The Finite Element Method

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1 The Finite Element Method 3D Problems Heat Transfer and Elasticity Read: Chapter 14 CONTENTS Finite element models of 3-D Heat Transfer Finite element model of 3-D Elasticity Typical 3-D Finite Elements

2 3-D HEAT TRANSFER Governing Equation x k x T x y Boundary Conditions k y T y z k z T z T = ˆT on Γ 1, T k x x n T x + k y y n T y + k z z n z + β(t T )=ˆq on Γ ( ) 2 where k x, k y and k z are conductivities of an orthotropic solid in the three coordinate directions, g is the internal heat generation per unit volume in a three-dimensional domain Ω, andˆt and ˆq are specified functions of position on the portions Γ 1 and Γ 2, respectively, of the surface Γ of the domain (see Fig.1); β is the convection coefficient and T is the ambient temperature. = g in Ω 3-D Problems 2

3 3-D HEAT TRANSFER (continued) ds n z nˆ n y Parts of the boundary n x Γ e z y DomainΩ Surface Γ Ω e x Weak Form 0= = w T k x T k y T k z Ω e x x y y z z w T k x Ω e x x + k w T y y y + k w T z z z wg dx + βwt ds w (q n + βt ) ds (3) Γ e Γ e A six-face 3-D finite element g dx 3-D Problems 3

4 3-D HEAT TRANSFER (continued) Finite element approximation T = n T j ψj e (x, y, z) j=1 Finite element model K e T e = f e + Q e where Kij e ψ e ψ i j e = k x Ω e x x + k ψ e ψ i j e y y y + k ψ e ψ i j e z z z + βψi e ψ j ds Γ e fi e = fψi e dx, Q e i = (q n + βt )ψi e ds Ω e Γ e ( ) dx 3-D Problems 4

5 3-D ELASTICITY Equations of Motion σ xx x σ xy x σ xz x + σ xy y + σ yy y + σ yz y + σ xz z + σ yz z + σ zz z 2 + f x = ρ u x t f y = ρ u y t f z = ρ u z t 2 Strain-Displacement Relations ε xx = u x x, 2ε xy = u x y + u y x, 2ε yz = u y z + u z y ε yy = u y y, ε zz = u z z 2ε xz = u x z + u z x 3-D Problems 5

6 3-D ELASTICITY (continued) Constitutive Relations σ xx σ yy σ zz σ xz σ yz σ xy = c 11 c 12 c c 12 c 22 c c 13 c 23 c c c c 66 ε xx ε yy ε zz 2ε xz 2ε yz 2ε xy The material axes are assumed coincide with the global axes and the material is orthotropic with respect to the global axes. Boundary Conditions t x σ xx n x + σ xy n y + σ xz n z = ˆt x t y σ xy n x + σ yy n y + σ yz n z = ˆt y t z σ xz n x + σ yz n y + σ zz n z = ˆt z ( ) on Γ σ or u = û on Γ u 3-D Problems 6

7 3-D ELASTICITY (continued) MATRIX FORM OF THE GOVERNING EQUATIONS Notation σ = D T = / x 0 0 / z 0 / y 0 / y 0 0 / z / x 0 0 / z / x / y 0 f x u x ε =, f = f y, u = u y f z u z σ xx σ yy σ zz σ xy σ xz σ yz Governing equations ε xx ε yy ε zz 2ε xz 2ε yz 2ε xy, D T σ + f = ρü ε = Du, σ = Cε 3-D Problems 7

8 3-D ELASTICITY (continued) Principle of virtual displacements (in matrix form) 0= (Dδu) T C (Du)+ρu T ü d dx (δu) T f dx (δu) T t ds Ω e Ω e Γ e Finite element approximation (in matrix form) u = Ψ = u x u y u z = Ψ, w = δu = δu x δu y δu z = Ψδ ψ ψ ψ n ψ ψ ψ n ψ ψ ψ n = { u 1 x u 1 y u 1 z u 2 x u 2 y u 2 z... u n x u n y u n z } T 3-D Problems 8

9 3-D ELASTICITY (continued) Finite Element Model 1 M e e + K e e = F e + Q e where K e = h e B T CB dx, M e = ρh e Ψ T Ψ e dx Ω e Ω e F e = Ψ T f dx, Q e = Ψ T t ds Ω e Γ e At each node ( uvw,, ) 4 1 ( ) D Problems 9

