FEA Course Lecture I Outline 100203 UCSD A

FEA Course Lecture I – Outline 10/02/03 - UCSD A) 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 1. 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 1

History of FEA (continued) 2. Archimedes had used a concept of ‘splitting a domain and reassembling it’ to calculate the volume of a wedge by breaking it into a series of triangles. 3. 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 2

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 3

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

1. FE Descitization: • Each line segment is an element. • Collection of these line segments is called a mesh. • Elements are connected at nodes. 2. Element Equations He = 2 R sin (q/2) 3. Assembly of Equations and Solution Pn = Sigma He (n=1, N) For q = 2 /n, He = 2 R sin ( /n), Pn = n 2 Rsin( /n) 4. Convergence • As n approaches infinity, P = 2 R • if x = 1/n Pn = 2 Rsin( x)/x • As n approaches infinity, x->0, • Limit (2 Rsin( x)/x) as x->0 = limit (2 Rcos( x)/1) = 2 5. Error Estimation Error, Ee = |Se – He| = R[2 /n – 2 Sin( /n)] Total Error = n. Ee = 2 R-Pn UCSD - FEA Course - Fall 2003 5

C) Some Examples of the Second Order Equations in 1 - Dimension, -d/dx(adu/dx) = q for 0 < x < L Primary Variable u Constant a Source term q Secondary Variable Qo Field Transverse Deflection of a Cable Transverse Deflection Tension in Cable Distributed Transverse Load Axial Force Axial Deformation of a bar Longitudinal Displacement EA (E= Young's Modulus, A = Cross Sectional Area) Friction or contact force on surface of bar Axial Force Heat Transfer Temperature Thermal Conductivity Heat Source Heat Flow Through Pipes Hydrostatic Pressure D 4/128 m (D- Diameter, m - viscosity) Flow Source (Generally Zero) Flow Rate Laminar Incompressible Flow through a Channel under Constant Pressure Gradient Velocity Viscosity Pressure Gradient Pressure Flow Through Porous Media Fluid Head Coefficient of Permeability Fluid Flux Flow (seepage) Electrostatics Electrostatic Potential Dielectric Constant Charge Density Electric Flux UCSD - FEA Course - Fall 2003 6

D) Some Examples of the Poisson Equation – . (k u) = f Field of Application Primary Variable u Material Constant K Source Variable f Secondary Variables d, du/dx, du/dy Heat Transfer Temperature T Conductivity k Heat Source Q Heat Flow q [comes from conduction k T/ n and convection h(T-T ) Irrotational Flow of an Ideal Fluid Stream Function y Velocity Potential f Density r Mass Production s (normally zero) Velocities: y / x = -v; y / y = u f / x = -v; f / y = u Groundwater Flow Piezometric Head f Permeability K Recharge Q Seepage: q = k f/dn Velocities: u = -k f/dx , v = -k f/dy Torsion of Members with Constant Cross. Section Stress Function Y k=1 G = Shear Modulus f=2 q = angle of twist per unit length Gqdf/dx = -syz Gqdf/dx = -sxz Electrostatics Scalar Potential f Dielectric Constant e Charge Density r Displacement Flux density Dn Magnetostatics Magnetic Potential f Permeability m Charge density r Magnetic Flux density Bn Transverse Deflection of Elastic Membranes Transverse deflection u Tension T in membrane Transversely distributed Load Normal force q [Both tables taken from J. N. Reddy's Book "Introduction to the Finite Element Method", J. N. Reddy, Mc. Graw Hill Publishers, 2 nd Edition, Page 71] UCSD - FEA Course - Fall 2003 7

E) Some Examples of Coupled Systems 1. Plane Elasticity dsx/dx + dsxy/dy + fx = rd 2 u/dt 2 dsxy/dx + dsy/dy + fy = rd 2 v/dt 2 2. Flow of Viscous Incompressible Fluids [Navier Stokes Equations] [Conservation of Linear Momentum] rdu/dt - d/dx(2 mdu/dx) - d/dy[m(du/dy + dv/dx ) ] + d. P/dx – fx = 0 rdv/dt - d/dx[m(dv/dy + du/dx ) ] - d/dy(2 mdu/dy) + d. P/dy – fy = 0 [Conservation of Mass] (du/dx + dv/dy) = 0 UCSD - FEA Course - Fall 2003 8

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 9

SOFTWARE-Specific Session: 1. Intro to ANSYS. Basic file operations. Simple plate problem. 2. Intro to FEMLAB. Fluid mechanics problem. Critical look at results. 3. Intro to software-specific issues. h-elements, p-Elements Homework 1 and Reading Assignments. UCSD - FEA Course - Fall 2003 10