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FE Course Lecture II – Outline UCSD - 10/09/03 1. Review of Last Lecture (I) • • • • • Formal Definition of FE: Basic FE Concepts Basic FE Illustration Some Examples of the Second Order Equations in 1Dimension Some Examples of the Poisson Equation – . (ku) = f and Some Examples of Coupled Systems 2. Intro to 1-Dimensional FEs [Beams and Bars]. 1. Fluid Mechanics Problem 2. Heat Transfer (Thermal) Problem 3. Beam/Bar problem Finite Elements Principles and Practices - Fall 03 1-Dimensional Finite Elements 1. Stiffness and Load Vector Formulations for mechanical, heat transfer and fluid flow problems. The system equation to be solved can be written in matrix form as: [K] {D} = {q} Where [K] is traditional known as the ‘stiffness’ or ‘coefficient’ matrix (conductance matrix for heat transfer, flow-resistance matrix for fluid flow), {D} is the displacement (or temperature, or velocity) vector and {q} is the force (or thermal load, or pressure gradient) vector. Finite Elements Principles and Practices - Fall 03 A) For heat transfer problem in 1-dimensional, we have: fx = -Kdt/dx [Fourier Heat Conduction Equation] Q = -KAdt/dx (where Q=A fx) [KT}{T} = {Q} [applicable for steady-state heat transfer problems] kA 1 1 T1 q1 L 1 1 T q 2 2 Tbase=100oC 1 Tamb=20oC 5 5 Finite Elements Principles and Practices - Fall 03 B) For fluid flow problem in 1-dimensional, we have: md2u/dy2 – dp/dx = 0 [KF}{u} = {P} [applicable for steady-state flow problems]. P – pressure gradient 1 u1 q1 L 1 1 u q 2 2 m 1 Finite Elements Principles and Practices - Fall 03 C) For stress problem in 1-dimensional, we have: kd2u/dx2 – q = 0 [KF}{u} = {F}. F – joint force. EA 1 1 d1 q1 L 1 1 d q 2 2 u=uo = 0 How about for a tube under pure torsion? How will the coefficients look like? Finite Elements Principles and Practices - Fall 03 Review of Analysis Results. E.g., stress distribution. Exact Vs FE solution. Error Estimation. SOFTWARE-Specific Session: Intro to software-specific issues. h-elements, p-Elements, adoptive meshing. Build 1D problem on ANSYS. Go through all steps. Thermal problem on ANSYS Bar problem on ANSYS Flow problem on ANSYS/FEMLAB. Homework 1 and Reading Assignments. Finite Elements Principles and Practices - Fall 03