Florian Frank

• PD Dr.
• Senior Software Engineer at Math2Market

A numerical method is formulated for the solution of the advective Cahn-Hilliard (CH) equation with constant and degenerate mobility in three-dimensional porous media with non-vanishing velocity on the exterior boundary. The CH equation describes phase separation of an immiscible binary mixture at constant temperature in the presence of a mass cons...

We consider an energy-based boundary condition to impose an equilibrium wetting angle for the Cahn-Hilliard-Navier-Stokes phase-field model on voxel-set-type computational domains. These domains typically stem from the micro-CT imaging of porous rock and approximate a (on {\mu}m scale) smooth domain with a certain resolution. Planar surfaces that a...

Many mathematical models of computational fluid dynamics involve transport of conserved quantities, which must lie in a certain range to be physically meaningful. The analytical or numerical solution u of a scalar conservation law is said to be bound-preserving if global bounds u∗ and u∗ exist such that u∗≤u≤u∗ holds in the domain of definition. Th...

A novel precondition operator for the Schur complement of the stationary Stokes equations is proposed. Numerical experiments demonstrate that its discrete version is superior to established precondition operators if the computational domain is a thin channel or contains such (e.g. porous media or filters). Discrete diffusion is added to the establi...

We present a solution strategy for computing the equilibrium density profiles of mixtures consisting of an arbitrary number of components and phases in a closed system at constant temperature. Our approach is based on the density gradient formulation of the Helmholtz energy in the canonical ensemble, which is a functional of the component densities...

Measuring the thickness of thin porous materials provides valuable insights into their structure, properties, and performance, including key properties such as porosity and permeability, and is highly beneficial for a range of industrial applications, particularly for ensuring effective quality control processes. A novel approach for estimating the...

In recent years, convolutional neural networks (CNNs) have experienced an increasing interest in their ability to perform a fast approximation of effective hydrodynamic parameters in porous media research and applications. This paper presents a novel methodology for permeability prediction from micro-CT scans of geological rock samples. The trainin...

In the past several years, convolutional neural networks (CNNs) have proven their capability to predict characteristic quantities in porous media research directly from pore-space geometries. Due to the frequently observed significant reduction in computation time in comparison to classical computational methods, bulk parameter prediction via CNNs...

In recent years, convolutional neural networks (CNNs) have experienced an increasing interest for their ability to perform fast approximation of effective hydrodynamic parameters in porous media research and applications. This paper presents a novel methodology for permeability prediction from micro-CT scans of geological rock samples. The training...

The aim of this paper is to develop suitable models for the phenomenon of cell blebbing, which allow for computational predictions of mechanical effects including the crucial interaction of the cell membrane and the actin cortex. For this sake we resort to a two phase-field model that uses diffuse descriptions of both the membrane and the cortex, w...

This work presents an enriched Galerkin (EG) discretization for the two-dimensional shallow-water equations. The EG finite element spaces are obtained by extending the approximation spaces of the classical finite elements by discontinuous functions supported on elements. The simplest EG space is constructed by enriching the piecewise linear continu...

The present work documents the current state of development for our MATLAB/GNU Octave-based open source toolbox FESTUNG (Finite Element Simulation Toolbox for UNstructured Grids). The goal of this project is to design a user-friendly, research-oriented, yet computationally efficient software tool for solving partial differential equations (PDEs). S...

Summary of the benchmark study and main conclusions

This is a data repository accompanying the paper "Numerical benchmark for flow in highly heterogeneous aquifers" by Cristian D. Alecsa, Imre Boros, Florian Frank, Peter Knabner, Mihai Nechita, Alexander Prechtel, Andreas Rupp, Nicolae Suciu. The benchmark consists of evaluating the performance of numerical methods by comparing solutions of the equ...

This article presents numerical investigations on accuracy and convergence properties of several numerical approaches for simulating steady state flows in heterogeneous aquifers. Finite difference, finite element, discontinuous Galerkin, spectral, and random walk methods are tested on two-dimensional benchmark flow problems. Realizations of log-nor...

This article presents numerical investigations on accuracy and convergence properties of several numerical approaches for simulating steady state flows in heterogeneous aquifers. Finite difference, finite element, discontinuous Galerkin, spectral, and random walk methods are tested on one- and two-dimensional benchmark flow problems. Realizations o...

