The 2nd WSEAS International Conference on APPLIED and THEORETICAL MECHANICS

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On dimensional reduction in multiscale, finite
element and atomistic, analysis in solid mechanics

• DUBRAVKA MIJUCA
• Department of Mechanics, Faculty of Mathematics
• University of Belgrade
• SERBIA
• dmijuca_at_matf.bg.ac.yu
• www.matf.bg.ac.yu/dmijuca

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Motive
To develop the reliable computational technique
to solve engineering problems that are complex in

• material,
• geometry,
• loading,
• scale of observations.

Incompressible cube
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Finite element method (FEM)
The most natural and comprehensive way to
discretize solid body.

• Primal only one solution variable
• Mixed two or more solution variables
• Multifield two or more physical fields (eg.
  Themo-mechanical, magneto-hydro-dynamics)
• Coupled all multifield procedures are semi or
  strongly coupled
• Multiscale different scale resolution of
  numerical calculation on the same model problem
• Bridged with numerical calculations on the
  atomistic scale

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Present FEM approach HC8/27MD/LJ

• Primal-mixed FEM
• Multifield transient heat transfer /
  elastostatics,
• Semi-coupled physical fields
• Multiscale mesh resolution,
• Full scale, three-dimensional in geometry,
  physical lows and constitutive relations,
• Bridged by semi-coupling with simulation on
  atomistic level.

It is already proven that present FEM approach is
reliable and time efficient.
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Multiscale analysis

• Continuum mechanics (CM) can be used alone only
  if there is inherent assumption that material
  properties vary continuously throughout the
  solid.
• Nowadays it is widely accepted that dual nature
  of the structure of matter, continuous when
  viewed at large length scales and discrete when
  viewed at an atomic scale, can be traced only by
  the multiscale materials modeling (MMM)
  approaches.
• Reliable numerical MMM approaches should
  harmonize continuum and atomistic analyses
  methods.

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Multiscale robustness
Definition (new, original) Some numerical (e.g.
FE) approach is multiscale robust if it retains
its robustness throughout the scale resolution of
the finite element mesh.

• There is a growing need to develop systematic
  modeling and simulation approaches to provide the
  accurate data about the state of stress, defect
  structure, thermal and mechanical performance
  throughout the geometrical scales

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Dimensional reduction

• Dimensional reduction is one of two generic
  techniques for the geometrical simplification of
  the considered physical problem
• The solid bodies with one or two axial dimensions
  much smaller than other are traditionally
  examined by plate, shell or beam theories based
  on dimensional reduction in the direction normal
  to middle surface, or beam's cross section,
  respectively.

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Main shortcomings of dimensional reduction

• Aspect-ratio restriction.
• Transitional error between FE of different
  dimensionalities.
• Inappropriate in analysis of coated bodies
• Inappropriate in multiscale analysis (very narrow
  finite elements required)
• Deterioration of the place and intensity of
  maximal stress results.

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Dimensional reduction
Dimensional reduction theories are only subsets
of the reliable full scale theories.

• It will be shown here that if we use reliable FE
  approach and assumptions from dimensional reduced
  theories (e.g. plate/shell theories) , the
  solution will converge to target result of that
  dimensional reduction theory (from bellow per
  displacement, from above per temperature).

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How to avoid dimensional reduction

• And stay time efficient ?

The imperative of computational mechanics in
order to prevail The coming crisis in
computational science (D.E. Post, Los Alamos
National Laboratory Report LA-UR-04-0388, 2004),
is to develop time efficient full scale 3d
numerical approaches.

• It implies
• Performance Challenge
• Programming Challenge
• Prediction Challenge

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Transient heat transfer in solids
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Elastostatics
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Primal-mixed finite element approach in transient
heat transfer
Subscript p is for prescribed
Subscript v is for unknown variable
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Time discretization
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Primal-mixed finite element approach in
elastostatics
Subscript p is for prescribed
Subscript v is for unknown variable
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Finite element configurations
Primal-mixed
Primal
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4
3
6
8
9
2
1
5
QUAD4
QUAD9
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System Matrix
Primal-mixed FEA
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Geometric Invariance

• In order to minimize the accuracy error and
  enable introduction of the stress constraints,
  tensorial character 1,2 of the present finite
  element equations is fully respected.

1 Draškovic Z (1988) On invariance of finite
element approximations, Mechanika Teoretyczna i
Stosowana 26597601 2 Bottasso C L, Borri M,
Trainelli L (2002) Geometric invariance,
Computational Mechanics 29 163169 DOI
10.1007/s00466-002-0329-8
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Reliability

• The proposed hexahedral finite element HC8/27
  passes low and high order convergence and
  efficiency tests in transient heat transfer and
  elastostatic analyses, that is, it is

• solvable,
• stable,
• accurate and
• efficient,
• Shortly it is reliable.

