A method for determining the parameters in a rheological model for viscoelastic materials by minimizing Tikhonov functionals
Rebecca Rothermel, Wladimir Panfilenko, Prateek Sharma, Anne Wald, Thomas Schuster, Anne Jung, Stefan Diebels
AA METHOD FOR DETERMINING THE PARAMETERS IN ARHEOLOGICAL MODEL FOR VISCOELASTIC MATERIALS BYMINIMIZING TIKHONOV FUNCTIONALS
REBECCA ROTHERMEL ∗ , WLADIMIR PANFILENKO ∗ , PRATEEK SHARMA § , ANNEWALD † , THOMAS SCHUSTER ∗‡ , ANNE JUNG § , AND
STEFAN DIEBELS § Abstract.
Mathematical models describing the behavior of viscoelastic materials are often basedon evolution equations that measure the change in stress depending on its material parameters suchas stiffness, viscosity or relaxation time. In this article, we introduce a Maxwell-based rheologicalmodel, define the associated forward operator and the inverse problem in order to determine thenumber of Maxwell elements and the material parameters of the underlying viscoelastic material.We perform a relaxation experiment by applying a strain to the material and measure the generatedstress. Since the measured data varies with the number of Maxwell elements, the forward operatorof the underlying inverse problem depends on parts of the solution. By introducing assumptionson the relaxation times, we propose a clustering algorithm to resolve this problem. We provide thecalculations that are necessary for the minimization process and conclude with numerical resultsby investigating unperturbed as well as noisy data. We present different reconstruction approachesbased on minimizing a least squares functional. Furthermore, we look at individual stress componentsto analyze different displacement rates. Finally, we study reconstructions with shortened data setsto obtain assertions on how long experiments have to be performed to identify conclusive materialparameters.
Key words. parameter identification, viscoelasticity, inverse problem, rheological model, solu-tion dependent forward operator, Tikhonov functional
MSC 2010:
1. Introduction.
Simulations are used to predict the behavior of a materialwithout conducting expensive experiments. In order to achieve accurate results froma simulation, the physical behavior as well as the properties of the material shouldbe captured by the simulation model. The behavior of the material can be inter-preted by the qualitative analysis of an experimental result obtained from materialtesting. However, the properties of the material need to be quantitatively deter-mined through comparison of the simulation and experimental results. With varia-tion and adjustment of the simulation results, the parameters of the model are op-timised to fit the experimental data. Thus, parameter identification in materialsscience is an important field of application for inverse problems with many differentpurposes. Among them are the identification of the stored energy function of hypere-lastic materials [3,17,24,35,36,42], the surface enthalpy dependent heat fluxes of steelplates [31,32], inverse scattering problems [6] or the refractive index through terahertztomography [41]. All these inverse problems are ill-posed in the sense that even smallperturbations in the measured data cause severe artifacts in the solution. This is whythe application of regularization methods is necessary. A standard approach is to usethe residual between simulations and measurements as a data fitting term, see [5, 18].Further authors conduct a sensitivity analysis to determine the possible deviations inthe parameters due to noise [12, 15, 16, 26]. ∗ DEPARTMENT OF MATHEMATICS, SAARLAND UNIVERSITY SAARBRÜCKEN,GERMANY § CHAIR OF APPLIED MECHANICS, SAARLAND UNIVERSITY SAARBRÜCKEN,GERMANY † INSTITUTE FOR NUMERICAL AND APPLIED MATHEMATICS, UNIVERSITY OFGÖTTINGEN, GERMANY ‡ EMAIL: [email protected] , CORRESPONDING AUTHOR1 a r X i v : . [ m a t h . NA ] F e b iscoelastic material behavior is exhibited by various polymeric materials such asadhesives, elastomers and rubber. The modeling of these materials requires param-eters that define the relaxation time spectrum, to model the loading rate dependentbehavior [4,19,33,37]. The reconstruction of these parameters do not only depend onthe precision, but also on the duration of the experiments. The numerical implemen-tation of a continuous relaxation time spectrum with discrete parameters represents afurther challenge. Therefore, the parameter identification of viscoelastic materials isin great demand and is used in many fields. Results for the identification of materialparameters of viscoelastic structures with a linear model can also be found in [8, 11].High-order discontinuous Galerkin methods for anisotropic viscoelastic wave equa-tions are found in [38]. There, the authors use a constitutive equation with memoryvariables. An overview of different methods for solving inverse problems and deeperinsights to regularization theory can be found in the standard textbooks [9,23,25,30].For nonlinear problems, such as the one presented in this article, these methods haveto be adapted correspondingly (see [22, 30, 39, 40]). Outline.
In Section 2 we describe the rheological model for viscoelastic materialson which the investigations in this article are based. Assuming that the number ofMaxwell elements are given the constitutive model can be solved analytically, whichis done in Section 3. Relying on this model the simulation of data is performed inSection 4. The task of computing the model parameter from measured stresses at cer-tain time instances is an inverse problem. In contrast to standard settings for inverseproblems, the forward operator depends in this case on parts of the solution, namelythe number of maxwell elements. In Section 5 we develop a clustering algorithm toovercome this difficulty. In Section 6 we perform a series of numerical experimentsusing exact as well as noisy data. Since the inverse problem is ill-posed we intro-duce regularization methods by minimizing corresponding Tikhonov functionals andprove that this stabilizes the solution process. Furthermore we demonstrate the ef-fect of shortening the data series and finally obtain assertions on its influence to thereconstruction result. A concluding section finishes the article.
