Inverse Analysis of Forming Processes based on FORGE environment S. Marie 1, a, R. Ducloux 1, b, P. Lasne 1, c, J. Barlier 1, d and L.

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1 Inverse Analysis of Forming Processes based on FORGE environment S. Marie 1, a, R. Ducloux 1, b, P. Lasne 1, c, J. Barlier 1, d and L. Fourment 2, e 1 Transvalor, Parc de Haute Technologie, 694 av du Dr Maurice Donat, Mougins, France 2 Mines ParisTech, CEMEF, CNRS UMR 7635, BP 207, Sophia Antipolis Cedex, France a stephane.marie@transvalor.com, b richard.ducloux@transvalor.com, c patrice.lasne@transvalor.com, d julien.barlier@transvalor.com, e lionel.fourment@mines-paristech.fr Keywords: Inverse analysis, Parameter identification, Automatic optimization, Parallel solving, Forge, Sheet forming, Stamping. Abstract. In the field of materials forming processes, the use of simulation coupled with optimization is a powerful numerical tool to support design in industry and research. The finite element software Forge, a reference in the field of the two-dimensional and three-dimensional simulation of forging processes, has been coupled to an automatic optimization engine. The optimization method is based on meta-model assisted evolutionary algorithm. It allows solving complex optimization problems quickly. This paper is dedicated to a specific application of optimization, inverse analysis. In a first stage, a range of reverse analysis applications are considered such as material rheological and tribological characterization, identification of heat transfer coefficients and, finally, the estimation of Time Temperature Transformation curves based on existing Continuous Cooling Transformation diagrams for steel quenching simulation. In a second part, a novel inverse analysis application is presented in the field of cold sheet forming, the identification of the material anisotropic constitutive parameters that allow matching with the final shape of the component after stamping. The advanced numerical methods used in this kind of complex simulations are described along with the obtained optimization results. This article shows that automatic optimization coupled with Forge can solve many inverse analysis problems and is a valuable tool for supporting development and design of metals forming processes. 1. Introduction Forge finite element software is used to simulate hot, warm and cold forming of axisymmetric and three-dimensional parts. It has been coupled to an automatic optimization engine and this work is focused on a specific application of this optimization algorithm, the inverse analysis of material parameters. 1.1 Forging simulation Forge software uses thermo-viscoplastic constitutive models under hot conditions and thermoelasto-vicoplastic constitutive models for warm and cold forming [1]. More specific material models such as those based on the anisotropic Hill criterion enable the prediction of residual stresses and geometrical dimensions at the end of forming [2]. Forge is a parallel code that is quite efficient in simulating very complex parts using a large number of cores as well as on entry level computers [3]. Computation time reduction due to parallelism as well the robustness of utilized numerical methods within its solver has made it possible to integrate an optimization module within the Forge solution.

2 1.2 Optimization Automatic optimization is the perfect complement to simulation. An automatic optimization method based on evolution strategies assisted by meta-modeling has been developed in Forge [4]. This optimization strategy is quite robust and makes it possible to solve the most complex optimization problems in the metal forming field [5, 6]. To solve optimization problems with reasonable time, a meta-model continuously improved along with the generations and is used to dramatically reduce the number of exact cost function evaluations. Exact cost function evaluation for an individual requires Forge calculations with associated parameter values. Evolution operators are used to set up next generations, each generation containing several different individuals. This optimization approach makes it easy to handle two levels of parallelization: the parallelization of the Finite Element software itself and the parallelization of the optimization algorithm. This latest parallelism level of parallelism is very efficient because within each generation all individuals can be simultaneously and independently evaluated on several computers. 1.3 Inverse analysis In the advanced framework of Forge, a potential feature of optimization is presented; it is called Inverse Analysis or Inverse Identification of constitutive laws; it is a direct application of the optimization algorithm. A simple method is proposed to implement this inverse analysis module. 1.4 Content of the article In a first part, a range of reverse analysis applications is presented, such as rheological and tribological characterization of materials, identification of heat transfer coefficients and estimation of Time Temperature Transformation curve based on existing Continuous Cooling Transformation diagram for steel quenching simulation. In second part, a novel inverse analysis application is introduced in the field of cold sheet metal forming. 2. Inverse analysis review applications The principle of the reverse analysis is as follow: different experimental curves being available from forming experiments such as compression or tensile tests, the optimization module is used to run several Forge simulations using various parameter values suggested by the optimization engine. The inverse analysis objective function is simply the squared difference between the experimental and numerical curves. An external macro spreadsheet is used to sample the curves of simulation results with the same values as those of the experimental values to calculate the cost function. Best parameters values found by the optimization engine minimize the gap with experimental curves. The optimization parameters are the coefficients of the material law to identify. The optimization is generally not constrained. 2.1 Reverse analysis for rheology identification After running a compression test in the workshop on a given material, one can obtain a test efforts based on displacement tools curve. Depending on the test curves workshop, one wishes to identify the rheological law that characterizes the material in question. This is a very complex problem because rheological law may have many parameters, which are unknown as we seek to identify. An example of law is Hansel Spittel with many rheological parameters very complex to identify for a given material: = + where,, are respectively stress, strain and strain rate and T is the temperature given in Celsius, m1 and m9 define the material's sensitivity to temperature, m5 term coupling temperature and strain, m8 term coupling temperature and strain rate, m2, m4, and m7 define the material's sensitivity to strain, m3 depends on the material's sensitivity to the strain rate. A, m1, m2, m3, m4, m5, m7, m8, m9 are the coefficients of the Hansel Spittel equation and are potential optimization unknown parameters. On the following sections we present rheological law identification examples with a number of reduced parameters.

