arXiv subjects

Eugenio Aulisa

Publications and source records attributed to Eugenio Aulisa.

At least 19 recordsLinked to original sources

arXiv2026

An Adaptive Finite Element Method for Marker-Driven Level-Set Transport on Hierarchical Meshes

This work presents a new adaptive finite-element framework for level-set transport, achieving high accuracy in kinematic interface transport problems relevant to interface-capturing methods for two-phase flows while reducing computational cost. This framework accommodates dynamic refinement and coarsening, and is compatible with standard finite-element data structures. The algorithm is applicable to both structured and unstructured discretizations in two and three dimensions. The method uses a tree-based hierarchical mesh with dynamic local refinement that tracks the evolving interface, concentrating resolution in a narrow band around the zero level set while preserving a coarse discretization elsewhere. The level-set field is updated through marker transport on adaptively refined meshes, enabling interface prediction and the construction of an evolving adaptive hierarchy. Concurrently, backward characteristic tracing provides accurate evaluation of the advected level-set field. Both operations employ an efficient multilevel marker and point-location algorithm to identify containing elements across refinement levels. Numerical experiments on two- and three-dimensional structured and unstructured grids, including quadrilateral, simplicial, wedge, and hexahedral meshes, subjected to severe interface deformations, demonstrate that the adaptive strategy achieves accuracy comparable to uniform discretizations at a substantially lower cost, while maintaining good conservation properties under aggressive adaptivity. Consequently, the proposed approach provides an efficient and flexible framework that can be naturally integrated into level-set-based multiphase flow solvers.

math.NA↗

arXiv2025

Red Blood Cells as Elastic Surfaces

We study red blood cells using the Helfrich-Canham functional: due to their lipid bilayer structure, RBCs are naturally modeled using the theory of elastic surfaces. In this study, we demonstrate that Cassinian ovals, except for the limiting case of the round sphere, do not solve the shape equation. We further discuss conditions under which they may serve as effective approximations.

math.DG↗

arXiv2022

Exact Subdomain and Embedded Interface Polynomial Integration in Finite Elements with Planar Cuts

The implementation of discontinuous functions occurs in many of today's state-of-the-art partial differential equation solvers. However, in finite element methods, this poses an inherent difficulty: efficient quadrature rules available when integrating functions whose discontinuity falls in the element's interior are for low order degree polynomials, not easily extended to higher order degree polynomials, and cover a restricted set of geometries. Many approaches to this issue have been developed in recent years. Among them one of the most elegant and versatile is the equivalent polynomial technique. This method replaces the discontinuous function with a polynomial, allowing integration to occur over the entire domain rather than integrating over complex subdomains. Although eliminating the issues involved with discontinuous function integration, the equivalent polynomial tactic introduces its problems. The exact subdomain integration requires a machinery that quickly grows in complexity when increasing the polynomial degree and the geometry dimension, restricting its applicability to lower order degree finite element families. The current work eliminates this issue. We provide algebraic expressions to exactly evaluate the subdomain integral of any degree polynomial on parent finite element shapes cut by a planar interface. These formulas also apply to the exact evaluation of the embedded interface integral. We provide recursive algorithms that avoid overflow in computer arithmetic for standard finite element geometries: triangle, square, cube, tetrahedron, and prism, along with a hypercube of arbitrary dimensions.

math.NA↗

arXiv2022

Approximation Methods for Geometric Regulation

In these notes we collect some results from several of the authors' works in order to make available a single source and show how the approximate geometric methods for regulation have been developed, and how the control design strategy has evolved from the theoretical methods, involving the regulator equations, to what we now call the regularized controller. In between these two extremes we developed, in a series of works, a fairly rigorous analysis of the regularization scheme leading to the regularized dynamic regulator equations and an iterative scheme that produces very accurate tracking and disturbance rejection control laws. In our most recent work we have extended dynamic regulator equations to what we now refer to as the regularized controller. This new formulation has only recently being applied to examples including linear and nonlinear delay equations.

math.OC↗

arXiv2021

Computational p-Willmore Flow with Conformal Penalty

The unsigned p-Willmore functional introduced in \cite{mondino2011} generalizes important geometric functionals which measure the area and Willmore energy of immersed surfaces. Presently, techniques from \cite{dziuk2008} are adapted to compute the first variation of this functional as a weak-form system of equations, which are subsequently used to develop a model for the p-Willmore flow of closed surfaces in $\mathbb{R}^3$. This model is amenable to constraints on surface area and enclosed volume, and is shown to decrease the p-Willmore energy monotonically over time. In addition, a penalty-based regularization procedure is formulated to prevent artificial mesh degeneration along the flow; inspired by a conformality condition derived in \cite{kamberov1996}, this procedure encourages angle-preservation in a closed and oriented surface immersion as it evolves. Following this, a finite-element discretization of both systems is discussed, and an application to mesh editing is presented.

