An upwind least square formulation for free surfaces calculation of viscoplastic steadystate metal forming problems
 Ugo Ripert^{1}Email author,
 Lionel Fourment^{2} and
 JeanLoup Chenot^{1}
https://doi.org/10.1186/s4032301500375
© Ripert et al. 2015
Received: 2 March 2015
Accepted: 25 June 2015
Published: 14 July 2015
Abstract
Despite using very large parallel computers, numerical simulation of some forming processes such as multipass rolling, extrusion or wire drawing, need long computation time due to the very large number of time steps required to model the steady regime of the process. The direct calculation of the steadystate, whenever possible, allows reducing by 10–20 the computational effort. However, removing time from the equations introduces another unknown, the steady final shape of the domain. Among possible ways to solve this coupled multifields problem, this paper selects a staggered fixedpoint algorithm that alternates computation of mechanical fields on a prescribed domain shape with corrections of the domain shape derived from the velocity field and the stationary condition v.n = 0. It focuses on the resolution of the second step in the frame of unstructured 3D meshes, parallel computing with domain partitioning, and complex shapes with strong contact restraints. To insure these constraints a global finite elements formulation is used. The weak formulation based on a Galerkin method of the v.n = 0 equation is found to diverge in severe tests cases. The least squares formulation experiences problems in the presence of contact restraints, upwinding being shown necessary. A new upwind least squares formulation is proposed and evaluated first on analytical solutions. Contact being a key issue in forming processes, and even more with steady formulations, a special emphasis is given to the coupling of contact equations between the two problems of the staggered algorithm, the thermomechanical and free surface problems. The new formulation and algorithm is finally applied to two complex actual metal forming problems of rolling. Its accuracy and robustness with respect to the shape initialization of the staggered algorithm is discussed, and its efficiency is compared to nonsteady simulations.
Keywords
Steadystate Free surface Least square formulation Upwind shift SUPG Contact Metal forming Rolling Finite elementsBackground
One or two decades ago, when computers resources were much lower, steadystate formulations have attracted considerable interest in the field of material forming simulation More precisely in the metal forming field, it was extensively applied for rolling processes. Early simulations were done for thick plates [1] and simple shape rolling [2]. They were extended to seamless tube rolling [3, 4] and complicated shape rolling [5] before being applied to more complex material behavior [6] and to rolling stand deformation [7].
This was often justified by the fact that it was not possible to simulate the unsteady problem in all its complexity at that time. Nowadays, few publications using these methods can be found in the open literature, either because software codes are now quite mature and currently used in R&D [8] or because present computer resources make full, timedependent simulations possible. However, some metal forming simulations still end up with computational times (several weeks or months) that are much too large for practical use by the engineer. This mainly occurs wherever resolution requires a very large number of time steps—such problems cannot therefore be accelerated by a larger number of processors. Moreover, as the efficiency of computing processors does not increase as it used to, perspectives to reduce the computational time of these problems are shrinking. Therefore, for appropriate problems, the steadystate formulations arise again as a way to reduce the computational times by one to two orders of magnitude. The work presented in this paper is carried out in the frame of material forming in order to adapt and generalize some steadystate formulations to today’s numerical methods (unstructured meshes and parallel computations) and problems complexity (geometries and robustness).
Semicontinuous metal forming processes such as rolling or wire drawing can be simulated by the same numerical methods that are utilized for more conventional processes, like an incremental formulation within a Lagrangian framework [9]. Such an approach is indispensable in studies of intrinsically transient phenomena such as end geometry, edging strategies, crocodiling, twisting… However, when it comes to long products, most of the process can be regarded as stationary, and the forming of product ends is of secondary interest. In order to reach this stationary regime by passing the initial transient regime, very long sections of the product must be considered and consequently very large finite element meshes. In addition, the time steps must be small enough to properly capture the local contact phenomena between the tools. These two features result in a dramatic increase of the computational time, which can range between several hours to several weeks even on highly parallel computers.
