Folgende Informationen dazu vom PTC Customer Support:
Mechanica Structure verwendet Serendipity Elemente.
Zienkiewicz, Methode der Finiten Elemente, 2. erweiterte und voellig neubearbeitete Auflage, Studienausgabe
7.4 Lagrange Rechteckelemente
... Obwohl sie sich leicht erzeugen lassen, ist die Brauchbarkeit dieser Klasse nicht nur dadurch begrenzt, dass eine grosse Anzahl innerer Knoten auftritt, sondern auch dadurch, dass die Polynome hoeherer Ordnung ein schlechtes Verhalten hinsichtlich der Kruemmungsanpassung aufweisen...
7.11 Rechteckige Prismen der Lagrangeschen Klasse
... Saemtliche in 7.4 gemachten Bemerkungen ueber innere Knoten und die Nachteile hinsichtlich des Kruemmungsverhaltens der Funktionen gelten auch hier. Dadurch wird ihre praktische Einsatzmoeglichkeit stark eingeschraenkt...
Bei den Serendipity Elementen scheint es keine Nachteile zu geben. Literatur zu 7.11:
Argyris, J.H: Continua and Discontinua. Proc. Conf. Matrix Methods in Structural Mechanics. Wright Patterson Air Forces Base. Ohio, Oct. 1965.
Anbei noch zwei TPIs
Gruss
Ronald Fuchs
Issue #: 31234
Version: 20.0
Datecode:
Machine: All
Status: Customer
Understanding Whether Pro/MECHANICA Uses Lagrange Elements or Serendipity Elements
Long Description:
Description
-----------
This document describes whether Pro/MECHANICA implements meshes using Lagrange
elements or Serendipity elements
Alternate Technique
-------------------
See Resolution below.
Resolution
----------
Pro/MECHANICA uses serendipity elements. The main reason is that Lagrange elements
are not hierarchical (that is, the basis functions at p-order p are not
a superset of the basis functions at a lower p). Implementing a solver is easier
for serendipity elements.
Though lagrange elements (which have internal nodes) are more accurate than serendipity elements
at lower polynomial order, the accuracy is about the same for higher p-order levels.
Issue #: 32403
Version: 20.0
Datecode:
Machine: All
Status: Customer
Pro/MECHANICA solid elements (Serendipity elements) and their characteristics
Long Description:
Description
-----------
What kind of solid elements are used by Pro/MECHANICA and what characteristics
do they have?
Alternate Technique
-------------------
See Resolution
Resolution
----------
Pro/MECHANICA uses serendipity elements. The main reason is that Lagrange
elements are not hierarchical (that is, the basis functions at p-order p are
not a superset of the basis functions at a lower p). Implementing a solver is
easier for serendipity elements. Though lagrange elements (which have internal
nodes) are more accurate than serendipity elements at lower polynomial order,
the accuracy is about the same for higher p-order levels. The accuracy is
comparable for the same number of degrees of freedom. But Lagrange elements
have more degrees of freedom than serendipity elements, for a given p. So, for
a given p, they are more accurate, but also more expensive.
For higher p-levels the variables do not represent a variable with physical
sense (e.g. translation at a point) any more. Here is an example for shape
functions with a polynomial order up to three:
n=1: u(x)=a_o + a_1*x
n=2: u(x)=a_0 + a_1*x + a_2*x^2
n=3: u(x)=a_0 + a_1*x + a_2*x^2 + a_3*x^3
The Pro/MECHANICA meshes are conforming so there is no partial overlay or gap
of faces or edges. The elements ensure exact continuity of displacements. The
Pro/MECHANICA solid elements guarantee monotonic convergence for strain energy
only like the most common finite element formulations. No formulation
guarantees monotonic convergence on any quantity other than total strain
energy. Pro/MECHANICA uses the Gauss integration algorithm to obtain the
stiffness matrices.
Note 1:
It is assumed that only solid elements are used. At the location of links the mesh
is not conforming any more and the displacements are not exactly continuous any more.
Note 2:
In rare cases integration errors may have effect on the monotonic convergence.
Reference:
"Finite Element Analysis", by Barna Szabo and Ivo Babuska,
John Wiley & Sons, 1991
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