Blank optimization in a stamping process—Influence of the geometry definition

Finite Elements in Analysis and Design 61 (2012) 75–84

Contents lists available at SciVerse ScienceDirect

Finite Elements in Analysis and Design journal homepage: www.elsevier.com/locate/finel

Blank optimization in a stamping process—Influence of the geometry definition R. de-Carvalho n, S. Silva, R.A.F. Valente, A. Andrade-Campos Department of Mechanical Engineering, Centre for Mechanical Technology and Automation, GRIDS Research Group, University of Aveiro, ´rio de Santiago, 3810-193 Aveiro, Portugal Campus Universita

a r t i c l e i n f o

abstract

Article history: Received 12 September 2011 Received in revised form 22 May 2012 Accepted 24 June 2012 Available online 20 July 2012

Nowadays initial geometry optimization methods are increasingly being adopted in order to solve complex mechanical plastic forming processes. This kind of approach can focus, in estimating the initial shape of a certain metallic specimen (or blank) in order to achieve a desired geometry after forming. In the present work the superplastic forming of a carter is described and studied in detail. After plastic forming it is possible to verify an undesirable non-homogeneous thickness distribution in the final geometry. A non-uniform thickness distribution of the initial blank can be proposed in order to obtain a regular final thickness of the sheet and avoid this problem. In the present paper the blank surface is modeled by means of a non-uniform rational B-spline (NURBS) surface, where the coordinates of specific NURBS control net vertices are chosen to be the optimization variables. The optimization procedure is carried out by combining a finite element (FE) software and a suitable optimization code. Four different studies were performed, differing in the number of control vertices that formulates the NURBS surface (16, 25, 36 and 49 control vertices were considered), in order to study the influence of the initial geometry definition. A geometry definition leading to better results is achieved, considering both computational cost and final result precision, and subsequent discussions are carried out. & 2012 Elsevier B.V. All rights reserved.

Keywords: Sheet metal forming Inverse methodology Initial geometry optimization NURBS parameterization Geometry influence

1. Introduction The interest of the stamping industry in numerical simulation of sheet metal forming, including inverse engineering approaches, is increasing. This fact occurs mainly because trial and error design procedures, commonly used in the past, are no longer economically competitive. The use of simulation codes is currently a common practice in the industrial forming environment, as the results typically obtained by means of the Finite Element Method (FEM) software codes are well accepted by both the industrial and scientific communities. In general terms, sheet metal forming is a complex deformation process controlled by parameters such as blank shape, tools’ geometry, sheet thickness values, blank holding force, friction, etc. [1]. Due to its complexity, and the higher combination level of all these input variables, optimization procedures are a fundamental tool in the proper design of the process parameters, and accounting for the prediction and correction of undesirable forming defects such as fracture, springback, wrinkling, shape deviations and unbalanced residual stresses [2]. Due to the high level and robustness of commercial FEM codes, it can be attractive

n

Corresponding author. Tel.: þ351 234370830. E-mail address: [email protected] (R. de-Carvalho).

0168-874X/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.finel.2012.06.009

to use FEM simulation results as objective function estimators in a more comprehensive optimization procedure. Regarding the modification of the shape of the monitored specimen, during the optimization process, two different optimization problems can be distinguished. If the optimization process tries to find a geometry or shape that is optimal, in the sense that it minimizes a certain objective while satisfying given constraints, the problem can be defined as a shape optimization problem. However, if the final geometry or shape is already known and the initial geometry/shape becomes the sought information, the problem is then defined as an inverse problem. This particular inverse problem is called initial shape design optimization problem. When applied to sheet metal forming processes, its aim for instance would be to estimate the initial shape of a specimen (or a blank) in order to achieve the desired geometry after the forming process. Within this context, the initial blank shape can be set as one of the most important process parameters to be considered, with a direct influence on the quality of the finished part. In this sense, accounting for an initial shape design procedure can effectively reduce the final cost and the product development time associated with the plastic forming operations [2]. Therefore, many initial blank design methodologies have been proposed in the literature to determine optimum initial blank shape profiles for diverse constraints and objective functions [1]. It is possible to categorize some of these methods in deformation path iteration

