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# Introduction to FEM

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### Introduction to FEM

1. 1. INTRODUCTION TO FINITE ELEMENT METHOD (FEM)
2. 2. 1. INTRODUCTION ENGINEERING DEPARTMENT: To develop new products and/or manufacturing processes Product calculation: - Component dimensioning - Design verification - Material selection Preliminary design: concept design INDUSTRIALISATION: - Development of manufacturing Process calculation: - Process parameters drawings - Manufacturing process definition - Development of manufacturing tools - Production - Tooling design 2
3. 3. 1. INTRODUCTION: CALCULATION METHODS - Analytic method: - Consists on the use of analytic equation to represent the behaviour of a physical problem (As exercises solved by hand in previous chapters or other subjects, heat transfer, material resistance, dynamics, vibrations,…) - Advantages: relatively fast to be solved - Limitations: Hard to represent complex phenomena in real components, not always applicable. 3
4. 4. 1. INTRODUCTION: CALCULATION METHODS - Numeric methods (Finite Element Methods FEM) - To divide a complex problem into many simple problems (elements) - Problem solution by numeric methods (Newton-Raphson) by using iterations and increments. - Advantages: Capability to solve complex problems. - Limitations: Time expensive resolution method, the use of computers is required. 4
5. 5. 1. INTRODUCTION: FORMULATION TYPES - Implicit: - Tries to obtain the structural equilibrium for each time increment. - More sophisticate algorithms higher time increments (FASTER). - High precision - Convergence problems when solving non-linear phenomena: hard variations in boundary condition, material behaviour, loads, contacts,… - Explicit: - Does not need iterations, just time increments (Does not try to get the exact solution) - No convergence problems - Utilizes constant time increments - High calculation time - Recommendable to solve non-linear problems. 5
6. 6. 1. INTRODUCTION: APLICATIONS Solid mechanics: - Structural linear calculations (linear static, linear dynamics) IMPLICIT - Plasticity range calculation (no-linear quasi-static or dynamics) EXPLICIT Fluid mechanics: - Linear calculation (wind tunnel example) IMPLICIT - Non-linear calculations (atmospheric phenomena, turbulence, wind,...) EXPLICIT Thermodynamics: (linear problems-IMPLICIT) Multiphysics: thermo mechanic, thermo fluidic, fluid structure interaction… 6
7. 7. 1. INTRODUCTION: APLICATIONS Solid mechanics: Structural static calculation Design vs. FEM Set-up IMPLICIT Aluminium sheet bulge-test 7
8. 8. 1. INTRODUCTION: APLICATIONS Solid mechanics: eigenvalues Trunk door IMPLIT First mode17Hz Solid mechanics: Forming processes Punching EXPLICIT 8
9. 9. 1. INTRODUCTION: APLICATIONS Solid mechanics: Forming processes 9
10. 10. 1. INTRODUCTION: APLICATIONS Solid mechanics: Machining process Vc=300 m.min-1 Vc≥600 m.min-1 EXPLICIT 10
11. 11. 1. INTRODUCTION: APLICATIONS Fluid mechanics: linear and non-linear examples Hurricane simulation Air flow simulation F1 IMPLICIT EXPLICIT 11
12. 12. 1. INTRODUCTION: APLICATIONS Thermodynamics: Turbine heat transfer simulation IMPLICIT Tube and die temperature pattern simulation IMPLICIT 12
