Note for OPTI 521 with Professor Burge at UA
Note for OPTI 521 with Professor Burge at UA
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Date Created: 02/06/15
Opti521 Report 1 Basic Concept and a simple example of FEM Michihisa Onishi NOV 14 2007 1 Introduction The Finite Element Method FEM was developed in 1950 for solving complex structural analysis problem in engineering especially for aeronautical engineering then the use of FEM have been spread out to various elds of engineering In solving a structural problem the fundamental continuum equation is set up for in nitesimal small elements of a bulk Since this fundamental equation usually results in the di erential equations or integral equations with some boundary condition it is not easy to get analytical solutions For this case the discrete analysis can be used to approximate the continuum problems with in nite degree of freedom DOF by using only nite degree of freedom The discrete analysis includes RayleighRitz Method Method of Weighted Residuals MWR Finite Differential Method FDM and Boundary Element Method BDM as typical examples FEM is also categorized in the discrete analysis The basic idea of discrete analysis is to replace the in nite dimensional linear problem with a nite dimensional linear problem using a nite dimensional subspace For the Finite Element Method a space of piecewise linear functions is taken to approximate the solutions An appropriate set of basis is usually referred to an element 2 Formulation of small displacement elastic problem Although the materials covered in this section is out of scope of the OPTl521 class we should discuss the basic concept of elastic problem For small deformation the basic equations for elastic problem are given by following equations 5 Equation of Equilibrium a F 0 where F is the body force per unit volume This equation simply represenm the equilibrium of the forces applied to the material b StrainiDisplacement Relationship Y 1 1 Bu 311 3 uJ 21 2 39 39 2 Bx Bx For the two dimensional plane stress problem with homogeneous isotropic material au av 1K Bu dY gg ax y ayy y 23x ay c Stressistrain Relationship 0 dwsd For ahomogeneous and isotropic material the stressstrain relationship can be greatly simpli ed Starting from Hook s theorem gaggeda al sfga wm 3 lineman the stressstrain relationship is given by E 039x m kvw Isy 442 M E 0 mil 7 105 IsZ s E a 17vs vs 5 x Z 1v172v Z x y Y E 1V nyy I nyz In G7x where G 21 I Matrix expression greatly simpli es the expression 639 D 5 D is called as Dmatrix 17v v v o o o a S v 17 v v o o o 7 5 v v 17v 0 o o 5 5 D L o o 1 2V 0 o a E 1 v1 7 2v 2 17 2V 7 7x o o o o 2 o In 7 1 7 o o o o 0 W W This is the most basic stressstrain relationship for homogeneous and isotropic materials For the two dimensional case plane stress problem the stressstrain relationship can be expressed as follows Starting from Hook s theorem 1 1 ex E Tx ivay 8y E039y 7147 E E Tx Wsx vsy 0y Wsy vsx E 171 TV G v WiT39 Using matrix expression 639 D 5 D Dmatrix 1 V 0 0 8X E D V l 0 039 039 8 8 1V1V 1 1 y y 0 0 T Txy 7V d Dynamic Boundary Condition Y1fonSwhereT oquotn j j nJ is the surface normal This condition should be applied to the forces on the surface of the bulk considered 9 Geometric Boundary Condition u 1tquotonSu 3 Element In the process of discretization an appropriate basis of piecewise functions is used The term element usually refers to a set of basis used in FEM Although there are many types of elements correspond to the various types of problems the simplest element triangle linear plane element is introduced here Inside the triangle element the displacement u is approximated by primary expression uxy 050 051x my Vxgty o 1x zy Let the displacement at each node of the element be uh v up VJ and uk vk then the following relations should be satis ed y 1YiquotLu l u1aoaix1wzy1 V1 o 1x1 zy1 u2w0w1x2w2y2 Vz o 1xz zyz u3wo051x3wzy3 V3 o 1x3 zy3 Angina kAum Solv1ng the equations for 1 z 391 and 2 we get 0 1 0 1H1012 y3uzy3 y1u301 yzlA 1 x y 052u1xzx3uzx3x1u3x1x2lA 1 1 where A 1 x2 y2 rV1Cy2y3vzy3y1V3y1yzA 1 x y 3 3 z vlxZ x3vzx3 x1v3x1 xzA we 71103352 yzx3uzy1x3 y3x1u3y2x1y1xzlA o V1y3x2 yzx3vzy1x3 y3x1v3y2x1y1xzlA Using the results the strain of the element can be expressed by au 3v 1 av Bu Ix E an 8y 2gt yxy w2 l Zaxdy Then the straindisplacement relation is given by g B 12 B is called as Bmatrix 1 yry 0 yryl 0 yryz 0 8 BZ 0 erz 0 eri 0 Xi xz 5 5y xrxg yryz xra yryl xrxz yryz 7xy T u u1 v1 u2 v2 M3 v3 The most important concept here is the Bmatrix which gives us the matrix of coefficient for the strain displacement relationship The final simultaneous equations can be obtained by this Bmatrix working with the D matrix and the principle of virtual work Similarly to the above development a physical quantity A can be approximated by Axy a0 a1xa2y N1141 N2142 N3A3 2 MA N A N1y2 y3xx3 x2yy3x2 y2x3A N2 y3 y1xxl xgyy1xg y3xllA N3 y1 yzxx2 x1yy2x1 y1xzA The matrix N is called the Shape Function 4 Variational Principle Although there are many methods for discretization such as collocation method and Galerkin method the