3 2.1.1 Reverse analysis for cold rheological law A first example is presented with cold rheological law. Cold Hansel Spittel law depending on material deformation only is written with three unknown parameters: = + where, are respectively strain and regularized strain coefficient, m1 defines the material's sensitivity to temperature and m2 defines the material's sensitivity to strain. We assume here that the law of material behavior does not depend on strain rate or temperature. In terms of optimization problem we have three unknown parameters A, m2 and and we define a range of variation between minimum and maximum values for each of them: 500 A 1000, 0.1 m2 0.5 and Following compression and tensile tests we have gotten two curves force versus displacement tools for material considered. Minimizables are squared difference between test and Forge simulation force versus displacement curves. The final weighted cost function is written as follow within the two least squares minimizables and calculated for compression and tensile simulations: =/++/+, =,, /, =,, where Y i denote experimental force values and y i interpolated force results simulation. To be sure that the optimum is found, total number of chained simulations was 60 with twice the number of parameters computations per generation to be optimized over 10 generations. It is important to note parallel computing aspects of this method because all individual evaluations are calculated simultaneously in each generation. Indeed for each unknown parameter optimization we have two completely independent calculations. Here for 3 parameters we have 6 independent calculations per generation. Therefore the cost of the CPU time is reduced to only 10 Forge calculations with 6 nodes of a parallel cluster. Total computing time for this inverse analysis was 20 minutes. Best parameters were found at iteration 39 of 60: = Final cost function is the sum of squared deviations between tests and simulation curves; Figure 1 shows that effort versus displacement curves for compression and traction are close between experiments and best Forge simulations. To go further, best fit should probably be obtained by second optimization within new parameters range around this first solution. / Fig. 1: Force versus displacement curve comparison between optimized Forge simulation (green), worst Forge simulation (red) and experiments (blue) for compression and tensile tests

4 2.1.2 Reverse analysis for more complex rheological law Like in the previous example we present identification of more complex Hansel Spittel rheological law depending on deformation, temperature and strain rate: = Sought are six unknown optimization parameters A, m1, m2, m3, m4 and m8. We define a range of variation between minimum and maximum values for each of them. Following four torsion tests done in the workshop we get four curves of torsion torque versus time for different temperatures and strain rate conditions (for temperature at 900 C, 1200 C, strain rate at 0.1s -1, 1s -1 ). First we have to define four Forge reference simulations for each temperature and strain rate with same condition as the workshop experiments. The final cost function is the sum of the four squared difference between the test curves and numerical curves results obtained by simulation Forge. 2.2 Reverse analysis for friction identification Another type of reverse analysis is friction identification. The goal is to identify friction law parameters fitting the lubricant used in the forging process. Friction laws can be considered as special rheological laws representing the interface between the material and tools. A typical example of tribological law is Tresca: = where, are respectively shear stress and Von Mises equivalent stress, is the coefficient of the Tresca law and is a potential optimization unknown parameter value between 0 and 1. We propose an identification method which is to set up the experimental ring test. We define ring test reference Forge simulation has the same as the experimental compression test ring. The unknown parameter is coefficient of Tresca Fiction law, the minimizable is a cost function representative of dimension of the ring. We choose a cost function representative of the final aimed shape. We have a cost function calculated within internal and external diameters of the final shape: =/++/+ = = where d is internal or external diameter of the final simulated or experimental shape for a specific value of the parameter. Fig. 2: Final shape of simulated ring test with internal and external diameter used for cost function With this method friction coefficient reflecting lubricant used for ring test is found. This inverse analysis can be done for more complex tribology such as coulomb-tresca or viscoplastic law with more unknown parameters coefficients of the friction law. 2.3 Reverse analysis for thermal identification Thermal exchange coefficients are often a tedious issue if one wishes to setup precise parameters for cooling, heating or quenching processes. In case of quenching there are different phases linked with the boiling of the cooling liquid. It can be interesting to use a model where thermal exchange coefficients would be a function of the surface temperature of the billet which obviously has a large influence on the boiling. Measure temperature profiles are obtained by several thermo-couple sensors and the goal is to identify Heat Transfer Coefficients (HTC) which fit the temperature versus time evolution.

