ON THE ISSUE OF MULTISCALE ROBUSTNESS IN COMPUTATIONAL MECHANICS

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ON THE ISSUE OF MULTISCALE ROBUSTNESS IN
COMPUTATIONAL 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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Alternative titles

• From Megastructure to Nanostructure analysis in
  One Simulation Framework
• From macro to micro insight of engineering
  structure in one simulation framework without
  submodelling

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Mathematical Models to trace the displacements of
nanoparticles

• Schrödinger most general
• Bohr-Oppenheimer decoupled Schrödinger
• Newtonian mechanics feasible to solve

Mijuca D, Mastilovic S (2005) A Novel One-To-One
Multiscale Approach to Computational Mechanics of
Materials, 1st International Workshop on
Nanoscience Nanotechnology IWON2005, November
1518, pp. 180-186
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Engineering approach

• (is) Historically simplified with the aim to
  shorten the time from idea to manufacturing,
  construction or real time decision making

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Novel engineering approaches

• Nonsimplified engineering calculations, HEXA8,
  450000 d.o.f., materially nonlinear, 4 iteration,
  app. 5minutes, PCDualCore1.8MhZ, Software Straus7

Megastructure
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Numerical methods referring to different length
scales
FEM
MD
1 Schmauder S Computational Mechanics Annu.
Rev. Mater. Res. Vol.32, 2002, pp.43765
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From Continuum to Nanomechanics

• Contemporary computer hardware resources and new
  mathematical and software tools to handle large
  scale computing, open doors for richer FE
  theories in which reliability is not compromised
  even if the model problem analyzed requires
  consideration through geometrical scales from
  continuum to micro to nanomechanics.

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Industrial Problems

• New materials, as composite sandwiched plates
  with foam in its core, and functionally graded
  materials,
• Analysis at higher temperatures (fire) when
  material properties drastically change,
• Fully 3d reliable numerical procedure toward
  incompressibility 5.
• Fiber-optic sensors in non-destructive testing
• Coated bodies

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Resent References

• Mijuca D (2004) On hexahedral finite element
  HC8/27 in elasticity. Computational Mechanics
  33(6)466-480
• Mijuca D, Mastilovic S (2005) A Novel One-To-One
  Multiscale Approach to Computational Mechanics of
  Materials, 1st International Workshop on
  Nanoscience Nanotechnology IWON2005, November
  1518, pp. 180-186
• Mijuca D, Ziberna A and Medjo B (2006) A Novel
  Primal-Mixed Finite Element Approach for Heat
  Transfer in Solids, Computational Mechanics,
  on-line first DOI 10.1007/s00466-006-0034-0
• Mijuca D (2006) On Dimensional Reduction in
  Multiscale, Finite Element and Atomistic,
  Analysis in Solid Mechanics, WSEAS Transactions
  On Applied And Theoretical Mechanics, Issue 1,
  Volume 1, pp.16-25,, ISSN 1991-8747
• Mijuca D (2006) On Primal-Mixed formulation in
  Elasticity and Thermoelasticity, Monograph,
  Faculty of Mathematics University of Belgrade
  (2006)

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Recent Citation List

• Cuomo M, Contrafatto L (2000)  Stress rate for
  elastoplastic models with internal variables
  based on augmented Lagrangian regularization.
  Intern. Journal of Solids and Structures 37
  3935-3964
• Zoran Draškovic (2002)  Numerical Comparison of
  The Scalar, Pseudoinvariant And Invariant
  Approach In The Derivation Of Finite Element
  Equations Of Motion In Curvilinear
  Coordinates. FACTA UNIVERSITATIS, Series Mech.
  Vol.312, pp.351 - 357
• Gómez Jáuregui Valentín (2004). Tensegrity
  Structures and their Application to Architecture,
  Queen's University, School of Architecture, Belfas
  t, Northern Ireland
• Icardi U, Atzori A (2004)   Simple, efficient
  mixed solid element for accurate analysis of
  local effects in laminated and sandwich
  composites, Advances in Engineering Software 35,
  pp 843859,
• V. A. Maksimyuk and I. S. Chernyshenko (2004)
  Mixed functionals in the theory of nonlinearly
  elastic shells. Volume 40, Number 11 / November,
   International Applied Mechanics, Springer New
  York ISSN 1063-7095 (Print) 1573-8582 (Online)
• K.S. Chavan, B.P. Lamichhane and B.I. Wohlmuth
  (2005) Locking-Free Finite Element Methods For
  Linear Nonlinear Elasticity In 2d And 3d. (2005)
  Preprint SFB404 2005-13
• Jim Wood (2005) D5603 - Procedural Benchmarks
  for Common Fabrication Details in Plate/Shell
  Structures, NAFEMS-FENet.
• U. Icardi (2006) Layerwise mixed element with
  sublaminates approximation and 3D zig-zag field,
  for analysis of local effects in laminated and
  sandwich composites. International Journal for
  Numerical Methods in Engineering. on line first

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Multiscale analysis

• There is a growing need to develop systematic
  modeling and simulation approaches for multiscale
  and multimaterials model problems in mega and
  microstructures in order to provide the accurate
  data of the subregions of different geometric
  scales

• about the state of stress,
• defect structure,
• thermo-mechanical performance.

