FEA Information Engineering Journal

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1 ISSN Volume 2, Issue 8, August 2013 FEA Information Engineering Journal R7 LS-DYNA 9 th European LS-DYNA Users Conference

2 FEA Information Engineering Journal Aim and Scope FEA Information Engineering Journal (FEAIEJ ) is a monthly published online journal to cover the latest Finite Element Analysis Technologies. The journal aims to cover previous noteworthy published papers and original papers. All published papers are peer reviewed in the respective FEA engineering fields. Consideration is given to all aspects of technically excellent written information without limitation on length. All submissions must follow guidelines for publishing a paper, or periodical. If a paper has been previously published, FEAIEJ requires written permission to reprint, with the proper acknowledgement give to the publisher of the published work. Reproduction in whole, or part, without the express written permissio of FEA Information Engineering Journal, or the owner of of the copyright work, is strictly prohibited. FEAIJ welcomes unsolicited topics, ideas, and articles. Monthly publication is limited to no more then five papers, either reprint, or original. Papers will be archived on For information on publishing a paper original or reprint contact editor@feaiej.com Subject line: Journal Publication Cover: Fig. 11 Coupled EM-thermal-ICFD application: Cooling of coils used for induced heating applications Paper: LS-DYNA R7: Conjugate heat transfer problems and coupling between the Incompressible CFD (ICFD) solver and the thermal solver, applications, results and examples. 2 Fea Information Engineering Journal August 2013

3 FEA Information Engineering Journal TABLE OF CONTENTS Volume 2, Issue No. 8 August 2013 Publications are to The th European LS-DYNA Users Conference - ARUP LS-DYNA R7: Conjugate heat transfer problems and coupling between the Incompressible CFD (ICFD) solver and the thermal solver, applications, results and examples. Iñaki Çaldichoury Facundo Del Pin Livermore Software Technology Corporation LS-DYNA R7: Coupled Multiphysics analysis involving Electromagnetism (EM), Incompressible CFD (ICFD) and solid mechanics thermal solver for conjugate heat transfer problem solving Iñaki Çaldichoury (1) Pierre L'Eplattenier (1) Facundo del Pin (1) Miro Duhovic (2) (1) Livermore Software Technology Corporation (2) Institut für Verbundwerkstoffe GmbH, LS-DYNA R7: Strong Fluid Structure Interaction (FSI) capabilities and associated meshing tools for the incompressible CFD solver (ICFD), applications and examples. Facundo Del Pin Iñaki Çaldichoury Livermore Software Technology Corporation LS-DYNA R7: Recent developments, application areas and validation results of the compressible fluid solver (CESE) specialized in high speed flows. Zeng-Chan Zhang Iñaki Çaldichoury Livermore Software Technology Corporation LS-DYNA R7: Update On The Electromagnetism Module (EM) Pierre L'Eplattenier Iñaki Çaldichoury Julie Anton Livermore Software Technology Corporation All contents are copyright to the publishing company, author or respective company. All rights reserved. 3 Fea Information Engineering Journal August 2013

4 LS-DYNA R7: Conjugate heat transfer problems and coupling between the Incompressible CFD (ICFD) solver and the thermal solver, applications, results and examples. Iñaki Çaldichoury Facundo Del Pin Livermore Software Technology Corporation 7374 Las Positas Road Livermore, CA Abstract LS-DYNA version R7 includes CFD solvers for both compressible and incompressible flows. The incompressible CFD solver (ICFD) may run as a stand alone CFD solver for pure thermal fluid problems or it can be strongly coupled using a monolithical approach with the LS-DYNA solid thermal solver in order to solve the complete conjugate heat transfer problem. This paper will focus on the thermal part of the ICFD solver and its associated features. Several results of thermal and conjugate heat transfer problems will be presented as well as some industrial applications for illustration and discussion purposes.

5 1- Introduction LS-DYNA version R7 double precision aims to solve complex multi-physics problems involving fluids, electromagnetism or chemistry interacting with the solid mechanics and thermal solvers of LS-DYNA. This paper will focus on the incompressible flow solver (ICFD) and more specifically on its thermal and conjugate heat transfer capabilities. Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy and heat between physical systems. The ICFD solver offers the possibility to solve and study the behavior of temperature flow in fluids. Potential applications are numerous and include refrigeration, air conditioning, building heating, motor coolants, defrost or even heat transfer in the human body. Furthermore, the ICFD thermal solver is fully coupled with the thermal solver using a monolithic approach which allows the solving of complex problems where both heated structures and flows are present and interact together. 2- Heat Equation and coupling with the thermal solver for solids The distribution of heat in a given region of fluid over time is described by a convection-diffusion equation also called heat equation: where α is the called thermal diffusivity and f is a potential source of heat. This formulation is incomplete if the appropriate set of boundary conditions and initial conditions is not specified. The user can specify the temperature or the heat flux on the boundaries resulting in Dirichlet of Neumann boundary conditions respectively: Furthermore, if no boundary condition is specified, the solver will automatically apply a Neumann condition: Let us note that, as in accordance with the incompressibility hypothesis, the temperature does not influence the flow s velocity. For specific applications involving free convection, the classic Boussinesq model has been introduced and is available to users. For the thermal coupling between the heat equation solved by the thermal solver in the structure and the heat equation solved in the fluid by the ICFD solver, a monolithic approach has been adopted. The coupling between the structure and the fluid is therefore very tight and strong at the fluid-structure interface. The resulting full system includes both the structural and the fluid temperature unknowns (See Figure 1) and is solved using a direct solver which may in some cases be computer-time consuming.

6 Figure 1 Vector of temperature unknowns when the fluid thermal solver and structure thermal solvers are coupled using a monolithically approach. 3-1 The analytical solutions 3- Validation of the conjugate heat transfer solver As the development of the different R7 solvers progresses, several verification, validation and benchmarking tests have been conducted both internally at LSTC and externally by beta testing users in order to track bugs and improve numerical accuracy. This section will present some of the results obtained on a conjugate heat transfer problem involving the flow in a parallel plane channel (2D problem) or a cylindrical channel (3D problem). In [1], the analytical solution of the conjugate heat transfer problem in a parallel plane channel has been studied by applying a periodic temperature boundary condition prescribed on the exterior face of the solid channel (. In [2], the problem has been extended to the axi-symmetric cylindrical channel case with a periodic prescribed temperature boundary condition. In all cases, the flow is considered laminar and fully hydrodynamically and thermally developed. Numerous industrial applications meet such conditions and are often encountered in nuclear reactor cooling designs, heat exchangers for Stirling-cycle machines or internally finned ducts. Figure 2 offers a sketch of the complete fluid-solid conjugate heat transfer problem. As in most fluid mechanics problems, it is often more convenient to work in dimensionless quantities: with the half height of the internal channel wall, the half height of the exterior channel wall, z the axial coordinate of the channel, the adimensional temperature, and the solid and fluid thermal conductivities respectfully, the angular frequency, U and the longitudinal component of the fluid velocity and its mean value and finally Pe the Peclet number. The temperature distribution has been obtained analytically in [1] and in [2] for the 2D and 3D cases respectfully by expressing the energy balance equation as a complex-valued hypergeometric confluent equation. The temperature profile can be written as: Where it has been shown in [1] that for the 2D case, and can be expressed as the real and complex parts of the complex valued function:

7 where is the confluent hypergeometric function and,, are complex constants that be calculated using the boundary conditions. For the 3D case, it has been shown in [2] that the solution can be written as : where Γ is the Gamma function, I and K are the first and second type Bessel functions and,, and are complex constants that be calculated using the boundary conditions. Figure 3 offers some examples of temperature distributions for different set of parameters for the 2D case. Figure 2 Sketch of the longitudinal section of the channel Figure 3 Analytical solution: dimensionless temperature distribution versus and for, a) Fluid domain only b) Fluid and Solid coupled domains.

