INVERSION: SIMPLE BEAM CALIBRATION¶

In this example, the measured deflections of a simply supported beam under a uniform load $p$ are used to calibrate the Young's Modulus $E$ of the beam material. Basic uncertainty on the applied $p$ is assumed. The calibration is carried out with default model/data discrepancy options.

Package imports¶

In [1]:
from uqpylab import sessions
import numpy as np

Start a remote UQCloud session¶

In [2]:
# Start the session
mySession = sessions.cloud()
# (Optional) Get a convenient handle to the command line interface
uq = mySession.cli
# Reset the session
mySession.reset()

Processing .
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 done!

 uqpylab.sessions :: INFO     :: This is UQ[py]Lab, version 1.00, running on https://uqcloud.ethz.ch. 
                                 UQ[py]Lab is free software, published under the open source BSD 3-clause license.
                                 To request special permissions, please contact:
                                  - Stefano Marelli (marelli@ibk.baug.ethz.ch).
                                 A new session (157415016f764591ba6def75feccdaec) started.

 uqpylab.sessions :: INFO     :: Reset successful.

Set the random seed for reproducibility¶

In [3]:
uq.rng(100, 'twister');

Forward model¶

The simply supported beam problem is shown in the following figure:

No description has been provided for this image

The forward model computes the deflection of the beam $V$ at mid-span location, which reads:

$$ V = \frac{ 5 p L^4 }{32 E b h^3}$$

This computation is carried out by the function uq_SimplySupportedBeam(X) supplied with UQLab. The input variables of this function are gathered into the $N \times M$ matrix X, where $N$ and $M$ are the number of realizations input variables, respectively. The variables are given in the following order:

  • $b$: beam width $(m)$
  • $h$: beam height $(m)$
  • $L$: beam length $(m)$
  • $E$: Young's modulus $(MPa)$
  • $p$: uniform load $(kN/m)$

Define the forward model as a MODEL object:

In [4]:
ModelOpts = {
    'Type' : 'Model',
    'mString': '(5/32)*(X(:, 5).*X(:, 3).^4)./(X(:, 4).*X(:, 1).*X(:, 2).^3)',
    'isVectorized': True
}
In [5]:
myModel = uq.createModel(ModelOpts)

Prior distribution of the model parameters¶

The prior information about the model parameters is gathered in a probabilistic model that includes both known and unknown parameters.

The geometrical dimensions $b$ (beam width), $h$ (beam height) and $L$ (beam length) are perfectly known:

  • $b = 0.15$ m
  • $h=0.3$ m
  • $L = 5$ m

The applied load $p$ is known up to some Gaussian measurement noise. The Young's modulus $E$, target of the calibration experiment, is given a lognormal prior distribution:

  • $p \sim \mathcal{N}(\mu_p = 1.2\times10^{-2}, \sigma_p = 6\times10^{-4})$
  • $E \sim \mathcal{LN}(\mu_E = 3\times10^4, \sigma_E = 4.5\times10^3)$

Specify these distributions as an INPUT object:

In [6]:
PriorOpts = {
  "Marginals": [
    {
      "Name": "b", # beam width
      "Type": "Constant",
      "Parameters": [0.15], # (m)
    },
    {
      "Name": "h", # beam height
      "Type": "Constant",
      "Parameters": [0.3], # (m)
    },
    {
      "Name": "L", # beam length
      "Type": "Constant",
      "Parameters": [5], # (m) 
    },
    {
      "Name": "E", # Young's modulus
      "Type": "LogNormal",
      "Moments": [30e9,4.5e9] # (Pa)
    },
    {
      "Name": "p",           # uniform load
      "Type": "Gaussian",
      "Moments": [12000,600] # (N/m)
    }
  ]
}
In [7]:
myPriorDist = uq.createInput(PriorOpts)

Constant model parameters in the prior indicate certainty about their value. These parameters will not be calibrated.

Measurement data¶

The measurement data consists of $N = 5$ independent measurements of the beam mid-span deflection. The data is stored in the column vector y:

In [8]:
myData = {
    "y": (np.array([12.84, 13.12, 12.13, 12.19, 12.67])/1000).tolist(),
    "Name": "Mid-span deflection",
}

Discrepancy model¶

By default, the Bayesian calibration module of UQLab assumes an independent and identically distributed discrepancy $\varepsilon\sim\mathcal{N}(0,\sigma^2)$ between the observations and the predictions for each data. The variance $\sigma^2$ of the discrepancy term is by default given a uniform prior distribution:

$$ \pi(\sigma^2) = \mathcal{U}(0,\mu_{\mathcal{Y}}^2), \quad \mathrm{with} \quad \mu_{\mathcal{Y}} = \frac{1}{N}\sum_{j=1}^{N}y_{j} \quad (\mathrm{here~equal~to}~0.01259)$$

Bayesian analysis¶

The options of the Bayesian analysis are specified with the following structure:

In [9]:
BayesOpts = {
    "Type" : "Inversion",
    "Data" : myData,
}

Run the Bayesian inversion analysis:

In [10]:
myBayesianAnalysis = uq.createAnalysis(BayesOpts)

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Print out a report of the results:

In [11]:
uq.print(myBayesianAnalysis)

%----------------------- Inversion output -----------------------%
   Number of calibrated model parameters:         2
   Number of non-calibrated model parameters:     3

   Number of calibrated discrepancy parameters:   1

%------------------- Data and Discrepancy
%  Data-/Discrepancy group 1:
   Number of independent observations:            5

   Discrepancy:
      Type:                                       Gaussian
      Discrepancy family:                         Scalar
      Discrepancy parameters known:               No

   Associated outputs:
      Model 1: 
         Output dimensions:                       1

%------------------- Solver
   Solution method:                               MCMC

   Algorithm:                                     AIES
   Duration (HH:MM:SS):                           00:01:12
   Number of sample points:                       3.00e+04

%------------------- Posterior Marginals
---------------------------------------------------------------------
| Parameter | Mean    | Std     | (0.025-0.97) Quant. | Type        |
---------------------------------------------------------------------
| E         | 2.4e+10 | 2.1e+09 | (2.1e+10 - 3e+10)   | Model       |
| p         | 1.2e+04 | 5.9e+02 | (1.1e+04 - 1.3e+04) | Model       |
| Sigma2    | 4.2e-06 | 1.3e-05 | (1e-07 - 3.8e-05)   | Discrepancy |
---------------------------------------------------------------------

%------------------- Point estimate
----------------------------------------
| Parameter | Mean    | Parameter Type |
----------------------------------------
| E         | 2.4e+10 | Model          |
| p         | 1.2e+04 | Model          |
| Sigma2    | 4.2e-06 | Discrepancy    |
----------------------------------------

%------------------- Correlation matrix (model parameters)
----------------------
|   |  E       p     |
----------------------
| E |  1       0.46  |
| p |  0.46    1     |
----------------------

Create a graphical representation of the results:

In [12]:
uq.display(myBayesianAnalysis);

Terminate the remote UQCloud session¶

In [13]:
mySession.quit()

 uqpylab.sessions :: INFO     :: Session 157415016f764591ba6def75feccdaec terminated.
Out[13]:
True