3 016 5 Journal of East China Normal University Natural Science No 3 May 016 : 1000-564101603-0009-1 1 1 1 00444; E- 00030 : Lagrange Lobatto Matlab : ; Lagrange; : O41 : A DOI: 103969/jissn1000-56410160300 High order interpolation function for surface contact problem FENG Yun-qing 1 HOU Lei 1 1 Department of Mathematics Shanghai University Shanghai 00444 China; Computational Sciences E-institute of Shanghai Universities Shanghai 00030 China Abstract: This paper mainly adopts Lagrange bicubic shape function to construct interpolation function and uses finite element method to solve the coupling equations of surface contact The Lobatto points are used to construct the interpolation nodes to avoid the Runge phenomenon Higher shape functions and two different numerical integration methods are adopted to improve the accuracy of the numerical solution According to the above analysis this article uses Matlab program to simulate the deformation and stress changes in surface contact problem Key words: finite element; Lagrange; orthogonal integration 0 : 015-05 : 117147 : E-mail: yqyzsh@16com : E-mail: houlei@shueducn
10 016 [1-] u t = 1 τ u u ρ τ t = η λ D 1 ελ λ exp τ 11 + τ τ u τ 01 + [ u τ + u τ T ] ξ[d τ + D τ T ] u xy Γ = u Γ u t=t0 = u 0 τ xy Γ = τ Γ τ t=t0 = τ 0 ρ ξ η λ Cauchy τ u P-T/T P-T/T Maxwell [-4] [15-6] Lagrange Lobatto [1] Gauss Gauss-Lobatto Gauss Gauss-Lobatto Gauss Gauss-Lobatto [7] Gauss Gauss-Lobatto 4 4 3 3 4 4 Gauss 4 4 Gauss-Lobatto [8-9] 1
3 : 11 Lobatto Lagrange16 [h h] Lobatto [10] ω 0 = 1 ω 1 = 1 x + 1 σ i = ω i+1 = σ i = i d [Ax] i i 1 dx i+1 1 i! hi 1 Ax = 1 x h [ 1] 4 Lobatto ω 5 = 1 8 7 5x4 6x + 1 5 5 5 5 1 [ 1] [ 1] Lobatto 4 4 11 16 1 [110] Fig11 11 The number of interpolation nodes of element mesh 1 Fig1 The number of interpolation nodes of global mesh Lagrange16
1 016 [ 1] Lobatto 4 Lagrange L 1 ξ = 5 ξ 1 ξ 1 8 5 L ξ = 5 5 5 ξ ξ 1 8 5 L 3 ξ = 5 5 5 ξ + ξ 1 8 5 L 4 ξ = 5 ξ 1 ξ + 1 8 5 Lagrange16 : ϕ 1 ξ η = L 1 ξl 1 η ϕ ξ η = L ξl 1 η ϕ 3 ξ η = L 3 ξl 1 η ϕ 4 ξ η = L 4 ξl 1 η ϕ 5 ξ η = L 4 ξl η ϕ 6 ξ η = L 4 ξl 3 η ϕ 7 ξ η = L 4 ξl 4 η ϕ 8 ξ η = L 3 ξl 4 η ϕ 9 ξ η = L ξl 4 η ϕ 10 ξ η = L 1 ξl 4 η ϕ 11 ξ η = L 1 ξl 3 η ϕ 1 ξ η = L 1 ξl η ϕ 13 ξ η = L ξl η ϕ 14 ξ η = L 3 ξl η ϕ 15 ξ η = L 3 ξl 3 η ϕ 16 ξ η = L ξl 3 η L i η Lagrange 16 Lobatto 49 49 Lagrange16 Matlab 01 01 u = iu 1 x y; t+ju x y; t u 1 x y; t u x y; t u x τ11 x y; t τ 1 x y; t y τ= τ 11 x y; t τ 1 x y; t τ 1 x y; t τ x y; t τ 1 x y; t τ x y; t τ τ 1 x y; t = τ 1 x y; t 01 u 1 t u t = 1 ρ = 1 ρ τ11 x + τ 1 u 1 u 1 x + u u 1 1 τ1 x + τ u u 1 x + u u