math.NA↗

arXiv2021

Efficient quadrature rules for finite element discretizations of nonlocal equations

In this paper we design efficient quadrature rules for finite element discretizations of nonlocal diffusion problems with compactly supported kernel functions. Two of the main challenges in nonlocal modeling and simulations are the prohibitive computational cost and the nontrivial implementation of discretization schemes, especially in three-dimensional settings. In this work we circumvent both challenges by introducing a parametrized mollifying function that improves the regularity of the integrand, utilizing an adaptive integration technique, and exploiting parallelization. We first show that the "mollified" solution converges to the exact one as the mollifying parameter vanishes, then we illustrate the consistency and accuracy of the proposed method on several two- and three-dimensional test cases. Furthermore, we demonstrate the good scaling properties of the parallel implementation of the adaptive algorithm and we compare the proposed method with recently developed techniques for efficient finite element assembly.

math.NA↗

arXiv2019

Tumor ablation due to inhomogeneous -- anisotropic diffusion in generic 3-dimensional topologies

We derive a full 3-dimensional (3-D) model of inhomogeneous -- anisotropic diffusion in a tumor region coupled to a binary population model. The diffusion tensors are acquired using Diffusion Tensor Magnetic Resonance Imaging (DTI) from a patient diagnosed with glioblastoma multiform (GBM). Then we numerically simulate the full model with Finite Element Method (FEM) and produce drug concentration heat maps, apoptosis regions, and dose-response curves. Finally, predictions are made about optimal injection locations and volumes, which are presented in a form that can be employed by doctors and oncologists.

physics.med-ph↗

arXiv2019

A computational study of preconditioning techniques for the stochastic diffusion equation with lognormal coefficient

We present a computational study of several preconditioning techniques for the GMRES algorithm applied to the stochastic diffusion equation with a lognormal coefficient discretized with the stochastic Galerkin method. The clear block structure of the system matrix arising from this type of discretization motivates the analysis of preconditioners designed according to a field-splitting strategy of the stochastic variables. This approach is inspired by a similar procedure used within the framework of physics based preconditioners for deterministic problems, and its application to stochastic PDEs represents the main novelty of this work. Our numerical investigation highlights the superior properties of the field-split type preconditioners over other existing strategies in terms of computational time and stochastic parameter dependence.

math.NA↗

arXiv2019

A field-split preconditioning technique for fluid-structure interaction problems with applications in biomechanics

We present a novel preconditioning technique for Krylov subspace algorithms to solve fluid-structure interaction (FSI) linearized systems arising from finite element discretizations. An outer Krylov subspace solver preconditioned with a geometric multigrid (GMG) algorithm is used, where for the multigrid level sub-solvers, a field-split (FS) preconditioner is proposed. The block structure of the FS preconditioner is derived using the physical variables as splitting strategy. To solve the subsystems originated by the FS preconditioning, an additive Schwarz (AS) block strategy is employed. The proposed field-split preconditioner is tested on biomedical FSI applications. Both 2D and 3D simulations are carried out considering aneurysm and venous valve geometries. The performance of the FS preconditioner is compared with that of a second preconditioner of pure domain decomposition type.

math.NA↗

arXiv2019

Geometric model of the fracture as a manifold immersed in porous media

In this work, we analyze the flow filtration process of slightly compressible fluids in porous media containing man made fractures with complex geometries. We model the coupled fracture-porous media system where the linear Darcy flow is considered in porous media and the nonlinear Forchheimer equation is used inside the fracture. We develop a model to examine the flow inside fractures with complex geometries and variable thickness, on a Riemannian manifold. The fracture is represented as the normal variation of a surface immersed in $\mathbb{R}^3$. Using operators of Laplace Beltrami type and geometric identities, we model an equation that describes the flow in the fracture. A reduced model is obtained as a low dimensional BVP. We then couple the model with the porous media. Theoretical and numerical analysis have been performed to compare the solutions between the original geometric model and the reduced model in reservoirs containing fractures with complex geometries. We prove that the two solutions are close, and therefore, the reduced model can be effectively used in large scale simulators for long and thin fractures with complicated geometry.

