Finite Element Method - Chapter 2: Plane problem - Nguyễn Thanh Nhã

artment of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã Plane problems Mechanical problem - Plane stress state - Plane strain state - Axisymmetric state Element form - Triangular element - Quadrilateral element Polynomial degree of the shape functions - Linear - Quadratic - Cubic - Type of the shape functions - Lagrange shape func. - serendipity shape func. - hierarchical shape func. 2.1. Basis equation of plane problem Finite Element Method Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã Geometry Plane structure / 2D continuum: - degeneration of the 3D continuum onto a middle surface with a thickness parameter Plane stress model: - Small thickness relative to other dimensions of the middle surface Plane strain model: - Infinite dimension in an direction Geometry description - Position vector of a material point on the middle surface  1 2 T X XX or   T X YX 2.1. Basis equation of plane problem Finite Element Method Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã Kinetics and Kinematics Fundamental assumption for realization of the degeneration of the 3D to 2D Plane stress VS plane strain Plane stress model 11 12 21 22 0 0 0 0 0              σ 33 23 13 0     11 22 12             σ 33 0 Note: 2.1. Basis equation of plane problem Finite Element Method Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã Kinetics and Kinematics Fundamental assumption for realization of the degeneration of the 3D to 2D Plane stress VS plane strain Plane strain model 11 12 21 22 0 0 0 0 0              ε 33 23 13 0     11 11 22 22 12 122                           ε 23 13 0  Note: 33 0  2.1. Basis equation of plane problem Finite Element Method Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã Second fundamental assumption about displacement field 1 1 1 2 2 2 1 2 ( , ) ( , ) u u X X u u X X    Displacement vector of a body:  1 2 1 1 2 2 1 2( , ) ( , ) ( , ) T X X u X X u X Xu Acceleration vector of a body:  1 2 1 1 2 2 1 2( , ) ( , ) ( , ) T X X u X X u X Xu Variational of disp. vector:  1 2 1 1 2 2 1 2( , ) ( , ) ( , ) T X X u X X u X X  u Dirichlet BCs:    *, , uu t t  X u X X Neumann BCs:  1 2 T b bb * * * 1 2 T t t   t Note: Plane stress: 3 0u Plane strain:3 0u  Kinetics and Kinematics 2.1. Basis equation of plane problem Finite Element Method Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã Strain – displacement relationship sym ε u In vector mode: ε D u 1 11 1 22 22 12 2 1 0 0 2 X u uX X X                                     1 2,X X  Strain components are constant along the thickness Kinetics and Kinematics 2.1. Basis equation of plane problem Finite Element Method Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã Constitutive equation Plane stress state esσ C ε 2 1 0 1 0 1 1 2 es E sym                   C Plane strain state evσ C ε    1 0 1 0 1 1 2 1 2 2 ev E sym                       C 2.1. Basis equation of plane problem Finite Element Method Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã Principle of virtual work The weak form for 3D structural problem *dV dV dV d                      u u ε σ u b u t The principle of virtual work for 2D problem /2 /2 3 3 /2 /2 /2 /2 * 3 3 /2 /2 h h A h A h h h A h h dX dA dX dA dX dA dX d                              u u ε σ u b u t Because all integrants are functions only of X1 and X2 * A A A hdA hdA hdA hd                   u u ε σ u b u t Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã 2.2. Finite element discretization 2.2. Finite element discretization Finite Element Method Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã Partitioning into finite elements Partitioning of 2D continuum Ω into finite domains Ωe 1 NE e e    i jwith if i j    The principle of virtual work holds for each finite element e int e e e dyn extW W W    Quadrilateral element Trianglular element 2.2. Finite element discretization Finite Element Method Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã Shape function of plane elements Shape function of plane elements: 2D polynomials        1 2 1 2 1 2 0 0 , , p p i jij i j           u u Complete polynomials for Triangular elements 1p  2p  3p  4p  1 1 2 2 1 2 21 2  5 1 4 1 2  3 2 1 2  2 3 1 2  1 4 1 2  5 2 2 2 1 2  3 1 2  3 1 3 1 2  4 2 2 1 2  2 1 2  3 1 3 2 0p  