Variability and sensitivity analysis of U-shaped deep drawn metal sheet

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Variability and sensitivity analysis of U-shaped deep drawn metal sheet

Jérémy Lebon, Guénhaël Le Quilliec, Rajan Filomeno Coelho, Piotr Breitkopf, Pierre Villon

To cite this version:

Jérémy Lebon, Guénhaël Le Quilliec, Rajan Filomeno Coelho, Piotr Breitkopf, Pierre Villon. Vari-

ability and sensitivity analysis of U-shaped deep drawn metal sheet. 11e colloque national en calcul

des structures, CSMA, May 2013, Giens, France. �hal-01717081�

CSMA 2013

11e Colloque National en Calcul des Structures 13-17 Mai 2013

Variability and sensitivity analysis of U-shaped deep drawn metal sheet

Jérémy LEBON ^1,2,∗ , Guénhaël Le Quilliec ² ,

Rajan FILOMENO COELHO ² , Piotr BREITKOPF ¹ , Pierre VILLON ¹

∗

[email protected]

1

Laboratoire Roberval, UMR 7337, Université de Technologie de Compiègne

2

Université Libre de Bruxelles (ULB), BATir Department CP 194/2, 50, avenue Franklin Roosevelt, B-1050 Brussels (BELGIQUE)

Résumé — The frame of this work is the variability and sensitivity analysis of after-springback shape of deep drawn metal sheet using numerical models. We focus our interest on the propagation of the random- ness in input (process or physical parameters) to the after-springback shape variability and sensitivity.

As potentially numerous samples on a small variation range on the input process and physical variables may be involved, the underlying model has to reach simultaneously high performances in terms of com- putational costs, and accurate prediction of after-springback shape for small variations of the inputs. We show that the use of a full Finite Element Model is inappropriate, and propose an alternative two-pronged methodology to address these challenges. The deep drawing simulation process is performed using an original low cost semi-analytical approach based on a Bending-Under-Tension model exhibiting a high accuracy for small random perturbations on the input parameters and low computational cost. The spring- back variability and sensitivity analysis of the process is based on an advanced use of the Polynomial Chaos Expansion from which the approximated probability density function and Sobol’ indices of the springback shape parameters are computed at low costs.

Mots clés — Springback variability, Sobol’ indices, semi-analytical bending under tension model, poly- nomial chaos expansion.

1 Introduction

The numerical estimation of variability and sensitivity of the after-springback shape with regards to some random input process and/or physical parameters (only aleatoric uncertainties are considered in this paper) is a challenging task [1, 2]. First of all, modeling the forming phase implies the assessment of many highly non linear phenomena (material non linearities, frictional contacts, ...). Moreover, po- tentially numerous samples on a small range of variations of the uncertain input variables may have to be accurately assessed by the numerical model (here a non-intrusive scheme for uncertainty propagation is considered with known p.d.f on the independent input variables)[3]. Then, the numerical model has to rise simultaneously two challenges : 1- its computational cost has to be as low as possible, 2- it has to provide a trustworthy answer to small variations of the input (the intrinsic numerical noise involved in the modeling of highly non linear phenomenon must be lower than the range of variation of the out- put). In this paper, we firstly show that the use of Finite Element Method (FEM) makes these challenges contradictory. In fact, FEM rapidly exhibits unaffordable computational costs when discretization error and numerical computation errors are being decreased.

In the first section, we illustrate this claim by modeling a classical case of 2D-U shaped metal forming

(Numisheet’93) [4] using a standard FEM legacy software [5]. Then, as an answer to the challenges

stated above, we propose in a second section a two-pronged approach combining a physics-based semi-

analytical model for a Bending-Under-Tension (B-U-T) description of the deep drawing process with

the Polynomial Chaos Expansion (PCE). The B-U-T model allows us to improve the numerical stability

of the forming model as well as to drastically decrease the computational cost. In a third section, the

PCE metamodel trained using the B-U-T model is presented in the context of uncertainty propagation

and sensitivity analysis : it is used to retrieve the p.d.f of the output springback shape functions and the

Sobol’ indices with a good accuracy for a low computational costs (avoiding Monte Carlo simulations).

