Patch test (finite elements)
Encyclopedia
The patch test in the finite element method
is a simple indicator of the quality of a finite element, developed by Bruce Irons
.
The patch test uses a partial differential equation
on a domain consisting from several elements set up so that the exact solution is known. Typically, in mechanics
, the prescribed exact solution consists of displacements that vary as linear functions in space (called a constant strain
solution). The elements pass the patch test if the finite element solution is the same as the exact solution.
It was long conjectured by engineers that passing the patch test is sufficient for the convergence of the finite element, that is, to ensure that the solutions from the finite element method converge to the exact solution of the partial differential equation
as the finite element mesh is refined. However, this is not the case, and the patch test is neither sufficient nor necessary for convergence.
Finite element method
The finite element method is a numerical technique for finding approximate solutions of partial differential equations as well as integral equations...
is a simple indicator of the quality of a finite element, developed by Bruce Irons
Bruce Irons (engineer)
Bruce Irons was an engineer and mathematician, known for his fundamental contribution to the finite element method, including the patch test and the frontal solver....
.
The patch test uses a partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
on a domain consisting from several elements set up so that the exact solution is known. Typically, in mechanics
Mechanics
Mechanics is the branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment....
, the prescribed exact solution consists of displacements that vary as linear functions in space (called a constant strain
Strain (materials science)
In continuum mechanics, the infinitesimal strain theory, sometimes called small deformation theory, small displacement theory, or small displacement-gradient theory, deals with infinitesimal deformations of a continuum body...
solution). The elements pass the patch test if the finite element solution is the same as the exact solution.
It was long conjectured by engineers that passing the patch test is sufficient for the convergence of the finite element, that is, to ensure that the solutions from the finite element method converge to the exact solution of the partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
as the finite element mesh is refined. However, this is not the case, and the patch test is neither sufficient nor necessary for convergence.