Faruk Omer Alpak; Alexander Samardžić; Florian Frank A distributed parallel direct simulator for pore-scale two-phase flow on digital rock images using a finite-volume-based implementation of the phase-field method Journal Article Journal of Petroleum Science and Engineering, 2017. @article{AlpakSF2017, title = {A distributed parallel direct simulator for pore-scale two-phase flow on digital rock images using a finite-volume-based implementation of the phase-field method}, author = {Faruk Omer Alpak and Alexander Samardžić and Florian Frank}, year = {2017}, date = {2017-11-12}, journal = {Journal of Petroleum Science and Engineering}, abstract = {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 for simulating two-phase flow with a large number of moving interfaces. The interfaces can be generated, transported or destroyed based on a thermodynamic Helmholtz free-energy minimization framework underpinning the governing equations. The phase-field method accurately conserves mass and is relatively straightforward to implement in conjunction with contact angle models that account for wettability on rock surfaces. The underlying free-energy minimization framework leads to the advective Cahn-Hilliard equation and modified Navier-Stokes equations that describe the phase-field model. We have implemented a particular variant of the phase-field method (PFM) into the computational core of a pore-scale multiphase flow simulator, namely PMFS-PFM, for the numerical simulations of incompressible flows of two immiscible fluid phases. The implementation was discussed in a previous paper for rectangular prism-shaped fully-connected domains, e.g., for simulating two-phase flow in a 2D slit or a 3D duct (Alpak et al., 2016). In this paper, we discuss the recent developments on PMFS-PFM. The main components of the new work are (I) implementation of support for inactive cells in PMFS-PFM by extending the original finite-volume method-based discretizations of the underlying partial differential equations (PDEs) in order to solve realistic 3D pore-scale flow problems on rock volumes stemming from imaging, and (II) enhancing the performance of the simulator through implementation of modern sparse linear solvers and distributed parallel computing. It has been shown that the simulations performed on complex pore-scale domains are consistent with the physics of the immiscible two-phase displacement. The parallel scalability of the code is reasonably well varying between 50% and 86% on the investigated test cases of varying complexity. Results indicate that the more disconnected the pore-scale domains, the lower the parallel efficiency. It has been noted that there is a possibility of improving the parallel efficiency by exploring various grid subdivisions.}, keywords = {}, pubstate = {published}, tppubtype = {article} } 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 for simulating two-phase flow with a large number of moving interfaces. The interfaces can be generated, transported or destroyed based on a thermodynamic Helmholtz free-energy minimization framework underpinning the governing equations. The phase-field method accurately conserves mass and is relatively straightforward to implement in conjunction with contact angle models that account for wettability on rock surfaces. The underlying free-energy minimization framework leads to the advective Cahn-Hilliard equation and modified Navier-Stokes equations that describe the phase-field model. We have implemented a particular variant of the phase-field method (PFM) into the computational core of a pore-scale multiphase flow simulator, namely PMFS-PFM, for the numerical simulations of incompressible flows of two immiscible fluid phases. The implementation was discussed in a previous paper for rectangular prism-shaped fully-connected domains, e.g., for simulating two-phase flow in a 2D slit or a 3D duct (Alpak et al., 2016). In this paper, we discuss the recent developments on PMFS-PFM. The main components of the new work are (I) implementation of support for inactive cells in PMFS-PFM by extending the original finite-volume method-based discretizations of the underlying partial differential equations (PDEs) in order to solve realistic 3D pore-scale flow problems on rock volumes stemming from imaging, and (II) enhancing the performance of the simulator through implementation of modern sparse linear solvers and distributed parallel computing. It has been shown that the simulations performed on complex pore-scale domains are consistent with the physics of the immiscible two-phase displacement. The parallel scalability of the code is reasonably well varying between 50% and 86% on the investigated test cases of varying complexity. Results indicate that the more disconnected the pore-scale domains, the lower the parallel efficiency. It has been noted that there is a possibility of improving the parallel efficiency by exploring various grid subdivisions. |
Florian Frank; Peter Knabner Convergence analysis of a BDF2/mixed finite element discretization of a Darcy–Nernst–Planck–Poisson system Journal Article ESAIM: Mathematical Modelling and Numerical Analysis (ESAIM: M2AN), 51 (5), pp. 1883–1902, 2017. @article{FK2014, title = {Convergence analysis of a BDF2/mixed finite element discretization of a Darcy–Nernst–Planck–Poisson system}, author = {Florian Frank and Peter Knabner}, doi = {10.1051/m2an/2017002}, year = {2017}, date = {2017-10-27}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis (ESAIM: M2AN)}, volume = {51}, number = {5}, pages = {1883–1902}, institution = {Department of Mathematics, University of Erlangen–Nürnberg}, abstract = {This paper presents an \textit{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 the solution of the underlying weak continuous problem is also a solution of a third problem for which an existence result is already established. Thereby a stability estimate arises, which provides an \textit{L}^{∞} bound of the concentrations/masses of the system. This bound is used as a level for a cut-off operator that enables a proper formulation of the fully discrete scheme. The error analysis copes without semi-discrete intermediate formulations and reveals convergence rates of optimal orders in time and space. Numerical simulations validate the theoretical results for lowest order finite element spaces in two dimensions.}, keywords = {}, pubstate = {published}, tppubtype = {article} } 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 the solution of the underlying weak continuous problem is also a solution of a third problem for which an existence result is already established. Thereby a stability estimate arises, which provides an L^{∞} bound of the concentrations/masses of the system. This bound is used as a level for a cut-off operator that enables a proper formulation of the fully discrete scheme. The error analysis copes without semi-discrete intermediate formulations and reveals convergence rates of optimal orders in time and space. Numerical simulations validate the theoretical results for lowest order finite element spaces in two dimensions. |
Christopher Thiele; Mauricio Araya-Polo; Faruk Omer Alpak; Béatrice Rivière; Florian Frank Inexact hierarchical scale separation: a two-scale approach for linear systems from discontinuous Galerkin discretizations Journal Article Computers and Mathematics with Applications, 74 (8), pp. 1769–1778, 2017. @article{ThieleEtAl2017IHSS, title = {Inexact hierarchical scale separation: a two-scale approach for linear systems from discontinuous Galerkin discretizations}, author = {Christopher Thiele and Mauricio Araya-Polo and Faruk Omer Alpak and Béatrice Rivière and Florian Frank}, url = {https://authors.elsevier.com/a/1Vwba3CDPPn4st https://github.com/lephlaux}, doi = {10.1016/j.camwa.2017.06.025}, year = {2017}, date = {2017-06-17}, journal = {Computers and Mathematics with Applications}, volume = {74}, number = {8}, pages = {1769–1778}, abstract = {\emph{Hierarchical scale separation} (HSS) is an iterative two-scale approximation method for large sparse systems of linear equations arising from \emph{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 \emph{decoupled} local fine-scale systems corresponding to the higher order solution components. This scheme then alternates between coarse-scale and fine-scale system solves until both components converge. The motivation of HSS is to promote parallelism by decoupling the fine-scale systems, and to reduce the communication overhead from classical linear solvers by only applying them to the coarse-scale system. We propose a modified HSS scheme (\emph{“inexact HSS”, “IHSS”}) that exploits the highly parallel fine-scale solver more extensively and only approximates the coarse-scale solution in every iteration thus resulting in a significant speedup. The tolerance of the coarse-scale solver is adapted in every IHSS cycle, controlled by the residual norm of the fine-scale system. Anderson acceleration is employed in the repeated solving of the fine-scale system to stabilize the scheme. We investigate the applicability of IHSS to systems stemming from the nonsymmetric interior penalty DG discretization of the Cahn–Hilliard equation, discuss its hybrid parallel implementation for large-scale simulations, and compare the performance of a widely used iterative solver with and without IHSS.}, keywords = {}, pubstate = {published}, tppubtype = {article} } 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-scale systems corresponding to the higher order solution components. This scheme then alternates between coarse-scale and fine-scale system solves until both components converge. The motivation of HSS is to promote parallelism by decoupling the fine-scale systems, and to reduce the communication overhead from classical linear solvers by only applying them to the coarse-scale system. We propose a modified HSS scheme (“inexact HSS”, “IHSS”) that exploits the highly parallel fine-scale solver more extensively and only approximates the coarse-scale solution in every iteration thus resulting in a significant speedup. The tolerance of the coarse-scale solver is adapted in every IHSS cycle, controlled by the residual norm of the fine-scale system. Anderson acceleration is employed in the repeated solving of the fine-scale system to stabilize the scheme. We investigate the applicability of IHSS to systems stemming from the nonsymmetric interior penalty DG discretization of the Cahn–Hilliard equation, discuss its hybrid parallel implementation for large-scale simulations, and compare the performance of a widely used iterative solver with and without IHSS. |