2 FEM limitations for small variations of input variables

2.1 Test case description

Die

Blank holder Blank holder

Die

Blank Punch

W

=52 r

=5 W

=50

55 6

55

L

/2=175

x z r

=5 h

Fig. 1 – Geometrical configuration of the modeled Numisheet’93 benchmark (left side, values in mm) and definition of the springback parameters ρ, β ₁ , β ₂ (right side)

We choose to model the B3 benchmark test problem of the conference Numisheet’93 [4] correspon- ding to the deep drawing of an 2D U-shaped metal sheet. The geometrical configuration of the numerical experiment is shown in Fig.1 (left). As the problem is symmetric only a half structure is modeled using appropriate symmetry boundary conditions. The blank is modeled using a single row of 175 first-order shell elements (S4R) with Simpson integration rules (7 integration points across the thickness). As the problem is essentially in plane strain state (the width of the blank is 35 mm and its thickness nominal value is 0.8 mm), corresponding boundary conditions are applied on each node. The blank is made of mild steel modeled as an elastic-plastic material. Isotropic elasticity and the Hill 48 [6] anisotropic yield criterion for the plasticity are considered. The following characteristic values are used : Young’s modu- lus= 206 GPa, Poisson’s ratio= 0.3, Yield strength= 167 MPa, Anisotropic yield criterion : r ₀₀ = 1.79, r ₄₅ = 1.51, r ₉₀ = 2.27. The tools (punch, blank holder and die) are modeled as rigid body surfaces. The contact occurring during forming phase is modeled using contact pairs. The punch velocity is taken here as 15 m/s and its displacement is s = 70 mm. The blank holder force is defined as F b = 2.45 kN and a mass of 5 kg is attached. To model the deep drawing process a two phase explicit-forming/implicit- springback approach is used.

2.2 Illustration and investigations on numerical instabilities

We uniformly vary the thickness of the blank from 0.7 mm to 0.9 mm with an 0.002 mm increment size. For each thickness value the springback shape parameters, the curvature ρ, the angles β ₁ and β ₂ are measured as shown in Fig.1 (right). The observed response in Fig.2(left) highlights numerical instabilities for small variations of the thickness parameter. A deeper insight into the sensibilities around the nominal value 0.8 mm. The Fig.2(right) shows that the FEM model may be used for thickness variations up to an order of ∆x ≈ 10 ⁻⁵ m hindering an accurate sensibility study.

Among the numerical instability sources one may identify the coarsity of the mesh, the contact algo- rithm, the round-off errors, as well as the through thickness integration errors. In [7], the authors show that the through thickness integration related error plays a leading role : increasing the number of inte- gration points and adapting their position across the thickness significantly reduce the global numerical noise. As an illustration, we model a slice Fig.3(left) of the metal sheet using shell elements and increa- sing the number of Gauss points across the thickness. The stress profile reaches numerical convergence Fig.3(middle) when the number of Gauss points is increased to around 200 (Fig.3 (right)).

To conclude, improving the accuracy of FEM in the scope of sensitivity and variability study, the

FEM mesh may be refined in every direction and the number of Gauss points increased. This may lead

to unaffordable computational costs for repeated computations. In the following, we propose to use a

model developed in [8] which allows us to circumvent simultaneously these both issues.

7 8 9 x 10

4

5 6 7

x

1/ l (x)

FEM Model

100^ï6 10^ï5 10^ï

0.2 0.4 0.6 0.8 1

6(x)

6 f(x) / 6(x)

`₁

`₂ l

Fig. 2 – Numerical instability for FEM simulations of the deep drawing process of 2-D U-shaped metal sheet under small variations of the thickness (m). On the left side, the obtained responses are depicted for ρ (similar results are obtained for other responses), on the right side the normalized numerical sensi- tivities of the model with regards to different order of magnitude of thickness variation is plotted for all responses (ρ, β ₁ ,β ₂ ).