10 TYPICAL 3-D FINITE ELEMENTS Linear tetrahedral element L 3 = L 1 = 0 3 L 4 = 0 2 u = a 0 + a 1 x + a 2 y + a 3 z {Ψ e } = L 1 L 2 L 3 L 4 Quadratic tetrahedral element {Ψ e } = D Problems 10 L 1 (2L 1 1) L 2 (2L 2 1) L 3 (2L 3 1) L 4 (2L 4 1) 4L 1 L 2 4L 2 L 3 4L 3 L 1 4L 1 L 4 4L 2 L 4 4L 3 L 4

11 TYPICAL 3-D FINITE ELEMENTS ζ Linear prism element L 3 = 0 η ξ L 2 = ζ = 1 ζ = L 1 = 0 3 Quadratic prism element u = a 0 + a 1 x + a 2 y + a 3 z + a 4 xz + a 5 yz L 1 (1 ζ) L {Ψ e } = 1 2 (1 ζ) L 3 (1 ζ) 2 L 1 (1 + ζ) L 2 (1 + ζ) L 3 (1 + ζ) 3-D Problems 11

12 TYPICAL 3-D FINITE ELEMENTS (cont ) Quadratic Prism Element {Ψ e } = 1 2 L 1 [(2L 1 1)(1 ζ) (1 ζ 2 )] L 2 [(2L 2 1)(1 ζ) (1 ζ 2 )] L 3 [(2L 3 1)(1 ζ) (1 ζ 2 )] L 1 [(2L 1 1)(1 + ζ) (1 ζ 2 )] L 2 [(2L 2 1)(1 + ζ) (1 ζ 2 )] L 3 [(2L 3 1)(1 + ζ) (1 ζ 2 )] 4L 1 L 2 (1 ζ) 4L 2 L 3 (1 ζ) 4L 3 L 1 (1 ζ) 2L 1 (1 ζ 2 ) 2L 2 (1 ζ 2 ) 2L 3 (1 ζ 2 ) 4L 1 L 2 (1 + ζ) 4L 2 L 3 (1 + ζ) 4L 3 L 1 (1 + ζ) 3-D Problems 12

13 TYPICAL 3-D FINITE ELEMENTS (cont ) ξ = +1 2 Linear brick element η = 1 5 ζ = +1 ξ = ζ η = +1 7 ξ 1 η 3 ζ = 1 4 nodes u = a 0 + a 1 x + a 2 y + a 3 z + a 4 yz + a 5 xz + a 6 xy + a 7 xyz (1 ξ)(1 η)(1 ζ) (1 + ξ)(1 η)(1 ζ) (1 + ξ)(1 + η)(1 ζ) {Ψ e } = 1 (1 ξ)(1 + η)(1 ζ) 8 (1 ξ)(1 η)(1 + ζ) (1 + ξ)(1 η)(1 + ζ) (1 + ξ)(1 + η)(1 + ζ) (1 ξ)(1 + η)(1 + ζ) Quadratic brick element 5 ζ ξ η D Problems 13

14 TYPICAL 3-D FINITE ELEMENTS (cont ) Quadratic Brick Element 5 ζ ξ η {Ψ e } = 1 8 (1 ξ)(1 η)(1 ζ)( ξ η ζ 2) (1 + ξ)(1 η)(1 ζ)(ξ η ζ 2) (1 + ξ)(1 + η)(1 ζ)(ξ + η ζ 2) (1 ξ)(1 + η)(1 ζ)( ξ + η ζ 2) (1 ξ)(1 η)(1 + ζ)( ξ η + ζ 2) (1 + ξ)(1 η)(1 + ζ)(ξ η + ζ 2) (1 + ξ)(1 + η)(1 + ζ)(ξ + η + ζ 2) (1 ξ)(1 + η)(1 + ζ)( ξ + η + ζ 2) 2(1 ξ 2 )(1 η)(1 ζ) 2(1 + ξ)(1 η 2 )(1 ζ) 2(1 ξ 2 )(1 + η)(1 ζ) 2(1 ξ)(1 η 2 )(1 ζ) 2(1 ξ)(1 η)(1 ζ 2 ) 2(1 + ξ)(1 η)(1 ζ 2 ) 2(1 + ξ)(1 + η)(1 ζ 2 ) 2(1 ξ)(1 + η)(1 ζ 2 ) 2(1 ξ 2 )(1 η)(1 + ζ) 2(1 + ξ)(1 η 2 )(1 + ζ) 2(1 ξ 2 )(1 + η)(1 + ζ) 2(1 ξ)(1 η 2 )(1 + ζ)

15 TYPICAL 3-D or SHELL FINITE ELEMENT MESHES 2-D Problems: 15

16 TYPICAL 3-D FINITE ELEMENT MESHES 2-D Problems: 16

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