ME232A: Introduction to Computational Mechanics I

Stanford University, Winter Quarter 2000-01

Professor Peter M. Pinsky

Announcements

Class Schedule:
Professor:
Peter M. Pinsky
Durand 275
Tel: 3-9327
pinsky@stanford.edu

Office hours:
Teaching assistant:
Haw-Ling Liew
Durand 283
Tel: 3-8089
hlliew@stanford.edu

Office hours:
Problem Sets and Computing Projects

Students will be assigned problem sets periodically (about seven sets) and three Matlab computing assignments. Prior knowledge of Matlab is not required. Instruction on Matlab and Stanford access will be discussed in class.

Graded homeworks will be returned in class. Unclaimed homeworks will be placed in the ME232 box located directly opposite the administration office of Division of Mechanics and Computation (second floor of Durand Building).

Course Grading
Overview

This course sequence (A,B) is designed to provide an introductory overview of modern computational methods for problem arising primarily in mechanics of solids and is intended to appeal to students from various engineering disciplines. In the first (A) course the basic concepts of the finite element method (FEM) and boundary element method (BEM) are outlined; providing a backgroud that will allow students to understand much of the technology behind these two important techniques and also allow them to formulate finite element and boundary element approaches for novel applications from 'first principles'.

The A course emphasizes the derivation of the basic equations of linear solid mechanics and explores how the FEM and BEM are developed for these equations. Specific problems that will be considered include membranes, steady heat conduction (diffusion) and problems in plane elasticity, i.e. plane stress, plane strain, axisymmetric elasticity, torsion. The conservation laws of solid mechanics are developed both in local form (as differential equations) and in variational form (e.g. principle of virtual displacements, principle of minimum potential energy). The variational forms of the governing equations are used as the starting point for develpoing the FEM and BEM approaches - providing an important connection between mechanics and computation.

A practical understanding of the FEM and BEM will not be complete without an appreciation of the data structures and coding methods that these methods employ. Students will develop a simple but effective code using Matlab and employ the Matlab PDE Toolbox for its pre- and post-processing features. In this way, students will be able to solve realistics problems based on their own coding.

The A course provides a good background for understanding the theory underlying commercial codes, for creating finite elememt models and making good 'modeling' choices. In the B course, selected advanced topics are discussed. These include nonlinear beahavior, the treatment of material constraints (e.g. incompressibility) and structural constraints (e.g. contact mechanics), elements for structural analysis, as well as modeling time-dependent problems.

Prerequisites

Introductory knowledge of strength of materials and matrix algebra. Continuum mechanics would be useful but not essential.

Guide to Primary Course Topics
  1. Problems in one dimension: (a) axially loaded bars and (b) heat diffusion; strong form and weak form (principle of virtual displacements) of the governing equations; finite element formulation ; element arrays, assembly and solution; finite element Matlab code.
  2. Review of linear continuum mechanics; conservation laws in local and global form; variational principles.
  3. Two-dimensional membrane problem; strong and weak forms; boundary conditions.
  4. General methods of approximation: collocation, least squares, Galerkin and boundary integral approaches.
  5. Finite element formulation; simple triangular and rectangular elements; elements arrays and assembly procedure; solution and computation of fluxes.
  6. Choice of finite element shape functions; completeness and notions of convergence; forming higher order elements.
  7. Solution of the heat diffusion problem in two dimensions using the boundary element method.
  8. Review of equations for two-dimensional elasticity (plane stress, plane strain and axisymmetry); symmetry properties; variational principles.
  9. Finite element formulation for plane elasticity; simple triangular elements; data structures and coding considerations.
  10. Boundary element formulation for plane elasticity.
  11. Isoparametric finite elements and numerical integration.
  12. Introduction to geometry engines, meshing, visualization and simulation-based design.
Text (Optional) Other Reading