Introduction

calabru is a model calibration package.

A basic working example is presented to demonstrate the package.

Let’s consider a simple beam model created in Python using PyCBA(https://ccaprani.github.io/pycba/notebooks/intro.html) module. We begin with some necessary imports:

import calabru as clb # calabru package
import numpy as np # for numerical arrays
import matplotlib.pyplot as plt # for plotting
from IPython import display # For images in notebook
import pycba as cba # for creating beam model
%matplotlib inline

The beam model represents a two span bridge, with a UDL of 20 kN/m acting on each span.

display.Image("../images/pycba_example.png",width=800)

For the calibration problem, we wish to update the model such that the simulated deflection at midspan of Span 1, when subjected to the 20kN/m UDL, matches closely with a known target measurement. We define this target measurement be the deflection obtained from the original example:

m = 1
milli = 1e-3
target_deflections = [- 2.06 * milli*m]

The model is updated by tuning the bending stiffness parameter (EI).

EI_start = [20 * 600e7 * 1e-6]

Workflow

A minimal workflow of calabru consist of the following steps:

1. Create a function handler for the model that will be updated.

def create_pycba(parameters):
    # init parameters according to PyCBA
    EI = parameters[0]
    LM = [[1, 1,  20, 0, 0], [2, 1, 20, 0, 0]]
    L = [7.5, 7.0]
    R = [-1, 0, -1, 0, -1, 0]

    # PyCBA commands
    beam_analysis = cba.BeamAnalysis(L, EI, R, LM)
    beam_analysis.analyze()

    # get the target response i.e. the maximum deflection at midspan of Span 1 (7.5m) - index = 44
    deflection_responses = [beam_analysis.beam_results.results.D[44]]

    return deflection_responses

As can be seen, create_pycba is a function handler which creates and analyzes the PyCBA beam model. The function handler must have the following feature: - Updating parameters as a list/array. - returns the model output that corresponds to the target measurements as a list/array.

For this example, parameters is the list input argument of the bending stiffness while deflection_responses is the list of deflection output from the beam model.

2. Invoke the ``ModelUpdating`` class.

update_process = clb.ModelUpdating(
                                    function_handle=create_pycba,
                                    param_list=EI_start,
                                    target_list=target_deflections,
                                    max_error=0.01,
                                    )

3. Run the class function to update model.

param_history, resp_history  = update_process.update_model()

where param_history is the updating history of the given model parameters, and resp_history is the history of differences between model and target measurements for each updating step.

Alternatively, we can directly access the class variables of update_process (primarily list/arrays). As such, the parameters and residuls can be use for post-processing later on (e.g., plotting).The typical outputs can be extracted as follows:

# for update history of parameters
update_process.param_update_history

[[120000.0],
 [164129.0074781965],
 [180289.11776900673],
 [180261.77743457866],
 [180264.5151988686]]

# for history of model differences
update_process.resp_diff_history

[[0.0010345365722656233], [0.0002025152882936033], [-2.839546441430439e-07]]

We can then compare with PyCBA’s example, here we get

EI_pycba = 30 * 600e7 * 1e-6
EI_updated = update_process.param_update_history[-1]

print("The target EI ={}".format(EI_pycba))
print("The updated EI ={:1f}".format(EI_updated[0]))

The target EI =180000.0
The updated EI =180264.515199

which is very close to the example’s orginal value for EI.