2D volume elements Integration points in the thickness direction

−4 −3 −2 −1 0 1 2 3 4

x 10⁻⁴

−1

−0.5 0 0.5 1 1.5x 10⁸

Thickness deep(m) σtt(MPa)

=10 I=50 I=100

I=200 I=400

0 50 200 300 400

10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰

Number of Gauss points through the thickness

Mean Square error

Fig. 3 – Evolution of the stress through the section for different numbers of integration points (thickness 0.8 mm).

3 Physics-based semi-analytical Bending-Under-Tension model

The B-U-T model considers the deep drawing process of a 2D U-shaped metal sheet as a 2D plane strain Bending-Under-Tension (B-U-T) forming process with negligible shear stress (the width of the sheet being sufficiently large) [8]. It is based on a semi-analytical approach which combines an analytical approach with finite element modeling. This allows to take benefit from material laws independence and to avoid time consuming and low convergence issues related to contact modeling and high number of degrees of freedom.This model is constructed in three steps. The first one consists in identifying a finite (usually small) numbers of regions of the metal sheet with an homogeneous loading state in the length direction. For the U-shaped sheet (Fig.4) 5 regions are identified. The behavior of a single typical section is sufficient to describe the behavior of the whole region. This section (or slice) may be modeled by a handful of 2D/3D solid elements or even a single shell element (Fig.4). The second step consists in applying the proper loading path to the corresponding regions.The entire loading path for each slice is divided into a sequence of loading states. For each sequence, particular boundary conditions have to be applied [8]. Finally, in the last step the whole model is integrated and the springback shape is reconstructed.

3.1 Retrieving the physical smoothness of the response

The same range of thickness values as in Fig.2 is considered. 4 nodes plate elements model each slice of the B-U-T model. 10, 50, 400 integration points across the sections are sequentially investigated. As shown on the Fig.5 (left), the through-thickness integration noise is put in evidence by the B-U-T model.

A more realistic smooth response is observed using 400 integration points.

Using the B-U-T model provides some advantages for the targeted variability and sensitivity studies.

First of all, the contact is modeled using an analytical uniform pressure applied on the lower or upper side

+ + + +

l0 h0 5×

Fig. 4 – Finite element description of one slice of the metal sheet

6 6 5 7 7.5 8 8 5 9 9.5 10

x 10^ï4 3 8

3.9 4 4.1 4 2 4 3 4.4

Thickness (m)

1/ l P

P

P

100⁻¹⁰ 10⁻⁹ 10⁻⁸ 10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴ 0.2

0.4 0.6 0.8 1

∆(x)

∆f(x)/∆(x)

β₁ β₂ ρ

S f

Sensitivity Limit of full FEM

Fig. 5 – Retrieved numerical stability with regards to small thickness variations for different number of through-thickness integration points. On the left side, the obtained responses are depicted for ρ. On the right side, the normalized numerical sensitivity limits of the B-U-T model compared to the full FEM are depicted for all responses.

of each slice and does not induce numerical noise. Then, considering this many integration points across the sections is still computationally affordable. Moreover, such a model allows us to carry out sensitivity study for a variation range [10 ⁻⁴ ; 10 ⁻⁷ ]m uncovered by the refined FEM model (Fig. 5(right)). Finally, a high enough number of integration points improves the smoothness of the output function and allows us to consider potentially lower order approximation with faster convergence rate. We thus retain the B-U-T model as a ”high fidelity model ” to construct the PCE surrogate to propagate the uncertainties.

4 An introduction to PCE

The PCE is a metamodel, that is intended to give an intrinsic representation of the stochastic beha- vior of a functional f (scalar random variable) that is defined as a function of an input random vector

ξ = {ξ ₁ ,ξ ₂ , . . . , ξ M }. We suppose that E[ f ² ] < ∞ and that the M coordinates of ξ are independent with

prescribed marginal probability density function f _ξ (ξ), with ξ ∈ Ω ^M , [9]. For the sake of simplicity, let us assume that ξ has independent Gaussian components. According to the Cameron Martin theorem [10], the PCE approximation of f is given by f (ξ) = ∑ ^∞ _j=0 γ j Ψ j (ξ), where the M dimensional Hermitian PCE basis {Ψ j } is constructed by tensor products of the one-dimensional polynomials. Practically, among all {Ψ _j , j ∈ N } only a finite number of terms is kept. A common truncation scheme consists in retai- ning only the terms whose degree does not exceed N yielding the reduced expansion (N ^th order PCE)

f ^PCE (ξ) ≈ ∑ ^P−1 _j=0 γ j Ψ j (ξ). In this case, the number P of terms in the expansion is then given by : P = (N + M)!

