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Fem 3d solid

The document discusses the finite element method for analyzing 3D solid structures. It describes the tetrahedron element, including its shape functions, strain matrix, and element matrices. It also briefly mentions the hexahedron element and higher order elements with curved surfaces. The key aspects covered are the derivation of shape functions for the tetrahedron element based on volume coordinates, the calculation of the strain matrix and element matrices, and two methods for evaluating the element matrices.

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1 FFinite Element Methodinite Element Method FEM FOR 3D SOLIDS for readers of all backgroundsfor readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 9:
2Finite Element Method by G. R. Liu and S. S. Quek CONTENTSCONTENTS  INTRODUCTION  TETRAHEDRON ELEMENT – Shape functions – Strain matrix – Element matrices  HEXAHEDRON ELEMENT – Shape functions – Strain matrix – Element matrices – Using tetrahedrons to form hexahedrons  HIGHER ORDER ELEMENTS  ELEMENTS WITH CURVED SURFACES
3Finite Element Method by G. R. Liu and S. S. Quek INTRODUCTIONINTRODUCTION  For 3D solids, all the field variables are dependent of x, y and z coordinates – most general element.  The element is often known as a 3D solid element or simply a solid element.  A 3D solid element can have a tetrahedron and hexahedron shape with flat or curved surfaces.  At any node there are three components in the x, y and z directions for the displacement as well as forces.
4Finite Element Method by G. R. Liu and S. S. Quek TETRAHEDRON ELEMENTTETRAHEDRON ELEMENT  3D solid meshed with tetrahedron elements
5Finite Element Method by G. R. Liu and S. S. Quek TETRAHEDRON ELEMENTTETRAHEDRON ELEMENT z=Z x=X z=Z y=Y w4 v4 u4 w2 u2 u2 w1 u1 v1 w3 u3 v3 i j l k1 = 4 = 2 = 3 = fsy fsz fsx Consider a four node tetrahedron element 1 1 1 2 2 2 3 3 3 4 4 4 node 1 node 2 node 3 node 4 e u v w u v w u v w u v w                           =                           d
6Finite Element Method by G. R. Liu and S. S. Quek Shape functionsShape functions ( , , ) ( , , )h ex y z x y z=U N d 1 2 3 4 1 2 3 4 1 2 3 4 node 1 node 2 node 3 node 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 N N N N N N N N N N N N     =      N 6447448 6447448 6447448 6447448 where Use volume coordinates (Recall Area coordinates for 2D triangular element) 1234 234 1 V V L P = 1=i 2=j 3=k 4=l P y z x
7Finite Element Method by G. R. Liu and S. S. Quek Shape functionsShape functions Similarly, 1234 123 4 1234 124 3 1234 134 2 ,, V V L V V L V V L PPP === Can also be viewed as ratio of distances 234 134 123124 1 2 3 4 1 234 1 234 1 234 1 234 , , ,P P PPd d dd L L L L d d d d − − −− − − − − = = = = 1=i 2=j 3=k 4=l P y z x 14321 =+++ LLLL since 1234123124134234 VVVVV PPPP =+++ (Partition of unity)
8Finite Element Method by G. R. Liu and S. S. Quek Shape functionsShape functions    = jkl i Li nodesremotetheat the0 nodehomeat the1 44332211 44332211 44332211 zLzLzLzLz yLyLyLyLy xLxLxLxLx +++= +++= +++= (Delta function property) 14321 =+++ LLLL                           =               4 3 2 1 4321 4321 4321 11111 L L L L zzzz yyyy xxxx z y x
9Finite Element Method by G. R. Liu and S. S. Quek Shape functionsShape functions Therefore, where                           =               z y x dcba dcba dcba dcba V L L L L 1 6 1 4444 3333 2222 1111 4 3 2 1 1 det , det 1 1 1 1 det 1 , det 1 1 1 j j j j j i k k k i k k l l l l l j j j j i k k i k k l l l l x y z y z a x y z b y z x y z y z y z y z c y z d y z y z y z        = = −                  = − = −           (Adjoint matrix) (Cofactors) i j k l i= 1,2 j = 2,3 k = 3,4 l = 4,1
10Finite Element Method by G. R. Liu and S. S. Quek Shape functionsShape functions             ×= l k j i l k j i l k j i z z z z y y y y x x x x V 1 1 1 1 det 6 1 (Volume of tetrahedron) )( 6 1 zdycxba V LN iiiiii +++==Therefore,
11Finite Element Method by G. R. Liu and S. S. Quek Strain matrixStrain matrix Since, ( , , ) ( , , )h ex y z x y z=U N d Therefore, ee BdLNdLU ===ε where NLNB                     ∂∂∂∂ ∂∂∂∂ ∂∂∂∂ ∂∂ ∂∂ ∂∂ == 0 0 0 00 00 00 xy xz yz z y x                     = 44 44 44 4 4 4 33 33 33 3 3 3 22 22 22 2 2 2 11 11 11 1 1 1 0 0 0 00 00 00 0 0 0 00 00 00 0 0 0 00 00 00 0 0 0 00 00 00 2 1 bd cd bc d c b bd cd bc d c b bd cd bc d c b bd cd bc d c b V B (Constant strain element)