13. 13. 2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION - Definition of a finite element Geometrical definition (element shape): - Composed by nodes Physic definition (element type): - Degrees of freedom (DOF) - Analytic formulation of the element (mechanical field resolution, thermal fields,…) z 6 Degrees of Freedom (DOF) u v w θx θy θz θz u x θx w θy v y u: Linear displacement in X v: Linear displacement in y w: Linear displacement in z θx: Rotation with respect X θy : Rotation with respect Y θz : Rotation with respect Z 13
14. 14. 2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION - Element types:      Interpolation order: linear (1st order) y quadratic (2nd order) Linear interpolation v2 v 1 v1 v1 Nodal displacement vector of a first * θ1 δ= v order beam element in 2D de 2 Quadratic interpolation x V(x)=mx+b v3 v2 x V(x)=ax2+bx+c θ2 14
15. 15. 2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION - Mechanical field calculation: Motion differential equation . .. [M]{δ} + [C]{δ}+[k]{δ} ={Fext} [M]: Mass matrix [C]: Damping matrix [k]: Stiffness matrix - Mechanical field: STATIC Acceleration = 0 Velocity = 0 . .. [M]{δ} + [C]{δ}+[k]{δ} ={Fext} {δ}: Displacement vector . {δ}: Velocity vector .. {δ}: Acceleration vector {Fext}: External load vector [k]{δ} ={Fext} 5 unknowns and 5 equations 15
16. 16. 2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION - Stress field calculation through nodal displacement. Strain through local displacement INTERPOLATION  displacement at any point of the structure  δ1     . { δ }e = [ N 1 , N 2 ,...., N n ]    .   δn  Generalised Hooke’s law  Stress calculation through local strain (1 − ν)  ν σx   σ   ν  y  σz  E    0  = τ xy  (1 − 2 ⋅ ν) ⋅ (1 + ν)   0 τ yz      τ zx     0  ν (1 − ν) ν ν 0 0 0 0 (1 − 2ν) 2 0 0 ν (1 − ν) 0 0 0 0 0 0 0 0 0 (1 − 2ν) 2 0  ε 0  x   ε  0  y   ε  0  z     γxy  0  γyz    (1 − 2ν) γzx     2  0       εx       εy       εz      =  γ xy         γ yz       γ zx           ∂ ∂x 0 0 ∂ ∂y 0 0 ∂ ∂y ∂ ∂x 0 ∂ ∂z ∂ ∂z 0 0     0     ∂  u ∂z     . v 0   w     ∂  ∂y    ∂  ∂x        { ε }= [ ∂ ] { δ } ε3 ε1 ε2 16
17. 17. 2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION - INTERPOLATION FUNCTIONS: Determination of the displacement at any point of the structure. {δ }e = displacement vector at any point in a {δ}e = [N]{δ* }e determined element {δ } = nodal displacement vector of a determined * e {δ }e  δ1  = [N1 , N 2 ,..., N n ]   δ n  element [N ] = interpolation function matrix [Ni ] = interpolation function of the nodal displacement a determined node i. {δ i } = displacement vector at a determined node i. So, the displacement at any point of a determined element is obtained: {δ } = [N1 ]{δ1}+ [N 2 ]{δ 2 }+ ... + [N n ]{δ n } Where [Nk] represents the contribution of node k’s displacement in the total displacement of any determined point. 17
18. 18. 2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION - STIFFNESS MATRIX: { } { } In order to cause the nodal displacements δ * and consequently deform the element it is necessary the presence of nodal forces f * Definition: The stiffness coefficient Kij represents the necessary force to apply to a certain degree of freedom i to obtain an unitary displacement of the degree of freedom j being 0 the influence in the displacement of the rest of the degrees of freedom n f i = ∑ K ij ⋅ δ j j =1 n= number of DOF K11 ⋅ δ1 + K12 ⋅ δ 2 + ... + K1n ⋅ δ n = f1  K n1 ⋅ δ1 + K n 2 ⋅ δ 2 + ... + K nn ⋅ δ n = f n { } = [K ] {δ } Writing in matrix form: f * * e e e [K]e = Stiffness matrix of the element 18
