Implementation of the Recursive Formula for Evaluation of the Strength of Daniels’ Model
The paper deals with the classical fiber bundle model
of equal load sharing, sometimes referred to as the Daniels’ bundle
or the democratic bundle. Daniels formulated a multidimensional
integral and also a recursive formula for evaluation of the
strength cumulative distribution function. This paper describes
three algorithms for evaluation of the recursive formula and also
their implementations with source codes in the Python high-level
programming language. A comparison of the algorithms are provided
with respect to execution time. Analysis of orders of magnitudes of
addends in the recursion is also provided.
[1] H.E. Daniels, The Statistical Theory of the Strength of Bundles of
Threads, Proc. Royal Soc. (London), 183A, 405–435, 1945.
[2] H.E. Daniels, The Maximum of a Gaussian Process Whose Mean Path
Has a Maximum, with an Application to the Strength of Bundles of
fibers, Advances in Applied Probability, 21 (2), 315–333, 1989.
[3] R.L. Smith, S.L. Phoenix, Asymptotic Distributions for the Failure
of Fibrous Materials Under Series-Parallel Structure and Equal
Load-Sharing, Journal of Applied Mechanics, 48, 75–82, 1981.
[4] D. Sornette, Elasticity and failure of a set of elements loaded in parallel.,
J. Phys. A: Math. Gen., 22, L243–L250, 1989. (Letter to the Editor).
[5] G. Rossum, et al., The Python Programming Language, http://python.
org, 1991.
[6] F. Johansson, et al., mpmath: a Python library for arbitrary-precision
floating-point arithmetic (version 0.18), http://mpmath.org, 2014.
[7] T. Oliphant, et al., NumPy: Open source scientific package for scientific
computing with Python, http://numpy.scipy.org, 2006.
[1] H.E. Daniels, The Statistical Theory of the Strength of Bundles of
Threads, Proc. Royal Soc. (London), 183A, 405–435, 1945.
[2] H.E. Daniels, The Maximum of a Gaussian Process Whose Mean Path
Has a Maximum, with an Application to the Strength of Bundles of
fibers, Advances in Applied Probability, 21 (2), 315–333, 1989.
[3] R.L. Smith, S.L. Phoenix, Asymptotic Distributions for the Failure
of Fibrous Materials Under Series-Parallel Structure and Equal
Load-Sharing, Journal of Applied Mechanics, 48, 75–82, 1981.
[4] D. Sornette, Elasticity and failure of a set of elements loaded in parallel.,
J. Phys. A: Math. Gen., 22, L243–L250, 1989. (Letter to the Editor).
[5] G. Rossum, et al., The Python Programming Language, http://python.
org, 1991.
[6] F. Johansson, et al., mpmath: a Python library for arbitrary-precision
floating-point arithmetic (version 0.18), http://mpmath.org, 2014.
[7] T. Oliphant, et al., NumPy: Open source scientific package for scientific
computing with Python, http://numpy.scipy.org, 2006.
@article{"International Journal of Mechanical, Industrial and Aerospace Sciences:70085", author = "Václav Sadílek and Miroslav Vořechovský", title = "Implementation of the Recursive Formula for Evaluation of the Strength of Daniels’ Model", abstract = "The paper deals with the classical fiber bundle model
of equal load sharing, sometimes referred to as the Daniels’ bundle
or the democratic bundle. Daniels formulated a multidimensional
integral and also a recursive formula for evaluation of the
strength cumulative distribution function. This paper describes
three algorithms for evaluation of the recursive formula and also
their implementations with source codes in the Python high-level
programming language. A comparison of the algorithms are provided
with respect to execution time. Analysis of orders of magnitudes of
addends in the recursion is also provided.", keywords = "Daniels bundle model, equal load sharing, Python,
mpmath.", volume = "9", number = "6", pages = "991-5", }