1p  2p  3p  4p  5p  2.2. Finite element discretization Finite Element Method Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã Shape function of plane elements Complete Bi-polynomials for Quadrilateral elements        1 2 1 2 1 2 0 0 , , p p i ji j i j                       u u Complete polynomials of degree p are derived by multiplication of two 1-D polynomials of degree p 1p  2p  3p  4p  1 1 2 2 1 2 21 2  5 1 4 1 2  3 2 1 2  2 3 1 2  1 4 1 2  5 2 2 2 1 2  3 1 2  3 1 3 1 2  4 2 2 1 2  2 1 2  3 1 3 2 0p  1p  2p  3p  4p  5p  Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã 2.3. Quadrilateral plane element 2.3. Quadrilateral plane element Finite Element Method Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã 1 2 1 1       Bilinear Lagrange elements Geometry – Isoparametric mapping 1 2 1 2 1 1      1 2 1 1       1 2 1 1        1 2 34 1X 2X 1eX 2eX 4eX 1 2 3 4 3eX 1 2 T X X   X Physical coordinates Natural coordinates An arbitrary materal point within the quadrilateral element is indentifiable by its physical and natural corrdinaters 1 2 T     ξ 2.3. Quadrilateral plane element Finite Element Method Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã The ansatz function of a node i can be derived by multiplication of the 1D Lagrange polynomials Bilinear Lagrange elements Ansatz functions    1 1 2 2 i iN and N  corresponding to this node Lagrange polynomials of order p: 1 1 1 1 1 1 1 ( ) kp i k i k k i N             1 1 ( ) 0 i k if i k N if i k                    1 1 1 1 1 2 1 1 2 2 1 2 1 2 1 2 , 1 1 1 1 1 1 2 2 4 N N N N                   ξ    11 1 1 1 1 2 N       12 2 2 1 1 2 N    1D shape functions for node 1: 2D shape function for node 1: 2.3. Quadrilateral plane element Finite Element Method Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã Bilinear Lagrange elements Bilinear shape functions Shape func. for node 1 1 2 4 3 2 1 1 1 1  1 1 2,N   12 2N   11 1N  1 2 4 3 2 1 1 1 1  2 1 2,N    21 1N   12 2N  Shape func. for node 2               1 1 1 1 1 2 1 1 2 2 1 2 1 2 1 2 , 1 1 1 1 2 2 1 1 4 N N N N                   ξ 2.3. Quadrilateral plane element Finite Element Method Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã Bilinear Lagrange elements Shape func. for node 3 Shape func. for node 4 1 2 4 3 2 1 1  3 1 2,N   1 2 4 3 2 1 1  4 1 2,N   Bilinear shape functions 2.3. Quadrilateral plane element Finite Element Method Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã Bilinear Lagrange elements 1 2 4 3 2 1 1e iu 2e iu 3e iu 4e iu  1 2,iu   1 2 4 3 2 1 1  1 1 2,N   1 2 4 3 2 1 1  2 1 2,N   1 2 4 3 2 1 1  3 1 2,N   1 2 4 3 2 1 1  4 1 2,N  Bilinear shape functions      1 1 2 1 1 1 4 N    ξ      2 1 2 1 1 1 4 N    ξ      3 1 2 1 1 1 4 N    ξ      4 1 2 1 1 1 4 N    ξ 2.3. Quadrilateral plane element Finite Element Method Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã Bilinear Lagrange elements Derivations of ansatz functions wrt the natural coords       1 1 ,1 2 1 1 1 4 N N         ξ ξ       2 2 ,1 2 1 1 1 4 N N        ξ ξ       3 3 ,1 2 1 1 1 4 N N        ξ ξ       4 3 ,1 2 1 1 1 4 N N         ξ ξ       1 1 ,2 1 2 1 1 4 N N         ξ ξ       2 2 ,2 1 2 1 1 4 N N         ξ ξ       3 3 ,2 1 2 1 1 4 N N        ξ ξ       4 4 ,2 1 2 1 1 4 N N        ξ ξ 2.3. Quadrilateral plane element Finite Element Method Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã Bilinear Lagrange elements Geometry 1 2 1 1       1 2 1 2 1 1      1 2 1 1       1 2 1 1        1 2 34 1X 2X 1eX 2eX 4eX 1 2 3 4 3eX           1 1 2 2 3 3 4 4 1 2 1 2 1 2 1 2 1 2 1 2 3 4 , T e e e e e e e e e T T T T T e e e e X X X X X X X X X X         X X X X X 1 2 ei ei ei X X        X           4 1 2 1 1 , NN ei i ei i i i N N        X X ξ X ξ X ξ X ξ Approximation of geometry 2.3. Quadrilateral plane element Finite Element Method Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã Bilinear Lagrange elements                         1 1 1 2 2 1 1 2 3 4 2 1 2 1 2 3 4 3 2 1 3 2 4 1 4 2 0 0 0 0 0 0 0 0 e e e e e e e N e e X X X X N N N N X X N N N N X X X X                                    ξX ξ X ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ Approximation of geometry     eNX ξ ξ X 2.3. Quadrilateral plane element Finite Element Method Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã Bilinear Lagrange elements 1 2 1 2 XX X X X