esaform_2015
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Esaform congress program 2015TRANSCRIPT
Efficient Updated-Lagrangian Simulations in Forming ProcessesDiego Canales1,a, Adrien Leygue1,b, Francisco Chinesta1,c, Elías Cueto2,d,EricFeulvarch3,e,Jean-Michel Bergheau3,f, Yannick Vincent4,g, Frederic Boitout4,h,
1GeM, Ècole Centrale de Nantes, 1 rue de la Nöe, 44300 Nantes, France2I3A, University of Zaragoza, María de Luna, 50018 Zaragoza, Spain
3LTDS, ENISE, 58, rue Jean Parot, 42023 Saint-Etienne, France4ESI Group, France
[email protected], [email protected],[email protected], [email protected], [email protected],
[email protected], [email protected],[email protected]
Keywords: PGD, updated-Lagrangian, Forming processes.
Abstract. A new efficient udated-Lagrangian strategy for numerical simulations of material formingprocesses is presented in this work. The basic ingredients are the in-plane-out-of-plane PGD-baseddecomposition and the use of a robust numerical integration technique (the Stabilized ConformingNodal Integration). This strategy is of general purpose, although it is especially well suited for plate-shape geometries. This paper is devoted to show the feasibility of the technique through some simplenumerical examples.
Introduction
The numerical simulation of material forming processes is a fundamental tool in the industry. It allowsreducing the design cycle time and optimizing the process without the need for costly and time consum-ing test campaigns. However, it is a very challenging task, even for the high performance computationavailable nowadays. The simulation of processes such as co-extrusion, friction stir welding (FSW) orresin transfer moulding (RTM) involves some inherent difficulties: thermomechanical coupled for-mulations, large deformations or the need for computing the thermomechanical path of the materialparticles to compute important quantities of interest (such as residual stresses or the microstructure).
Despite the fact that accurate simulations tools are available, the actual industry demands moreefficient solutions to improve its competitiveness. With this aim, efforts have been made to applydifferent reduced order modeling methods to these simulations. Among others, we can highlight thecontributions of the Proper Generalized Decomposition in this field [REF libro procesos].
Proposed strategy
The proposed strategy can be seen as the natural extension of the in-plane-out-of-plane based on theProper Generalized Decomposition (PGD) for an updated-Lagrangian framework. This technique,developed by B.Bognet et al [REF], has been used in Eulerian frameworks to simulate solids an non-Newtonian fluids [REF] with impressive saving of computational cost compared with traditional FEmethods.
The main idea of our approach is to take advantage of updated-Lagrangian methods, very conve-nient for material forming simulations, but with a significant reduction of its computational complexityusing the in-plane-out-of-plane PGD-based decomposition. The material particles positions and thephysical values are projected into a plane and into an axis. Using these nodal projections two func-tional spaces are constructed, 2D and 1D respectively. Then, the thermomechanical problem is solvedas a succession of problems 2D and 1D in these spaces thanks to the in-plane-out-of-plane PGD-based
formulation. In the 2D space the projected mesh could be, eventually, very distorted. For a better per-formance of the PGD, the Stabilized Conforming Nodal Integration (SCNI) is introduced to computethe 2D PGD operators. Once the solution is obtained, the primary variables can be reconstructed in thematerial particles of the 3D domain and update their positions. This reconstruction does not involveany interpolation or projection stage, avoiding the numerical diffusion and other numerical difficul-ties. After the updating step, these steps are repeated until the end of the simulation. In the Fig. 1, ageneral scheme of the strategy is shown.
Fig. 1: General scheme of the proposed strategy
The main ingredients of the proposed strategy, the in-plane-out-of-plane decomposition and theSCNI are presented in detail below. A general flow model for material forming processes with linearand non-linear behaviors laws is used.The In-plane-out-of-plane Decomposition in an Updated-Lagrangian Framework. Let us de-scribe this updated-Lagrangian strategy through a generic model of a material forming process whichinvolves a material flow, such as co-extrusion or FSW. In first approximation, we neglect the elasticbehavior assuming a purely viscoplastic constitutive equation. This assumption is known as the flowformulation in the forming processes community [REF].
