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Final

# MECH 4604 Study Guide - Final Guide: Linear Combination, Collocation, Galerkin Method

Department
Mechanical Engineering
Course Code
MECH 4604
Professor
Andrew Speirs
Study Guide
Final

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CHP 1. 7-step finite element analysis: Preprocessing Phase (Model Definition): 1. Create and discretize the solution domain into finite elements; i.e. subdivide the problem into nodes and elements; 2. Assume a shape function to represent the physical behaviour of
an element; i.e. an approximate continuous function is assumed to represent the solution of an element (i.e. the distribution of the field variable); 3. Develop equations for an element (i.e. relationship between field and flow variables); 4. Assemble the elements to
present the entire problem; construct the global stiffness matrix; 5. Apply boundary conditions, initial conditions, and loading; Solution Phase: 6. Solve a set of linear or nonlinear algebraic equations simultaneously to obtain nodal results, such as displacement values
at different nodes in a solid mechanics problem or temperature values at different nodes in a heat transfer problem, Post-Processing Phase: 7. Obtain important information, such as deformed shape, distribution of stress, strain, distribution of temperature and heat
fluxes, etc.
CHP 2. Understand how to formulate finite element problems:
Stress Problem

Direct Method (simple
problems only):



For varying cross sections, use an
average cross-sectional area
between two nodes.





  
  
  
  
  
  

Heat Problem

Convection (use dummy node)
Conduction
Direct Stiffness Method


 
 

Elemental Stiffness Matrix
 
 
  
  
Global Stiffness Matrix
 
 
 
  



 
 



 
 

Apply BC’s

Substitution:


 
 
 
 
 
Displacement BC
Homogeneous:

 
 
   
  
Displacement BC
Non-homogeneous:

 
 
 
 
Delete trivial rows/columns and
solve:

 
 
 
Post-Process




Use the force at node j to get the
correct sign


Minimum Total Potential Energy
Method 


























Weighted Residuals
1. Assume a solution that approximates the actual distribution of the DOF variable
2. Show that the solution satisfies the boundary conditions
3. Find an expression for the residual (e.g. using differential equation of the physical
problem)
4. Minimize the residual (in some sense)




1. Collocation Method
Set = 0 at arbitrary locations.
One location for each unknown
coefficient in assumed solution. Solve
for unknown coefficients to obtain
variation in DOF variable(s).
2. Subdomain Method
Coefficients chosen so that integral of residual is 0 over
selected ranges
• One range for each unknown coefficient in assumed
solution
Over each range we require

Solve for unknown coefficients
3. Galerkin Method
Define trial functions
 
i.e. we need one trial function for each independent function
of etc. so that assumed solution is a
linear combination of trial functions
4. Least squares method
Simplifies to 

Applies over the whole element 
Perform for each unknown coefficient , then solve for all unknown coefficients to
obtain











