Introduction to finite elements April 13 2009 Outline Introduction to Finite Elements Review midterm Introduction to finite elements Basic approaches to finite elements Larry Caretto Mechanical Engineering 501AB Will start material originally scheduled for April 22 Parallel reading for this week is pages 711 to 739 in Hoffman Seminar in Engineering Analysis April 13 2009 Example application in one dimension 2 Midterm Results Sketch the wave equation solution u x t at times ct 0 1 0 2 and 0 3 for f x u x 0 as shown at the right Number of students 12 Maximum possible 100 Mean 45 1 Median 43 5 Standard deviation 28 2 Grade distribution 1 f x 0 5 0 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 Shapes at ct 0 1 1 u x t Functions 0 5 u x t 0 f x ct f x ct 0 5 0 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 x 1 0 0 1 0 2 0 3 0 4 0 5 x 0 6 0 7 0 8 0 9 1 3 Problem One II Outline solution for potential u x y t 2u 2u u 2 2 0 x L 0 y H t 0 t y x u 0 y t 0 u L y t U 1 u x 0 t 0 These are inverse of original initial condition Shapes at ct 0 3 Functions 0 f x ct w1 x y and w2 x y each have only one nonhomogenous boundary condition 0 5 0 3 0 4 0 5 0 6 u x H t U 2 v x y t is diffusion equation solution with homogenous boundary conditions w x y is Laplace equation solution requiring superposition w x y w1 x y w2 x y 0 5 0 2 u x y 0 U 0 u x y t v x y t w x y 1 0 1 4 Problem Two For ct 0 3 must account for periodic extensions of initial conditions for x 0 and x 1 entering region 0 x 1 0 1 Shapes at ct 0 2 1 f x ct f x ct 2 10 14 26 35 40 47 65 71 73 75 83 f x ct u x t 0 9 x u x t f x ct f x ct 2 Functions Problem One 0 7 0 8 0 9 1 x 5 ME 501B Engineering Analysis 6 1 Introduction to finite elements April 13 2009 Problem Two II Problem Two II Solutions in notes v x y t Cnme n 2 m 2 t L H n 1 m 1 4U 1 w2 x y n 0 n x m y sin sin L H 2n 1 x 2n 1 y sin sinh L L w1 x y n 0 2n 1 sinh 2n 1 H L 4U 2 2n 1 y 2n 1 x sin sinh H H 2n 1 L 2n 1 sinh H Need to use solutions for w x y to find coefficients in v x y t solution C pq Solution for w2 x y found by substituting x and y L and H and U1 for U2 in w1 solution 4 HL HL p x q y sin dxdy L H U 0 w1 x y w2 x y sin 00 7 8 Problem Three Problem Three II Replace each fk in equation below by its Taylor series to order of the error f i fi 2 After the algebra f i 2 8 f i 1 8 f i 1 f i 2 2 8 8 2 f i h 32 8 8 32 f i h 5 L f i O h 4 12h 12h 5 12h f i 2 8 f i 1 8 f i 1 f i 2 12h Get first derivative of ex at x 0 for h 0 1 and h 0 05 and find the order of the error f 2h 2 f i 2h 3 f i 2h 4 f i 2h 5 f i f i 2h i 2 3 4 5 f h 2 f i h 3 f i h 4 f i h 5 f i 1 f i f i h i 2 3 4 5 f h 2 f i h 3 f i h 4 f i h 5 f i 1 f i f i h i 2 3 4 5 f i 2 f i f i 2h 2 3 4 e 0 2 0 1 8e 0 0 1 8e 0 0 1 e 0 2 0 1 0 999996662696 12 0 1 f i e0 2 0 05 8e 0 0 05 8e 0 0 05 e 0 2 0 05 0 99999979160465 12 0 05 n f i 2 h f 2h f 2h f 2 h i i i 2 3 4 5 f i 5 log 2 log 1 log 2 08395 x10 7 log 3 3373 x10 6 4 001 log h2 log h1 log 0 05 log 0 1 9 Why Finite Elements Finite Element Methods A different approach for converting differential equations into algebraic equations More complicated approach than finite differences Capable of handling complex geometry more accurately than finite differences Will not cover finite volume a melding of the two Uses concept of interpolation polynomials Introduces gradients naturally 11 ME 501B Engineering Analysis 10 Designed for 2D and 3D geometries Can use for 1D case as example Basic idea is to divide region into small elements line area volume Use interpolating polynomial for each element Represent both geometry independent variables and dependent variable Polynomials called basis functions or shape functions 12 2 Introduction to finite elements April 13 2009 Finite Element Methods II Finite Element Analyses Start analysis for region Look at set of small elements in region Assemble analyses for individual elements is into a set of nodal equations for the entire region Result is set of algebraic equations for the dependent variable at nodes that are points on elements Converts differential equations into a set of algebraic equations at distinct points How do we represent differential equation in terms of polynomials Variational approach Rayleigh Ritz method formulates problem as the maximum or minimum of a function integral Apply directly to problems in solid mechanics governed by a variational principle Hamilton s principle Method of weighted residuals Applies to general differential equations Least squares approaches 13 14 Method of Weighted Residuals MWR uses shape functions N i x or Ni x for each node u ui i x in region to write approxi 1 imate solution for true u x Have to find all the ui values Write differential equation as L u b 0 Weighting functions wi x for individual nodes in region that satisfy this equation w L u b d 0 i i 1 K N Choice for wi determines final form of algebraic equations solved 2u 2u wi 2 2 Q dydx 0 i 0 …