2. A mathematical model for material behavior.
We consider relaxationexperiments in which a strain is applied to a material. For a given displacement rate η and maximum strain value ¯ ε we write the strain in a time interval t ∈ [0 , T ] as ε ( t ) = (cid:26) η · t, ≤ t ≤ ¯ εη ¯ ε, ¯ εη < t ≤ T. (2.1)The function is illustrated in Figure 2.1 and describes the following procedure. Thematerial is stretched at a displacement rate η until we reach a maximum strain value¯ ε . This happens accordingly at time t = ¯ εη . Afterwards the applied strain is keptconstant.We describe the stress in the material by a rheological multi-parameter modelcombining several Maxwell elements. Two opposing properties are essentially deter-mining the time evolution of a material under strain, elasticity and viscosity. Elasticityis modeled by a spring, whereas viscosity is modeled by a damper. The series com-position of a spring and a damper yields a Maxwell element. We use one of the mostcommon combinations to represent the material, a parallel combination of a springwith different Maxwell elements [14, 21, 33]. A parallel combination of these elementsensures that the entire relaxation spectrum of the material can be represented by themodel. = 0 t = T Time [s] S t r a i n [ % ] ¯ ε t = ¯ εη Figure 2.1: Strain-time curve ε with displacement rate η and maximum strain value¯ ε µ j=1 j=2 j=3 µ µ µ τ τ τ Figure 2.2: Rheological Model with three Maxwell elementsThe number of Maxwell elements can extend to any number n with a relaxationtime τ j in each of the dampers and a stiffness of the spring µ j in each of the Maxwellelements. In Figure 2.2 we see an illustration with three Maxwell elements. Thedeformation across the damper and across the spring in a Maxwell element changesaccording to the relaxation time of the damper. The Maxwell springs reach the origi-nal undeformed position and do not produce any stress after fully relaxing. Therefore,with a single Maxwell element the model can only simulate one stiffness for the entireduration of relaxation of the material. However, viscoelastic materials show differ-ent stiffness at different relaxation times [14, 37]. The entire relaxation spectrum canbe divided into different regions such as the flow region, the entanglement region,the transition region and the glassy region [1]. The arrangement, the length andthe entanglement of the polymeric chains on the molecular level explains the vary-ing stiffness during the relaxation spectrum. A shorter chain relaxes faster than alonger chain and hence after its relaxation does not contribute to the stiffness of thematerial [2]. Moreover, if all the Maxwell elements relax to zero stresses then theequilibrium position is attained and the single spring with the stiffness µ representsthe basic stiffness of the material, which ensures that the material has a stiffness in ts unperturbed position and does not behave like a fluid. The deformation of thematerial represented by the strain value ε is divided into an elastic component ε ej and an inelastic component ε ij in each of the Maxwell elements. The elastic compo-nent corresponds to the spring and the non-elastic component to the damper. As thestretching of the damper is dependent on its relaxation time an evolution equationbased on the energy conservation is used to model the change in the inelastic strainwith time [20, 29]. For small deformations the evolution equation is given by(2.2) ˙ ε ij ( t ) = ε ( t ) − ε ij ( t ) τ j / , where ˙ ε ij represents the time derivative. The total stress produced in the system isthen given by the sum of stresses induced in each of the springs. Assuming a linearelastic behavior of the springs. The stress is given by(2.3) σ ( t ) = µ ε ( t ) + n (cid:88) j =1 µ j ( ε ( t ) − ε ij ( t )) . With this information we have now all ingredients available to construct the forwardoperator as F n : R n +1 ≥ × N → L ([0 , T ])(2.4) ( µ, µ , . . . , µ n , τ , . . . τ n , n ) σ, (2.5)where n is the number of Maxwell elements and σ is the stress defined by equation(2.3). We note that the forward operator depends on parts of the solution at thispoint since we require the unknown number of Maxwell elements n to calculate thestress. We will solve this problem in Section 5.At this point, the inelastic stresses are included as a constraint in the formulationof the inverse problem. In the following section we will see that these constraints canbe directly included to obtain an explicit formulation of the forward operator.