Xiaoqun Mu; Florian Frank; Faruk Omer Alpak; Walter G. Chapman Stabilized density gradient theory algorithm for modeling interfacial properties of pure and mixed systems Journal Article Fluid Phase Equilibria, 435 , pp. 118–130, 2017. @article{MuFrankAlpakChapman2017, title = {Stabilized density gradient theory algorithm for modeling interfacial properties of pure and mixed systems}, author = {Xiaoqun Mu and Florian Frank and Faruk Omer Alpak and Walter G. Chapman}, url = {http://www.sciencedirect.com/science/article/pii/S0378381216305787}, doi = {10.1016/j.fluid.2016.11.024}, year = {2017}, date = {2017-03-15}, journal = {Fluid Phase Equilibria}, volume = {435}, pages = {118–130}, abstract = {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" (SDGT) algorithm is introduced in our work to solve DGT equations for multiphase pure and mixed systems. This algorithm makes it possible to calculate interfacial properties accurately at any domain size larger than the interface thickness without choosing a reference substance or assuming the functional form of the density profile. As part of DGT inputs, the perturbed chain statistical associating fluid theory (PC-SAFT) equation of state (EoS) was employed for the first time with the SDGT algorithm. PC-SAFT has excellent performance in predicting liquid phase properties as well as phase behaviors. The SDGT algorithm with the PC-SAFT EoS was tested and compared with experimental data for several systems. Numerical stability analyses were also included in each calculation to verify the reliability of this approach for future applications.}, keywords = {}, pubstate = {published}, tppubtype = {article} } 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" (SDGT) algorithm is introduced in our work to solve DGT equations for multiphase pure and mixed systems. This algorithm makes it possible to calculate interfacial properties accurately at any domain size larger than the interface thickness without choosing a reference substance or assuming the functional form of the density profile. As part of DGT inputs, the perturbed chain statistical associating fluid theory (PC-SAFT) equation of state (EoS) was employed for the first time with the SDGT algorithm. PC-SAFT has excellent performance in predicting liquid phase properties as well as phase behaviors. The SDGT algorithm with the PC-SAFT EoS was tested and compared with experimental data for several systems. Numerical stability analyses were also included in each calculation to verify the reliability of this approach for future applications. |
Raphael Schulz; Nadja Ray; Florian Frank; Hari Mahato; Peter Knabner Strong solvability up to clogging of an effective diffusion–precipitation model in an evolving porous medium Journal Article European Journal of Applied Mathematics, 28 (2), pp. 179–207, 2017. @article{SRFMK2015, title = {Strong solvability up to clogging of an effective diffusion–precipitation model in an evolving porous medium}, author = {Raphael Schulz and Nadja Ray and Florian Frank and Hari Mahato and Peter Knabner}, doi = {10.1017/S0956792516000164}, year = {2017}, date = {2017-02-27}, journal = {European Journal of Applied Mathematics}, volume = {28}, number = {2}, pages = {179–207}, institution = {Department of Mathematics, University of Erlangen–Nürnberg}, abstract = {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 a first step, we consider a parametrization of the underlying pore geometry by a single parameter, e.g. by a generalized “radius” or the porosity. Then, the level set equation transforms to an ordinary differential equation for the parameter. For such an idealized setting, the degeneration of the diffusion tensor with respect to porosity is illustrated with numerical simulations. The second part and main objective of this article is the analytical investigation of the resulting coupled partial differential equation–ordinary differential equation model. In the case of non-degenerating coefficients, local-in-time existence of at least one strong solution is shown by applying Schauder’s fixed point theorem. Additionally, non-negativity, uniqueness, and global existence or existence up to possible closure of some pores, i.e. up to the limit of degenerating coefficients, is guaranteed.}, keywords = {}, pubstate = {published}, tppubtype = {article} } 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 a first step, we consider a parametrization of the underlying pore geometry by a single parameter, e.g. by a generalized “radius” or the porosity. Then, the level set equation transforms to an ordinary differential equation for the parameter. For such an idealized setting, the degeneration of the diffusion tensor with respect to porosity is illustrated with numerical simulations. The second part and main objective of this article is the analytical investigation of the resulting coupled partial differential equation–ordinary differential equation model. In the case of non-degenerating coefficients, local-in-time existence of at least one strong solution is shown by applying Schauder’s fixed point theorem. Additionally, non-negativity, uniqueness, and global existence or existence up to possible closure of some pores, i.e. up to the limit of degenerating coefficients, is guaranteed. |
Faruk Omer Alpak; Béatrice Rivière; Florian Frank A Phase-field method for the direct simulation of two-phase flows in pore-scale media using a non-equilibrium wetting boundary condition Journal Article Computational Geosciences, 20 (5), pp. 881–908, 2016. @article{ARF2016, title = {A Phase-field method for the direct simulation of two-phase flows in pore-scale media using a non-equilibrium wetting boundary condition}, author = {Faruk Omer Alpak and Béatrice Rivière and Florian Frank}, url = {http://link.springer.com/article/10.1007%2Fs10596-015-9551-2}, doi = {10.1007/s10596-015-9551-2}, year = {2016}, date = {2016-10-01}, journal = {Computational Geosciences}, volume = {20}, number = {5}, pages = {881–908}, abstract = {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 sequentially, fall short in rigorously capturing crucial flow phenomena at the pore scale. Direct simulation techniques are needed that account for the full coupling between capillary and viscous flow phenomena. Consequently, there is a strong demand for robust and effective numerical methods that can deliver high-accuracy, high-resolution solutions of pore-scale flow in a computationally efficient manner. Direct simulations of pore-scale flow on imaged volumes can yield important insights about physical phenomena taking place during multi-phase, multi-component displacements. Such simulations can be utilized for optimizing various enhanced oil recovery (EOR) schemes and permit the computation of effective properties for Darcy-scale multi-phase flows. We implement a phase-field model for the direct pore-scale simulation of incompressible flow of two immiscible fluids. The model naturally lends itself to the transport of fluids with large density and viscosity ratios. In the phase-field approach, the fluid-phase interfaces are expressed in terms of thin transition regions, the so-called diffuse interfaces, for increased computational efficiency. The conservation law of mass for binary mixtures leads to the advective Cahn–Hilliard equation and the condition that the velocity field is divergence free. Momentum balance, on the other hand, leads to the Navier–Stokes equations for Newtonian fluids modified for two-phase flow and coupled to the advective Cahn–Hilliard equation. Unlike the volume of fluid (VoF) and level-set methods, which rely on regularization techniques to describe the phase interfaces, the phase-field method facilitates a thermodynamic treatment of the phase interfaces, rendering it more physically consistent for the direct simulations of two-phase pore-scale flow. A novel geometric wetting (wall) boundary condition is implemented as part of the phase-field method for the simulation of two-fluid flows with moving contact lines. The geometric boundary condition accurately replicates the prescribed equilibrium contact angle and is extended to account for dynamic (non-equilibrium) effects. The coupled advective Cahn–Hilliard and modified Navier–Stokes (phase-field) system is solved by using a robust and accurate semi-implicit finite volume method. An extension of the momentum balance equations is also implemented for Herschel–Bulkley (non-Newtonian) fluids. Non-equilibrium-induced two-phase flow problems and dynamic two-phase flows in simple two-dimensional (2-D) and three-dimensional (3-D) geometries are investigated to validate the model and its numerical implementation. Quantitative comparisons are made for cases with analytical solutions. Two-phase flow in an idealized 2-D pore-scale conduit is simulated to demonstrate the viability of the proposed direct numerical simulation approach.}, keywords = {}, pubstate = {published}, tppubtype = {article} } 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 sequentially, fall short in rigorously capturing crucial flow phenomena at the pore scale. Direct simulation techniques are needed that account for the full coupling between capillary and viscous flow phenomena. Consequently, there is a strong demand for robust and effective numerical methods that can deliver high-accuracy, high-resolution solutions of pore-scale flow in a computationally efficient manner. Direct simulations of pore-scale flow on imaged volumes can yield important insights about physical phenomena taking place during multi-phase, multi-component displacements. Such simulations can be utilized for optimizing various enhanced oil recovery (EOR) schemes and permit the computation of effective properties for Darcy-scale multi-phase flows. We implement a phase-field model for the direct pore-scale simulation of incompressible flow of two immiscible fluids. The model naturally lends itself to the transport of fluids with large density and viscosity ratios. In the phase-field approach, the fluid-phase interfaces are expressed in terms of thin transition regions, the so-called diffuse interfaces, for increased computational efficiency. The conservation law of mass for binary mixtures leads to the advective Cahn–Hilliard equation and the condition that the velocity field is divergence free. Momentum balance, on the other hand, leads to the Navier–Stokes equations for Newtonian fluids modified for two-phase flow and coupled to the advective Cahn–Hilliard equation. Unlike the volume of fluid (VoF) and level-set methods, which rely on regularization techniques to describe the phase interfaces, the phase-field method facilitates a thermodynamic treatment of the phase interfaces, rendering it more physically consistent for the direct simulations of two-phase pore-scale flow. A novel geometric wetting (wall) boundary condition is implemented as part of the phase-field method for the simulation of two-fluid flows with moving contact lines. The geometric boundary condition accurately replicates the prescribed equilibrium contact angle and is extended to account for dynamic (non-equilibrium) effects. The coupled advective Cahn–Hilliard and modified Navier–Stokes (phase-field) system is solved by using a robust and accurate semi-implicit finite volume method. An extension of the momentum balance equations is also implemented for Herschel–Bulkley (non-Newtonian) fluids. Non-equilibrium-induced two-phase flow problems and dynamic two-phase flows in simple two-dimensional (2-D) and three-dimensional (3-D) geometries are investigated to validate the model and its numerical implementation. Quantitative comparisons are made for cases with analytical solutions. Two-phase flow in an idealized 2-D pore-scale conduit is simulated to demonstrate the viability of the proposed direct numerical simulation approach. |