8 3-1 The numerical solutions Figure 4 offers a view of the mesh used for both the 2D and 3D cases. For this analysis, several parameters will be varied such as the Peclet number, the angular frequency, thermal conductivities and so forth. For the 2D case, the continuity of temperature at the interface is well insured by the numerical simulation as can be observed on Figure 5. Figure 6 offers a qualitative comparison between the analytical and the numerical temperature profiles at the solid-fluid interface. In Figure 7a), the dimensionless temperature distribution at the solid-fluid interface is reported and a comparison is made with the analytical solution for different Peclet numbers. A higher Peclet number yields smaller amplitude at the interface. This is consistent as a higher Peclet number value implies more temperature and advection and on the other hand, an infinitely small Pecklet number would mean that the fluid has no influence on the solid temperature distribution. Figure 7b), shows that while the period of the axial temperature distribution strongly differs in the three considered cases the oscillation amplitude does not display strong differences. Figure 7c) shows again the dimensionless temperature distribution at the interface for three different values of γ. As expected, a higher γ value yields a temperature distribution closer to the boundary condition profile imposed at η = σ. Finally Figure 7d) shows the temperature profiles at different η along the channel. For all η values, the progressive alignment of the numerical solution with the analytical solution can be observed i.e the progressive establishment of the developed thermal profile. As a conclusion, all figures show an excellent agreement with the analytical solutions. As for the 2D case, the continuity of temperature between the solid and the fluid can be distinctly observed on Figure 8 for the 3D problem. Figure 9 further confirms the consistent behavior of the numerical solution. As expected, a higher Pe yields a higher temperature amplitude at the interface, a higher periodicity at the boundary impacts the frequency at the interface without impacting the amplitude, a lower thermal conductivity ratio gives a lower temperature at the interface as well as a bigger solid thickness which allows more temperature diffusion through the solid. The numerical solutions are in excellent agreement with the analytical solutions. Figure 4 a) 3D surface mesh view, b) 2D mesh view Figure 5 Numerical solution: temperature distribution in the fluid solid domain for.

9 Figure 6 Dimensionless temperature: Qualitative comparison between the numerical and analytical results for the dimensionless temperature at the fluid-solid interface for. a) b) c) d) Figure 7 2D case : dimensionless temperature distribution between the analytical solution (in Blue) and the numerical solution: a), b) c) in the hydrodynamically and thermally developed region at the fluid solid interface for a) and, b) and, c) and d) Starting from the inlet with and

10 Figure 8 Numerical solution: temperature distribution in the fluid solid domain for. a) 3D cut view, b) Channel cut view Figure 9 3D case: dimensionless temperature distribution between the analytical solution (in Blue) and the numerical solution: a), b) c) d) in the hydrodynamically and thermally developed region at the fluid solid interface for and, b) and, c) and, d) and. 4- Further applications for the conjugate heat transfer solver As seen previously, the ICFD solver can be used to solve conjugate heat transfer problems involving flow in pipes and cooling of systems and structures. Figure 10 features an example where the solver is being used to solve a stamping application. A workpiece gets stamped against the die, and the fluid flowing through a snake shaped pipe through the die is responsible for the subsequent cooling of both the workpiece and the die. Figure 11 shows another example of a problem currently under investigation where the conjugate heat transfer solver is being used to solve a coupled problem involving the thermal and the

11 EM solvers. Due to the Electromagnetic Joule heating, a sparse coil gets heated up and the flow running through the center of it is used to cool it down in order to prevent it from melting. Figure 10 Stamping application, temperature fringes: a) Initial state with hot workpiece b) Workpiece moved to the right and pressed against the die. Coolant liquid flowing through the die active. c) Workpiece moved back to initial position. Cooling of the die and the workpiece. D) Final state, cooled workpiece and die. Figure 11 Coupled EM-thermal-ICFD application: Cooling of coils used for induced heating applications

12 References [1] A. B. a. E. D. S. a. G. C. a. P. D'Agaro, «Conjugate forced convection heat transfer in a plane channel : Longitudinally periodic regime,» International Journal of thermal Sciences, n 47, pp , [2] M. R. A. a. N. Tabari, «Two dimensional analytical solution of the laminar foced convection in a circular duct with periodic boundary condition,» Journal of thermodynamics, 2012.

13 LS-DYNA R7: Coupled Multiphysics analysis involving Electromagnetism (EM), Incompressible CFD (ICFD) and solid mechanics thermal solver for conjugate heat transfer problem solving Iñaki Çaldichoury (1) Pierre L'Eplattenier (1) Facundo del Pin (1) Miro Duhovic (2) (1) Livermore Software Technology Corporation 7374 Las Positas Road Livermore, CA (2) Institut für Verbundwerkstoffe GmbH, Erwin-Schrödinger-Str., Gebäude Kaiserslautern, Germany Abstract LS DYNA R7 s new modules and capabilities include: two fluid mechanics (CFD) solvers for incompressible (ICFD) and compressible flows (CESE) and an Electromagnetism solver (EM). The objective of these solvers is not only to solve for their particular domain of physics but to make full use of LS-DYNA capabilities and material library in order to solve coupled multiphysics. This paper will present how the EM solver can solve inductive heating problems, the problematic that arises when cooling the heated materials and/or coils is needed and how the ICFD solver can be used in conjunction in order to solve the complete EM-conjugate heat transfer problem. For illustration purposes, an industrial application studied at the Institut für Verbundwerkstoffe (See Advances in simulating the processing of materials by electromagnetism induction paper) will be introduced and discussed.