5 Reference cooling simulation is defined such as the real cooling test and we have to set up sensors in the simulation at the same thermo-couple position. Optimization parameters are exchange coefficient for different temperatures values. To improve the model, surface temperatures of the billet are defined as unknown optimization parameters; it permits to estimate points of inflection temperature in the quenching process. Final cost function is the sum of the quadratic differences between experimental and numerical curves for all thermocouples. A part containing 3 thermocouples has been quenched, the purpose being to find 5 thermal exchange parameters at 5 unknown surface part temperatures which would give the best fit. In this example, we have 10 unknown parameters and we have chosen a specific range of variation for each parameter: ,80 120, , , where (in W/(m². K) unity) is thermal exchange coefficient for surface part temperature (in C unity), between two temperatures linear interpolation is made. Results of experiment and computed temperatures for 3 thermocouples are presented in Figure 3, different colors are for each thermo-couple, lines with markers are experimental data, dotted lines are initial values, and continuous lines are for best reverse HTC. Fig. 3: Experimental and computed temperatures versus time evolution It shows that at least the inverse analysis fit is much better than the one obtained with the original set. For this optimization we have used 30 generations of 20 individuals. Total number of simulations was 600. Due to the parallel computing aspects of this method many individual evaluations have been calculated simultaneously in each generation because for 10 parameters there are 20 independent calculations per generation. Total computing time for the inverse analysis was 10 hours using 10 nodes of a parallel cluster machine. 2.4 TTT diagram identification from CCT diagram Anew for quenching simulation an inverse analysis application is presented concerning the preprocessing Forge database for steel material properties in heat treatment process. It is a computer aided method to identify Time Temperature Transformation diagram (TTT diagram) with assumed Continuous Cooling Transformation curves (CCT curves) and specific composition of the material. An isothermal transformation diagram is valid if the temperature is held constant during the transformation with rapid cooling to that temperature which is very difficult to achieve in the metal industry. Because it is more convenient to cool materials at a certain rate than to cool quickly and hold at a certain temperature, the continuous cooling transformation phase diagram which represents types of phase changes occurring in a material at different cooling rates is generally used by industrials for heat treating steel.

6 Forge database pre-processor contains software tool based on Kirkaldy method that can compute TTT diagram from chemical composition and grain size material [7]. From this reference TTT diagram, CCT curves are calculated but they are sometimes far from the CCT curves provided by customers. For accurate Forge quenching simulation, it is then necessary to modify TTT curve parameters for closer cooling curves. TTT from CCT diagram evaluation by a trials errors manual method is very time-consuming. In order to reduce the necessary time involved, a computer inverse analysis application computing automatically TTT from CCT curves has been developed. The principle of the optimization is as follow. A reference TTT diagram is first evaluated from the chemical composition of steel and it gives range evaluation of optimization parameters. The optimization is focused on the following order, the transformation time for ferrite, pearlite, bainite and austenite at the end. The 50% austenite is not optimized. Transformation start curves for ferrite, pearlite, bainite and then curves to 90% for pearlite and bainite are fixed one after the other with three optimization parameters according to a Kirkaldy model: = + where and are respectively transformation temperature and asymptotic temperature in Kelvin for different phases and,, are optimization parameters. The inverse analysis is decoupled in five optimization problems with 3 parameters for each phase. For each optimization cost function is the quadratic difference between experimental and numerical CCT cooling curves obtained. For these 5 optimizations we impose 15 calculations per generation and 10 generations; there are 750 calculations to perform. For faster response time optimization engine of the inverse analysis has been modified and is close to experimental design. The interpolation degree of meta-model has been decreased to promote exploration of the parameters optimization domain. Thanks to this, total optimization time is less than 15 minutes. An example of steel alloy TTT diagram based on existing CCT experimental curves has been computed within this automatic inverse analysis preprocessor. In the next Figure 4, CCT results of optimized TTT diagram are compared to the experimental CCT curves; transformation phase of ferrite, pearlite and bainite are presented. Grey lines are for experimental data, dotted lines are for austenite, color lines are for optimized curves. Fig. 4: Optimized and experimental CCT diagram CCT derived from optimal TTT diagram is close to experimental data. To go further, the model should probably be improved including others intermediate transformation phases.