The balance of momentum and mass equations without inertia and the assumed incompressibilityof the flow read:
∇ · σ = 0, ∇ · v = 0 (1)To study the evolution and distribution of temperatures, the rigid-plastic material equations are
coupled with the following heat transfer equation:
∇ · (k∇T ) + r − (ρcpT ) = 0 (2)
The rate of heat generation due to plastic deformation is calculated as
r = βσ : d (3)
where β, the fraction of mechanical energy transformed to heat and d the strain rate tensor. Togetherwith these equations, a behavior law and the appropriate boundary conditions should be considered.The mechanical and thermal problem will be solved iteratively.
Considering a plate-shape domain Ξ = Ω× I, this technique assumes the next separated approx-imation of the velocity field
v(x, y, z) =
u(x, y, z)v(x, y, z)w(x, y, z)
≈N∑i=1
uixy(x, y) · ui
z(z)vixy(x, y) · viz(z)wi
xy(x, y) · wiz(z)
, (4)
where uxy(x, y), vxy(x, y) and wxy(x, y) are function of the in-plane coordinates whereas uz(z), viz(z)and wz(z) are functions involving the thickness coordinate. Similarly, the strain rate tensor can beexpressed in separated way. For intensive variables, such as the viscosity, an SVD decompositionshould be perform in order to obtained its in-plane-out-of-plane separation:
µ(x, y, z) =M∑k=1
µxyk (x, y)µz
k(z). (5)
Two different behavior laws have been considered, a linear behavior,∇p = ∇ · (µ · ∇v)∇ · v = 0
(6)
where µ is the fluid viscosity, and a Power Law,∇p = ∇ · T∇ · v = 0
(7)
where the extra-stress tensor for power-law fluids writes:
T = 2K · dn−1 · d (8)
withK and n two rheological parameters. The equivalent strain rate d given by:
d =√2(d : d) (9)
where " : " denotes the tensor product twice contracted.To circumvent the issue related to stablemixed formulations (LBB conditions) within the separated
representation used in what follows, we consider a penalty formulation that modifies the mass balanceby introducing a penalty coefficient λ small enough
∇ · v+ λ · p = 0. (10)
The Stabilized Conforming Nodal Integration. In the proposed strategy, after updating the nodalposition, their orthogonal projections into a plane and an axis constitute the nodal position of theinterpolant spaces used for the PGD solution. One problem we have face is that the 2D mesh will be,eventually, very distorted and the numerical integration of the discrete operators may be not accurateenough.
Chen et al. [?] introduced the SCNI technique to perform an accurate nodal integration in meshlessmethods. However, it is very interesting that SCNI can be incorporated into traditional FE interpolantsto produce a very robust method to deal with distorted mesh.
The SCNI is based on the assumed strain method, in which a modified gradient is introduced atthe integration point (node):
∇v(xi) =1
Ai
∫Ωi
∇v(x) dΩ, (11)
where xi are the coordinates of node ni. The cell Ωi is one element of any partition of the domain.Typically a Voronoi tessellation is used.
The modified strain rate tensor is given by
d(xi) =1
Ai
∫Ωi
d(x)dΩ =1
Ai
∫Ωi
∂u∂x
12
(∂u∂y
+ ∂v∂x
)12
(∂u∂z
+ ∂w∂x
)12
(∂u∂y
+ ∂v∂x
)∂v∂y
12
(∂v∂z
+ ∂w∂y
)12
(∂u∂z
+ ∂w∂x
)12
(∂v∂z
+ ∂w∂y
)∂w∂z
dΩ. (12)
Applying the divergence theorem, it results:
d(xi) =1
Ai
∫Γi
u(x)nx12(u(x)ny + v(x)nx)
12(u(x)nz + w(x)nx)
12(u(x)ny + v(x)nx) v(x)ny
12(v(x)nz + w(x)ny)
12(u(x)nz + w(x)nx)
12(v(x)nz + w(x)ny) w(x)nz
dΓ.