3 : 13 τ 1 t τ 11 t τ t = η λ = η λ u 1 x 1 λ τ 11 ε τ 11 + τ τ 11 + ξ u 1 x τ 11 + u 1 τ 11 [ ξ u 1 ξ u x u 1 λ τ ε τ 11 + τ τ x + u ] τ 1 u 1 τ + ξ u [ τ + ξ u x ξ u 1 = η u1 λ + u 1 x λ τ 1 ε τ 11 + τ τ 1 [ + 1 ξ u x ξ ] [ u 1 u1 τ 11 + 1 ξ x + [ 1 ξ u1 ξ u x ] τ x + u ] τ 1 u 1 τ 1 τ 11 τ x + u + u ] τ 1 τ 1 3 4 5 u τ u H τ H 01 H V h H : V h = span{ϕ 1 x y ϕ x y ϕ 16 x y} V h ϕ m x ym = 1 16 ϕ m x y ϕ m ξ η H V h ũ V h τ V h Lũ ϕ m = f ϕ m m = 1 16 6 L τ ϕ m = f ϕ m m = 1 16 7 L 01 x y t e = [ 1] [ 1] Lagrange16 1 49 Matlab u 1 x m y m ; t n t = t n x ũ 1 ũ τ 11 τ 1 τ ϕ m : ũ 1 x y; t n = u 1 mϕ m x y τ 11 x y; t n = τ 11 mϕ m x y 8
14 016 : ũ 1 t x y; tn = ũ 1 x x y; tn = τ 11 x y; tn = 16 u 1t mϕ m x y u 1 mϕ mx x y τ 11 mϕ my x y 9 6 7 ũ 1 ũ τ 11 τ 1 τ 8 9 1 5 {A} n 16 16{u 1t } n 16 1 = 1 ρ {B 1} n 16 16{τ 11 } n 16 1 + {B } n 16 16{τ 1 } n 16 1 16 16 u 1 m{e 1 m} n 16 16 u m{e m} n 16 16 {u 1 } n 16 1 {u 1 } n 16 1 {A} n 16 16 {u t} n 16 1 =1 ρ {B 1} n 16 16 {τ 1} n 16 1 + {B } n 16 16 {τ } n 16 1 16 16 u 1 m{e 1 m} n 16 16 u m{e m} n 16 16 {u } n 16 1 {u } n 16 1 10 11 {A} n 16 16 {τ 11t} n 16 1 = η λ {B 1} n 16 16 {u 1} n 16 1 1 λ {A}n 16 16 {τ 11} n 16 1 16 ε τ 11 m + τ m{cm} n 16 16 {τ 11 } n 16 1 16 + ξ + ξ 16 u 1 m{e 1 m} n 16 16 + 16 u 1 m{c 1 m} n 16 16 u m{e m} n 16 16 u 1 m{c m} n 16 16 ξ {τ 11 } n 16 1 {τ 11 } n 16 1 u m{c 1 m} n 16 16 {τ 1 } n 16 1 1
3 : 15 {A} n 16 16{τ t } n 16 1 = η λ 1 λ {A}n 16 16{τ } n 16 1 16 ε τ 11 m + τ m{cm} n 16 16 {τ } n 16 1 16 + ξ + ξ u 1 m{e 1 m} n 16 16 + 16 16 u m{c m} n 16 16 u m{e m} n 16 16 u m{c 1 m} n 16 16 ξ {τ } n 16 1 {τ } n 16 1 u 1 m{c m} n 16 16 {τ 1 } n 16 1 13 {A} n 16 16{τ 1t } n 16 1 = η λ {B } n 16 16{u 1 } n 16 1 + {B 1 } n 16 16{u } n 16 1 1 λ {A}n 16 16{τ 1 } n 16 1 16 ε τ 11 m + τ m{cm} n 16 16 {τ } n 16 1 + 16 u 1 m{e 1 m} n 16 16 + 1 ξ + 1 ξ + u m{c 1 m} n 16 16 ξ u 1 m{c 1 m} n 16 16 + 1 ξ u 1 m{c m} n 16 16 ξ {A} 16 16 = {B 1 } 16 16 = ϕ 1 ϕ 1 dxdy ϕ 1 ϕ 16 dxdy ϕ 1x ϕ 1 dxdy ϕ 1x ϕ 16 dxdy u m{e m} n 16 16 {τ 1 } n 16 1 u 1 m{c m} n 16 16 u m{c m} n 16 16 u m{c 1 m} n 16 16 ϕ 16 ϕ 1 dxdy ϕ 16 ϕ 16 dxdy ϕ 16x ϕ 1 dxdy ϕ 16x ϕ 16 dxdy {τ 11 } n 16 1 {τ 1 } n 16 1 {τ } n 16 1 14