Balthasar Reuter; Vadym Aizinger; Manuel Wieland; Florian Frank; Peter Knabner FESTUNG: A MATLAB/GNU Octave toolbox for the discontinuous Galerkin method, Part II: Advection operator and slope limiting Journal Article Computers & Mathematics with Applications, 72 (7), pp. 1896–1925, 2016, ISBN: 0898-1221. @article{FESTUNG2, title = {FESTUNG: A MATLAB/GNU Octave toolbox for the discontinuous Galerkin method, Part II: Advection operator and slope limiting}, author = {Balthasar Reuter and Vadym Aizinger and Manuel Wieland and Florian Frank and Peter Knabner}, url = {http://www.sciencedirect.com/science/article/pii/S0898122116304606}, doi = {10.1016/j.camwa.2016.08.006}, isbn = {0898-1221}, year = {2016}, date = {2016-08-25}, journal = {Computers & Mathematics with Applications}, volume = {72}, number = {7}, pages = {1896–1925}, abstract = {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 comprehensively documented. Particular attention was paid to maintaining a direct mapping between discretization terms and code routines as well as to supporting the full code functionality in GNU Octave. The present work focuses on a two-dimensional time-dependent linear advection equation with space/time-varying coefficients, and provides a general order implementation of several slope limiting schemes for the DG method.}, keywords = {}, pubstate = {published}, tppubtype = {article} } 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 comprehensively documented. Particular attention was paid to maintaining a direct mapping between discretization terms and code routines as well as to supporting the full code functionality in GNU Octave. The present work focuses on a two-dimensional time-dependent linear advection equation with space/time-varying coefficients, and provides a general order implementation of several slope limiting schemes for the DG method. |
Florian Frank; Balthasar Reuter; Vadym Aizinger; Peter Knabner FESTUNG: A MATLAB/GNU Octave toolbox for the discontinuous Galerkin method, Part I: Diffusion operator Journal Article Computers & Mathematics with Applications, 70 (1), pp. 11–46, 2015, ISSN: 0898-1221. @article{FESTUNG, title = {FESTUNG: A MATLAB/GNU Octave toolbox for the discontinuous Galerkin method, Part I: Diffusion operator}, author = {Florian Frank and Balthasar Reuter and Vadym Aizinger and Peter Knabner}, url = {http://www1.am.uni-erlangen.de/FESTUNG}, doi = {10.1016/j.camwa.2015.04.013}, issn = {0898-1221}, year = {2015}, date = {2015-01-01}, journal = {Computers & Mathematics with Applications}, volume = {70}, number = {1}, pages = {11–46}, abstract = {This is the first 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 provide a rapid prototyping package for application development using DG methods. The implementation relies on fully vectorized matrix/vector operations and is carefully documented; in addition, a direct mapping between discretization terms and code routines is maintained throughout. The present work focuses on a two-dimensional time-dependent diffusion equation with space/time-varying coefficients. The spatial discretization is based on the local discontinuous Galerkin formulation. Approximations of orders zero through four based on orthogonal polynomials have been implemented; more spaces of arbitrary type and order can be easily accommodated by the code structure.}, keywords = {}, pubstate = {published}, tppubtype = {article} } This is the first 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 provide a rapid prototyping package for application development using DG methods. The implementation relies on fully vectorized matrix/vector operations and is carefully documented; in addition, a direct mapping between discretization terms and code routines is maintained throughout. The present work focuses on a two-dimensional time-dependent diffusion equation with space/time-varying coefficients. The spatial discretization is based on the local discontinuous Galerkin formulation. Approximations of orders zero through four based on orthogonal polynomials have been implemented; more spaces of arbitrary type and order can be easily accommodated by the code structure. |
Nadja Ray; Tycho van Noorden; Florian Frank; Peter Knabner Multiscale modeling of colloid and fluid dynamics in porous media including an evolving microstructure Journal Article Transport in Porous Media, 95 (3), pp. 669–696, 2012, ISSN: 0169-3913. @article{RvNFK2012, title = {Multiscale modeling of colloid and fluid dynamics in porous media including an evolving microstructure}, author = {Nadja Ray and Tycho van Noorden and Florian Frank and Peter Knabner}, doi = {10.1007/s11242-012-0068-z}, issn = {0169-3913}, year = {2012}, date = {2012-01-01}, journal = {Transport in Porous Media}, volume = {95}, number = {3}, pages = {669–696}, publisher = {Springer}, abstract = {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 matrix due to (non-)DLVO forces is included as an additional transport mechanism. Fluid flow is described by incompressible Stokes equations with interaction energy as forcing term. Attachment and detachment processes are modeled by a surface reaction rate. The evolution of the underlying microstructure is captured by a level set function. The crucial point in completing this model is to set up appropriate boundary conditions on the evolving solid–liquid interface. Their derivation is based on mass conservation. As a result of an averaging procedure by periodic homogenization in a level set framework, on the macroscale we obtain Darcy’s law and a modified averaged convection–diffusion equation with effective coefficients due to the evolving microstructure. These equations are supplemented by microscopic cell problems. Time- and space-dependent averaged coefficient functions explicitly contain information of the underlying geometry and also information of the interaction potential. The theoretical results are complemented by numerical computations of the averaged coefficients and simulations of a heterogeneous multiscale scenario. Here, we consider a radially symmetric setting, i.e., in particular we assume a locally periodic geometry consisting of circular grains. We focus on the interplay between attachment and detachment reaction, colloidal interaction forces, and the evolving microstructure. Our model contributes to the understanding of the effects and processes leading to porosity changes and pore clogging from a theoretical point of view.}, keywords = {}, pubstate = {published}, tppubtype = {article} } 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 matrix due to (non-)DLVO forces is included as an additional transport mechanism. Fluid flow is described by incompressible Stokes equations with interaction energy as forcing term. Attachment and detachment processes are modeled by a surface reaction rate. The evolution of the underlying microstructure is captured by a level set function. The crucial point in completing this model is to set up appropriate boundary conditions on the evolving solid–liquid interface. Their derivation is based on mass conservation. As a result of an averaging procedure by periodic homogenization in a level set framework, on the macroscale we obtain Darcy’s law and a modified averaged convection–diffusion equation with effective coefficients due to the evolving microstructure. These equations are supplemented by microscopic cell problems. Time- and space-dependent averaged coefficient functions explicitly contain information of the underlying geometry and also information of the interaction potential. The theoretical results are complemented by numerical computations of the averaged coefficients and simulations of a heterogeneous multiscale scenario. Here, we consider a radially symmetric setting, i.e., in particular we assume a locally periodic geometry consisting of circular grains. We focus on the interplay between attachment and detachment reaction, colloidal interaction forces, and the evolving microstructure. Our model contributes to the understanding of the effects and processes leading to porosity changes and pore clogging from a theoretical point of view. |
Florian Frank; Nadja Ray; Peter Knabner Numerical investigation of homogenized Stokes–Nernst–Planck–Poisson systems Journal Article Computing and Visualization in Science, 14 (8), pp. 385–400, 2011, ISSN: 1432-9360. @article{FRK2013, title = {Numerical investigation of homogenized Stokes–Nernst–Planck–Poisson systems}, author = {Florian Frank and Nadja Ray and Peter Knabner}, doi = {10.1007/s00791-013-0189-0}, issn = {1432-9360}, year = {2011}, date = {2011-01-01}, journal = {Computing and Visualization in Science}, volume = {14}, number = {8}, pages = {385–400}, publisher = {Springer Berlin Heidelberg}, abstract = {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 provide closed-form expressions for effective coefficients. Our aim is to study numerically the convergence of the models for vanishing microstructure, i. e., the behavior for ε → 0, where ε is the characteristic ratio between pore diameter and size of the porous medium. To this end, we propose a numerical scheme capable of solving the fully coupled microscopic SNPP system and also the corresponding averaged systems. The discretization is performed fully implicitly in time using mixed finite elements in two space dimensions. The averaged models are evaluated using simulation results and their approximation errors in terms of ε are estimated numerically.}, keywords = {}, pubstate = {published}, tppubtype = {article} } 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 provide closed-form expressions for effective coefficients. Our aim is to study numerically the convergence of the models for vanishing microstructure, i. e., the behavior for ε → 0, where ε is the characteristic ratio between pore diameter and size of the porous medium. To this end, we propose a numerical scheme capable of solving the fully coupled microscopic SNPP system and also the corresponding averaged systems. The discretization is performed fully implicitly in time using mixed finite elements in two space dimensions. The averaged models are evaluated using simulation results and their approximation errors in terms of ε are estimated numerically. |