principle of virtual work is widely used to formulate the FEM for continuum elastic problems It requires that the energy of the system in equilibrium should be minimized or at least locally minimized For the FEM this principle states that Voyagde IVEamp4IdV Isgfamp41d5 o 5410onSu While the expression of the principle of virtual work seems difficult and complicated what the equation means is really simple It requires that the energy stored in the body should be equal to the energy provided by the applied body force and the surface force Working with the element introduced in section 3 and using the equations 639 D 839 if B 12 The equation above can be rewrite as 5U W 6U JV68T039dV IV68TDgdV W jV uTFdV L uTTdS If we assume that the total variational strain energy and the total virtual work can be given by summing the variational strain energy and the virtual work of each element respectively ie 50 2 a1 and W 25W suf x rn represen he mrth elernen where r700quot J39V 52 DsdV 17 5W J39V 5u 17w Lu Jur1 dS 17 Then he followlng equauons can be ob arned WW Jltugt Kltugt 5W 7 5ltugtrlt1 vgt 111 5ltugtrlt1 gt where K jw781rDBdV m LL Nr 17gtdV 1 1J 70Nrlt7gtd5 Srnce In above equauons u can be arbmary chosen he govemlng equauon of each elernen ls grven by K H 1quot Thls expresslon represen s exacny he Hook39s law whlch 15 extended o he rnul r drrnensrons The nal equauon for he syslern consldered can be cons ruc ed by he sum or he governlng equauons of all elernen s 5 Simple Example Now conslder he slmple example shown on he nght The coordna es of each porn are grven by Node100 Node21o 6 5 Node 3 20 Nodem 21 Node511 Node2o1 9 Node 1 and Node 6 are xed on he wall D 3 The force A 1N IS applled o he Node 4 as shown In he gure 1 2 3 lts ep 1gt Preparation 5r calculating Dematrix The tner plane slrarn lrnear elernen should be chosen slnce r represen s he physrcs consldered here bener han he o her elernen s For srrnplrcny assurne ha he Drmztnx ls gven by 1 0 5 0 no e ha hls Drmzmx does no seern o be reallsllc D 0 5 1 0 For alurrnnurn E7OGPa and 0 23 Drmztnx ls gven 0 0 1 New fe 22xer 22 u u quotDE u u uzss l an n l an u an l u 74gtltluwwuzz l u 1 15 also assumed ha he governing equauon for he syslern ls grven by K190 F ltStep 2gt Constructing the elements Four element can be constructed by given six nodes Element 1 Node 1 gt 2 gt 5 Element 2 Node 2 gt 3 gt 4 Element 3 Node 4 gt 5 gt 2 Element 2 Node 5 gt 6 gt 1 ltStep 3gt Calculating the Bmatrix and the Kmatrix The Bmatrix is given by yry 0 yryl 0 yryz 0 0 xrn 0 er1 0 erz xrxg yryz xra yryl X1 Xz yryz For the element 1 the Bmatrix can be expressed as follows 1 0 0 71 o 1 o o o 1 05 0 A1 1 01 BB o o o 71 o 1 DD 05 1 0 1 1 1 0 71 71 1 1 0 0 0 1 1 0 71 05 0 705 0 1 1 71 71 0 1 2 715 71 05 05 71 715 2 1 71 0 71 71 1 1 0 705 0 05 71 0 1 KK1B1T 1171181 Therefore the governing equation K u f for the element 1 can be expressed by 1 0 71 05 0 705 u f 0 1 1 71 71 0 v g 71 1 2 715 71 05 15 7 f2 05 71 715 2 1 71 39 v2 7 g2 0 71 71 1 1 0 u5 f 705 0 05 71 0 1 v5 g5 By repeating the same procedure the governing equations for each element can be obtained The followings shown below are the result ltE12menmgt ltE12m2nl2gt 1111Wu2v2u3v3uAIvAIuSIvSIuEIvE 11mmuzwzu wau4v4u5v5uev5 ltE12m2m3gt ltE12memw4gt 111Wu2lv2lu lv3uAv4u5v5uEvE u1v1u2v2u3v3uAv4u5v5uEvE m 10 y Ur1r11 ltStep 4gt Adding up all elements As a result of adding up all equations the following equation can be obtained ltSumgt u15v15u2 ltStep 5gt Boundary conditions The boundary conditions for this example are Constraints 111 V1 U6 Vs 0 0 Force f4 g4 0 391 Then the system equation can be rewrite as follows where f1 g1 and f5 g5 are the reaction forces due to the constrains applied on Node 1 and 6 ltStep 6gt Formulation of the system equations Following the procedure explained above the nal equations for the system can be rewrite as follows For the displacement u 4 15 1 05 0 15 2 15 u 0 15 4 1 1 15 0 15 2 v2 0 1 1 2 15 1 05 0 0 u 0 05 1 15 2 1 1 0 0 v3 0 0 15 1 1 2 0 1 05 u4 0 15 0 05 1 0 2 1 1 v4 1 2 15 0 0 1 1 4 15 us 0 15 2 0 0 05 1 15 4 v 0 For the reaction force F 2 V2 10500000 15u3 f1 1 10000 150v3 g1 0 00000 112142 00000005 1 v4 g6 5 V For the strain 39 B12Bu For the stress 39 639D D8 6 Summary While only the basic concept and the simplest example are introduced in this report the FEM is widely used in various engineering fields such as nonlinear problem natural vibration analysis large displacement region thermal analysis uid dynamics and even an electromagnetic field analysis and then there are many types of elements related to the specific physical quantities Even though it is impossible to give a full explanation of FEM in this report good optomechanical engineer should know the basic concept and various applications of FEM which enable us to expand the capability of dealing optomechanical issues Reference 1 K Washizuka H Miyamoto Y Yamada Y Yamamoto T Kawai Handbook of the Finite Element Method Baifukan October 2001 2 I Burge Class note of OPTl521 College of Optical Science University of Arizona 2007
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