# A Backwards-Tracking Lagrangian-Eulerian Method for Viscoelastic Two-Fluid Flows
## 5 Conclusions  
In this work, a new Lagrangian-Eulerian method for the simulation of viscoelastic free surface flow has been proposed. The method was developed from a previously proposed method, in which the fluid momentum and continuity equations were solved on a stationary Eulerian octree grid, while the viscoelastic constitutive equation was solved along the trajectories of Lagrangian nodes which were convected by the flow. The main improvements in this work was the introduction of a backwards-tracking procedure for solving the viscoelastic constitutive equation, as well an extension to the simulation of viscoelastic free surface flow with the volume of fluid method.  
The backwards-tracking procedure allowed for the storage of the viscoelastic stresses in a structured arrangement. Consequently, the unstructured interpolation of stresses as well as the addition and deletion of Lagrangian nodes, which was required in the previous method, was eliminated in favor of a significant reduction of the computational cost and increased robustness. Furthermore, no additional stabilization method, such as both sides  
Figure 19: Viscoelastic jet buckling simulation with \(\lambda=0.1\,\mathrm{s}\).  
diffusion or reformulation of the constitutive equation, was employed for any of the flows studied in this work.  
The proposed method was validated with analytic solutions for a planar Poiseuille flow of Oldroyd-B fluids. The transient, as well steady flow, solutions were in excellent agreement with analytic solutions and the steady solution converged to the analytic solution with second order accuracy. Furthermore, the proposed method was evaluated for two types of viscoelastic free surface flow. The viscoelastic die swell effect was simulated with the Oldroyd-B model and the predicted swell ratios were in good agreement with the numerical data from the literature. Planar jet buckling of a highly viscoelastic fluid was also simulated, demonstrating the capability of the method for an additional flow case as well as the use of adaptive grid refinements.  
The computational performance was not subject to detailed study in this work. However, as per the construction of the new method, the computational cost for the viscoelastic stress calculation was significantly reduced when compared to the previous Lagrangian-Eulerian method. Furthermore, it has previously been shown that the Lagrangian formulation of the constitutive equation is highly suitable for parallel calculations, including for GPU-acceleration. This also applies for the new method.  
In conclusion, the new method is an important step in the development of efficient and useful tools for simulating industrial processes involving viscoelastic fluids. In future research, the proposed Lagrangian-Eulerian method will be used for the numerical investigation of viscoelastic free surface flows with moving objects, including for viscoelastic adhesive application and parts assembly, as well as for additive manufacturing processes.  
Conceptualization, S.I. and A.M.; Methodology, S.I. and A.M.; Software, S.I. and A.M.; Validation, S.I.; Writing--original draft preparation, S.I.; Writing--review and editing, A.M., R.K., F.E.; Supervision, A.M., R.K., F.E.; Funding acquisition, F.E. All authors have read and agreed to the published version of the manuscript.  
This research has been partly carried out in a Centre for Additive Manufacturing--Metal (CAM2) in a joint project financed by Swedish Governmental Agency of Innovation Systems (Vinnova), coordinated by Chalmers University of Technology. The work has also been supported in part by Vinnova through the FFI Sustainable Production Technology program and in part by the Chalmers Area of Advance Production. The support is gratefully acknowledged.  
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.