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Multifield modeling of electromagnetic metal forming processes
Journal of Materials Processing Technology 177 (2006) 270–273
Multifield modeling of electromagnetic metal forming processes J. Unger a,∗ , M. Stiemer b , B. Svendsen a , H. Blum b a
Leonhard-Euler-Str. 5, University of Dortmund, D-44227 Dortmund, Germany b Vogelpothsweg 87, University of Dortmund, D-44227 Dortmund, Germany
Abstract The purpose of this work is the formulation of a thermo-magneto-mechanical multifield model and presentation of the numerical strategies applied. In particular, this model is used to simulate electromagnetic sheet metal forming processes (EMF). In this process, deformation of the workpiece is driven by the interaction of a current generated in the workpiece by a magnetic field generated by a coil adjacent to the workpiece. The interaction of these two fields results in a material body force known as Lorentz force. Up to now, modeling approaches found for EMF in the literature are restricted to the axisymmetric case. For real industrial applications however, the modeling of three-dimensional forming operations becomes crucial for an effective process design. The implementation of such a 3D model still represents work in progress. Results shown in this paper are restricted to the axisymmetric case. © 2006 Elsevier B.V. All rights reserved. Keywords: Electromagnetic forming processes; Magneto-thermomechanical coupling; Staggered solution algorithm; Finite element method
1. Introduction
2. Model formulation
Electromagnetic forming is a dynamic, high strain-rate forming method in which strain-rates of ≥103 s−1 are achieved. In this process, deformation of the workpiece is driven by the interaction of a current generated in the workpiece with a magnetic field generated by a coil adjacent to the workpiece. In particular, the interaction of these two fields results in a material body force, i.e., the Lorentz force and the electromotive power, representing an additional supply of momentum and energy to the material. On the other hand the EM-system is sensitively influenced by the spatio-temporal evolution of the mechanical structure. In recent years considerable effort has been undertaken to simulate such coupled processes. However, approaches tested so far were mainly restricted to 2D or axisymmetric geometries [1–3]. Yet, it is the 3D modeling capability that is required to advance effectively in the design of industrial EMF processes. In this work modeling and simulation approaches for such a 3D model are discussed. Since their implementation still represents work in progress results for the axisymmetric forming set-up are presented here.
The coupled multifield model for electromagnetic forming of interest here, represents a special case of the general continuum thermodynamic formulation for inelastic non-polarizable and non-magnetizable materials given in [4]. Further details of the algorithmic realization of the model are discussed in [5,6]. In particular, this special case is based on the quasi-static approximation to Maxwell’s equations, in which the wave character of the electromagnetic (EM) fields is neglected. In this case, the unknown fields of interest are the motion field , the scalar potential χ and the vector potential a determining in particular the magnetic field in the usual fashion. Assuming Dirichlet boundary conditions for all fields and homogeneous Neumann boundary conditions, one derives the weak field relations: 0= (ρr ξ¨ − lr )ξ ∗ + KF−T ∇ξ ∗ , Br T 0 = {˙a − L a}a∗ + {χ − av}div a∗ (1) R R + κEM curl a curl a∗ , 0= ∇χ∇χ∗ , R
∗
Corresponding author. Tel.: +49 231 755 5717; fax: +49 231 755 2688. E-mail address: [email protected] (J. Unger).
0924-0136/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2006.03.190
for , a, and χ, respectively. Here, * , a* , and χ∗ represent the corresponding test fields. Further, R represents a fixed region
J. Unger et al. / Journal of Materials Processing Technology 177 (2006) 270–273
271
Fig. 1. Different meshing strategies. Left: fictitious boundary method with inaccurate modeling of boundaries. Right: improvement by means of ALE approach.