math.AP↗

arXiv2018

Monolithic coupling of implicit material point method with finite element method

A monolithic coupling between the material point method (MPM) and the finite element method (FEM) is presented. The MPM formulation described is implicit, and the exchange of information between particles and background grid is minimized. The reduced information transfer from the particles to the grid improves the stability of the method. Once the residual is assembled, the system matrix is obtained by means of automatic differentiation. In such a way, no explicit computation is required and the implementation is considerably simplified. When MPM is coupled with FEM, the MPM background grid is attached to the FEM body and the coupling is monolithic. With this strategy, no MPM particle can penetrate a FEM element, and the need for computationally expensive contact search algorithms used by existing coupling procedures is eliminated. The coupled system can be assembled with a single assembly procedure carried out element by element in a FEM fashion. Numerical results are reported to display the performances and advantages of the methods here discussed.

physics.comp-ph↗

arXiv2018

Fluid-structure interaction simulations of venous valves: a monolithic ALE method for large structural displacements

Venous valves are bicuspidal valves that ensure that blood in veins only flows back to the heart. To prevent retrograde blood flow, the two intraluminal leaflets meet in the center of the vein and occlude the vessel. In fluid-structure interaction (FSI) simulations of venous valves, the large structural displacements may lead to mesh deteriorations and entanglements, causing instabilities of the solver and, consequently, the numerical solution to diverge. In this paper, we propose an Arbitrary Lagrangian-Eulerian (ALE) scheme for FSI simulations designed to solve these instabilities. A monolithic formulation for the FSI problem is considered and, due to the complexity of the operators, the exact Jacobian matrix is evaluated using automatic differentiation. The method relies on the introduction of a staggered in time velocity %in the discretization scheme to improve stability, and on fictitious springs to model the contact force of the valve leaflets. Since the large structural displacements may compromise the quality of the fluid mesh as well, a smoother fluid displacement, obtained with the introduction of a scaling factor that measures the distance of a fluid element from the valve leaflet tip, guarantees that there are no mesh entanglements in the fluid domain. To further improve stability, a Streamline Upwind Petrov Galerkin (SUPG) method is employed. The proposed ALE scheme is applied to a 2D model of a venous valve. The presented simulations show that the proposed method deals well with the large structural displacements of the problem, allowing a reconstruction of the valve behavior in both the opening and closing phase.

math.NA↗

arXiv2018

Construction of h-refined continuous finite element spaces with arbitrary hanging node configurations and applications to multigrid algorithms

We present a novel approach for the construction of basis functions to be employed in selective or adaptive h-refined finite element applications with arbitrary-level hanging node configurations. Our analysis is not restricted to $1$-irregular meshes, as it is usually done in the literature, allowing our results to be applicable to a broader class of local refinement strategies. The proposed method does not require the solution of any linear system to obtain the constraints necessary to enforce continuity of the basis functions and it can be easily implemented. A mathematical analysis is carried out to prove that the proposed basis functions are continuous and linearly independent. Finite element spaces are then defined as the spanning sets of such functions, and the implementation of a multigrid algorithm built on these spaces is discussed. A spectral analysis of the multigrid algorithm highlights superior convergence properties of the proposed method over existing strategies based on a local smoothing procedure. Finally, linear and nonlinear numerical examples are tested to show the robustness and versatility of the multigrid algorithm.

math.NA↗

arXiv2017

Fracture Model Reduction and Optimization for Forchheimer Flows in Reservoir

In this study, we analyze the flow filtration process of slightly compressible fluids in fractured porous media. We model the coupled fractured porous media system, where the linear Darcy flow is considered in porous media and the nonlinear Forchheimer equation is used inside the fracture. Flow in the fracture is modeled as a reduced low dimensional BVP which is coupled with an equation in the reservoir. We prove that the solution of the reduced model can serve very accurately to approximate the solution of the actual high-dimensional flow in reservoir fracture system, because the thickness of the fracture is small. In the analysis we consider two types of Forchhemer flows in the fracture: isotropic and anisotropic, which are different in their nature. Using method of reduction, we developed a formulation for an optimal design of the fracture, which maximizes the capacity of the fracture in the reservoir with fixed geometry. Our method, which is based on a set point control algorithm, explores the coupled impact of the fracture geometry and beta-Forchheimer coefficient.

math.AP↗

arXiv2017

Convergence estimates for multigrid algorithms with SSC smoothers and applications to overlapping domain decomposition