19. 19. 2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION DETERMINATION OF THE STIFFNESS MATRIX OF A FINITE ELEMENT Relation between nodal forces an nodal displacements: {f }= [K ]{δ } * * Based on CAPLEYRON theory, the external work of the nodal forces is represented: w= 1 *T * {δ } {f } 2 w= 1 *T {δ } [K ]{δ * } 2 1 T σ The internal deformation energy caused by the nodal displacements: u = ∫ {ε } { }⋅ dv 2 As: {ε } = ∂{δ } = ∂[N ]{ * }= [B]{ * } 1 δ δ }T   δ * u = ∫ { [T [D ][ }⋅ dv δ * B]  B]{  2v T {σ } = [D]{ε } = [D][B]{δ * } T {ε } Being  w = u { } [K ]{δ } 1 * δ 2 T * { } 1 = δ* 2 [K ] = ∫ [B ]T [D][B ]⋅ dv T {σ }   * T  ∫ [B ] [D ][B ]⋅ dv  δ   v  { } STIFFNESS MATRIX v 19
20. 20. 2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION - DETERMINATION OF THE STIFFNESS MATRIX IN GLOBAL COORDINATES Transformation matrix a x [T ] = bx   cx  az   bz  cz   ay by cy From local coordinate system of the element To global coordinate system {δ }= [T ]{δ } * {f }= [T ]{f } {f }= [T ] {f } * * {f }= [K ]{δ } * * T * * {f }= [K ]{δ } * * * {f }= [T ] {f }= [T ] [K ]{δ }= [T ] [K ][T ]{δ } * T * T [K ] = [T ]T [K ][T ] * T * [K] in GLOBAL coord. system 20
21. 21. 2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION - INTERPOLATION FUNCTION OF A TRUSS ELEMENT Truss element: y - 2 node one-dimensional element (2DOF) u1 - Only allows to calculate tractive-compressive condition Nodal displacement vector {δ }= {u ,u } T 1 1 i u (x ) L u2 2 j x x , y , z Local axis i, j Element nodes 2 Determination of the interpolation function u1 , u2 Nodal displacements 2 G.D.L  1st order equation u ( x ) = a0 + a1.x u (0) = u1 u (l ) = u2 }u = a + a l} u1 = a0 2 a0 = u1 1 1 a1 = − u1 + u2 l l } 0 u1  u = [N1 , N 2 ]  u2  u1  1 0  a0   =   u 2  1 l  a1    1 a0  1 l  =  a1  l − 1 0 u1    1 u2   N1 = 1 − x l 0 1  − 1 1  u1  = 1 − x , x  u1  u = [1, x ]      u2    l  l  u2       l l N2 = x l 21
22. 22. 2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION - STIFFNESS MATRIX OF A TRUSS ELEMENT Truss element: 2DOF y u1 u2 1 2 L {δ } * x ,y x u1  =  u2  x ,y u   x ux ,y = [N1 N2 ] 1  = 1 − u2   l u1   ∂N1 ∂N2  u1   1 1  u1  ∂u  ∂  , ε x = =  [N1 , N2 ]  =   = − ,   u2   ∂x ∂x  u2   l l  u2  ∂x  ∂x        [B ] x N2 = l x l Local coordinate system [B ] Stiffness matrix obtaining formula: [k ] = ∫ [B] [D][B].dv [N ] = [N1 ,N2 ] N1 = 1 − x  u1    l  u2   T v 1  1  1 −  − 2 l  l  1 1  l2 l dx = E.S 1 − 1 K = ∫  E − , .dv = S.E ∫  . 1 1 1   l l l − 1 1   v  0 − 2 2   l   l     l [] 22
23. 23. 2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION - STIFFNESS MATRIX OF A TRUSS ELEMENT [k ] = EL.S −11  e  − 1 1  Stiffness matrix of TRUSS element in local coordinate system - RIGIDITY MATRIX OF A TRUSS ELEMENT IN GLOBAL COORD. SYSTEM  y y Displacement vector in global axis:  u1  x ϑ x cosθ u1   sinθ v   1 u  =   0  2 v2  X ,Y  0    − sinθ cosθ 0 0 cosθ 0 sinθ 0 0  u1  0  0      − sinθ  v1   cosθ  0  x ,y   23
