FEA Course Lecture I – Outline 10/02/03 - UCSD Formal Definition of FEA:

1. FEA Course Lecture I – Outline • 10/02/03 - UCSD • Formal Definition of FEA: • An approximate mathematical analysis tool to study the behavior of a continua (or a system) to an external influence such as stress, heat, pressure, magnetic filed etc. This involves generating a mathematical formulation of the physical process followed by a numerical solution of the mathematics model. • History of FEA • Greek Mathematicians were the “first” to use the basic principles of FE to solve a physical problem (i.e., finding the area of a circle, or find the value of Pi) • A =  * R • C = 2R (More on this later). UCSD - FEA Course - Fall 2003

2. History of FEA (continued) • Archimedes had used a concept of ‘splitting a domain and re-assembling it’ to calculate the volume of a wedge by breaking it into a series of triangles. • Modern FEA – as we know it. • 1941 – Hrenikoff – Framework method for Plane elastic medium represented as collection of bars and beams; • 1943 – Courant solved a St. Venant’s Torsion Problem through an assemblage of triangular elements; • 1956 – Turner, Cough, Martin and Topp [UC Berkeley/Aerospace]; • 1960 –Clough was the first to use the formal name of “Finite Elements”. UCSD - FEA Course - Fall 2003

3. Basic Concept: Division of a given domain into a set of simple sub-domains called finite elements accompanied with polynomial approximations of solution over each element in terms of nodal values. Assembly of element equation with inter-element continuity of solution and balance of force considered. What are Finite Elements? Any geometric shape that allows computation of solutions (with approximation) or provides necessary relations among the values of solution at selected points (called nodes) of the sub-domain. UCSD - FEA Course - Fall 2003

4. Se q R B) Basic Illustration: Approximation of Circumference of a Circle UCSD - FEA Course - Fall 2003

5. FE Descitization: • Each line segment is an element. • Collection of these line segments is called a mesh. • Elements are connected at nodes. • Element Equations • He = 2R sin (q/2) • Assembly of Equations and Solution • Pn = Sigma He (n=1, N) • For q = 2/n, He = 2R sin (/n), Pn = n2Rsin(/n) • Convergence • As n approaches infinity, P = 2R • if x = 1/n • Pn = 2Rsin(x)/x • As n approaches infinity, x->0, • Limit (2Rsin(x)/x) as x->0 = limit (2Rcos(x)/1) = 2 • Error Estimation • Error, Ee = |Se – He| • = R[2/n – 2Sin(/n)] • Total Error = nEe • = 2R-Pn UCSD - FEA Course - Fall 2003

6. C) Some Examples of the Second Order Equations in 1- Dimension, -d/dx(adu/dx) = q for 0 < x < L UCSD - FEA Course - Fall 2003

7. D) Some Examples of the Poisson Equation – . (ku) = f [Both tables taken from J. N. Reddy's Book "Introduction to the Finite Element Method", J.N. Reddy, McGraw Hill Publishers, 2nd Edition, Page 71] UCSD - FEA Course - Fall 2003

8. E) Some Examples of Coupled Systems • Plane Elasticity • dsx/dx + dsxy/dy + fx = rd2u/dt2 • dsxy/dx + dsy/dy + fy = rd2v/dt2 • Flow of Viscous Incompressible Fluids [Navier Stokes Equations] • [Conservation of Linear Momentum] • rdu/dt - d/dx(2mdu/dx) - d/dy[m(du/dy + dv/dx ) ] + dP/dx – fx = 0 • rdv/dt - d/dx[m(dv/dy + du/dx ) ] - d/dy(2mdu/dy) + dP/dy – fy = 0 • [Conservation of Mass] • (du/dx + dv/dy) = 0 UCSD - FEA Course - Fall 2003

9. System Level Modeling System Level Modeling – Reduced-order macro models are converted into simulation templates where the physically correct result can be further optimized with system level trade-offs. Sample elector-mechanical library elements [Courtesy Coventor] UCSD - FEA Course - Fall 2003

10. SOFTWARE-Specific Session: • Intro to ANSYS. Basic file operations. Simple plate problem. • Intro to FEMLAB. Fluid mechanics problem. Critical look at results. • Intro to software-specific issues. h-elements, p-Elements • Homework 1 and Reading Assignments. UCSD - FEA Course - Fall 2003