12Finite Element Method by G. R. Liu and S. S. Quek Element matricesElement matrices e T T e eV dV V= =∫k B cB B cB 11 12 13 14 21 22 23 24 31 32 33 34 41 42 43 44 d d e e T e V V V Vρ ρ      = =        ∫ ∫ N N N N N N N N m N N N N N N N N N N where           = ji ji ji ij NN NN NN 00 00 00 N
13Finite Element Method by G. R. Liu and S. S. Quek Element matricesElement matrices 1 2 3 4 ! ! ! ! d 6 ( 3)!e m n p q eV m n p q L L L L V V m n p q = + + + +∫ Eisenberg and Malvern [1973] : 2 0 0 1 0 0 1 0 0 1 0 0 2 0 0 1 0 0 1 0 0 1 0 2 0 0 1 0 0 1 0 0 1 2 0 0 1 0 0 1 0 0 2 0 0 1 0 0 1 0 2 0 0 1 0 0 1 2 0 0 1 0 020 2 0 0 1 0 2 0 0 1 . 2 0 0 2 0 2 e e V sy ρ                  =                     m
14Finite Element Method by G. R. Liu and S. S. Quek Element matricesElement matrices Alternative method for evaluating me: special natural coordinate system z x z=Z y i j l k 1 = 4 = 2 = 3 = ξ=0 ξ=1 ξ=1 ξ=constant P Q
15Finite Element Method by G. R. Liu and S. S. Quek Element matricesElement matrices z x z=Z y i j l k 1 = 4 = 2 = 3 = η=0 η=0 η=1 η=constant P
16Finite Element Method by G. R. Liu and S. S. Quek Element matricesElement matrices z x z=Z y i j l k 1 = 4 = 2 = 3 = ζ=1 ζ=1 ζ=1 ζ=0 ζ=constant P QR
17Finite Element Method by G. R. Liu and S. S. Quek Element matricesElement matrices z x z=Z y i j l k 1 = 4 = 2 = 3 = ξ=0 η=0 ζ=1 ξ=1 η=0 ζ=1 ξ=1 η=1 ζ=1 ζ=0 ζ=constant P [xP=η(x3−x2)+x2, yP=η(y3−y2)+y2,0] O B B [xB=ξ(xP −x1)+x1, yB=ξ[η(yP−y1)+y1],0] O [x=(1−ζ)(x4−xB)+xB, y=(1−ζ)(y4−yB)+yB, z=(1−ζ)z4] ξ=constant ζ=constant 0 )( )( 223 223 = +−= +−= P P P z yyyy xxxx η η 0 )()()( )()()( 1122311 1122311 = +−+−=+−= +−+−=+−= B PB PB z yyyyyyyyy xxxxxxxxx ξξηξ ξξηξ 4 321214444 321214444 )1( )()()()( )()()()( zz yyyyyyyyyyy xxxxxxxxxxx B B ζ ξζξζζζ ξζξζζζ −= −−−+−−=−−= −−−+−−=−−= )1( )1( )1( 4 3 2 1 ζ ηξζ ξηζ ζξ −= −= = −= N N N N
18Finite Element Method by G. R. Liu and S. S. Quek Element matricesElement matrices Jacobian:                   ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = ζ ζ ζ η η η ξ ξ ξ z y x z y x z y x J 2 4 312141313121 312141313121 6 00 ]det[ ξζξηξξζηζζ ξηξξζηζζ V z yyyyyy xxxxxx −=++−+ ++−+ =J 1 1 1 0 0 0 d det d d d e T T e V Vρ ρ ξ η ζ= =∫ ∫ ∫ ∫m N N N N [J] 11 12 13 14 1 1 1 21 22 23 242 0 0 0 31 32 33 34 41 42 43 44 6 d d de eV ρ ξζ ξ η ζ      = −        ∫ ∫ ∫ N N N N N N N N m N N N N N N N N
19Finite Element Method by G. R. Liu and S. S. Quek Element matricesElement matrices l f f f l sz sy sx e d][ 43 T ∫           = − Nf z=Z x=X z=Z y=Y w4 v4 u4 w2 u2 u2 w1 u1 v1 w3 u3 v3 i j l k1 = 4 = 2 = 3 = fsy fsz fsx For uniformly distributed load: { } { } { } { } { } { }                                                           = × × × × × × − 13 13 13 13 13 13 43 2 1 0 0 0 0 0 0 f sz sy sx sz sy sx e f f f f f f l
20Finite Element Method by G. R. Liu and S. S. Quek HEXAHEDRON ELEMENTHEXAHEDRON ELEMENT  3D solid meshed with hexahedron elements P P’ P’’P’’’
21Finite Element Method by G. R. Liu and S. S. Quek Shape functionsShape functions eNdU = 1 2 3 4 5 6 7 8 displacement components at node 1 displacement components at node 2 displacement components at node 3 displacement components at node 4 displacement co e e e e e e e e e             =               d d d d d d d d d mponents at node 5 displacement components at node 6 displacement components at node 7 displacement components at node 8 1 1 1 ( 1, 2, ,8)ei u v i w     = =      d L 1 7 5 8 6 4 2 0 z y x 3 0 fsz fsy fsx [ ]87654321 NNNNNNNNN = )8,,2,1( 00 00 00 L=           = i N N N i i i iN
22Finite Element Method by G. R. Liu and S. S. Quek Shape functionsShape functions 4(-1, 1, -1) (1, -1, 1)6 (1, -1, -1)2 1 7 5 8 6 4 2 0 z y x 3 0 fsz fsy fsx 8(-1, 1, 1) 7 (1, 1, 1) (-1, -1, 1)5 (-1, -1, -1)1 3(1, 1, -1) ξ η ζ ii i ii i ii i zNz yNy xNx ),,( ),,( ),,( 8 1 8 1 8 1 ζηξ ζηξ ζηξ ∑ ∑ ∑ = = = = = = )1)(1)(1( 8 1 iiiiN ζζηηξξ +++= (Tri-linear functions)
23Finite Element Method by G. R. Liu and S. S. Quek Strain matrixStrain matrix [ ]87654321 BBBBBBBBB = whereby                     ∂∂∂∂ ∂∂∂∂ ∂∂∂∂ ∂∂ ∂∂ ∂∂ == 0 0 0 00 00 00 xNyN xNzN yNzN zN yN xN ii ii ii i i i ii LNB Note: Shape functions are expressed in natural coordinates – chain rule of differentiation ee BdLNdLU ===ε