14 1- Introduction LS-DYNA R7 double precision includes an Electromagnetics (EM) solver as well as a CFD solver for incompressible flows (ICFD). The objective of these new solvers is not only to solve for their particular domain of physics but to make full use of LS-DYNA capabilities and material library in order to solve coupled multiphysics problems. So far, several applications involving the EM solver coupled with either the solid mechanics solver or the thermal solver or both have been solved and presented in [1], [2] or [3]. On the other hand, the ICFD solver is also capable of solving Fluid Structure interaction problems (FSI) or conjugate heat transfer problems as was shown in [4], [5] and [6]. The current paper aims to bring the multiphysics concept to the next step by solving an industrial application involving Electromagnetic heating and fluid cooling. Therefore, both the EM and the ICFD solvers will be used and coupled through the LS- DYNA thermal solver. This paper will begin by introducing the induced heating capabilities of the EM solver and the conjugate heating capabilities of the ICFD solver. It will then proceed by introducing the background of the industrial application studied and finally show the numerical model and some first results when compared to the experiment. 2- The EM solver for induced heating applications The Electromagnetism (EM) module solves the Maxwell equations in the Eddy current (induction-diffusion) approximation [1]. This is suitable for cases where the propagation of electromagnetic waves in air (or vacuum) can be considered as instantaneous which is the case in most industrial magnetic metal welding, forming or inductive heating applications. The EM solver is coupled with the structural mechanics solver (the Lorentz forces are added to the mechanics equations of motion), and with the structural thermal solver (the Ohmic heating is added to the thermal solver as an extra source of heat) thus allowing the simulation of moving coils and the heating or deformation of work pieces. The EM fields are solved using a Finite Element Method (FEM) for the conductors and a Boundary Element Method (BEM) for the surrounding air/insulators. Thus no air mesh is necessary. Among its various features, the EM solver in LS-DYNA includes an inductive heating solver. It was introduced in order to solve the computer cost issue arising when high frequency currents, thus very small time steps, were combined with long simulation runs (typically, an AC current with a frequency ranging from khz to MHz and a total time for the process in the order of a few seconds). The induction heating solver works the following way: it assumes a current which oscillates very rapidly compared to the total time of the process. The following assumption is made: a full eddy-current problem is solved over two full periods with a "micro" EM time step, see Figure 1. An average of the EM fields during the full period as well as the joule heating is computed. It is then assumed that the properties of the material (heat capacity, thermal conductivity, magnetic permeability) and mostly the electrical conductivity which drives the flow of the current and the joule heating do not change for the next periods of the current within the macro EM time step chosen. As all the properties are largely temperature dependent, the assumption can therefore be considered accurate as long as the temperature does not change too much. During these periods, no EM computation is performed; only the averaged joule heating term is added to the thermal solver. However, as the temperature and thus the electrical conductivity changes together with all the other material properties mentioned, the EM fields need to be updated accordingly so another full eddy current resolution is computed for a full period of the current giving new averaged EM fields (introducing a macro EM time step). In this way the solver can efficiently solve inductive heating problems for both the cases of a static or moving coil. In the present paper, it will be used in order to simulate the heating of a coil.

15 Figure 1 Graphical representation of the electromagnetic field and resulting joule heating calculation scheme implemented in the LS-DYNA inductive heating solver

16 3- The ICFD solver for conjugate heat transfer applications Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy and heat between physical systems. By solving the heat equation, the ICFD solver offers the possibility to solve and study the behavior of temperature flow in fluids. Potential applications are numerous and include refrigeration, air conditioning, building heating, motor coolants, defrost or even heat transfer in the human body. In the current paper, it will be used in order to solve the coil cooling. For the thermal coupling between the heat equation solved by the thermal solver in the structure and the heat equation solved in the fluid by the ICFD solver, a monolithic approach has been adopted. The coupling between the structure and the fluid is therefore very tight and strong at the fluid-structure interface. The resulting full system includes both the structural and the fluid temperature unknowns (See Figure 2) and is solved using a direct solver which is very robust but may in some cases be computer-time consuming. Figure 2 Vector of temperature unknowns when the fluid thermal solver and structure thermal solvers are coupled using a monolithically approach.

17 4- Joining and welding applications using induced heating As the requirements for lightweight designs keep increasing in the automotive industry, the use of thermoplastic based composite materials for welding and joining during the vehicle body construction will become more and more frequently encountered (See Figure 3). The technology to provide quick and efficient joining by robotic means for both metal to composite and composite to composite thermoplastic parts has been in development at the Institute für Verbundwerkstoffe (IVW) over the last 15 years, in the form of robotic composite welding through electromagnetic induction. Indeed, it is now known that in woven carbon fiber reinforcement structures, eddy current joule heating can produce enough heat in the composite to allow for thermal bonding to occur without the use of susceptors. With the correct selection of electromagnetic, thermal and mechanical parameters the composite can be melted precisely in a small volume of material close to the bond line so that the two parts can be joined effectively without any detrimental effects of overheating and deconsolidation. In order to study different parameter set ups and in order to predict the heating of workpieces by moving coils, the EM solver is being used. An example of such a study can be seen in Figure 4. Further results and examples of the uses of the EM solver for solving such induced heating applications can be found in [7] and [8]. The only physics currently missing from the current numerical simulations is fluid dynamics and the cooling of the coil. Indeed, during the experimental set-up, a cold water flow passes through the sparse coil in order to maintain the coil temperature at acceptable levels. Without this cooling process, the coil would certainly melt. It is therefore of particular interest to be able to simulate and correctly predict the behavior of the cooling process in order to prevent potential damages to coils and work tools. Figure 3 Examples of automotive components that could or have already benefitted from induction welding (left). Induction welding Kuka robot at the Institute für Verbunwerkstoffe (right) joining the two thermoformed components of the BMW M-series front bumper beam

18 Figure 4 Graphical representation of the electromagnetic field and resulting joule heating calculation scheme implemented in the LS-DYNA inductive heating solver 5.1 EM-thermal simulation 5- The coupled EM-thermal-ICFD model For this work, a pancake coil geometry will be used with an approximate radius of 0.05 m. The tube diameter of the coil is approximately 8 mm and its sparse inner radius (through which the coolant is supposed to flow) is of 6mm. Its material is copper and its structural, thermal and electrical properties are given in Table 1. The mesh size considered through the thickness of the coil is approximately 0.6 mm (See Figure 5). During the preliminary tests, an EM-thermal only problem was run with a current frequency of 289 khz and an Amplitude of A. The influence of temperature on the electrical conductivity is taken into account. As could be expected, the maximum coil temperature rises quickly and reaches over 1085 C at its central area after only 15 seconds which is the coil s melting point (See Figure 6). This justifies the use of a water flow though the coil for cooling purposes. Table 1 Coil physical parameters Density Heat Capacity Thermal Conductivity Electrical Conductivity 8960 kg/(m^3) 385 J/(Kg.K) 390 W/(m.K) 5.998e7 S/m

19 Figure 5 Coil geometry and mesh. a) Global view, b) Zoom on inner section Figure 6 EM-thermal only maximum temperature increase after 15 seconds at A 5.2 Adding ICFD coupling For the ICFD model, the mesh size used will be approximately 0.6 mm with four elements added in the anisotropic direction of the boundary layer which results in about 10 elements through the thickness of the pipe and 1.5 M elements in the fluid volume mesh (See Figure 7). The inlet velocity corresponds to a flow rate of 1 l/min with a temperature of 23 degrees. The thermal properties of the liquid will be those of water. The water flow will be directed from the center outwards. We will work under the assumption that the flow is laminar. The effects of the coolant can be quickly observed as the coil s temperature now reaches equilibrium instead of continuously rising. In Figure 8a), it can be observed that the correct representation of the boundary layer is of tremendous importance since the temperature diffusion gradient in the fluid seems to be located primarily in that zone. The good behavior of the conjugate heat transfer solver can be further noted when comparing the temperature fringes between the solid and the fluid that are consistent at the interface even if the geometries do not match perfectly (See Figure 8 b)). Qualitatively, it can be observed that the temperature is at its highest point at the center of the coil. This is consistent with the experimental

20 images extracted from a thermal camera that show a high temperature zone at the coil s center (See Figure 9). In the ICFD coupled model, the EM solver is switched off after 8 second as steady state has been reached and therefore this source of heat is removed. Consequently and in accordance with the experimental behavior, the coil starts cooling. A comparison between an experimental point taken in the hottest zone of the coil and its numerical equivalent shows the good agreement between the experiment and the simulation (See Figure 10). As a conclusion, it seems possible to track down the effects of cooling on the coil s temperature through the use of the combined LS-DYNA EM-thermal-ICFD solvers. Further tests will be conducted at various current frequencies and amplitudes in order to confirm the observed consistent behavior. Figure 7 Fluid mesh: zoom on the inflow section Figure 8 Temperature fringes : a) Fluid only, zoom on central part. b) Fluid and structure zoom on coil section

