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A modification of the semi-analyticinversion method: determination of theyield stress and a comparison with theparametrization algorithmSalih Tatar a & Zahir Muradoğlu b
a Zirve University, Department of Mathematics , Gaziantep ,Turkeyb Kocaeli University, Department of Mathematics , TurkeyPublished online: 16 May 2013.
To cite this article: Salih Tatar & Zahir Muradoğlu , Inverse Problems in Science and Engineering(2013): A modification of the semi-analytic inversion method: determination of the yield stress anda comparison with the parametrization algorithm, Inverse Problems in Science and Engineering,DOI: 10.1080/17415977.2013.797413
To link to this article: http://dx.doi.org/10.1080/17415977.2013.797413
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Inverse Problems in Science and Engineering, 2013http://dx.doi.org/10.1080/17415977.2013.797413
A modification of the semi-analytic inversion method: determination ofthe yield stress and a comparison with the parametrization algorithm
Salih Tatara∗ and Zahir Muradoglub
aDepartment of Mathematics, Zirve University, Gaziantep, Turkey; bDepartment of Mathematics,Kocaeli University, Turkey
(Received 2 July 2012; final version received 16 April 2013)
In this study, an effective modification of the semi-analytic inversion method ispresented. The semi-analytic inversion method is developed to solve an inversecoefficient problem arising in materials science instead of the parametrizationmethod as a different and stronger method. The inverse coefficient problem isrelated to reconstruction of the unknown coefficient g = g(ξ2), ξ2: = |∇u|2,from the nonlinear equation −∇.
(g(|∇u|2)∇u
) = 2ϕ, x ∈ � ⊂ R2. The semi-
analytic inversion method has some advantages. The first distinguishable featureof this method is that it uses only a few measured output data to determinethe whole unknown curve, whereas the parametrization algorithm uses manymeasured output data for the determination of only some part of the unknowncurve. The second distinguishable feature of this method is its well-posedness.In the semi-analytic inversion method, the algorithm for determination of theyield stress, which is one of the main unknowns of the inverse problem, is verycomplicated. That is why we need to modify this algorithm. The demonstratednumerical results for different engineering materials also show that the modifiedsemi-analytic inversion method allows us to determine the elastoplastic param-eters of a kind of engineering materials with high accuracy, even various noiselevels.
Keywords: yield stress; semi-analytic inversion method; torque; plasticityfunction; measured output data; power-law material
Nomenclature
Symbols
g plasticity functionG modulus of rigiditygh approximate function of gκ strain hardening exponent
κh approximate value of κ
ν Poisson coefficientu Prandtl stress functionE Young’s modulusϕ angle of twist per unit length
�ϕ angle step
*Corresponding author. Email: [email protected]
© 2013 Taylor & Francis
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2 S. Tatar and Z. Muradoglu
� cross-sectionT theoretical value of torque
wh piecewise uniform meshT experimental value of torqueξ2
0 yield stressξ2
0h approximate value of the yield stressγ noise level
g0 shear complianceεg absolute errorδg relative error
1. Introduction
The problems that involve determination of the unknown coefficients in partial differentialequations from some additional conditions (also called measured output data) are wellknown in mathematical literature as the inverse coefficient problems. These additionalconditions may be given on the boundary of the domain, at the final time or on the wholedomain i.e., non-local condition. The inverse problems require defining related to directproblems. As it is known, a direct problem aims to find a solution that satisfies given anordinary and partial differential equation and related to initial and boundary conditions.In some problems, the main ordinary and partial differential equation and the initial andboundary conditions are not sufficient to obtain the solution, but, instead some additionalconditions are required. Such problems are called the inverse problems. A problem is saidto be well-posed or properly posed in the sense of Hadamard if it has the following threeproperties: there exists a solution of the problem (existence), there is at most one solutionof the problem (uniqueness) and the solution depends continuously on the data (stability).If at least one of these properties does not hold, then the equation is called ill-posed orimproperly posed. In this context, another definition of the inverse problems can be givenas follows: If one of two problems which are inverse to each other is ill-posed, we call it theinverse problem and the other one the direct problem. Inverse problems are often ill-posed.