76

R. de-Carvalho et al. / Finite Elements in Analysis and Design 61 (2012) 75–84

approaches [3,4], backward tracing methods [5], ideal forming [6], inverse approach (IA) [7,8], slip line field method [9,10], geometrical mapping [11], sensitivity analysis method [12,13], and, finally, trial and error formulations still based on FEM [14–16]. The initial blank design optimization problem is generally solved using a methodology that couples a FEM software with an optimization algorithm. If a commercial FEM software is used, and considering that usually these software codes are black-box featured, an interface program can be developed in order to (i) send the information to the FEM software, (ii) execute the FEM program, (iii) retrieve the results, (iv) calculate the objective function and constraints, (v) send the information to the optimization algorithm and (vi) obtain new and improved optimization variables that would be sent again, directly or indirectly, to the FEM software. If a gradient-based optimization algorithm is used along with a black-boxed software, following these general steps described, the calculation of the derivatives can be accomplished, for instance, by the finite-differences method [17,18]. This 6-step typical methodology seems to be straightforward. However, generally FEM software codes do not directly accept optimization variables as input information. These optimization variables can define a geometry (or a shape of a piece) that must be previously defined and transformed into nodes (and elements) in order to be properly used in the numerical simulation. This task, called geometry parameterization and discretization, is not straightforward and can influence the entire optimization process as well as their success. The shape (geometry) of three-dimensional pieces can be defined by their surfaces. Therefore, shape optimization or initial geometry problems can be reduced to surface definition optimization problems where optimization variables are defined concerning the selected surface type. The mathematical surface representation can be performed explicitly, implicitly or parametrically. Explicit formulations are useful in many applications but are axis dependent, cannot adequately represent multiple-valued functions and cannot be used where a constraint involves an infinite derivative. Implicit formulations, on the other hand, are able to represent multiple-valued functions but are still axis dependent. The nonuniform rational B-spline (NURBS) surfaces are parametric surface representations, such as Be´zier surfaces, that can be defined by a relatively small set of variables (control net vertices) and are appropriate to this kind of problem [19]. Nevertheless, their use and definition can still influence the final optimized result. Therefore, in the present work this approach is adopted, and the sensitivity of the obtained results to the parametric surface is studied for initial geometry problems [17]. In doing so, the superplastic forming of a metallic carter [1] is used as benchmark problem, considering four distinct NURBS control vertices distributions and an improved objective function, and leading to a more robust and comprehensive approach when compared to preliminary results by the authors [17]. Although being a conceptual problem, the idea is that it could drive the formulation and systematization of a mathematical inverse methodology that could be extrapolated to other industrial benchmarks.

2. Parametric surface definition—NURBS During the past decades, Computer Aided Design (CAD) tools have increasingly played an important role for a variety of applications in many fields, such as industrial design and manufacture, electrical and mechanical engineering, robotics, computer vision, image processing, computer graphics, biomedicine, etc., either for functional or aesthetic reasons [20,21]. Regarding the parametric representations, splines are the common choice in the geometric design of curves and surfaces.

A spline surface is defined by a set of two or more coordinate positions called control points. Splines can be considered as interpolation splines or approximation splines, in case the surface passes through all the control vertices or if the surface passes near the control vertices, respectively. There are different sorts of splines including Cardinal Splines, Kochanec–Bartlet Splines, Bezier Splines, B-Splines and NURBS. Be´zier Splines, B-Splines and NURBS curves and surfaces are, among these, the most popular form of parametric representations. NURBS are the ones with greater flexibility and precision, being therefore assumed as the standard in the computer graphics industry and in computer aided design [19], allowing for an accurate description of complex surfaces. Historically, the first approach to appear was the Be´zier formulation, followed by the B-spline formulation and then the NURBS formulation, in the search for greater flexibility and precision. Furthermore, B-splines and NURBS formulation can be considered as generalizations of Be´zier formulation, being manipulated in similar ways as Be´zier formulation [22]. The mathematical formulation of these surfaces is based in the so-called basis functions. The Be´zier surfaces are based on Bernstein basis, which limit their flexibility since the degree of the surface (in each parametric direction) is one degree lower than the number of control vertices in that direction, and also due to the global nature of the Bernstein basis. In other words, this means that changing a single control vertex can affect the entire shape of the surface. On the other hand, B-spline basis can include the Bernstein basis as a special case. This approach is generally non-global, meaning that each vertex is associated with a unique basis (support) function, and as a consequence affecting the shape of a curve only over a restricted range of parameter values. In the B-spline basis the maximum possible order of the surface in each parametric direction is equal to the number of control polygon vertices in that direction [19,22]. Rational representations of B-splines surfaces are possible by means of NURBS, since these surfaces represent special cases of general rational B-spline surfaces. To develop a NURBS surface it is necessary to define four variables: (i) the surface degree in the directions u and w (k and l respectively); (ii) the control points; (iii) the knot vector and (iv) the weighting factors. The knot vector influences the basis functions values and, as a consequence, the shape of the NURBS surface. The weighting factors are associated to the control vertices and can also be used to influence the shape of the NURBS. A NURBS surface projected into three-dimensional space can be defined as Pn þ 1 Pm þ 1 i¼1 j ¼ 1 hij Bij N ik ðuÞM jl ðwÞ Q ðu,wÞ ¼ Pn þ 1 Pm þ 1 ð1Þ i1 J ¼ 1 hij N ik ðuÞMðwÞ where Q(u, w) are the cartesian coordinates for the QUOTE parametric space. Bij are the three-dimensional control net vertices, Nik(u) and Mjl(w) are the non-rational B-spline basis functions and hij are the weighting factors [19]. A NURBS surface can use a tensorial product of open knot vectors that may be uniform—if the individual knot values are evenly spaced—or non-uniform when this distribution is not constant. The open character of these vectors belongs to the fact that the vector has multiplicity of knot values at the ends equal to the order k of the B-spline basis function.