7 3. Reverse analysis on stamping process A new inverse analysis approach in the field of cold sheet forming is presented. It consists in identifying the material anisotropic constitutive parameters that allow matching final shape of a component after stamping. The cold drawing process consists in punching a thin sheet metal alloy maintained at constant effort by two hold-down pressure pad (Figure 5). The sheets of aluminum alloys have great interest because they allow reducing weight of final part. But they are difficult to shape because of their anisotropic behavior that must be mastered or at least anticipated. The very marked anisotropy of alloy sheet is due to lamination operation before; it modifies material s behavior during forming and Figure 6 illustrates low and high anisotropic with the well-known ears effect on the final stamping part. Fig. 5: Stamping process of aluminum part Fig. 6: Low and high anisotropy effect Anisotropic Hill criterion has been developed in Forge software to simulate ears effect in stamping. We propose an identification method to find material anisotropic constitutive parameters fitting the final stamping shape with exact ears dimension. 3.1 Advanced numerical methods The cold forging technique is difficult because it requires a good mastery of the process to give the expected output in term of part quality and shape control; thus cold forging simulation implies some difficulties which are overcame in Forge by advanced numerical methods. Elasticity is taken into account thanks to an elasto-viscoplastic model [2]. After forming process with large displacements, part spring back phenomenon is accurately simulated by a steady elastic unloading. Anisotropy is taken into account by anisotropic Hill criterion [8]. CPU time reduction is important for optimization particularly for cold sheet forming simulation which can be computation time consuming. This is due to thin sheet geometry and 3D complex geometry with high curvature disadvantageous for meshes dimension and therefore for computation time. For right balance between accuracy and computation time, meshes are automatically adapted by advanced topological tetrahedral remesher using anisotropic mesh adaptation techniques [9]. Anisotropic mesh permits to preserve several elements in the thickness and to decrease the number of nodes for thin sheet part. Remeshing is local; it saves CPU time and it preserves accuracy. The mesh is automatically adapted below multi-criteria which are the geometry (automatic tools curvature adaptation and thickness) and finite element error estimation based on velocity or temperature gradient. Fig. 7: Anisotropic mesh for thin geometry and local remeshing for automatic curvature adaptation generates accurate mesh with limited number of elements All of these Forge solver capabilities are used for the stamping process optimization.

8 3.2 Reverse anisotropic behavior Material's anisotropic behavior is modeled based on the Hill criterion which is derived directly from the Von Mises criterion considering anisotropy in different directions: = where is stress tensor, is yield stress, F, G, H, L, M, N are constants characteristic of the current state of anisotropy. The coefficients of the Hill criterion can be written through relationships with Lankford coefficients r0, r90, r45: =,=,=.+ + +,=.+. ++,== For isotropic behaviour, Hill parameters F, G, H are equal to 0.5 and L, M, N are equal to 1.5, Lankford coefficients r0, r90, r45 are equal to unity. Traditionally tensile tests in different directions will give the Lankford parameters. Hill criterion is then deducted from Lankford parameter values. However Lankford identification tests can be a tricky method also we propose an easy inverse analysis numerical method with Forge optimization module to find these anisotropic Lankford parameters values. 3.3 Parameters and minimizables for identification We want identify anisotropic Lankford parameters r0, r90 and r45 fitting the final stamping form with ears shown in Figure 8. In the rheological data file, coefficient values r0, r90 and r45 are replaced by 3 unknown optimization parameters with a range of variation between 0.1 and 2. The objective is to fit as well as possible final dimensions of the actual ear piece shown in the Figure 8. Optimization minimizables are based on height measurements of backside ear and right side ear; they are evaluated on final heights reached in these two areas of developing ears; they are the absolute difference between simulated and real heights of final shape in rear and right area illustrated in the Figure 8; the final cost function is the sum of these two computed minimizables. Fig. 8: Objective of optimization is to target the real ears of the final stamping shape, minimizables of optimization are difference between aimed and simulated ears height 3.4 Optimization results Within evolution strategies method initial evaluations have a great incidence on optimization convergence. Also Forge optimization engine requires the first individual to be isotropic with unit Lankford parameters and the first generation to be experimental design with farthest values each other. It permits to test boundary values of parameters in the optimization. As shown in Figure 9, for each individual value of set parameters r0, r90 and r45 we obtain specific simulated final shape of the stamping process. As simulation results we have obtained different ears positions, with two ears in front and behind, with two ears on the sides, with four ears on transverse positions, or with four ears in the right place but too pronounced. Fig. 9: Various forms of final shape created by optimization Fig. 10: Real and optimal simulated shape of the stamping process

