An effective inverse analysis tool for parameter identification of anisotropic material models

International Journal of Mechanical Sciences 77 (2013) 130–144 Contents lists available at ScienceDirect International Journal of Mechanical Science...

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International Journal of Mechanical Sciences 77 (2013) 130–144

Contents lists available at ScienceDirect

International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci

An effective inverse analysis tool for parameter identiﬁcation of anisotropic material models Gabriella Bolzon n, Marco Talassi Department of Civil and Environmental Engineering, Politecnico di Milano, piazza Leonardo da Vinci 32, 20133 Milano, Italy

art ic l e i nf o

a b s t r a c t

Article history: Received 2 August 2012 Received in revised form 7 February 2013 Accepted 9 September 2013 Available online 4 October 2013

Effective inverse analysis tools rest on the synergetic combination of even complex experimental and numerical procedures. This is the case of methodologies recently developed for the identiﬁcation of parameters entering anisotropic elastic–plastic constitutive laws on the basis of data collected from indentation tests. The values of the sought material properties are inferred from a discrepancy minimization procedure, which entails the repeated simulation of the test. In this situation, the overall computing burden can be signiﬁcantly reduced, without compromising the accuracy of the results, replacing traditional ﬁnite element approaches by approximated analytical models, based on the interpolation by radial basis functions of a few numerical results ﬁltered by proper orthogonal decomposition. The effectiveness of this technique is demonstrated in the present study with speciﬁc reference to earlier investigated elastic–perfectly-plastic materials and is further veriﬁed on the more general case of hardening constitutive models. The accuracy of the identiﬁcation results and the role of the likely occurrence of multiple solutions are discussed. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Parameter identiﬁcation Anisotropic material models Hardening plasticity Indentation tests Proper orthogonal decomposition (POD) Radial basis function (RBF)

1. Introduction Anisotropy is a characteristic feature of several natural and artiﬁcial structural materials like wood [1], bone [2,3], and tissues [4]. Anisotropy can be also induced by fabrication methods like extrusion or lamination of metal sheets and pipes [5–10], by the presence of ﬁbres reinforcing composites and laminates [11–13], by the growth mechanisms of thin ﬁlms, coatings and single crystals [14,15], by the synthesis of porous metals [16] as well as by damaging phenomena [17]. A spreading approach to the mechanical characterization of materials is based on instrumented indentation test [18], which constitutes an almost mandatory approach in the case of biological, possibly living structures [2–4] or for investigations at the micron scale [19,20]. This simple experiment consists of pressing a diamond or hard steel tip of different geometry against the sample to be investigated. The curves relating the force exerted on the material surface versus the penetration depth of the tip represent the data most widely exploited to characterization purposes. This information reﬂects the material properties on average and can be correlated to a scalar measure of the distribution of elastic moduli in anisotropic solids [2,11,15,21–23]. The estimation of each material property requires additional data, which may consist

n

Corresponding author. Tel.: þ 39 02 2399 4319; fax: þ 39 02 2399 4300. E-mail address: [email protected] (G. Bolzon).

0020-7403/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijmecsci.2013.09.009

of the output of complementary tests performed with different geometry of the indenter tip [24,25] and/or of measurements concerning the deformation left on the material surface [13,26], which can be mapped after the removal of the indentation tool by different equipments, depending on the characteristic size of the imprint [27–29]. In fact, material anisotropy and other directiondependent quantities, like residual stresses [30,31], entail the loss of axis-symmetry of the residual imprint produced by conical or spherical tips conforming with Standards [32,33]. Usually, the indentation test is simulated by non-linear ﬁnite element techniques and the material parameters are recovered through the minimization of an objective function, which quantiﬁes the discrepancy between the experimental information and the corresponding output of the computations [34]. The optimization process is based on iterative algorithms, which require a number of function evaluations and entail computational costs fast increasing with the number of unknown parameters. Therefore, most investigations concern problems with a few unknowns and models are reduced as much as possible. Thus, isotropic elastic response and relatively small anisotropy ratios are common assumption [13,24,25]. Transversely isotropic metals obeying elastic–perfectly-plastic constitutive models governed by six independent parameters (three elastic moduli and three yield limits) have been investigated in [26] but the exhaustive analysis of situations involving so many parameters is usually prevented by the huge computational burden associated to traditional simulation techniques. This drawback can

G. Bolzon, M. Talassi / International Journal of Mechanical Sciences 77 (2013) 130–144

be dramatically alleviated by some accurate analytical approximation of the system response based on model reduction techniques of proven usefulness in several application ﬁelds [35–43]. The purpose of the present contribution is to assess the merit and the limitations of an approach based on the combination of proper orthogonal decomposition (POD) and radial basis functions (RBF). The identiﬁcation problems considered for the present demonstrative purpose concern the class of transversely isotropic materials already investigated in [26]. Pseudo-experimental data are produced by ﬁnite element (FE) models of the test. Random disturbances added to the numerical results replace the measurement noise. The possibility of recovering the values of the constitutive parameters, which deﬁne the input of the simulated experiment, is discussed.

2. Constitutive model Linear elasticity and hardening plasticity obeying Hill′s constitutive model [44] are usually adopted to describe the mechanical response of anisotropic (orthotropic) metal materials. The general formulation is reduced to the case of transversal isotropy about the x-axis (in the x–y–z reference system) for the present demonstrative purposes. The considered situation is typical of the mechanical response of rolled sheet [5–8], tubes [9,10], whisker-reinforced alloys [13] and nano-porous metal foams used for scaffold production [16]. The elastic behaviour is governed by 3 independent elastic moduli (Ey , Ey ¼ Ez and Gxy ¼ Gxz ) to be identiﬁed starting from indentation results. The lateral contraction ratios νyz and νxy ¼ νxz (being νij =Ei ¼ νji =Ei , i; j ¼ x; y; z) are supposed to be a priori known, due to their known poor inﬂuence on indentation results [28,45], while the shear modulus 2Gyz ¼ Ey =ð1 þ νyz Þ due to transversal isotropy. The elastic threshold is deﬁned by a limit value of Hill′s equivalent stress, namely: seq ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ kx ðsyy szz Þ2 þky ðszz sxx Þ2 þ kz ðsxx syy Þ2 þkxy s2xy þkyz s2yz þ kzx s2zx