3. Description of the case study to be analyzed An important problem in sheet metal forming is the design of the initial blank geometry. There is interest in studying the influence of the initial geometry on the optimization process. In

R. de-Carvalho et al. / Finite Elements in Analysis and Design 61 (2012) 75–84

order to perform this study, a problem of a superplastic forming of a carter [23] was chosen. In Fig. 1 it is possible to observe the geometry of the die and the metal blank considered in the present work. The material used in the blank is assumed to have an elastic– viscoplastic behavior, with properties that roughly represent the 2004 based commercial superplastic aluminum alloy Supral 100 (Al–6Cu–0.4Zr), at a temperature of 470 1C. The material presents a Young’s modulus of about 71 GPa and a Poisson’s ratio of 0.34. The flow stress is assumed to follow a power law strain hardening function, given by [24] cr

e_ ¼ ðAq~ n ½ðmþ 1Þecr m Þ1=m þ 1

ð2Þ  11

where A, m and n are material parameters equal to 3.41  10 , cr _ 2 and 0, respectively. Also in the equation, e represents the uniaxial equivalent creep strain rate, which is equal to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2=3e_ cr : e_ cr , q~ is the uniaxial equivalent deviatoric stress and

ecr is the equivalent creep strain. For this problem, initially the blank thickness was considered uniform and equal to 4 mm. Considering this scenario, after the whole blank is formed it is possible to verify that the strain fields are not uniform in the whole surface, thus leading to a nonhomogeneous thickness distribution in the final geometry (Fig. 2). This non-uniform thickness distribution can lead to thinning zones where subsequent rupture is prone to happen. To avoid this problem, and in the context of the inverse analysis in the present work, an initial undeformed blank with non-uniform thickness is ideally proposed in order to obtain a regular thickness profile of the final component, even with non-homogeneous strains. This non-uniform thickness blank, a tailored-blank, can be achieved by means of, for instance, metal casting (foundry) or forging operations. Therefore, the final uniform thickness profile can be defined a-priori and set up to a specific value, which for this study, was chosen to be equal to 4 mm.

77

4.1. FE mesh size The chosen finite element size is a numerical parameter that, usually, has a noticeable influence on the solution outcome. A finer mesh can improve the results accuracy when compared to a coarse mesh, but a finer mesh is computationally more expensive. Therefore, an optimum balance between the finite element size and the computational requirements should be achieved [25]. To perform the present analysis, the forming process was carried out with six different meshes that differ on the total number of elements for the discretization of the blank mesh. Distinct meshes consisting of 1845, 2750, 3300, 5032, 6478 and 7380 elements with uniform distribution were considered. The element type considered was C3D8 (8-node, trilinear hexahedral with full integration) [24]. Fig. 3 shows the evolution of the pressure, needed for the blank forming, as a function of the z-displacement (forming direction), as measured at the blank geometric centre. Fig. 4 shows the evolution of the pressure needed for the maximum z-displacement with the number of finite elements and Fig. 5 represents the evolution of the simulation time with the number of finite elements. Considering the graph from Fig. 3 it is possible to conclude that the influence of the element size in the applied pressure has no significant difference for the studied meshes. The pressure needed to

4. Sensitivity analysis on numerical parameters A sensitivity analysis on the process numerical parameters assumes an important role on the understanding and improvement of a modeling process, since it is essential to determine the influence of such parameters in the process, in order to improve the results [25]. In this section two sensitivity studies are presented, namely, the influence of the finite element size in the results of the numerical simulation of forming process, as well as the influence of the blank surface definition in the optimization process.