3. Analytical solution of the forward problem.
For the solution of theforward problem, we first want to focus on the calculation of the inelastic componentof the strain value ε ij , which is not explicitly given as a so-called internal variable ofthe system. An evolution equation such as (2.2) is often solved by numerical methodslike the Crank-Nicolson method. However, these methods have the disadvantage ofapproximation errors. Since we are dealing with an ill-posed inverse problem, it isdesirable to avoid such errors in the calculation of the forward problem. To thisend, we discuss the analytical solution of the evolution equation and, using this, theforward problem in the next section.Apart from the mechanical application [29], the evolution equation (2.2) is alsoused in other applications like Magnetic Particle Imaging [7] to model relaxation. Itssolution can be formulated analytically as ε ij ( t ) = t (cid:90) ε (˜ t ) 2 τ j exp (cid:18) − t − ˜ tτ j (cid:19) d ˜ t. By inserting the piecewise linear strain (2.1) we obtain the following result:
Proposition 3.1. he inelastic strain ε ij of the j -th Maxwell element with the corresponding relax-ation time τ j at time t ∈ [0 , T ] is given by ε ij ( t ) = τ j η exp (cid:16) − τ j t (cid:17) + ηt − τ j η, ≤ t ≤ ¯ εητ j η exp (cid:16) − τ j t (cid:17) (cid:104) − exp (cid:16) ετ j η (cid:17)(cid:105) + ¯ ε, ¯ εη < t ≤ T. (3.1) Proof . We first consider 0 ≤ t ≤ ¯ εη and can thus use ε ( t ) = ηt , ε ij ( t ) = 2 τ j η exp (cid:18) − τ j t (cid:19) t (cid:90) ˜ t exp (cid:18) τ j ˜ t (cid:19) d ˜ t = 2 τ j η exp (cid:18) − τ j t (cid:19) τ j (cid:20) exp (cid:18) τ j t (cid:19) (cid:18) τ j t − (cid:19) + 1 (cid:21) , where we used t (cid:82) ˜ t exp (cid:0) a ˜ t (cid:1) d ˜ t = a (exp ( at ) ( at −
1) + 1) for a ∈ R in the last step.The remainder is only a matter of transformation, ε ij ( t ) = τ j η exp (cid:18) − τ j t (cid:19) τ j t exp (cid:18) τ j t (cid:19) − τ j η exp (cid:18) − τ j t (cid:19) exp (cid:18) τ j t (cid:19) + τ j η exp (cid:18) − τ j t (cid:19) = τ j η exp (cid:18) − τ j t (cid:19) + ηt − τ j η. Next we examine the case ¯ εη < t ≤ T , where we have to split the integral in two partsto be able to use the corresponding value for ε ( t ), ε ij ( t ) = 2 τ j exp (cid:18) − τ j t (cid:19) η ¯ εη (cid:90) ˜ t exp (cid:18) τ j ˜ t (cid:19) d ˜ t + ¯ ε t (cid:90) ¯ εη exp (cid:18) τ j ˜ t (cid:19) d ˜ t . For each integral we then get ¯ εη (cid:90) ˜ t exp (cid:18) τ j ˜ t (cid:19) d ˜ t = τ j (cid:20) exp (cid:18) ετ j η (cid:19) (cid:18) ετ j η − (cid:19) + 1 (cid:21) , t (cid:90) ¯ εη exp (cid:18) τ j ˜ t (cid:19) d ˜ t = τ j (cid:20) exp (cid:18) τ j t (cid:19) − exp (cid:18) ετ j η (cid:19)(cid:21) . nserting these results yields ε ij ( t ) = 2 τ j exp (cid:18) − τ j t (cid:19) (cid:34) η τ j (cid:20) exp (cid:18) ετ j η (cid:19) (cid:18) ετ j η − (cid:19) + 1 (cid:21) + ¯ ε τ j (cid:20) exp (cid:18) τ j t (cid:19) − exp (cid:18) ετ j η (cid:19)(cid:21)(cid:35) = 2 τ j exp (cid:18) − τ j t (cid:19)(cid:34) η τ j ετ j η exp (cid:18) ετ j η (cid:19) − η τ j (cid:18) ετ j η (cid:19) + η τ j ε τ j (cid:18) τ j t (cid:19) − ¯ ε τ j (cid:18) ετ j η (cid:19)(cid:35) = exp (cid:18) − τ j t (cid:19) (cid:20) exp (cid:18) ετ j η (cid:19) (cid:16) ¯ ε − τ j η − ¯ ε (cid:17) + τ j η (cid:18) τ j t (cid:19) ¯ ε (cid:21) = − τ j η (cid:18) − τ j t + 2¯ ετ j η (cid:19) + τ j η (cid:18) − τ j t (cid:19) + ¯ ε = τ j η (cid:18) − τ j t (cid:19) (cid:20) − exp (cid:18) ετ j η (cid:19)(cid:21) + ¯ ε. We use this result to obtain the stress responses of the individual Maxwell ele-ments. This is also useful for an analysis of different displacement rates η in Section6.3. Proposition 3.2.
The stress σ j of the j -th Maxwell element with j > and thecorresponding stiffness µ j and relaxation time τ j at time t ∈ [0 , T ] is given as σ j ( t ) = µ j τ j η (cid:16) − exp (cid:16) − τ j t (cid:17)(cid:17) , ≤ t ≤ ¯ εη − µ j τ j η (cid:16) − exp (cid:16) − ετ j η (cid:17)(cid:17) exp (cid:16) − τ j t (cid:17) , ¯ εη < t ≤ T. (3.2) The stress of the single spring, denoted by σ j =0 , can be specified with the correspondingstiffness µ as σ ( t ) = (cid:26) µηt, ≤ t ≤ ¯ εη µ ¯ ε, ¯ εη < t ≤ T. According to these calculations, the total stress can be written as the sum of thestresses of the single spring and the Maxwell elements, i.e. σ ( t ) = n (cid:88) j =0 σ j ( t ) = µηt + n (cid:80) j =1 µ j τ j η (cid:16) − exp (cid:16) − τ j t (cid:17)(cid:17) , ≤ t ≤ ¯ εη µ ¯ ε − n (cid:80) j =1 µ j τ j η (cid:16) − exp (cid:16) − ετ j η (cid:17)(cid:17) exp (cid:16) − τ j t (cid:17) , ¯ εη < t ≤ T. (3.3) Proof . We start with the stress of the single spring. This is calculated by σ ( t ) = µε ( t ). Inserting (2.1) for the different time intervals gives us the stress as shownabove. The stress of the Maxwell elements is calculated by σ j ( t ) = µ j ( ε ( t ) − ε ij ( t ))as given in equation (2.3). By considering the different time intervals we can insertboth, the strain ε ( t ) and the inelastic strain ε ij ( t ) of the Maxwell element. Hence, for0 ≤ t ≤ ¯ εη we obtain σ j ( t ) = µ j (cid:18) ηt − τ j η exp (cid:18) − τ j t (cid:19) − ηt + τ j η (cid:19) = µ j τ j η (cid:18) − exp (cid:18) − τ j t (cid:19)(cid:19) . imilarly, for ¯ εη < t ≤ T , we obtain σ j ( t ) = µ j (cid:18) ¯ ε − τ j η (cid:18) − τ j t (cid:19) (cid:20) − exp (cid:18) ετ j η (cid:19)(cid:21) − ¯ ε (cid:19) = − µ j τ j η (cid:18) − exp (cid:18) − ετ j η (cid:19)(cid:19) exp (cid:18) − τ j t (cid:19) . The representation of the total stress (3.3) follows directly by inserting (3.2) in (2.3).Since the stress can be expressed in terms of the material parameters, we are ableto formulate the forward operator (2.4). It is still solution-dependent, because we stillneed the number of Maxwell elements to calculate the stress, but this is unknown.Before we address this topic, we will briefly discuss the simulation of data, where theabove results will be helpful.