In the continuity of the Lagrangian approach, the steady regime can be incrementally computed using an Arbitrary Lagrangian or Eulerian (ALE) formulation inside the socalled Eulerian window, as in [11] so benefiting from the computational domain reduction and mesh adaptation [12]. However, the approach being basically the same (lagrangian step followed by mesh repositioning and field transfer), it still requires many time increments to properly propagate the domain deformation from the input plane to the output plane. Therefore the computational time is of the same order of magnitude (time reduction is only about 30% in [9] for the test case investigating in “Applications to metal forming problems” and 15% for a U channel with 6 stands [13]). Only the direct computation of the steady regime through directly solving steadystate problems equations allows reducing computing time by an order of magnitude or more [10, 14, 15].
In [16], Tortorelli and Balagangadhar introduced the reference frame concept that allows extending the ALE framework to steadystate problems. Within a displacementbased formulation, the domain geometry is directly computed by solving a multifield problem and integrating state variables on the undeformed configuration. A simplified formulation was applied for welding [15], and also for drawing and rolling processes in 2D [17] using a penalty contact formulation instead of the more specific contact treatment initially developed in [16].
Within a velocitybased formulation, the multifield steadystate problem was directly solved by Ellwood et al. [18] for simple polymer extrusion process. This fully coupled resolution is achieved by adding free surface conditions (29) to the mechanical problem and thus considering geometry corrections in addition to velocities and pressures. However, due to strong couplings between material flow computation, free surface computation and contact with tools constraining free surface correction a decoupled approach is preferred in literature [2, 5, 19]. It consists of a staggered fixedpoint algorithm where the thermomechanical equations and convection equations are first solved for a given and fixed shape, and then the domain shape is corrected according to the newly computed material flow. Few iterations generally allow finding the domain shape, provided that the initial shape is not too distant from the final shape [4]. All these approaches only differ in the way the domain free surface is corrected, either by streamline integration, one by one, or by resolution of a global free surface problem.
The first approach is almost exclusively based on structured meshes, the nodes of which can be aligned along the streamlines, thus providing an easy and accurate way to integrate shape corrections [6]. Contact is a recurrent problem of these formulations. Contact equations are left aside during streamlines integration to avoid any discontinuities. A next step is therefore needed to project nodes on the tools using simple [20] or complex smoothing operations [5, 6].
The second approach is more general and applies both for unstructured meshes and complex flows (where using a structured mesh aligned with streamlines is inconceivable) [21]. In [22], the domain shape computation is regarded as a convection problem, namely the integration of the normal velocity along the free surface. Alternatively, as in [19], the domain correction can be directly computed from the velocity field by solving the free surface equation (see Eq. (29)), either using a least square formulation [19] or a Galerkin formulation [2].
The aim of the present work is to develop a steadystate formulation compatible with a velocitybased formulation, parallel computations and unstructured meshes. It is also expected to be sufficiently robust to handle all kinds of complex geometries met in metal forming applications. Therefore, the iterative fixed point approach is used to ensure compatibility with velocity formulation, and the free surface correction is computed through a global resolution to ensure compatibility with unstructured meshes and parallel computations.
“First step: position of the thermalmechanical problem and velocity field computation” presents the simple forming problem over a known domain; it consists of the thermomechanical equations of the metal forming problem. “Second step: free surface correction” gives a general description of the computation of the free surface correction for a known material velocity field with a nonnull normal velocity, and presents different weak formulations used in literature. Section 4 introduces the new weak formulation for the free surface correction that shows necessary to handle complex contact conditions in 3D, along with the coupling of contact equations between the two alternatively solved problems (simple forming problem and free surface correction). Resolution of simple analytical problems in “Evaluation on analytical tests” assesses the quality and robustness of several variants of the proposed formulation and “Applications to metal forming problems” applies the most promising one to actual metal forming problems.
First step: position of the thermalmechanical problem and velocity field computation
In this section, the shape of the computational domain \(\Omega\) is supposed to be known as its contact surface \(\Gamma_{c}\), (the part of \(\Gamma = \partial \Omega\) that is both geometrically in contact with the forming tools and undergoing a negative contact pressure). The resolution of the thermomechanical equations aims at finding the steadystate velocity field associated to this domain and contact surface.
Conservation equation and boundary conditions
Conservation equations: mechanics
The constitutive equation is written for s, the deviatoric part of \(\varvec{\sigma}\) and follows the Norton–Hoff model (2).