Fig. 2. Three-dimensional-EMF process: schematic of forming set-up with an elliptic tool coil and formed sheet metal and modeling domains of the sheet Br and the EM-field R. Axisymmetric EMF process: schematic of forming set-up with a circular tool coil.
in Euclidean point space containing the system under consideration in which the electromagnetic fields exist and on whose boundary the boundary conditions for these fields are specified. Here, the system consists of the workpiece, the tool coil, and air (see Fig. 2). As such, R contains in particular the fixed reference configuration Br , and all subsequent (i.e., deformed) configurations, of the workpiece. Here, F := ∇ represents the deformation gradient, and L := ∇v the spatial velocity gradient. Further, κEM represents the magnetic diffusivity, v the spatial velocity field, ρr the referential mass density, K the Kirchhoff stress, and lr = det(F)j × b the Lorentz force in terms of the magnetic flux b = curl a and the current density j. Note that (1)2,3 follow from Maxwell’s equations, while (1)1 represents the weak form of the momentum balance. The above weak field relations are completed by the thermodynamically consistent formulation of the adiabatic thermo-elasto-viscoplastic material model [4]. Consider next the finite element discretization of (1). The difference in electromagnetic and thermo mechanical timescales together with the distinct nature of the fields involved (i.e., Eulerian in the electromagnetic case, Lagrangian in the thermo mechanical large deformation context), argue for a staggered numerical solution procedure in two different meshes resulting in the following algorithmic system: fn (xn+1 , a˜ n+1 ) = 0, en (xn+1 , a˜ n+1 ) = 0,
(2) A starting value a˜ n+1 of the nodal vector potential array is computed for the measured amperage in the tool coil at time tn+1 and the known mechanical state of the system at time tn via (2)2 . (3) From a˜ n+1 , a corresponding value ln+1 for the Lorentz force is obtained. Using this, the system consisting of (2)1 is solved via Newton–Raphson iteration (i.e., at fixed a˜ n+1 ) to obtain xn+1 . (4) Proceed to next time step tn+1 = tn + tn,n+1 and proceed with (2). Else, if tn ≥ ts , terminate the simulation, where ts is the total simulation time. Besides its physical motivation, the staggered algorithm offers the possibility to apply custom solutions for both the mechanical, as well as the EM-system. Here, on the mechanical side an effective continuum shell formulation is applied to minimize the computational effort [7]. On the electromagnetic side, in contrast to the axisymmetric case, the Coulomb gauge
(2)
in terms of the arrays xn+1 and a˜ n+1 of time-dependent system nodal positions and vector potential values at instance tn+1 . The solution of the mechanical part of (2) involves in particular the consistent linearization required for the Newton–Raphson iteration in the context of large deformation inelastic problems. In detail, the staggered algorithm procedure consists of the following steps: (1) Initialize a˜ 0 , x0 and their time derivatives and proceed to (4).
Fig. 3. Development of the radial component of the magnetic flux in the sheet metal plate and surrounding region at 22 s (above) and 80 s (below).
272
J. Unger et al. / Journal of Materials Processing Technology 177 (2006) 270–273
Fig. 4. Development of the plastic deformation and Mises equivalent stress.
condition is generally not satisfied and the electromagnetic fields are not regular for standard boundary conditions. To ensure that the corresponding finite element solution reflects this lack of smoothness, a penalty method or a least-square approach can be applied. In the simulation presented here, N´ed´elec elements [8] are employed to overcome this difficulty. These are based on averaged degrees-of-freedom with respect to the element edges instead of discrete degrees-of-freedom at the element nodes. The discussion of the basic modeling aspects is concluded with a brief discussion of data transfer and meshing characteristics. In previous finite element models of the EMF process of the authors the nodal positions of the EM finite element mesh was fixed in space at any time. This required a relatively fine discretization in the region where the workpiece is moving to minimize the error induced by elements containing Gaussian points in both domains (see Fig. 1 left). Such elements suffer from their inability to represent sharp transitions imposed by the different EM-properties of the surrounding air and the workpiece. This approach is known from fluid structure interaction as the fictitious boundary method. A more recent approach applies an incrementally progressing mesh-updating algorithm [6] and is based on the arbitrary Lagrangian Eulerian solution scheme (ALE). Here, the position of the nodal coordinates of the EM-mesh is determined by means of solving a Laplace problem. On the surface of the workpiece, tool coil and boundary of the computational domain of the EM-system Dirichlet boundary conditions for the Laplace problem are applied: at each instance the boundary of the workpiece moves and prescribes a displacement increment for the new nodal positions of the EM-mesh such that the nodes of both meshes match at the boundary of the workpiece. As a result, at any time, mechanical and EM-mesh exhibit a consistent morphology as depicted in Fig. 1 right. 