Fabian Brunner; Florian Frank; Peter Knabner FV upwind stabilization of FE discretizations for advection–diffusion problems Inproceedings Finite Volumes for Complex Applications VII – Methods and Theoretical Aspects, pp. 177–185, Springer, 2014. @inproceedings{BFK2014, title = {FV upwind stabilization of FE discretizations for advection–diffusion problems}, author = {Fabian Brunner and Florian Frank and Peter Knabner}, doi = {10.1007/978-3-319-05684-5_16}, year = {2014}, date = {2014-01-01}, booktitle = {Finite Volumes for Complex Applications VII – Methods and Theoretical Aspects}, journal = {Finite Volumes for Complex Applications VII}, volume = {77}, pages = {177–185}, publisher = {Springer}, abstract = {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 advective term. Numerical experiments indicate that the upwind-mixed method is competitive with the finite volume method. It prevents the appearance of spurious oscillations and produces nonnegative solutions for strongly advection-dominated problems, while the amount of artificial diffusion is lower than that of the finite volume method. This makes the method attractive for applications in which too much numerical diffusion is critical and may lead to false predictions; e.g., if highly nonlinear reactive processes take place only in thin interaction regions.}, keywords = {}, pubstate = {published}, tppubtype = {inproceedings} } 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 advective term. Numerical experiments indicate that the upwind-mixed method is competitive with the finite volume method. It prevents the appearance of spurious oscillations and produces nonnegative solutions for strongly advection-dominated problems, while the amount of artificial diffusion is lower than that of the finite volume method. This makes the method attractive for applications in which too much numerical diffusion is critical and may lead to false predictions; e.g., if highly nonlinear reactive processes take place only in thin interaction regions. |
Peter Knabner; Florian Frank; Joachim Hoffmann; Serge Kräutle; Stephan Oßmann; Alexander Prechtel Entwicklung, Zuverlässigkeit und Effizienz reaktiver Mehrkomponententransportmodelle Incollection Systemanalyse, Modellierung und Prognose von Natural Attenuation-Prozessen um Untergrund, Synopse des KORA-Themenverbund 7, pp. 195–233, 2008, ISSN: 1611-5627, (DGFZ, Dresden). @incollection{KFHKOP2008, title = {Entwicklung, Zuverlässigkeit und Effizienz reaktiver Mehrkomponententransportmodelle}, author = {Peter Knabner and Florian Frank and Joachim Hoffmann and Serge Kräutle and Stephan Oßmann and Alexander Prechtel}, url = {http://www.natural-attenuation.de/download/kora-projekt-70-dgfz-dresden.pdf}, issn = {1611-5627}, year = {2008}, date = {2008-01-01}, booktitle = {Systemanalyse, Modellierung und Prognose von Natural Attenuation-Prozessen um Untergrund, Synopse des KORA-Themenverbund 7}, pages = {195–233}, note = {DGFZ, Dresden}, keywords = {}, pubstate = {published}, tppubtype = {incollection} } |
Christopher Thiele; Mauricio Araya-Polo; Faruk Omer Alpak; Béatrice Rivière; Florian Frank Society of Petroleum Engineers, 2017. @conference{Thiele2017IHSS, title = {Inexact hierarchical scale separation: an efficient linear solver for discontinuous Galerkin discretizations}, author = {Christopher Thiele and Mauricio Araya-Polo and Faruk Omer Alpak and Béatrice Rivière and Florian Frank}, url = {https://www.onepetro.org/conference-paper/SPE-182671-MS}, doi = {10.2118/182671-MS}, year = {2017}, date = {2017-02-20}, journal = { SPE Reservoir Simulation Symposium. SPE-182671-MS}, publisher = {Society of Petroleum Engineers}, abstract = {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 large-scale simulations, and compare its performance to a widely used iterative solver. The solution of the linear systems, in particular in massively parallel applications, is a known performance bottleneck in direct numerical approaches. HSS splits the linear system into a coarse-scale system of reduced size corresponding to the local mean values of the DG solution, and a set of decoupled local fine-scale systems corresponding to the higher order components of the DG solution. The scheme then alternates between coarse-scale and fine-scale system solves until both components converge, employing a standard iterative solver for the coarse-scale system and direct solves for the set of small fine-scale systems, which allow for an optimal parallelization. The motivation of HSS is to increase parallelism by decoupling the linear systems, therefore reducing the communication overhead within sparse matrix-vector multiplications of classical iterative solvers. Providing some mild assumptions on the underlying DG basis functions, the above-mentioned splitting can be done on the resulting linear systems (i. e. without knowledge of the numerical scheme), which further motivates the development of the HSS scheme as a blackbox solver for DG discretizations. We propose a modified HSS algorithm ("inexact HSS," IHSS") that shifts computation to the highly parallel fine-scale solver, and thus reduces global synchronization. The key result is that the IHSS scheme significantly speeds up the linear system solves and outperforms a standard GMRES solver (up to 9x speedup for some configurations). A hybrid parallel IHSS solver has been implemented using the Trilinos package. Its convergence for linear systems from the Cahn-Hilliard problem is verified, and its performance is compared to a standard iterative solver from the same package. In the future, IHSS may possibly be used as a blackbox solver to speed up DG based simulations, e.g., of reservoir flow or multicomponent transport.}, keywords = {}, pubstate = {published}, tppubtype = {conference} } 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 large-scale simulations, and compare its performance to a widely used iterative solver. The solution of the linear systems, in particular in massively parallel applications, is a known performance bottleneck in direct numerical approaches. HSS splits the linear system into a coarse-scale system of reduced size corresponding to the local mean values of the DG solution, and a set of decoupled local fine-scale systems corresponding to the higher order components of the DG solution. The scheme then alternates between coarse-scale and fine-scale system solves until both components converge, employing a standard iterative solver for the coarse-scale system and direct solves for the set of small fine-scale systems, which allow for an optimal parallelization. The motivation of HSS is to increase parallelism by decoupling the linear systems, therefore reducing the communication overhead within sparse matrix-vector multiplications of classical iterative solvers. Providing some mild assumptions on the underlying DG basis functions, the above-mentioned splitting can be done on the resulting linear systems (i. e. without knowledge of the numerical scheme), which further motivates the development of the HSS scheme as a blackbox solver for DG discretizations. We propose a modified HSS algorithm ("inexact HSS," IHSS") that shifts computation to the highly parallel fine-scale solver, and thus reduces global synchronization. The key result is that the IHSS scheme significantly speeds up the linear system solves and outperforms a standard GMRES solver (up to 9x speedup for some configurations). A hybrid parallel IHSS solver has been implemented using the Trilinos package. Its convergence for linear systems from the Cahn-Hilliard problem is verified, and its performance is compared to a standard iterative solver from the same package. In the future, IHSS may possibly be used as a blackbox solver to speed up DG based simulations, e.g., of reservoir flow or multicomponent transport. |
Florian Frank; Chen Liu; Faruk Omer Alpak; Mauricio Araya-Polo; Béatrice Rivière Society of Petroleum Engineers, 2017. @conference{Frank2017SPE, title = {A discontinuous Galerkin finite element framework for the direct numerical simulation of flow on high-resolution pore-scale images}, author = {Florian Frank and Chen Liu and Faruk Omer Alpak and Mauricio Araya-Polo and Béatrice Rivière }, url = {https://www.onepetro.org/conference-paper/SPE-182607-MS}, doi = {10.2118/182607-MS}, year = {2017}, date = {2017-02-20}, journal = {SPE Reservoir Simulation Symposium. SPE-182606-MS}, publisher = {Society of Petroleum Engineers}, abstract = {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, i.e. at scales where the Darcy velocity homogenization starts to break down. The simulator is one of the key components of a molecule-to-reservoir truly multiscale modeling workflow. A Helmholtz free-energy driven, thermodynamically based diffuse-interface method is used for the effective simulation of a large number of advecting interfaces, while honoring the interfacial tension. The advective Cahn–Hilliard (mass balance) and Navier–Stokes (momentum balance) equations are coupled to each other within the phase-field framework. Wettability on rock-fluid interfaces is accounted for via an energy-penalty based wetting (contact-angle) boundary condition. Individual balance equations are discretized by use of a flexible discontinuous Galerkin (DG) method. The discretization of the mass balance equation is semi-implicit in time; momentum balance equation is discretized with a fully-implicit scheme, while both equations are coupled via an iterative operator splitting approach. We discuss the mathematical model, DG discretization, and briefly introduce nonlinear and linear solution strategies. Numerical validation tests show optimal convergence rates for the DG discretization indicating the correctness of the numerical scheme. Physical validation tests demonstrate the consistency of the mass distribution and velocity fields simulated within our framework. Finally, two-phase flow simulations on two real pore-scale images demonstrate the utility of the pore-scale simulator. The direct pore-scale numerical simulation method overcomes the limitations of pore network models by rigorously taking into account the flow physics and by directly acting on pore-scale images of rocks without requiring a network abstraction step or remeshing. The proposed method is accurate, numerically robust, and exhibits the potential for tackling realistic problems.