21 Figure 9 Comparison for coil temperature between a) Experimental results from thermal camera, b) Numerical simulation results Temperature ( C) Time (s) Figure 10 Heating and cooling. Temperature behavior: comparison between experimental point p1 and equivalent numerical location

22 References [1] I. Çaldichoury et P. L'Eplattenier, «Update On The Electromagnetism Module In LS-DYNA,» chez 12th LS-DYNA Users Conference, Detroit, [2] I. Çaldichoury and P. L'Eplattenier, "Simulation of a Railgun: A contribution to the validation of the Electromagnetism module in LS-DYNA v980," in 12th International LS-DYNA Users Conference, Detroit, [3] I. Çaldichoury and P. L'Eplattenier, "Validation Process of the Electromagnetism(EM) Solver in LS- DYNA v980 : the TEAM test cases," in 12th International LS-DYNA Users Conference, Detroit, [4] F. Del Pin et I. Çaldichoury, «LS-DYNA R7 : Recent Developments, Application Areas and Validation Process of the Incompressible CFD solver in LS-DYNA,» 12th LS-DYNA Internation conference, June [5] I. Çaldichoury et F. Del Pin, «LS-DYNA R7 : Conjugate heat transfer problem : coupling between the Incompressible CFD (ICFD) solver and the thermal solver, applications, results and examples,» 9th European LS-DYNA Conference, June [6] F. Del Pin et I. Çaldichoury, «LS-DYNA R7 : Strong Fluid Structure Interaction (FSI) capabilities and associated meshing tools for the incompressible CFD (ICFD) solver, applications and examples,» 9th European LS-DYNA Conference, June [7] M. Duhovic, L. Moser, P. Mitschang, M. Maier et al, «Simulating the Joining of Composite Materials by Electromagnetic Induction,» 12th International LS-DYNA conference, June [8] M. Duhovic, P. Mitschang, M. Maier et al, «Advances in Simulating the Processing of Composite Materials by Electromagnetic Induction,» 9th European LS-DYNA Conference, June 2013.

23 LS-DYNA R7: Strong Fluid Structure Interaction (FSI) capabilities and associated meshing tools for the incompressible CFD solver (ICFD), applications and examples. Facundo Del Pin Iñaki Çaldichoury Livermore Software Technology Corporation 7374 Las Positas Road Livermore, CA Abstract LS-DYNA version R7 includes CFD solvers for both compressible and incompressible flows. The solvers may run as standalone CFD solvers or they could be coupled to the LS-DYNA solid mechanics and thermal solvers for fluid structure interaction (FSI) and conjugate heat transfer problems. This paper will focus on the Incompressible CFD solver in LS-DYNA (ICFD) and its Fluid-solid interaction capabilities (FSI). Fluid structure interaction problems occur in physics whenever the flow over a structure causes deformation or displacement which in turn may influence the way how the fluid behaves. One of the solver s main features is the implementation of a robust strong FSI coupling which opens a wide new range of applications in the range of aerodynamics, hydrodynamics, hemodynamics and so forth. Several examples will be provided for illustration and discussion. The ICFD solver is the first in LS-DYNA to make use of a new volume mesher that takes surface meshes bounding the fluid domain as input. For FSI problems that involve big displacements, the volume mesher algorithms need to be robust and flexible. Some of the latest developments and mesh control tools that are made available for the user will therefore also be introduced. _

24 1- Introduction Fluid structure interaction problems occur in physics whenever the flow over a structure causes deformation which in turn, may influence the way the fluid behaves. In order to accurately representing these interactions, the fluid part and the structural part evolve in a coupled system where fluid forces applied on the solid and solid displacements through the fluid interact at each time step. This opens a very wide range of analysis and applications such as the fatigue of airplane wings, airfoils or turbine blades for the aerospace and turbo machinery industry, the study of galloping, vibrations and fluttering of flaps, bridges or similar structures, the analysis of artificial heart valve openings or other prosthesis for the bio-medical industry and so on. LS-DYNA version R7 double precision includes the incompressible flow solver (ICFD) which can run as a stand-alone CFD solver or be coupled with the structural and thermal solvers of LS-DYNA. One of the solver's main features is the implementation of a robust strong FSI coupling that be triggered through the use of LS-DYNA's implicit solver for the solid mechanics part. The main focus of this paper will be a description of its mechanism and potential applications. 2- Types of FSI coupling Fluid-structure interaction problems involving an incompressible viscous flow and elastic non-linearstructure have been solved in the past using different methods. The monolithic approach considers the fluid and the solid as a single domain with the fluid and solid equations solved together in a coupled way. However, solving the pressure together with the rest of the unknowns (typically velocities or displacements) is too expensive from the computational point of view: the non-linear system to be solved is large and ill-conditioned with always non-defined positive matrices. A second approach would be to segregate the pressure from the velocity in the monolithic scheme. This would still imply that the fluid and solid equations are solved in a coupled way in the same system and would therefore require an implementation of the solid equations directly in the ICFD solver. This would not be a practical solution and would contradict the objective of the present solver which is to make fully use of LS-DYNA's mechanical solver capabilities in order to solve complex fluid structure interaction problems. A third approach would be by using a partitioned (or staggered) method ( [1], [2]) where the fluid and solid equations are uncoupled and that therefore allows using specifically designed codes on the different domains and offer significant benefits in terms of efficiency: smaller and better conditioned subsystems are solved instead of a single problem. It is the method adopted by the present solver in order to solve FSI problems. In the partitioned approach, two schemes are distinguished: loosely (or weakly) ([17]) coupled scheme or strongly coupled scheme ( [3], [4]). Both are available when using the ICFD solver in LS-DYNA. Loosely coupled schemes require only one solution of either field per time step in a sequentially staggered manner and are thus particularly appealing in terms of efficiency. However, they tend to become unstable when the "added-mass effect" is significant ( [5]). In fluid mechanics, added mass or virtual mass is the inertia added to a system because an accelerating or decelerating body must move some volume of surrounding fluid as it moves through it, since the object and fluid cannot occupy the same physical space simultaneously. However the name "added-mass effect" _

25 has been used in the literature to indicate the numerical instabilities that typically occur in the internal flow of an incompressible fluid whose density is close to the structure density. The added mass effect therefore does usually not occur in aero-elasticity problems as the solid's density is a lot higher than the fluid's density, but it becomes very important in several other applications such as bio-mechanics where the materials are normally muscles and arteries and the fluid is blood. Strongly coupled schemes require the convergence of the fluid and solid variables at the interface and give, after an iterative process, the same results as non-partitioned schemes. However, they are also subject to the "added mass effect" resulting in a non-convergence of the solution. Special stabilization techniques must therefore be developed in order to diminish its influence and aim for a wider range of applications for fluid structure interactions. Figure 1 gives a summary of the different FSI couplings possible, those implemented in the ICFD solver and the influence of the "added mass effect". Both loose and strong coupling are available to the user but in this paper the focus will be brought on the strong coupling only. Figure 1 Summary of the different FSI couplings and the problems induced by the added mass effect. 3- Strong FSI coupling in LS-DYNA R Resolution scheme Both the ICFD solver and the implicit solver for solid mechanics are two independent solvers that both usually use a distinct time step. However, when the FSI coupling is activated, the strong coupling mechanism will be automatically triggered. In that case, the smallest time step of the two solvers will be used in order to solve the coupled problem. During the initial resolution step, the ICFD solver will solve the fluid domain using the initial velocity and pressure conditions. The forces applied on the structure will then be applied on the solid and recognized as pressure load boundary conditions. The solid will use those results to solve its own solid domain. It will then give back the displacements of its boundary nodes to the fluid. The solid and the fluid then go into a convergence procedure (Newton Loop) where a certain number of iterations are conducted until a convergence criterion is reached. The two solvers are then in agreement regarding the solid fluid interface boundary conditions (pressure and displacements). After _

