Sandro Manservisi

Interface reconstruction with least-squares fit and split advection in three-dimensional Cartesian geometry

In this paper we present and analyze different volume-of-fluid (VOF) reconstruction and advection algorithms that approximate the interface separating two immiscible fluids in the three-dimensional space. The paper describes the improvement of the reconstruction when a least-square fit algorithm, which minimizes a distance functional, is applied. Its performance is tested for several smooth surfaces against other simple reconstruction methods. Then Eulerian, Lagrangian and mixed split advection schemes are presented and analyzed. In particular, one advection method is discussed that conserves mass exactly for a divergence-free velocity field, thus allowing computations to machine precision.

Analysis and Approximation of the Velocity Tracking Problem for Navier--Stokes Flows with Distributed Control

We consider the mathematical formulation, analysis, and the numerical solution of a time-dependent optimal control problem associated with the tracking of the velocity of a Navier--Stokes flow in a bounded two-dimensional domain through the adjustment of a distributed control. The existence of optimal solutions is proved and the first-order necessary conditions for optimality are used to derive an optimality system of partial differential equations whose solutions provide optimal states and controls. Semidiscrete-in-time and fully discrete space-time approximations are defined and their convergence to the exact optimal solutions is shown. A gradient method for the solution of the fully discrete equations is examined, as are its convergence properties. Finally, the results of some illustrative computational experiments are presented.

A mixed markers and volume-of-fluid method for the reconstruction and advection of interfaces in two-phase and free-boundary flows

In this work we present a new mixed markers and volume-of-fluid (VOF) algorithm for the reconstruction and advection of interfaces in the two-dimensional space. The interface is described by using both the volume fraction function C, as in VOF methods, and surface markers, which locate the interface within the computational cells. The C field and the markers are advected by following the streamlines. New markers are determined by computing the intersections of the advected interface with the grid lines, then other markers are added inside each cut cell to conserve the volume fraction C. A smooth motion of the interface is obtained, typical of the marker approach, with a good volume conservation, as in standard VOF methods. In this article we consider a few typical two-dimensional tests and compare the results of the mixed algorithm with those obtained with VOF methods. Translations, rotations and vortex tests are performed showing that many problems of the VOF technique can be solved and a good accuracy in the geometrical motion and mass conservation can be achieved.

The Velocity Tracking Problem for Navier--Stokes Flows With Boundary Control

We present some systematic approaches to the mathematical formulation and numerical approximation of the time-dependent optimal control problem of tracking the velocity for Navier--Stokes flows in a bounded, two-dimensional domain with boundary control. We study the existence of optimal solutions and derive an optimality system from which optimal solutions may be determined. We also define and analyze semidiscrete-in-time and full space-time discrete approximations of the optimality system and a gradient method for the solution of the fully discrete system. The results of some computational experiments are provided.

On the properties and limitations of the height function method in two-dimensional Cartesian geometry

In this study we define the continuous height function to investigate the approximation of an interface line and its geometrical properties with the height function method. We show that in each mixed cell the piecewise linear interface reconstruction and the approximation of the derivatives and curvature based on three consecutive height function values are second-order accurate. We also discuss the quadratic reconstruction and fourth-order accurate expressions of the normal and curvature. We present a hierarchical algorithm to compute the normal vector and curvature of an interface line with the height function method that switches automatically between second- and fourth-order approximations and that can be applied also when the local radius of curvature is of the order of the grid spacing.

Numerical Analysis of Vanka-Type Solvers for Steady Stokes and Navier-Stokes Flows

We consider Vanka-type smoothers for solving Stokes and Navier-Stokes problems. In each iteration step, this smoother requires the solution of several small local subproblems over finite element blocks. It is shown that for particular choices for the blocks, the algorithm always converges to the solution of the Stokes problem and, under suitable conditions, to the solution of the Navier-Stokes problem. The convergence properties are analyzed and numerical examples are presented.

A geometrical predictor-corrector advection scheme and its application to the volume fraction function

We present a multidimensional Eulerian advection method for interfacial and incompressible flows in two-dimensional Cartesian geometry. In the scheme we advect the grid nodes backwards along the streamlines to compute the pre-images of the grid lines. These pre-images are approximated by continuous, piecewise-linear lines. The enforcement of the discrete version of the incompressibility constraint is a very important issue to determine correctly the flux polygons and to reduce considerably the integration, discretization and interpolation numerical errors. The proposed method compares favorably with other previous multidimensional advection methods as long as the initial interface line is well reconstructed. Conversely, we show that when the interface is very fragmented the total numerical error is completely dominated by the reconstruction error and in these conditions it is very difficult to assess which advection scheme is the most reliable one.

Simulation of axisymmetric jets with a finite element Navier-Stokes solver and a multilevel VOF approach

A multilevel VOF approach has been coupled to an accurate finite element Navier-Stokes solver in axisymmetric geometry for the simulation of incompressible liquid jets with high density ratios. The representation of the color function over a fine grid has been introduced to reduce the discontinuity of the interface at the cell boundary. In the refined grid the automatic breakup and coalescence occur at a spatial scale much smaller than the coarse grid spacing. To reduce memory requirements, we have implemented on the fine grid a compact storage scheme which memorizes the color function data only in the mixed cells. The capillary force is computed by using the Laplace-Beltrami operator and a volumetric approach for the two principal curvatures. Several simulations of axisymmetric jets have been performed to show the accuracy and robustness of the proposed scheme.

A variational inequality formulation of an inverse elasticity problem

Abstract We present some systematic approaches to the mathematical formulation and numerical resolution of an optimal control problem in linear elasticity. The objective of the optimization is to match a desired displacement by controlling the Youngs modulus so as to minimize a quadratic functional. Theoretical results are presented in the general framework of linear elastic theory which lead to a variational inequality. Also, we define and analyze a finite element approximation of the optimality system and a gradient method for the solution of the discrete variational inequality. Finally, numerical experiments for the simulation of a simplified model for the “sag bending process” in the manufacturing of automobile windscreens are discussed.

A MARKER-VOF ALGORITHM FOR INCOMPRESSIBLE FLOWS WITH INTERFACES

In this paper, we present a three-dimensional (3D) reconstruction algorithm for Cartesian grids and a split advection algorithm which is based on a two-dimensional (2D) Eulerian-Lagrangian scheme that conserves mass exactly for incompressible flows. In the Volume-of-Fluid/Piecewise Linear Interface Calculation (VOF/PLIC) method a linear function in every grid cell cut by the interface approximates the free surface or the surface between two immiscible phases. The reconstruction is not continuous, and not accurate in regions with high surface curvature or when the interface develops thin filaments. Therefore, we have developed a new 2D mixed markers and VOF algorithm that follows the motion of a smooth interface with a good conservation of volume. Results are shown for flows with nonconstant vorticity.Copyright