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Elasticity/Rayleigh-Ritz method

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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.