# Numerical Simulation of KBKZ Integral Constitutive Equations in Hierarchical Grids
## 5 Discussion  
The work aims to present the numerical simulation of the KBKZ integral constitutive equations for incompressible and transient complex flows. We used a new solver HiGTree/HiGFlow lately developed by [PERSON] et al. [23]. In this solver, we have implemented the methodology described in Section 3.1 to simulate viscoelastic flows modeled by integral constitutive equations. Initially, the numerical technique was verified by refined mesh in channel flows. Using the FLUID \(S1\) (see Table 2), we performed nine simulations using non-refined and refined meshes. For comparison purposes, a mesh was chosen and the simulation using the OpenFOAM solver [26] was performed. In these simulations, we can see that, although the methodologies used in HiGFlow and OpenFOAM are quite different (the first uses finite differences and the second uses finite volume), we obtained very similar results in both systems. We also verified that the errors decrease with the mesh refinement and that the order of convergence of the velocity was around two, as expected. A classic problem in the simulation of integral viscoelastic flows is known as 4:1 abrupt contraction and, thus, the literature for this problem is extensive. We chose to check our methodology for the four values of \(De\) presented in [PERSON] [14]. In addition to the comparison with the results of this author, we performed the simulations using the OpenFOAM solver [26] and also compared our results with the experimental ones from [PERSON] [15] and with the numerical results from FreeFlow [3]. We know that, in the contraction region, there are singularities and numerical techniques that might exhibit different behaviors. Although the values obtained by us in this work differ somewhat from the values obtained by [PERSON] [14] or OpenFOAM [26], they were quite comparable to the experimental values of [PERSON] [15]. Thus, we verified that the methodology presented here is capable of simulating complex flows in transient fluid regimes governed by integral constitutive models using the rapid technique of finite differences in hierarchical meshes with local refinement. The computational efficiency of the models has been previously studied [3,21,26]. However, it is worth mentioning that the methods used in the present work used fewer integration points (deformation fields) compared to the points used in the early work of [PERSON] et al. [28] and are similar to those used more recently by [PERSON] and [PERSON] [32]. This improvement is due mainly to the distinct methodologies adopted to obtain the integration points, which allow efficient distribution of the elapsed time. Although the results  
Figure 13: Streamlines of the contraction flow problem with \(De=2.07\). (**a**) OpenFOAM and (**b**) HiGFlow. presented here are two-dimensional, our methodology is still able to simulate flows in higher dimensions (the user just has to specify the dimension (2, 3 or higher) in the input data file). Our future work will be to present simulations in three dimensions for classic problems. In three dimensions, for example, integral models are still computationally expensive, as there is a need to store and connect a fixed \(N\) number of fields to each cell. Therefore, we will also work on ways to improve or modify the integral calculation--for instance, as was done in the recent work of [PERSON] [32]. Funding acquisition, A.C.; Methodology, J.B., M.S.B.d.A., R.T.L. and A.C.; Project administration, A.C.; Software, J.B., M.S.B.d.A. and A.C.; Supervision, A.C.; Validation, J.B. and M.S.B.d.A. ; Writing--original draft, J.B., M.S.B.d.A., R.T.L. and A.C.; Writing--review & editing, H.A.C.S. All authors have read and agreed to the published version of the manuscript. All authors are grateful for the financial support from the Brazilian Petroleum Agency (ANP)/Petrobras, grant 0050.0075367.12.9, and from the Sao Paulo Research Foundation (FAPESP), grants 2013/07375-0, 2017/21105-6 and 2020/02990-1.