MECH593 Introduction to Finite Element Methods Finite Element Analysis of 2-D Problems Dr. Wenjing Ye

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 ----- Ω 𝜕 𝜕𝑥 𝜅𝑤 𝜕𝑇 𝜕𝑥 𝑑𝑥𝑑𝑦= Γ 𝜅𝑤 𝜕𝑇 𝜕𝑥 𝑛 𝑥 𝑑𝑠 Ω 𝜕 𝜕𝑦 𝜅𝑤 𝜕𝑇 𝜕𝑦 𝑑𝑥𝑑𝑦= Γ 𝜅𝑤 𝜕𝑇 𝜕𝑦 𝑛 𝑦 𝑑𝑠 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 ----- Ω 𝜕𝑤 𝜕𝑥 𝜅 𝜕𝑇 𝜕𝑥 + 𝜕𝑤 𝜕𝑦 𝜅 𝜕𝑇 𝜕𝑦 −𝑤𝑄 𝑥,𝑦 𝑑𝑥𝑑𝑦 − Γ 𝑤 𝜅 𝜕𝑇 𝜕𝑥 𝑛 𝑥 + 𝜅 𝜕𝑇 𝜕𝑦 𝑛 𝑦 𝑑𝑠 =0 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 (T3) T1 Derivation of linear triangular shape functions: T2 Let T3 Interpolation properties Same

FEM Implementation of 2-D Heat Conduction – Shape Functions linear triangular element – local (area) coordinates T1 𝜙 1 = 1 𝑥 𝑦 2 𝐴 𝑒 𝑥 2 𝑦 3 − 𝑥 3 𝑦 2 𝑦 2 − 𝑦 3 𝑥 3 − 𝑥 2 = 𝐴 1 𝐴 𝑒 =𝜉 A2 A3 A1 T2 𝜙 2 = 1 𝑥 𝑦 2 𝐴 𝑒 𝑥 3 𝑦 1 − 𝑥 1 𝑦 3 𝑦 3 − 𝑦 1 𝑥 1 − 𝑥 3 = 𝐴 2 𝐴 𝑒 =𝜂 T3 T1 𝜉=1 𝜙 3 = 1 𝑥 𝑦 2 𝐴 𝑒 𝑥 1 𝑦 2 − 𝑥 2 𝑦 1 𝑦 1 − 𝑦 2 𝑥 2 − 𝑥 1 = 𝐴 3 𝐴 𝑒 =1−𝜉−𝜂=𝜁 𝜂=1 T2 T3 𝜉=0 f1 f2 f3 𝜂=0

FEM Implementation of 2-D Heat Conduction – Shape Functions quadratic triangular element (T6) – local (area) coordinates T1 T4 T6 𝜙 1 =𝜉 2𝜉−1 𝜙 2 =𝜂 2𝜂−1 𝜙 3 =𝜁 2𝜁−1 𝜙 4 =4𝜉𝜂 𝜙 5 =4𝜂𝜁 𝜙 6 =4𝜉𝜁 T2 T5 T3 T1 𝜉=1 𝜂=1 T4 T6 𝜉=0.5 T2 T3 T5 𝜉=0 𝜂=0 Serendipity Family – nodes are placed on the boundary for triangular elements, incomplete beyond quadratic

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 (Q4) Let h 𝜙 1 = 1 4 1−𝜉 1−𝜂 4 3 𝜙 2 = 1 4 1+𝜉 1−𝜂 x 𝜙 3 = 1 4 1+𝜉 1+𝜂 1 2 𝜙 4 = 1 4 1−𝜉 1+𝜂 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

Formulation of 2-D 4-Node Rectangular Element – Bi-linear Element (Q4) Physical domain (physical element) Reference domain (master element) h h 3 4 4 3 x x y 1 1 2 2 x 𝑥= 𝜙 1 𝑥 1 + 𝜙 2 𝑥 2 + 𝜙 3 𝑥 3 + 𝜙 4 𝑥 4 𝑦= 𝜙 1 𝑦 1 + 𝜙 2 𝑦 2 + 𝜙 3 𝑦 3 + 𝜙 4 𝑦 4

FEM Implementation of 2-D Heat Conduction – Element Equation Weak Form of 2-D Model Problem ----- Ω 𝑒 𝜕𝑤 𝜕𝑥 𝜅 𝜕𝑇 𝜕𝑥 + 𝜕𝑤 𝜕𝑦 𝜅 𝜕𝑇 𝜕𝑦 −𝑤𝑄 𝑥,𝑦 𝑑𝑥𝑑𝑦 + Γ 𝑒 𝑤 𝑞 𝑛 𝑑𝑠 =0 𝑇= 𝑗=1 𝑛 𝑇 𝑗 𝜙 𝑗 Assume approximation: and let w(x,y)=fi(x,y) as before, then Ω 𝑒 𝜕 𝜙 𝑖 𝜕𝑥 𝜅 𝜕 𝜕𝑥 𝑗=1 𝑛 𝑇 𝑗 𝜙 𝑗 + 𝜕 𝜙 𝑖 𝜕𝑦 𝜅 𝜕 𝜕𝑦 𝑗=1 𝑛 𝑇 𝑗 𝜙 𝑗 − 𝜙 𝑖 𝑄 𝑥,𝑦 𝑑𝑥𝑑𝑦 + Γ 𝑒 𝜙 𝑖 𝑞 𝑛 𝑑𝑠 =0 𝑗=1 𝑛 𝐾 𝑖𝑗 𝑇 𝑗 = Ω 𝑒 𝜙 𝑖 𝑄 𝑥,𝑦 𝑑𝑥𝑑𝑦 − Γ 𝑒 𝜙 𝑖 𝑞 𝑛 𝑑𝑠 where

Linear Triangular Element Equation 𝑗=1 𝑛 𝐾 𝑖𝑗 𝑇 𝑗 = Ω 𝑒 𝜙 𝑖 𝑄 𝑥,𝑦 𝑑𝑥𝑑𝑦 − Γ 𝑒 𝜙 𝑖 𝑞 𝑛 𝑑𝑠 where 𝒍 𝒊𝒋 is the length vector from the ith node to the jth node.

Assembly of Stiffness Matrices 𝐹 𝑖 𝑒 = Ω 𝑒 𝜙 𝑖 𝑄 𝑥,𝑦 𝑑𝑥𝑑𝑦 − Γ 𝑒 𝜙 𝑖 𝑞 𝑛 𝑑𝑠 = 𝑗=1 𝑛 𝑒 𝐾 𝑖𝑗 𝑒 𝑇 𝑗 𝑒 𝑈 1 = 𝑇 1 1 , 𝑈 2 = 𝑇 2 1 + 𝑇 1 2 , 𝑈 3 = 𝑇 3 1 + 𝑇 4 2 , 𝑈 4 = 𝑇 2 2 , 𝑈 5 = 𝑇 3 2 ,

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: CD: convection DA and BC: 0.6 m C B 0.4 m y x