3. Application Results shown here are obtained on the basis of an axisymmetric model. Simulations have been carried out for a sheet metal plate consisting of the aluminium alloy AA 6005. The identification of the dynamic viscoplastic material parameters for this material has been carried out with the help of experimental data of the dynamic expansion of metal tubes via a maximum likelihood estimation using finite element simulations together with experimental data [9]. The experimental set-up for the electro-
magnetic sheet metal forming process of interest here is depicted in Fig. 2 right. Here, a sheet is placed on top of a flat circular tool coil and fixed with a blank holder. During the forming process, the tool coil current is measured and used as input data for the simulation. The simulated magnetic field development is shown in Fig. 3. In particular, the distribution of the radial component br of the magnetic flux during forming at 22 s (above) and 80 s (below) is shown. The radial component of b drives the forming operation resulting in an axial body force component. Initially, close to the tool coil, the equilibrium values of a and so the Lorentz force are extremely high. When the sheet-plate is accelerated away from the tool coil, however, it moves into a region where a is nearly zero. Forming stages for the plate at various instances are shown in Fig. 4. The contours represent the development of the accumulated inelastic deformation εP and the yield stress. Maximum values of the strain-rate reached here are on the order of 104 s−1 . At the beginning of the process, the center of the plate remains at rest, whereas at a radius of about r = 21 mm, the plate experiences high Lorentz forces and begins to accelerate. In later stages, the center of the plate is then pulled along by the rest of the plate and accelerated via predominantly inertial forces, resulting in the cap shaped structure at the end of the process with maximal inelastic strain in the top of the cap. The accompanying yield stress development shown in Fig. 4 displays clearly the corresponding maximal increase of the yield stress in the center of the late. The above-discussed results were obtained on the basis of 2D axisymmetric forming processes. Acknowledgement The authors wish to thank the Deutsche Forschungsgemeinschaft (DFG) for financial support of the work reported on here under the contact FOR 443/3-1. References [1] A. El-Azab, M. Garnich, A. Kapoor, Modeling of the electromagnetic forming of sheet metals: state-of-the-art and future needs, J. Mater. Process. Technol. 142 (2003) 744–754. [2] G. Fenton, G.S. Daehn, Modeling of electromagnetically formed sheet metal, J. Mater. Process. Technol. 75 (1998) 6–16. [3] D. Oliveira, M. Worswick, Electromagnetic forming of aluminum alloy sheet, J. Phys. IV France 110 (2003) 293–298.
J. Unger et al. / Journal of Materials Processing Technology 177 (2006) 270–273 [4] B. Svendsen, T. Chanda, Continuum thermodynamic formulation of models for electromagnetic thermoinelastic materials with application to electromagnetic metal forming, Continuum Mech. Thermodyn. 17 (2005) 1–16. [5] M. Stiemer, J. Unger, H. Blum, B. Svendsen, Algorithmic formulation and numerical implementation of coupled electromagnetic–inelastic continuum models for electromagnetic metal forming, Int. J. Numer. Meth. Eng., published online. [6] M. Stiemer, J. Unger, H. Blum, B. Svendsen, Fully coupled 3D simulation of electromagnetic forming, in: Proceedings of the 2nd Interna-
273
tional Conference on High Speed Forming (ICHSF), Dortmund, 2006, 63–72. [7] S. Reese, B. Svendsen, M. Stiemer, J. Unger, M. Schwarze, H. Blum, On a new finite element technology for electromagnetic metal forming processes, Arch. Appl. Mech. 74 (2005) 834–845. [8] J.C. N´ed´elec, A new family of mixed finite elements in P3 , Numer. Math. 50 (1986) 57–81. [9] J. Unger, M. Stiemer, A. Brosius, B. Svendsen, H. Blum, M. Kleiner, Model calibration and inverse error propagation with application to electromagnetic forming, in preparation.