X                          Jacobi transformation - The calculation of strain vector ε requires that the displacement components be devivated wrt physical cords X - Displacement components and approximation of the position vector are expressed as functions of natural coordinates  Transformation relation between derivatives wrt physical and natural coords 1,2; 1,2   1 2 1 1 1 1 1 2 2 2 2 2 X X X X X X                                                             J ξ ξ X 2.3. Quadrilateral plane element Finite Element Method Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã Bilinear Lagrange elements Jacobi transformation The rule for derivating functions in natural coords wrt physical cords is obtained by inversion of the Jacobi matrix   0J ξ 1      J ξ X ξ Inversion of the Jacobi matrix     2 2 2 11 1 1 2 1 1 X X X X                     J ξ J ξ   1 2 1 2 1 2 2 1 X X X X               J ξDeterminant of the Jacobi matrix 2.3. Quadrilateral plane element Finite Element Method Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã Jacobi transformation The derivatives of physical cords wrt natural coords:    , 1 NN ei i i X X N          ξ ξ The derivatives of function in natural coords wrt physical coords    , ,( ) eX        X ξ ξ N ξ X 1 2 1 2X X X X                              Definition of the inverse of the Jacobi matrix 1 2 1 1 1 1 1 2 2 2 2 2 X X X X X X                                                       1      J ξ X ξ Bilinear Lagrange elements 2.3. Quadrilateral plane element Finite Element Method Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã Jacobi transformation Transformation of a surface element dA in natural coords  1 2 1 2dA dX dX d d   J ξ 2dX 1dX 1 2dA d d X X 1dξ 2dξ 1 2 1 2 dA d d d d     ξ ξ Physical coordinates Natural coordinates (see more in text book!) Bilinear Lagrange elements 2.3. Quadrilateral plane element Finite Element Method Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã Approximation of element quantities Approximations of displacements, variations and second time derivative of displacements:       e u ξ u ξ N ξ u       e   u ξ u ξ N ξ u       e u ξ u ξ N ξ u 1 1 2 2 3 3 4 4 1 2 1 2 1 2 1 2 T e e e e e e e e eu u u u u u u u   u 1 1 2 2 3 3 4 4 1 2 1 2 1 2 1 2 T e e e e e e e e eu u u u u u u u           u 1 1 2 2 3 3 4 4 1 2 1 2 1 2 1 2 T e e e e e e e e eu u u u u u u u   u Bilinear Lagrange elements 2.3. Quadrilateral plane element Finite Element Method Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã Approximation of element quantities Approximations of displacements, variations and second time derivative of displacements:       e u ξ u ξ N ξ u       e   u ξ u ξ N ξ u       e u ξ u ξ N ξ u 1 1 2 2 3 3 4 4 1 2 1 2 1 2 1 2 T e e e e e e e e eu u u u u u u u   u 1 1 2 2 3 3 4 4 1 2 1 2 1 2 1 2 T e e e e e e e e eu u u u u u u u           u 1 1 2 2 3 3 4 4 1 2 1 2 1 2 1 2 T e e e e e e e e eu u u u u u u u   u 1 2 3 4 1 1 eu 1 2 eu 4 2 eu 4 1 eu 3 2 eu 3 1 eu 2 2 eu 2 1 eu 2u 1u Bilinear Lagrange elements 2.3. Quadrilateral plane element Finite Element Method Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã Strain vector approximation of element 1 2 1 1 1 2 11 11 2 22 22 1 2 2 12 1 2 1 2 2 1 2 2 1 1 1 2 0 0 2 X X u uX X X X X X                                                                               D ε D u              e e    ε ξ ε ξ D ξ u D ξ N ξ u B ξ u Displacements – strain relationship: Strain vector approximation Bilinear Lagrange elements 2.3. Quadrilateral plane element Finite Element Method Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã Strain vector approximation of element     eε ξ B ξ u      B ξ D ξ N ξ   :B ξ Differential operator (B-operator) :eu Linear mapping of element displacement vector B-operator for element node i                   1 2 ,1 ,2 1 1 1 2 ,1 ,2 2 2 1 2 1 2 ,1 ,2 ,1 ,2 2 2 1 1 0i i i i i i i i i N N X X N N X X N N N N X X X X                                        ξ ξ B ξ ξ ξ ξ ξ ξ Bilinear Lagrange elements 2.3. Quadrilateral plane element Finite Element Method Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã Approximation of internal virtual work The internal virtual work Approximation of internal virtual work       1 1 int 1 2 1 1 eW C hd d         ε ξ ε ξ J ξ       1 1 int 1 2 1 1 e e T T eW C hd d         u B ξ B ξ u J