Among the inverse problems, the inverse coefficient problems are quite interestingand receive much attention in mathematical literature, especially, the torsion problemsin linear elasticity. However, the inverse problems in torsional deformation are less wellknown and have received relatively little attention in mathematical literature as well asin engineering literature. The quasi-static mathematical model of torsional creep withinJ2- deformation theory of the plasticity is given in [1]. In this model, one seeks the solutionu(x), x ∈ � ⊂ R
2, of the following nonlinear boundary value problem:{−∇.(g(|∇u|2)∇u
) = 2ϕ, x ∈ � ⊂ R2,
u(x) = 0, x ∈ ∂�,(1)
where �: = (0, a)× (0, b), a, b > 0 is the cross-section of a bar, ϕ is the angle of twist perunit length, g = g(ξ2), ξ2 = |∇u|2 is the plasticity function and u(x) is the Prandtl stressfunction. The nonlinear boundary value problem (1) represents an elastoplastic torsion of astrain hardening bar. According to J2- deformation theory, the plasticity function satisfiesthe following conditions [2–4]:
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Inverse Problems in Science and Engineering 3
⎧⎪⎪⎨⎪⎪⎩
0 < c0 ≤ g(ξ2) ≤ c1, ∀ξ2 ∈ [ξ∗2, ξ∗2], ξ∗2 > 0,
g′(ξ2) ≤ 0,
g(ξ2) + 2ξ2g′(ξ2) ≥ γ0 > 0,
∃ξ20 ∈ (ξ∗2, ξ∗2) g(ξ2) = g0, ∀ξ2 ∈ [
ξ∗2, ξ∗2].
(2)
Now we define the parameters in (2). First, we define ξ20 = max
x∈�|∇u(x)|2. In materials
science, it corresponds to the yield stress which is the maximum stress or force per unit areawithin a material that can arise before the onset of permanent deformation. When stressesup to the yield stress are removed, the material resumes its original size and shape. In otherwords, there is a temporary shape change that is self-reversing after the force is removed,so that the object returns to its original shape. This kind of deformation is called pure elasticdeformation. On the other hand, irreversible deformations are permanent even after stresseshave been removed. One type of irreversible deformation is pure plastic deformation. Forsuch materials, the yield stress marks the end of the elastic behaviour and the beginning ofthe plastic behaviour. For any angle ϕ > 0, all points of the bar have non-zero strain intensitywhich means the conditon ξ∗2 > 0 in (2) makes sense. Furthermore, g0 = 1/G is the shearcompliance, G = E/(2(1 + ν)) is the modulus of rigidity, E > 0 is the Young’s modulus,ν ∈ (0, 0.5) is the Poisson coefficient, and c0, c1, γ0 are some positive constants. ThePoisson coefficient for some materials such as aluminum, bronze, copper, ice, magnesium,molybdenum, monel metal and nickel silver are 0.334, 0.34, 0355, 0.33, 0.35, 0.307, 0.315,0.322 respectively. Since the Poisson coefficient of the aforementioned materials are around0.3, it is assumed to be 0.3 throughout this paper. We note that changing of this value affectnumerical calculations but does not affect the applicability of the algorithm.
The mechanical properties of a material describe how it will react to physical forces.Mechanical properties occur as a result of the physical properties inherent to each material,and are determined through a series of standardized mechanical tests. Mechanical propertiesare also used to help classify and identify material. The most common properties consideredare strength, ductility, hardness, elastic shear modulus, yield stress and strain hardeningexponent. There are many studies to determine these properties in literature. For examplein [5], the authors are proposed a polycrystalline approach in order to model the elastic-plastic behavior of an austenitic-ferritic stainless steel. This study concerns the predictionof the mechanical behaviour of an austenitic-ferritic stainless steel whose microstructuralheterogeneity is strongly marked. In [6], the authors investigate the mechanical behaviourof a mortar and five polymer modified mortars (PMMs) with different polymer contents.The mechanical characterization of the materials is based on compression and three-pointbending tests. The determination of material parameters used in internal variable constitutivemodels is studied in [7]. In order to determine the best-suited material parameter set, twooptimization approaches are used: a gradient-based method and a continuous evolutionaryalgorithm method. Both methods are used to determine the 12 material parameters neededfor an AA1050-O aluminium alloy. Also the authors compare these two methods, in termsof the cost function and the convergence rate, in the case of an AA1050-O aluminiumalloy. An application of inverse approach to experimental indentation data in order todetermine combined hardening models parameters is given in [8]. The authors show thatthe inverse analysis applied to experimental cyclic indentation curves can give a quite goodapproximation of the monotonic stress–strain curve and the beginning of cyclic tensile test.Also it is indicated that the model used in the paper is not sufficient to take into account
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4 S. Tatar and Z. Muradoglu