16 016 {B } 16 16 = {E 1 m} 16 16 = {E m} 16 16 = ϕ 1y ϕ 1 dxdy {Cm} 16 16 = {C 1 m} 16 16 = {C m} 16 16 = ϕ 1y ϕ 16 dxdy ϕ m ϕ 1x ϕ 1 dxdy ϕ m ϕ 1x ϕ 16 dxdy ϕ m ϕ 1y ϕ 1 dxdy ϕ m ϕ 1y ϕ 16 dxdy ϕ m ϕ 1 ϕ 1 dxdy ϕ m ϕ 1 ϕ 16 dxdy ϕ mx ϕ 1 ϕ 1 dxdy ϕ mx ϕ 1 ϕ 16 dxdy ϕ my ϕ 1 ϕ 1 dxdy ϕ my ϕ 1 ϕ 16 dxdy ϕ 16y ϕ 1 dxdy ϕ 16y ϕ 16 dxdy ϕ m ϕ 16x ϕ 1 dxdy ϕ m ϕ 16x ϕ 16 dxdy ϕ m ϕ 16y ϕ 1 dxdy ϕ m ϕ 16y ϕ 16 dxdy ϕ m ϕ 16 ϕ 1 dxdy ϕ m ϕ 16 ϕ 16 dxdy ϕ mx ϕ 16 ϕ 1 dxdy ϕ mx ϕ 16 ϕ 16 dxdy ϕ my ϕ 16 ϕ 1 dxdy ϕ my ϕ 16 ϕ 16 dxdy ϕ mx ϕ my ϕ m x y 10 14 1 5 [15] 1 10 10 {A} n 16 16 Euler { } n u1 = {Fτ 11 τ 1 } n 16 1 t {Gu 1 u } n 16 16 {u 1} n 16 1 15 16 1 1 t {A}n 16 16 {u 1} n+1 16 1 1 t {A}n 16 16 {u 1} n 16 1 = {Fτ 11 τ 1 } n 16 1 {Gu 1 u } n 16 16 {u 1} n 16 1 16
3 : 17 Crank-Nicolson 1 t {A}n 16 16 {u 1} n+1 16 1 1 t {A}n 16 16 {u 1} n 16 1 = {Fτ 11 τ 1 } n 16 1 1 {Gu 1 u } n 16 16 {u 1} n+1 16 1 + 1 {u 1} n 16 1 17 Adams : {A} n 16 16{u 1 } n+1 16 1 = {A}n 16 16{u 1 } n 16 1 + 3 t {Fτ 11 τ 1 } n 16 1 {Gu 1 u } n 16 16 {u 1} n 16 1 t {Fτ 11 τ 1 } n 16 1 {Gu 1 u } n 16 16 {u 1} n 16 1 18 1 4 3 Gauss-Lobatto Gauss Gauss-Lobatto Gauss-Lobatto Gauss [81] Lagrange16 3 3 4 4 Gauss 4 4 Gauss-Lobatto 4 01 Matlab [0 ] [0 ] Dyna h = 05 41 9 Gauss t u 10 u 0 9 Gauss 4 ; Lagrange-Euler Lagrange-Crank-Nicolson Lagrange-Adams t = 1 t = 43 16 Gauss 44 16 Gauss
18 016 t = 1 t = t = 3 41 9 Gauss t = 1 u 10 = 01 u 0 = 0 Fig 41 Gauss integration with 9 points t = 1 u 10 = 01 u 0 = 0 Fig 4 4 9 Gauss The stress distribution calculated by Gauss integration with 9 points of three kinds of mesh 43 16 Gauss t = 1 u 10 = 01 u 0 = 0 Fig 43 Gauss integration with 16 points t = 1 u 10 = 01 u 0 = 0 45 16 Gauss-Lobatto 46 Lagrange-Euler Lagrange-Crank-Nicolson Lagrange- Adams 16 Gauss-Lobatto Lagrange- Euler Lagrange-Adams t = 1 t = Lagrange-Crank-Nicolson t = 1 t = t = 3
13Ï ¾ : Ep ¼ê L ÀÂ-E K 19 ã 44 16 : Gauss È éan«ª AåCzã Fig 44 The stress distribution calculated by Gauss integration with 16 points of three kinds of mesh ã 45 Fig 45 16 : Gauss-Lobatto È t = 1 u10 = 01 u0 = 0 Gauss-Lobatto integration with 16 points t = 1 u10 = 01 u0 = 0 ã 46 16 : Gauss-Lobatto È éan«ª AåCzã Fig 46 The stress distribution calculated by Gauss-Lobatto integration with 16 points of three kinds of mesh 5 Ø æ^k {é mcþ?1lñ Ì $^ Lagrange 16 :Vng/¼ê E ¼ê Ün«ØÓ k é mcþ?1lñ ª ÍÜ 01 ê éfýý ê È æ^øóa Èúª?1ê [ ª uy 9 : Gauss È Ú 16 : Gauss-Lobatto È J Ø ; 16 : Gauss È Ð
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