Elasticity/Rayleigh-Ritz method

The Rayleigh-Ritz method

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The potential energy functional has the form

 

The standard method of finding an approximate solution to the mixed boundary value problem is to minimize   over a restricted class of functions (the Rayleigh-Ritz method), by assuming that

 

where   are functions that are chosen so that they vanish on   and   is a function that approximates the boundary displacements on  . The constants   are then chosen so that they make   a minimum.


Suppose,

 

Then,

 

where,

 

To minimize   we use the relations

 

to get a set of   equations which provide us with the values of  .


This is the approach taken for the displacement-based finite element method. If, instead, we choose to start with the complementary energy functional, we arrive at the stress-based finite element method.