In this paper we study convergence estimates for a multigrid algorithm with smoothers of successive subspace correction (SSC) type, applied to symmetric elliptic PDEs. First, we revisit a general convergence analysis on a class of multigrid algorithms in a fairly general setting, where no regularity assumptions are made on the solution. In this framework, we are able to explicitly highlight the dependence of the multigrid error bound on the number of smoothing steps. For the case of no regularity assumptions, this represents a new addition to the existing theory. Then, we analyze successive subspace correction smoothing schemes for a set of uniform and local refinement applications with either nested or non-nested overlapping subdomains. For these applications, we explicitly derive bounds for the multigrid error, and identify sufficient conditions for these bounds to be independent of the number of multigrid levels. For the local refinement applications, finite element grids with arbitrary hanging nodes configurations are considered. The analysis of these smoothing schemes is cast within the far-reaching multiplicative Schwarz framework.

math.NA↗

arXiv2017

A monolithic ALE Newton-Krylov solver with Multigrid-Richardson-Schwarz preconditioning for incompressible Fluid Structure Interaction

In this paper we study a monolithic Newton-Krylov solver with exact Jacobian for the solution of incompressible FSI problems. A main focus of this work is on the use of geometric multigrid preconditioners with modified Richardson smoothers preconditioned by an additive Schwarz algorithm. The definition of the subdomains in the Schwarz smoother is driven by the natural splitting between fluid and solid. The monolithic approach guarantees the automatic satisfaction of the stress balance and the kinematic conditions across the fluid-solid interface. The enforcement of the incompressibility conditions both for the fluid and for the solid parts is taken care of by using inf-sup stable finite element pairs without stabilization terms. A suitable Arbitrary Lagrangian Eulerian (ALE) operator is chosen in order to avoid mesh entanglement while solving for large displacements of the moving fluid domain. Numerical results of two and three-dimensional benchmark tests with Newtonian fluids and nonlinear hyperelastic solids show a robust performance of our fully incompressible solver especially for the more challenging direct-to-steady-state problems.

physics.comp-ph↗

arXiv2015

Well Productivity Index for Compressible Fluids and Gases

In this paper we discuss the notion of the diffusive capacity for the generalized Forchheimer flow of fluid through porous media. The diffusive capacity is an integral characteristic of the flow motivated by the engineering notion of the productivity index (PI), Dake 1983, Raghavan 1993, Christopher et al. 2014. The PI characterizes the well capacity with respect to drainage area of the well and in general is time dependent. We study its time dynamics for two types of fluids: slightly compressible and strongly compressible fluid (ideal gas). In case of the slightly compressible fluid the PI stabilizes in time to the specific value, determined by the so-called pseudo steady state solution, Aulisa et al. 2009, 2011, 2012. Here we generalize our results from Aulisa et al. 2012 on long term dynamics of the PI in case of arbitrary order of the nonlinearity of the flow. In this paper we study the mathematical model of the PI for compressible gas flow for the first time. In contrast to slightly compressible fluid this functional mathematically speaking is not time-invariant. At the same time it stays "almost" constant for a long period of time, but then it rapidly blows up as time approaches the certain critical value. This value depends on the initial data (initial reserves) of the reservoir. The "greater" are the initial reserves, the larger is this critical value. We present numerical and analytical results for the time asymptotic of the PI and its stability with respect to the initial data. Using comparison theorems for porous media equation from Vázquez 2007 we obtain estimates between the PI's for the original gas flow and auxiliary flow with a distributed source. The latter one generates the time independent PI, and can be calculated using formula similar to one in case of slightly compressible fluid.

math.AP↗

arXiv2013

Upscaling of Nonlinear Forchheimer Flows

In this work we propose upscaling method for nonlinear Forchheimer flow in highly heterogeneous porous media. The generalized Forchheimer law is considered for incompressible and slightly-compressible single-phase flows. We use recently developed analytical results Aulisa et al. (2009) and write the resulting system in terms of a degenerate nonlinear flow equation for the pressure with the nonlinearity that depends on the pressure gradient. The coarse scale parameters for the steady state problem are determined so that the volumetric average of velocity of the flow in the domain on fine scale and on coarse scale are close enough. A flow-based coarsening approach is used, where the equivalent permeability tensor is first evaluated following the streamline of the existing linear cases, and successively modified in order to take into account the nonlinear effects. Compared to previous works Durlofsky and Karimi-Fard (2009) and Peszynska et al. (2009), our approach relies on recent analytical results of Aulisa et al. (2009) and combines it with rigorous mathematical upscaling theory for monotone operators. The developed upscaling algorithm for nonlinear steady state problems is effectively used for variety of heterogeneities in the domain of computation. Direct numerical computations for average velocity and productivity index justify the usage of the coarse scale parameters obtained for the special steady state case in the fully transient problem. Analytical upscaling formulas in stratified domain are obtained for the nonlinear case. They correlate with high accuracy with numerical results.

math.NA↗