ð1Þ

where in the present hypothesis: ! 1 2 1 1 kx ¼ ; ky ¼ kz ¼ 2 2 R2y R2x 2Rx kxy ¼ kzx ¼

1 R2xy

;

kyz ¼

3 R2y

ð2Þ

The dimensionless parameters Rx , Ry and Rxy represent the ratios between the yield limits (assumed equal in tension and compression) under uniaxial and shear loading along the main Y material axes. Therefore: sp ¼ Rx sY ; sYy ¼ Ry sY ¼ Rz sY ¼ sYz ; sYxy ¼ x ﬃﬃﬃ sYzx ¼ Rxy sY ; while sYyz ¼ sYy = 3 in the isotropy plane. The hardening material response is described by the exponential rule: ε n eq seq sY Y r 0 ð3Þ ε where the equivalent strain measure, conjugate to seq (1), reads: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 2 kx kx εeq ¼ r x εyy εzz þ r y εzz εxx þr z εxx εyy þ r xy ε2xy þr yz ε2yz þ r zx ε2zx kz ky

ð4Þ ky kx ; ry ¼ rz ¼ ð2kx þ ky Þ2 ð2kx þky Þ2 2 2 2 r xy ¼ r zx ¼ ¼ ; r yz ¼ kxy kzx kyz

rx ¼

ð5Þ

131

while the material constants sY and εY represent the value of seq and εeq at the onset of yielding, to be identiﬁed on the basis of indentation results together with the coefﬁcients Rx , Ry , Rxy and with the hardening exponent n ( o1). Expressions (1) and (4) reduce to those corresponding to the classical Hencky–Huber–von pﬃﬃﬃ Mises formulation when Rx ¼ Ry and, contemporarily, Rxy ¼ Ry = 3.

3. Inverse analysis for parameter identiﬁcation In instrumented indentation, a tip made of hard materials is pressed against the surface of the component to be analysed by a force, which is continuously increased up to a maximum and then brought back to zero in a controlled mode. During this process, the penetration of the tip is monitored and a sequence of N h depth values hmi (i¼ 1, 2,…, N h ) is sampled and recorded for N h pre-ﬁxed values of the force. The geometry of the residual deformation left on the material surface is mapped at the removal of the indenter by means of contact or optical instruments (microscopes or proﬁlometers) operating at different scales [27–29]. The out-of-plane displacement components umj (j¼1, 2,…, Nu) relevant to a pre-selected regular grid of points lying on the indented area are stored as further data to be exploited by the identiﬁcation procedure. The simulation of the test, performed for given values of the constitutive parameters (elastic moduli Ex , Ey , Gxy , yield limits sYx , sYy , sYxy and the hardening exponent n, in the present context) collected by vector z, returns the computed penetration depth hci ðzÞ at the pre-ﬁxed Nh load levels and the distribution of residual displacements ucj ðzÞ. The sought material properties are recovered by the minimization of the discrepancy between corresponding measurements and computations, expressed by the norm: ωðzÞ ¼

2 1 Nh hmi hci ðzÞ 1 Nu umj ucj ðzÞ 2 þ ∑ ∑ Nh i ¼ 1 hmax Nu j ¼ 1 umax

ð6Þ

The factors hmax and umax in (6) represent normalization terms, here assumed to coincide with the maximum absolute value attained by the corresponding distributions hmi and umj . However, when a proper statistical characterization of real measurements is available, these normalization coefﬁcients can be replaced by weights on force and displacement components correlated with the variances that quantify the experimental noise [34]. In the present applications, the entries of the vector z^ producing the minimum discrepancy ωðz^ Þ are sought by a gradientbased algorithm available in the popular optimization tool [46]. In view of the possible convergence toward local minimum points, the search is repeated 10 times for each exercise, starting from 10 different randomly deﬁned initialization vectors. While this provision does not guarantee to identify the absolute minimum point, a wide exploration in the domain of the material properties permits to single out the main characteristics of the objective function (6) and to infer the reliability of the identiﬁcation results. Each material calibration exercise requires to evaluate the discrepancy (and, hence, to perform a new simulation of the test with varied input vector z) hundreds of times, according to the computational experience documented for instance in [26]. Therefore, a good balance between accuracy and effectiveness of the simulation tool represents a key issue for practical applications of the considered identiﬁcation procedure. This goal can be achieved implementing the modelling provision illustrated in the following.

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are fully constrained, an assumption that does not inﬂuence the system response since the material volume affected by indentation is about half mm for the investigated range of constitutive properties, listed in Table 2. The indenter tip is assumed to be rigid, after having veriﬁed that its elastic properties produce almost negligible effects on both the indentation curves and the residual deformation. The characteristics of the interface between the indentation tool and the specimen is the same assumed in Model A and in former computations [26]. The realism of the numerical output is improved by considering the smooth relationship between the contact pressure p and the gap g illustrated by the sketch in Fig. 3 and described by the analytical expressions: 8 if g 4 g 0 <0 p¼ ð7Þ g eð1 ðg=g0 ÞÞ 1 ifg r g 0 : p0 1 g e1

