MECH300H Introduction to Finite Element Methods Lecture 7 Finite Element Analysis of 2-D Problems

2-D Discretization Common 2-D elements:

2-D Model Problem with Scalar Function - Heat Conduction Governing Equation in W Boundary Conditions Dirichlet BC: Natural BC: Mixed BC:

Weak Formulation of 2-D Model Problem Weighted - Integral of 2-D Problem ----- Weak Form from Integration-by-Parts -----

Weak Formulation of 2-D Model Problem Green-Gauss Theorem ----- where nx and ny are the components of a unit vector, which is normal to the boundary G.

Weak Formulation of 2-D Model Problem Weak Form of 2-D Model Problem ----- EBC: Specify T(x,y) on G NBC: Specify on G where is the normal outward flux on the boundary G at the segment ds.

FEM Implementation of 2-D Heat Conduction – Shape Functions Step 1: Discretization – linear triangular element T1 Derivation of linear triangular shape functions: T3 Let T2 Interpolation properties Same

FEM Implementation of 2-D Heat Conduction – Shape Functions linear triangular element – area coordinates T1 A2 A3 A1 T3 T2 f1 f2 f3

Interpolation Function - Requirements Interpolation condition Take a unit value at node i, and is zero at all other nodes Local support condition fi is zero at an edge that doesn’t contain node i. Interelement compatibility condition Satisfies continuity condition between adjacent elements over any element boundary that includes node i Completeness condition The interpolation is able to represent exactly any displacement field which is polynomial in x and y with the order of the interpolation function

Formulation of 2-D 4-Node Rectangular Element – Bi-linear Element Let Note: The local node numbers should be arranged in a counter-clockwise sense. Otherwise, the area Of the element would be negative and the stiffness matrix can not be formed. f2 f1 f4 f3

FEM Implementation of 2-D Heat Conduction – Element Equation Weak Form of 2-D Model Problem ----- Assume approximation: and let w(x,y)=fi(x,y) as before, then where

FEM Implementation of 2-D Heat Conduction – Element Equation

Assembly of Stiffness Matrices

Imposing Boundary Conditions The meaning of qi: 3 3 1 1 1 2 2 3 3 1 1 1 2 2

Imposing Boundary Conditions Consider Equilibrium of flux: FEM implementation:

Calculating the q Vector Example:

2-D Steady-State Heat Conduction - Example AB and BC: CD: convection DA: 0.6 m C B 0.4 m y x