4. Data simulation.
We simulate a relaxation experiment where a strain isapplied to a material. This strain is completely determined by the parameters η (dis-placement rate) and ¯ ε (maximum strain). So, if we choose these two parameters and atime period [0 , T ], we have completely determined the strain using the representation(2.1). This function can then be supposed to be known for all t ∈ [0 , T ]. In Figure4.1, we see strains for three different displacement rates η and the maximum strain¯ ε = 20 %. The fastest displacement rate is 10 mm/s with a maximum strain beingattained after 2 seconds. For the slowest displacement rate of 1 mm/s, the 20% strainis only achieved after 20 seconds. We chose T = 100 seconds.0 20 40 60 80 100 120Time [s]02468101214161820 S t r a i n [ % ] η For the simulation of the stress function we then choose the number of Maxwellelements n and the material parameters ( µ, µ , . . . , µ n , τ , . . . τ n ). Thus, we can solvethe forward problem with equation (3.3) obtaining the stress function σ . In Figure4.2 we see the stress functions corresponding to the strains with different displace-ment rates η of Figure 4.1. Here, we have selected one spring and three Maxwell lements with the material parameters µ = 10, ( µ , τ ) = (4 , . µ , τ ) = (7 , . µ , τ ) = (1 , µ and µ j have the unit MPa and the unit ofthe relaxation times of the different dampers are seconds. The relaxation times wereselected in three different decades to simulate a relaxation time spectrum, whereas thestiffness for each of the elements were selected randomly to ensure no correlation be-tween the relaxation time and the stiffness. Since the Maxwell elements are connectedin parallel in the rheological model as shown in Figure 2.2, the order of the Maxwellelements is not important. This is also evident in the forward model (3.3), since wesimply have a sum of the stress contributions of the individual Maxwell elements andthe single spring. So we stick to the convention to always number them according tothe order of the relaxation times.0 20 40 60 80 100 120Time [s]050100150200250300350 S t r e ss [ M P a ] σ we have to discretize with respect to time. We choose m + 1 as the number of time steps t i = i · Tm for i = 0 , . . . , m in the interval[0 , T ] and thus also have a discretized representation of the stress function with( σ ( t ) , . . . , σ ( t m )) ∈ R m +1 . To avoid an inverse crime, one should choose differentdiscretizations for the direct and inverse problem. But since we can analyticallycompute the forward problem depending on the number of Maxwell elements n , thisproblem is not apparent. In Section 6 we discuss results of experiments and showvarious scenarios that lead to reconstructions of different accuracy concerning thematerial parameters.For any ill-posed inverse problem, reconstruction is considerably more difficult incase of noisy data σ δ . In this regard, we will also see different results in Section 6.We simulate noise by adding scaled standard normally distributed random numbersto the discretized stress vector such that k σ − σ δ k < δ with noise level δ >
0. A noiselevel δ = 0 corresponds to unperturbed data.
5. Solving the inverse problem.
In this section we develop a clustering al-gorithm, which allows us to overcome the fact that the forward operator depends on he unknown number of Maxwell elements. Currently we cannot compute the forwardoperator without a fixed number of Maxwell elements. Together with the ambigu-ity of our forward operator, this contributes to the ill-posedness of the problem. Toillustrate the mentioned ambiguity, we will consider the following example. Example 5.1.