24. 24. 2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION - STIFFNESS MATRIX OF A TRUSS ELEMENT IN GLOBAL COORD. SYSTEM Naming µ = senϑ   λ = cosϑ  [K ]e = [T ]T [K ] [T ] λ − µ 0 0  1   E.S  µ λ 0 0  0 [K ]e = l 0 0 λ − µ   − 1   0 0 µ λ  0  λ2 λµ  µ2 E.S  µλ [K ]e =  2 l −λ − λµ  − λµ − µ 2  − λ2 − λµ λ2 µλ − λµ  2 −µ  λµ   2 µ   0 − 1 0  λ 0 0 0  − µ  0 1 0  0  0 0 0  0 µ 0 λ 0 0 λ 0 −µ 0 0  µ  λ Stiffness matrix of a TRUSS element in GLOBAL coordinate system 24
25. 25. 2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION - INTERPOLATION FUNCTION OF A BEAM ELEMENT Beam element: y - 2 node unidimensional element (4DOF) v1 - Only allows to calculate behind loading condition z Beam element interpolation function: N v   1   =  dN1 ϑ    dx 2 3   x  x 1 − 3  + 2  , v   l l  = ϑ   x x2  −6 2 +6 , l l  N2 N3 dN 2 dN 3 dx dx v1  N4   ϑ1  dN 4     v2  dx    ϑ2    x 2 x3 x−2 + 2 , l l x x2 1− 4 + 3 2 , l l v2 ϑ1 ϑ2  x v1  ϑ    =  1 v2  ϑ2    {δ } T 2 3 x 2 x 3  v1   x  x 3  − 2  , − + 2    l l  ϑ1  l l   x x2 x x 2  v2  6 2 −6 3 , − 2 + 3 2   l l l l  ϑ2  25
26. 26. 2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION - INTERPOLATION FUNCTION OF A COMPLETE BEAM ELEMENT Complete Beam element: y - 2 node one-dimensional element (6DOF) v1 - Only allows to calculate behind loading condition Complete Beam element interpolation function:  u  N1    v  =  0 ϑ     0    x 1 − l u      v  = 0 ϑ       0  0 N3 dN3 dx 0 2 x x 1 − 3  + 2  l l x x2 −6 2 +6 l l 0 N4 dN4 dx N2 0 0 0 3 x2 x3 x −2 + 2 l l x x2 1−4 +3 3 l l v2 ϑ1 ϑ2 u 2 u1 x z u1   v1  0     ϑ1  N6     dN6  u2  dx  v2    ϑ2    0 N5 dN5 dx x l 0 0 {δ } T 0 2 x x 3  − 2  l l x x2 6 2 −6 3 l l u1  v   1 ϑ    =  1 u2  v2      ϑ2  3 u   1  0  v1   x 2 x 3  ϑ1    − + 2   l l  u2  x x 2  v  − 2 + 3 2  2  l l  ϑ   2 26
27. 27. 2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION - STIFFNESS MATRIX OF A BEAM ELEMENT BEAM element: 4 DOF dv dθ d 2v εx = −y = − y 2 being θ = dx dx dx Beam deflection ( { }) { } d 2v d2 ε x = − y 2 = − y 2 [N ] δ * = [B ] δ * dx dx {δ } * x, y  v1  θ    =  1 v  2 θ 2  x , y   [N ] = [N1 , N2 , N3 , N4 ] Local coordinate system [B] = − y − 62 + 12 x3 , − 4 + 6 x2 ,   l l l l 6 x 2 x − 12 3 , − + 6 2  l2 l l l  [k ] = ∫ [B]T [D][B]dv = v x  6 − l 2 + 12 l 3    − 4 + 6 x  l  l x 4 x 6 x 2 x l 2  6 2 = E∫   − 2 + 12 3 , − + 6 2 , 2 − 12 3 ,− + 6 2  dx ∫ y ds 6 x l l l l l l l l  s 0  − 12 3   l 2 l   2 x  − + 6 2  l   l 27
28. 28. 2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION - STIFFNESS MATRIX OF A BEAM ELEMENT Stiffness matrix of BEAM element in local coordinate system - STIFFNESS MATRIX OF A BEAM ELEMENT IN GLOBAL COORD. SYSTEM  y y Displacement vector in global axis:  u1  x ϑ x cosθ u1   sinθ v   1 u  =   0  2 v2  X ,Y  0    − sinθ cosθ 0 0 0 0 cosθ sinθ 0  u1    0  v 1    − sinθ  u2   cosθ  v2    x ,y 28
29. 29. 2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION - STIFFNESS MATRIX OF A BEAM ELEMENT IN GLOBAL COORD. SYSTEM Naming µ = senϑ   λ = cosϑ  λ − µ  0 [k ]e = E.Iz  0 0   0 12 2  l3 µ   − 12 λ  l3  − 6 µ 2 [k ]e = E.I z  l  − 12  3 µ2  l 12  3 λµ l − 6  l3 µ  12 2 λ l3 6 λ l2 12 λµ l3 − 12 2 λ l3 6 l2 µ 0 0 0 λ 0 0 0 0 0 0 0 1 0 0λ 0− µ 0 0 0 µ λ 0 0  0  0  0   0 0   0  0  0   0 1    0  4 l 6 12 2 µ µ 2 l l3 −6 − 12 λ λµ 2 l l3 2 −6 µ l l 0 12 l3 6 l2 0 − 12 l3 6 l2 12 2 λ l3 −6 µ l2 0 0 0 − 12 6 0 3 2 l l −6 4 0 l2 l 0 0 0 −6 12 0 3 l2 l −6 2 0 2 l l               4  l 0  6   l2   2   l   0   −6   l2    4   l  λ −µ 0 0 0 0 µ λ 0 0 0 0  0 0 0 0 0 0 0 0 1 0 0 0 0 0 0  λ − µ 0 µ λ 0  0 0 1  Stiffness matrix of a BEAM element in GLOBAL coordinate system 29
