Finite Element Method - Chapter 1: Introduction to FEM - Nguyễn Thanh Nhã

ain idea of FEM 1.1. Fundamentals of FEM for structure Finite Element Method Department of Engineering Mechanics – HCMUT 2016 Main idea of FEM 1.1. Fundamentals of FEM for structure Finite Element Method Department of Engineering Mechanics – HCMUT 2016 Real model Discreted structure Displacement field in element e Displacement at nodes i, j k, l of element e Vector of shape functions “finite element” Main idea of FEM 1.1. Fundamentals of FEM for structure Finite Element Method Department of Engineering Mechanics – HCMUT 2016 FEM in Modeling and Simulation Main idea of FEM 1.1. Fundamentals of FEM for structure Finite Element Method Department of Engineering Mechanics – HCMUT 2016 Truss, bar, link Beam Plate Axisymmetric element Solid 3D Plane element Shell Main idea of FEM 1.1. Fundamentals of FEM for structure Finite Element Method Department of Engineering Mechanics – HCMUT 2016 Main idea of FEM 1.1. Fundamentals of FEM for structure Finite Element Method Department of Engineering Mechanics – HCMUT 2016 Main idea of FEM 1.1. Fundamentals of FEM for structure Finite Element Method Department of Engineering Mechanics – HCMUT 2016 CAD PRE-PROCESSING FEM POST-PROCESSING GOOD? YES NO START FINISH Main idea of FEM Department of Engineering Mechanics – HCMUT 2016 1.2. Strong form of boundary value problem 1.2. Strong form of boundary value problem Finite Element Method Department of Engineering Mechanics – HCMUT 2016 Displacement – Strain relationship 1 2 3 11 22 33 1 2 3 1 2 1 3 2 3 12 21 13 31 23 32 2 1 3 1 3 2 ; ; 1 1 1 ; ; ; 2 2 2 u u u X X X u u u u u u X X X X X X                                                 Cauchy formulation: 1X 2X 3X 1u 2u 3u O 1.2. Strong form of boundary value problem Finite Element Method Department of Engineering Mechanics – HCMUT 2016 1 2 3 11 22 33 1 2 3 1 2 1 3 2 3 12 21 13 31 23 32 2 1 3 1 3 2 ; ; 1 1 1 ; ; ; 2 2 2 u u u X X X u u u u u u X X X X X X                                                 Cauchy formulation In tensor mode 1,1 1,2 2,1 1,3 3,1 1,2 2,1 2,2 2,3 3,2 , , 1,3 3,1 2,3 3,2 3,3 1 1 ( ) ( ) 2 2 1 1 1 ( ) ( ) ( ) 2 2 2 1 1 ( ) ( ) 2 2 i j j i u u u u u u u u u u u u u u u u u                 Displacement – Strain relationship 1.2. Strong form of boundary value problem Finite Element Method Department of Engineering Mechanics – HCMUT 2016  11 22 33 12 23 132 2 2 T       In vector mode 1 11 2 22 1 333 2 12 3 2 123 13 3 2 3 1 0 0 0 0 0 0 2 0 2 2 0 0 X X u X u uX X X X X X                                                       11 12 13 22 23 33sym                1 2 3 Tu u u u ε D u : differential operatorD sym ε u Displacement – Strain relationship 1.2. Strong form of boundary value problem Finite Element Method Department of Engineering Mechanics – HCMUT 2016 Momentum balance equations 11 12 13 1 1 1 2 3 21 22 23 2 2 1 2 3 31 32 33 3 3 1 2 3 b u X X X b u X X X b u X X X                                            1 11,1 12,2 13,3 1 1 21,1 22,2 23,3 2 1 31,1 32,2 33,3 3 u b u b u b                                In tensor mode ,i i ij i ij j j u b b X         div  u σ b 11,1 12,2 13,321,1 22,2 23,3 , 31,1 32,2 33,3 ij jdiv                      σ Displacement – Strain relationship 1.2. Strong form of boundary value problem Finite Element Method Department of Engineering Mechanics – HCMUT 2016 Momentum balance equations  11 22 33 12 23 13 T     σ In vector mode 11 221 2 3 1 1 33 2 2 122 1 3 3 3 23 3 2 1 13 0 0 0 0 0 0 0 0 0 X X Xu b u b X X X u b X X X                                                               D σ b    11 12 13 22 23 33sym              σ  1 2 3 Tb b bb : differential