For a material described by two Maxwell elements with relaxationtimes τ = τ and stiffnesses µ , µ we have ε i = ε i . We furthermore consideranother material described by the same basic stiffness but only one Maxwell elementwith relaxation time ˜ τ = τ = τ and stiffness ˜ µ . If additionally ˜ µ = µ + µ applies,there is no possibility to distinguish these two materials by means of the stress createdby our forward operator. The two stress-time curves are exactly the same with nopossibility to reconstruct the parameters unambiguously. This shows, how to construct different tuples ( µ, µ , . . . , µ n , τ , . . . , τ n , n ), suchthat F n ( µ, µ , . . . , µ n , τ , . . . τ n , n ) = σ for a specific σ ∈ L ([0 , T ]). For this reason we make further requirements. A commonassumption is that the occurring relaxation times lie in different decades [13]. Forinstance, if τ l ∈ [10 , l ∈ { , . . . , N } , then τ j / ∈ [10 , j ∈ { , . . . , l − , l + 1 , . . . N } . This information will help us to solve the ill-posed problem.We set a maximum number of Maxwell elements N , such that the unknown actualnumber of Maxwell elements n is smaller or equal to N . Then, we reconstruct thedesired parameters ( µ, µ , . . . , µ N , τ , . . . τ N ) with a minimization algorithm, obtainingthe stiffness parameter for the single spring as well as stiffnesses and relaxation timesfor N Maxwell elements. Of course, at this point there could be too many parameters,because the number of Maxwell elements could be too large. Nevertheless these valuesare helpful. They do not fulfill the requirement of having the relaxation times indifferent decades yet. Therefore, we apply a clustering algorithm to these parameters,which clusters them according to this requirement. That means our reconstructionconsists of two parts: • minimization algorithm with N Maxwell elements, • clustering algorithm which reduces N to n Maxwell elements.We describe this in more detail in the following.First we will explain the minimization process. At this point we are faced withthe following problem: We have applied a known strain to a material, measured thegenerated stress and want to deduce the parameters of the material. To this end weminimize the residual R = k F N ( µ, µ , . . . , µ N , τ , . . . τ N , N ) − σ k . (5.1)This minimization process is done with N Maxwell elements. Since the problem is non-linear with respect to the relaxation times, we use an iterative solution method withseveral starting values for the stiffnesses and relaxation times ( µ (0) , µ (0)1 , . . . , µ (0) N , τ (0)1 , . . . τ (0) N ).The starting values are distributed over the possible range of parameter values. Theusage of several starting values is necessary because of the nonlinearity of the problemwhich in general causes the existence of several local minima. For each of those start-ing values we then apply the forward operator to these parameters and calculate (5.1).This gives us an estimate of how well our initial values match the desired parameters.We change the starting values and the process starts again in a new step. This way he residual R ( k ) = (cid:13)(cid:13)(cid:13) F N (cid:0) µ ( k ) , µ ( k )1 , . . . , µ ( k ) N , τ ( k )1 , . . . τ ( k ) N , N (cid:1) − σ (cid:13)(cid:13)(cid:13) with the iteration index k = 0 , , . . . , K is recalculated in each step, which allowsus to evaluate the changes in the parameters. For this process we use the function’lsqnonlin’ and ’MultiStart’ provided in the MATLAB Optimization Toolbox [27].Next we look at the clustering algorithm mentioned before. For a decade [10 k , k +1 ]with any k ∈ N we consider the index set J k := { j ∈ { , . . . , N } : τ j ∈ [10 k , k +1 ] } . Of course, depending on the application, other intervals instead of decades can alsobe of interest. All pairs ( µ j , τ j ), j ∈ J k , must be assigned to one Maxwell element. Ifwe cluster the relaxation times, and thus the Maxwell elements, into index sets, thenumber of non-empty index sets gives us the actual number n of Maxwell elements inthe material. There are two ways to continue:1) Start the minimization process again with the actual number of Maxwellelements n .2) Calculate suitable parameters (˜ µ k , ˜ τ k ) with ˜ τ k ∈ [10 k , k +1 ] from the alreadyreconstructed parameters through˜ µ k := (cid:88) j ∈ J k µ j , ˜ τ k := (cid:88) j ∈ J k µ j ˜ µ k τ j . (5.2)Point 2) has of course the advantage that there is no need for a new elaborate min-imization process. The drawback is that there is no analytic way to combine theparameters, so that the exact same stress curve is obtained by applying the forwardoperator. But, since we deal with a numerical approximation of the parameters, thecalculation of the new parameters from already reconstructed ones can result in anaccidental improvement. Example 5.2.
We look at different cases and how they are treated by the clus-tering algorithm.1. Let J k = { , , ∈ { , . . . , N } : τ j ∈ [10 k , k +1 ] } with τ = τ = τ and N = 6 . It is obvious that this should actually be a single Maxwell element.Consider the forward operator with σ ( t ) = µ ε ( t ) + N (cid:88) j =1 µ j (cid:0) ε ( t ) − ε ij ( t ) (cid:1) . Since ε ij varies only in τ j for different Maxwell elements, it follows that ε i = ε i = ε i leading to σ ( t ) = µ ε ( t ) + ( µ + µ + µ )( ε ( t ) − ε i ( t )) + (cid:88) j =2 , , µ j ( ε ( t ) − ε ij ( t )) . (5.3) Therefore, it is reasonable to choose the new values as (˜ µ k , ˜ τ k ) = ( µ + µ + µ , τ ) . his is done by the clustering algorithm as ˜ µ k = (cid:88) j ∈ J k µ j = µ + µ + µ , ˜ τ k = (cid:88) j ∈ J k µ j ˜ µ k τ j = µ + µ + µ µ + µ + µ τ = τ .
2. In the next case we want to illustrate the calculation of the relaxation timesin the clustering algorithm. Let J k = { , , ∈ { , . . . , } : τ j ∈ [10 k , k +1 ] } with ( µ , τ ) = (7 , , ( µ , τ ) = (1 , . and ( µ , τ ) = (2 , . , i.e. τ ≈ τ ≈ τ . Then the stiffness values serve as a weighting of how important thecorresponding relaxation value is. Regarding the forward operator (5.3) weobserve that, under these assumptions, the values of ε ( t ) − ε ij ( t ) , t ∈ [0 , T ] ,for j = 1 , , are quite similar, especially considering that other relaxationtimes lie in different decades. Thus, if one assumes that ε i ≈ ε i ≈ ε i applies,we see that the largest contribution among these three to the total stress isprovided by the Maxwell element ( µ , τ ) , since µ (cid:29) µ > µ . Therefore, wecan assume that τ = 4 provides a good approximation to the actual relaxationtime. In order to include this weighting in more complicated examples, theclustering algorithm calculates ˜ τ k as follows, instead of just calculating themean value of all relaxation times, ˜ µ k = (cid:88) j ∈ J k µ j = µ + µ + µ = 10 , ˜ τ k = (cid:88) j ∈ J k µ j ˜ µ k τ j = 710 · · . · . . = 4 . In the first case of our examples, this recalculation of the parameters by the clusteringalgorithm would not change anything in the generated stress curve.