T is the temperature, \(\dot{\varepsilon }\) the strain rate tensor, \(\dot{\bar{\varepsilon }}\) the equivalent strain rate, \(\bar{\varepsilon }\) the equivalent strain (3) and \(K(\bar{\varepsilon },T)\) the material consistency, which is here supposed to only depend on T and \(\bar{\varepsilon }\) to model temperature softening and strain hardening or softening.
Domain upstream and downstream boundaries

stressbased the stress vector in these boundary planes is set to the average tension stress at each point (FE node), the shear component being zero:$$\begin{aligned} {\varvec{\sigma}} \cdot {\varvec{n}} = {{{\varvec{F}}^{tension} }/ S}\end{aligned}$$(7)

velocitybased the inplane velocity is cancelled (8) and the normal velocity (in the main flow direction) is imposed homogeneous on the Sect. (9); there remains one equation to write to obtain a wellposed problem, the integral of the stress normal to the section is set equal to the imposed tension force (10).$${\varvec{v}}  ( {{\varvec{v}} \cdot {\varvec{n}}})\,{\varvec{n}} = 0$$(8)$$\begin{aligned} \varvec{v}_{i} \cdot \varvec{n} = \varvec{v}_{j} \cdot \varvec{n}\quad \forall i,j \in \Gamma_{inlet} \,\,\,\left( {\Gamma_{outlet} } \right) \end{aligned}$$(9)$$\begin{aligned} \int_{{\Gamma_{inlet} }}^{{}} {{\varvec{\sigma}}\,{\varvec{n}}\,dS} = {\varvec{F}}_{inlet}^{tension} \end{aligned}$$(10)
Here, the first (stressbased) formulation is used and furthermore F ^{ tension } = 0 for simplicity.
This set of equations is discretized using tetrahedral finite element with the quasilinear P1^{+}/P1 “mini element” interpolation [25] satisfying the compatibility condition [26]. Contact is enforced by a penalty method using a nodetofacet formulation [24].
Conservation equation: energy
On the inlet surface, the initial temperature is imposed (either homogeneous, or a map obtained by computing former history…).
State variables equations
\(\bar{\varepsilon }\) (and more generally any state variable, such as e.g. pertaining to microstructural, metallurgical models) and \(\dot{\bar{\varepsilon }}\) are computed at integration points so their finite element discretization are not continuously interpolated. With the P1^{+}/P1 quasilinear interpolation used here, \(\dot{\bar{\varepsilon }}\) can be regarded as piecewise per element or P0. In order to avoid using a more complex and not necessarily more accurate discontinuous Galerkin method as in [15], it is preferred to project the P0 equivalent strain rate \(\dot{\bar{\varepsilon }}\) onto a P1 continuous linear mapping in order to use the standard SUPG formulation to solve Eq. (15) and then to use the P1 linear continuous interpolation of \(\bar{\varepsilon }\) to compute the equivalent strain values at integration points. Referring to [28], this method provides accurate results and allows using the same resolution method as for the temperature equation.
Thermomechanical coupling
However, the algorithm is not suited for more complex thermomechanical behaviors such as Friction Stir Welding (FSW), continuous casting or elastoplastic behaviors. Some changes have to be made to help the convergence. Adding simple relaxation methods for reducing potentially large changes can be highly beneficial. For even more stronger coupling behavior, sub fixed points should be added to insure the thermomechanical convergence at each iteration [6].
Second step: free surface correction
In this section, the velocity field \({\mathbf{v}}\) is supposed to be known and to be the solution of the steadystate flow on the considered domain, which needs to be corrected in order to satisfy the free surface condition (18).
Governing equations
When the flow is steady, the velocity field is tangential to the surface \(\Gamma\) of the domain \(\Omega\); streamlines of surface particles are on the surface and there is no material flow through this surface. In other words, the normal component of the velocity field is null on \(\Gamma\), which provides the fundamental equation for the stationarity of the free surface (18).
It can be noticed that \(\varvec{u}(\varvec{t})\) is linear with respect to t, if t has a fixed and constant direction, and that it is quadratic with respect to t in the general case.
