Antonio Cervone

A volumetric formulation of the surface tension force within the eXtended Finite Element Method (XFEM) for two-phase flows is presented. This approach was first introduced for the standard Finite Element Method (FEM). Here, it is extended to the XFEM framework to capture the pressure discontinuity arising from the presence of interfacial forces. The method relies on an extension of the curvature to the whole computational domain, enabling the surface tension force to be treated as a regular volumetric term rather than as a singular interfacial contribution. The resulting system exactly satisfies the Young--Laplace law in equilibrium condition. The coupled Navier--Stokes system is stabilized by means of face-oriented ghost-penalties, while the cut integrals are evaluated using a recent quadrature rule approach which does not rely on triangulation of the cut sections and simplifies the extension from two- to three-dimensional computations. Also, a new harmonic extension strategy for the extended curvature is proposed. Finally, the numerical model is tested against benchmark cases driven by surface tension in order to assess the accuracy of the method. Static configurations are reproduced with high accuracy in realistic numerical settings, with spurious currents correctly vanishing. Damped oscillations are also reproduced and compared with both analytical and numerical reference results.

In this work, we present a numerical approach for solving optimal control problems for fluid heat transfer applications with a mixed optimality system: an FEM code to solve the adjoint solution over a precise restricted admissible solution set and an open-source well-known code for solving the state problem defined over a different one. In this way, we are able to decouple the optimality system and use well-established and validated numerical tools for the physical modeling. Specifically, two different CFD codes, OpenFOAM (finite volume-based) and FEMuS (finite element-based), have been used to solve the optimality system, while the data transfer between them is managed by the external library MEDCOUPLING. The state equations are solved in the finite volume code, while the adjoint and the control are solved in the finite element code. Two examples taken from the literature are implemented in order to validate the numerical algorithm: the first one considers a natural convection cavity resulting from a Rayleigh–Bénard configuration, and the second one is a conjugate heat transfer problem between a fluid and a solid region.

In this work we present a numerical approach for solving optimal control problems for fluid heat transfer applications with an emphasis on the coupling of multiple codes. Specifically, two different CFD codes, OpenFOAM (finite volume based) and FEMuS (finite element based), have been used to solve the optimality system, while the data transfer between them is managed by the external library MEDCOUPLING. The state equations are solved in the finite volume code, while the adjoint and the control are solved in the finite element one. Two examples taken from the literature are implemented in order to validate the numerical algorithm: the first one considers a natural convection cavity resulting from a Rayleigh-Bénard configuration, and the second one is a Conjugate Heat Transfer problem between a fluid and a solid region.

The Volume-of-Fluid (VOF) method is widely used for multiphase flow simulations, where the VOF function implicitly represents the interface through the volume fraction field. The height function (HF) method on a Cartesian mesh integrates the volume fractions of a column of cells across the interface. A stencil of three consecutive heights and centered finite differences computes the unit normal n and the curvature κ with second-order convergence with grid refinement. The interface line can cross more than one cell of the column and the value of the geometrical properties of the interface should be interpolated in the cut cells. The continuous height function is used to show that a constant approximation of the two geometrical properties across the column provides first-order convergence, while linear or quadratic interpolations provide second-order convergence. The numerical results agree with the theoretical development presented in this study.

In this paper, the optimal control theory is applied to a temperature optimization problem by coupling finite element and finite volume codes. The optimality system is split into the state and adjoint system. The direct problem is solved by the widely adopted finite volume OpenFOAM code and the adjoint-control equation using a variational formulation of the problem with the in-house finite element FEMuS code. The variational formulation of the problem is the natural framework for accurately capturing the control correction while OpenFOAM guarantees the accuracy of the state solution. This coupling is facilitated through the open-source MED and MEDCoupling libraries of the SALOME platform. The code coupling is implemented with the MED libraries and additional routines added in the FEMuS and OpenFOAM codes. We demonstrate the accuracy, robustness, and performance of the proposed approach with examples targeting different objectives using distributed and boundary controls in each case.

In this paper, the optimal control theory is applied to a temperature optimization problem by coupling finite element and finite volume codes. The optimality system is split into the state and adjoint system. The direct problem is solved by the widely adopted finite volume OpenFOAM code and the adjoint-control equation using a variational formulation of the problem with the in-house finite element FEMuS code. The variational formulation of the problem is the natural framework to capture accurately the control correction while OpenFOAM guarantees the accuracy of the state solution. This coupling is facilitated through the open-source MED and MEDCoupling libraries of the SALOME platform. The code coupling is implemented with the MED libraries and additional routines added in the FEMuS and OpenFOAM codes. We demonstrate the accuracy, robustness and performance of the proposed approach with examples targeting different objectives using distributed and boundary controls in each case.

This work focuses on the well-known issue of mass conservation in the context of the finite element technique for computational fluid dynamic simulations. Specifically, non-conventional finite element families for solving Navier–Stokes equations are investigated to address the mathematical constraint of incompressible flows. Raviart–Thomas finite elements are employed for the achievement of a discrete free-divergence velocity. In particular, the proposed algorithm projects the velocity field into the discrete free-divergence space by using the lowest-order Raviart–Thomas element. This decomposition is applied in the context of the projection method, a numerical algorithm employed for solving Navier–Stokes equations. Numerical examples validate the approach’s effectiveness, considering different types of computational grids. Additionally, the presented paper considers an interface advection problem using marker approximation in the context of multiphase flow simulations. Numerical tests, equipped with an analytical velocity field for the surface advection, are presented to compare exact and non-exact divergence-free velocity interpolation.

The volume of fluid (VOF) method is a popular technique for the direct numerical simulations of flows involving immiscible fluids. A discrete volume fraction field evolving in time represents the interface, in particular, to compute its geometric properties. The height function method (HF) is based on the volume fraction field, and its estimate of the interface curvature converges with second-order accuracy with grid refinement. Data-driven methods have been recently proposed as an alternative to computing the curvature, with particular consideration for a well-balanced input data set generation and symmetry preservation. In the present work, a two-layer feed-forward neural network is trained on an input data set generated from the height function data instead of the volume fraction field. The symmetries for rotations and reflections and the anti-symmetry for phase swapping have been considered to reduce the number of input parameters. The neural network can efficiently predict the local interface curvature by establishing a correlation between curvature and height function values. We compare the trained neural network to the standard height function method to assess its performance and robustness. However, it is worth noting that while the height function method scales perfectly with a quadratic slope, the machine learning prediction does not.

This work focuses on the well-known issue of mass conservation in the context of the finite element technique for computational fluid dynamic simulations. Specifically, non-conventional finite element families for solving Navier-Stokes equations are investigated to address the mathematical constraint of incompressible flows. Raviart-Thomas finite elements are employed for the achievement of a discrete free-divergence velocity. In particular, the proposed algorithm projects the velocity field into the discrete free-divergence space by using the lowest-order Raviart-Thomas element. This decomposition is applied in the context of the projection method, a numerical algorithm employed for solving Navier-Stokes equations. Numerical examples validate the approach’s effectiveness, considering different types of computational grids. Additionally, the presented paper considers an interface advection problem using marker approximation, in the context of multiphase flow simulations. Numerical tests, equipped with an analytical velocity field for the surface advection, are presented to compare exact and non-exact divergence-free velocity interpolation.