6. Numerical results.
In this section we show numerical results of our derivedminimization and clustering algorithm. For the evaluation of our method we generatesimulated data for known material parameters that serve as ground truth. We choosea material described by one spring for basic elasticity and n = 3 Maxwell elementsand the following material parameters:Table 6.1: Selected material parameters to simulate data j µ j [MPa] 10 4 7 1 τ j [s] - 0.2 3.7 25By solving the forward operator analytically (see Proposition 3.2) we obtain thestress-time curve of the material, which we observe in the range [0 , .1. Exact data. It will be shown that in case of exact data we can determinethe material parameters very reliably. Table 6.2 shows the effects of the clusteringalgorithm. While the minimization algorithm determines stiffnesses and relaxationtimes for the given maximum number of Maxwell elements (here N = 5), the clusteringalgorithm can combine them accordingly and thus obtain the actual number n = 3.Here we used the complete data set for t ∈ [0 , µ j are given in MPa and the τ j in seconds j = 0 1 2 3 4 5reconstructed values µ j τ j - 0.200 3.695 3.706 3.706 25.000after clustering µ j τ j - 0.200 3.700 25.000While after the calculation of the minimization algorithm there is only one Maxwellelement with τ ∈ [0 ,
1) and τ ∈ [10 , , τ = 3 .
695 is closest to the current value of 3 . µ (cid:29) µ > µ , which allows the clustering algorithm to use the weighting (5.2) leadingto better results. We continue by investigating noisy data σ δ . These aregenerated by adding scaled standard normally distributed numbers to our discretizedstress vector σ , such that k σ − σ δ k < δ with noise level δ >
0. For better comparabilitywe specify the relative noise level δ rel by k σ − σ δ kk σ δ k < δ rel . In our calculations we use the Euclidean norm and a relative noise level of δ rel ≈ N = 5. To better assess the effect ofthe disturbance, we used several data sets for this purpose. That is, we have thesame relative noise level δ rel , but different random numbers, which leads to slightlydifferent data sets. We perform our reconstruction for each of these different data setsto identify possible weaknesses.We see that the reconstruction results differ, as expected, significantly from theexact values. However, it is obvious that the reconstructions of ( µ , τ ) show a tremen-dous error, while the other parameters are computed rather stably. This confirmsconsiderations of the authors in [8] where the authors have proven that the recon-struction of small relaxation times always is severely ill-conditioned and the condition able 6.3: Approximated material parameters with similar noisy data, where the µ j are given in MPa and the τ j in seconds j = 0 1 2 3exact values µ j
10 4 7 1 τ j - 0.2 3.7 25experiment 1 µ j τ j - 0.019903 3.7734 27.795experiment 2 µ j τ j - 0.29222 3.6908 24.2965experiment 3 µ j τ j - 0.01344 3.6444 26.633experiment 4 µ j τ j - 0.0993 3.6626 24.0342deteriorates as τ →
0. Since stiffness and relaxation time are always to be consid-ered in pairs, we see deviations in τ also in the corresponding stiffness µ . As eachMaxwell element has to provide a certain part to the total stress, too small values inone parameter are compensated by higher values in the other parameter to approachthe total stress.0 10 20 30 40 50 60 700.10.20.30.40.50.60.70.8 exactcomputed r e l a x a t i o n t i m e τ [ s ] stiffness µ [MPa]Figure 6.1: Spread of the material parameters ( µ , τ ) for 100 runs with different noiseseeds, but same noise level δ rel for η = 10 mm/sWe repeat this experiment with 100 different perturbed data sets with the samenoise level and plot the values of ( µ , τ ) in Figure 6.1. We see that there is a very largevariance in values spread over 100 different experiments. The values are distributedup to results such as ( µ , τ ) = (66 . , . µ , τ ) = (1 . , . here are also results that come close to the exact values. The median of all valuesis (0 . , . µ , τ )) is most affected. Thisconfirms stability investigations obtained in [8]. So far we focused on exper-iments with the fastest displacement rate of 10 mm/s. In this subsection we discussslower displacement rates.Table 6.4: Clustered material parameters with noisy data ( η = 1 mm/s) where the µ j are given in MPa and the τ j in seconds j = 0 1 2 3exact values µ j
10 4 7 1 τ j - 0.2 3.7 25experiment 1 µ j τ j - 0.088687 4.2318 27.263experiment 2 µ j τ j - - 3.4762 23.689experiment 3 µ j τ j - 0.82841 6.3049 3000.5experiment 4 µ j τ j - 0.885029 3.9963 23.423We compare Table 6.3 with Table 6.4, in which we used a displacement rateof 10 mm/s and 1 mm/s, respectively. Again, we use N = 5 for the minimizationand a relative noise level of about 1%, but four different noise seeds. It is obviousthat the results are much worse, specifically the smallest relaxation time and thecorresponding stiffness are approximated poorly. Again, this has to be expected sincethe reconstruction of small relaxation times is severely ill-conditioned. In a secondrun, the algorithm was not even able to find Maxwell elements with relaxation timesin three different decades.For a better understanding of why the results are worse compared to the resultsfor the faster displacement rate of 10 mm/s, we look at the different stress-time curvesthat a displacement rate of 1 mm/s (Figure 6.2) and 10 mm/s (Figure 6.2) generate.For each of the curves, we show the contribution of each element separately. Thatmeans σ is the stress generated by the single spring while σ , , is the stress of thethree Maxwell elements. The sum of these individual stresses yields the total stress, cf.