Fuzzy dynamics of multibody systems with polymorphic uncertainty in the material microstructure

Abstract

This paper deals with the fuzzy dynamics of multibody systems with polymorphic uncertainty in the material microstructure. Macroscopic material properties are obtained using fuzzy-stochastic FEM based computational homogenization. In particular, the spectral stochastic local FEM is utilized to simulate a representative volume element of the microstructure. Forward dynamics of the macroscopic system is modeled using the Graph Follower algorithm. Thereby we propagate the uncertainty from the lowest level of material microstructure to the highest level of multibody dynamics. Differences in the propagation of epistemic and aleatoric uncertainties to the macroscale and their influence on the multibody dynamics are discussed. A particular example of a multibody system used in this paper is a multistory frame, whereby the considered heterogeneous material is a cement-based concrete.

Appendix A: Short introduction to fuzzy set theory

The history of fuzzy numbers began in 1965 with the introduction of fuzzy sets [3] which are an extension of the classical set theory based on the notion of different grades of membership.

Let us briefly repeat some basic definitions from fuzzy set theory [3]. In the case of a fuzzy set $\tilde{\mathcal {P}}$ the grade of membership of p is defined by the membership function $\mu _{\tilde{\mathcal {P}}}(p) \in [0,~1]$. Here $\mu _{\tilde{\mathcal {P}}}(p)=1$ means that the element p entirely belongs to the set $\tilde{\mathcal {P}}$, $\mu _{\tilde{\mathcal {P}}}(p)=0$ means that p is not a member of the set $\tilde{\mathcal {P}}$. In the case of a conventional set $\mathcal {P}$ the membership function of an element p may have only two values $\mu _{\mathcal {P}}(p)\in \{0,~1\}$, i.e. the element can only belong to or not belong to the set $\mathcal {P}$.

For practical applications, a few very important types of fuzzy sets are fuzzy numbers, fuzzy intervals, crisp numbers, and crisp intervals. A fuzzy number $\tilde{a}$ is the convex fuzzy set over the universal set $\mathbb {R}$ with the membership function $\mu _{\tilde{a}}(p) \in [0,~1]$, where $\mu _{\tilde{a}}(p)=1$ only for one single value of $p=\bar{a}$ called the modal value. The fuzzy interval $\tilde{A}$ is the convex fuzzy set defined similarly to the fuzzy number, however with the difference that $\mu _{\tilde{A}}(p)=1$ holds for some interval called modal interval $\bar{A}$. A crisp interval A can be considered as the fuzzy set of points such that $\mu _{A}(p)=1$, if $p\in A$, and $\mu _{A}(p)=0$ otherwise. The crisp number a is then the fuzzy set with the membership function given by the Kronecker delta function $\mu _a(p)=\delta (p,a)$. Figure 36 represents from left to right: crisp number $p=1$, crisp interval [2,  3], symmetric triangular fuzzy number with $\bar{p}=4.5$, fuzzy interval with the modal interval $p\in [6.5,~7.5]$, and the arbitrary non-convex subnormal fuzzy set with nonzero membership function on the interval $p\in [9,~11]$.

Fig. 36

Membership function plotted for (from left to right): crisp number, crisp interval, symmetric triangular fuzzy number, fuzzy interval, and the arbitrary non-convex subnormal fuzzy set

Zadeh’s extension principle is used to perform unary and binary arithmetical operations of fuzzy numbers. Due to the high complexity of calculations performed using the extension principle, an alternative approach was proposed in the literature. The fuzzy numbers are reduced to sets of nested intervals for different degrees of membership, i.e. $\alpha$-cuts (Fig. 37). These intervals are also called intervals of confidence [10]. Lower and upper bounds for every quantity of interest are then evaluated for every $\alpha$-cut. Collection of intervals of confidence for output quantity results naturally in the reconstruction of the output’s membership function.

Fig. 37

Triangular fuzzy number with modal value m decomposed into 6 $\alpha$-cuts

In most cases, two optimization problems must be solved for every $\alpha$-cut. However, if the evaluation of the system is costly, the optimization approach becomes too expensive. As an alternative one may use the extended transformation method [10].