Locking (FEM)

Under Locking (from the English for "locking" or "stiffening") is meant in the finite element method (FEM), the phenomenon that element formulations can react with certain types of load too stiff and thus not have the correct deformations in response to calculate the load .

Known for this are in particular thrust locking and volume locking , which stand for the fact that in thrust conditions or loads that want to change the volume of a body, the calculated reaction is mapped too stiffly or not at all.

Causes and explanation

Especially with finite elements with linear shape functions it is known that this type of shape functions does not contain all terms of a (complete) polynomial , see Pascal's triangle . This explains the deficit in the mathematical imaging performance of these elements.

An example of this is the classic 4-node element with a typical approach function

{\ displaystyle N_ {3} = {\ frac {1} {4}} (1+ \ xi) (1+ \ eta) = {\ frac {1} {4}} (1+ \ xi + \ eta + \ xi \ eta)}

,

with which the bending line of a beam can only be calculated with a large numerical error: It is known that the bending line of a cantilever under a point load can be described with the Bernoulli assumptions by a function with a quadratic and cubic component:

{\ displaystyle w (x) = {\ frac {F} {6 \, EI}} (3lx ^ {2} -x ^ {3})}

.

The example for shows that for a complete description of a quadratic polynomial ( parabola ) at least the terms and are missing. From the terms for representing a cubic polynomial , or there is no part at all, see Pascal's triangle . These missing terms are the cause of the unsatisfactory quality of this formulation, if you want to specifically describe questions that depend on the shear modulus . ${\ displaystyle N_ {3}}$ ${\ displaystyle \ xi ^ {2}}$ ${\ displaystyle \ eta ^ {2}}$ ${\ displaystyle \ xi ^ {3}}$ ${\ displaystyle \ eta ^ {3}}$ ${\ displaystyle \ xi ^ {2} \ eta}$

literature

TJR Hughes: The Finite Element Method, Linear Static and Dynamic Finite Element Analysis . New Jersey: Prentice-Hall 1987
OC Zienkiewicz, RL Taylor: The Finite Element Method; Volume 1; The Basis, fifth edition ., Oxford, Butterworth-Heinemann 2000