operatorD 1.2. Strong form of boundary value problem Finite Element Method Department of Engineering Mechanics – HCMUT 2016 Stress – strain relationship (Hook’s law)  11 11 22 22 33 33 12 12 23 23 13 13 2 0 0 0 2 0 0 0 2 0 0 0 20 0 20 2sym                                                        σ C ε σ Cε In vector mode 1.2. Strong form of boundary value problem Finite Element Method Department of Engineering Mechanics – HCMUT 2016 System of equations div  u σ b σ Cε sym ε u ( )symdiv    u b C u System of equations Static problem ( )symdiv   C u b 0 1.2. Strong form of boundary value problem Finite Element Method Department of Engineering Mechanics – HCMUT 2016 Boundary conditions *( , ) ( , )t t   σ X n t X X Neumann Boundary Condition (Stress) Dirichlet Boundary Condition (Displacement) *( , ) ( , ) ut t  u X n u X X  *t u * 0u   Department of Engineering Mechanics – HCMUT 2016 1.3. Weak form of boundary value problem 1.3. Weak form of boundary value problem Finite Element Method Department of Engineering Mechanics – HCMUT 2016 Principle of Virtual Work Rewrite the strong form ( )symdiv    C u b u - In general, the solution of this differential equation is not possible analytically Some integral principles of mechanics + the principle of virtual displacements or the principle of virtual work - Approximation methods (FEM and others) are used to find an approximate solution. - Numerical methods do not solve directly the strong form of differential equation but solve its integral over the domain (weak form). + the principle of virtual forces + the principle of the minimum of total potential (static problem) + Hamilton's principle of continuum (dynamic problem) 1.3. Weak form of boundary value problem Finite Element Method Department of Engineering Mechanics – HCMUT 2016 - For structure problem, test function vector is chosen as the virtual displacement - Special test function has the following properties: - In the Principle of Virtual Work, the strong form of the differential equation as well as the static BCs are scalarly multiplied by a vector-valued test function and integrated over the domain u + satisfies the geometrical BCsu 0 uu X    + satisfies the field conditions sym u   + is infinitesimal + is arbitrary u u u u Principle of Virtual Work 1.3. Weak form of boundary value problem Finite Element Method Department of Engineering Mechanics – HCMUT 2016 Rewrite the balance momentum and the static BCs *  σ n t 0div   u σ b 0 Multiplication by the test function , integration over the volume and respectively over the Neumann boundary u *( ) ( ) 0dV div dV dA                    u u b u σ u σ n t Apply Gausses' theorem: ( )div dV dA dA                u σ u σ n u σ n *( ) : ( ) 0dV dV dA dA                           u u b u σ u σ n u σ n t Integral equation Principle of Virtual Work 1.3. Weak form of boundary value problem Finite Element Method Department of Engineering Mechanics – HCMUT 2016 *( ) : ( ) 0dV dV dA dA                           u u b u σ u σ n u σ n t : :  u σ ε σ *:dV dV dV dA                    u u ε σ u b u tWeak form equation In energy conservation form intdyn extW W W    dynW dV      u u int : TW dV dV         ε σ u D CD u * extW dV dA            u b u t Virtual work of inertial forces internal forces external forces Principle of Virtual Work 1.3. Weak form of boundary value problem Finite Element Method Department of Engineering Mechanics – HCMUT 2016 - Dirichlet BCs are strongly fulfilled - Neumann BCs and the equilibrium equation are fulfilled only weakly - Principle of virtual work only gives approximate solutions - The weak formulation forms the basis for FEM Principle of Virtual Work