all phenomenons coming in the steels behaviour under cyclic indentation. Experimentalapplications of two different approaches for interpretation of instrumented indentationexperiments are proposed in [9]. The first approach is the application of methodologydeveloped for spherical indentation based on models. The second approach is the applicationof inverse algorithm based on the minimization between experiments and simulated data.Also it is shown that inverse approach leads to a good prediction of the elastic modulusfor metals with a face-centred cubic crystal structure. An inverse analysis approach is usedto identify the modified Cam-Clay parameters from a pressuremeter curve in [10]. Thenumerical process implemented in this study is based on the interaction of two numericaltools; an optimization code (SiDoLo) and a general finite element code (CESAR-LCPC).In [11], the authors investigate the mechanical behaviour of a fluoro-elastomer. Severaltension and compression cyclic loading and relaxation tests have been performed to examinethe non-linear stress–strain behaviour, the loading rate-dependent response, and hysteresisphenomena during cyclic tests of this material. A hybrid algorithm which combines agenetic algorithm and a gradient-based algorithm is used for the identification of materialparameters in [12]. In [13], a simple theory model is proposed, which included threesteps: dimensionless analysis, finite element modelling and data fitting, to characterizethe elastic-plastic properties of thin film materials on elastic-plastic substrates. A newmethod to evaluate the in-plane anisotropic plastic properties of engineering steels by singlespherical indentation is proposed in [14]. This new method deals with materials that obeythe work hardening law and have in-plane anisotropy of the yield stress in orthogonaldirections. Various approaches are proposed to determine the material properties of power-law materials, based on dimensionless analysis and the concept of a representative strainin [15]. In this work, non-linear optimization algorithms are developed and integrated withfinite element (FE) analysis to determine and improve the accuracy of the elastic-plasticmechanical properties of a power-law material without the concept of dimensionless analysisand a representative strain. The optimization approach shows that a unique set of fourkey material properties of a given material (Young’s modulus, Poisson’s ratio, yield stressand work hardening exponent) can be determined from the loading–unloading indentationcurve of only a single indentation curve, without the need for using results from two ormore indentations. In this study, we modify the inversion method which is developed fordetermination of the the elastic shear modulus, the yield stress and the strain hardeningexponent of a wide class of engineering materials. This determination problem arises in aninverse coefficient problem for the nonlinear boundary value problem (1). We define theinverse coefficient problem. For this aim, let u = u(x) be the solution of the non-linearboundary value problem (1). Then the theoretical value of the torque is defined as follows[1,16]:
T (ϕ) = 2∫
�
u(x; g;ϕ)dx . (3)
The inverse coefficient problem consists of determining the plasticity function g = g(ξ2)
in (1), from the experimentally given values T (ϕ) of the torque. For this purpose, twonumerical methods have been developed in the literature: The parametrization method [17]and the semi-analytic inversion method (also called the fast algorithm).[18] We refer thereaders to the next sections for details of these methods. Here, a new modification of thesemi-analytic inversion method is given, especially for determination of the yield stress.
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Inverse Problems in Science and Engineering 5
The remainder of this paper comprises four sections: In Section 2, the parametrizationand the fast algorithms are explained. In Section 3, a new algorithm to determine the yieldstress is derived and adapted to the original semi-analytic inversion method. Numericalexamples are presented in Section 4. The final section of the paper contains discussions andcomments on the forthcoming studies.
2. Parametrization of the unknown curve g(ξ2) and the fast algorithm
For many engineering materials, the plasticity function g(ξ2) has the following form[18–20]:
g(ξ2) ={
1/G, ξ2 ≤ ξ20 ,
1/G(ξ2/ξ2
0
)0.5(κ−1), ξ2
0 < ξ2, κ ∈ [0, 1], (4)
which corresponds to the Ramberg-Osgood curve σi = σ0(ei/e0)κ . Within the range of
J2- deformation theory, the plasticity function (4) describes the elastoplastic properties ofwide class of engineering materials, where κ ∈ [0, 1] is the strain hardening exponent. Thevalue κ = 1 and κ = 0 correspond to pure elastic and pure plastic cases, respectively.Evidently, this function satisfies all conditions (2). We want to draw attention to the repre-sentation g(ξ2) of the plasticity function. Since the relationship between the intensities ofshear stress and shear strain is given by the equation S = g(ξ2)ξ , we use the notation g(ξ2)
instead of g(ξ) for the plasticity function.The parametrization algorithm consists of discretization of the unknown curve (4), in
the following form [21]:
gh(ξ2) =⎧⎨⎩
β0 = 1/G, ξ2 ∈ (0, ξ2
0
],
β0 − β1(ξ2 − ξ2
0
), ξ2 ∈ (
ξ20 , ξ2
1
],
β0 − ∑M−1m=1 βm
(ξ2
m − ξ2m−1
) − βM(ξ2 − ξ2
M−1
), ξ2 ∈ (
ξ2M−1, ξ
2M
].