4. Simulation of the indentation test The effect of the material parameters on the indentation results can be evaluated by FE simulations, taking into account the material and geometric non-linearity associated with large plastic deformation and frictional contact [4,11–13,20,24–26]. The numerical analyses relevant to the present work have been performed by a commercial code [47], in the large displacement framework under the hypothesis that elastic strains are much smaller than the plastic contribution. Two different model problems have been considered. The ﬁrst one (henceforth named ‘Model A’) coincides with that implemented in former computations devoted to the analysis of thin aluminium foils by the traditional FE approach [26], with assumed elastic–perfectly plastic constitutive response and idealized sharp indenter tip. Model A is exploited in the present study for numerical comparison purposes. The second model problem (‘Model B’) refers to the more realistic situation of hardening materials subjected to indentation tests conforming to Rockwell Standards [33]. Model B is used in the present contribution to assess the performances of the proposed parameter identiﬁcation methodology over a wider range of metal properties. In both cases, indentation is assumed to be performed in the anisotropic plane x–y, richer in information relevant to each material direction and more easily accessible in practice, for instance in the case of thin metal sheet or tubes [6–10]. A three-dimensional representation of the problem is therefore required. Model A considers a cylindrical portion of the sampled material, 100 μm radius and 60 μm height, discretized into 1728 solid (8-node) elements with reduced Gauss-integration rule and hourglass control mode. The mesh, represented in Fig. 1, is deﬁned by a total of 2184 nodes, 114 on the indented surface. The displacements of the bottom nodes are ﬁxed, while they are free on the lateral surfaces. These boundary conditions are rather immaterial since the maximum penetration depth in the simulated experiments is of few microns and most deformation is exhausted at about 20 μm from the indenter axis for the considered range of material properties. The idealized sharp conical (681 opening angle) tip [24,26,48] is modelled as a deformable elastic isotropic body, with Young modulus E¼1140 GPa and Poisson ratio v¼ 0.07, typical values for diamond. The contact surface between the indentation tool and the specimen is characterized by Coulomb friction without dilatancy, with assumed friction coefﬁcient equal to 0.15. Model B considers a spherical region of the solid to be investigated. The blunt shape of the indenter tip deﬁned by Rockwell Standards for metals (sphero-conical with 1201 opening angle and 200 μm radius of the spherical end [29,33,49]) replaces the earlier idealized geometry. The maximum value of the applied load in the simulated experiment (2 kN) is also consistent with Standards. The modelled domain has 5 mm radius and is discretized into 2750 linear solid elements (2 2 2 Gauss point integration scheme) deﬁned by a total 3401 nodes, 421 on the indented surface, as visualized in Fig. 2. The displacements of the nodes situated on the curved boundary surface

y

0

Thus, the interaction between the deformable solid and the rigid tip occurs when g o g 0 while a small interpenetration is allowed for p4p0 . The hard contact condition is recovered as g 0 -0 and p0 -0. Otherwise, suitable values of parameters g 0 and p0 are suggested by the size of the elements entering into contact under the indenter tip and by the yield limits. For the present applications, g 0 ¼ 4 μm and p0 ¼ 200 MPa are assumed. This provision allows to remove the already evidenced oscillations exhibited by the indentation curves produced by Model A simulations (see Fig. 4 and Ref. [26]), essentially due to the progressive enforcement of the hard contact condition at the nodes of the roughly discretized indented surface during the loading process. The average computing time of one of these traditional FE analyses, to be eventually performed hundreds times to inverse analysis purposes, is of the order of 10–20 min. However, the overall performances of the proposed identiﬁcation procedure can be dramatically improved by exploiting model reduction techniques based on the expected correlation of the output of these repetitive computations. The combination of proper orthogonal decomposition (POD) and interpolation by radial basis functions (RBFs) produces an optimal compromise between the accuracy of the results and the efﬁciency of the computations. The procedure is synthetically illustrated in the next section; details can be found in [42,43].

5. POD-RBF approximation An initial set of Ns different combinations of the constitutive properties is generated in the region of interest. The corresponding values are collected by vectors zk (k ¼ 1; :::; N s ). In the present application, the entries of zk are randomly generated within the limits deﬁned in Tables 1 and 2 for Model A and Model B, respectively. Notice that the Ns admissible parameter sets, schematically visualized by dots in Fig. 5 (Model A), Figs. 6 and 7 (Model B), do not span all the investigated space due to the physical constraint of positivedeﬁniteness of the quadratic forms (1) and (4).

x

x

z Fig. 1. FE discretization of the indented solid and a lateral view of the mesh under the indenter tip: model A.

G. Bolzon, M. Talassi / International Journal of Mechanical Sciences 77 (2013) 130–144

133

x

x

y

z Fig. 2. FE mesh exploited for the simulation of indentation tests conforming with Rockwell standard (model B) and a detail of the discretization under the indenter tip.

Table 1 Range of the investigated material properties associated to the simulation Model A. 30 rEx r 80 GPa

20r Ey r 50 GPa

10r Gxy r 25 GPa

20 rsYx r 80 MPa

20r sYy r 40 MPa

10r sYxy r 40 MPa

Table 2 Range of the investigated material properties associated to the simulation Model B. 150r Ey r250 GPa

150 rEy r250 GPa

70r Gxy r 110 GPa

200r sYx r1300 MPa

200 rsYy r 1300 MPa

200r sYxy r 1300 MPa

0r nr 0.477

Snapshots can be referred to an optimally reduced orthonormal reference system Φ, such that ΦT ¼ Φ 1 , where Φ ¼ U U V UΛ 1=2

ð8Þ

In the above expression (8), matrix V stores the retained eigenvectors associated to matrix D, while the diagonal matrix named Λ 1=2 collects normalization terms consisting of the inverse of the square root of each retained eigenvalue. Thus, the computed values of the penetration depth and of the residual deformation can be evaluated with the desired accuracy by the combination of the elementary contributions: U ¼ ΦUA

ð9Þ

where matrix A ¼ Φ U U collects ampliﬁcation coefﬁcients. The linear transformation (9) can be generalized to different parameter sets, deﬁned within the region of interest and not included in the initial selection z, by means of the interpolation functions g k collected in vector g, which depend on the Euclidean distance jjz zk jj in the normalized parameter space. Thus T

uðzÞ ¼ Φ U aðzÞ ¼ Φ U B U gðjjz zk jjÞ

ð10Þ

The interpolation coefﬁcients stored in matrix B are recovered from the training set z, such that: U ¼ uðzÞ ¼ Φ U B U gðzÞ

ð11Þ

Therefore, see (9) Fig. 3. Hard (left) and smooth (right) contact conditions.