}, keywords = {}, pubstate = {published}, tppubtype = {conference} } 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, i.e. at scales where the Darcy velocity homogenization starts to break down. The simulator is one of the key components of a molecule-to-reservoir truly multiscale modeling workflow. A Helmholtz free-energy driven, thermodynamically based diffuse-interface method is used for the effective simulation of a large number of advecting interfaces, while honoring the interfacial tension. The advective Cahn–Hilliard (mass balance) and Navier–Stokes (momentum balance) equations are coupled to each other within the phase-field framework. Wettability on rock-fluid interfaces is accounted for via an energy-penalty based wetting (contact-angle) boundary condition. Individual balance equations are discretized by use of a flexible discontinuous Galerkin (DG) method. The discretization of the mass balance equation is semi-implicit in time; momentum balance equation is discretized with a fully-implicit scheme, while both equations are coupled via an iterative operator splitting approach. We discuss the mathematical model, DG discretization, and briefly introduce nonlinear and linear solution strategies. Numerical validation tests show optimal convergence rates for the DG discretization indicating the correctness of the numerical scheme. Physical validation tests demonstrate the consistency of the mass distribution and velocity fields simulated within our framework. Finally, two-phase flow simulations on two real pore-scale images demonstrate the utility of the pore-scale simulator. The direct pore-scale numerical simulation method overcomes the limitations of pore network models by rigorously taking into account the flow physics and by directly acting on pore-scale images of rocks without requiring a network abstraction step or remeshing. The proposed method is accurate, numerically robust, and exhibits the potential for tackling realistic problems. |
Christopher Thiele; Mauricio Araya-Polo; Dimitar Stoyanov; Florian Frank; Faruk Omer Alpak Asynchronous hybrid parallel SpMV in an industrial application Conference 2016 International Conference on Computational Science and Computational Intelligence (CSCI), IEEE, 2016. @conference{Thiele2016IEEE, title = {Asynchronous hybrid parallel SpMV in an industrial application}, author = {Christopher Thiele and Mauricio Araya-Polo and Dimitar Stoyanov and Florian Frank and Faruk Omer Alpak}, url = {http://ieeexplore.ieee.org/document/7881519}, doi = {10.1109/CSCI.2016.0226}, year = {2016}, date = {2016-12-15}, booktitle = {2016 International Conference on Computational Science and Computational Intelligence (CSCI)}, pages = {1196–1201}, publisher = {IEEE}, abstract = {Direct numerical simulations of pore-scale flows are nowadays feasible due to advances in pore-scale imaging of rock samples obtained during hydrocarbon exploration and production, HPC availability, and numerical algorithms. The Cahn–Hilliard equation governs the separation of a two-component mixture into distinct bulk phases in the pore space. The efficient solution of the linear systems arising from the discretization of the Cahn–Hilliard equation is our subject of this study. The underlying spatial discretization is based on the mixed formulation using the interior penalty discontinuous Galerkin method. The resulting nonlinear system is linearized by the Newton–Raphson method, which requires the solution of multiple large sparse linear systems at each time step. The sparse linear systems are in turn solved by an iterative linear solver. In this work, we investigate parallel iterative solvers for large sparse linear systems and their suitability for practical direct pore-scale simulation problems. We focus our efforts on two main aspects: software infrastructure that supports this kind of computations and their scalability and performance. Based on our review of the main software frameworks for distributed linear algebra, we decided to implement the simulator with Trilinos. In order to investigate possible performance gains from asynchronicity in iterative solvers, we compared Trilinos against GaspiLS, a solver based on the GASPI PGAS specification. GaspiLS overlaps communication and computation during sparse matrix-vector products. The asynchronous solver performs well and scales better than the synchronous ones for most test problems, especially for large linear systems distributed over many computational nodes (up to 23% faster and 10% higher parallel efficiency in these cases). Trilinos is an established, comprehensive framework that provides a reliable and efficient platform for complex PDE solvers. GaspiLS on the other hand shows good performance in its early development stages and aims to extend its asynchronous approach to additional solvers, preconditioners, and features.}, keywords = {}, pubstate = {published}, tppubtype = {conference} } Direct numerical simulations of pore-scale flows are nowadays feasible due to advances in pore-scale imaging of rock samples obtained during hydrocarbon exploration and production, HPC availability, and numerical algorithms. The Cahn–Hilliard equation governs the separation of a two-component mixture into distinct bulk phases in the pore space. The efficient solution of the linear systems arising from the discretization of the Cahn–Hilliard equation is our subject of this study. The underlying spatial discretization is based on the mixed formulation using the interior penalty discontinuous Galerkin method. The resulting nonlinear system is linearized by the Newton–Raphson method, which requires the solution of multiple large sparse linear systems at each time step. The sparse linear systems are in turn solved by an iterative linear solver. In this work, we investigate parallel iterative solvers for large sparse linear systems and their suitability for practical direct pore-scale simulation problems. We focus our efforts on two main aspects: software infrastructure that supports this kind of computations and their scalability and performance. Based on our review of the main software frameworks for distributed linear algebra, we decided to implement the simulator with Trilinos. In order to investigate possible performance gains from asynchronicity in iterative solvers, we compared Trilinos against GaspiLS, a solver based on the GASPI PGAS specification. GaspiLS overlaps communication and computation during sparse matrix-vector products. The asynchronous solver performs well and scales better than the synchronous ones for most test problems, especially for large linear systems distributed over many computational nodes (up to 23% faster and 10% higher parallel efficiency in these cases). Trilinos is an established, comprehensive framework that provides a reliable and efficient platform for complex PDE solvers. GaspiLS on the other hand shows good performance in its early development stages and aims to extend its asynchronous approach to additional solvers, preconditioners, and features. |
Max Grossman; Mauricio Araya-Polo; Faruk Omer Alpak; Florian Frank; Jan Limbeck; Vivek Sarkar Analysis of sparse matrix-vector multiply for large sparse linear systems Conference ECMOR XV – 15th European Conference on the Mathematics of Oil Recovery, 2016. @conference{Grossman2016Analysis, title = {Analysis of sparse matrix-vector multiply for large sparse linear systems}, author = {Max Grossman and Mauricio Araya-Polo and Faruk Omer Alpak and Florian Frank and Jan Limbeck and Vivek Sarkar }, url = {http://www.earthdoc.org/publication/publicationdetails/?publication=86246}, doi = {10.3997/2214-4609.201601798}, year = {2016}, date = {2016-08-29}, booktitle = {ECMOR XV – 15th European Conference on the Mathematics of Oil Recovery}, abstract = {Advances in pore-scale imaging (e.g., micro-CT scanning), increasing the 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. Discretization of the partial differential equations that govern the physics of multiphase fluid flow and transport and phase separation gives rise to large sparse linear systems for practical direct pore-scale simulation problems. Of particular interest to us is a linear system arising from the discretization of the Cahn–Hilliard equation—a fourth order nonlinear parabolic partial differential equation that governs the separation of a two-component mixture into distinct bulk phases in the pore space. The underlying spatial discretization is performed using the discontinuous Galerkin method. The resulting nonlinear system is solved by the use of the Newton–Raphson method, which requires the solution of multiple large sparse linear systems over the course of several nonlinear solver iterations. The sparse linear systems are in turn solved by the use of an iterative linear solver. Iterative linear solvers approach the solution process by gradually requiring the computation of sparse matrix-vector (SpMV) products. SpMV products often emerge as a computational bottleneck for the simulation of large problems, since they are extremely memory bound. In this work, we perform a quantitative and qualitative evaluation of techniques for performing SpMV on large matrices. We evaluate different SpMV software implementations (frameworks and kernels) on absolute performance, performance portability, and programmability across a range of state-of-the-art hardware platforms: x86-based, Intel MIC and NVIDIA GPU-accelerated. We examine their behavior at several different dataset scales, from matrices on the order of megabytes up to dozens of gigabytes. In general, we find that no implementation meets all needs and that failing to rigorously evaluate all options at each change in application characteristics (e.g. matrix sizes) can result in significant losses in performance. For example, we find that for a 5GB matrix wrt to a naive multi-threaded x86 baseline the highest performing GPU kernel runs about 1.75x faster, and the highest performing x86 kernel runs about 1.29x faster.