Two-phase flow with viscosity contrast at the pore scale is modeled by a time-dependent Cahn–Hilliard–Navier–Stokes model and belongs to the class of diffuse interface method. The model allows for moving contact line and varying wettability. The numerical scheme utilizes an efficient pressure-correction projection algorithm, in conjunction with int...

Permeability estimation of porous media from directly solving the Navier–Stokes equations has a wide spectrum of applications in petroleum industry. In this paper, we utilize a pressure-correction projection algorithm in conjunction with the interior penalty discontinuous Galerkin scheme for space discretization to build an incompressible Navier–St...

In this paper, we derive a theoretical analysis of nonsymmetric interior penalty discontinuous Galerkin methods for solving the Cahn–Hilliard equation. We prove unconditional unique solvability of the discrete system and derive stability bounds with a generalized chemical energy density. Convergence of the method is obtained by optimal a priori err...

Many mathematical models of computational fluid dynamics involve transport of conserved quantities which must lie in a certain range to be physically meaningful. The analytical or numerical solution u of a scalar conservation law is said to satisfy a maximum principle (MP) if global bounds umin and umax exist such that umin ≤ u ≤ umax holds in the...

Modeling the droplet nucleation process requires a molecular-scale approach to de- scribe the interfacial tension (IFT) of spherical interfaces. Density gradient theory (DGT), also referred to as square gradient theory in some publications, has been widely used to compute the IFT of many pure and mixed systems at the molecular scale. However, the a...

Permeability estimation of porous media from direct solving Navier-Stokes equation has a wide spectrum of applications in petroleum industry. In this paper, we utilize a pressure-correction projection algorithm in conjunction with the interior penalty discontinuous Galerkin scheme for space discretization to build an incompressible Navier-Stokes si...

Permeability estimation of porous media from direct solving Navier--Stokes equation has a wide spectrum of applications in petroleum industry. In this paper, we utilize a pressure-correction projection algorithm in conjunction with the interior penalty discontinuous Galerkin scheme for space discretization to build an incompressible Navier--Stokes...

This is the fourth installment in our series on implementing the discontinuous Galerkin (DG) method as an open source MATLAB /GNU Octave toolbox. Similarly to its predecessors, this part presents new features for application developers employing DG methods and follows our strategy of relying on fully vectorized constructs and supplying a comprehens...

Advances in pore-scale imaging, increasing availability of computational resources, and developments in numerical algorithms have started rendering direct pore-scale numerical simulations of multiphase flow on pore structures feasible. In this paper, we describe a two-phase-flow simulator that solves mass- and momentum-balance equations valid at th...

We consider an energy-based boundary condition to impose an equilibrium wetting angle for the Cahn-Hilliard-Navier-Stokes phase-field model on voxel-set-type computational domains. These domains typically stem from the micro-CT imaging of porous rock and approximate a (on {\mu}m scale) smooth domain with a certain resolution. Planar surfaces that a...

The phase-field method is a versatile and robust technique for modeling interfacial motion in multiphase flows in pore-scale media. The method provides an effective way to account for surface effects by use of diffuse interfaces. The resulting model significantly simplifies the numerical implementation of mass transport and momentum balance solvers...

This paper presents an a priori error analysis of a fully discrete scheme for the numerical solution of the transient, nonlinear Darcy–Nernst–Planck–Poisson system. The scheme uses the second order backward difference formula (BDF2) in time and the mixed finite element method with Raviart–Thomas elements in space. In the first step, we show that th...

Hierarchical scale separation (HSS) is an iterative two-scale approximation method for large sparse systems of linear equations arising from discontinuous Galerkin (DG) discretizations. HSS splits the linear system into a coarse-scale system of reduced size corresponding to the local mean values of the solution, and a set of decoupled local fine-sc...

Hierarchical scale separation (HSS) is a new approach to solve large sparse systems of linear equations arising from discontinuous Galerkin (DG) discretizations. We investigate its applicability to systems stemming from the nonsymmetric interior penalty DG discretization of the Cahn-Hilliard equation, discuss its hybrid parallel implementation for...

Advances in pore-scale imaging, increasing availability of computational resources, and developments in numerical algorithms have started rendering direct pore-scale numerical simulations of multiphase flow on pore structures feasible. In this paper, we describe a two-phase flow simulator that solves mass and momentum balance equations valid at the...