Each training vector zk deﬁnes the input material characteristics for a FE analysis, which simulates the indentation test carried out under load control up to a pre-deﬁned maximum value of the force and then back to zero. These computations return the measurable quantities that deﬁne the discrepancy function (6), namely N h tip penetration depths hci ðzk Þ at N h pre-ﬁxed load levels and N u values of the residual out-of-plane displacements ucj ðzk Þ, corresponding to a selected grid of points on the indented surface. These information sets, named snapshots in the present context, are stored in matrices Uh ½Nh N s and Uu ½Nu N s , in ordered correspondence with the vectors zk collected by matrix z. Due to their physical meaning, data collected by either snapshot matrix (Uh and Uu , in the following simply indicated by U) are correlated. This occurrence can be veriﬁed by the eigen-analysis of the square matrix D ¼ UT U U, which returns most eigenvalues equal to either zero or very small numbers [50,51]. Eigen-pairs can be computed in a sequence, starting from the largest eigenvalue and stopping the search on the basis of the comparative evaluation of the accuracy of the results associated to different truncation schemes, as reported in the following.

A ¼ B UG

ð12Þ

where matrix G gathers the vector values gðzk Þ. The analytical representation (10) of the system response permits to carry out inverse analysis exercises within 1 min computing time, whereas the traditional FE approach may require several hours. Furthermore, the reduced model (10) produces some regularization of the simulated results, which are illustrated in the following. In the present applications, interpolation functions coincide with the RBFs [52,53]: g k ðzÞ ¼ e jjz zk jj

ð13Þ

6. Numerical results 6.1. Model problem A The above introduced POD-RBF approach has been veriﬁed ﬁrst on the identiﬁcation problems earlier considered in the numerical study [26], where all simulations were performed by a traditional FE

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Fig. 4. Comparison among the indentation curves calculated by the FE method (Model A) and the corresponding POD-RBF approximations based on 2 and on 5 retained eigen-modes; parameter values are listed in Table 5 for the graphs on the left, in Table 6 for the graphs on the right.

Fig. 5. Spatial distribution of the randomly generated constitutive properties (elastic moduli and yield limits) related to Model A: training (dots) and veriﬁcation (stars) sets.

approach on the basis of the discretization shown in Fig. 1 (Model A). The material behaviour was assumed elastic–perfectly plastic, characterized by six independent material parameters (three elastic moduli and three yield limits). The target material properties are listed in Tables 5 and 6, while the corresponding pseudo-experimental data are represented in Fig. 8 (target curves). For each case, the tip penetration depth is sampled at N h ¼102 load increments or decrements, for each

Fig. 6. Spatial distribution of the randomly generated elastic moduli considered for the simulation of indentation tests in Model B.

0.12 mN up to the maximum 6 mN force and then back to zero. This information is stored for further use in inverse analysis together with N u ¼ 2500 out-of-plane displacements, relevant to a grid of 50 50 points at equally spaced intervals deﬁning an area of 17 17 μm2 on the specimen surface, conforming with

G. Bolzon, M. Talassi / International Journal of Mechanical Sciences 77 (2013) 130–144

135

Table 3 Eigenvalue sequence relevant to the informative content of the indentation curves and of the geometry of the residual imprint generated by the simulation Model A. Italics evidence the value corresponding to the ﬁrst neglected eigen-pair. Indentation curves

Imprint geometry

3.445E þ 05 2.654E þ 01 1.262E þ 01 3.249E þ 00 2.681E þ 00 1.458E þ00

3.385Eþ 05 1.161E þ 04 7.326E þ03 1.184E þ03 4.461E þ02 3.295Eþ 02 1.481E þ 02

6.848E þ01 3.767E þ 01 2.408Eþ 01 1.561E þ 01 1.107Eþ01 8.800Eþ 00

Table 4 Eigenvalue sequence relevant to the informative content of the indentation curves and of the geometry of the residual imprint generated by the simulation Model B. Italics evidence the value corresponding to the ﬁrst neglected eigen-pair. Indentation curves

Imprint geometry

2.176E þ09 3.664E þ 05 1.266E þ04 5.407Eþ03 4.482Eþ 03

8.419E þ 08 4.539E þ 07 1.826E þ 07 1.031E þ 07 3.240Eþ06 2.336Eþ06 1.121E þ06

8.016E þ 05 6.322E þ 05 3.845E þ 05 3.413E þ 05 2.394E þ 05 2.019E þ 05 1.726E þ 05

1.210E þ05 1.041E þ 05 8.974E þ04 7.548E þ 04 5.451E þ 04 4.455E þ04 4.308Eþ 04

Table 5 Target and identiﬁed parameter values produced by the inverse analysis procedure based on the POD-RBF approximation of the simulation model A: application 1. Discrepancy

[MPa] 18.000

–

21,844 14,421 10,751

36.820 35.262 36.398

30.058 29.732 29.663

17.960 18.381 19.073

9.02E 06 1.31E 05 1.52E 05

Gxy [MPa]

42,000

36,000

Curves and imprint No noise 48,988 35,115 67,094 29,997 77,458 46,828 Noisy data

49,036 77,458 67,171 77,458 59,850

35,142 21,654 30,009 46,824 29,096

21,880 12,130 14,446 10,751 10,751

36.805 34.276 35.236 36.384 35.228

30.064 29.558 29.730 29.666 30.172

17.953 19.106 18.387 19.075 19.027

1.67E 05 1.95E 05 2.10E 05 2.31E 05 2.35E 05

Imprint only No noise 73,670 41,750 59,461 49,526

32,275 36,525 34,990 33,533

15,761 16,699 10,751 22,043

34.981 35.101 34.801 36.204

29.554 29.440 29.722 29.752

18.241 18.446 19.021 18.169

3.21E 06 3.48E 06 5.54E 06 6.04E 06

32,481 36,496 33,802

15,762 16,735 21,926

35.034 35.180 36.307

29.558 29.461 29.797

18.208 18.423 18.175

1.08E 05 1.09E 05 1.36E 05

Noisy data

the selection performed in [26]. Eventually, the numerical output is corrupted by disturbances that replace the experimental noise and consist of a randomly generated addendum deﬁned within the range 72.5% of the original value of each measurable quantity.

sYxy

[MPa] 30.000

Ey [MPa]