Contact condition
This unilateral contact condition is not sufficient because it does not prevent the complete release of all contact conditions during the free surface correction [20] leading to oscillations or even divergence. It proves necessary to enforce a bilateral contact condition (25) on \(\Gamma_{c}^{FSC}\), the contact surface of \(\Gamma\). It is defined using both the sign of contact normal stress associated to the current velocity field and the penetrations occurred after the free surface correction (its definition is presented into more details in “Contact conditions transmission between velocity field computation and free surface correction”).
Weak formulation
It can be noticed that the LS formulation naturally extends to the case where t is a vector with several degrees of freedom. Both SUPG and LS formulations provide non symmetric weights (see Figure 2) which enable them to handle the convection terms.
Volume mesh regularization
New formulation for free surface correction
As discussed in “Weak formulation”, the Galerkin formulation is limited to specific and rather simple configurations; both Galerkin and SUPG formulations are well adapted when the correction t is a scalar, while the LS formulation allows an easy extension to 3D problems when t is a vector. However, studies of analytical problems (see “Evaluation on analytical tests”) show that the LS formulation is not converging to the expected solution when contact is taken into account and when the flow is significantly oriented. Therefore, the LS formulation should be improved to be applied to 3D forming problems. Two modifications are presented hereafter.
Upwind least square formulation
It should be noticed that the matrices symmetry of the LS formulation is consequently lost with these new formulations.
Full 2 degrees of freedom (DoF) formulation
Contact conditions transmission between velocity field computation and free surface correction
At the first iteration, \(\Gamma_{c}^{init}\) is defined geometrically from the initial domain shape, by the contact distance (41). This contact zone is then used for the resolution of the velocity field calculation to apply unilateral contact (5) and friction (6) conditions. From the penalty formulation, with \(\rho_{c}\) the contact penalization factor, the resolution allows computing contact forces \(\lambda_{k}^{VC}\) at any node k (42). During the free surface correction, these contact forces define the zone \(\Gamma_{c}^{FSC}\)(42) where a sliding bilateral contact condition is applied (25). It should be noted from Eq. (23) that the nonpenetration equations are always enforced for free surface nodes. Symmetrically to the velocity field calculation, a pseudocontact force \(\lambda_{k}^{FSC}\) can be defined at any node k (43) after the free surface correction. The contact surface \(\Gamma_{c}^{VC}\) is now defined by (43) for the next iterations.
It is worth mentioning for the calculation of \(\lambda_{k}^{FSC}\) in Eq. (43) that the free surface correction equations are not mechanical equations, so that it is more questionable to consider \(\lambda_{k}^{FSC}\) as a real contact force; it is rather related to the accuracy of the free surface movement. Consequently, it shows necessary to introduce a numerical pseudoadhesion coefficient \(\varepsilon\) in the definition of \(\lambda_{k}^{FSC}\). In practice, \(\varepsilon\) is taken equal to 2% of the mesh size.
Evaluation on analytical tests
Presentation of the tests
DoF test cases
In order to first compare the different weak formulations, only one degree of freedom per node is authorized, in the direction normal to the flattened mesh. Thus the Laplacian regularization (39) can be omitted. 2D meshes of respectively 6000 and 36,000 nodes are used to evaluate the finite element convergence of the methods tested.
DoF test case with edges
DoF test cases with contact
Results for test cases without contact
Error (%) for nodes position displacement on analytical test cases for different refinement of 2D meshes
Mesh size  Gaussian function  Sine function  3D example  

Coarse  Fine  Coarse  Fine  Coarse  Fine  
Galerkin  No convergence  Not tested  
SUPG  0.08 (0.91)  0.02 (0.38)  1.69 (0.21)  0.42 (0.08)  
LS  0.80 (6.42)  0.22 (2.98)  7.41 (1.48)  1.67 (0.56)  51.9 (353)  62.1 (945) 
LS_supg  0.22 (2.36)  0.04 (0.85)  5.19 (0.57)  1.85 (0.26)  2.7 (9.6)  0.5 (3.37) 
LS_sl  0.18 (3.78)  0.04 (2.47)  4.84 (0.56)  1.84 (0.28)  2.6 (10.6)  1.6 (11.5) 
DoF test cases
On these test cases, the Galerkin method (28a) fails to converge (see Table 1). LS_supg and LS_sl (resp. (33) and (34)) are either as accurate (sine function) or more accurate (Gaussian function) as the LS formulation. The SUPG formulation (28b) is the most accurate because it has the best nodal accuracy on the sheet edges, whereas least squares methods tend to smooth the solution. In other words, these nodal boundary conditions are more favorable to Galerkinlike formulations.