N!M! (1)

4.1 Computing the set of PCE coefficients

The bottleneck of the PCE approximation lies in the computation of its coefficients. Among the methods surveyed in the literature, the direct integration methods (quadrature/cubature) based on the orthogonality of the polynomials are exact but computationally expensive in relatively high dimensions.

An alternative to these methods consists in finding the best set of PCE coefficients γ = [γ ₀ , . . . , γ P−1 ] ^>

by minimizing the residual error in the least square sense (regression based approach) :

γ = argmin(kf(ξ) − Ψ(ξ)γk ² ) (2)

yielding

γ = (Ψ(ξ)Ψ(ξ) ^> ) ⁻¹ Ψ(ξ)f(ξ) (3)

where



 

 

 

 

 

 

f(ξ) = [ f (ξ ⁽¹⁾ ), . . . , f (ξ ^(Q) )] ^> ,

Ψ =







Ψ ₀ (ξ ⁽¹⁾ ) . . . Ψ P−1 (ξ ⁽¹⁾ )

.. . .. .

Ψ 0 (ξ ^(Q) ) . . . Ψ P−1 (ξ ^(Q) )







and ξ ⁽ⁱ⁾ ,i ∈ { 1, ..., Q} represent Q > P samples of the random vector ξ. When the sampling is random, the optimal Q number of realizations needed to assess the coefficients with a good accuracy is still an open research topic, but an empirical rule [11] proposes Q ≥ (M − 1) × P. To improve the efficiency of this approach one may choose among the roots of the (N + 1) ^th orthogonal polynomial ordered by increasing norm until the rank of the information matrix Ψ(ξ)Ψ(ξ) ^> reaches P.

4.2 Statistical measures and sensitivity analysis using PCE coefficients

Once the set of coefficients {γ ₀ , ..., γ P−1 } one may take full advantage of the orthogonality property of the polynomial basis to compute the statistical moments of f by analytically post-treating set of coefficients, avoiding Monte Carlo simulations. For example, the first two statistical moments are :

E [ f ] = γ ₀ σ ² ( f ) =

P−1

∑

j=1

E (Ψ ² _j )γ ² _j (4) Moreover a direct access to the Sobol’ indices S i

,i

,...,i

with regards to the subset of variables {ξ _i

,ξ i

, . . . ,ξ i

} ⊆ {ξ ₁ ,ξ ₂ , . . . , ξ M } is also analytically provided [12, 13]. In the following let us denotes

u = {i ₁ ,i ₂ , . . . , i s } . S _u denotes the s-order sensitivity index where s=card(u) represents which amount of

the total variance is due to the uncertainties in the set of input parameters ξ u . An approximation of the Sobol’ decomposition of f is written in the PCE approximation of f denoted f ^PCE . The Sobol’ indices S _u is given by

S _u ≈ ∑ β∈β

γ ² _β E [Ψ ² _β ]

σ ² ( f ) (5)

where β _u ⊂ { 1, . . . , P} refers to the subset of indices which selects among the polynomials {Ψ _j , j = 1, ...P} those involving the set of variables {ξ i

,ξ i

, . . . ,ξ i

} and only them. (Note that ∑ u S _u = 1). One may count 2 ^M − 1 sensitivity indices.

Practically one also often uses the total sensitivity indices S T

which evaluate the total effect of the parameter i on the variance. They are defined by :

S _T

= ∑

i∈u

S _u (6)

4.3 Notes on reducing the number in PCE

The computational cost of the PCE is related to the number of calls needed to accurately assess the

PCE coefficients. The number of these coefficients drastically increase with the PCE order and/or the

number of variables. The PCE order mainly depends on by the smoothness of the output function with

regards to the uncertain input parameters [13]. Thus, improving the smoothness of the output function may lead to substantial reduction of computational effort.