(5)
The parametrization algorithm focuses on the unknown parameters βm appearing in (5).At each mth state, the algorithm aims to find the parameters βm , using the measured outputdata Tm that corresponds to the angle ϕm . For convenience of the readers, we give theparametrization algorithm. For this purpose, assume that (ϕ0,T0) is a given measured outputdata for the pure elastic case. Then the parametrization algorithm is presented as follows:
The parametrization algorithm:
(P1) Choose iterations β(1)0 , β
(2)0 so that the following conditions hold:
Th[β
(1)0
]> T0 > Th
[β
(2)0
].
(P2) Use next iteration β(3)0 = (
β(1)0 +β
(2)0
)/2, and calculate the theoretical value of the
torque Th[β
(3)0
].
(P3) Determine the next iteration by using (14) as follows:if Th
[β
(3)0
]< T0, then β
(4)0 = (
β(3)0 + β
(2)0
)/2,
if Th[β
(3)0
]> T0, then β
(4)0 = (
β(3)0 + β
(1)0
)/2.
(P4) Calculate∣∣Th[G(3)] − T0
∣∣.(P5) If
∣∣Th[β(3)0 ] − T0
∣∣ < εT , then β0 = β(3)0 . Otherwise, continue the steps (P2)–(P4).
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6 S. Tatar and Z. Muradoglu
(P6) Repeat the process until the following stopping condition∣∣Th[β(n)0 ] − T0
∣∣ < εT is fulfilled.
For each the pure plastic torsion case, for the determination of the unknown parametersβm , a similar algorithm can be used (one just needs to write β
(n)m instead of β
(n)0 in the
parametrization algorithm). Although the parametrization algorithm is used for numericalsolution of some class of inverse coefficients problems, it has some disadvantages. The firstone is that the application of this algorithm requires lots of measured output data. This isof course undesirable since getting these data needs time and costs. The second is the ill-posedness of the algorithm. This situation is illustrated in [21] and a regularization methodis offered.
For all these reasons a new method i.e., the semi-analytic inversion method is introducedin [18]. The semi-analytic inversion method is based on the determination of the three mainunknowns of (4), namely the elastic shear modulus G > 0, the yield stress ξ2
0 and the strainhardening exponent κ ∈ [0, 1]. The first distinguishable feature of this algorithm is that ituses only a few values of the data (ϕi ,Ti ). Furthermore, the new method determines wholeof the unknown curve by using these a few data. The second distinguishable feature of thisalgorithm is its well-posedness. But, in the semi-analytic inversion method the algorithmthat is used for determination of the yield stress is complicated and it needs many parametersto be determined before applying it. According to this algorithm, the yield stress is foundusing the following steps:
Algorithm for determination of the yield stress:
(YS1) Choose the iteration ϕ(1)0 such that ϕ
(1)0 > ϕ0.
(YS2) Use the synthetic output data T (1)0 = δ1T0, δ1 = ϕ0/ϕ
(1)0 and apply the parametriza-
tion algorithm to find β(1)0 = 1/G(1).
(YS3) Calculate the relative error δG(1) =∣∣∣∣(Gh − G(1))/Gh
∣∣∣∣.(YS4) (a) If δG(1) < εG , use the next iteration ϕ
(2)0 > ϕ
(1)0 .
(b) If δG(1) > εG , use the next iteration ϕ(2)0 = 0.5
[ϕ
(1)0 + ϕ0
].
(YS5) Use the synthetic output data T (2)0 = δ2T0, δ2 = ϕ0/ϕ
(2)0 and apply YS2 and YS3
for the new output data.
Repeat the process YS1–YS5 until the following conditions
δG(n) < εG , δG(n+1) > εG , δξ(n)0 :=
∣∣∣∣ξ20
(n) − ξ20
(n+1)
∣∣∣∣0.5
[ξ2
0(n) + ξ2
0(n+1)
] < εξ are fulfilled.