Fig. 2. Final non-uniform thickness distribution predicted by the FE analysis.

Fig. 1. (a) Blank and (b) die geometry definitions [22].

78

R. de-Carvalho et al. / Finite Elements in Analysis and Design 61 (2012) 75–84

Pressure for the maximum displacement (MPa)

Fig. 3. Influence of the element size in the pressure applied.

4.2. Initial geometry definition

92 91 90 89 88 87 86 1000

2000

3000

4000

5000

6000

7000

8000

Number of finite elements Fig. 4. Influence of the number of finite elements in the pressure needed for the maximum displacement.

Duration of the simulation (min)

120

100 80

60

40 20 0

0

1000

2000

3000

4000

5000

6000

7000

8000

Number of finite elements Fig. 5. Evolution of the simulation time with the number of finite elements.

achieve the maximum displacement seems to have converged considering the results from Fig. 4. However, it was for meshes 2 and 4 that less pressure was needed in order to achieve the same displacement. Based on this fact, and from the analysis of Fig. 5 (where it is noticeable that the CPU time increases with the number of elements), mesh 2 was chosen as the reference one to be adopted in the following. The mesh 2 has 55  50  1 (2750) elements, respectively in the highest, lowest dimension and thickness.

In order to understand the initial geometry influence in the optimization process the blank upper surface was defined by means of a NURBS surface, where the z-coordinates of specific NURBS control vertices were taken as optimization variables. Considering the symmetry conditions of this problem, only one quarter of the die and the blank was considered during the FE simulation. The blank mesh was achieved in order to have a good description of the mechanical problem allied to the less CPU time for the FE simulation (direct problem), as detailed in the previous section. An efficient combination of these two factors is important in this sort of problems in order to reduce the time required to solve the inverse problem. Regarding the initial geometry influence, four different studies were performed. These studies differ in the number of control vertices that formulates the NURBS surface, thus leading to different locations of the NURBS control vertices. The studies in the present work considered 9, 16, 25 and 36 optimization variables, corresponding to 16, 25, 36 and 49 control vertices, respectively, in the NURBS surface formulation. In all studies the distribution of the control vertices were considered to be uniform in the mesh, as seen in Fig. 6. The control vertices located in the free edges of the blank (as an example for the first study d4, d8, d12, d13, d14, d15, d16) are not optimized in this example. A uniform open knot vector was assumed, and the weighting factors hij were considered to be equal to one in the formulation of the NURBS surface. This parametric definition allows for a sensitivity study focusing on the influence of the number and the localization of the control nodes in the optimization process. The algorithm applied in the NURBS surface implementation is listed in Table 1. The presented optimization procedure was performed using a combination of a finite element commercial program and an optimization software code. The FE analysis was performed with Abaqus [23], while the optimization process was carried out with the Sdl optimization software [18,26,27]. The communication between these two programs was ensured by a FORTRAN interface, where the new geometry is parameterized and then the relevant input variables written in the finite element input file. This numerical procedure can run with no operator intervention

R. de-Carvalho et al. / Finite Elements in Analysis and Design 61 (2012) 75–84

79

Fig. 6. NURBS control vertices distribution: (a) 16, (b) 25, (c) 36 and (d) 49 control vertices.