30. 30. 2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION - STIFFNESS MATRIX OF A COMPLETE BEAM ELEMENT IN GLOBAL COORD. SYSTEM Naming µ = senϑ   λ = cosϑ  12 EI 2 EA 2 µ + λ L3 L − 12 EI EA µλ + µλ 3 L L − 12 EI 2 EA 2 λ + µ L3 L 6 EI µ L2 6 EI λ L2 sy. 4 EI L = TT k e T Ke = − 12 EI 2 EA 2 µ − λ L3 L 12 EI EA µλ − µλ L3 L − 6 EI µ L2 12 EI EA µλ − µλ L3 L − 12 EI 2 EA 2 λ − µ L3 L 6 EI λ L2 6 EI µ L2 − 6 EI λ L2 2 EI L 12 EI 2 EA 2 µ + λ L3 L − 12 EI EA µλ + µλ L3 L 6 EI µ L2 12 EI 2 EA 2 λ + µ L3 L − 6 EI λ L2 4 EI L λ = cos θ µ = sin θ 30
31. 31. 2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION - DETERMINATION THE EQUIVALENT NODAL LOAD VECTOR In a real problem different type of external loads can be found: - Punctual forces - Moments - Distributed loads For FEM modelling all external load should be applied in the element nodes   - Punctual forces - Moments - Distributed loads NECESITY TO OBTAIN AN EQUIVALENT SYSTEM BASED IN NODAL LOADS f = f* 31
32. 32. 2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION - DETERMINATION THE EQUIVALENT NODAL LOAD VECTOR The external work due to all the external load applied to the system is given by 1 T w1 = ∫ {δ } { f }⋅ ds 2s By using the interpolation functions: w1 = { } [N ] { f }⋅ ds = 1 {δ } ∫ [N ] { f }⋅ ds 2 1 δ* 2∫ s T * T T T s Thus the work of the equivalent system can be written as: w2 = w1 = w2 { } {f } 1 * δ 2 T * { } ∫ [N ] { f }⋅ ds = 1 {δ } {f } 2 1 * δ 2 T * T T * s {f }= ∫ [N ] { f }⋅ ds T * s 32
33. 33. 2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION DETERMINATION OF THE STRESS/STRAIN CONDITION: Once, the nodal displacement vector of the studied system is solved the stress/strain condition at any point can be obtained. STEP 1: STRAIN DETERMINATION AT A CERTAIN POINT  δ1     . { δ }e = [ N 1 , N 2 ,...., N n ]    .   δn  Determination of the elongation at the selected point {ε } = [∂ ]{δ } {ε } = ∂[N ]{δ * }= [B]{δ * }   εx   ε    y  εz      = γ xy   γ yz      γ zx       ∂ ∂x 0 0 ∂ ∂y 0 ∂ ∂z 0 ∂ ∂y 0 ∂ ∂ ∂x ∂z 0 0   0   ∂ u  ∂ z    v  0     w ∂  ∂y  ∂  ∂x  Strain vector determination 33
34. 34. 2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION DETERMINATION OF THE STRESS/STRAIN CONDITION: STEP 2: STRESS DETERMINATION AT A CERTAIN POINT The relation between the strain and the stress in the linear elastic domain is given by the generalised Hooke’s law: [ ] [ ] [ ] 1 σ x − υ (σ y + σ z ) E 1 ε y = σ y − υ (σ z + σ x ) E 1 ε z = σ z − υ (σ x + σ y ) E εx = 2(1 + υ ) τ xy G E τ 2(1 + υ ) τ yz γ yz = yz = G E τ 2(1 + υ ) γ zx = zx = τ zx G E γ xy = τ xy = Generalized Hooke’s law LAMÉ ' s _ law : G = E  For isotropic materials 2(1 + υ ) 34
35. 35. 2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION DETERMINATION OF THE STRESS/STRAIN CONDITION: STEP 2: STRESS DETERMINATION AT A CERTAIN POINT The relation between the strain and the stress in the linear elastic domain is given by the generalized Hooke’s law: 0 0 0  υ (1 − υ ) υ  υ (1 − υ ) υ 0 0 0  ε  x   σ x   υ σ  0 0  ε  υ (1 − υ ) 0 y  y     1 − 2υ   0 σ z  0 0  0  ε z  E  0         =  2  τ xy  (1 − 2υ )(1 + υ )   γ xy    1 − 2υ    0 τ yz  0 0 0  0  γ yz         2   γ   τ zx     1 − 2υ   zx   0 0 0 0    0 2     {σ } = [D]{ε } 35