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Detection of branching points in noisy processes

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Abstract

Processes in engineering mechanics often contain branching points at which the system can follow different physical paths. In this paper a method for the detection of these branching points is proposed for processes that are affected by noise. It is assumed that a bundle of process records are available from numerical simulations or from experiments, and branching points are concealed by the noise of the process. The bundle of process records is then evaluated at a series of discrete values of the independent process coordinates. At each discrete point of the process, the associated point set of process values is investigated with the aid of cluster analysis. The detected branching points are verified with a recursive algorithm. The revealed information about the branching points can be used to identify the physical and mechanical background for the branching. This helps to better understand a mechanical system and to design it optimal for a specific purpose. The proposed method is demonstrated by means of both a numerical example and a practical example of a crashworthiness investigation.

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Abbreviations

b :

Branching point

C :

Cluster (point set)

\({\tilde{C}}\) :

Fuzzy cluster (discrete fuzzy set)

|C|:

Cardinality of cluster C

\({d\left(\underline{z}_{1},\underline{z}_{2}\right)}\) :

Distance between \({\underline{z}_{1}}\) and \({\underline{z}_{2}}\)

\({\mathbb{\underline{D}}}\) :

Real-valued domain of process coordinates

f, →:

Mapping

G :

Quality measure for cluster configuration

L U , L SC :

Length of record for check of uncertainty and silhouette coefficient, respectively

M :

Set

O(.):

Set of outliers

p(.):

Representative process record

P(.):

Process group

SC :

Silhouette coefficient

U :

Uncertainty

v :

Velocity

\({\underline{x}(.)}\) :

Mechanical input quantities

\({\underline{z}(.)}\) :

Process variables

X, Z:

Random variables

X ~ (.):

Specification of the distribution of X according to (.)

\({\underline{\mathbb{Z}}}\) :

Real-valued domain of process variables

|:

For which the following holds

ε :

Noise

\({\underline{\theta}}\) :

Process coordinates

μ :

Membership value according to fuzzy set theory

τ :

Single process coordinate

(a, b, c, . . .):

Vector with elements a, b, c, . . .

(a, b):

Open interval with limits a and b

(z, μ):

Value pair with elements z and μ

[a, b]:

Closed interval with bounds a and b

{a, b, c}:

Set with elements a, b, c

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Authors and Affiliations

  1. Department of Civil Engineering, National University of Singapore, Singapore, Singapore

    Michael Beer

  2. Dynamore GmbH, Stuttgart, Germany

    Martin Liebscher

Authors
  1. Michael Beer
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  2. Martin Liebscher
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Correspondence to Michael Beer.

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Beer, M., Liebscher, M. Detection of branching points in noisy processes. Comput Mech 45, 363–374 (2010). https://doi.org/10.1007/s00466-009-0458-4

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