Table 1 NURBS algorithm [19]. Algorithm applied in the NURBS surface implementation Specify the number of control vertices in the u and w directions. Specify the order of the surface in each of the u and w directions. Specify the number of isoparametric lines in each of the u and w directions. Specify the control net. Calculate the uniform open knot vector in the u direction. Calculate the uniform open knot vector in the w direction. For each parametric u value Calculate the basis functions Ni,k(u). For each parametric value, w Calculate the basis functions, Mj,l(w). Calculate the Sum function. For each control vertex in the u direction For each control vertex in the w direction Calculate the surface point, Q(u, w). end loop end loop end loop end loop

once it has started. Fig. 7 illustrates the numerical procedure performed in order to achieve the optimum blank shape, which is based on inverse and incremental approach in combination with the optimization algorithm. The initial process parameters (tools geometry, mechanical properties of the blank sheet, blank holder force, etc.) are fixed during the optimization process. Also, and as stated before, an initial value for the blank thickness is given (4 mm) which, when combined with the NURBS surface formulation, will allow to iteratively re-design the initial blank shape. The parameterized initial blank shape is then exported to the FE code Abaqus in order to perform the numerical simulation of the stamping plastic forming process, in this case with the use of an implicit algorithm (Abaqus Standard). After the first simulation, the calculation of the deviations between desired and modeled

Fig. 7. Blank shape optimization procedure.

part is performed considering the objective function. If the stopping criterion is achieved then all the process could be stopped and the optimum blank shape is assumed to have been found. Otherwise the blank shape is modified through the combination of the NURBS surface formulation and the optimization algorithm. Afterwards, the new blank is remeshed and used for the second simulation procedure. The same calculation is repeated for the following steps until the stopping criterion is finally achieved.

80

R. de-Carvalho et al. / Finite Elements in Analysis and Design 61 (2012) 75–84

The success of the optimization procedure strongly depends on the correct definition of the objective function, which turns out to be essential to the efficient determination of the optimization variables effectively minimizing the objective function [26,28]. One of the most used objective functions consists of the sum of squares and for this specific problem can be defined as Pnsnodes ðei eopt Þ2 min F ¼ i ¼ 1 nsnodes subjected to : 1 o eini o 9

ð3Þ

in which nsnodes is the number of surface nodes, ei are the observable relevant variables (in the present context, the final thickness of each element of the blank) and eopt corresponds to the optimum (required) final thickness (that is, 4 mm). Also, eini is the set of the optimization variables that represent the initial thickness of the blank, i.e. the thickness before the forming process. Fig. 8 shows the procedure adopted for the final thicknesses calculation. In this work, a Levenberg–Marquardt gradient algorithm [29,30] was used to minimize the objective function. It was considered as the stooping criterion, a maximum number of 300 iterations or, as an alternative, a stagnation value of 1  10  30, in terms of objective function value between each iteration. The objective function sensibilities are calculated by means of a

Fig. 8. Final thicknesses calculation.

forward finite differences scheme, and the step size considered is 5  10  3.

5. Results and discussion In this section the results that synthesize the current study are presented and discussed. In Fig. 9 it is possible to observe the evolution of the normalized objective function value with the iteration number, for the different analyses. In Fig. 10 the evolution of the optimization variables along z-coordinate with the iteration’s evolution is presented, for the study with 9 optimization variables. In Table 2 the values of the optimization variables, along with the objective function for the best iteration of each study, are presented. In this work the best iteration is the iteration which has reached the lowest value in terms of objective function. Regarding the results presented in Fig. 9, as well as the values of the objective function from Table 2, it is possible to verify that the four studies considered have achieved good results in terms of objective function values. The objective function values show decreases that vary from 69.7%, for the 25 optimization vertices, to 81.2%, for the 16 optimization vertices, relative to the initial trial simulation (considering a blank with a uniform 4 mm thickness). It is possible to observe that the best value of objective function was achieved for the study with 16 optimization variables. However, for this study, the optimization method has needed twice as many iterations to reach stagnation. Also, a tendency for the evolution of the objective function, with the optimization variables number, was not clearly verified. The evolution of the optimization variables z-coordinate with the iteration number was presented only for the study considering 9 optimization variables, being the evolution of the z-coordinate for the other similar studies. It is possible to observe that the highest variation in this values occur in the first iterations. Fig. 11 presents the contour graphs for the thickness of the blank (in millimeters). Both initial and final thicknesses are presented for the best iteration of each study. In Fig. 12, the blank thickness range evolution with the number of optimization variables is presented. Observing the results presented in Fig. 11 it is possible to conclude that increasing the number of optimization variables and, consequently, the number of control vertices allows the

Fig. 9. Evolution of the normalized objective function value with the iterations.

R. de-Carvalho et al. / Finite Elements in Analysis and Design 61 (2012) 75–84

81

Fig. 10. Evolution of the optimization variables z-coordinate with the iterations for the study with 9 optimization variables.