9 The nearest part of the final target form was found for the fourth individual in the sixth generation of optimization and Figure 10 shows that the optimal simulated shape is very close to the actual shape. The minimal final cost function was close to naught with very small relative differences in term of final size earrings, 0.1% for rear ear and 0.2% for right side ear. Best Lankford parameters values were: r0 = 0.93, r90 = 0.92 and r45 = 0.42 It gives Hill parameters values: F = 0.524, G = 0.518, H = 0.482, L = M = N = The anisotropic coefficients were identified Lankford values close to unity in the two main directions and the lateral component is highly anisotropic. Thanks to the adaptive remeshing computation time has been divided by a factor of 2.5, which allows optimization within a reasonable time for this type of complex calculation. We have imposed 8 calculations per generation and 10 generations of optimization; thus there were 80 Forge calculations. Each Forge simulation was performed in parallel within 6 cores of a Linux cluster and 8 calculations of the same generation were made simultaneously. Thanks to the parallel computing the inverse analysis was performed on a Linux cluster of 48 cores within 10 hours. 4. Conclusion Forge reverse analysis module, with its large application domain, can be used for development and design assistance of metals forming processes. Inverse analysis handles many different situations and is a good way to identify material properties or process parameters. It can be applied for axisymmetric and three-dimensional metal forming processes. This unique software feature is easy to use, fast on parallel machine and is very efficient for developing innovative forging designs. References [1] Wagoner R.H., Chenot J.-L., Metal forming analysis, Cambridge Univ. Press, Cambridge, 2001 [2] Chenot J.-L., Fourment L., Ducloux R., Wey E., Finite element modelling of forging and other metal forming processes, 13th ESAFORM Conference on Material Forming, Brescia, Italie, 2010 [3] Coupez T., Marie S., From a Direct Solver to a Parallel Iterative Solver in 3-D Forming Simulation, The International Journal of Supercomputer Applications and High Performance Computing, volume 11, number 4, pages , Winter 1997 [4] Emmerich M., Giotis A., Ozdemir M., Bäck T., Giannakoglou K., Metamodel-assisted evolution strategies, In Parallel Problem Solving from Nature VII, pages , 2002 [5] Ducloux R., Marie S., Monnereau D., Behr N., Fourment L., Ejday M., Automatic optimization techniques applied to a large range of industrial test cases using metamodel assisted Evolutionary Algorithm, Numiform 2010, Pohang, Korea, June 13-17, 2010 [6] Fourment L., Ducloux R., Marie S., Ejday M., Monnereau D., Masse T., Montmitonnet P., Mono and Multi Objective optimization techniques applied to a large range of industrial test cases using metamodel assisted evolutionary algorithms, Numiform 2010, Pohang, Korea, 2010 [7] Aliaga C., Massoni E., Louin J.C., Denis S., 3D Finite element simulation of residual stresses and distortions of cooling steel workpieces, Proceedings of the 3rd International Conference on Quenching and Control of Distorsion, page 288, 24-26, Prague, République Tchèque, mars 1999 [8] Liu Z. G., Lasne P., Massoni E., Formability study of magnesium alloy AZ31B, The 8th International Conference and Wokshop on Numerical Simulation of 3D Sheet Metal Forming Processes (Numisheet 2011), AIP Conference Proceedings, Volume 1383, pp (2011) [9] Coupez T., Metric construction by length distribution tensor and edge based error for anisotropic adaptive meshing, J. of Computational Physics 230 (2011)