Target

Fig. 7. Spatial distribution of the randomly generated parameters (initial yield limits and hardening exponent) governing the material response in the plastic range in Model B.

sYy

16,000

sYx [MPa] 35.000

Ex [MPa]

73,189 42,206 49,639

The discrepancy function (6) is minimized within the limits deﬁned in Table 1 by a gradient-based ﬁrst order optimization algorithm selected in the optimization package [46]. Due to the expected existence of local minimum points, the search is repeated 10 times, starting from different entries of the initialization vector z0 , as introduced in Section 3. The POD-RBF approximation of the system response considered for the present identiﬁcation exercises is trained on the results of Ns ¼ 500 FE analyses, performed once for all for the random distribution of parameter values indicated by dots in Fig. 5. It is

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Table 6 Target and identiﬁed parameter values produced by the inverse analysis procedure based on the POD-RBF approximation of the simulation model A: application 2. sYy

sYxy

Discrepancy

20,000

sYx [MPa] 60.000

[MPa] 30.000

[MPa] 26.000

–

21,074 20,895 17,232

60.054 59.762 60.132

29.939 29.974 30.082

26.034 26.058 26.275

4.54E 06 4.70E 06 5.21E 06

Ex [MPa]

Ey [MPa]

Gxy [MPa]

72,000

36,000

Curves and imprint No noise 39,017 36,217 66,601 34,241 33,050 44,231

Target

39,171 66,129 77,458 77,458

35,737 34,130 40,791 21,920

21,001 20,915 10,751 10,762

60.034 59.748 60.170 59.826

29.915 29.953 30.464 29.672

26.056 26.095 25.495 27.327

1.06E 05 1.07E 05 1.52E 05 1.76E 05

Imprint only No noise 40,896 32,742 77,458 39,829 76,195

36,699 46,780 32,819 48,486 48,486

20,942 16,863 20,751 11,129 14,310

60.031 60.280 59.892 59.713 60.145

29.935 29.914 30.021 29.881 30.118

25.996 26.564 26.272 26.267 25.954

1.39E 06 1.65E 06 1.86E 06 3.27E 06 3.99E 06

36,595 34,796 47,439 22,940

20,920 20,699 13,846 10,751

60.094 59.851 60.160 59.841

29.923 29.962 30.138 29.890

25.970 26.028 25.904 26.757

7.78E 06 7.95E 06 1.04E 05 1.17E 05

Noisy data

Noisy data

41,882 65,289 77,458 77,458

worth noticing that convergence of the minimization procedure implemented and reported in [26] for the same problem was achieved in about 10 iterations: the 10 different initializations considered herein would then require some 700 direct computations for each application, since the derivatives of the discrepancy function with respect to the sought parameters are evaluated by a ﬁnite difference scheme at each iteration. The number and the distribution of the training points in the parameter space affects the accuracy of the non-linear amplitude function aðzÞ, which combines the basis vectors stored in matrix Φ (see (10)). In turn, the number of bases (retained eigen-modes) inﬂuences the accuracy and the ﬁltering capability of the POD truncation. These selections are made on the basis of some preliminary accuracy evaluation; for details, see e.g. [42,43]. In the present study, ﬁve basis vectors are retained to reconstruct the indentation curves, 12 for describing the imprint (see Table 3). Notice the fast initial decrease of the eigenvalues relevant to the indentation curves. This feature reﬂects the existing correlation among these data and, hence, their relatively low informative content about the variability of the several parameters, which deﬁne the problem of interest. The indentation curves obtained as the output of the FE simulations carried out for the target parameters listed in Tables 5 and 6 are compared in Fig. 4 with the corresponding POD-RBF approximations based on two and on ﬁve retained eigen-pairs. The main observable difference consists of the reduction of the oscillations exhibited by the loading branch of the indentation curves, related to the already commented enforcement of the contact conditions. Either all available pseudo-experimental information or geometry data only constitute the input of the discrepancy function (6). The comparison between the results obtained in the two cases permits to evaluate the relative informative content of the imprint, which represents the input source that reﬂects the directionality of the material properties. In all analysed situations, the 10 different initialization vectors z0 drive the convergence of the optimization procedure toward different minimum points z^ , which correspond to comparable small values of ωðz^ Þ (see Tables 5 and 6). Correspondingly, the simulation of the test performed by all identiﬁed parameter sets reproduces

the pseudo-experimental response with almost the same accuracy, as evidenced for instance by the graphs in Fig. 8, which refer to the case of noisy data collected from both indentation curves and residual imprints. It is worth recalling that the FE model exploited to generate the pseudo-experimental data does not coincide with the analytical POD-RBF approximation implemented to the inverse analysis purposes. On the other hand, models are not even expected to exactly reproduce the reality. Results reported in Tables 5 and 6 show that the yield limits are always accurately captured, while the scatter on the largest elastic modulus (Ex) can be much larger. This outcome is somewhat expected due to the large amount of irreversible deformation that develops under the idealized sharp geometry of the indenter tip in the analysed situations, which correspond to a relatively small ratio between the yield limit and the elastic modulus under the hypothesis of perfect plasticity. Notice the almost negligible elastic recovery at unloading visualized by the graphs of the indentation curves in Fig. 8. Table 7 compares the errors reported in [26] with those recovered from the present investigation: the results are almost equivalent but the POD-RBF based procedure is much less sensitive to the random addendum, which replaces the experimental noise. This outcome is justiﬁed by the role of the numerical disturbances, which affect the FE output and are ﬁltered by the POD-RBF approximation. The substantial equivalence between identiﬁcation results recovered through traditional FE simulations or analytical PODRBF approximation can be appreciated from the graphs in Fig. 9, which visualize the discrepancy function ωðzÞ (6) evaluated for: z ¼ αz^ 1 þ ð1 αÞz^ 2