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New definition

• The new definition of the multiscale robustness
  in computational mechanics, is introduced in
  order to support theoretical settings of
  multiscale analysis, that is bridging of
  simulation on atomistic scale with simulation on
  the scale of continuum mechanics, and vice versa.

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Multiscale Robustness

• Some finite element approach is multiscale robust
  if it retains its reliability regardless of the
  geometrical scale of the finite element used for
  the discretization of the material domain.

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Multiscale robustness test

• The necessary and sufficient solvability test
  must be fulfilled regardless of geometrical
  multiscale resolution of finite element
  discretization.

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Motive
To develop reliable multiscale FE procedure for
solving engineering and bioengineering problems
that are complex in

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

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Transient heat transfer in solids
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Elastostatics
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Present FEM approach

• primal-mixed FEM FEMIX HC8/27(MD/LJ)
• multifield transient heat transfer /
  elastostatics,
• semi-coupled physical fields (via temperature),
• multiphisics,
• multiscale,
• Coordinate independent
• full scale, three-dimensional in geometry,
  physical lows and constitutive relations,
• bridged by semi-coupling (via displacement) with
  simulation on atomistic level
• Residual stress input straightforwardly enabled.

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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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Primal-mixed finite element approach in
elastostatics
Subscript p is for prescribed
Subscript v is for unknown variable
Initial stress/strain, Residual stress,
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Finite element family HC
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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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Multiscale robustness

• Necessary conditions for solvability
• Sufficient conditions for solvability. Eigenvalue
  analysis

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Eigenvalues for different one finite element
configurations
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(Why not) Dimensional Reduction
Dimensional reduction (geometrical
simplification) is one of two generic techniques
for physical problem idealization

• Solid bodies with one or two axial dimensions
  much smaller than other, are traditionally
  examined by plate, shell or beam theories.

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Fully 3D FE procedure
Present procedure is fully 3D per

• geometry
• loading
• constitutive relation

No dimensional reduction.
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Main flaws of dimensional reduction

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

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Full scale analysis vs. simplified

• Beam, Plate and Shell theories can be also
  simulated by the Full 3d approaches.
• If we enter dimensional reduction assumptions in
  the present full scale FE approach, the app.
  solution will converge theirs target results.

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Clamped square plate
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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 Plane theory Constraints Case
B
Transversal shear stresses are suppressed on the
physical boundaries in order to simulate the
plate theory.
The stress concentration point is on the clamped
edge.
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Clamped 3D Plate Displacement Convergence
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Clamped 3D Plate Conclusions

• 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.
• 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 plate FEA multiscale
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Plate with a hole (one quarter)
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Plate with a hole
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Plate with a hole
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Deformed configurations
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Long steel Coated Shaft
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Thermo-mechanical barrier coating

Five model problems with decreasing thickness of
coating and bond is examined.
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Convergence per decreasing of coating thickness
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Nuclear Reactor Core Primal Approach
False heat flux result in the first iteration per
time, Lower maximal heat flux result.
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Nuclear Reactor Core Primal-mixed Approach
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Execution time
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Molecular dynamics
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Nanoidentation- Bridging the scales
FE HC8/9
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Wind turbine rotor blade design

• Vesta turbine design
• Layered composite
• 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, CFD

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Failure assessment
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FE analysis
Delamination
Debonding
Crack
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Pressure load from CFD
Model without gear box
Model with gear box
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Full multiscale 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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Aramide Composite Material
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Fiber Optic Sensor
Specimen size 12x105mmx7.3mm, Optic Sensor Core
d0.0625mm
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Rotary Pellet Plant Kiln Straus7
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Paper in press

• ON ADVANCED HSL SOLUTION ROUTINES IN PRIMAL-MIXED
  FEA THERMO-MECHANICAL ANALYSIS
• Iain Duff (President of SIAM) , Dubravka Mijuca

Computational Science and Engineering
Department, Atlas Centre, Rutherford Appleton
Laboratory, Oxon OX11 0QX Faculty of
Mathematics, University of Belgrade, Studentski
trg 16, 11000 Belgrade, Serbia
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Paper in press

• ON A NEW FULLY 3D MULTIFIELD FINITE ELEMENT
  APPROACH IN A SEMI-COUPLED THERMO-MECHANICAL
  ANALYSIS
• Dubravka Mijuca

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PRIMACY OF COMPUTATIONAL MECHANICS IN
COMPUTATIONAL SCIENCE

• The Mechanics exclusively give us insight in the
  motion and deformation of particles of both
  solids and fluids.

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Computational Mechanics Students

• Team while working on a NEC SX6i supercomputers
  at Japan - Serbia Center for Simulation Science
• www.center.bg.ac.yu/simulation/

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Reactor Engineering
With Prof. samu Motojima,  Director General of
the NIFST NINS Japan and Prof. Milos Skoric from
the NIFS Japan Nikola Tesla Museum Belgrade
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Special Session Finite element simulation of the
high risk constructions, experiences andproblems
2nd WSEAS International Conference on APPLIED and
THEORETICAL MECHANICS (MECHANICS '06), Venice,
Italy, November 20-22, 2006. www.worldses.org/conf
erences/2006/venice/mechanics
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AD ASTRA PER ASPERA