24Finite Element Method by G. R. Liu and S. S. Quek Strain matrixStrain matrix ζζζζ ηηηη ξξξξ ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ z z Ny y Nx x NN z z Ny y Nx x NN z z Ny y Nx x NN iiii iiii iiii Chain rule of differentiation                   ∂ ∂ ∂ ∂ ∂ ∂ =                   ∂ ∂ ∂ ∂ ∂ ∂ z N y N x N N N N i i i i i i J ζ η ξ ⇒ where                   ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = ζ η ξ ζ η ξ ζ η ξ z z z y y y x x x J
25Finite Element Method by G. R. Liu and S. S. Quek Strain matrixStrain matrix 8 8 8 1 1 1 ( , , ) , ( , , ) , ( , , )i i i i i i i i i x N x y N y z N zξ η ζ ξ η ζ ξ η ζ = = = = = =∑ ∑ ∑Since, or 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1                   ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ = = = = = = = = = ζ η ξ ζ η ξ ζ η ξ i i i i i i i i i i i i i i i i i i i i i i i i i i i N z N z N z N y N y N y N x N x N x J 1 1 1 2 2 23 5 6 7 81 2 4 3 3 3 4 4 43 5 6 7 81 2 4 5 5 5 6 6 61 2 3 4 5 6 7 8 7 7 7 8 8 8 x y z x y zN N N N NN N N x y z x y zN N N N NN N N x y z x y zN N N N N N N N x y z x y z ξ ξ ξ ξ ξξ ξ ξ η η η η η η η η ζ ζ ζζ ζ ζ ζ ζ    ∂ ∂ ∂ ∂ ∂∂ ∂ ∂    ∂ ∂ ∂ ∂ ∂∂ ∂ ∂    ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ =   ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂    ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂    ∂ ∂ ∂∂ ∂ ∂ ∂ ∂    J               
26Finite Element Method by G. R. Liu and S. S. Quek Strain matrixStrain matrix                   ∂ ∂ ∂ ∂ ∂ ∂ =                   ∂ ∂ ∂ ∂ ∂ ∂ − ζ η ξ i i i i i i N N N z N y N x N 1 J                     ∂∂∂∂ ∂∂∂∂ ∂∂∂∂ ∂∂ ∂∂ ∂∂ == 0 0 0 00 00 00 xNyN xNzN yNzN zN yN xN ii ii ii i i i ii LNB Used to replace derivatives w.r.t. x, y, z with derivatives w.r.t. ξ, η, ζ
27Finite Element Method by G. R. Liu and S. S. Quek Element matricesElement matrices 1 1 1 T T 1 1 1 d det[ ]d d d e e V A ξ η ζ + + + − − − = =∫ ∫ ∫ ∫k B cB B cB J Gauss integration: ),,(d)d,( 1 1 1 1 1 1 1 1 1 jjikji n i m j l k fwwwfI ζηξηξηξ ∑∑∑∫ ∫ ∫ = = = + − + − + − == 1 1 1 1 1 1 d det d d d e T T e V Vρ ρ ξ η ζ − − − = =∫ ∫ ∫ ∫m N N N N [J]
28Finite Element Method by G. R. Liu and S. S. Quek Element matricesElement matrices For rectangular hexahedron: det eabc V= =[J]                           = 88 7877 686766 58575655 4847464544 282726252433 28272625242322 1817161514131211 . m mm mmm mmmm mmmmm mmmmmm mmmmmmm mmmmmmmm m sy e
29Finite Element Method by G. R. Liu and S. S. Quek Element matricesElement matrices (Cont’d) where ζηξρ ζηξρ ζηξρ ddd 00 00 00 ddd 00 00 00 00 00 00 ddd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1           =                     = = ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ − − − − − − − − − ji ji ji j j j i i i jiij NN NN NN abc N N N N N N abc abc NNm
30Finite Element Method by G. R. Liu and S. S. Quek Element matricesElement matrices (Cont’d) or           = ij ij ij ij m m m 00 00 00 m where )1)(1)(1( 8 d)1)(1(d)1)(1(d)1)(1( 64 ddd 3 1 3 1 3 1 1 1 1 1 1 1 1 1 1 1 jijiji jijiji jiij hab abc NNabcm ζζηηξξ ρ ζζζζζηηηηηξξξξξ ρ ζηξρ +++= ++++++= = ∫∫∫ ∫ ∫ + − + − + − + − + −
31Finite Element Method by G. R. Liu and S. S. Quek Element matricesElement matrices (Cont’d) E.g. 216 8)111)(111)(111( 8 3 1 3 1 3 1 33 abcabc m ρρ ×=××+××+××+= 216 1 216 2 216 4 216 8 46352817 184538276857473625162413 483726155814786756342312 8877665544332211 abc mmmm abc mmmmmmmmmmmm abc mmmmmmmmmmmm abc mmmmmmmm ρ ρ ρ ρ ==== ============ ============ ========
32Finite Element Method by G. R. Liu and S. S. Quek Element matricesElement matrices (Cont’d)                           = 8 48. 248 4248 42128 242148 1242248 21244248 216 sy abc ex ρ m Note: For x direction only (Rectangular hexahedron)
33Finite Element Method by G. R. Liu and S. S. Quek Element matricesElement matrices l f f f l sz sy sx e d][ 43 T ∫           = − Nf 1 7 5 8 6 4 2 0 z y x 3 0 fsz fsy fsx { } { } { } { } { } { }                                                           = × × × × × × − 13 13 13 13 13 13 43 2 1 0 0 0 0 0 0 f sz sy sx sz sy sx e f f f f f f l For uniformly distributed load:
34Finite Element Method by G. R. Liu and S. S. Quek Using tetrahedrons to form hexahedronsUsing tetrahedrons to form hexahedrons  Hexahedrons can be made up of several tetrahedrons 1 5 6 8 1 4 3 8 1 2 3 4 5 7 8 3 1 6 8 6 3 2 1 6 3 6 7 8Hexahedron made up of 5 tetrahedrons:
35Finite Element Method by G. R. Liu and S. S. Quek Using tetrahedrons to form hexahedronsUsing tetrahedrons to form hexahedrons 1 2 3 4 5 7 8 6 1 2 4 5 8 6 2 3 7 8 6 4 1 4 5 6 1 2 46 5 8 6 4 Break into three Hexahedron made up of six tetrahedrons:  Element matrices can be obtained by assembly of tetrahedron elements
36Finite Element Method by G. R. Liu and S. S. Quek HIGHER ORDER ELEMENTSHIGHER ORDER ELEMENTS  Tetrahedron elements 1 9 8 7 10 2 5 6 3 4 5 2 3 6 1 3 7 1 2 8 1 4 9 2 4 10 3 4 (2 -1) for corner nodes 1,2,3,4 4 4 4 for mid-edge nodes 4 4 4 i i iN L L i N L L N L L N L L N L L N L L N L L = = =  =  =   =  =  =  10 nodes, quadratic:
37Finite Element Method by G. R. Liu and S. S. Quek HIGHER ORDER ELEMENTSHIGHER ORDER ELEMENTS  Tetrahedron elements (Cont’d) 20 nodes, cubic: 1 2 9 9 5 1 1 3 11 1 1 42 2 9 9 6 3 1 3 12 4 1 42 2 9 9 7 1 1 2 13 22 2 9 8 2 1 22 9 9 2 2 32 9 10 3 2 32 (3 1)(3 2) for corner nodes 1,2,3,4 (3 1) (3 1) (3 1) (3 1) (3 1) (3 1) (3 1) (3 1) (3 1) i i i iN L L L i N L L L N L L L N L L L N L L L N L L L N L L N L L L N L L L N L L L = − − = = − = − = − = − = − = − = − = − = − 2 4 9 14 4 2 42 9 15 3 3 42 9 16 4 3 42 17 2 3 4 18 1 2 3 19 1 3 4 20 1 2 4 for edge nodes (3 1) (3 1) (3 1) 27 27 for center surface nodes 27 27 L N L L L N L L L N L L L N L L L N L L L N L L L N L L L      = −  = −  = −  =  =   =  =  1 13 12 7 15 2 9 6 3 4 5 8 10 11 14 16 17 18 19 5 20
38Finite Element Method by G. R. Liu and S. S. Quek HIGHER ORDER ELEMENTSHIGHER ORDER ELEMENTS  Brick elements Lagrange type: i(I,J,K) (0,0,0) η ξ ζ (n,m,p) (n,0,0) (n,m,0) (nd =(n+1)(m+1)(p+1) nodes) 1 1 1 ( ) ( ) ( )D D D n m p i I J K I J KN N N N l l lξ η ς= = 0 1 1 1 0 1 1 1 ( )( ) ( )( ) ( ) ( ) ( )( ) ( )( ) ( ) n k k n k k k k k k k k n l ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ − + − + − − − − − = − − − − − L L L L where
39Finite Element Method by G. R. Liu and S. S. Quek HIGHER ORDERHIGHER ORDER ELEMENTSELEMENTS  Brick elements (Cont’d) Serendipity type elements: 4(-1, 1, -1) (1, -1, 1)6 (1, -1, -1)2 8(-1, 1, 1) 7 (1, 1, 1) (-1, -1, 1)5 (-1,-1,-1)1 3(1, 1, -1) ξ η ζ 9(1,0,-1) 10(0,1,-1) 11(-1,0,-1) 12(0-1,-1) 13 14 3 15 16 17 18 19 20 1 8 21 4 21 4 (1 )(1 )(1 )( 2) for corner nodes 1, , 8 (1 )(1 )(1 ) for mid-side nodes 10,12,14,16 (1 )(1 j j j j j j i j j j j N j N j N ξ ξ η η ς ς ξ ξ η η ς ς ξ η η ς ς η ξ = + + + + + − = = − + + = = − + L 21 4 )(1 ) for mid-side nodes 9,11,13,15 (1 )(1 )(1 ) for mid-side nodes 17,18,19,20 j j j j j j N j ξ ς ς ς ξ ξ η η + = = − + + = 20 nodes, tri-quadratic:
40Finite Element Method by G. R. Liu and S. S. Quek HIGHER ORDER ELEMENTSHIGHER ORDER ELEMENTS  Brick elements (Cont’d) 2 2 21 64 29 64 1 3 29 64 (1 )(1 )(1 )(9 9 9 19) for corner nodes 1, , 8 (1 )(1 9 )(1 )(1 ) for side nodes with , 1 and 1 (1 )(1 9 j j j j j j j j j j j j N j N N ξ ξ η η ς ς ξ η ς ξ ξ ξ η η ς ς ξ η ς η η = + + + + + − = = − + + + = ± = ± = ± = − + L 1 3 29 64 1 3 )(1 )(1 ) for side nodes with , 1 and 1 (1 )(1 9 )(1 )(1 ) for side nodes with , 1 and 1 j j j j j j j j j j j j j N η ξ ξ ς ς η ξ ς ς ς ς ξ ξ η η ς ξ η + + = ± = ± = ± = − + + + = ± = ± = ± 32 nodes, tri-cubic: ξ η ζ
41Finite Element Method by G. R. Liu and S. S. Quek ELEMENTS WITH CURVEDELEMENTS WITH CURVED SURFACESSURFACES 1 4 98 7 10 2 5 6 3 7 18 16 12 15 14 11 13 5 17 19 20 6 109 8 2 1 4 3 9 8 7 10 2 5 6 3 1 4 13 7 18 16 12 15 14 11 5 17 19 20 6 10 9 8 2 1 4 3 ξ η ζ
42Finite Element Method by G. R. Liu and S. S. Quek CASE STUDYCASE STUDY  Stress and strain analysis of a quantum dot heterostructure Material E (Gpa) υ GaAs 86.96 0.31 InAs 51.42 0.35 GaAs substrate GaAs cap layer InAs wetting layer InAs quantum dot
43Finite Element Method by G. R. Liu and S. S. Quek CASE STUDYCASE STUDY
44Finite Element Method by G. R. Liu and S. S. Quek CASE STUDYCASE STUDY 30 nm 30 nm
45Finite Element Method by G. R. Liu and S. S. Quek CASE STUDYCASE STUDY
46Finite Element Method by G. R. Liu and S. S. Quek CASE STUDYCASE STUDY

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Fem 3d solid

  • 1.
    1 FFinite Element MethodiniteElement Method FEM FOR 3D SOLIDS for readers of all backgroundsfor readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 9:
  • 2.