}, keywords = {}, pubstate = {published}, tppubtype = {conference} } Advances in pore-scale imaging (e.g., micro-CT scanning), increasing the 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. Discretization of the partial differential equations that govern the physics of multiphase fluid flow and transport and phase separation gives rise to large sparse linear systems for practical direct pore-scale simulation problems. Of particular interest to us is a linear system arising from the discretization of the Cahn–Hilliard equation—a fourth order nonlinear parabolic partial differential equation that governs the separation of a two-component mixture into distinct bulk phases in the pore space. The underlying spatial discretization is performed using the discontinuous Galerkin method. The resulting nonlinear system is solved by the use of the Newton–Raphson method, which requires the solution of multiple large sparse linear systems over the course of several nonlinear solver iterations. The sparse linear systems are in turn solved by the use of an iterative linear solver. Iterative linear solvers approach the solution process by gradually requiring the computation of sparse matrix-vector (SpMV) products. SpMV products often emerge as a computational bottleneck for the simulation of large problems, since they are extremely memory bound. In this work, we perform a quantitative and qualitative evaluation of techniques for performing SpMV on large matrices. We evaluate different SpMV software implementations (frameworks and kernels) on absolute performance, performance portability, and programmability across a range of state-of-the-art hardware platforms: x86-based, Intel MIC and NVIDIA GPU-accelerated. We examine their behavior at several different dataset scales, from matrices on the order of megabytes up to dozens of gigabytes. In general, we find that no implementation meets all needs and that failing to rigorously evaluate all options at each change in application characteristics (e.g. matrix sizes) can result in significant losses in performance. For example, we find that for a 5GB matrix wrt to a naive multi-threaded x86 baseline the highest performing GPU kernel runs about 1.75x faster, and the highest performing x86 kernel runs about 1.29x faster. |
Max Grossman; Christopher Thiele; Mauricio Araya-Polo; Florian Frank; Faruk Omer Alpak; Vivek Sarkar A survey of sparse matrix-vector multiplication performance on large matrices Conference Rice Oil & Gas High Performance Computing Workshop, 2016. @conference{GrossmanEtAl2016Survey, title = {A survey of sparse matrix-vector multiplication performance on large matrices}, author = {Max Grossman and Christopher Thiele and Mauricio Araya-Polo and Florian Frank and Faruk Omer Alpak and Vivek Sarkar}, url = {http://arxiv.org/abs/1608.00636}, year = {2016}, date = {2016-03-03}, booktitle = {Rice Oil & Gas High Performance Computing Workshop}, abstract = {One of the main sources of sparse matrices is the discretization of partial differential equations that govern continuum- physics phenomena such as fluid flow and transport, phase separation, mechanical deformation, electromagnetic wave propagation, and others. Recent advances in high performance computing area have been enabling researchers to tackle increasingly larger problems leading to sparse linear systems with hundreds of millions to a few tens of billions of unknowns. Iterative linear solvers are popular in large-scale computing as they consume less memory than direct solvers. Contrary to direct linear solvers, iterative solvers approach the solution gradually requiring the computation of sparse matrix- vector (SpMV) products. The evaluation of SpMV products can emerge as a bottleneck for computational performance within the context of the simulation of large problems. In this work, we focus on a linear system arising from the discretization of the Cahn–Hilliard equation, which is a fourth order nonlinear parabolic partial differential equation that governs the separation of a two-component mixture into phases [3]. The underlying spatial discretization is performed using the dis- continuous Galerkin method and Newton’s method. A number of parallel algorithms and strategies have been evaluated in this work to accelerate the evaluation of SpMV products.}, keywords = {}, pubstate = {published}, tppubtype = {conference} } One of the main sources of sparse matrices is the discretization of partial differential equations that govern continuum- physics phenomena such as fluid flow and transport, phase separation, mechanical deformation, electromagnetic wave propagation, and others. Recent advances in high performance computing area have been enabling researchers to tackle increasingly larger problems leading to sparse linear systems with hundreds of millions to a few tens of billions of unknowns. Iterative linear solvers are popular in large-scale computing as they consume less memory than direct solvers. Contrary to direct linear solvers, iterative solvers approach the solution gradually requiring the computation of sparse matrix- vector (SpMV) products. The evaluation of SpMV products can emerge as a bottleneck for computational performance within the context of the simulation of large problems. In this work, we focus on a linear system arising from the discretization of the Cahn–Hilliard equation, which is a fourth order nonlinear parabolic partial differential equation that governs the separation of a two-component mixture into phases [3]. The underlying spatial discretization is performed using the dis- continuous Galerkin method and Newton’s method. A number of parallel algorithms and strategies have been evaluated in this work to accelerate the evaluation of SpMV products. |
Florian Frank; Chen Liu; Alessio Scanziani; Faruk Omer Alpak; Béatrice Rivière An energy-based contact angle boundary condition on jagged surfaces for the Cahn–Hilliard equation Technical Report 2017. @techreport{FrankAR2017CA, title = { An energy-based contact angle boundary condition on jagged surfaces for the Cahn–Hilliard equation}, author = {Florian Frank and Chen Liu and Alessio Scanziani and Faruk Omer Alpak and Béatrice Rivière}, url = {https://arxiv.org/abs/1711.05815}, year = {2017}, date = {2017-11-15}, abstract = {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 micrometer scale) smooth domain with a certain resolution. Planar surfaces that are perpendicular to the main axes are naturally approximated by a layer of voxels. However, planar surfaces in any other directions and curved surfaces yield a jagged/rough surface approximation by voxels. For the standard Cahn-Hilliard formulation, where the contact angle between the diffuse interface and the domain boundary (fluid-solid interface/wall) is 90 degrees, jagged surfaces have no impact on the contact angle. However, a prescribed contact angle smaller or larger than 90 degrees on jagged voxel surfaces is amplified in either direction. As a remedy, we propose the introduction of surface energy correction factors for each fluid-solid voxel face that counterbalance the difference of the voxel-set surface area with the underlying smooth one. The discretization of the model equations is performed with the discontinuous Galerkin method, however, the presented semi-analytical approach of correcting the surface energy is equally applicable to other direct numerical methods such as finite elements, finite volumes, or finite differences, since the correction factors appear in the strong formulation of the model.}, keywords = {}, pubstate = {published}, tppubtype = {techreport} } 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 micrometer scale) smooth domain with a certain resolution. Planar surfaces that are perpendicular to the main axes are naturally approximated by a layer of voxels. However, planar surfaces in any other directions and curved surfaces yield a jagged/rough surface approximation by voxels. For the standard Cahn-Hilliard formulation, where the contact angle between the diffuse interface and the domain boundary (fluid-solid interface/wall) is 90 degrees, jagged surfaces have no impact on the contact angle. However, a prescribed contact angle smaller or larger than 90 degrees on jagged voxel surfaces is amplified in either direction. As a remedy, we propose the introduction of surface energy correction factors for each fluid-solid voxel face that counterbalance the difference of the voxel-set surface area with the underlying smooth one. The discretization of the model equations is performed with the discontinuous Galerkin method, however, the presented semi-analytical approach of correcting the surface energy is equally applicable to other direct numerical methods such as finite elements, finite volumes, or finite differences, since the correction factors appear in the strong formulation of the model. |
Florian Frank; Chen Liu; Faruk Omer Alpak; Steffen Berg; Béatrice Rivière Direct numerical simulation of flow on pore-scale images Unpublished 2017. @unpublished{FrankEtAl2017Direct, title = {Direct numerical simulation of flow on pore-scale images}, author = {Florian Frank and Chen Liu and Faruk Omer Alpak and Steffen Berg and Béatrice Rivière}, year = {2017}, date = {2017-08-31}, keywords = {}, pubstate = {published}, tppubtype = {unpublished} } |
Florian Frank; Faruk Omer Alpak; Chen Liu; Béatrice Rivière 2016. @unpublished{FALR2016, title = {A discontinuous Galerkin discretization of the advective Cahn–Hilliard equation on porous domains stemming from micro-CT imaging}, author = {Florian Frank and Faruk Omer Alpak and Chen Liu and Béatrice Rivière}, url = {https://arxiv.org/abs/1610.03457}, year = {2016}, date = {2016-10-11}, abstract = {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 constraint and dissipation of free energy. Porous media/pore-scale problems specifically entail high-resolution images of rocks in which the solid matrix and pore spaces are fully resolved. The interior penalty discontinuous Galerkin method is used for the spatial discretization of the CH equation in mixed form, while a semi-implicit convex–concave splitting is utilized for temporal discretization. The spatial approximation order is arbitrary, while it reduces to a finite volume scheme for the choice of elementwise constants. The resulting nonlinear systems of equations are reduced using the Schur complement and solved via Newton's method. The numerical scheme is first validated using numerical convergence tests and then applied to a number of fundamental problems for validation and numerical experimentation purposes including the case of degenerate mobility. First-order physical applicability and robustness of the numerical method are shown in a breakthrough scenario on a voxel set obtained from a micro-CT scan of a real sandstone rock sample.}, keywords = {}, pubstate = {published}, tppubtype = {unpublished} } 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 constraint and dissipation of free energy. Porous media/pore-scale problems specifically entail high-resolution images of rocks in which the solid matrix and pore spaces are fully resolved. The interior penalty discontinuous Galerkin method is used for the spatial discretization of the CH equation in mixed form, while a semi-implicit convex–concave splitting is utilized for temporal discretization. The spatial approximation order is arbitrary, while it reduces to a finite volume scheme for the choice of elementwise constants. The resulting nonlinear systems of equations are reduced using the Schur complement and solved via Newton's method. The numerical scheme is first validated using numerical convergence tests and then applied to a number of fundamental problems for validation and numerical experimentation purposes including the case of degenerate mobility. First-order physical applicability and robustness of the numerical method are shown in a breakthrough scenario on a voxel set obtained from a micro-CT scan of a real sandstone rock sample. |
