SELFLEARNING FINITE ELEMENTS FOR INVERSE ESTIMATION OF THERMAL CONSTITUTIVE MODELS

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SELF-LEARNING FINITE ELEMENTS FOR INVERSE
ESTIMATION OF THERMALCONSTITUTIVE MODELS

• Wilkins Aquino and John C. Brigham
• School of Civil and Environmental Engineering
• Cornell University
• Ithaca, NY

8th US Congress on Computational Mechanics,
Austin, TX. July 2005
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Outline

• Motivation
• Neural networks and material modeling
• Self-learning finite elements
• Example
• Concluding remarks

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Motivation

• Need to monitor heat and moisture state of
  reinforced concrete structure remotely
• Difficulties with aging concrete
• Changes in internal microstructure due to damage
  and aging
• Heterogeneous (macro scale)
• Nonlinear response
• These difficulties translate into
• Changing thermal and mass transport properties
• Changing degree of anisotropy

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Neural Networks and Constitutive Models

• Neural networks are nonlinear mappings that can
  generate very complex behavior.
• NN are adaptive, imprecision tolerant, and can
  represent very general maps.
• Ghaboussi et al. introduced the concept of
  modeling material behavior using neural nets.
• Ghaboussi, J., Garret, J.H., and Wu, X (1991)
  Knowledge-base modeling of material behavior
  using neural networks, J. Eng. Mech. ASCE.
  117(1),132-153.

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Neural Networks (2)
Reed and Marks, Neural Smithing, 1999
Feedforward neural network thermal model
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Neural Networks (3)

• NN are trained with a set of known inputs and
  outputs. After successful training NN are able to
  generalize and predict behavior outside of the
  training set.
• There are many different training algorithms for
  NN based on conventional and non-conventional
  optimization theories.

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Self-Learning Finite Element Analysis

• How can we use NN and FEA to solve inverse
  material identification problems?
• NN can be used as the material model in finite
  element analysis.
• But, how can we train a NN with data (e.g. heat
  fluxes, temperature gradients, and tempratures)
  that we dont have?
• Self-learning techniques are used for this
  purpose.

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Inverse Steady State Heat Transfer Problem

• Determine thermal conductivity of material from
  steady-state temperature field.

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Neural Network Thermal Models
Constitutive Model for Heat Flow
Neural network representation of constitutive
behavior. Notice that Fouriers law need not be
valid in this representation.
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Implementation of Neural Network Thermal Model in
Finite Element Scheme

• Finite element formulation of the forward
  steady-state heat transfer problem using neural
  network material model

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How is NN Trained?

• A comprehensive data set (i.e. temperature
  gradients, temperature, fluxes) is not known a
  priori.
• An idea Can we take advantage of data produced
  during iterative finite element analyses?
• Indeed, this is the concept that self-learning
  methods use.

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Self-learning Numerical Methods For Inverse
Material Identification

• Construct a numerical model (e.g. FE) of the
  system.
• Define material behavior in FE model using a
  neural-network representation.
• Pretrain NN model using an arbitrary, but
  physically meaningful data set.
• The neural-network material model is trained in
  successive analyses of the system using internal
  data (e.g. stress and strain, temperature
  gradients and fluxes, etc.).
• This data is generated by the neural network
  itself inside the numerical model.

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Self-learning Algorithm (2)
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Self-learning Algorithm (3)

• Main concepts
• Fluxes obtained from first finite element
  analysis are in weak equilibrium with external
  loads.
• Correction in temperature gradient and
  temperature fields occurs during second finite
  element analysis.
• The set formed these two analyses contains new
  information about the real material behavior.
• These information is learned by the neural
  network in the training part of the algorithm.

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Plate ExampleTemperature Dependent Conductivity
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Plate Example (4)Error Behavior
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Y
X
Heat Flux in X-Direction
Heat Flux in Y-Direction
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K11 in conductivity tensor
K22 in conductivity tensor
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K21
K12
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System with Three Rows of Sensors and Gaussian
Noise
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Average Error in NN Estimates After Training
Process
1RF one Row of sensors- no noise 3RF three rows
of sensors-no noise 5RF five rows of sensors-no
noise N indicates same cases with noise added
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Concluding Remarks

• The presented self-learning method provides a
  means to inversely estimate entire constitutive
  models.
• It can be very useful in applications where high
  adaptability of a material model is needed (e.g.
  system is sampled over time and material model
  needs to be updated).
• This method can be readily extended to
  multiphysics problems.