Finally, the value ξ20h := 0.5
[ξ2
0(n) + ξ2
0(n+1)
]is assumed to be an approximate value of the
yield stress ξ20 . Note that choosing the parametersϕ
(1)0 , ϕ
(2)0 , . . ., εG , εξ , δG are very sensitive
and affect the iteration number and consequently the precision of the solution provided bythe algorithm. Also in each step the error δG(n) needs to be calculated. Thus, this algorithmcontains many parameters to be determined before applying it. Therefore, we need a new
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Inverse Problems in Science and Engineering 7
algorithm which reduces the number of the required parameters to determine the yield stressto minimum. According to semi-analytic inversion method, the other two main parametersof the unknown function (4) are found as follows: The modulus of rigidity is determinedby applying the parametrization algorithm just once for the measured output data (ϕ0,T0).Having found approximate value of the modulus of rigidity and the yield stress, the strainhardening exponent is determined by formula (7), (see below). This formula is due to theexplicit form of the plasticity function given by (4). Although both the parametrization andthe semi-analytic inversion methods are used for the functions in the form (4), some similaralgorithms may be developed for another type of functions that satisfy all condition (2).
The rest of the paper is devoted to establish the new algorithm to find the yield stressand comparison of the numerical results with the parametrization algorithm. We note thatthe other steps of the semi-analytic inversion method remain the same.
3. Determination of the yield stress by a new algorithm
In this section, we develope a new algorithm for determination of the yield stress. For thispurpose, let (ϕ1,T1) and (ϕ2,T2) be given measured output data for the pure elastic andthe pure plastic cases respectively. By applying the parametrization algorithm for the pair(ϕ1,T1), the elastic shear modulus G > 0 can be found. The approximate value of the elasticshear modulus is denoted by Gh > 0. For the present, we assume that the approximate valueof the yield stress is known and denoted by ξ2
0h . By using the parametrization algorithm onemore time, the following function can be constructed (see Figure 2):
g(ξ2) ={
β0 = 1/Gh, ξ2 ∈ (0, ξ2
0h
],
β0 − β1(ξ2 − ξ2
0h
), ξ2 ∈ (
ξ20h, ξ2
1h
],
(6)
where ξ21h = max
�|∇u|2 and it can be found by solving the nonlinear direct problem for the
input data (6) and ϕ = ϕ1. We use a finite difference scheme to solve the nonlinear directproblem with these inputs and refer the readers to the next section for details.
Since the point (ξ21h, g(ξ2
1h)) needs to be on the plasticity curve g = g(ξ2), from theexplicit form (4) of the plasticity function, we obtain the following equation
g(ξ21h) = 1/Gh
(ξ2
1h/ξ20h
)0.5(κh−1), ξ2
1h > ξ20h,
for the determination of the approximate value κh of the unknown strain hardeningexponent κ ∈ [0, 1]. This equation implies ln
(g(ξ2
1h) · Gh) = 0.5(κh − 1) ln
(ξ2
1h/ξ20h
).
Hence, the following formula is obtained to get approximate value of the strain hardeningexponent κ [18]:
κh = 1 + ln(g(ξ2
1h
) · Gh)
ln(ξ2
1h/ξ20h
) . (7)
The denominator of the above fraction shows that the value ξ21h can not be chosen very close
to ξ20h . This means the angle step �ϕ := ϕ2 − ϕ1 needs to be chosen large enough. This
assertion is also confirmed by the computational experiments below. The main differencebetween the new method and the semi-analytic inversion method becomes more visiblein the determination process of the yield stress. In addition to the pair (ϕ2,T2), the pair(ϕ3,T3) also is given in pure plastic case as an additional condition. In this procedure, wewill often use the formula (7). This procedure consists of the following two algorithms (alsosee Figure 1):
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8 S. Tatar and Z. Muradoglu
Figure 1. The schematical illustration of the Algorithms 1 and 2.
Figure 2. The function g(ξ2) defined by (6).
Algorithm 1:
(A1.1) Apply the parametrization algorithm for (ϕ1,T1) and find Gh .(A1.2) Solve linear direct problem ϕ = ϕ1 and G = Gh . Then find ξ2
0h = max� |∇u|2.(A1.3) Suppose ξ2
0 = ξ20h and apply the formula (7) for the pairs (ϕ1,T1) and (ϕ3,T3).
Then find the value of κh .(A1.4) Solve the nonlinear direct problem (1) for the found values of κh, ξ2
0 , Gh, ϕ3. Thenfind the value of T3.
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Inverse Problems in Science and Engineering 9
(A1.5) Apply (A1.1)–(A1.4) for (ϕ2,T2) and (ϕ3,T3) (equivalently apply (A1.1)–(A1.4)
by assuming ϕ1 = ϕ2,T1 = T2). Then find ˜T3.