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Numerical examples
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Clamped square plate
Clamped 3D Plate, HC8/9
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Clamped 3D Plate Full theory Case A
Full theory.
The stress concentration point is inside the
plate.
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Clamped 3D Plate Like Plane theory Case B
Transversal shear stresses are suppressed on the
physical boundaries.
The stress concentration point is on the clamped
edge.
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Primal Approach, QUAD4 and QUAD9
There is no substantial difference in primal FEA
QUAD4 and QUAD9 stress results.
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Clamped 3D platecomparison per stress component
S22
Primal FEA
Primal-mixed FEA
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Plate with a hole
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Plate with a holes
Primal FEA
Primal-mixed FEA
? The primal 2d FEA is insensitive to the
presence of the hole.
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Plate with a holes
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Clamped plate - stress

• Without holes The stress calculated by HC8/27 is
  smaller than by QUAD4
• With holes The stress calculated by HC8/27 is
  bigger than by QUAD4
• Primal-Mixed FE 3D approach contribute to the
  cleaver designing in which the structure is of
  less weight and reinforced only over the stress
  concentration points. (At least we will not drill
  the hole in these regions).
• If we use primal FEA we will underestimate real
  stress around the hole

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Clamped 3D Plate - dimensional reduction
consequence per stress
Conclusion zeroing the stress components on
clamped edge upon the assumptions of the plane
theory, we may simulate the plane theory itself.

• Nevertheless, transversal shear stress component
  can not be neglected, because it smears the real
  stress picture, in which stress concentration
  point is inside the plate in the vicinity of
  clamped edge, and not on the clamped edge itself.
• In addition, the maximal stress result is lower
  than one obtained by the plate theory assumptions.

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Clamped 3D Plate Displacement Convergence
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Clamped 3D Plate dimensional reduction
consequence per displacement
In Case A target results converge from below to
the solution little bit lower than one obtained
by the Kirchhoffs plate theory In Case B target
result instantly converge to the solution little
bit lower than one obtained by the Kirchhoffs
plate theory.

• Conclusion The plate theory is to conservative
• Explanation unrealistic neglecting of the
  transverse shear stress component, minimize shear
  stress influence which softness the finite
  element solution, so it may instantly undergo
  apparently higher deflections

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Time efficiency per number of d.o.f.
Pentium IV 2.4GHz 2MB RAM SCSI 2x36GB
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Thermo-mechanical barrier coating

Five model problems with decreasing thickness of
coating and bond is examined.
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Results per thickness of coating
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Rotor Blade

• Hypothetical wind turbine rotor blade example
• The blade is of glass fibre /epoxy matrix
  pre-pregs
• The gear box is made of fibre/epoxy
• The adhesive is made of epoxy
• External load - gravitational

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Pressure load from CFD
Model without gear box
Model with gear box
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Numerical results
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Full scale analysis
18789 dofs per displacement vector and stress
tensor 76.36 sec on the notebook 1.4GHz Intel
Pentium (R) 512MB RAM
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Nanoidentation- Bridging the scales

• Bridging of continuum (finite element) and
  atomistic (molecular dynamics) mechanics is more
  realistic and accurate if continuum approach is
  based on reliable fully three-dimensional
  numerical approach

HC8/9
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A Cylindrical Concrete Vessel for Storing the
Core of a Nuclear Reactor

• The walls of the cylinder have tubular cooling
  vents, which carry a cooling fluid.
• Heat flow rate through the walls over a period of
  5 hours.

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Nuclear Reactor Core
Primal Approach False result in the first
iteration per time, lower maximal heat flux result
Primal-mixed Approach
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Block under compression - Robustness
E240.56595979 N/mm2 a1 mm P4 N/mm2
n0.3
n0.499899987
Reliable results for the present example are only
provided by the B-Bar method.
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Conclusion

• Dimensional reduction can be avoid by the use
  time efficient fully 3d FE approach based on a
  primal-mixed formulation in termomechanics.

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Primacy of mechanics
Primacy of mechanics in Computational Science
should be always defended by the fact that

• (colloquially) ultimately we all want to know how
  some peace of material is deformed under applied
  forces
• (academically) the state of displacement and
  stress fields of considered solid body under the
  external thermo-mechanical forces help us in
  identification of its structural integrity and
  life