High order interpolation function for surface contact problem

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1 Journal of East China Normal University Natural Science No 3 May 016 : ; E : Lagrange Lobatto Matlab : ; Lagrange; : O41 : A DOI: /jissn High order interpolation function for surface contact problem FENG Yun-qing 1 HOU Lei 1 1 Department of Mathematics Shanghai University Shanghai China; Computational Sciences E-institute of Shanghai Universities Shanghai 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 : : : yqyzsh@16com : houlei@shueducn

2 [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 Gauss 4 4 Gauss-Lobatto [8-9] 1

3 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 = x4 6x [ 1] [ 1] Lobatto [110] Fig11 11 The number of interpolation nodes of element mesh 1 Fig1 The number of interpolation nodes of global mesh Lagrange16

4 1 016 [ 1] Lobatto 4 Lagrange L 1 ξ = 5 ξ 1 ξ L ξ = ξ ξ L 3 ξ = ξ + ξ L 4 ξ = 5 ξ 1 ξ 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 Lagrange16 Matlab 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

5 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 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 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 = L τ ϕ m = f ϕ m m = L 01 x y t e = [ 1] [ 1] Lagrange 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

6 : 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 ũ 1 ũ τ 11 τ 1 τ {A} n 16 16{u 1t } n 16 1 = 1 ρ {B 1} n 16 16{τ 11 } n {B } n 16 16{τ 1 } n u 1 m{e 1 m} n u m{e m} n {u 1 } n 16 1 {u 1 } n 16 1 {A} n {u t} n 16 1 =1 ρ {B 1} n {τ 1} n {B } n {τ } n u 1 m{e 1 m} n u m{e m} n {u } n 16 1 {u } n {A} n {τ 11t} n 16 1 = η λ {B 1} n {u 1} n λ {A}n {τ 11} n ε τ 11 m + τ m{cm} n {τ 11 } n ξ + ξ 16 u 1 m{e 1 m} n u 1 m{c 1 m} n u m{e m} n u 1 m{c m} n ξ {τ 11 } n 16 1 {τ 11 } n 16 1 u m{c 1 m} n {τ 1 } n

7 3 : 15 {A} n 16 16{τ t } n 16 1 = η λ 1 λ {A}n 16 16{τ } n ε τ 11 m + τ m{cm} n {τ } n ξ + ξ u 1 m{e 1 m} n u m{c m} n u m{e m} n u m{c 1 m} n ξ {τ } n 16 1 {τ } n 16 1 u 1 m{c m} n {τ 1 } n {A} n 16 16{τ 1t } n 16 1 = η λ {B } n 16 16{u 1 } n {B 1 } n 16 16{u } n λ {A}n 16 16{τ 1 } n ε τ 11 m + τ m{cm} n {τ } n u 1 m{e 1 m} n ξ + 1 ξ + u m{c 1 m} n ξ u 1 m{c 1 m} n ξ u 1 m{c m} n ξ {A} = {B 1 } = 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 {τ 1 } n 16 1 u 1 m{c m} n u m{c m} n u m{c 1 m} n 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

8 {B } = {E 1 m} = {E m} = ϕ 1y ϕ 1 dxdy {Cm} = {C 1 m} = {C m} = 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 [15] {A} n Euler { } n u1 = {Fτ 11 τ 1 } n 16 1 t {Gu 1 u } n {u 1} n t {A}n {u 1} n t {A}n {u 1} n 16 1 = {Fτ 11 τ 1 } n 16 1 {Gu 1 u } n {u 1} n

9 3 : 17 Crank-Nicolson 1 t {A}n {u 1} n t {A}n {u 1} n 16 1 = {Fτ 11 τ 1 } n {Gu 1 u } n {u 1} n {u 1} n Adams : {A} n 16 16{u 1 } n = {A}n 16 16{u 1 } n t {Fτ 11 τ 1 } n 16 1 {Gu 1 u } n {u 1} n 16 1 t {Fτ 11 τ 1 } n 16 1 {Gu 1 u } n {u 1} n Gauss-Lobatto Gauss Gauss-Lobatto Gauss-Lobatto Gauss [81] Lagrange Gauss 4 4 Gauss-Lobatto 4 01 Matlab [0 ] [0 ] Dyna h = Gauss t u 10 u 0 9 Gauss 4 ; Lagrange-Euler Lagrange-Crank-Nicolson Lagrange-Adams t = 1 t = Gauss 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

= 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

10 t = 1 t = t = 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 Gauss The stress distribution calculated by Gauss integration with 9 points of three kinds of mesh Gauss t = 1 u 10 = 01 u 0 = 0 Fig 43 Gauss integration with 16 points t = 1 u 10 = 01 u 0 = 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

integration with 16 points of three kinds of mesh ã 45 Fig

Gauss-Lobatto integration with 16 points t = 1 u10 = 01 u0

stress distribution calculated by Gauss-Lobatto

${é mcþ?1lñ Ì $^ Lagrange 16 :Vng/¼ê E ¼ê Ün«ØÓ k é mcþ?$

11 13Ï ¾ : Ep ¼ê L ÀÂ-E K 19 ã : 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 : Gauss-Lobatto È t = 1 u10 = 01 u0 = 0 Gauss-Lobatto integration with 16 points t = 1 u10 = 01 u0 = 0 ã : 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

12 0 016 Lagrange16 6 [ ] [ 1 ] [D] : 014 [ ] PHAN-THIEN N A nonlinear network viscoelastic model [J] Journal of Rheology : [ 3 ] TANNER R I Engineering Rheology [M] London: Oxford University Press 000 [ 4 ] THIEN N P TANNER R I A new constitutive equation derived from network theory [J] Journal of Non Newtonian Fluid Mechanics 1977: [ 5 ] [M] : 1999 [ 6 ] LIN Q ZHOU J Superconvergence in high order galerkin finite element methods [J] Computer Methods in Applied Mechanics and Engineering : [ 7 ] [M] : 1997 [ 8 ] HELENBROOK B T On the existence of explicit hp-finite element methods using Gauss-Lobatto integration on the triangle [J] SIAM Journal on Numerical Analysis : [ 9 ] XU Y On Gauss-Lobatto integration on the triangle [J] SIAM Journal on Numerical Analysis : [10] [M] : 006 [11] HOU L NASSEHI V Evaluation of stress-effective flow in rubber mixing [J] Nonlinear Analysis Theory Methods and Applications : [1] BOYD J P A numerical comparison of seven grids for polynomial interpolation on the interval [J] Computers and Mathematics with Applications : :

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