Florian Frank University of Erlangen–Nürnberg, 2013. @phdthesis{Frank2013, title = {Numerical studies of models for electrokinetic flow and charged solute transport in periodic porous media}, author = {Florian Frank}, url = {https://opus4.kobv.de/opus4-fau/frontdoor/index/index/docId/4025}, year = {2013}, date = {2013-01-01}, school = {University of Erlangen–Nürnberg}, abstract = {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 field-scale simulations that resolve the geometry of the solid matrix at the pore scale are not feasible in practice, a major interest lies in the quality assessment of corresponding averaged models. Depending on the chosen scaling, the different averaged models under investigation reasonably describe to a greater or lesser extent the effective macroscopic behavior of the phenomena considered. The underlying partial differential equations include effective tensors, the closed-form expression of which is provided by averaging of the solutions of auxiliary problems. These so-called cell problems are defined on small domains reflecting the periodic geometry of the solid matrix. The main objectives are both the qualitative and the quantitative investigation of homogenization processes by means of an extensive numerical study, i.e., of the convergence properties of the SNPP systems for vanishing microstructure. To this end, numerical schemes are proposed that are capable of solving accurately and efficiently the non-stationary, fully coupled/nonlinear SNPP system and also the corresponding averaged systems. The discretization is performed fully implicitly in time, while using mixed finite elements in two space dimensions, which are locally mass conservative with respect to the concentration of charged particles. The schemes are of optimal order in the discretization parameters, which is demonstrated numerically and also shown rigorously by an a priori error estimate for the overall discretization error. Subsequently, the thesis proceeds with the numerical realization of an extension to the SNPP system allowing for attachment and detachment processes on the surface of the considered locally periodic solid matrix. The resulting evolving microstructure has an impact on the liquid flow and thus consequently on the solute transport. The corresponding two-scale model, which contains these inter-scale dependencies, is approached numerically using mixed finite elements on both scales. Simulations illustrate the interplay between solute transport, evolving microstructure, and liquid flow.}, keywords = {}, pubstate = {published}, tppubtype = {phdthesis} } 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 field-scale simulations that resolve the geometry of the solid matrix at the pore scale are not feasible in practice, a major interest lies in the quality assessment of corresponding averaged models. Depending on the chosen scaling, the different averaged models under investigation reasonably describe to a greater or lesser extent the effective macroscopic behavior of the phenomena considered. The underlying partial differential equations include effective tensors, the closed-form expression of which is provided by averaging of the solutions of auxiliary problems. These so-called cell problems are defined on small domains reflecting the periodic geometry of the solid matrix. The main objectives are both the qualitative and the quantitative investigation of homogenization processes by means of an extensive numerical study, i.e., of the convergence properties of the SNPP systems for vanishing microstructure. To this end, numerical schemes are proposed that are capable of solving accurately and efficiently the non-stationary, fully coupled/nonlinear SNPP system and also the corresponding averaged systems. The discretization is performed fully implicitly in time, while using mixed finite elements in two space dimensions, which are locally mass conservative with respect to the concentration of charged particles. The schemes are of optimal order in the discretization parameters, which is demonstrated numerically and also shown rigorously by an a priori error estimate for the overall discretization error. Subsequently, the thesis proceeds with the numerical realization of an extension to the SNPP system allowing for attachment and detachment processes on the surface of the considered locally periodic solid matrix. The resulting evolving microstructure has an impact on the liquid flow and thus consequently on the solute transport. The corresponding two-scale model, which contains these inter-scale dependencies, is approached numerically using mixed finite elements on both scales. Simulations illustrate the interplay between solute transport, evolving microstructure, and liquid flow. |
Florian Frank Hydrogeochemical multi-component transport—mineral dissolution and precipitation with consideration of porosity changes in variably-saturated porous media Masters Thesis University of Erlangen–Nürnberg, 2008. @mastersthesis{Frank2008, title = {Hydrogeochemical multi-component transport—mineral dissolution and precipitation with consideration of porosity changes in variably-saturated porous media}, author = {Florian Frank}, year = {2008}, date = {2008-01-01}, school = {University of Erlangen–Nürnberg}, keywords = {}, pubstate = {published}, tppubtype = {mastersthesis} } |
Frank Florian; Reuter Balthasar; Aizinger Vadym Poster: FESTUNG – The Finite Element Simulation Toolbox for UNstructured Grids Presentation Sep 22, 2017, (SIAM TAMES: Texas Applied Mathematics and Engineering Symposium, Sep 21–23, 2017, Austin, TX, USA). @misc{FESTUNG2017Poster, title = {Poster: FESTUNG – The Finite Element Simulation Toolbox for UNstructured Grids}, author = {Frank Florian and Reuter Balthasar and Aizinger Vadym}, url = {https://www1.am.uni-erlangen.de/FESTUNG}, year = {2017}, date = {2017-09-22}, abstract = {FESTUNG is a free and open source MATLAB/GNU Octave toolbox with primary uses as a research and education tool. It implements the Discontinuous Galerkin Finite Element method and offers a rapid prototyping framework for application and algorithm development. Careful documentation, a clear code structure, and intuitive interfaces make the usage of the software package simple. By combining this simplicity with fully vectorized matrix/vector operations that deliver the optimal performance for MATLAB/GNU Octave applications, FESTUNG represents a very attractive software platform for small to medium-sized problems. The current set of features includes among others support for unstructured triangular grids, Local- and Hybridized-DG discretizations, higher order explicit and implicit Runge-Kutta time stepping methods, vertex based hierarchical slope limiters for arbitrary polynomial approximation spaces, a generic solver formulation that allows for easy coupling of multi-physics problems, and VTK/Tecplot-compatible output formats. Example implementations of advection- and diffusion-type operators are readily available on GitHub.}, note = {SIAM TAMES: Texas Applied Mathematics and Engineering Symposium, Sep 21–23, 2017, Austin, TX, USA}, keywords = {}, pubstate = {published}, tppubtype = {presentation} } FESTUNG is a free and open source MATLAB/GNU Octave toolbox with primary uses as a research and education tool. It implements the Discontinuous Galerkin Finite Element method and offers a rapid prototyping framework for application and algorithm development. Careful documentation, a clear code structure, and intuitive interfaces make the usage of the software package simple. By combining this simplicity with fully vectorized matrix/vector operations that deliver the optimal performance for MATLAB/GNU Octave applications, FESTUNG represents a very attractive software platform for small to medium-sized problems. The current set of features includes among others support for unstructured triangular grids, Local- and Hybridized-DG discretizations, higher order explicit and implicit Runge-Kutta time stepping methods, vertex based hierarchical slope limiters for arbitrary polynomial approximation spaces, a generic solver formulation that allows for easy coupling of multi-physics problems, and VTK/Tecplot-compatible output formats. Example implementations of advection- and diffusion-type operators are readily available on GitHub. |
Christopher Thiele; Mauricio Araya-Polo; Faruk Omer Alpak; Béatrice Rivière; Florian Frank Poster: Inexact hierarchical scale separation: an efficient linear solver for discontinuous Galerkin discretizations Presentation Feb 20, 2017, (SPE Reservoir Simulation Conference, February 20–22, 2017, Montgomery, Texas, USA). @misc{Poster2017IHSS, title = {Poster: Inexact hierarchical scale separation: an efficient linear solver for discontinuous Galerkin discretizations}, author = {Christopher Thiele and Mauricio Araya-Polo and Faruk Omer Alpak and Béatrice Rivière and Florian Frank}, year = {2017}, date = {2017-02-20}, abstract = {Hierarchical scale separation (HSS) is a two-scale approximation method for large sparse linear systems arising from discontinuous Galerkin discretizations. HSS splits the 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-scale systems corresponding to the higher order solution components. This reduces the communication overhead, e.g., within sparse matrix-vector multiplications of classical iterative solvers and allows for an optimal parallelization of the fine-scale solves. We propose a modified algorithm (“inexact HSS”) that shifts the workload to the parallel fine-scale solver to reduce global synchronization, resulting in a significant speedup. We investigate its applicability to systems stemming from the nonsymmetric interior penalty DG discretization of the Cahn–Hilliard equation, discuss its parallel implementation for large-scale simulations, and investigate its performance.}, note = {SPE Reservoir Simulation Conference, February 20–22, 2017, Montgomery, Texas, USA}, keywords = {}, pubstate = {published}, tppubtype = {presentation} } Hierarchical scale separation (HSS) is a two-scale approximation method for large sparse linear systems arising from discontinuous Galerkin discretizations. HSS splits the 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-scale systems corresponding to the higher order solution components. This reduces the communication overhead, e.g., within sparse matrix-vector multiplications of classical iterative solvers and allows for an optimal parallelization of the fine-scale solves. We propose a modified algorithm (“inexact HSS”) that shifts the workload to the parallel fine-scale solver to reduce global synchronization, resulting in a significant speedup. We investigate its applicability to systems stemming from the nonsymmetric interior penalty DG discretization of the Cahn–Hilliard equation, discuss its parallel implementation for large-scale simulations, and investigate its performance. |
