12/21/2001Numerical methods in continuum mechanics1 Continuum Mechanics On the scale of the object to be studied the density and other fluid properties will vary smoothly from one point to another Gas at normal pressure and low pressure Perfect fluid Newtonian fluid Shear forces are linearly proportional to velocity gradients Basic concepts

12/21/2001Numerical methods in continuum mechanics2 Conservation of Mass equation Conservation of Momentum Steady state equations Basic Equations

12/21/2001Numerical methods in continuum mechanics3 Simplified general steady state equation Assumptions in order to get simple equations Simplified General Equations

12/21/2001Numerical methods in continuum mechanics4 Finite Element Approach Methods of weighted residuals Terminology Trial function Test function Collocation Method w i = ( x – x i ), i = 1,2,………..,n where x i is a point within the domain Least Squares method w i = i =1,2 ……n ; where R is the residual and a i is an unknown coefficient in the trial function Galerkin’s approach w i = i= 1,2, ………….n ;where is the selected trial function

12/21/2001Numerical methods in continuum mechanics5 Galerkin and Weak Formulation Relation between test and trial functions Orthogonality - Minimum Error Definition of Weak formulation Advantages

12/21/2001Numerical methods in continuum mechanics6 Piecewise polynomial interpolation X i-1 XiXi X i+1 x Shape functions Piecewise polynomials and shape functions

12/21/2001Numerical methods in continuum mechanics7 Isoparametric Elements Four node element

12/21/2001Numerical methods in continuum mechanics8 Shape functions for bilinear isoparametric element These shape functions are defined in terms of the normalized natural domain Local and Global coordinates

12/21/2001Numerical methods in continuum mechanics9 Mapping from Local to Global Node to node Computing derivatives

12/21/2001Numerical methods in continuum mechanics10 Gauss Legendre quadrature Function description ab f(x) Sampling points weighting coefficients 2n-1 ; n

12/21/2001Numerical methods in continuum mechanics11 Solution to simplified equations Quadrilateral isoparametric element Same shape functions and mapping discussed before Element jacobi Assembling matrices from element to Global Global Jacobian Iterative loop for non linear term

12/21/2001Numerical methods in continuum mechanics12 Numerical Terminology & Formulation [201], [202], [211], [222], [212], [221]

12/21/2001Numerical methods in continuum mechanics13 Numerical Terminology & Formulation (i, j) (i+1, j+1) (i-1, j) (i+1, j-1)(i-1, j-1) (i, j-1) (i, j+1) (i+1, j) (i-1, j+1)

12/21/2001Numerical methods in continuum mechanics14 Tasks ahead To develop numerical solution to Navier Stokes equation in all coordinate systems Solve Navier stokes with non linear part with all possible boundary conditions