After getting T3 and ˜T3, we check whether a relation such as T3 < T3 <˜T3 (or
T3 > T3 >˜T3) is satisfied. Then we can find the approximate value of the yield stress ξ2
0 byusingAlgorithm 2. If such a relation does not exist, we can not apply it. This situation appearsto be a disadvantage of the new Algorithm. But later we will see that this situation does not
occur in the real world. Although Algorithm 2 is valid for the former case (T3 < T3 <˜T3),
a similar algorithm can be constructed for the latter case (T3 > T3 >˜T3).
Algorithm 2:
(A2.1) ϕ10 = ϕ1+ϕ2
2 .(A2.2) Solve the direct problem for ϕ = ϕ1
0 and find corresponding torque T . Apply
(A1.1)–(A1.4) for the pairs
⟨(ϕ1,T1), (ϕ1
0 ,T 10 )
⟩and
⟨(ϕ1
0 ,T 10 ), (ϕ2,T2)
⟩. Then
find T3,˜T3.
(A2.3) If |T3 − T | < ε then solve the direct problem for ϕ10 , calculate ξ2
0 = max�
|∇u|2and get ξ2
0h by ξ20h = ξ2
0 .(A2.4) If T3 − T > 0 then apply (A2.1)–(A2.3) for (ϕ1, ϕ
10).
(A2.5) If T3 − T < 0 then apply (A2.1)–(A2.3) for (ϕ10 , ϕ2).
Note that Algorithm 2 also involves Algorithm 1. Though there is no mathematical reason,
we give a physical explanation why the relation T3 < T3 <˜T3 should hold. Since ϕ2 > ϕ1
and we treat (ϕ2,T2) as if it is given in the pure plastic case, the solution of the directproblem in the pure elastic case gives ξ2
(0h)1< ξ2
0 < ξ2(0h)2
(see step 2 in Figure 1) which
means T3 < T3 <˜T3 holds. We denote ξ2
0 by ξ1(0h)1
and ξ2(0h)2
for ϕ1 and ϕ2, respectively.The main idea of these two algorithms is to decide the interval that includes the real anglethat exactly corresponds to the yield stress. After finding this angle approximately, we candetermine the approximate value of the yield stress as it is explained in the Algorithm 2.The new fast algorithm differs from the original one while in finding the yield stress. Theother steps remain the same. However, this modification is very useful and very effective.Because there is no parameter to choose in the algorithms except ε > 0 which appears inthe Algorithm 2. This is very important simplification for the nature of inverse problemssince each choice of these parameters restricts us in the solution process. For example, in[21] the influence of the stopping parameter ε > 0 on identifiability is shown to be verylarge, i.e. even very small changes of the parameter ε > 0 lead to large deviations in thesolution of the inverse problem.
4. Numerical examples with noise free and noise data
In many physical and engineering applications, two types of materials are encountered, i.e.stiff and soft materials with the parameters (E = 210 G Pa, ξ2
0 = 0.027) and(E = 110 G Pa, ξ2
0 = 0.020), respectively. In all numerical examples, κ = 0.2 isused. Noise free synthetic data for these types of materials are generated by solving the
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Table 1. Synthetic noise free data for the stiff and the soft materials .
E = 210 Gpa m = 1 m = 2 m = 3 m = 4 m = 5
ϕm × 103 2.2 3.3 3.6 3.7 3.85ξ2
m × 102 1.3 4.0 6.6 7.9 10.0Tm × 102 2.48 3.75 4.21 4.41 4.7
E = 110 Gpa m = 1 m = 2 m = 3 m = 4 m = 5ϕm × 103 4.4 5.58 5.97 6.28 6.6ξ2
m × 102 1.4 3.6 5.2 7.1 9.8Tm × 102 2.59 3.35 3.68 4.0 4.39
direct problem (1) and precise values of torque are found by calculating the integral (3)numerically. For the numerical solution of the nonlinear problem (1), the following finitedifference scheme is used [22]
− h2
h1
[gi+1/2, j
vi+1, j − vi, j
h1− gi−1/2, j
vi, j − vi−1, j
h1
]
− h1
h2
[gi, j+1/2
vi, j+1 − vi, j
h2− gi, j−1/2
vi, j − vi, j−1
h2
]= 2ϕ,
(8)
where (x1,i , x2, j ) ∈ wh , wh := {(x1,i , x2, j ) : x1,i = ih1/(N1 − 1), x2, j = jh2/(N2 − 1),
i = 2, N1 − 1, j = 2, N2 − 1}
is the piecewise uniform mesh with mesh steps hm =lm/(Nm − 1), m = 1, 2, l1 = a, l2 = b; vi, j := u(n)(x1,i , x2, j ) are the nodal values of thefunction (nth iteration). The coefficients gp,q are defined as follows:
gi±1/2, j := g
(∣∣∇u(n−1)(x1,i±1/2, x2, j )∣∣2
), gi, j±1/2 := g
(∣∣∇u(n−1)(x1,i , x2, j±1/2)∣∣2
).