ð14Þ

where z^ 1 and z^ 2 represent the ﬁrst two solutions listed in Table 5, which are obtained for α ¼ 1 and 0, respectively. In this interval, the discrepancy function evaluated on the basis of FE modelling for 0.1 increments of the parameter α presents the minimum value ωðz^ 3 Þ ¼ 8:36 10 6 at the intermediate point z^ 3 , which corresponds to α ¼ 0:5, i.e. to the parameter set: Ex ¼58041 MPa, Ey ¼ 32,556 MPa, Gxy ¼18,133 MPa, sYx ¼ 36.041 MPa, sYy ¼29.895 MPa, sYxy ¼18.171 MPa. Once again, the yield limits are estimated quite accurately, while a further set of elastic moduli is recovered, due to the limited sensibility of the indentation output to the reversible material response. The graphs in Fig. 10 show that the shift between the minimum points recovered by either FE or POD-RBF modelling depends primarily on the demanding approximation of the wavy input curves. The visualized differences are rather negligible in view of applications to real situations. 6.2. Model problem B The effectiveness of the parameter identiﬁcation procedure summarized in Section 3 is further veriﬁed considering the case of hardening plasticity characterized by seven independent material parameters (three elastic moduli, three yield limits and a common hardening exponent) deﬁned within the limits listed in Table 2, which includes the mechanical characteristics of most structural metals. The analytical representation of the system response provided by (10) is trained on the results of N s ¼ 794 different combinations of randomly distributed parameter values, indicated by dots in Figs. 6 and 7. Notice that the generated points are denser for lower values of the hardening coefﬁcient, deﬁned through a random extraction of the variable 0 rη r1 , such that: n¼

e3η 1 40

ð15Þ

In the POD-RBF approximation of the corresponding FE output, indentation curves are reconstructed starting from the four basis

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Fig. 8. Comparison among the target indentation curves and the main imprint proﬁles (along x- and y-axis) produced by the FE Model A and the corresponding ones recalculated by the POD-RBF approximation (5 retained modes for the curve; 12 for the imprint) using the parameter values listed in Table 5 (graphs on the left) and in Table 6 (graphs on the right), identiﬁed on the basis of all available information with noisy data.

vectors associated to the eigenvalues listed in Table 4 while imprints are described by the combination of 20 normalized elementary deformation modes, larger in number than in the previous examples. In fact, compared to the sharp conical idealization, the actual proﬁle of the Rockwell tip reduces the correlation of geometrical data [54]. Further six parameter combinations (numbered and indicated by stars in Figs. 6 and 7) are randomly generated for veriﬁcation purposes. The corresponding constitutive properties, listed in Tables 8–13 (target values), produce the uniaxial responses along the principal material directions represented in Fig. 11. Each parameter set constitutes the input of a traditional FE analysis carried out by the simulation Model B, which returns the pseudoexperimental information considered in the present identiﬁcation procedure. Numerical disturbances that replace the experimental noise consist of a random addendum (generated within the range 72.5 μm) of each considered measurable quantity. Some

representative indentation curves and the main proﬁles of the residual deformation are visualized in Figs. 12–14. The discrepancy function (6) is deﬁned by Nh ¼ 102 tip penetration values (for each 40 N increasing/decreasing loading step, up to and back from 2 kN maximum force) and by Nu ¼2500 out-ofplane displacements of a regular grid of 50 50 points deﬁning an area of 1.5 1.5 mm2 over the indented specimen surface. Also in this case, the inverse analysis exercises consider both “exact” and noisy input data, consisting of either all available pseudo-experimental information or the sole geometry of the residual imprints, which can be produced by standard hardness testers and has proved to be quite effective in practice in the case of isotropic solids [29]. The minimum search, started from 10 different initialization vectors, converges toward different but practically equivalent solutions, which correspond to similar small values of the

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Table 7 Minimum, maximum and average (in brackets) absolute error on the identiﬁed parameter values listed in Tables 5 and 6; bold ﬁgures refer to the results reported in [26]. Ex [%]

Ey [%]

Gxy [%]

sYx [%]

sYy [%]

sYxy [%]

Case 1: no noise Curves and imprint Imprint only

15 17–84 (54) 1–75 (34)

3 2–30 (4) 1–10 (5)

6 10–37 (2) 1–38 (2)

1 1–5 (3) 0–3 (1)

1 0–1 (1) 1–2 (1)

1 0–6 (3) 1–6 (3)

Case 1: noisy data Curves and imprint Imprint only

92 17–84 (58) 0–74 (31)

9 2–40 (10) 1–10 (5)

8 10–37 (1) 1–37 (13)

1 1–5 (1) 0–4 (1)

1 0–1 (1) 1–2 (1)

1 0–6 (4) 1–2 (1)

Case 2: no noise Curves and imprint Imprint only

13 7–54 (36) 6–55 (26)

2 1–23 (6) 2–35 (18)

1 4–14 (1) 4–44 (16)

1 0 0

1 0 0

1 0–1 (0) 0–2 (1)

Case 2: noisy data Curves and imprint Imprint only

56 8–46 (10) 8–42 (9)

23 1–39 (8) 2–36 (2)

10 5–46 (21) 3–46 (17)

1 0 0

1 0–2 (0) 0

1 0–5 (1) 0–3 (1)

Fig. 9. Discrepancy function evaluated by FE modelling (rhombs) and by the corresponding POD-RBF analytical approximation (continuous line). The ﬁrst and the second solution listed in Table 5 correspond to α ¼ 1 and 0, respectively.

