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QA00000009389e QA00000009389e - DS Support Knowledge Knowledge Base / Question Answer
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Knowledge Base Information
Understanding negative eigenvalue messages
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Available Languages English &egative eigenvalue messages are generated during t(e solution -ro"ess w(en t(e s'stem matri. is being de"om-osed) $(e messages "an be issued for a variet' of reasons some asso"iated wit( t(e -('si"s of t(e model and ot(ers asso"iated wit( numeri"al issues) An e.am-le of t(e message t(at is issued is: ***WARNING: THE SYSTEM MATRIX HAS 16 NEGATIVE EIGENVALUES. IN AN EIGENVALUE EXTRACTION STEP THE NUMBER OF NEGATIVE EIGENVALUES IS THIS MAY BE USED TO CHECK THAT EIGENVALUES HAVE NOT BEEN MISSED. NOTE: THE LANCZOS EIGENSOLVER APPLIES AN INTERNAL SHIFT WHICH WILL RESULT IN NEGATIVE EIGENVALUES. IN A DIRECT-SOLUTION STEADY-STATE STEADY-STATE DYNAMIC ANALYSIS NEGATIVE EIGENVALUES ARE EXPECTED. A STATIC ANALYSIS CAN BE USED TO VERIFY THAT THE SYSTEM IS STABLE. IN OTHER CASES NEGATIVE EIGENVALUES MEAN THAT THE SYSTEM MATRIX IS NOT POSITIVE DEFINITE: FOR EXAMPLE A BIFURCATION !BUCKLING" LOAD MAY HAVE BEEN EXCEEDED. NEGATIVE EIGENVALUES MAY ALSO OCCUR IF #UADRATIC ELEMENTS ARE USED TO DEFINE CONTACT SURFACES.
P('si"all' negative eigenvalue messages are often asso"iated wit( a loss of stiffness or solution uniqueness eit(er in t(e form of a material instabilit' or t(e a-- li"ation of loading be'ond a bifur"ation -oint 0-ossibl' "aused b' a modeling error1) During t(e iteration -ro"ess t(e stiffness matri. "an t(en be assembled in a state w(i"( is far from equilibrium w(i"( "an "ause t(e warnings to be issued)
&umeri"all' negative eigenvalues "an be asso"iated wit( modeling te"(niques t(at ma2e use of Lagrange multi-liers to enfor"e "onstraints or lo"al numeri"al instabilities t(at result in t(e loss of stiffness for a -arti"ular degree of freedom) Most negative eigenvalue warnings asso"iated wit( Lagrange multi-liers are su--ressed3 t(e e."e-tions are w(en quadrati" t(ree dimensional elements are used to define "onta"t surfa"es or w(en ('brid elements are used in a geometri"all' nonlinear simulation and undergo large deformations)
Mat(emati"all' t(e a--earan"e of a negative eigenvalue means t(at t(e s'stem matri. is not - ositive definite) If t(e basi" statement of t(e finite element -roblem is written as: 56 7 8K95.6 t(en a -ositive definite s'stem matri. 8K9 will b e nonsingular and satisf' 5.6$8K95.6 ; < for all non=ero 5.6) $(us w(en t(e s'stem matri. is -ositive definite an' dis-la"ement t(at t(e model e.-erien"es will -rodu"e -ositive strain energ')
In addition to t(e "auses s(own in t(e warning message some situations in w(i"( negative eigenvalue messages "an a--ear in"lude: > Bu"2ling anal'ses in w(i"( t(e -rebu"2ling res-onse is not stiff and linear elasti") In t(is "ase t(e negative eigenvalues often -oint to s-urious modes) ,emember t(at t(e formulation of t(e bu"2ling -roblem is -redi"ated on t(e res-onse of t(e stru"ture being stiff and linear elasti" -rior to bu"2ling) > Unstable Unstable material material res-onse: res-onse: ◦ A ('-erelasti" material going unstable at (ig( values of strain ◦ %nset of -erfe"t -lasti"it' ◦ ?ra"2ing of "on"rete or ot(er material failure t(at "auses softening of t(e material res-onse > $(e use of anisotro-i" elasti"it' wit( s(ear s(ear moduli t(at are unrealisti"all' unrealisti"all' ver' mu"( lower t(an t(e dire"t moduli) In t(is "ase ill"onditioning ma' o""ur triggering negative eigenvalues during s(earing d eformation) > A non-ositive definite s(ell s(ell se"tion stiffness defined in a U@!&S routine) > $(e use of a -retension node t(at is not "ontrolled "ontrolled b' using t(e B%U&DA, o-tion and t(e "om-onents "om-onents of t(e stru"ture are not 2inemati"all' "onstrained) In t(is "ase t(e stru"ture "ould fall a-art due to t(e -resen"e of rigid bod' modes) $(e warning messages t(at result ma' in"lude one related to negative eigenvalues) > Some a--li"ations of ('drostati" ('drostati" fluid fluid elements) > ,igid bod' bod' modes due due to errors in modeling modeling))
&egative eigenvalue warnings will sometimes be a""om-anied b' ot(er warnings addressing su"( t(ings as e."essive element distortion or magnitude of t(e "urrent strain in"rement) In "ases w(ere t( e anal'sis will not "onverge resolution of t(e non"onvergen"e will often eliminate t(e negative eigenvalue warnings as well)
or anal'ses t(at do "onverge "arefull' "(e"2 t(e results if t(e warnings a--ear in "onverged iterations) A "ommon "ause of negative eigenvalue warnings is t(e assembl' of t(e stiffness matri. about a nonequilibrium state) In t(ese instan"es t(e warnings will normall' disa--ear wit( "ontinued iteration a nd if t(ere are no warnings in an' iterations t(at (ave "onverged warnings t(at a--ear in non"onverged iterations ma' safel' be negle"ted) If t(e warnings a--ear in "onverged iterations (owever t(e solution must be "(e"2ed to ma2e sure it is -('si"all' realisti" and a""e-table) It ma' be t(e "ase t(at a solution satisf'ing t(e toleran"e for "onvergen"e (as been found for t(e model w(ile it is in a nonequilibrium state)