Pore-scale simulation is both computationally challenging and a key element of hydrocarbon exploration and production ([1], [2], [3]). The subject of this study is the efficient solution of the linear systems arising from the discretization of the Cahn–Hilliard equation ([4], [5]), which governs the separation of a two-component fluid mixture. We i...

Density gradient theory (DGT) allows fast and accurate determination of surface tension and density profile through a phase interface. Several algorithms have been developed to apply this theory in practical calculations. While the conventional algorithm requires a reference substance of the system, a modified "stabilized density gradient theory" (...

A numerical method is formulated for the solution of the advective Cahn-Hilliard (CH) equation with constant and degenerate mobility in three-dimensional porous media with non-vanishing velocity on the exterior boundary. The CH equation describes phase separation of an immiscible binary mixture at constant temperature in the presence of a mass cons...

This is the second in a series of papers on implementing a discontinuous Galerkin (DG) method as an open source MATLAB / GNU Octave toolbox. The intention of this ongoing project is to offer a rapid prototyping package for application development using DG methods. The implementation relies on fully vectorized matrix / vector operations and is compr...

Advances in pore-scale imaging (e.g., μ-CT scanning), increasing availability of computational resources, and recent developments in numerical algorithms have started rendering direct pore-scale numerical simulations of multi-phase flow on pore structures feasible. Quasi-static methods, where the viscous and the capillary limit are iterated sequent...

Discretization of the partial differential equations that govern the physics of multi-phase multi-component fluid flow and transport gives rise to large sparse linear systems for practical pore-scale simulation. In this work, we focus on a linear system arising from the discretization of the Cahn-Hilliard equation that governs the separation of a t...

Density gradient theory (DGT) allows fast and accurate determination of surface tension and density profile through a phase interface. Several algorithms have been developed to apply this theory in practical calculations. While the conventional algorithm requires a reference substance of the system, a modified "stabilized density gradient theory" (...

In the first part of this article, we extend the formal upscaling of a diffusion–precipitation model through a two-scale asymptotic expansion in a level set framework to three dimensions. We obtain upscaled partial differential equations, more precisely, a non-linear diffusion equation with effective coefficients coupled to a level set equation. As...

We contribute a third-party survey of sparse matrix-vector (SpMV) product performance on industrial-strength, large matrices using: (1) The SpMV implementations in Intel MKL, the Trilinos project (Tpetra subpackage), the CUSPARSE library, and the CUSP library, each running on modern architectures. (2) NVIDIA GPUs and Intel multi-core CPUs (supporte...

This is the second in a series of papers on implementing a discontinuous Galerkin (DG) method as an open source Matlab / GNU Octave toolbox. The intention of this ongoing project is to offer a rapid prototyping package for application development using DG methods. The implementation relies on fully vectorized matrix / vector operations and is compr...

This is the first in a series of papers on implementing a discontinuous Galerkin method as a MATLAB / GNU Octave toolbox. The main goal is the development of techniques that deliver optimized computational performance combined with a compact, user-friendly interface. Our implementation relies on fully vectorized matrix / vector operations and is ca...

We apply a novel upwind stabilization of a mixed hybrid finite element method of lowest order to advection–diffusion problems with dominant advection and compare it with a finite element scheme stabilized by finite volume upwinding. Both schemes are locally mass conservative and employ an upwind-weighting formula in the discretization of the advect...

We consider the dynamics of dilute electrolytes and of dissolved charged particles within a periodic porous medium at the pore scale, which is described by the non-stationary Stokes–Nernst–Planck–Poisson (SNPP) system. Since simulations that resolve the geometry of the solid matrix at the pore scale are not feasible in practice, a major interest li...

We consider colloidal dynamics and single-phase fluid flow within a saturated porous medium in two space dimensions. A new approach in modeling pore clogging and porosity changes on the macroscopic scale is presented. Starting from the pore scale, transport of colloids is modeled by the Nernst–Planck equations. Here, interaction with the porous mat...

We consider charged transport within a porous medium which at the pore scale can be described by the non-stationary Stokes–Nernst–Planck–Poisson (SNPP) system. We state three different homogenization results using the method of two-scale convergence. In addition to the averaged macroscopic equations, auxiliary cell problems are solved in order to p...