(3.3). The maximum strain is 20 %, which is attained at 20 seconds and 2 seconds fora displacement rate of 1 mm/s and 10 mm/s, respectively. This makes a big differencefor the individual stresses of the Maxwell elements. In both cases the maximum stressof the single spring is achieved at 200 MPa, but at different time instances. However,the maximum stress of the Maxwell elements at a slower displacement rate of 1 mm/sis much lower than for a faster displacement rate. In both cases, the maximum σ ( t ) σ ( t ) σ ( t ) σ ( t ) S t r e ss [ M P a ] Figure 6.2: Individual stress components for a displacement rate of 10 mm/sis attained at t = ¯ ε/η . In Table 6.5, we see the different values listed. At a slowerdisplacement rate, the stress values of the Maxwell elements are non-zero over a longertime period, which can be seen particularly well for the first Maxwell element. Butsince the values are so much smaller than at a higher displacement rate, these smallvalues are covered by noise much more easily. Therefore, it is advisable to use higherdisplacement rates. Based on this, we revert to a displacement rate of 10 mm/s inupcoming examples.Table 6.5: Maximum values of individual stress components [MPa] with slow and fastdeformation rates attained at t = ¯ ε/η j = 1 j = 2 j = 3 η = 1 σ j (¯ ε/η ) 0.4 12.94 9.9 η = 10 σ j (¯ ε/η ) 4 85.57 18.48 As a result of our investigations in Section6.2 we realized that small errors in the data can lead to large errors in the solution.This is a typical behavior of ill-posed inverse problems, which must be compensatedby means of regularization techniques. Therefore, instead of continuing to use as costfunction the residual (5.1) only, we will add a regularization term and minimize k F ( µ, µ , . . . , µ N , τ , . . . τ N ) − σ δ k + λ k ( µ, µ , . . . , µ N , τ , . . . τ N ) k , (6.1)where λ > Tikhonov-Phillipsregularization , a standard approach in the field of inverse problems (see [10,22,28,34]). σ ( t ) σ ( t ) σ ( t ) σ ( t ) S t r e ss [ M P a ] Figure 6.3: Individual stress components for a displacement rate of 1 mm/sIn Table 6.6, we see the comparison of two reconstructions with and without theregularization term. Again we used N = 5 and the clustering algorithm for theseresults. We see a significant improvement on the first Maxwell element, which waspreviously most affected by the disturbed data. As expected, too large values aresuppressed in the stiffness, but at the same time τ attains larger values. However,we also see a deterioration in the values of the third Maxwell element. Since largevalues are penalized by the regularization term, this mainly affects the relaxation time τ = 25 s. The values are lower and we get a higher stiffness µ = 1 . τ = 22 . µ j are given in MPa and the τ j in seconds j = 0 1 2 3exact values µ j
10 4 7 1 τ j - 0.2 3.7 25without µ j τ j - 0.1096 3.7056 25.3752with µ j τ j - 0.1607 3.5549 22.2532To avoid this, we test another penalty term. We minimize k F ( µ, µ , . . . , µ N , τ , . . . τ N ) − σ δ k + λµ . (6.2)Since µ is just a non-negative number in R , we can omit the norm. The idea is tohave the same positive effects as the classical Tikhonov-Phillips regularization, but ithout the negative influence on the largest relaxation time and thus the associatedstiffness. In Figures 6.4 and 6.5 we compare all three methods. As in Section 6.2 weevaluate the different approaches in 100 experiments with the same noise level butdifferent noise seeds. The clustered results are displayed in Figures 6.4 and 6.5. Ineach box the central marker indicates the median, the lower and upper end of thebox mark the 25th and 75th percentile, respectively. The outer boundaries mark thehighest and smallest values. Outliers are marked by the red ’+’ symbols.(a) (b) (c)00.10.20.30.40.50.60.7 r e l a x a t i o n t i m e τ Figure 6.4: Smallest relaxation time determined from 100 runs with (a) no regular-ization, (b) classical Tikhonov-Phillips regularization and (c) adjusted regularizationtermWe see that we still manage to achieve a reasonable reconstruction of τ withthe adjusted regularization term (see Figure 6.4 (c)). While τ was permanentlyreconstructed too small in the classical Tikhonov-Phillips regularization, we cannotobserve such attenuation in (c), since we no longer penalize large values except µ (see Figure 6.5). the last set of numerical experiments examines theeffects of using a smaller number of data. This is interesting in view of shortening theduration of the experiment or sampling the data at a lower frequency. It is particularlyrelevant for the reconstruction of Maxwell elements with high relaxation times. Sincewe do not know these, we cannot estimate how long it takes for the stress contributionof this element to decay to zero. In other words: It is unclear at which time a materialfinds its equilibrium. This is why in the following we will test reconstructions fromshortened data sets in order to be able to make an assertion about the accuracy ofthese reconstructions.We again work with noisy data ( δ rel ≈ N = 5, for the minimization and usethe adapted penalty term (6.2). Up to now, we have used a time interval of [0 , . a) (b) (c)2530354045 r e l a x a t i o n t i m e τ Figure 6.5: Largest relaxation time for 100 runs with (a) no regularization, (b) classicalTikhonov-Phillips regularization and (c) adjusted regularization term6.6. We focus on the basic elasticity µ and stiffness µ of the Maxwell element withslowest relaxation time and furthermore on the smallest and largest relaxation times τ and τ . The behavior of the central Maxwell element is comparable to the thirdone, which is why we omit a detailed discussion.The basic elasticity µ of the single spring can perfectly be identified after only25 seconds for a displacement rate of 10 mm/s. For the slower rate of 1 mm/s, 55seconds of the experiment are necessary for a reliable identification. The minimal timeto conclusively identify the stiffness µ of the spring in the third Maxwell element takes40 seconds for a rate of 10 mm/s and 50 seconds for a rate of 1 mm/s, respectively.The identification of the relaxation times