26 convergence is reached, the solver then proceeds to the next time step where this procedure is repeated. Figure 2 sums up this strong coupling mechanism. Figure 2 Strong FSI Interaction Scheme 3.2- Mesh handling In a CFD only analysis, the moving reference frame is fixed in space and a full Eulerian formulation is achieved. However, in cases of Fluid Structure Interaction (FSI) problems, the fluid boundary nodes between the solid and the fluid are Lagrangian and the whole fluid mesh domain deforms with the structure. This approach allows a strong and exact imposition of the solid boundary conditions on the fluid. _

27 The solid and fluid geometry must match at the interface but not necessarily the meshes. Figure 3 illustrates the mesh behavior on a FSI case. The ICFD solver uses an automatic volume mesh generator for the fluid domain. However, robust tools are necessary in order to cope with the severe mesh deformations and distortions that can occur in FSI cases due to the structure s displacements. By default, the solver will apply a remeshing of the entire fluid domain when an element gets inverted. The user has also the possibility to force a remeshing if elements that are too distorted are detected. Further advanced meshing tools are made available to the user. For instance the user can trigger an automatic remeshing where the solver will use an a-posteriori error estimator to compute a new mesh size bounded by the user to satisfy a maximum perceptual error based on the work by [6]. For every nodal value, it is therefore possible to associate an error value. As a result, if too many nodes reach the critical value set by the user, the solver will apply a remeshing and refine in the zones where the error was high. Figure 4 gives an example where this option is being used. Figure 3 NACA airfoil profile oscillating in fluid. Incoming flow velocity from left boundary Figure 4 Incoming flow from the left boundary causing the bending of elastic flag and adaptive remeshing in zones of interest 4- Validation test case: the Turek problem In this example we consider the flow in a 2D channel past a cylinder with an attached elastic "flag". This is the FSI benchmark problem proposed by Turek and Hron [7]. This problem combines the two singlephysics problem of the flow past a cylinder with a "flag" and the deformation of a finite-thickness cantilever _

28 beam due to its interaction with the fluid. It is a challenging problem as it tackles some of the main issues when dealing with fully coupled FSI problems namely large mesh deformations and the fluid s density being close to the density of the structure which usually results in critical instabilities. Figure 5 offers a sketch of the problem with and with the following cylinder center coordinates: (x=0.2,y=0.2). A mesh size of will be used resulting in a element mesh. Table 1 gives a summary of the parameters of three FSI cases that can be run. A strong FSI coupling is mandatory in order to solve the cases where the fluid and the solid s densities are equal. Table 2 shows the result for the FSI1 case regarding the lift and drag values. In that case the flag's displacements are very small and hard to reproduce which explains that the lift is slightly different from the reference result. For the FSI2 and FSI3 cases, periodic solutions can be observed with the largest deformation occurring in the FSI2 case as can be observed on Figure 6. Frequent re-meshing is needed. The oscillation period is of 0.5 and 0.19 for the FSI2 and FSI3 cases respectfully which is in good agreement with the results of [7]. Figure 7, Figure 8, and Table 2 show the good agreement between the present analysis and the reference results. It is interesting to observe that a better agreement is found in the FSI3 case regarding the lift values. One possible explanation could be that the wall effects become more important in the FSI2 case. Table 1 Parameter settings Figure 5 Test case sketch FSI 1 FSI 2 FSI 3 Solid Density Poisson Coefficient Young Modulus 1.4e6 1.4e6 5.6e6 Fluid Density Fluid Viscosity Average inflow velocity Reynolds number _

29 Figure 6 Flag deformation in the FSI2 case Figure 7 FSI 2 Lift Drag and tip displacements _

30 Figure 8 FSI 3 Lift, Drag and tip displacements Table 2 Comparison between current analysis and reference results by [7] X-displacements ( 10e-3) Y-displacements ( 10e-3) Lift Drag FSI 1 Present analysis Reference results FSI 2 Present analysis ± ± ± ±80.91 Reference results ± ± ± ±73.75 FSI 3 Present analysis -2.76± ± ± ±33 Reference results -2.69± ± ± ± Further application examples 5.1- Non-inertial reference frames for wind turbine applications One of the features of the solver is to define a rotating non-inertial reference frame. The reference frame considered here is rotating at a constant speed and thus it is undergoing acceleration with respect to the stationary frame. This rotating frame is then called a non-inertial reference frame. Potential applications include wind turbines, turbomachinery and every other rotating problem that admits a steady state solution. _

31 This feature was used in combination with the strong coupling FSI feature in order to study the deformation and torque of the wind turbine blades. The geometry used was similar to the one by [8] and the results were consistent and of the same order of magnitude (See Figure 9). Figure 9 Pressure isosurface in the non-inertial reference frame and torque applied to the blades 5.2- Heart valve opening and closing Another important application for the strong FSI coupling feature are heart valves and hemomechanics. The example shown on Figure 10 is currently under investigation in collaboration with Hossein Mohammadi of McGill University. It features an artificial heart valve with three leaflets. The three leaflets are initially shut but then open under the blood pressure. They are then forced shut again but a very strong counter pressure (See Figure 11). The complexity of this lies in the very low Young s modulus of the leaflets, their low density compared to the blood s density and their strong and rapid displacements which implies strong re-meshing capabilities. The blood could potentially be represented by a non- Newtonian material by using the Power law model for Non-Newtonian fluids included in the solver. _

32 Figure 10 Heart valve: Von Mises stresses and isosurfaces of pressure Figure 11 Blood pressure gradient applied between the inflow and the outflow of the arterial heart valve _

33 References [1] N. M, T. S, W. W et R. E, «Robustness and efficiency aspects for computational fluid structure interaction,» Computational Science and High Performance Computing II, pp , [2] P. S et F. C, «Partitioned procedures for the transient solution of coupled aeroelastic problems,» Comp. Meth Appl. Meth. Eng., vol. 190, pp , [3] F. C, C. K et F. C, «Partitionaed analysis of coupled mechanical systems,» Comp. Meth. Appl. Mech. Eng., vol. 190, pp , [4] D. W et P. D, «A computational framework for fluid-structure interaction element formulation and applications,» Comp. Meth Appl. Mech. Eng., vol. 195, pp , [5] C. P, F. G. J et N. F, «Added-mass effect in the design of partitioned algorithms for fluid structure problems,» Comp. Meth. Appl. Mech. Eng., vol. 194, pp , [6] Z. O et Z. J, «A simple error estimator and adaptive procedure for practical engieering analysis,» International Jounral for Numerical Methods in Engineering, vol. 24, pp , [7] T. S et H. J, «Proposal for Numerical Benchmarking of Fluid Structure Interaction between an elastic object and a laminar incompressible flow,» bungartz & m. schaefer vol 53, _

34 LS-DYNA R7: Recent developments, application areas and validation results of the compressible fluid solver (CESE) specialized in high speed flows. Zeng-Chan Zhang Iñaki Çaldichoury Livermore Software Technology Corporation 7374 Las Positas Road Livermore, CA Abstract LS-DYNA version R7 includes CFD solvers for both compressible and incompressible flows. The compressible flow solver is based on the CESE method, a novel numerical method for solving conservation laws. It has many nontraditional features such as space-time conservation, second order accuracy for flow variables and a powerful shock wave capturing strategy. This paper will focus on some advanced features of the solver namely its FSI capabilities. Several potential industrial applications will be presented such as airbag openings, piston type applications and turbomachines. Some results on high speed supersonic flows will also be presented for illustration and discussion purposes.