(13)After the discretization, the derivates of the shape functions are computed through the value of the
shape functions itself in certain integration points. These integration points are distributed along theboundaries of the elements of the partition.
Validation of the strategy
A simple model of co-extrusion. Let us assume a simple model of co-extrusion, two immisciblefluids, with very different viscosities, entering in a squared pipe. The interface between the fluids is,at the enter, in the middle of the section as it is shown in the figure . But this is not an equilibriumsituation, the less viscous fluid advances faster and, due the conservation of mass, the interface movesto diminish the effective flow section.
Fig. 2: Scheme of the co-extrusion
In the Fig. 3, the equilibrium position of the interface for two fluids with a given viscosity ratio ispresented. This plot has been obtained solving the analytical solution of two Stokes' flow between twoparallel plates. On the upper and bottom plates the sticking condition was imposed and in the interfacethe velocities and tangential forces were equaled.
The results, even when only a few nodes were used (around 800), are in very good agreement withthe qualitative expected behavior. In the Fig. 4, it can be seen that the flow section of the blue fluidhas diminished in approximately the predicted quantity.FSW kinematic like example.FSW is a solid state welding technique which since its invention in1991 is of great interest to the industry [?].The FSW welding process is conceptually simple. A non-consumable rotating tool with a specially designed pin and shoulder is inserted into the abutting edgesof sheets or plates to be joined and traversed along the line of joint. The tool heats the workpiece antits stir movement produces the joint.
In this work a simplified model of this process has been tested with the proposed strategy. Evenwhen neither the real technological parameters and geometry of the tools are not implemented, we
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0.4
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0.5
.
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Fig. 3: Analytical solution of the position of the interface
Fig. 4: PGD solution at the equilibrium
proved the feasibility of the numerical algorithm: we can solve a 3D flow model in an updated-Lagrangian framework with a computational cost in the order of several 2D problems. Moreover,the thermomechanical history of the material particles is computed directly from the simulation. Forinstance, we can study the evolution of the viscosity of a particle during the process. In the Fig. 5,we can observe how the viscosity of the particle decrease of the when it passes through the area ofthe tool, where the strain ratios are high due to the rotation of the pin. A picture of this simulation ispresented in the Fig. 6, where the system of reference is attached to the pin axis.
Fig. 5: Evolution of the viscosity a material particle near the pin
Fig. 6: FSW example
Conclusions and perspectives
The simulation of material forming processes is one of these scenarios where the traditional simula-tions have many difficulties. These issues are not only numerical but also determined by the require-ments of the most advanced industries. Recently, the PGD has been demonstrated as a very efficientreduced order modeling method and a particularly efficient solver for the simulation of material form-ing processes. In this work we show that it is possible to combine the PGD with robust interpolants tocreate efficient updated-Lagrangian simulations suitable for material forming processes.
The examples presented just prove the feasibility of the technique but does not represent any par-ticular simulation at the moment. The consideration of more realistic and complex processes withspecific industrial application is a work in progress.
References
[1] Dj.M. Maric, P.F. Meier and S.K. Estreicher: Mater. Sci. Forum Vol. 83-87 (1992), p. 119
[2] M.A. Green: High Efficiency Silicon Solar Cells (Trans Tech Publications, Switzerland 1987).
[3] Y. Mishing, in: Diffusion Processes in Advanced Technological Materials, edtied by D. GuptaNoyes Publications/William Andrew Publising, CityplaceNorwich, StateNY (2004), in press.
[4] G. Henkelman, G.Johannesson and H. Jnsson, in: Theoretical Methods in Condencsed PhaseChemistry, edited by S.D. Schwartz, volume 5 of Progress in Theoretical Chemistry and Physics,chapter, 10, Kluwer Academic Publishers (2000).