Multifield modeling of electromagnetic metal forming processes J. Unger a,∗ , M. Stiemer b , B. Svendsen a , H. Blum b a
Leonhard-Euler-Str. 5, University of Dortmund, D-44227 Dortmund, Germany b Vogelpothsweg 87, University of Dortmund, D-44227 Dortmund, Germany
Abstract The purpose of this work is the formulation of a thermo-magneto-mechanical multifield model and presentation of the numerical strategies applied. In particular, this model is used to simulate electromagnetic sheet metal forming processes (EMF). In this process, deformation of the workpiece is driven by the interaction of a current generated in the workpiece by a magnetic field generated by a coil adjacent to the workpiece. The interaction of these two fields results in a material body force known as Lorentz force. Up to now, modeling approaches found for EMF in the literature are restricted to the axisymmetric case. For real industrial applications however, the modeling of three-dimensional forming operations becomes crucial for an effective process design. The implementation of such a 3D model still represents work in progress. Results shown in this paper are restricted to the axisymmetric case. © 2006 Elsevier B.V. All rights reserved. Keywords: Electromagnetic forming processes; Magneto-thermomechanical coupling; Staggered solution algorithm; Finite element method
1. Introduction
2. Model formulation
Electromagnetic forming is a dynamic, high strain-rate forming method in which strain-rates of ≥103 s−1 are achieved. In this process, deformation of the workpiece is driven by the interaction of a current generated in the workpiece with a magnetic field generated by a coil adjacent to the workpiece. In particular, the interaction of these two fields results in a material body force, i.e., the Lorentz force and the electromotive power, representing an additional supply of momentum and energy to the material. On the other hand the EM-system is sensitively influenced by the spatio-temporal evolution of the mechanical structure. In recent years considerable effort has been undertaken to simulate such coupled processes. However, approaches tested so far were mainly restricted to 2D or axisymmetric geometries [1–3]. Yet, it is the 3D modeling capability that is required to advance effectively in the design of industrial EMF processes. In this work modeling and simulation approaches for such a 3D model are discussed. Since their implementation still represents work in progress results for the axisymmetric forming set-up are presented here.
The coupled multifield model for electromagnetic forming of interest here, represents a special case of the general continuum thermodynamic formulation for inelastic non-polarizable and non-magnetizable materials given in [4]. Further details of the algorithmic realization of the model are discussed in [5,6]. In particular, this special case is based on the quasi-static approximation to Maxwell’s equations, in which the wave character of the electromagnetic (EM) fields is neglected. In this case, the unknown fields of interest are the motion field , the scalar potential χ and the vector potential a determining in particular the magnetic field in the usual fashion. Assuming Dirichlet boundary conditions for all fields and homogeneous Neumann boundary conditions, one derives the weak field relations: 0= (ρr ξ¨ − lr )ξ ∗ + KF−T ∇ξ ∗ , Br T 0 = {˙a − L a}a∗ + {χ − av}div a∗ (1) R R + κEM curl a curl a∗ , 0= ∇χ∇χ∗ , R
∗
Corresponding author. Tel.: +49 231 755 5717; fax: +49 231 755 2688. E-mail address: [email protected] (J. Unger).
0924-0136/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2006.03.190
for , a, and χ, respectively. Here, * , a* , and χ∗ represent the corresponding test fields. Further, R represents a fixed region
J. Unger et al. / Journal of Materials Processing Technology 177 (2006) 270–273
271
Fig. 1. Different meshing strategies. Left: fictitious boundary method with inaccurate modeling of boundaries. Right: improvement by means of ALE approach.