35 1- Introduction The new compressible fluid solve included in LS-DYNA R7 double precision is based upon the Space- Time Conservation Element and Solution Element Method (CESE). The CESE method, originally proposed in [1] and further developed in [2] and [3] is a novel integral numerical method for solving conservation laws and includes many nontraditional features such as: - Local and global space-time conservation of the solution which limits the diffusion of the solution and loss of precision. - Second order scheme for both flow variables and their spatial derivatives for better solution accuracy. - Novel supersonic shock capturing strategy which does not involve any Riemann solver resulting in less calculation costs. Furthermore, the objective of these new solvers included in LS-DYNA R7 is not only to solve for their particular domain of physics but to make full use of LS-DYNA s capabilities and material library in order to solve coupled multiphysics. Consequently, the CESE solver has been extended in LS-DYNA to solve fluid structure interaction problems (FSI). In such cases, the solid can be modeled as a classic Lagrangian solid mechanics part while the fluid flow is based on Eulerian frame. This current paper will begin by introducing the CESE scheme applied to a one-dimensional case for illustration purposes. Several validation cases will be shown. It will then proceed to the FSI capabilities and will expend on some validation cases and potential industrial applications. 2- The CESE resolution scheme 2.1 1D example For simplification purposes, we will consider a one dimension form of the 1D convection equation based on the work by [1] : With constant advection term. This results in the element spatial discretization shown in Figure 1a) with points ( ) at various spatial location and at a given time n ( ). The first step of the CESE scheme is to consider time as another additional spatial coordinate thus forming the two dimensional Euclidean space (See Figure 1b)). Let us now define the following current density vector in : It can then be shown that by using Gauss divergence theorem in the Space-time the convection-diffusion equation gives:, the integral form of

36 Where is the boundary of an arbitrary space-time region in, is the normal area of a surface element on and is the space-time flux of h leaving the region V through the surface element ds. It is therefore possible to build in an elemental volume where space and time are conserved locally and treated in a unified way. This will be the tenet for the construction of CE elements. For the moment, let us build a solution element as in Figure 1c) that represents the interior of the space time region center on a given point of coordinates ( ) in. The variations within that SE will be considered small enough so that the solution can be expressed by the following Taylor series expansion: The time and spatial derivatives can be related by using the flow convection-diffusion convection : Consequently, two unknowns, and its spatial derivative are left to be able to compute the solution anywhere within the space time region centered around ( ). In order to provide the two equations mandatory to close the system, Figure 1d) shows the two CEs that will be defined. The integral form of the conservation is then applied on those two CEs resulting in two equations which allow to solve the system and advance through time, resulting in the previously described nontraditional CESE features.

37 2.2 Stabilization methods Figure 1 CESE method : 1D resolution steps The previously described scheme is stable for inviscid flows with no discontinuities. However, for viscous flows and in order to solve shock wave flows, it is necessary to introduce some numerical diffusion for stabilization. Instead of using the two CE- and CE+ in order to solve the system for (or simply written ) and its spatial derivative (or ), the spatial derivative will be estimated by a weighing technique using and determined by the previous timestep solution (See SE- and SE+ in Figure 2). Compared to the exact resolution of a diffusive and thus stable solution is obtained. Only one unknown is left therefore only one CE is needed to solve. Figure 2 sums up the modified stabilized scheme. The user can choose between a combination of the central difference and another weighted expression in order to express function of and or a simple relaxing procedure. More detail can be found in the short theory manual available on the LSTC website.

38 2.3 Boundary conditions Figure 2 Weighting technique for stable calculation Several boundary conditions are available to the user. It is possible to impose pressure, density temperature and velocity or to define non-reflective boundaries, reflective boundaries or solid walls. At the domain boundaries, the solver will extend the mesh domain by one layer and use the conditions defined by the user as input in this new element layer. This will then be used by the neighboring elements for solving. Figure 3 shows an example for different boundary conditions. Non reflective boundary conditions are used in order to define far field boundary conditions. For the solid wall and reflective boundaries, the normal velocity component is defined in opposite direction to the incoming velocity such as to be exactly zero at the interface (free slip condition). On top of that, for the solid wall condition, the tangent component is defined in the opposite direction such as to be null at the interface (non-slip condition). For inviscid flows, the solid wall boundary condition acts similarly to the reflective boundary condition. 2.4 Validation problems Figure 3 Boundary condition implementation Several validation cases for the CESE numerical scheme can be found on the LSTC website. Figure 4, Figure 5 and Figure 6 show some results obtained for a shock wave diffraction around a corner, an incoming supersonic flow around a step (M>3) and the shock wave diffraction patterns forming behind a supersonic wedge (M>1.3).

39 Figure 4 Diffraction of a shock wave on a sharp corner. Comparison with experimental results [4]. Figure 5 Incoming supersonic flow against a step. Superposition of shock wave patterns with results by [5]

40 Figure 6 Density isocontours forming around a supersonic wedge. Comparison with experimental pictures. 3- Fluid Structure Coupling (FSI) 3.1 Resolution method The CESE solver can be coupled with the solid mechanics LS-DYNA solver in order to solve fluid structure interaction problems. Since both solvers admit their own CFL condition on the timestep, the most constraining one from both domains will be used in FSI problems. This way, the fluid and structural solvers advance simultaneously in time. At each timestep, the fluid solver will communicate pressure forces on the structure that will act as exterior loads while the structure will give back its displacements and updated nodal velocities (See Figure 7). The structure is immersed in the fluid domain. Therefore both meshes are independent and the interface will be automatically tracked by the solver. Figure 8 features a 2D example with a structural beam moving through a 2D fluid mesh with a velocity. After the structural solver has communicated the nodal positions of the solid at time, the first step would be for the CESE solver to track which fluid elements are closest to the structure and perform a sorting procedure (shown in light green in Figure 8). Then, in order to calculate the solution of those elements, the neighbors that are blocked by the solid will be treated as solid wall boundary conditions. For example, in Figure 8, the fluid element S1 sees two solid wall neighbors while S2 only sees one. Finally, a searching procedure based on the fluid mesh size is used in order to determine which fluid elements are close to the solid element and an average of the pressure values will then be computed and transferred to the structural solver acting as an exterior load (in light pink in Figure 8). The solver will automatically know on which side of the solid face those fluid elements are located by using the solid element normal. This way, no leakage can occur. In order for this searching procedure to be able to correctly capture neighboring fluid elements, it is advised to use a finer mesh for the fluid than for the solid.