DoF test case with edges: effect of mesh smoothing
The convergence of the LS formulation is significantly affected by the Laplacian smoothing for 3D examples (Table 1). Nonlinear corrections decrease significantly at each iteration, making the convergence really slow. Even after 60 nonlinear iterations, the solution is far from the exact solution (Figure 10 right) and the convergence appears to be blocked. The introduction of an upwind shift solves this nonconvergence issue. Ls_supg and Ls_sl formulations give accurate results within less than 20 iterations (Table 1): the surface is precisely retrieved, the edges are perfectly preserved and the mesh is properly regularized (Figure 10).
Interaction with contact
The LS formulation is unable to compute the expected shape (see Figure 11). Although this formulation is sensitive to the flow direction, it is not sensitive to its orientation so the contact information is transported both downwind and upwind. The contact equations are satisfied but the contact area is underestimated; the normal component of the velocity is minimized all over the domain but the formulation does not succeed to cancel it; an unexpected shape is consequently obtained (see Figure 11). On the other hand, contact and free surface conditions are perfectly satisfied all over the domain for all the upwind formulations, not only the SUPG formulation but also for the newly LS_supg and LS_ls formulations. It is worth mentioning SUPG and LS_supg tend to create really slight oscillations on the free surface before the contact area. These oscillations are proportional to the discontinuities introduced by contact constraints, which can be regarded as Dirichlet conditions. For nonfully upwind formulations some strategies exist to prevent these problems [31]. However, these oscillations can be neglected in forming processes applications where the material flow and the contact conditions are not fully incompatible; they do not introduce any discontinuities.
Summary
To sumup, Galerkin formulation does not converge on the test cases chosen (the problem solution is not unique) while SUPG formulation provides the most accurate results on such analytical functions. LSbased formulations are properly converging toward the exact solution with decreasing finite element mesh size, and their accuracy is quite satisfactory. These formulations can be easily extended to 3D problems with a correction vector (rather than a scalar in a prescribed direction) where they make it possible to properly preserve the problem shape edges without requiring specific techniques to detect and deal with these surface singularities, a difficult and hazardous issue. When contact comes into play, the original LS formulation has a nonphysical behavior resulting from the lack of a direction for contact corrections propagation. Introducing an upwind shift, either with the LS_supg or LS_sl formulation allows fixing this issue while preserving or even improving the accuracy of the LS formulation. Therefore, the following forming applications will be studied using LS_sl. Even if LS_supg seems better, a full upwind formulation is preferred for avoiding issues with contact like the original LS formulation.
Applications to metal forming problems
Introduction
Rolling of thick plate
A thick metal sheet is rolled as presented in Figure 13. The tools are cylinders with a 600 mm diameter and separated by 18 mm. Their rotational velocity is about 27.5 rpm. This problem includes an edge, the displacement of which should not be restricted to the mesh surface normal direction. Consequently, a 1 DoF approach would fail and a more general free surface formulation is required. Two symmetry planes are used to reduce the problem size (Figure 13). A coarse mesh of about 5000 nodes and a finer one with 15,000 nodes (Figure 13) are used for the computations.
Rolling of long product
Convergences and speedsup
Volume loss (%)
Thick plate  Shape rolling  

Incremental  0.115  0.40 
Lam3  0.008  0.38 
Steadystate (coarse)  0.260  0.45 
Steadystate (fine)  0.023  1.15 
CPU times for the simulation of thick plate rolling
Number of nodes  Number of iterations/increments  CPU time  Speed up  

Incremental  10,000 → 16,000  280  2 h 57 min  1 
Lam3  12,000  28  5 min  35 
Steadystate  15,000  40  22 min  8 
CPU times for the simulation of “oval to square” rolling
Number of nodes  Number of iterations/increments  CPU time  Speed up  

Incremental  20,000 → 37,000  1099  24 h 45 min  1 
Lam3  12,000  100  38 min  39 
Steadystate  37,000  40  1 h 10 min  21 
Sensitivity to domain initialization
In steadystate formulations, initialization of the computational domain may become tricky because of its influence on the convergence. Less iterations are needed if the initial geometry is closer to the solution, but also convergence may not be reached if the initial geometry is too far or not adequate as in the “oval to square” example with Lam3 or as mentioned by Mori [10]. It is important to figure out that the formulation developed in this article is based on a small correction approach. From an engineering standpoint, it is crucial to have a robust method that is able to converge from almost any reasonable initial solution satisfying the basic boundary conditions.