    2Finite Element Methodby G. R. Liu and S. S. Quek CONTENTSCONTENTS  INTRODUCTION  TETRAHEDRON ELEMENT – Shape functions – Strain matrix – Element matrices  HEXAHEDRON ELEMENT – Shape functions – Strain matrix – Element matrices – Using tetrahedrons to form hexahedrons  HIGHER ORDER ELEMENTS  ELEMENTS WITH CURVED SURFACES
  • 3.
    3Finite Element Methodby G. R. Liu and S. S. Quek INTRODUCTIONINTRODUCTION  For 3D solids, all the field variables are dependent of x, y and z coordinates – most general element.  The element is often known as a 3D solid element or simply a solid element.  A 3D solid element can have a tetrahedron and hexahedron shape with flat or curved surfaces.  At any node there are three components in the x, y and z directions for the displacement as well as forces.
  • 4.
    4Finite Element Methodby G. R. Liu and S. S. Quek TETRAHEDRON ELEMENTTETRAHEDRON ELEMENT  3D solid meshed with tetrahedron elements
  • 5.
    5Finite Element Methodby G. R. Liu and S. S. Quek TETRAHEDRON ELEMENTTETRAHEDRON ELEMENT z=Z x=X z=Z y=Y w4 v4 u4 w2 u2 u2 w1 u1 v1 w3 u3 v3 i j l k1 = 4 = 2 = 3 = fsy fsz fsx Consider a four node tetrahedron element 1 1 1 2 2 2 3 3 3 4 4 4 node 1 node 2 node 3 node 4 e u v w u v w u v w u v w                           =                           d
  • 6.
    6Finite Element Methodby G. R. Liu and S. S. Quek Shape functionsShape functions ( , , ) ( , , )h ex y z x y z=U N d 1 2 3 4 1 2 3 4 1 2 3 4 node 1 node 2 node 3 node 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 N N N N N N N N N N N N     =      N 6447448 6447448 6447448 6447448 where Use volume coordinates (Recall Area coordinates for 2D triangular element) 1234 234 1 V V L P = 1=i 2=j 3=k 4=l P y z x
  • 7.
    7Finite Element Methodby G. R. Liu and S. S. Quek Shape functionsShape functions Similarly, 1234 123 4 1234 124 3 1234 134 2 ,, V V L V V L V V L PPP === Can also be viewed as ratio of distances 234 134 123124 1 2 3 4 1 234 1 234 1 234 1 234 , , ,P P PPd d dd L L L L d d d d − − −− − − − − = = = = 1=i 2=j 3=k 4=l P y z x 14321 =+++ LLLL since 1234123124134234 VVVVV PPPP =+++ (Partition of unity)
  • 8.
    8Finite Element Methodby G. R. Liu and S. S. Quek Shape functionsShape functions    = jkl i Li nodesremotetheat the0 nodehomeat the1 44332211 44332211 44332211 zLzLzLzLz yLyLyLyLy xLxLxLxLx +++= +++= +++= (Delta function property) 14321 =+++ LLLL                           =               4 3 2 1 4321 4321 4321 11111 L L L L zzzz yyyy xxxx z y x
  • 9.
    9Finite Element Methodby G. R. Liu and S. S. Quek Shape functionsShape functions Therefore, where                           =               z y x dcba dcba dcba dcba V L L L L 1 6 1 4444 3333 2222 1111 4 3 2 1 1 det , det 1 1 1 1 det 1 , det 1 1 1 j j j j j i k k k i k k l l l l l j j j j i k k i k k l l l l x y z y z a x y z b y z x y z y z y z y z c y z d y z y z y z        = = −                  = − = −           (Adjoint matrix) (Cofactors) i j k l i= 1,2 j = 2,3 k = 3,4 l = 4,1
  • 10.
    10Finite Element Methodby G. R. Liu and S. S. Quek Shape functionsShape functions             ×= l k j i l k j i l k j i z z z z y y y y x x x x V 1 1 1 1 det 6 1 (Volume of tetrahedron) )( 6 1 zdycxba V LN iiiiii +++==Therefore,
  • 11.
    11Finite Element Methodby G. R. Liu and S. S. Quek Strain matrixStrain matrix Since, ( , , ) ( , , )h ex y z x y z=U N d Therefore, ee BdLNdLU ===ε where NLNB                     ∂∂∂∂ ∂∂∂∂ ∂∂∂∂ ∂∂ ∂∂ ∂∂ == 0 0 0 00 00 00 xy xz yz z y x                     = 44 44 44 4 4 4 33 33 33 3 3 3 22 22 22 2 2 2 11 11 11 1 1 1 0 0 0 00 00 00 0 0 0 00 00 00 0 0 0 00 00 00 0 0 0 00 00 00 2 1 bd cd bc d c b bd cd bc d c b bd cd bc d c b bd cd bc d c b V B (Constant strain element)
  • 12.
    12Finite Element Methodby G. R. Liu and S. S. Quek Element matricesElement matrices e T T e eV dV V= =∫k B cB B cB 11 12 13 14 21 22 23 24 31 32 33 34 41 42 43 44 d d e e T e V V V Vρ ρ      = =        ∫ ∫ N N N N N N N N m N N N N N N N N N N where           = ji ji ji ij NN NN NN 00 00 00 N
  • 13.
    13Finite Element Methodby G. R. Liu and S. S. Quek Element matricesElement matrices 1 2 3 4 ! ! ! ! d 6 ( 3)!e m n p q eV m n p q L L L L V V m n p q = + + + +∫ Eisenberg and Malvern [1973] : 2 0 0 1 0 0 1 0 0 1 0 0 2 0 0 1 0 0 1 0 0 1 0 2 0 0 1 0 0 1 0 0 1 2 0 0 1 0 0 1 0 0 2 0 0 1 0 0 1 0 2 0 0 1 0 0 1 2 0 0 1 0 020 2 0 0 1 0 2 0 0 1 . 2 0 0 2 0 2 e e V sy ρ                  =                     m
  • 14.
    14Finite Element Methodby G. R. Liu and S. S. Quek Element matricesElement matrices Alternative method for evaluating me: special natural coordinate system z x z=Z y i j l k 1 = 4 = 2 = 3 = ξ=0 ξ=1 ξ=1 ξ=constant P Q
  • 15.