41 Figure 7 FSI resolution 3.2 Validation problem Figure 8 Solid-fluid Interface tracking In order to validate the FSI algorithm, the piston problem described by [6] will be considered. This case features a gas contained in a 1D chamber closed on its right hand side by a moving piston and on its left by a fixed wall (spring-back system). For the purpose of this test case, the 1D problem will be moved to the equivalent 3D problem. The piston is of mass, rigidity, unstreched length, at rest under pressure length and the piston displacement is. The piston is initially loaded so that at initial time it will compress the gas chamber. The reflective pressure wave will then push back the piston thus triggering the spring-back FSI problem. The objective of this test case is to study the piston's response and interaction with the fluid by looking at its displacements function of time. The gas chamber is at initial pressure and density and. Since the structure needs to be fully immersed in the fluid, the fluid properties outside the chamber need also to be considered. Atmospheric conditions will be used here.

42 The parameters chosen will be taken from [6] and the case of will be studied. The mesh size will also be chosen in order to match the reference simulation by [6] i.e 0.1 m in the X-direction for the reference length (See Figure 10). Figure 11a) shows the different pressure isocontours at a given time t during the gas compression. Figure 11b) shows the oscillation response of the piston function of time. The results are in good agreement with [6] regarding oscillation frequency and amplitude. It is to be noted that the present simulation offers slightly more damping effects. This is due to the fact that the reference simulation did not consider the gas outside the chamber and its interaction with the piston. The CESE solver also includes moving mesh capabilities which can be applied to this piston problem. While this feature will be detailed during the presentation, it will not be further described in this paper. Figure 9 Piston case sketch Figure 10 Test case mesh

43 Figure 11 a) Pressure isosurface during piston compression. b) Piston displacements function of time 3.3 Further industrial applications A few examples involving the CESE scheme and FSI include supersonic inflows or strong pressure waves causing structure deformation of displacement (See Figure 12), Airbag openings (See Figure 13) and transonic flows around turbomachines (See Figure 14). Figure 12 FSI case of a shell being blown away by supersonic flow

44 Figure 13 Airbag opening. Courtesy of TAKATA Corporation Figure 14 FSI case involving transonic flow around turbomachine

45 References [1] S. Chang, «The Method of Space Time Conservation Element and Solution Element-A new approach for solving the Navier Stokes and Euler equations,» Journal of Computational Physics, vol. 119, p. 195, [2] Z. Zhang, S. Chang et S. Yu, «A Space time Conservation Element and Solution Element for Solving the two and Three Dimensional Unsteady Euler equations using Quadrilateral and Haxagonal Meshes,» Journal of Computational Physics, vol. 175, pp , [3] S. Chang, «Courant Number insensitive CESE Schemes,» AIAA paper, [4] T. O.I.K, «Shock wave diffraction around a 90 degree sharp corner,» chez 18th ISSW, [5] W. P et C. P, «The numerical simulation of two dimensional fluid flow with strong shocks,» Journal of Computational Physics, vol. 54, n % , [6] E. Lefrançois et J.-P. Boufflet, «An Introduction to Fluid Structure Interaction : Application to the piston problem,» Society for industrial and applied mechanics, vol. 52, n %14, pp , [7] I. Çaldichoury et P. L'Eplattenier, «Update On The Electromagnetism Module In LS-DYNA,» chez 12th LS-DYNA Users Conference, Detroit, 2012.

46 LS-DYNA R7: Update On The Electromagnetism Module (EM) Pierre L'Eplattenier Iñaki Çaldichoury Julie Anton Livermore Software Technology Corporation 7374 Las Positas Road Livermore, CA Abstract An electromagnetism module is being developed in LS-DYNA version R7 double precision for coupled mechanical/thermal/electromagnetic simulations. The physics, numerical methods and capabilities of this module will be introduced. Some examples of industrial applications will be presented. These include magnetic metal forming, bending and welding in different configurations, high pressure generation for equation of state studies and material characterization, induction heating, resistive heating, short circuits due to crashes, electromagnetic launchers, ring expansions, magnetic levitation and so forth. Additionally, magnetic material capabilities are currently available for beta testing and will also be discussed in this paper.

47 1- Introduction An electromagnetism (EM) module is under development in LS-DYNA in order to perform coupled mechanical/thermal/electromagnetic simulations [1], [2].This module allows us to introduce some source electrical currents into solid conductors, and to compute the associated magnetic field, electric field, as well as induced currents. These fields are computed by solving the Maxwell equations in the eddy-current approximation. The Maxwell equations are solved using a Finite Element Method (FEM) [3], [4] for the solid conductors coupled with a Boundary Element Method (BEM) [5] for the surrounding air (or insulators). Both the FEM and the BEM are based on discrete differential forms (Nedelec-like elements [4], [6]). The solver exists in Serial and MPP versions and is available in the R7 double precision release. Electromagnetic Metal Forming (EMF) has historically been the main application of this solver, but newer developments now allow other processes that could be simulated. These include induced heating, resistive heating with the possibility of added of contact resistances, the coupling of a time dependent external field with conductors and so forth. This paper will describe the main physics of the solver along with application examples. It will then focus on the current state of development of magnetic material capabilities. 2-1 Presentation of the physics 2- The Electromagnetics Solver The Electromagnetic solver focuses on the calculation and resolution of the so-called Eddy currents and their effects on conducting pieces. Eddy current solvers are also sometimes called induction-diffusion solvers in reference to the two combined phenomena that are being solved. In Electromechanics, induction is the property of an alternating of fast rising current in a conductor to generate or induce a voltage and a current in both the conductor itself (self-induction) and any nearby conductors (mutual or coupled induction). The self-induction in conductors is then responsible for a second phenomenon called diffusion or skin effect. The skin effect is the tendency of the fast-changing current to gradually diffuse through the conductor s thickness such that the current density is largest near the surface of the conductor (at least during the current s rise time). Solving those two coupled phenomena through the Maxwell equations allows calculating the electromechanic force called the Lorentz force and the Joule heating energy which are then used in forming, welding, bending and heating applications among many others. Let Ω be a set of multiply connected conducting regions. The surrounding insulator exterior regions will be called Ω e. The boundary between Ω and Ω e is called Γ, and the (artificial) boundary on Ω at the end of the meshing region (hence where the conductors are connected to an external circuit) is called Γ c. In the following, we will denote n as the outward normal to surfaces Γ or Γ c. The electrical conductivity, permeability and permittivity are called σ, µ and ε respectively. In Ω e, we have σ = 0 and µ = µ 0. We solve the Maxwell equations in the so-called low frequency or eddy-current approximation, which is valid for good enough conductors with low frequency varying fields such that the condition ε 0 E t σe, where E is the electric field, is satisfied [1]. When using a vector potential A and scalar potential Φ representation and using the Gauge condition σa = 0 [7], we end up with the following system to solve [1], [2] : And : σ Φ =0 (1)