An almost optimal initialization of the computational domain can be obtained by the “extrusion” method (Figure 12), which was used in previous cases. For the purpose of testing the approach robustness, two nonoptimal methods are analyzed.
Similarly, the three different initializations are used to study the robustness of the algorithm for the second and more complex application problem. Figure 21 left shows that here again, the algorithm converges towards almost the same solution whatever the initial geometry, even if it is very far from the final solution, as shown in Figure 21 top–right with the forged initial shape. This robustness property mainly results from the mesh regularization introduced inside the free surface algorithm; it acts like a relaxation and allows a progressive convergence toward the final shape, as shown in Figure 21 bottom–right.
Conclusion
An iterative approach for the search of continuous processes steady state has been presented in this paper. The aim was to reduce computational times encountered with incremental formulations by searching directly the steady state with a staggered algorithm. A staggered fixedpoint algorithm is used where a first velocity field is solved for on a fixed and known geometry, and then a domain correction is performed. The second stage is controlled by a free surface calculation that needs to be solved in the most general way in terms of mesh type (unstructured), geometries complexity and parallel computation compatibility. A global resolution using the finite elements method and two DoF satisfies all imposed constraints.
Simple analytical test cases with only one DoF evidence the purely convective aspect of the free surface problem. Thus, shape functions for the variational problem have to take into account the flow direction. Two new weak formulations based on the LS method, for facilitating the transition to a 2 DoF algorithm, and using SUPG advantages are developed in this study: LS_supg and LS_sl (streamlinelike). However, LS_sl takes benefit from its fully upwind formulation to eliminate the oscillations which could occur with imposed contact constrains.
Using a two DoF formulation in the cross section helps to generalize the free surface algorithm by handling mesh singularities, like edges where displacement directions cannot be easily guessed. To obtain a solution, a Laplacian operator is added to the free surface problem introducing a mesh regularization. The final algorithm retrieves the geometry with accuracy while preserving edges and mesh quality.
Contact equations are considered during the free surface calculation by using both unilateral and sliding bilateral conditions. The second condition prevents a complete loss of contact and is possible through the mesh regularization. A contact analysis is then crucial to enforce the coupling between the two stages of the staggered fixedpoint algorithm. Detection of compressed nodes is performed progressively by using specific and complementary data from both mechanical and geometrical resolution stages.
The developed algorithms were applied with success on various analytical test cases and for material forming processes like shape rolling. Accuracies are similar to the unsteady formulation and important speedsup are gained, ranging from 10 to 20. The method appears to be robust even when using on purpose nonoptimal initial domains to increase the convergence difficulty.
However, the two DoF formulation for the free surface computation has too much freedom leading to some issues: relatively slow convergence of the free surface calculation, parasite displacements after both geometry and contact are stabilized. Adding more control upon each DoF by making a differentiation between surface and edge nodes during the free surface calculation would correct these problems, as the first ones need one DoF to be optimal whereas the last ones require a second DoF. This is subject of a future paper.
Declarations
Authors’ contributions
LF designed the study. With JLC, they supervised and guided the work, and offered appreciated insight on the industrial software Forge3. UR developed the iterative algorithm, ran the different test cases and conducted the detailed analysis. LF and UR drafted the manuscript, and JLC helped its reviewing. All authors read and approved the final manuscript.
Acknowledgements
Support for this work has been provided by the “ForgeALE” project gathering several companies of the metal forming industry (AscometalCREAS, AREVACEZUS, Aubert&Duval, Industeel and Ugitech). Their financial and technical support is gratefully acknowledged. Helps from Transvalor, in particular C. Béraudo and E. Perchat, during the development of a steady version of Forge^{®} were much appreciated. We also want to thank Pierre Montmitonnet for helpful comments while writing the paper.
Compliance with ethical guidelines
Competing interests The authors declare that there are no conflicts of interest.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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