Fig. 2. Three-dimensional-EMF process: schematic of forming set-up with an elliptic tool coil and formed sheet metal and modeling domains of the sheet Br and the EM-field R. Axisymmetric EMF process: schematic of forming set-up with a circular tool coil.
in Euclidean point space containing the system under consideration in which the electromagnetic fields exist and on whose boundary the boundary conditions for these fields are specified. Here, the system consists of the workpiece, the tool coil, and air (see Fig. 2). As such, R contains in particular the fixed reference configuration Br , and all subsequent (i.e., deformed) configurations, of the workpiece. Here, F := ∇ represents the deformation gradient, and L := ∇v the spatial velocity gradient. Further, κEM represents the magnetic diffusivity, v the spatial velocity field, ρr the referential mass density, K the Kirchhoff stress, and lr = det(F)j × b the Lorentz force in terms of the magnetic flux b = curl a and the current density j. Note that (1)2,3 follow from Maxwell’s equations, while (1)1 represents the weak form of the momentum balance. The above weak field relations are completed by the thermodynamically consistent formulation of the adiabatic thermo-elasto-viscoplastic material model [4]. Consider next the finite element discretization of (1). The difference in electromagnetic and thermo mechanical timescales together with the distinct nature of the fields involved (i.e., Eulerian in the electromagnetic case, Lagrangian in the thermo mechanical large deformation context), argue for a staggered numerical solution procedure in two different meshes resulting in the following algorithmic system: fn (xn+1 , a˜ n+1 ) = 0, en (xn+1 , a˜ n+1 ) = 0,
(2) A starting value a˜ n+1 of the nodal vector potential array is computed for the measured amperage in the tool coil at time tn+1 and the known mechanical state of the system at time tn via (2)2 . (3) From a˜ n+1 , a corresponding value ln+1 for the Lorentz force is obtained. Using this, the system consisting of (2)1 is solved via Newton–Raphson iteration (i.e., at fixed a˜ n+1 ) to obtain xn+1 . (4) Proceed to next time step tn+1 = tn + tn,n+1 and proceed with (2). Else, if tn ≥ ts , terminate the simulation, where ts is the total simulation time. Besides its physical motivation, the staggered algorithm offers the possibility to apply custom solutions for both the mechanical, as well as the EM-system. Here, on the mechanical side an effective continuum shell formulation is applied to minimize the computational effort [7]. On the electromagnetic side, in contrast to the axisymmetric case, the Coulomb gauge
(2)
in terms of the arrays xn+1 and a˜ n+1 of time-dependent system nodal positions and vector potential values at instance tn+1 . The solution of the mechanical part of (2) involves in particular the consistent linearization required for the Newton–Raphson iteration in the context of large deformation inelastic problems. In detail, the staggered algorithm procedure consists of the following steps: (1) Initialize a˜ 0 , x0 and their time derivatives and proceed to (4).
Fig. 3. Development of the radial component of the magnetic flux in the sheet metal plate and surrounding region at 22 s (above) and 80 s (below).
272
J. Unger et al. / Journal of Materials Processing Technology 177 (2006) 270–273
Fig. 4. Development of the plastic deformation and Mises equivalent stress.
condition is generally not satisfied and the electromagnetic fields are not regular for standard boundary conditions. To ensure that the corresponding finite element solution reflects this lack of smoothness, a penalty method or a least-square approach can be applied. In the simulation presented here, N´ed´elec elements [8] are employed to overcome this difficulty. These are based on averaged degrees-of-freedom with respect to the element edges instead of discrete degrees-of-freedom at the element nodes. The discussion of the basic modeling aspects is concluded with a brief discussion of data transfer and meshing characteristics. In previous finite element models of the EMF process of the authors the nodal positions of the EM finite element mesh was fixed in space at any time. This required a relatively fine discretization in the region where the workpiece is moving to minimize the error induced by elements containing Gaussian points in both domains (see Fig. 1 left). Such elements suffer from their inability to represent sharp transitions imposed by the different EM-properties of the surrounding air and the workpiece. This approach is known from fluid structure interaction as the fictitious boundary method. A more recent approach applies an incrementally progressing mesh-updating algorithm [6] and is based on the arbitrary Lagrangian Eulerian solution scheme (ALE). Here, the position of the nodal coordinates of the EM-mesh is determined by means of solving a Laplace problem. On the surface of the workpiece, tool coil and boundary of the computational domain of the EM-system Dirichlet boundary conditions for the Laplace problem are applied: at each instance the boundary of the workpiece moves and prescribes a displacement increment for the new nodal positions of the EM-mesh such that the nodes of both meshes match at the boundary of the workpiece. As a result, at any time, mechanical and EM-mesh exhibit a consistent morphology as depicted in Fig. 1 right. 