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湖南大学机械与运载工程学院CollegeofMechanical&VehicleEngineering,HunanUniversity三维问题的有限元方法湖南大学机械与运载工程学院CollegeofMechanical&VehicleEngineering,HunanUniversity2020/3/24空间实体单元•特点–思路和平面问题、一维问题完全相同–采用位移法构造形函数•实体单元的主要类型–四面体,长方体,直边六面体,曲边六面体湖南大学机械与运载工程学院CollegeofMechanical&VehicleEngineering,HunanUniversity2020/3/24四面体常应变单元z=Zx=Xz=Zy=Yw4v4u4w2u2u2w1u1v1w3u3v3ijlk1=4=2=3=fsyfszfsx111222333444node1node2node3node4euvwuvwuvwuvwd湖南大学机械与运载工程学院CollegeofMechanical&VehicleEngineering,HunanUniversity2020/3/24四面体常应变单元•形函数(,,)(,,)hexyzxyzUNd123412341234node1node2node3node4000000000000000000000000NNNNNNNNNNNNN类似于常应变三角形单元,采用什么样的坐标系呢?12342341VVLP1=i2=j3=k4=lPyzx湖南大学机械与运载工程学院CollegeofMechanical&VehicleEngineering,HunanUniversity2020/3/24四面体常应变单元•形函数123412341234124312341342,,VVLVVLVVLPPP怎样确定体积坐标?23413412312412341234123412341234,,,PPPPddddLLLLdddd1=i2=j3=k4=lPyzx14321LLLL是否满足形函数的性质?1234123124134234VVVVVPPPP湖南大学机械与运载工程学院CollegeofMechanical&VehicleEngineering,HunanUniversity2020/3/24四面体常应变单元•形函数jkliLinodesremotetheatthe0nodehomeatthe1443322114433221144332211zLzLzLzLzyLyLyLyLyxLxLxLxLx(Delta条件)14321LLLL432143214321432111111LLLLzzzzyyyyxxxxzyx湖南大学机械与运载工程学院CollegeofMechanical&VehicleEngineering,HunanUniversity2020/3/24四面体常应变单元•形函数因此zyxdcbadcbadcbadcbaVLLLL161444433332222111143211det,det1111det1,det111jjjjjikkkikkllllljjjjikkikkllllxyzyzaxyzbyzxyzyzyzyzcyzdyzyzyz(Adjointmatrix)伴随阵(Cofactors)余因子ijkli=1,2k=3,4l=4,1湖南大学机械与运载工程学院CollegeofMechanical&VehicleEngineering,HunanUniversity2020/3/24四面体常应变单元•形函数lkjilkjilkjizzzzyyyyxxxxV1111det61四面体体积)(61zdycxbaVLNiiiiii湖南大学机械与运载工程学院CollegeofMechanical&VehicleEngineering,HunanUniversity2020/3/24四面体常应变单元•应变矩阵(,,)(,,)hexyzxyzUNdeeBdLNdLUNLNB000000000xyxzyzzyx44444444433333333322222222211111111100000000000000000000000000000000000021bdcdbcdcbbdcdbcdcbbdcdbcdcbbdcdbcdcbVB湖南大学机械与运载工程学院CollegeofMechanical&VehicleEngineering,HunanUniversity2020/3/24四面体常应变单元•刚度和质量阵eTTeeVdVVkBcBBcB11121314212223243132333441424344ddeeTeVVVVNNNNNNNNmNNNNNNNNNNjijijiijNNNNNN000000N湖南大学机械与运载工程学院CollegeofMechanical&VehicleEngineering,HunanUniversity2020/3/24四面体常应变单元•质量阵的积分1234!!!!d6(3)!emnpqeVmnpqLLLLVVmnpqEisenbergandMalvern[1973]:20010010010020010010010200100100120010010020010010200100120010020200102001.200202eeVsym湖南大学机械与运载工程学院CollegeofMechanical&VehicleEngineering,HunanUniversity2020/3/24•求单元质量阵的另类方法–特殊的自然坐标系统zxz=Zyijlk1=4=2=3==0=1=1=constantPQ四面体常应变单元zxz=Zyijlk1=4=2=3==0=0=1=constantPzxz=Zyijlk1=4=2=3==1=1=1=0=constantPQR湖南大学机械与运载工程学院CollegeofMechanical&VehicleEngineering,HunanUniversity2020/3/24四面体常应变单元0)()(223223PPPzyyyyxxxx0)()()()()()(11223111122311BPBPBzyyyyyyyyyxxxxxxxxx4321214444321214444)1()()()()()()()()(zzyyyyyyyyyyyxxxxxxxxxxxBB1234(1)(1)(1)NNNNzxz=Zyijlk1=4=2=3==0=0=1=1=0=1=1=1=1=0=constantP[xP(x3x2)+x2,yP(y3y2)+y2,0]OBB[xB(xPx1)+x1,yB[(yPy1)y1],0]O[x=(1)(x4xB)xB,y=(1)(y4yB)yB,z=(1)z4]=constant=constant湖南大学机械与运载工程学院CollegeofMechanical&VehicleEngineering,HunanUniversity2020/3/24四面体常应变单元Jacobian:zyxzyxzyxJ24312141313121312141313121600]det[VzyyyyyyxxxxxxJ111000ddetdddeTTeVVmNNNN[J]1112131411121222324200031323334414243446dddeeVNNNNNNNNmNNNNNNNN湖南大学机械与运载工程学院CollegeofMechanical&VehicleEngineering,HunanUniversity2020/3/24六面体单元PP’P’’P’’’湖南大学机械与运载工程学院CollegeofMechanical&VehicleEngineering,HunanUniversity2020/3/24六面体单元•形函数eNdU12345678displacementcomponentsatnode1displacementcomponentsatnode2displacementcomponentsatnode3displacementcomponentsatnode4displacementcoeeeeeeeeedddddddddmponentsatnode5displacementcomponentsatnode6displacementcomponentsatnode7displacementcomponentsatnode8111(1,2,,8)eiuviwd17586420zyx30fszfsyfsx87654321NNNNNNNNN)8,,2,1(000000iNNNiiiiN湖南大学机械与运载工程学院CollegeofMechanical&VehicleEngineering,HunanUniversity2020/3/24六面体单元•坐标转换4(-1,1,-1)(1,-1,1)6(1,-1,-1)217586420zyx30fszfsyfsx8(-1,1,1)7(1,1,1)(-1,-1,1)5(-1,-1,-1)13(1,1,-1)iiiiiiiiizNzyNyxNx),,(),,(),,(818181)1)(1)(1(81iiiiN(Tri-linearfunctions)湖南大学机械与运载工程学院CollegeofMechanical&VehicleEngineering,HunanUniversity2020/3/24六面体单元87654321BBBBBBBBB000000000xNyNxNzNyNzNzNyNxNiiiiiiiiiiiLNBeeBdLNdLU如何推导B矩阵?(形函数在自然坐标系下得到)湖南大学机械与运载工程学院CollegeofMechanical&VehicleEngineering,HunanUniversity2020/3/24六面体单元zzNyyNxxNNzzNyyNxxNNzzNyyNxxNNiiiiiiii
本文标题:有限元程序设计--第八章 三维问题的有限元方法
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