The approximation error of this scheme on uniform mesh is O(h2) where h2 = h21 + h2
2.Table 1 shows the computational results for the stiff and the soft materials. In this table, m =1 and m = 2, 3, 4, 5 correspond to the pure elastic and the pure plastic cases, respectively.
The solution process of the inverse problem for the stiff material by the modifiedsemi-analytic inversion method is presented below step by step. First by applying parametri-zation algorithm for the pair (ϕ1,T1), the approximate value of G is found to be Gh = 80.50.The linear direct problem is solved for Gh = 80.50 and ξ2
0 is found to be ξ20 = 0.013. For
the pairs (ϕ1,T1) and (ϕ3,T3), the function g(ξ2) defined by (6) is obtained as follows:
g(ξ2) ={
0.0124, ξ2 ≤ 0.013,
0.0124 − 0.1110(ξ2 − 0.013), ξ2 > 0.013.(9)
The linear direct problem is solved with the input data (9) and ξ21h is found to be ξ2
1h = 0.067.By using formula (7), κh and T3 are found to be 0.2275 and 0.0388, respectively. For the
pairs (ϕ2,T2) and (ϕ3,T3), κh and ˜T3 are found to be 0.2346 and 0.0463, respectively.
Consequently, we have the relation T3 < T3 <˜T3. Then, by applying Algorithm 2, after
three iterations, ξ20h is found to be ξ2
0h = 0.0252. By repeating the similar steps, ξ20h is found
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Inverse Problems in Science and Engineering 11
to be ξ20h = 0.0185 for the soft material. These values will be accepted as the approximate
value of the yield stress for the stiff and the soft materials, respectively. In the semi-analyticinversion method, the yield stress for the stiff and the soft materials are found to be 0.026and 0.019. These approximate values seem more precise than the approximate values foundby the modified semi-analytic inversion method. But the semi-analytic inversion methodfinds these values by choosing the parameters, appearing in the algorithm, very small. Thismeans that we need more time since the algorithm requires many iterations.
To compare these obtained results, the parametrization algorithm is also applied forthe determination of the plasticity function g(ξ2). The reconstructed plasticity function isplotted in the Figure 3 by the dotted line. It is obvious from the figure that the optimal anglesteps �ϕ are 1.65 × 10−3 and 2.20 × 10−3 for the stiff and the soft materials respectively.We note that a smaller angle step is found for the stiff material. This result depends on theexperiments given in Table 1. Thus, it may not be always true that smaller angle step isneeded for the stiff material. However there are some relations between the parameters ofengineering materials. For instance, in the pure elastic case an increase of the parameterG > 0 implies an increase of the rigidity of a material, and as a result, leads to decrease ofthe torque for a fixed angle of twist ϕ (see [17,18,21] for details). Tables 2 and 3 show theapproximate values of κ for each given pair (ϕ,T ). Table 4 shows that the absolute and therelative errors defined by εg = ∥∥ g − gh
∥∥∞ and δg = ∥∥ (g − gh)/g∥∥∞, respectively, for
noise free inverse problem where gh is the approximate solution.In practice, the torque T can be given with some measurement error γ , i.e. with noise
data Tγ = T + γ T . Three types of noise levels with γ = +0.03, γ = −0.05 and
Figure 3. The reconstructed curve gh(ξ2) for different values of angle steps for the stiff (left figure)and the soft (right figure) materials, (noise free data).
Table 2. The approximate values of κ corresponding to each plastic case for the stiff material.
(ϕ × 103,T × 102) �ϕ × 103 gh(ξ2) ξ21h gh(ξ2
1h) κh
(3.30, 3.75) 1.1 0.0124 − 0.1385(ξ2 − 0.0252) 0.030 0.0117 0.35(3.60, 4.21) 1.4 0.0124 − 0.0900(ξ2 − 0.0252) 0.058 0.0094 0.34(3.70, 4.41) 1.5 0.0124 − 0.0800(ξ2 − 0.0252) 0.070 0.0088 0.33(3.85, 4.71) 1.65 0.0124 − 0.0670(ξ2 − 0.0252) 0.092 0.0073 0.30
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Table 3. The approximate values of κ corresponding to each plastic case for the soft material.