Table 2 Values of the optimization variables (in millimeters) and the objective function for the best iteration. Op.var.

d1

d2

d3

d4

d5

d6

d7

d8

d9

d10

d11

d12

d13

9 vertices 16 vertices 25 vertices 36 vertices

5.2448 5.1889 3.7866 4.5273

5.9266 5.2901 4.4368 4.0089

5.4551 5.5974 6.0688 4.7542

– 5.3058 4.3885 4.7814

5.1874 – 3.5395 5.6966

6.2075 5.5495 – 6.0772

5.659 6.0632 4.6669 –

– 6.4116 5.9201 5.0606

5.5125 5.2089 6.8653 4.5223

5.5826 – 6.1069 7.8016

5.9584 5.5134 4.8017 6.6657

– 4.4786 – 4.9714

– 5.8311 6.2191 4.9598

Op.var. d14

d15

d16

d17

d18

d19

d20

d21

d22

d23

d24

d25

d26

– 5.4825 5.5936 4.0746

– – 5.0101 7.4977

– 4.9893 5.777 4.8431

– 6.0191 5.7575 4.2511

– 5.1159 – 6.2052

– 5.5671 6.0147 –

– – 5.0224 6.7544

– – 5.8394 4.3991

– – 3.2367 5.3244

– – 6.8665 6.8045

– – – 5.3746

– – 4.4438 5.5439

– – 4.2894 –

Op.var. d27

d28

d29

d30

d31

d32

d33

d34

d35

d36

d37

d38

d39

F (%)

– – 5.7769 4.6045

– – 4.9559 4.0831

– – 5.1153 5.9371

– – – 4.002

– – – 5.6371

– – – 4.0436

– – – -

– – – 6.6808

– – – 4.4406

– – – 6.3958

– – – 5.1132

– – – 6.1315

– – – 3.7955

22.4 18.8 30.3 23.3

formulation of a more complex shape for the initial surface of the blank. Furthermore, in Fig. 12 it is possible to observe that if the number of optimization variables increases, then the thickness range also increases for the initial and final blanks. Thickness range was considered to be the difference between the higher and the lower values of thickness for a specific blank. Considering the final thickness of the optimized blanks (Fig. 11(b), (d), (f) and (h)) and comparing these with the final thickness of the blank that have an initial thickness of 4 mm (Fig. 2) it is possible to observe that the minimum thickness value had increased for all the optimized blanks. It was for the study considering 9 optimization variables that the highest values of the minimum thickness were observed. However, comparing this fact with the results concerning the objective function value evolution, it is seen that when dealing with 16 optimization variables then a minimum value of objective function was achieved.

It is possible to conclude that the increase from 9 to 16 optimization variables leads to better results, when considering the value of the objective function. The increase to 25 and 36 optimization variables, on the other side, has led to a more complex and flexible initial surfaces of the blank. However, and taking into account the objective function results, no improvements were inferred. This fact may occur due to fact that, when increasing the number of optimization variables, the vertices are not granted to be located in the same places for the distinct meshes, considering that they can be in more critical areas. These results and discussion is for this specific problem. Each stamping problem has different thinning zones where subsequent rupture is prone to happen, therefore ideal optimization variables location varies between different problems. For other mechanical problems this method can be considered, however the initial geometry definition should be analyzed in detail.

82

R. de-Carvalho et al. / Finite Elements in Analysis and Design 61 (2012) 75–84

Fig. 11. Blank thickness of the best iteration for the: (a) initial and (b) final blank for 9 optimization variables, (c) initial and (d) final blank for 16 optimization variables, (e) initial and (f) final blank for 25 optimization variables, (g) initial and (h) final blank for 36 optimization variables.

6. Conclusions A sensitivity study was performed considering the influence of the finite element mesh size in the forming process and the initial geometry definition in the optimization process. Considering the mesh studied, a blank mesh that has a good description of the mechanical problem allied to the less CPU time for the FE simulation was achieved.

