Meshless numerical formulation for analysis of shell-like structures

doctoral thesis

• Title (english): Bezmrežna numerička formulacija za analizu ljuskastih konstrukcija
• Author: Tomislav Jarak
• Mentor: Jurica Sorić (mentor)
• Committee member: Zdenko Tonković (predsjednik povjerenstva)
• Committee member: Jurica Sorić (član povjerenstva)
• Committee member: Jan Sladek (član povjerenstva); strani drzavljanin: Nije dostupno
• Granter: University of Zagreb; Faculty of Mechanical Engineering and Naval Architecture; Zagreb
• Defense date and country: 2010-06-28, Croatia
• Scientific / art field, discipline and subdiscipline: TECHNICAL SCIENCES; Mechanical Engineering; General Mechanical Engineering (Construction)
• Universal decimal classification (UDC): 631/632 - Agriculture. Plant diseases. Plant protection; 51 - Mathematics ; 53 - Physics

Abstract

Meshless computational methods for the analysis of plate and shell structures are
proposed in this thesis. The developed algorithms are based on the local PetrovGalerkin approach. A shell is considered as a three dimensional (3-D) solid continuum,
and the solid-shell concept, which allows the implementation of complete 3-D material
models, is employed. Geometry of the shell is described by employing a mapping
technique, whereby the shell middle surface is defined mathematically exactly.
Discretization is carried out by the couples of nodes located on the upper and lower
surfaces of the structure. The governing equations are the local weak forms (LWF) of
the 3-D equilibrium equations, which are written over the local sub-domains
surrounding the node couples. The approximation of all unknown field variables is
carried out by using the Moving Least Squares (MLS) approximation scheme in the inplane directions, while simple polynomials are applied in the thickness direction. Both
the purely displacement-based (primal) and mixed formulations are proposed and
special attention is given to the elimination of locking effects.
Two different primal formulations are presented where only the displacement field
is approximated. In both cases, the Poisson’s thickness locking effect is circumvented
by adopting the hierarchical quadratic interpolation for the transversal displacement
component. The transversal shear locking phenomenon is alleviated by applying a
sufficiently high order of the in-plane MLS functions.
In the mixed approach, appropriate strain and stress components are approximated
separately from the displacement field. The nodal strain and stress values are then
expressed in terms of the approximated displacements, and a global system of equations
containing only the unknown nodal displacement variables is obtained. In the
formulation for plates, thickness locking is eliminated by modifying the nodal values of
the normal transversal strain component, while the transversal normal stress is
approximated instead of the transversal normal strain in the algorithm for curved shells.
In the thin structural limit, transversal shear locking is efficiently suppressed by means
of the separate strains approximation. It is theoretically proved that the mixed approach
is numerically more efficient than the proposed primal meshless formulations.
The numerical efficiency of the derived algorithms is demonstrated by numerical
examples.

Abstract (croatian)

Predloženi su novi bezmrežni algoritmi za numeričku simulaciju procesa
deformiranja ploča i ljusaka u linearno-elastičnom području. Prikazane formulacije
temelje se na bezmrežnoj lokalnoj Petrov-Galerkinovoj (MLPG) metodi. Ljuske su
razmatrane kao deformabilna trodimenzijska tijela primjenom tzv. solid-shell koncepta
koji omogućava ugradnju potpunih trodimenzijskih materijalnih modela. Geometrija
ljusaka opisana je parametrizacijom srednje plohe, pri čemu je srednja ploha opisana
matematički egzaktno. Diskretizacija je provedena pomoću parova čvorova, koji se
nalaze na gornjoj, odnosno donjoj plohi ljuske. Jednadžbe ravnoteže zadovoljene su u
obliku lokalnih slabih formi uz primjenu Petrov-Galerkinovog principa. Testne funkcije
su opisane jednostavnim polinomima u smjeru normale na srednju plohu, dok se u
ravnini tangentnoj na srednju plohu koriste Heavisideove step funkcije. Nepoznate
veličine polja u svim predloženim algoritmima aproksimirane su primjenom metode
pomičnih najmanjih kvadrata (MLS funkcije) u tangentnoj ravnini, a u smjeru normale
jednostavnim polinomima.
Predložene su dvije formulacije temeljene na metodi pomaka. Poissonov locking
eliminiran je primjenom hijerarhijske kvadratne interpolacije za komponentu pomaka u
smjeru normale na srednju plohu. Poprečni posmični locking ublažen je primjenom
dovoljno visokog stupnja baze MLS funkcija.
U mješovitim formulacijama su uz polje pomaka direktno aproksimirane i neke od
komponenata tenzora deformacija i naprezanja. Nepoznate čvorne vrijednosti
deformacija i naprezanja izračunavaju se iz aproksimiranih pomaka pomoću
odgovarajućih kolokacijskih postupaka. Na taj način dobiva se zatvoreni globalni sustav
jednadžbi u kojem su nepoznanice samo čvorni pomaci. Poissonov locking je eliminiran
modificiranjem čvornih vrijednosti za poprečnu normalnu komponentu deformacije u
formulaciji za ploče, dok je u algoritmu za ljuske to postignuto direktnim
aproksimiranjem poprečne normalne komponente naprezanja. Poprečni posmični
locking učinkovito je otklonjen direktnim aproksimiranjem komponenata tenzora
deformacije koje djeluju u tangentnoj ravnini na srednju plohu. Pokazano je teorijski i
eksperimentalno da je mješoviti MLPG pristup superioran u odnosu na formulacije koje
su temeljene na metodi pomaka. Točnost i učinkovitost predloženih algoritama
pokazani su odgovarajućim numeričkim primjerima.

• Keywords: Meshless methods; Meshless Local Petrov-Galerkin method; Moving Least Squares approximation; plates; shells; Poisson’s thickness locking; transversal shear locking
• Keywords (croatian): bezmrežne metode; bezmrežna lokalna Petrov-Galerkinova metoda; metoda pomičnih najmanjih kvadrata; ploče; ljuske; Poissonov locking; poprečni posmični locking
• Language: english
• URN:NBN: urn:nbn:hr:235:047021
• Study programme: Title: Mechanical Engineering and Naval Architecture; Study programme type: university; Study level: postgraduate; Academic / professional title: doktor/doktorica znanosti, područje tehničkih znanosti (doktor/doktorica znanosti, područje tehničkih znanosti)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2020-06-19 12:20:23