Moreover, when increasing the dimension space, the computational cost to compute the full PCE grows exponentially. Some methods inspired from model selection schemes have been proposed to de- crease the number of terms (sparse PCE) in the expansion while preserving a high enough accuracy [14].

Least Angle Regression Stagewise (LARS) is the most efficient method according to [15].

We here propose to use a LARS-based algorithm based to build a sparse approximation of the PCE.

Let us first define the J ^th -sparse approximation of the output function f by : f ˜ _A ^J (ξ) = ∑

j∈ A

γ j ψ j (ξ) (7)

where A is a sparse index set with card(A) = J and J ≤ P.

Computational details on the LARS algorithm may be found in [15, 14]. Roughly speaking, each step of the algorithm allows us to add one term to the basis. The term to be added is chosen according to its correlation with the current residual. The best obtained model f _A

is chosen to give a low leave-one-out error [14].

Once the best model is obtained, the sparse coefficients values are computed by regression. If the number Q of available simulations is too low to reach the condition Q ≥ (M − 1) × P then the design of experiment may be enriched and the algorithm ran again.

5 Results

We demonstrate here the validity of the proposed approach on the sensitivity study of the springback shape of a 2D U-shaped deep drawn metal sheet. The following probabilistic model is considered in which all variables are considered as independent Gaussian variables with a standard deviation arbitrarily fixed at 3% of the mean value for each variable (Table 1).

Parameters Mean Std dev

Thickness 0.8 mm 3%

Young’s modulus 7.05e10 Pa 3%

Yield Strength 1.80e8 Pa 3%

Poisson’s coefficients 0.342 3%

Friction coefficients 1.50e-1 3%

Radius of the punch 1.00e-2 mm 3%

Radius of the matrix 1.00e-2 mm 3%

Clamp force 3.00e-1 N 3%

Tableau 1 – Stochastic input normal random variables for the springback shape parameters study

5.1 Variability results

In this study, we choose an a priori N = 5 ^th order PCE with 8 variables yielding to P = 1287 po- lynomial terms. Q = (M − 1) × P ≈ 9000 simulations would be necessary to compute the whole set of coefficients using the random regression approach. To reduce the computational cost, we apply the LARS strategy proposed in the previous section with a budget of simulation of 1000 simulations (which is even lower that the total number of terms contained in the PCE). A more general insight is given in Table 2 which exhibits the number of polynomial terms with regards to their degree in the LARS PCE expansion. We show a posteriori that considering this level on the empirical error, a 4 ^th order polynomial approximation is enough to assess the springback variability of responses β 1 and β 2 while for ρ, only a sparse 3 ^rd PCE approximation is needed.

Considering the obtained histograms (using 1000 Monte Carlo simulations on the B-U-T model and the PCE model) in Fig.6 a good agreement for each response is observed.

5.2 Sensitivity results

The results listed in Table 3 show that for the response ρ and β 1 the most important variables are the

Young modulus, the thickness and the yield strength. These variables all together accounts for more than

Polynomial degree Number of terms in PCE ρ β

β

0 1 1 1

1 8 6 8

2 33 28 31

3 69 92 97

4 0 254 245

5 0 0 0

Tableau 2 – Proportion of terms sorted according to their degree in LARS based PCE for each response.

8 8 5 9 9 5 10 10 5 11 11 5

0 0 02 0 04 0 06 0 08 0 1 0 12 0 14

l

Frequencies

Sparse PCE based p d f Exact p d f

1090 110 111 112 113 114 115 116

0 05 0 1 0 15 0 2

`₁

Frequencies

Sparse PCE based p d f Exact p d f

89 50 90 90 5 91 91 5 92 92 5

0 05 0 1 0 15 0 2

`₂

Frequencies

Sparse PCE based p d f Exact p d f

Fig. 6 – Obtained histograms

Responses ρ β1 β2

Parameters deg=2 deg=3 MC (2000 samples) deg=2 deg=3 MC (2000 samples) deg=2 deg=3 MC (2000 samples)