The influence of the geometry definition in a carter forming process was presented. Four studies were performed with differences in the number and location of control vertices that formulate the NURBS surface. The studies considered 9, 16, 25 and 36 optimization variables that correspond respectively to 16, 25, 36 and 49 control vertices in the NURBS surface formulation. In all the studies the location of the control vertices were considered to be uniform in the mesh. The four studies had

R. de-Carvalho et al. / Finite Elements in Analysis and Design 61 (2012) 75–84

83

Fig. 12. Blank thickness range of the best iteration.

achieved good values for objective function, having a minimum decrease of 69.7% for the 25 optimization vertices and a maximum decrease of 81.2% for the 16 optimization vertices, relative to the initial simulation that considered a blank with a uniform 4 mm thickness. It was for the study with 16 optimization variables that the best value of objective function of 18.8% was achieved.

Acknowledgments This work was co-financed by the Portuguese Foundation for Science and Technology via project PTDC/EME-TME/118420/2010 and by FEDER via the ‘‘Programa Operacional Factores de Competitividade’’ of QREN with COMPETE reference: FCOMP-01-0124FEDER-020465. One of the authors, Raquel de-Carvalho, was supported by the grant SFRH/BD/45588/2008 for scientific research from the Portuguese Science and Technology Foundation. Both supports are gratefully acknowledged. References [1] R. Padmanabhan, M.C. Oliveira, A.J. Baptista, J.L. Alves, L.F. Menezes, Blank design for deep drawn parts using parametric NURBS surfaces, J. Mater. Process. Technol. 209 (2009) 2402–2411, http://dx.doi.org/10.1016/ j.jmatprotec.2008.05.035. [2] M. Azaouzi, S. Belouettara, G. Rauchsa, A numerical method for the optimal blank shape design, Mater. Des. 32 (1011) 756–765, http://dx.doi.org/10. 1016/j.matdes.2010.07.027. [3] S.H. Park, J.W. Yoon, D.Y. Yang, Y.H. Kim, Optimum blank design in sheet metal forming by the deformation path iteration method, Int. J. Mech. Sci. 41 (1999) 1217–1232, http://dx.doi.org/10.1016/S0020-7403(98)00084-8. [4] A. Fazli, B. Arezoo, A comparison of numerical iteration based algorithms in blank optimization, Finite Elem. Anal. Des. 50 (2012) 207–216, http://dx.doi.o rg/10.1016/j.finel.2011.09.011. [5] T.W. Ku, H.J. Lim, H.H. Choi, S.M. Hwang, B.S. Kang, Implementation of backward tracing scheme of the FEM to blank design in sheet metal forming, J. Mater. Process. Technol. 111 (2001) 90–97, http://dx.doi.org/10.1016/ S0924-0136(01)00518-0. [6] K. Chung, J.W. Yoon, O. Richmond, Ideal sheet forming with frictional constraints, Int. J. Plasticity 16 (2000) 595–610, http://dx.doi.org/10.1016/ S0749-6419(99)00068-6. [7] H. Naceur, Y.Q. Guo, J.L. Batoz, Blank optimization in sheet metal forming using an evolutionary algorithm, J. Mater. Process. Technol. 151 (2004) 183–191, http://dx.doi.org/10.1016/j.jmatprotec.2004.04.036.

