Transcription of Method of Finite Elements I: Shape Functions
1 Adrian EggerMethod of Finite Elements I: Shape FunctionsWhy Shape Functions ? Discretization leads to solution in the nodes, but no information concerning the space in between Shape Functions required to approximate quantities between nodes Underlying assumption of how quantities are distributed in an element (stiffness, mass, element loads; displacements, strains, stress, internal forces, etc.) Geometry transformation3/24/20152 Adrian Egger | FEM I | FS 2015 ?What can Shape Functions be used for? to interpolate between discrete nodal quantities continuous across element = ( ) = 1( ) 1+ 2( ) 23/24/20153 Adrian Egger | FEM I | FS 2015 What can Shape Functions be used for? to discretize continuous quantities to nodal continuous across element discrete nodal quantities3/24/20154 = =0 ( ) = =0 011 111 021 121 ( ) = 2 212 2 212 1 1 2 2 Adrian Egger | FEM I | FS 2015 Alternative way to derive loading vector Recap: We calculate the solution in the nodes What is the influence of element loading in the nodes We must fix the element such that reaction forces develop in the nodal DOF we are interested in!
2 Equivalent to solving differential equation = 3/24/20155 Adrian Egger | FEM I | FS 2015 How to derive Shape Functions Interpolation Functions are generally assumed!(within certain parameters and restrictions) Minimal amount of continuity / differentiability Etc. Wish to implement this repetitive task as easily as possible, computer implementation using highly optimized numerical schemes, and thus natural coordinates (r,s,t) are introduced ranging from -1 < r,s,t< Egger | FEM I | FS 2015 Derivation of Shape Functions :Bar element (I) a relationship for r(x). We choose -1 < r < an appropriate Shape function A at each DOF by substituting values of r 3/24/20157 Adrian Egger | FEM I | FS 2015 Derivation of Shape Functions :Bar element (II) the previous into previous Shape Functions (as a function of r )3/24/20158 Adrian Egger | FEM I | FS 2015 Derivation of Shape Functions :Beam element (I) a relationship for r(x).
3 We choose 0 < r < an appropriate Shape function polynomial3/24/20159 Adrian Egger | FEM I | FS 2015 Derivation of Shape Functions :Beam element (II) an expression linking displacements and A at each DOF by substituting values of r 3/24/201510 Adrian Egger | FEM I | FS 2015 Derivation of Shape Functions :Beam element (III) the previous into previous Shape Functions (as a function of r )3/24/201511 Adrian Egger | FEM I | FS 2015 Questions3/24/201512 Adrian Egger | FEM I | FS 2015