48 where ȷ s is a divergence free source current density. σ A t + 1 μ A + σ Φ = ȷ s (2) Equations (1) and (2) are projected over Nedelec-like basis functions, resulting in the linear systems: S 0 (σ)φ = 0 (4) M 1 (σ) a t + S1 1 μ a = (5) D01 (σ)φ + Sa where A and Φ have been decomposed on their corresponding 1-for and 2-for basis functions a and Φ and S 0, S 1, M 1 and D 01 are FEM matrices (see [1], [6] and for details). The outside term SA is solved using a BEM method [5], for which details can be found in [1]. 2-2 Coupling with the mechanics and thermal Once the EM fields have been computed, the Lorentz force F = ȷ B is evaluated at the nodes and added to the mechanical solver [2]. The mechanical and electromagnetic solvers each have their own time step. For a typical EMF simulation, the mechanical time step is about 10 times smaller than the electromagnetic one. At this point, the explicit mechanical solver of LS-DYNA is used when coupled with electromagnetism. The mechanical module computes the deformation of the conductors and the new geometry is used to compute the evolution of the EM fields in a Lagrangian way. The Joule heating power term j2 is added to the thermal solver which uses its own time step to update the σρ temperature. Several thermal models are available, isotropic, orthotropic, isotropic with phase change and so forth. The temperature can be used in turn in an electromagnetic equation of state to update the electromagnetic parameters, mainly the conductivity σ. At this time, a Burgess model [8] has been introduced, as well as a simpler Meadon model, and a tabular model where a load curve defines σ versus the temperature [2]. 2-3 Coupling with external circuits The eddy current solver can be coupled with one or several external circuits. At this point, each circuit can be either an imposed current through a segment set, where the current versus time is defined by a load curve; an imposed voltage drop between two segment sets, again with a load curve defining the voltage versus time; or an R, L, C circuit. Imposed currents are dealt with global constraints on the BEM system, whereas imposed voltages or R, L, C are taken care of with Dirichlet constraints on the scalar potential Φ in the FEM system [1], [2]. Segment sets, through which the flux of the current density versus time is computed, can be used as Rogowski coils. It is also possible as an option, to apply an external field on a given conductor. This feature can for example be useful in cases where the user knows or has a good idea of the magnetic field generated by a coil on a workpiece. This way, the whole coil does not need to be modeled which can save a lot of calculation time. 2-4 Examples of applications

49 Figure 1 shows an example of an industrial application for metal forming. This study conducted in collaboration with M. Worswick and J. Imbert from the University of Waterloo, Ontario Canada, features a metal sheet undergoing plastic deformation and being forced on a conical die (only 1/2 of the die and the work piece are represented) by strong magnetic forces generated by the coil s high density magnetic field and the induced currents of the workpiece. The main objective of this study was to predict the final shape of the metal sheet. Details on the experimental setup as well as experiment/simulations comparisons can be found in [1]. Figure 3features an example of an industrial application of the inductive heating solver. This solver is being used in induced heating problems involving larger time scales (typically a few seconds) and very high frequency currents. In this test case, a steel plate is moving at a constant velocity while being heated by a set of coils. More details about the inductive heating solver as well as industrial applications can be found in [9]. For cases where induction-diffusion effects can be neglected a special solver called the resistive heating solver can be used. Since no BEM effects need to be calculated, this solver is very fast and very high time steps can be used. In such cases, no Lorentz forces are computed but Joule heating is still taken into account. One application example would be the study of short-circuits in car batteries due to crash or impacts. A contact capability has been introduced in the EM module to handle electromagnetism contact between two conductors. One on the applications of this new capability is rail-gun simulations. In a rail gun, the electromagnetic forces created by an electrical current are used to accelerate a projectile between two conductor rails at supersonic speeds, as shown on Figure 5. The contact capability allows simulating the sliding contact between the rails and the projectile. Other industrial applications for this feature include electromagnetic welding cases where two conductive metal pieces come into brutal contact with each other as shown in Figure 6. More details about rail gun simulations can be found in [11]. Figure 1 Magnetic metal forming example where an Al plate is formed against a conical die by EM forces

50 Figure 2 Temperature fringes of an inductive heating example where a plate moves through and is being heated by five coils Figure 3 Rail gun model: Current flowing between rails and projectile generates magnetic field (fringes) and Lorentz forces that accelerate the projectile

51 Figure 4 Current density fringes of a Welding test case between two metal pieces. Conducted in collaboration with the University of Waterloo, Canada 3-1 Magnetic materials 3- Magnetic material capabilities So far all the conductors were considered nonmagnetic materials. This means that their permeability is considered equal to the vacuum permeability (μ 0 = μ material ). Certain type of conductors exhibit magnetization behavior in response to an applied magnetic field. Such materials are called magnetic materials. Magnets are a special case of magnetic materials where no source magnetic field is needed to reach a magnetized state. The degree of magnetization that a material obtains in response to an applied magnetic field is expressed represented by μ with: B = μ H where B is the magnetic flux density and H the magnetic field intensity. For nonmagnetic materials μ is equal to μ 0 which is the permeability of free space i.e a measure of the amount of resistance encountered when forming a magnetic field in a classical vacuum. For magnetic materials however, μ is different from the vacuum permeability μ 0. Magnetic materials can be further divided into linear magnetic materials (paramagnets, diamagnets) and nonlinear magnetic materials (ferromagnets) (See Figure 5). In practice magnetic materials can be encountered in applications such as generators, motors or flux concentrators. In some cases, the magnetization process is very fast compared to the diffusion of the Eddy Currents. In order to save calculation time, it is therefore interesting to consider the solver in an already initially magnetized state or steady state. Consequently, there are currently two new features under development in the EM solver: the implementation of magnetic materials where the whole transient magnetization process is solved as well as a so called magneto-static solver where the magnetic materials would be directly considered in a magnetized state.

52 Figure 5 μ_p for paramagnetic materials, μ_d for diamagnetic material, _f for ferromagnetic material 3-2 Numerical issues One difficulty with non-linear materials is that the numerical system becomes non-linear, i.e. the matrices in (5), namely S 1 1, depends on the solution through µ. In an eddy current problem, if the time μ step is sufficiently small, one can build the matrices using the solution of the previous time step and solve a linear system. However, when the time step becomes too large, iterations on the non-linearity are needed. Another difficulty which arises for eddy current problems with large time step and even more for magnetostatics is that the system (5) becomes singular which requires special treatment. Finally, still for large time step eddy current or magnetostatic problems, the traditional method used by the EM solver with iterations between the FEM and BEM systems (so called preconditioned Richardson s iterations) does not converge anymore since the spectral radius of the global (FEM+BEM) matrix is larger than 1. We thus are developing a new method to solve the coupled system using the Generalized Minimal RESidual method (GMRES). The system we are dealing with is indeed non symmetric and GMRES, a Krylov s subspace based method, is suitable for such systems. Using this method a larger range of problems involving magnetic problems has been solved and we are now working on improving its efficiency by adding preconditioners. 3-3 Availability in LS-DYNA R7 The GMRES method is already available as a beta version for linear and nonlinear eddy current as well as for linear magnetostatic problems. It should soon be available for nonlinear magnetostatic problems. Once it will have passed several internal validation tests, it will be released in the official R7 version. In the meantime, users are free to try it out as beta and send back feedback. 4- Conclusion The Electromagnetism module of LS-DYNA was presented. The electromagnetic fields are computed by solving the Maxwell equations in the eddy-current approximation, using a FEM for the conductors coupled with a BEM for the surrounding air and insulators. The module can be used both in serial and in MPP [13]. The eddy-current solver is the main one, from which further solvers are derived for such as the induced heating and resistive heating solvers for special applications.