is more difficult than of the stiffnesses.The relaxation times τ of the third Maxwell element can be precisely determinedafter 45 seconds for a rate of 10 mm/s and after 70 seconds for the slower rate of1 mm/s, respectively. After that the result gets worse with every second the dataset is shortened. There are not sufficient data points left to perform a reliable re-construction. The variation in the values for the fastest relaxation time τ is muchlarger than for τ and also very sensitive to noise. Experimental observations show ingeneral that, if the displacement rate is too high, the third Maxwell element with theslowest relaxation time behaves like an equilibrium spring since there is not enoughtime to relax during loading. The stiffness of the third element superimposes withthe equilibrium spring. However, during relaxation the equilibrium spring and thethird Maxwell element can be easily distinguished from each other. Hence, the fasterloading rate of 10 mm/s can predict µ and µ even with 30 s of the data whereas,for the 1 mm/s loading rate, at least 50 s of data is required for accurate prediction.On the other hand the prediction of τ is easier with the 10 mm/s data because, ifthe displacement rate is too low, the first Maxwell element with the fastest relaxationtime starts to relax during the loading itself. This is shown for a rate of 1 mm/s inFigure 6.6. A satisfying identification is not possible during the entire experimentalduration of 100 seconds. However, for the faster rate of 10 mm/s τ it can already be s t i ff n e ss µ [ M P a ] exactcomputed duration time T [s] s t i ff n e ss µ [ M P a ] exactcomputed100 90 80 70 60 50 40 30 204567891011duration time T [s] s t i ff n e ss µ [ M P a ]
100 90 80 70 60 50 40 30 204567890123 exactcomputed exactcomputed s t i ff n e ss µ [ M P a ] duration time T [s]100 90 80 70 60 50 40 30 204560123duration time T [s]100 90 80 70 60 50 40 30 20 r e l a x a t i o n t i m e τ [ s ] r e l a x a t i o n t i m e τ [ s ] duration time T [s]100 90 80 70 60 50 40 30 200.50.60.70.80.90.10.20.30.4exactcomputed r e l a x a t i o n t i m e τ [ s ] duration time T [s]100 90 80 70 60 50 40 30 2030352025 duration time T [s] r e l a x a t i o n t i m e τ [ s ] , T ] of the experiment on the materialparameters µ, µ , τ and τ for a displacement rate of 10 mm/s (left) and 1 mm/s(right), respectively. etermined after 50 seconds.As a summary of the observations above and for a better understanding, Figure 6.7again shows the stress-time curves for the three different displacement rates, but nowincluding the time in the experiments to identify the individual material parameters.The parameters are not of the same order for the different rates. For instance, for adisplacement rate of 1mm/s the relaxation time τ can be reconstructed first, while at10mm/s the stiffness µ is most stable to data shortening. Thus, no general statementcan be made about a correlation of the material parameters and the duration of theexperiment without considering conditions such as the displacement rate. S t r e ss [ M P a ] Time [s] 1mm/s10mm/s0 20 40 60 80 10010 30 50 70 90050100150200250300350 µµ τ τ µ µ τ τ Figure 6.7: Synthetic stress-time data produced by a spring combined with a threeMaxwell element model including the duration of the experiment to conclusively iden-tify the stiffnesses and relaxation times for displacement rates of 1mm/s (red) and10mm/s (blue)
7. Conclusion and future work.
In this paper, we presented an algorithmfor the reconstruction of material parameters in a Maxwell-based rheological multi-parameter model for viscoelastic materials. After a short introduction to the materialmodel, we defined the forward operator and proposed an analytical way to solve theevolution equation with respect to relaxation times. This allowed us to determinean analytical solution of the forward operator with regard to the number of Maxwellelements.Since the number of Maxwell elements is not known, we suggested a methodto overcome the solution-dependence of the forward operator by using a clusteringalgorithm. For the parameter identification, we considered different approaches basedon minimizing a least squares functional. In experiments we have seen that thesimple approach to minimize the residual is not sufficient for perturbed data andespecially the material parameters of the Maxwell element with shortest relaxation ime is very susceptible to noise. By analyzing the stress-time curve decomposedinto the single spring and Maxwell element stresses, we were able to give a clearrecommendation towards higher displacement rates. This analysis also contributed tothe understanding of why the small relaxation times with their corresponding stiffnessare that sensitive to noise.Subsequently, we considered various regularization methods such as the classicalTikhonov-Phillips approach. We conclude that a specific regularization adapted tothe smallest relaxation time is preferable.Investigations on shortened data sets demonstrate that our approach for mate-rial parameter identification shows great advantages for performing real experimentssince it yields suggestions on the minimum duration of the experiments. This is veryimportant, because typical relaxation experiments can last days or weeks until thebasic elasticity can be determined using conventional evaluation methods. Whetherthe duration of an experiment is sufficient for the identification of further parametersis not obvious by conventional approaches. In that sense our new approach might beof significant interest for the identification of parameters in materials science.Future research is focused on validating the presented algorithms on real datasets. Furthermore, it might be interesting to apply other numerical regularizationmethods, such as the Landweber iteration or Newton type methods. REFERENCES[1]
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