ξ Since ,e eu u do not depend on ξ       1 1 int 1 2 1 1 e e T T e e e eW C hd d               u B ξ B ξ J ξ u u k u 2.3. Quadrilateral plane element Finite Element Method Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã Approximation of internal virtual work Element stiffness matrix of a bilinear plane quadrilateral element Using Gaussian integral (2 x 2 integration points)       1 1 1 2 1 1 e T TC hd d      k B ξ B ξ J ξ         2 2 1 2 1 2 1 2 1 2 1 1 , , , ,e i j T i j i j i j i j i j h            k B CB J 2.3. Quadrilateral plane element Finite Element Method Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã Approximation of dynamic virtual work The internal virtual work of inertial forces Approximation virtual work of inertial forces       1 1 1 2 1 1 e dynW hd d          u ξ u ξ J ξ       1 1 1 2 1 1 e e T T e e e e dynW hd d                u N ξ N ξ J ξ u u m u Element mass matrix of a bilinear plane quadrilateral element       1 1 1 2 1 1 e T T hd d       m N ξ N ξ J ξ Using Gaussian integral (2 x 2 integration points)         2 2 1 2 1 2 1 2 1 2 1 1 , , , ,e i j T i j i j i j i j i j h             m N N J 2.3. Quadrilateral plane element Finite Element Method Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã Approximation of virtual external work Natural coordinatesPhysical coordinates 1X 2X 1 2 3 4 1 2 T X X   X  *2 1 2,t X X 1 2 3 4  1 2,X Xb 1 2 1 1       1 2 1 2 1 1      1 2 1 1       1 2 1 1        1 2 34 1 2 T     ξ  *2 1 2,t    1 2, b 2.3. Quadrilateral plane element Finite Element Method Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã Approximation of virtual external work Element vector of volume forces Volume loads       1 1 1 2 1 1 e e T e e ext pW hd d          u N ξ b ξ J ξ u r       1 1 1 2 1 1 e T p hd d       r N ξ b ξ J ξ 1 2 1 2 34  1 2, b 2.3. Quadrilateral plane element Finite Element Method Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã Approximation of virtual external work Example for element boundary THREE Boundary loads 4 4 * * 1 1 e ie ext i ext i i W hd hd W                  u t u t    3 *1 1 3,1 ,1eextW hd          u t 1X 2X 1 2 3 4  *2 1 2,t X X 1 2 3 4 3d 1dX 2dX 1 2 1 1       1 2 1 2 1 1      1 2 1 1       1 2 1 1        1 2 34  *2 1,1t  3d 2.3. Quadrilateral plane element Finite Element Method Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã Approximation of virtual external work Using Pythagore theorem Boundary loads 2 2 2 3 1 2d dX dX   The toal differentials 1 2 1 1 2 1 X X X dX d d d                    The transformation relation of line element 3d 2 2 2 21 2 3 1 1 1 X X d d                        3 3 1 1,1d d    J 1 2 2 2 1 2 3 1 1 X X d                       2.3. Quadrilateral plane element Finite Element Method Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã Approximation of virtual external work Boundary loads Positon vector of the element edge 3d       4 4 1 1 1 1 3 ,1 ,1 ,1ei ei i i X N N       X X Derivative of the position vector wrt natura coord 1       4 1 ,1 1 ,1 1 31 ,1 ,1 ,1ei i e i N           X X N X Horizontal & Vertical component of the edge element 3d      1 3 3 4 41 ,1 1 ,1 1 1 1 ,1 ,1 ,1e e X dX d X N X N d                 3 4 1 1 1 2 2 e eX X d         2.3. Quadrilateral plane element Finite Element Method Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã Approximation of virtual external work Boundary loads Jacobi determinant  3 1,1J   1/2 2 2 3 4 3 4 3 1 1 1 2 2 1 1 1 1 ,1 2 2 2 2 e e e eX X X X                    J     1/2 2 2 3 4 3 4 1 1 2 2 1 2 e e e eX X X X        External virtual work on the boundary 3      3 1 * 1 1 3 1 1 1 ,1 ,1 ,1extW h d          u t J       1 * 1 1 3 1 1 3 1 ,1 ,1 ,1e T e enh d           u N t J u r 2.3. Quadrilateral plane element Finite Element Method Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã Approximation of virtual external work Boundary loads The consistent equivalent load of the boundary load       1 * 3 1 1 3 1 1 1 ,1 ,1 ,1e Tn h d      r N t J External virtual work of an element 4 4 4 1 1 1 i e e e e e e ext ext ni ni n i i i W W                u r u r u r 2.3. Quadrilateral plane element Finite Element Method Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã Approximation of virtual external work Exercise: Derive the boundary load vectors for the three edges: 1 2 4, ,  