(ϕ × 103,T × 102) �ϕ × 103 gh(ξ2) ξ21h gh(ξ2
1h) κh
(5.58, 3.35) 1.18 0.0238 − 0.2900(ξ2 − 0.0185) 0.033 0.0196 0.33(5.97, 3.68) 1.57 0.0238 − 0.2230(ξ2 − 0.0185) 0.048 0.0172 0.31(6.28, 4.00) 1.88 0.0238 − 0.1800(ξ2 − 0.0185) 0.066 0.0153 0.30(6.60, 4.39) 2.20 0.0238 − 0.1390(ξ2 − 0.0185) 0.095 0.0130 0.27
Table 4. The absolute and relative errors for the noise free inverse problem.
E = 210 Gpa Paramet. Fast alg.
εg 4.4 × 10−4 1.7 × 10−4
δg 6.0 × 10−2 1.5 × 10−2
E = 110 Gpa Paramet. Fast alg.εg 4.6 × 10−4 4.3 × 10−4
δg 4.2 × 10−2 1.9 × 10−2
Figure 4. The reconstructed curve gh(ξ2) for �ϕ = 1.65 × 10−3 for the stiff material (left figure )and the reconstructed curve gh(ξ2) for �ϕ = 2.20 × 10−3 for the soft material (right figure), (noisedata).
γ = ±0.07 are used in computational experiments. The results are presented in Figure 4.Table 5 shows the absolute and the relative errors for three different levels of the inverseproblem. By comparing the numerical results in the case of noise, we can conclude thatthe relative error δg corresponding to the modified semi-analytic inversion method is lessthan the relative error corresponding to the parametrization algorithm. Note that the resultsobtained for the soft material are almost the same in terms of accuracy, absolute and relativeerrors.
Finally, we want to emphasize that this study will constitute a background for futureapplications, i.e. in solving some kinds of inverse problems that arise physics and materialsscience numerically.
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Inverse Problems in Science and Engineering 13
Table 5. The absolute and relative errors for the noise inverse problem.
γ (For stiff) εg (Paramet.) εg (Fast alg.) δg (Paramet.) δg (Fast alg.)
+0.03 9.3 × 10−4 8.5 × 10−4 1.3 × 10−1 1.0 × 10−1
−0.05 2.0 × 10−3 1.0 × 10−3 2.7 × 10−1 1.9 × 10−1
±0.07 2.2 × 10−3 1.3 × 10−3 2.8 × 10−1 2.1 × 10−1
γ (For soft) εg (Paramet.) εg (Fast alg.) δg (Paramet.) δg (Fast alg.)+0.03 1.2 × 10−3 1.0 × 10−3 9.3 × 10−2 8.6 × 10−2
−0.05 2.5 × 10−3 2.3 × 10−3 2.0 × 10−1 1.7 × 10−1
±0.07 2.6 × 10−3 2.4 × 10−3 2.2 × 10−1 1.9 × 10−1
5. Concluding remarks
Anew method is given for determination of the yield stress for power hardening engineeringmaterials. It may be viewed as a very effective modification to the semi-analytic inversionmethod. The modification is just related to determination of yield stress and does not affectwell-posedness of the semi-analytic inversion method. Thus, the modified semi-analyticinversion method is also well-posed. One of the advantages of this method is that it doesnot require complicated calculations and many parameters. Since each choice of theseparameters affect the numerical calculations (some of them cause ill-posedness of themethod), the new method solves this problem using a minimum number of parameters.The presented numerical examples show that the modified semi-analytic inversion methodallows us to determine the plasticity function with high accuracy, even for acceptable noiselevels. The authors of this paper consider to apply the presented method to different problemsin physical and mechanical science such as the bending problem of a plate in upcomingstudies. In addition, whether the presented analyses can be exploited for non-dimensionalparameters in fluid mechanics, such as Reynolds number, Mach number, Froude number,Weber number and Strouhal number is also an interesting problem. Such non-dimensionalparameters are used for geometric scaling, and for developing dynamic similitude in ex-perimental processes. Furthermore, the authors consider to study applicability of both theparametrization and semi-analytic inversion methods if there is no assumption about theplastic response of the material.
AcknowledgementsThe research has been partially supported by the Scientific and Technological Research Council ofTurkey (TUBITAK), also by Zirve University Research Fund. The authors thank the reviewers fortheir very careful reading and for pointing out several mistakes as well as for their useful commentsand suggestions.
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