 0.266399E-06 15 -0.109851E-07 0.250872E-06 16 -0.134591E-07 0.231293E-06 17 -0.157275E-07 0.207928E-06 18 -0.177441E-07 0.181109E-06 19 -0.194496E-07 0.151226E-06 20 -0.207653E-07 0.118702E-06 21 -0.216040E-07 0.839384E-07 22 -0.219679E-07 0.472553E-07 23 0.880218E-10 0.284767E-06 24 0.692376E-10 0.277941E-06 25 0.114365E-10 0.266684E-06 26 -0.465731E-10 0.251155E-06 27 -0.970630E-10 0.231563E-06 28 -0.138932E-09 0.208177E-06 29 -0.164633E-09 0.181326E-06 30 -0.152922E-09 0.151395E-06 31 -0.666807E-10 0.118832E-06 32 0.123645E-09 0.841476E-07 33 0.348329E-09 0.480186E-07 34 0.290915E-08 0.284602E-06 35 0.568377E-08 0.277879E-06 36 0.834360E-08 0.266667E-06 37 0.108786E-07 0.251158E-06 38 0.132569E-07 0.231578E-06 39 0.154393E-07 0.208202E-06 40 0.173892E-07 0.181356E-06 41 0.190758E-07 0.151427E-06 42 0.204751E-07 0.118880E-06 43 0.215547E-07 0.843177E-07 44 0.222280E-07 0.486261E-07 45 0.574813E-08 0.284125E-06 46 0.113488E-07 0.277503E-06 47 0.167731E-07 0.266342E-06 48 0.219493E-07 0.250870E-06 49 0.268003E-07 0.231329E-06 50 0.312466E-07 0.207996E-06 51 0.352050E-07 0.181199E-06 52 0.385867E-07 0.151326E-06 53 0.412948E-07 0.118853E-06 54 0.432291E-07 0.844282E-07 55 0.443139E-07 0.490316E-07 56 0.674212E-07 0.144602E-07 57 0.671167E-07 0.492551E-07 58 0.656890E-07 0.844716E-07 59 0.628552E-07 0.118731E-06 60 0.587505E-07 0.151068E-06 61 0.535817E-07 0.180830E-06 62 0.475278E-07 0.207534E-06 63 0.407356E-07 0.230789E-06 64 0.333304E-07 0.250265E-06 65 0.254261E-07 0.265684E-06 66 0.171343E-07 0.276794E-06 67 0.859263E-08 0.283334E-06 68 0.000000E+00 0.284827E-06 69 0.000000E+00 0.286008E-06 70 0.000000E+00 0.286719E-06 71 0.000000E+00 0.287047E-06 72 0.000000E+00 0.287056E-06 73 0.000000E+00 0.286799E-06 74 0.000000E+00 0.286329E-06 75 -0.885255E-08 0.283169E-06 76 -0.173308E-07 0.276147E-06 77 -0.255863E-07 0.264886E-06 78 -0.335116E-07 0.249422E-06 79 -0.409742E-07 0.229934E-06 80 -0.478466E-07 0.206687E-06 81 -0.540147E-07 0.180019E-06 82 -0.593918E-07 0.150326E-06 83 -0.639501E-07 0.118017E-06 84 -0.677704E-07 0.833474E-07 85 -0.710866E-07 0.457882E-07 86 -0.747848E-07 0.000000E+00 87 -0.465382E-07 0.548193E-08 88 -0.222980E-07 0.932727E-08 89 0.304139E-09 0.118962E-07 90 0.223686E-07 0.134189E-07 91 0.445858E-07 0.141584E-07 ELEMENT NO FXX FYY FXY 1 157.925903 12.320086 30.722723 2 244.916443 10.354716 -18.482588 3 154.291000 -7.813887 25.617579 4 234.212860 8.184300 -22.303001 5 148.891205 -8.920999 26.413481 6 228.046448 7.901186 -21.196423 7 142.116760 -9.326011 27.260603 8 218.950287 7.711358 -19.060856 9 133.377869 -9.444216 27.557730 10 206.202560 7.427414 -16.457104 11 122.508499 -9.351587 27.297066 12 189.942871 7.087284 -13.584266 13 109.509796 -9.037952 26.502115 14 170.565689 6.812156 -10.659593 15 94.421280 -8.453288 25.119614 16 148.878525 6.860985 -8.205246 17 77.223526 -7.504423 23.100119 18 126.537170 7.460763 -7.244399 19 57.667561 -6.346229 20.778383 20 106.236877 7.176868 -8.922749 21 36.062096 -6.891784 20.385952 22 89.043343 -9.605721 -9.351143 23 23.895409 -22.666592 29.659601 24 58.582275 -213.823380 39.055367 25 76.311859 4.718694 27.912863 26 157.642044 10.904240 -19.313572 27 76.519691 -5.359186 29.121048 28 157.267761 7.034226 -18.720295 29 