discrepancy function. The average value and the standard deviation of the estimated material properties are listed in Tables 8–13. Results are practically unaffected by the perturbation of the pseudo-experimental data and the difference between the inverse analysis results and the target value of the parameters is within 20% in most cases. The graphs of Fig. 11 visualize the stress strain curves relevant to uniaxial tests performed along the principal material directions reconstructed for the solution set, which corresponds to the minimum discrepancy. The agreement with the target ones is extremely good for Test 5 and Test 6 in all analysed situations; the same holds true for Test 1 when the whole available information is exploited by the inverse analysis procedure. In the case of Test 2, the material response is nearly isotropic and Rockwell indentation produces an almost axis-symmetric imprint, visualized by the proﬁles along x and y directions drawn in Fig. 12. Notice the good ﬁt of the pseudo-experimental information with the results returned by the simulation with the identiﬁed parameter values, although the corresponding uniaxial curves represent upper and lower bounds of the target ones (see Fig. 11). Large but almost immaterial errors affect the identiﬁed values of parameter n relevant to Test 6 (see Table 13). In fact, the target n value is extremely small and an almost perfect plastic response is

recovered for all estimates. This feature is common to all considered low hardening materials, while accuracy of the estimates increases for large n; see e.g. the ﬁgures reported in Table 9 (Test 2) and Table 12 (Test 5). Strong anisotropy is evidenced by the imprint proﬁles visualized in Fig. 13, produced by the parameter values listed in Table 10 (Test 3): sinking-in is exhibited along x direction whilst pronounced piling-up occurs along y. In this case, the ratio between the yield limits is extremely large, namely: sYx =sYy ﬃ 1:5;sYxy =sYy ﬃ 3:1. However, the accuracy of the results recovered by the simulation model with calibrated material properties might be acceptable, depending on the dispersion of experimental input (see e.g. [29]): 14–19% overestimation of the material strength along x compensates the corresponding underestimation along y (13–22% error on average), whilst the identiﬁed sYxy value is affected by at most 5% error. The imprint proﬁles drawn in Fig. 14, relevant to Test 4, evidence that a quite strong anisotropic output is produced also by the constitutive properties listed in Table 11, although the parameter values relevant to x and y directions are not so markedly different (Ex =Ey ﬃ sYx =sYy ﬃ 1:2). In this case, however, the yield stress sYxy is almost the double of the value associated to nearly isotropic conditions, considered in [13,25] for simpliﬁcation purposes. The graphs in Fig. 14 exhibit the biggest observable discrepancy in the comparison between the pseudo-experimental information relevant to the imprint geometry and the corresponding simulation with the identiﬁed parameter values. Correspondingly, the errors affecting the estimated parameter values are the largest encountered in the explorative campaign concerning Model B. Some improvement may be likely achieved by a more accurate description of the residual deformation, i.e. by a further reﬁned POD-RBF approximation in the corresponding region of the parameter domain. On the other hand, a unique solution was recovered from the data relevant to the imprint only in this particular case (see Table 11).

7. Closing remarks An extensive numerical study has been performed in order to assess the performances of a combined numerical–experimental procedure based on indentation test, conceived for the identiﬁcation of parameters entering the anisotropic elastic–plastic Hill’s constitutive law. This systematic investigation was made possible by the considerable reduction in computing times gained by a POD-RBF analytical approximation of the investigated

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Fig. 10. Comparison among the target indentation curve and those evaluated by FE modelling and by the POD-RBF analytical approximation in correspondence of the minimum points of the graphs in Fig. 9.

Table 8 Test 1: target and recovered (average value and standard deviation) material parameters in the case of ‘exact’ and noisy data; italics evidence the error in percentage on the mean value.

Target

Ex [GPa]

Ey [GPa]

Gxy [GPa]

sYx [MPa]

sYy [MPa]

sYxy [MPa]

n

Discrepancy ( 10 5)

189

186

75.6

486

295

672

0.029

–

1857 23 1 1857 23 1

88.9 7 9.9 18 89.0 7 10.0 18

4427 81 9 4377 81 10

306 730 4 3117 29 5

698 7 36 4 698 7 35 4

0.0197 0.007 34 0.018 70.008 38

1.79–7.72

1787 16 4 1937 28 4

88.9 7 11.1 18 94 75.9 24

4157 60 15 4747 68 2

3197 26 8 3117 47 5

71 716 6 685 748 2

0.0167 0.010 45 0.0177 0.011 41

1.55–7.00

Curves and imprint Exact 1887 31 1 Noisy 1887 31 1 Imprint only Exact 1977 27 4 Noisy 169 723 11

5.67–11.8

4.49–5.63

Table 9 Test 2: target and recovered (average value and standard deviation) material parameters in the case of ‘exact’ and noisy data; italics evidence the error in percentage on the mean value.

Target

Ex [GPa]

Ey [GPa]

Gxy [GPa]

sYx [MPa]

sYy [MPa]

sYxy [MPa]

n

Discrepancy ( 10 5)

231

201

95.2

908

1152

732

0.230

–

216 716 7 203 7 15 1

87.0 7 11.1 9 91.8 712.0 4

999 7 273 10 821 7385 10

10277 153 11 9377 240 19

604 745 17 584 7 55 20

0.2917 0.031 27 0.2777 0.028 20

0.80–1.14

2187 25 8 220 7 28 9

87.9 7 12.7 8 88.7 7 11.5 7

942 7 276 4 886 7 273 2

10247 175 11 1040 7181 10

616 754 16 655.6 7 66 10

0.287 70.032 25 0.2707 0.032 17

Curves and imprint Exact 208 730 10 Noisy 2117 23 9 Imprint only Exact Noisy

203 727 12 1937 26 16

16.0–18.5

0.63–1.07 13.3–13.7

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Table 10 Test 3: target and recovered (average value and standard deviation) material parameters in the case of ‘exact’ and noisy data; italics evidence the error in percentage on the mean value.

Target

Ex [GPa]

Ey [GPa]

Gxy [GPa]

sYx [MPa]

sYy [MPa]

sYxy [MPa]

n

Discrepancy ( 10 5)

205

199

96.4

556

382

1200

0.049

–

205 7 21 3 2157 19 8

937 4.9 4 90.3 7 1.5 6

653 7 36 17 6337 1 14

302 7 55 21 3337 33 13

11667 79 3 1143 7103 5

0.059 7 0.015 20 0.0657 0.016 33

1.06–2.41

205 7 22 3 205 7 22 3

937 5.0 4 937 5.0 4

654 7 39 18 660 738 19

298 7 63 22 294 7 64 23

11687 81 3 1169 779 3

0.059 7 0.015 20 0.058 7 0.014 18

Curves and imprint Exact 222 7 31 8 Noisy 2157 40 5

Imprint only Exact Noisy

223 7 32 9 223 7 33 9

7.74–8.12

0.98–2.21 6.62–7.90

Table 11 Test 4: target and recovered (average value and standard deviation) material parameters in the case of ‘exact’ and noisy data; italics evidence the error in percentage on the mean value.