Young’ modulus S₁ 0.3333 0.3748 0.3892 0.3806 0.4199 0.3823 0.0270 0.0320 0.0312

Thickness S₂ 0.3299 0.2806 0.2600 0.2947 0.2462 0.2756 0.0050 0.0135 0.0234

Poisson S₃ 0.0205 0.0138 0.0120 0.0236 0.0161 0.134 0.0017 0.0003 0.0020

Die radius S₄ 0.0003 0.0013 0.0017 0.0002 0.0011 0.0015 0.9124 0.8948 0.9002

Yield strength S₅ 0.2934 0.2903 0.2897 0.2670 0.2566 0.2345 0.0004 0.0034 0.0145

Tool radius S₆ 0.0001 0.0029 0.0033 0.0108 0.0228 0.0339 0.0035 0.0022 0.0056

Friction coefficient S₇ 0.0004 0.0008 0.0012 0.0005 0.0008 0.0011 0.0000 0.0000 0.0001

Blank holder forces S8 0.0000 0.0009 0.0014 0.0000 0.0006 0.0013 0.0000 0.0013 0.0104

Young’ modulus ST1 0.3365 0.3947 0.4157 0.3837 0.4411 0.4452 0.0270 0.0320 0.0407

Thickness S_T2 0.3421 0.2961 0.2843 0.3066 0.2613 0.2297 0.0050 0.0135 0.0252

Poisson ST3 0.0213 0.0198 0.0201 0.0246 0.0222 0.0189 0.0017 0.0003 0.0013

Die radius S_T4 0.0095 0.0054 0.0050 0.0094 0.0051 0.0045 0.9124 0.8948 0.9028

Yield strength S_T5 0.3004 0.3064 0.3093 0.2752 0.2742 0.2801 0.0004 0.0034 0.0025

Tool radius S_T6 0.0038 0.0084 0.0092 0.0155 0.0286 0.0297 0.0035 0.0022 0.0037

Friction coefficient S_T7 0.0079 0.0063 0.0072 0.0071 0.0064 0.0071 0.0000 0.0000 0.0000

Blank holder forces S_T8 0.0005 0.0051 0.0079 0.0006 0.0046 0.0053 0.0000 0.0013 0.0034

Tableau 3 – Sensitivity results for each response obtained using a 2 ^nd and 3 ^rd order PCE compared to 2000 Monte Carlo simulations

95% of the uncertainties on the total variance. The total sensitivity indices relative to these variables are close values to the sensitivity indices denoting that interaction of these variables with other is negligible.

Considering the β ₂ response the die radius is the most important variable.

Finally one may add that all these results depend on the hypothesis made on the distribution of the input random variables. A realistic input stochastic model is of paramount importance.

6 Conclusion and prospects

In this paper we proposed an original two-pronged approach to assess the springback variability and

to perform sensitivity analysis using an advanced use of the Polynomial Chaos Expansion combined with

a physics based Bending-Under-Tension model applied to a 2-D deep drawn metal sheet. This combi-

nation allows us to circumvent typical cost issues and numerical instability from full FEM simulations (contact modeling, through thickness integration). The use of such a model allows us to reach a sufficient numerical stability for small variations of the random parameters by increasing the number of Gauss points through the thickness. This permits to improve the smoothness of the output function with regards to the uncertain parameters. Consequently, the stochastic metamodel is used to perform the variability study and the Sobol’ sensitivity analysis, namely PCE, to efficiently propagate the uncertainties through the model. We demonstrate our results on the deep drawing process of a 2D U-shaped metal sheet and use here up to 8 random variables.

In a more general way, our approach demonstrates that the use of simplified physics based model for large strain forming process allows to reduce the numerical instability and makes possible an accurate and low cost sensitivity study. It opens the way to the use of other types of physics-based metamodels such as POD/PGD approaches to smooth the output function with regards to the uncertain input parameters.

Further works need to be done to incorporate a more reliable interpretation of the results by refining the stochastic input model on which the interpretation of the results entirely relies.

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