Theory

Introduction

sladnalsd

Mesh Generation

askldaskld

Finite Element Preliminaries

TODO - update for Tri3 elements…

Element Type

dlsfkldsf

Isoparametric Representation

sdlfkjdslf

Shape Functions

sdlkfjdsf

Cartesian Partial Derivatives

dakljfdslkf

Numerical Integration

akdlskfsdf

Plane-Stress Elements

Constitutive matrix:

\[\begin{split}\textbf{D} = \begin{bmatrix} \lambda + 2 \mu & \lambda & 0 \\ \lambda & \lambda + 2 \mu & 0 \\ 0 & 0 & \mu \\ \end{bmatrix}\end{split}\]

where \(\lambda\) and \(\mu\) and the Lamé parameters:

\[\begin{split}\lambda &= \frac{E \nu}{(1 + \nu)(1-2 \nu)} \\ \mu &= \frac{E}{2 (1 + \nu)} \\\end{split}\]

Local stiffness matrix:

\[\textbf{k}_{\rm e} = t \int_\Omega \textbf{B}^{\rm T} \textbf{D} \textbf{B} d \, \Omega\]

equates to:

\[\begin{split}\textbf{k}_{\rm e} = \sum_{i=1}^n w_i \textbf{B}^{\rm T} \textbf{D} \textbf{B} J_i t \\\end{split}\]

Stiffness Matrix Assembly

aslkdjaskld

Boundary Conditions

asdkjasdl

Prescribed Nodal Displacement

asdasd

\[\begin{split}\textbf{K}[\rm{dof}, :] = 0 \\ \textbf{K}[\rm{dof}, \rm{dof}] = 0 \\ \textbf{f}[\rm{dof}] = \textbf{u} \\\end{split}\]

Nodal Spring

asdasd

\[\textbf{K}[\rm{dof}, \rm{dof}] = k\]

Nodal Load

asdasd

\[\textbf{f}[\rm{dof}] = P\]

Post-Processing

asdasd

Nodal Forces

asdasd

\[\textbf{f} = \textbf{K} \textbf{u}\]

Nodal Stresses

Note about gauss points vs. nodal points - Felippa

\[\boldsymbol{\sigma} = \textbf{D} \textbf{B} \textbf{u}\]