Identification of constitutive parameters for fractional viscoelasticity

https://doi.org/10.1016/j.cnsns.2013.05.019Get rights and content

Highlights

  • Proposing a model for parameters identification of fractional viscoelastic materials.

  • Identification results are not sensitive to noisy measurement responses.

  • Different space measurement points take no significant impact on results.

  • The ant number of ant colony optimization algorithm is as big as possible.

Abstract

This paper develops a numerical model to identify constitutive parameters in the fractional viscoelastic field. An explicit semi-analytical numerical model and a finite difference (FD) method based numerical model are derived for solving the direct homogenous and regionally inhomogeneous fractional viscoelastic problems, respectively. A continuous ant colony optimization (ACO) algorithm is employed to solve the inverse problem of identification. The feasibility of the proposed approach is illustrated via the numerical verification of a two-dimensional identification problem formulated by the fractional Kelvin–Voigt model, and the noisy data and regional inhomogeneity etc. are taken into account.

Introduction

The fractional derivative has become an extremely adequate tool to model mechanical properties of viscoelastic materials [1], because it is such an intimate descriptor of viscoelastic materials behavior that only a small number of parameters are enough to accurately represent a particular material [1], [2], [3], [4], [5]. An important problem connected with the fractional rheological models, as mentioned by Lewandowski [6], is the estimation of the model parameters from experimental data. As a matter of fact, there are a number of literatures related to this issue. However, these work seem to be mainly driven by determining parameters of simple fractional viscoelastic devices (such as viscoelastic dampers, dynamic vibration isolators etc. [6], [7], [8], [9], [10], [11], [12], [13]), or by identifying fractional viscoelastic parameters of simple and homogeneous structures (such as beams [14]), instead of the parameters identification of a homogeneous/inhomogeneous fractional viscoelastic field. On the other hand, these work were mostly carried out in the frequency domain. When measurement data are limited and polluted, an error may arise in the integral transform from time domain to the frequency domain, as shown in the Section 4.

With the above consideration, this paper attempts to tackle with two issues, i.e.

  • 1.

    The identification of constitutive parameters for fractional viscoelastic fields, instead of for simple devices or simple homogeneous structures.

  • 2.

    The identification of fractional constitutive parameters in the time domain, instead of in the frequency domain.

In the Section 1, a Finite Element (FE) equation is derived for the direct problem, which can be solved by the Laplace transform or FD method for 2-D homogenous and regionally inhomogeneous fractional viscoelastic fields.

In Sections 2 The governing equations of 2-D direct fractional viscoelastic problem, 3 The inverse problem for identification of fractional viscoelastic constitutive parameters, the identification of fractional viscoelastic constitutive parameters is treated as an optimization problem that is solved by an ACO algorithm [15].

The ACO algorithm needs the solution provided in the Section 1, but does not need derivatives of solutions with respect to constitutive parameters. These derivatives are demanded for the gradient based algorithm, and seem unease to calculate accurately due to a difficulty caused by the Gamma function. Although the ACO algorithm has successfully been applied to solve various kinds of inverse problems [16], [17], [18], [19], it seems first time to be used to solve inverse fractional viscoelastic problems.

In the Section 4, an identification problem of fractional viscoelastic constitutive parameters for a plate with a rectangular opening is investigated. The impacts of regional inhomogeneity, noisy data, and spatial arrangement of measurement points etc. on the solution are taken into account. The solutions obtained by the proposed approach are less sensitive to the noisy data given in this paper, and different spatial arrangements of measurement points seem no significant impact on the solution. Numerical results indicate that the proposed approach is available for the identification of constitutive parameters of homogeneous/regionally inhomogeneous fractional viscoelastic fields in the time domain.

Section snippets

The governing equations of 2-D direct fractional viscoelastic problem

The equilibrium equation of two-dimensional quasi-static problems is given by [20]LTσ+b=0xVThe relationship between the strain and displacement is described byε(t)=Lu(t)L=x10x20x2x1Twhere σ(t)={σx1(t),σx2(t),σx1x2(t)}T and ε(t)={εx1(t),εx2(t),εx1x2(t)}T stand for the vectors of stress and strain, respectively, x = {x1, x2} represents the vector of coordinates, b refers to the vector of the body force, and u(t)={ux1(t),ux2(t)}T represents the vector of displacements.

The boundary conditions

The inverse problem for identification of fractional viscoelastic constitutive parameters

The inverse problem in this paper mainly involves the combined identification of ακ, υκ, E0κ and E1κ, the solution can be obtained by minimizing a cost functional defined byJ(Φ)=n=1tedl=1ki=12u¯xi(x1l,x2l,tn,Φ)-u¯xir(x1l,x2l,tn)u¯xir(x1l,x2l,tn)2where ΦT={{ακ}T,{υκ}T,{E0κ}T,{E1κ}T}, {ακ}T = {α1, α2, …, ακ, ..., αMT}, {υκ}T = {υ1, υ2, …, υκ, …, υMT}, {E0κ}T={E01,E02,,E0κ,,E0MT}, {E1κ}T={E11,E12,,E1κ,,E1MT}. u¯xir(x1l,x2l,tn) and u¯xi(x1l,x2l,tn,Φ) refer to measured and estimated values of displacement

Implementation of ant colony algorithm

The major computing steps of the Gridding Partition based continuous ACO algorithm [25] include

  • Step 1

    • Set error bound λ

    • Set initial lower and upper bounds ϕ1-,ϕ¯l(l=1,2,3,,M)M is the number of variables to be identified

    • Set number of the ants K

    • Set number of the grids Nop

    • Set pheromone value ϑ0

    • Set evaporation rate of the pheromone ρ(ρ(0.5,1))

    • Set maximum number of iteration NLS

    • Set number of iteration kstep = 0

  • Step 2

    • Divide [ϕl-,ϕl¯] into Nop parts with the sizehl=ϕ1¯-ϕl-Nop

    • Set ϑjlkstep=ϑ0j=1,2,,(Nop+1

Numerical verification and discussion

For simplicity, all the computing parameters are assumed dimensionless.

Let us consider a plate with a rectangular opening as shown in Fig. 1 that also gives a description of FE mesh with 128 8-node elements. The size of computing domain is 10 × 10.

At first, let us examine a Laplace transformation with limited and polluted measurement data in the time domain at x1 = 3, x2 = 3, the computing parameters are f1 = 1 × 106, f2 = 2 × 105, α = 0.3217, E01=3.53×108, E11=3.462×109, υ = 0.15 ũx1r(3,3,sh) is given by [26]ũx1

Conclusions

The major contribution of this paper includes

  • 1.

    A numerical model in the time domain is presented to identifying unknown constitutive parameters for homogeneous/regionally inhomogeneous fractional viscoelastic fields, instead of for simple fractional viscoelastic devices or simple homogeneous fractional viscoelastic structures.

  • 2.

    An explicit semi-analytical numerical solution is derived for the direct homogenous fractional viscoelastic problem, which provides a quick solution of the direct problem.

Acknowledgments

The research leading to this paper is funded by NSF (10421002, 10772035, 10721062, 11072043), NKBRSF (2010CB832703).

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