Target

Ex [GPa]

Ey [GPa]

Gxy [GPa]

sYx [MPa]

sYy [MPa]

sYxy [MPa]

n

Discrepancy ( 10 5)

186

155

89.6

412

335

426

0.014

–

2137 41 37 2137 41 37

88.0 7 10.4 2 88.0 7 10.4 2

312 71 24 3137 1 24

266 7 18 21 268 7 18 20

3757 33 12 3757 33 12

0.0197 0.006 36 0.0187 0.006 29

6.13–8.70

182 17 183 18

81.5 9 81.4 9

287 30 284 31

249 26 252 25

356 16 356 16

0.005 64 0.005 64

Curves and imprint Exact 1947 49 4 Noisy 1947 49 4

Imprint only Exact Noisy

153 18 154 17

8.48–11.1

5.72 7.77

Table 12 Test 5: target and recovered (average value and standard deviation) material parameters in the case of ‘exact’ and noisy data; italics evidence the error in percentage on the mean value.

Target

Ex [GPa]

Ey [GPa]

Gxy [GPa]

sYx [MPa]

sYy [MPa]

sYxy [MPa]

n

Discrepancy ( 10 5)

244

209

87.6

1236

922

697

0.285

–

2077 10 1 2177 15 4

86.17 13.0 2 83.17 9.1 5

10157 287 18 994 7 327 20

890 7 101 3 908 7 117 2

741 7108 6 7657 109 10

0.288 7 0.052 1 0.283 7 0.058 1

0.53–1.29

2197 25 5 1967 35 6

86.0 714.9 2 89.7 713.1 2

10377 278 16 10377 357 16

892 7 128 3 889 7 133 4

7117 71 2 761 781 9

0.298 7 0.032 5 0.285 7 0.049 0

Curves and imprint Exact 2077 30 15 Noisy 1927 32 21 Imprint only Exact Noisy

2017 28 18 1957 24 20

14.6–15.0

0.42–1.05 12.6–14.9

Table 13 Test 6: target and recovered (average value and standard deviation) material parameters in the case of ‘exact’ and noisy data; italics evidence the error in percentage on the mean value.

Target

Ex [GPa]

Ey [GPa]

Gxy [GPa]

sYx [MPa]

sYy [MPa]

sYxy [MPa]

n

Discrepancy ( 10 5)

173

241

76.1

1108

792

1296

0.007

–

2197 13 9 220 7 16 9

90.6 7 13.7 19 92.0 7 12.4 21

11477 33 4 11367 64 3

7417 68 6 7337 73 7

12337 116 5 12147 113 6

0.015 70.023 114 0.021 70.021 200

0.52–1.69

2157 13 11 2197 15 9

94.6 7 13.6 24 90.6 7 14.0 19

11627 29 5 11537 36 4

7167 75 10 729 780 8

12077 136 7 1222 7 126 6

0.020 70.027 186 0.0187 0.025 157

Curves and imprint Exact 2007 30 16 Noisy 2017 30 16 Imprint only Exact Noisy

2017 35 16 2017 30 16

9.68–10.7

0.43–1.60 7.82–8.98

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141

Fig. 11. Comparison between target ("experimental") and recovered stress strain curves corresponding to uniaxial loading along the principal material directions (x: continuous lines; y: dashed lines).

system response, tuned on the results of a number of non-linear FE simulations, performed once for all for each experimental setup. The particular case of transversally isotropic hardening structural metals characterized by six or seven independent parameters has been analysed. Despite the introduced simpliﬁcations, the dimension of the inverse analysis problems considered in the present contribution is much larger than those reported in the existing literature on the subject and is prone to be further generalized. The results of the performed investigations concerning elastic– perfectly plastic material response are consistent with those published in a previous work based on a traditional FE approach. Most parameter values are accurately estimated although some signiﬁcant error affects the larger elastic moduli as a consequence of the amount of irreversible deformation, which develops under the assumed sharp indenter tip and shields the reversible contribution. Computations, not reported here for brevity, show that this drawback is largely mitigated by the bluntness of the geometry of the indenter tips suggested by Standards.

Rockwell geometry has been exploited in further analyses concerning hardening metal response. Several physical situations characterized by different combinations of constitutive parameters deﬁned over a wide range of admissible values have been analysed. In most cases, the difference between the estimates of the material properties and the target value of the parameters is within 20%, with a signiﬁcant gain with respect to the results reported for instance in [13,25] despite the improved generality of the material model considered in the present contribution. In all investigated cases, the present inverse analysis procedure showed to be rather robust with respect to artiﬁcially added numerical disturbances, which replace the experimental noise in the computations. On the other hand, the extensive numerical study pointed out that uniqueness of the identiﬁed material properties is not always ensured in the considered context. In almost all analysed cases, several parameter sets were found, which reproduce the pseudo-experimental data quite accurately. A similar situation is evidenced in Ref. [25], despite the much more restrictive assumptions on the material response introduced there.

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Fig. 12. Comparison between the input noisy information and the corresponding data recalculated by the POD-RBF reduced numerical model for Test 2 with the identiﬁed parameter set corresponding to minimum discrepancy.

Fig. 13. Comparison between the input noisy information and the corresponding data recalculated by the POD-RBF reduced numerical model for Test 3 with the identiﬁed parameter set corresponding to minimum discrepancy.

The optimal compromise between the accuracy of the results and the efﬁciency of computations based on the POD-RBF approach permits to single out the main characteristics of the discrepancy function by a wide exploration in the domain of the sought material

properties, thus providing a substantial support to the assessment of parameter calibration methodologies based on indentation tests and to the evaluation of the reliability of the resulting estimates.

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143

Fig. 14. Comparison between the input noisy information and the corresponding data recalculated by the POD-RBF reduced numerical model for Test 4 with the identiﬁed parameter set corresponding to minimum discrepancy.

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An effective inverse analysis tool for parameter identification of anisotropic material models

An effective inverse analysis tool for parameter identification of anisotropic material models

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