    15Finite Element Methodby G. R. Liu and S. S. Quek Element matricesElement matrices z x z=Z y i j l k 1 = 4 = 2 = 3 = η=0 η=0 η=1 η=constant P
  • 16.
    16Finite Element Methodby G. R. Liu and S. S. Quek Element matricesElement matrices z x z=Z y i j l k 1 = 4 = 2 = 3 = ζ=1 ζ=1 ζ=1 ζ=0 ζ=constant P QR
  • 17.
    17Finite Element Methodby G. R. Liu and S. S. Quek Element matricesElement matrices z x z=Z y i j l k 1 = 4 = 2 = 3 = ξ=0 η=0 ζ=1 ξ=1 η=0 ζ=1 ξ=1 η=1 ζ=1 ζ=0 ζ=constant P [xP=η(x3−x2)+x2, yP=η(y3−y2)+y2,0] O B B [xB=ξ(xP −x1)+x1, yB=ξ[η(yP−y1)+y1],0] O [x=(1−ζ)(x4−xB)+xB, y=(1−ζ)(y4−yB)+yB, z=(1−ζ)z4] ξ=constant ζ=constant 0 )( )( 223 223 = +−= +−= P P P z yyyy xxxx η η 0 )()()( )()()( 1122311 1122311 = +−+−=+−= +−+−=+−= B PB PB z yyyyyyyyy xxxxxxxxx ξξηξ ξξηξ 4 321214444 321214444 )1( )()()()( )()()()( zz yyyyyyyyyyy xxxxxxxxxxx B B ζ ξζξζζζ ξζξζζζ −= −−−+−−=−−= −−−+−−=−−= )1( )1( )1( 4 3 2 1 ζ ηξζ ξηζ ζξ −= −= = −= N N N N
  • 18.
    18Finite Element Methodby G. R. Liu and S. S. Quek Element matricesElement matrices Jacobian:                   ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = ζ ζ ζ η η η ξ ξ ξ z y x z y x z y x J 2 4 312141313121 312141313121 6 00 ]det[ ξζξηξξζηζζ ξηξξζηζζ V z yyyyyy xxxxxx −=++−+ ++−+ =J 1 1 1 0 0 0 d det d d d e T T e V Vρ ρ ξ η ζ= =∫ ∫ ∫ ∫m N N N N [J] 11 12 13 14 1 1 1 21 22 23 242 0 0 0 31 32 33 34 41 42 43 44 6 d d de eV ρ ξζ ξ η ζ      = −        ∫ ∫ ∫ N N N N N N N N m N N N N N N N N
  • 19.
    19Finite Element Methodby G. R. Liu and S. S. Quek Element matricesElement matrices l f f f l sz sy sx e d][ 43 T ∫           = − Nf z=Z x=X z=Z y=Y w4 v4 u4 w2 u2 u2 w1 u1 v1 w3 u3 v3 i j l k1 = 4 = 2 = 3 = fsy fsz fsx For uniformly distributed load: { } { } { } { } { } { }                                                           = × × × × × × − 13 13 13 13 13 13 43 2 1 0 0 0 0 0 0 f sz sy sx sz sy sx e f f f f f f l
  • 20.
    20Finite Element Methodby G. R. Liu and S. S. Quek HEXAHEDRON ELEMENTHEXAHEDRON ELEMENT  3D solid meshed with hexahedron elements P P’ P’’P’’’
  • 21.
    21Finite Element Methodby G. R. Liu and S. S. Quek Shape functionsShape functions eNdU = 1 2 3 4 5 6 7 8 displacement components at node 1 displacement components at node 2 displacement components at node 3 displacement components at node 4 displacement co e e e e e e e e e             =               d d d d d d d d d mponents at node 5 displacement components at node 6 displacement components at node 7 displacement components at node 8 1 1 1 ( 1, 2, ,8)ei u v i w     = =      d L 1 7 5 8 6 4 2 0 z y x 3 0 fsz fsy fsx [ ]87654321 NNNNNNNNN = )8,,2,1( 00 00 00 L=           = i N N N i i i iN
  • 22.
    22Finite Element Methodby G. R. Liu and S. S. Quek Shape functionsShape functions 4(-1, 1, -1) (1, -1, 1)6 (1, -1, -1)2 1 7 5 8 6 4 2 0 z y x 3 0 fsz fsy fsx 8(-1, 1, 1) 7 (1, 1, 1) (-1, -1, 1)5 (-1, -1, -1)1 3(1, 1, -1) ξ η ζ ii i ii i ii i zNz yNy xNx ),,( ),,( ),,( 8 1 8 1 8 1 ζηξ ζηξ ζηξ ∑ ∑ ∑ = = = = = = )1)(1)(1( 8 1 iiiiN ζζηηξξ +++= (Tri-linear functions)
  • 23.
    23Finite Element Methodby G. R. Liu and S. S. Quek Strain matrixStrain matrix [ ]87654321 BBBBBBBBB = whereby                     ∂∂∂∂ ∂∂∂∂ ∂∂∂∂ ∂∂ ∂∂ ∂∂ == 0 0 0 00 00 00 xNyN xNzN yNzN zN yN xN ii ii ii i i i ii LNB Note: Shape functions are expressed in natural coordinates – chain rule of differentiation ee BdLNdLU ===ε
  • 24.
    24Finite Element Methodby G. R. Liu and S. S. Quek Strain matrixStrain matrix ζζζζ ηηηη ξξξξ ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ z z Ny y Nx x NN z z Ny y Nx x NN z z Ny y Nx x NN iiii iiii iiii Chain rule of differentiation                   ∂ ∂ ∂ ∂ ∂ ∂ =                   ∂ ∂ ∂ ∂ ∂ ∂ z N y N x N N N N i i i i i i J ζ η ξ ⇒ where                   ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = ζ η ξ ζ η ξ ζ η ξ z z z y y y x x x J
  • 25.