3. Application Results shown here are obtained on the basis of an axisymmetric model. Simulations have been carried out for a sheet metal plate consisting of the aluminium alloy AA 6005. The identification of the dynamic viscoplastic material parameters for this material has been carried out with the help of experimental data of the dynamic expansion of metal tubes via a maximum likelihood estimation using finite element simulations together with experimental data [9]. The experimental set-up for the electro-
magnetic sheet metal forming process of interest here is depicted in Fig. 2 right. Here, a sheet is placed on top of a flat circular tool coil and fixed with a blank holder. During the forming process, the tool coil current is measured and used as input data for the simulation. The simulated magnetic field development is shown in Fig. 3. In particular, the distribution of the radial component br of the magnetic flux during forming at 22 s (above) and 80 s (below) is shown. The radial component of b drives the forming operation resulting in an axial body force component. Initially, close to the tool coil, the equilibrium values of a and so the Lorentz force are extremely high. When the sheet-plate is accelerated away from the tool coil, however, it moves into a region where a is nearly zero. Forming stages for the plate at various instances are shown in Fig. 4. The contours represent the development of the accumulated inelastic deformation εP and the yield stress. Maximum values of the strain-rate reached here are on the order of 104 s−1 . At the beginning of the process, the center of the plate remains at rest, whereas at a radius of about r = 21 mm, the plate experiences high Lorentz forces and begins to accelerate. In later stages, the center of the plate is then pulled along by the rest of the plate and accelerated via predominantly inertial forces, resulting in the cap shaped structure at the end of the process with maximal inelastic strain in the top of the cap. The accompanying yield stress development shown in Fig. 4 displays clearly the corresponding maximal increase of the yield stress in the center of the late. The above-discussed results were obtained on the basis of 2D axisymmetric forming processes. Acknowledgement The authors wish to thank the Deutsche Forschungsgemeinschaft (DFG) for financial support of the work reported on here under the contact FOR 443/3-1. References [1] A. El-Azab, M. Garnich, A. Kapoor, Modeling of the electromagnetic forming of sheet metals: state-of-the-art and future needs, J. Mater. Process. Technol. 142 (2003) 744–754. [2] G. Fenton, G.S. Daehn, Modeling of electromagnetically formed sheet metal, J. Mater. Process. Technol. 75 (1998) 6–16. [3] D. Oliveira, M. Worswick, Electromagnetic forming of aluminum alloy sheet, J. Phys. IV France 110 (2003) 293–298.
J. Unger et al. / Journal of Materials Processing Technology 177 (2006) 270–273 [4] B. Svendsen, T. Chanda, Continuum thermodynamic formulation of models for electromagnetic thermoinelastic materials with application to electromagnetic metal forming, Continuum Mech. Thermodyn. 17 (2005) 1–16. [5] M. Stiemer, J. Unger, H. Blum, B. Svendsen, Algorithmic formulation and numerical implementation of coupled electromagnetic–inelastic continuum models for electromagnetic metal forming, Int. J. Numer. Meth. Eng., published online. [6] M. Stiemer, J. Unger, H. Blum, B. Svendsen, Fully coupled 3D simulation of electromagnetic forming, in: Proceedings of the 2nd Interna-
273
tional Conference on High Speed Forming (ICHSF), Dortmund, 2006, 63–72. [7] S. Reese, B. Svendsen, M. Stiemer, J. Unger, M. Schwarze, H. Blum, On a new finite element technology for electromagnetic metal forming processes, Arch. Appl. Mech. 74 (2005) 834–845. [8] J.C. N´ed´elec, A new family of mixed finite elements in P3 , Numer. Math. 50 (1986) 57–81. [9] J. Unger, M. Stiemer, A. Brosius, B. Svendsen, H. Blum, M. Kleiner, Model calibration and inverse error propagation with application to electromagnetic forming, in preparation.