[8] J. Wang, A. Goel, F. Yang, J. Gau, Blank optimization for sheet metal forming using multi-step finite element simulations, Int. J. Adv. Manuf. Technol. 40 (2009) 709–720. [9] X. Chen, R. Sowerby, Blank development and the prediction of earing in cup drawing, Int. J. Mech. Sci. 8 (1996) 509–516. [10] M.H. Parsa, P.H. Matin, M.M. Mashhadi, Improvement of initial blank shape for intricate products using slip line field, J. Mater. Process. Technol. 145 (2004) 21–26. [11] R. Sowerby, J.L. Duncan, E. Chu, The modeling of sheet metal stampings, Int. J. Mech. Sci. 28 (1986) 415–430, http://dx.doi.org/10.1016/0020-7403(86)90062-7. [12] H. Shim, K. Son, K. Kim, Optimum blank shape design by sensitivity analysis, J. Mater. Process. Technol. 104 (2000) 191–199, http://dx.doi.org/10.1016/ S0924-0136(00)00556-2. [13] K. Son, H. Shim, Optimal blank shape design using the initial velocity of boundary nodes, J. Mater. Process. Technol. 134 (2003) 92–98, http://dx.doi.org/10.1016/ S0924-0136(02)00927-5. [14] V. Pegada, Y.W. Chun, S. Santhanam, An algorithm for determining the optimal blank shape for the deep drawing of aluminium cups, J. Mater. Process. Technol. 125–126 (2002) 743–750, http://dx.doi.org/10.1016/S09240136(02)00382-5. [15] M. Azaouzi, H. Naceur, A. Delame´zie re, J.L. Batoz, S. Belouettar, An heuristic optimization algorithm for the blank shape design of high precision metallic parts obtained by a particular stamping process, Finite Elem. Anal. Des. 44 (2008) 842–850, http://dx.doi.org/10.1016/j.finel.2008.06.008. [16] M. Azaouzi, A. Delame´zie re, J.L. Batoz, S. Belouettar, D. Sibaud, Design of the blank contour for high precision metallic parts obtained by stamping, In: Proceedings of the Seventh International Conference on Numerical Simulation of 3D Sheet Metal Forming Processes, NUMISHEET, Switzerland, Interlaken, 2008, pp. 705–708. [17] R. de-Carvalho, S. Silva, R.A.F. Valente, A. Andrade-Campos, The geometry definition influence in inverse analysis—application to Carter forming process, in: Proceedings of the 14th International ESAFORM Conference on Material Forming, Belfast, Ireland, 2011, pp. 65–70, http://dx.doi.org/10. 1063/1.3589493. [18] A. Andrade-Campos, Development of an optimization framework for parameter identification and shape optimization problems in engineering, Int. J. Manuf. Mater. Mech. Eng. 1 (2011) 57–79. [19] D.F. Rogers, An Introduction to NURBS with historical perspective, Academic Press, 2001. [20] E. Dimas, D. Briassoulis, 3D geometric modelling based on NURBS: a review, Adv. Eng. Softw. 30 (1999) 741–751, http://dx.doi.org/10.1016/S09659978(98)00110-0. [21] H. Pottmann, S. Leopoldseder, M. Hofer, T. Steiner, W. Wang, Industrial geometry: recent advances and applications in CAD, Comput. Aided Des. 37 (2005) 751–756, http://dx.doi.org/10.1016/j.cad.2004.08.013. [22] Y. Bazilevs, V.M. Calo, J.A. Cottrell, J.A. Evans, T.J.R. Hughes, S. Lipton, M.A. Scott, T.W. Sederberg, Isogeometric analysis using T-splines, Comput. Methods Appl. Mech. Eng. 199 (2010) 229–263. [23] ABAQUS 6.7, User Manual, Simulia Inc., Dassault Syste´mes, 2007. [24] J.P. Kleinermann, J.P. Ponthot, Parameter identification and shape/process optimization in metal forming simulation, J. Mater. Process. Technol. 139 (2003) 521–526, http://dx.doi.org/10.1016/S0924-0136(03)00530-2.

84

R. de-Carvalho et al. / Finite Elements in Analysis and Design 61 (2012) 75–84

[25] M.C. Oliveira, R. Padmanabhan, A.J. Baptista, J.L. Alves, L.F. Menezes, Sensitivity study on some parameters in blank design, Mater. Des. 30 (2009) 1223–1230. [26] R. de-Carvalho, R.A.F. Valente, A. Andrade-Campos, On the objective function evaluation in parameter identification of material constitutive models— Single-point or FE analysis, in Proceedings of the 13th ESAFORM Conference on Material Forming, Brescia, Italy , 2010, pp. 33–36; http://dx.doi.org/10. 1007/s12289-010-0700-9. [27] R.A.F. Valente, A. Andrade-Campos, J.F. Carvalho, P.S. Cruz, An integrated methodology for parameter identification and shape optimization in metal forming and structural application, Optim. Eng. 12 (2011) 129–152, http://dx .doi.org/10.1007/s11081-010-9126-y.

[28] A. Andrade-Campos, R. de-Carvalho, R.A.F. Valente, Novel criteria for determination of material model parameters, Int. J. Mech. Sci. 54 (2012) 294–305, http://dx.doi.org/10.1016/j.ijmecsci.2011.11.010. [29] D. Marquardt, An algorithm for least-squares estimation of nonlinear parameters, J. Soc. Ind. Appl. Math. 11 (1962) 431–441. [30] R. de-Carvalho, R.A.F. Valente, A. Andrade-Campos, Optimization strategies for non-linear material parameters identification in metal forming problems, Comput. Struct. 89 (2010) 246–255, http://dx.doi.org/10.1016/j.compstruc. 2010.10.002.