    25Finite Element Methodby G. R. Liu and S. S. Quek Strain matrixStrain matrix 8 8 8 1 1 1 ( , , ) , ( , , ) , ( , , )i i i i i i i i i x N x y N y z N zξ η ζ ξ η ζ ξ η ζ = = = = = =∑ ∑ ∑Since, or 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1                   ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ = = = = = = = = = ζ η ξ ζ η ξ ζ η ξ i i i i i i i i i i i i i i i i i i i i i i i i i i i N z N z N z N y N y N y N x N x N x J 1 1 1 2 2 23 5 6 7 81 2 4 3 3 3 4 4 43 5 6 7 81 2 4 5 5 5 6 6 61 2 3 4 5 6 7 8 7 7 7 8 8 8 x y z x y zN N N N NN N N x y z x y zN N N N NN N N x y z x y zN N N N N N N N x y z x y z ξ ξ ξ ξ ξξ ξ ξ η η η η η η η η ζ ζ ζζ ζ ζ ζ ζ    ∂ ∂ ∂ ∂ ∂∂ ∂ ∂    ∂ ∂ ∂ ∂ ∂∂ ∂ ∂    ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ =   ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂    ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂    ∂ ∂ ∂∂ ∂ ∂ ∂ ∂    J               
  • 26.
    26Finite Element Methodby G. R. Liu and S. S. Quek Strain matrixStrain matrix                   ∂ ∂ ∂ ∂ ∂ ∂ =                   ∂ ∂ ∂ ∂ ∂ ∂ − ζ η ξ i i i i i i N N N z N y N x N 1 J                     ∂∂∂∂ ∂∂∂∂ ∂∂∂∂ ∂∂ ∂∂ ∂∂ == 0 0 0 00 00 00 xNyN xNzN yNzN zN yN xN ii ii ii i i i ii LNB Used to replace derivatives w.r.t. x, y, z with derivatives w.r.t. ξ, η, ζ
  • 27.
    27Finite Element Methodby G. R. Liu and S. S. Quek Element matricesElement matrices 1 1 1 T T 1 1 1 d det[ ]d d d e e V A ξ η ζ + + + − − − = =∫ ∫ ∫ ∫k B cB B cB J Gauss integration: ),,(d)d,( 1 1 1 1 1 1 1 1 1 jjikji n i m j l k fwwwfI ζηξηξηξ ∑∑∑∫ ∫ ∫ = = = + − + − + − == 1 1 1 1 1 1 d det d d d e T T e V Vρ ρ ξ η ζ − − − = =∫ ∫ ∫ ∫m N N N N [J]
  • 28.
    28Finite Element Methodby G. R. Liu and S. S. Quek Element matricesElement matrices For rectangular hexahedron: det eabc V= =[J]                           = 88 7877 686766 58575655 4847464544 282726252433 28272625242322 1817161514131211 . m mm mmm mmmm mmmmm mmmmmm mmmmmmm mmmmmmmm m sy e
  • 29.
    29Finite Element Methodby G. R. Liu and S. S. Quek Element matricesElement matrices (Cont’d) where ζηξρ ζηξρ ζηξρ ddd 00 00 00 ddd 00 00 00 00 00 00 ddd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1           =                     = = ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ − − − − − − − − − ji ji ji j j j i i i jiij NN NN NN abc N N N N N N abc abc NNm
  • 30.
    30Finite Element Methodby G. R. Liu and S. S. Quek Element matricesElement matrices (Cont’d) or           = ij ij ij ij m m m 00 00 00 m where )1)(1)(1( 8 d)1)(1(d)1)(1(d)1)(1( 64 ddd 3 1 3 1 3 1 1 1 1 1 1 1 1 1 1 1 jijiji jijiji jiij hab abc NNabcm ζζηηξξ ρ ζζζζζηηηηηξξξξξ ρ ζηξρ +++= ++++++= = ∫∫∫ ∫ ∫ + − + − + − + − + −
  • 31.
    31Finite Element Methodby G. R. Liu and S. S. Quek Element matricesElement matrices (Cont’d) E.g. 216 8)111)(111)(111( 8 3 1 3 1 3 1 33 abcabc m ρρ ×=××+××+××+= 216 1 216 2 216 4 216 8 46352817 184538276857473625162413 483726155814786756342312 8877665544332211 abc mmmm abc mmmmmmmmmmmm abc mmmmmmmmmmmm abc mmmmmmmm ρ ρ ρ ρ ==== ============ ============ ========
  • 32.
    32Finite Element Methodby G. R. Liu and S. S. Quek Element matricesElement matrices (Cont’d)                           = 8 48. 248 4248 42128 242148 1242248 21244248 216 sy abc ex ρ m Note: For x direction only (Rectangular hexahedron)
  • 33.
    33Finite Element Methodby G. R. Liu and S. S. Quek Element matricesElement matrices l f f f l sz sy sx e d][ 43 T ∫           = − Nf 1 7 5 8 6 4 2 0 z y x 3 0 fsz fsy fsx { } { } { } { } { } { }                                                           = × × × × × × − 13 13 13 13 13 13 43 2 1 0 0 0 0 0 0 f sz sy sx sz sy sx e f f f f f f l For uniformly distributed load:
  • 34.
    34Finite Element Methodby G. R. Liu and S. S. Quek Using tetrahedrons to form hexahedronsUsing tetrahedrons to form hexahedrons  Hexahedrons can be made up of several tetrahedrons 1 5 6 8 1 4 3 8 1 2 3 4 5 7 8 3 1 6 8 6 3 2 1 6 3 6 7 8Hexahedron made up of 5 tetrahedrons:
  • 35.
    35Finite Element Methodby G. R. Liu and S. S. Quek Using tetrahedrons to form hexahedronsUsing tetrahedrons to form hexahedrons 1 2 3 4 5 7 8 6 1 2 4 5 8 6 2 3 7 8 6 4 1 4 5 6 1 2 46 5 8 6 4 Break into three Hexahedron made up of six tetrahedrons:  Element matrices can be obtained by assembly of tetrahedron elements
  • 36.
