P. Krysl 11/13/00

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Nonlinear Finite Element Simulation To do it
or not to do it, that is not the question.

• Petr Krysl

http//Hogwarts.ucsd.edu/pkrysl
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Outline

• Models, approximation, errors.
• Shell modeling w/subdivision surfaces.
• Adaptive discretizations.
• Optimal models.
• Conclusions.

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Modeling
Physical problem
Equations, assumptions
Mathematical idealization
Discretization(finite elements),algorithms
(solvers)
Numerical approximation
Accurate solutionof mathematical model?
Agreement w/ physics of problem?
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Predictive?
Experiment
Simulation
From OBrien et al. 1999

• Is this simulation capability predictive?

Not in this case non-physical material
properties had been used to make it look right
(computer graphics ).
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Is the world linear?

• Linearization is a (very useful) approximation of
  the way real world processes work.
• Nonlinear simulations are just a more
  sophisticated way of modeling.
• If used to predict how things work, the
  prediction will be more accurate.
• Accuracy should be thought of in terms of
  uncertainties.

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Are computers creative?

• Linear matrix equation Axb
• Solution x is not fundamentally new.
• Understanding.
• Computers transform.
• What error is involved in the transformation?
• Choose the right algorithm.
• Quantify the error.

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Modeling
Physical problem
Conceptual errors
Equations, assumptions
Mathematical idealization
Discretization errors
Discretization(finite elements),algorithms
(solvers)
Numerical approximation
Transformationerrors
Accurate solutionof mathematical model?
Agreement w/ physics of problem?
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Simulation Errors

• Errors
• Modeling
• Conceptual (omitted physics, range, )
• Approximation
• Discretization (space, time) and
• Transformation (finite-precision arithmetics!)
• Error control
• Quantify (error estimation), and
• Minimize (adaptivity).

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Example Modeling errors
Static force GgM 0.8Fy
Support
Wire
Mass M
w

• Range of validity (elasticity) and
• Neglected effect (damping).

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Quality Assurance
It is impossible to validate a computer code,
only a single simulations may be validated.
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Selected Topics in Modeling

• Thin Shells
• Discretization
• Adaptivity
• Model reduction (optimal models)

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Shell Analysis
Severe inelastic deformation, contact,
fragmentation, varied spatial and time scales.
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Subdivision Surfaces

• Sequence of control grids which all lead to the
  same limit surface.

• Properties
• Compactly supported basis
• C1
• Nested approximations (multiresolution).

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Subdivision Shell Elements
Subdivision well suited for Kirchhoff-Love
approximation of thin shells.
Cirak, Ortiz, Schröder 1999
Pinched Hemisphere Benchmark
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Nonlinear Problems
Cirak, Grinspun, Krysl, Ortiz, Schröder 2000
Airbag inflation
Thin Film delamination
Plastic buckling
Crumpling
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Adaptive Approx. Spaces

• Adaptive Spatial Multiresolution
• Remeshing
• Local refinement
• FEM Split finite elements, but ensure
  compatibility
• Constraints
• Lagrangian multipliers or penalty methods
• Irregular splitting of neighboring elements

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Conceptual Hierarchy

• Infinite sequence of globally refined spaces
• Mesh is globally refined to form
  and so on
• Strict nesting of

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1D Example

• One dimensional grid
• Subset is a particular refined basis

Nodes associated with active basis functions
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Refinement Implementation

• To refine replace the nodal patch to be refined
  by other, finer patches select appropriately
  degrees of freedom.

1D linear
Subdivision surface
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A Little Bit of Motivation

• Big, 10,000 degree-of-freedom model, right?

Wrong, 18 degree-of-freedom model, visually
indistinguishable from the big one.
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Cost of Simulations

• Asymptotic behaviour
• Example linear statics
• Assembly of K O(N), solution of Kxf O(N2).
• Total cost aN2 bN c
• Fixed budget (given grid)
• Select modes as linear combinations of hat
  functions

optimal
log(Error)
log(modes)
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Optimal Representation

• Example Mechanical system w/ 2 dofs

Optimal coordinates in configuration space
best-fit linear subspace
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Example Lead-plug damper
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RC Bridge under seismic loads

• El Centro accelerogram
• Isotropic damage model

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RC Bridge continued
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Grid Convergence Studies
Nonlinear analyses need answers with some
indication of accuracy. Approach establish
convergence by a series of analyses on refined
grids.
Full FE model between 20 and 4,320 hexahedral
elements. Offset line load, triangular loading
pulse, J-2 plasticity with combined hardening.
Model problem Elastoplastic arch
Displacement under load h1, 1/2, 1/3, 1/4,
1/5, 1/6
Contours of equivalent plastic strain
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Reduced Model Lower Cost
Approximate optimal representation on a finer
grid by re-interpolation of modes on a coarse
grid.
h1/5 to h1/6
Run time
Improved solution on a finer grid costs only a
fraction of the cost of solution on the coarser
grid.
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Monte Carlo Simulation
Notched plate with uncertain material properties
under low-cycle fatigue load
Contour of plastic deformation at the end of the
load cycling
Displacement of the loaded face
Copper plate 24x20x2 mm. Finite element model of
1/8 of the plate 415 hexahedra. J-2
elastoplastic material. Elastic and hardening
modulus, yield stress are uncertain with standard
deviation 0.025.
Evolution of the equivalent plastic strain at the
notch
Statistics of the response due to uncertainty?
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Monte Carlo Simulation cont
Monte Carlo Vary the material properties for
each repeated run. Problem may have to run
thousands of simulations. Hence use an optimal
(reduced) model.
Plastic strain at the notch mean value
Reduced dynamic model 3 Ritz modes yield
accuracy of plastic strain better than 2.
Plastic strain at the notch standard deviation
Full FE model 3,250 CPU sec Reduced FE model
170 CPU sec
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Engineering Design
Customer Preferences
Technical Specifications
Design Synthesis
Refinement
Design Description
Prototype Hardware
Function/ Behavior
Fabrication
Operation
Modeling and Simulation
Manufacturing
Product
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Design Space Exploration
Point-by-point
Design Variable Space
Performance Variable Space
Set-based
Inexpensive tangent approximation with reduced
models.
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Conclusions

• Robust, efficient, and scalable simulations in
  constant (and increasing) demand.
• Much more attention needs to be paid to
  verification validation.
• More attention should be given to
• Integration with design and manufacturing and
• Synergy with experiments.