74.013268 -9.656651 30.019243 30 151.885300 6.013625 -16.684847 31 70.360985 -10.330386 31.778349 32 145.115494 5.631763 -13.196098 33 65.807121 -10.267996 33.371666 34 136.338379 5.322870 -9.142807 35 60.234138 -9.934524 34.520279 36 125.400490 5.073721 -4.852313 37 53.443108 -9.352308 35.147041 38 112.336319 5.060810 -0.564904 39 45.096653 -8.419412 35.167702 40 97.212753 5.470625 3.392227 41 34.675465 -7.066862 34.522320 42 79.883148 5.761829 6.840075 43 21.830235 -5.876630 33.468239 44 59.278076 1.687056 11.004463 45 8.268250 -8.539404 32.956196 46 32.138313 -26.463701 22.104101 47 2.560837 -32.393398 30.020575 48 -3.419106 -158.912048 52.707214 49 -2.436311 -0.108350 26.052177 50 77.201515 9.156313 -20.680470 51 -0.783453 -6.478128 29.389797 52 78.639763 5.215799 -17.590981 53 -0.551637 -10.660508 31.869452 54 76.675636 3.623309 -14.256392 55 -0.796064 -11.908232 34.215569 56 73.029945 2.982471 -10.067174 57 -0.992169 -11.857497 36.376221 58 68.382523 2.578192 -5.304001 59 -1.128397 -11.357370 38.194935 60 62.686062 2.295735 -0.201771 61 -1.410914 -10.554439 39.633434 62 55.771549 2.262021 5.138405 63 -2.197704 -9.359666 40.685757 64 47.285736 2.499788 10.717977 65 -3.922763 -7.766502 41.343445 66 36.484940 1.958855 16.865788 67 -6.558257 -6.670453 41.546238 68 22.174541 -4.159215 24.812128 69 -8.218885 -10.252495 40.415695 70 3.837137 -30.641903 36.662498 71 -5.298054 -32.473328 30.255367 72 -12.645240 -108.241676 45.984890 73 -80.327049 -2.613984 24.191494 74 -1.097677 6.568793 -22.296230 75 -77.881805 -8.824903 28.458561 76 1.070117 2.767521 -18.238537 77 -75.485687 -12.580400 31.878490 78 1.795860 1.048860 -14.178474 79 -72.307762 -13.807447 34.406170 80 1.636196 0.223941 -9.812724 81 -67.993042 -13.735342 36.466198 82 1.316365 -0.342472 -5.017928 83 -62.503033 -13.136997 38.220577 84 0.984228 -0.819548 0.213650 85 -55.939785 -12.231685 39.746696 86 0.474411 -1.150391 5.953345 87 -48.465042 -10.961782 41.114109 88 -0.603190 -1.406191 12.420296 89 -40.259987 -9.356592 42.310013 90 -2.881718 -2.573744 20.004845 91 -31.273268 -8.265688 42.844421 92 -6.902014 -8.385126 28.993351 93 -20.705254 -11.152402 40.502155 94 -11.642939 -27.331755 37.810944 95 -9.238629 -26.849737 27.280319 96 -11.824013 -65.024971 34.570160 97 -158.170792 -2.442602 21.511524 98 -79.047531 3.768292 -24.462851 99 -155.973572 -11.653852 27.006340 100 -76.064873 0.237977 -20.280874 101 -151.793533 -14.944118 30.428455 102 -73.223198 -1.295055 -16.364956 103 -145.140030 -15.712955 32.310612 104 -69.976875 -2.180934 -12.525989 105 -136.170319 -15.451409 33.534714 106 -65.826309 -2.927631 -8.405723 107 -124.933037 -14.777358 34.470387 108 -60.608913 -3.689085 -3.825435 109 -111.365227 -13.864719 35.258514 110 -54.383499 -4.468897 1.399084 111 -95.304100 -12.672667 35.988167 112 -47.338806 -5.344087 7.574921 113 -76.533539 -11.197052 36.632446 114 -39.759769 -6.861498 15.044297 115 -55.051147 -9.927115 36.510925 116 -31.805851 -10.922228 23.434090 117 -31.953234 -10.951775 32.962830 118 -22.727345 -21.238634 29.280420 119 -11.255016 -18.938660 19.930035 120 -10.372881 -32.507412 21.496132 121 -235.434433 1.376922 17.061792 122 -157.484863 0.978824 -27.237322 123 -237.326660 -15.027872 25.975975 124 -153.