    36Finite Element Methodby G. R. Liu and S. S. Quek HIGHER ORDER ELEMENTSHIGHER ORDER ELEMENTS  Tetrahedron elements 1 9 8 7 10 2 5 6 3 4 5 2 3 6 1 3 7 1 2 8 1 4 9 2 4 10 3 4 (2 -1) for corner nodes 1,2,3,4 4 4 4 for mid-edge nodes 4 4 4 i i iN L L i N L L N L L N L L N L L N L L N L L = = =  =  =   =  =  =  10 nodes, quadratic:
  • 37.
    37Finite Element Methodby G. R. Liu and S. S. Quek HIGHER ORDER ELEMENTSHIGHER ORDER ELEMENTS  Tetrahedron elements (Cont’d) 20 nodes, cubic: 1 2 9 9 5 1 1 3 11 1 1 42 2 9 9 6 3 1 3 12 4 1 42 2 9 9 7 1 1 2 13 22 2 9 8 2 1 22 9 9 2 2 32 9 10 3 2 32 (3 1)(3 2) for corner nodes 1,2,3,4 (3 1) (3 1) (3 1) (3 1) (3 1) (3 1) (3 1) (3 1) (3 1) i i i iN L L L i N L L L N L L L N L L L N L L L N L L L N L L N L L L N L L L N L L L = − − = = − = − = − = − = − = − = − = − = − 2 4 9 14 4 2 42 9 15 3 3 42 9 16 4 3 42 17 2 3 4 18 1 2 3 19 1 3 4 20 1 2 4 for edge nodes (3 1) (3 1) (3 1) 27 27 for center surface nodes 27 27 L N L L L N L L L N L L L N L L L N L L L N L L L N L L L      = −  = −  = −  =  =   =  =  1 13 12 7 15 2 9 6 3 4 5 8 10 11 14 16 17 18 19 5 20
  • 38.
    38Finite Element Methodby G. R. Liu and S. S. Quek HIGHER ORDER ELEMENTSHIGHER ORDER ELEMENTS  Brick elements Lagrange type: i(I,J,K) (0,0,0) η ξ ζ (n,m,p) (n,0,0) (n,m,0) (nd =(n+1)(m+1)(p+1) nodes) 1 1 1 ( ) ( ) ( )D D D n m p i I J K I J KN N N N l l lξ η ς= = 0 1 1 1 0 1 1 1 ( )( ) ( )( ) ( ) ( ) ( )( ) ( )( ) ( ) n k k n k k k k k k k k n l ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ − + − + − − − − − = − − − − − L L L L where
  • 39.
    39Finite Element Methodby G. R. Liu and S. S. Quek HIGHER ORDERHIGHER ORDER ELEMENTSELEMENTS  Brick elements (Cont’d) Serendipity type elements: 4(-1, 1, -1) (1, -1, 1)6 (1, -1, -1)2 8(-1, 1, 1) 7 (1, 1, 1) (-1, -1, 1)5 (-1,-1,-1)1 3(1, 1, -1) ξ η ζ 9(1,0,-1) 10(0,1,-1) 11(-1,0,-1) 12(0-1,-1) 13 14 3 15 16 17 18 19 20 1 8 21 4 21 4 (1 )(1 )(1 )( 2) for corner nodes 1, , 8 (1 )(1 )(1 ) for mid-side nodes 10,12,14,16 (1 )(1 j j j j j j i j j j j N j N j N ξ ξ η η ς ς ξ ξ η η ς ς ξ η η ς ς η ξ = + + + + + − = = − + + = = − + L 21 4 )(1 ) for mid-side nodes 9,11,13,15 (1 )(1 )(1 ) for mid-side nodes 17,18,19,20 j j j j j j N j ξ ς ς ς ξ ξ η η + = = − + + = 20 nodes, tri-quadratic:
  • 40.
    40Finite Element Methodby G. R. Liu and S. S. Quek HIGHER ORDER ELEMENTSHIGHER ORDER ELEMENTS  Brick elements (Cont’d) 2 2 21 64 29 64 1 3 29 64 (1 )(1 )(1 )(9 9 9 19) for corner nodes 1, , 8 (1 )(1 9 )(1 )(1 ) for side nodes with , 1 and 1 (1 )(1 9 j j j j j j j j j j j j N j N N ξ ξ η η ς ς ξ η ς ξ ξ ξ η η ς ς ξ η ς η η = + + + + + − = = − + + + = ± = ± = ± = − + L 1 3 29 64 1 3 )(1 )(1 ) for side nodes with , 1 and 1 (1 )(1 9 )(1 )(1 ) for side nodes with , 1 and 1 j j j j j j j j j j j j j N η ξ ξ ς ς η ξ ς ς ς ς ξ ξ η η ς ξ η + + = ± = ± = ± = − + + + = ± = ± = ± 32 nodes, tri-cubic: ξ η ζ
  • 41.
    41Finite Element Methodby G. R. Liu and S. S. Quek ELEMENTS WITH CURVEDELEMENTS WITH CURVED SURFACESSURFACES 1 4 98 7 10 2 5 6 3 7 18 16 12 15 14 11 13 5 17 19 20 6 109 8 2 1 4 3 9 8 7 10 2 5 6 3 1 4 13 7 18 16 12 15 14 11 5 17 19 20 6 10 9 8 2 1 4 3 ξ η ζ
  • 42.
    42Finite Element Methodby G. R. Liu and S. S. Quek CASE STUDYCASE STUDY  Stress and strain analysis of a quantum dot heterostructure Material E (Gpa) υ GaAs 86.96 0.31 InAs 51.42 0.35 GaAs substrate GaAs cap layer InAs wetting layer InAs quantum dot
  • 43.
    43Finite Element Methodby G. R. Liu and S. S. Quek CASE STUDYCASE STUDY
  • 44.
    44Finite Element Methodby G. R. Liu and S. S. Quek CASE STUDYCASE STUDY 30 nm 30 nm
  • 45.
    45Finite Element Methodby G. R. Liu and S. S. Quek CASE STUDYCASE STUDY
  • 46.
    46Finite Element Methodby G. R. Liu and S. S. Quek CASE STUDYCASE STUDY