Florian Frank; Béatrice Rivière; Faruk Omer Alpak Poster: Discontinuous Galerkin approximation of the Cahn–Hilliard equation with degenerate mobility and contact angle condition Presentation May 10, 2016, (8th International Conference on Porous Media & Annual Meeting of InterPore, Cincinnati, Ohio, USA). @misc{Frank2016Poster, title = {Poster: Discontinuous Galerkin approximation of the Cahn–Hilliard equation with degenerate mobility and contact angle condition}, author = {Florian Frank and Béatrice Rivière and Faruk Omer Alpak}, year = {2016}, date = {2016-05-10}, abstract = {Phase-field models are becoming increasingly popular in hydrodynamics for modeling multi-phase fluid flow below Darcy scale. Instead of sharp interfaces, phase transitions appear as diffuse finite-thickness transition regions. The Cahn–Hilliard (CH) equation describes phase separation (the alignment of a system into spatial domains predominated by one of the two components) of an immiscible binary mixture at constant temperature in the presence of a mass constraint and dissipation of free energy. It is a stiff, fourth-order, nonlinear parabolic partial differential equation, which may serve as a prototype phase-field problem as intermediate step towards models that take other or additional phenomena into account, e.g., miscibility or multiple components. We apply a discontinuous Galerkin (DG) discretization of the CH equation with constant and degenerate mobility that is specialized for voxel grids stemming from micro-CT imaging. The scheme is of arbitrary order in space and of first order in time.}, note = {8th International Conference on Porous Media & Annual Meeting of InterPore, Cincinnati, Ohio, USA}, keywords = {}, pubstate = {published}, tppubtype = {presentation} } Phase-field models are becoming increasingly popular in hydrodynamics for modeling multi-phase fluid flow below Darcy scale. Instead of sharp interfaces, phase transitions appear as diffuse finite-thickness transition regions. The Cahn–Hilliard (CH) equation describes phase separation (the alignment of a system into spatial domains predominated by one of the two components) of an immiscible binary mixture at constant temperature in the presence of a mass constraint and dissipation of free energy. It is a stiff, fourth-order, nonlinear parabolic partial differential equation, which may serve as a prototype phase-field problem as intermediate step towards models that take other or additional phenomena into account, e.g., miscibility or multiple components. We apply a discontinuous Galerkin (DG) discretization of the CH equation with constant and degenerate mobility that is specialized for voxel grids stemming from micro-CT imaging. The scheme is of arbitrary order in space and of first order in time. |
Chen Liu; Béatrice Rivière; Faruk Omer Alpak; Florian Frank Poster: Discontinuous Galerkin approximation of the compressible Navier–Stokes equation at pore scale Presentation Mar 02, 2016, (Oil & Gas HPC Conference, Rice University, Houston, TX, USA). @misc{Liu2016Poster, title = {Poster: Discontinuous Galerkin approximation of the compressible Navier–Stokes equation at pore scale}, author = {Chen Liu and Béatrice Rivière and Faruk Omer Alpak and Florian Frank}, url = {http://frank.ink/wp-content/uploads/2016/03/posterLiuRAF2016lowQ.pdf}, year = {2016}, date = {2016-03-02}, abstract = {Variations of the Navier–Stokes equation are standard models for the description of viscous liquid and gas flow, used in many industrial fields such as automobile, aerospace, and hydrocarbon production industries. An application in the latter is the direct simulation of fluid transport through porous media at the pore scale, more precisely, on spatial domains that resolve the geometry of porous matrices of rocks. This poster presents a discontinuous Galerkin (DG) method for the compressible Navier–Stokes equation defined on voxel sets representing the pore space of rock samples at micrometer scale. The method exhibits optimal convergence and the simulated velocity fields compare well against the ones yielded by analytical solutions for simple geometries. The DG-based simulator also delivers intuitive velocity fields for complex pore geometries. }, note = {Oil & Gas HPC Conference, Rice University, Houston, TX, USA}, keywords = {}, pubstate = {published}, tppubtype = {presentation} } Variations of the Navier–Stokes equation are standard models for the description of viscous liquid and gas flow, used in many industrial fields such as automobile, aerospace, and hydrocarbon production industries. An application in the latter is the direct simulation of fluid transport through porous media at the pore scale, more precisely, on spatial domains that resolve the geometry of porous matrices of rocks. This poster presents a discontinuous Galerkin (DG) method for the compressible Navier–Stokes equation defined on voxel sets representing the pore space of rock samples at micrometer scale. The method exhibits optimal convergence and the simulated velocity fields compare well against the ones yielded by analytical solutions for simple geometries. The DG-based simulator also delivers intuitive velocity fields for complex pore geometries. |
Florian Frank; Nadja Ray; Kai Uwe Totsche; Peter Knabner Poster: Numerical simulations of a homogenized Stokes–Nernst–Planck–Poisson–problem Presentation Feb 14, 2011, (International Workshop on Analytical and Numerical Methods for Multiscale Systems, IWR and MATCH, Heidelberg, Germany). @misc{Poster2011, title = {Poster: Numerical simulations of a homogenized Stokes–Nernst–Planck–Poisson–problem}, author = {Florian Frank and Nadja Ray and Kai Uwe Totsche and Peter Knabner}, url = {http://frank.ink/wp-content/uploads/2015/09/poster2011.pdf}, year = {2011}, date = {2011-02-14}, abstract = {We perform the homogenization of the non-stationary Stokes–Nernst–Planck–Poisson system using two-scale convergence. Due to the specific nonlinear coupling of the underlying equations special attention has to be paid when passing to the two-scale limit. The resulting homogenized system consists of averaged macroscopic equations as well as cell problems that deliver suitable effective parameters. We perform numerical simulations of both the Stokes–Nernst–Planck–Poisson system and the derived fully coupled two-scale problem. The discretization is done in two space dimensions using mixed finite elements. Furthermore, we study the breakthrough curves for the microscopic and the homogenized problem for different choices of the small scale parameter ε. }, note = {International Workshop on Analytical and Numerical Methods for Multiscale Systems, IWR and MATCH, Heidelberg, Germany}, keywords = {}, pubstate = {published}, tppubtype = {presentation} } We perform the homogenization of the non-stationary Stokes–Nernst–Planck–Poisson system using two-scale convergence. Due to the specific nonlinear coupling of the underlying equations special attention has to be paid when passing to the two-scale limit. The resulting homogenized system consists of averaged macroscopic equations as well as cell problems that deliver suitable effective parameters. We perform numerical simulations of both the Stokes–Nernst–Planck–Poisson system and the derived fully coupled two-scale problem. The discretization is done in two space dimensions using mixed finite elements. Furthermore, we study the breakthrough curves for the microscopic and the homogenized problem for different choices of the small scale parameter ε. |
Florian Frank; Nadja Ray; Kai Uwe Totsche; Peter Knabner Poster: Numerical simulations of a two-scale model for fluid flow and colloidal transport in porous media Presentation Oct 10, 2010, (International Workshop on Multiscale Modeling, Simulation and Optimization, Fraunhofer IISB, Erlangen, Germany). @misc{Poster2010, title = {Poster: Numerical simulations of a two-scale model for fluid flow and colloidal transport in porous media}, author = {Florian Frank and Nadja Ray and Kai Uwe Totsche and Peter Knabner}, year = {2010}, date = {2010-10-10}, abstract = {The modeling of coupled fluid flow and complex transport processes in porous media deals with different time and/or spatial scales. In the case of colloidal transport sorption and coagulation induced by electrostatic interaction on the pore scale change the microstructure of the porous medium and thus the fluid flow. A direct numerical approach is not feasible since microscopic heterogeneities are not resolveable in general on macroscopic domains. Via homogenization techniques however an effective macroscopic model description for the exact heterogeneous model can be derived. Numerical convergence of a representative heterogeneous model to the effective macroscopic two-scale model for vanishing microstructure is studied in the case of the flow problem. This confirms the theoretical statement of the convergence behavior in the limit process. Furthermore, simulation results of a homogenized colloidal transport model which incorporates electrostatic interaction among and beween colloidal particles and charged surfaces on the microscale are presented.}, note = {International Workshop on Multiscale Modeling, Simulation and Optimization, Fraunhofer IISB, Erlangen, Germany}, keywords = {}, pubstate = {published}, tppubtype = {presentation} } The modeling of coupled fluid flow and complex transport processes in porous media deals with different time and/or spatial scales. In the case of colloidal transport sorption and coagulation induced by electrostatic interaction on the pore scale change the microstructure of the porous medium and thus the fluid flow. A direct numerical approach is not feasible since microscopic heterogeneities are not resolveable in general on macroscopic domains. Via homogenization techniques however an effective macroscopic model description for the exact heterogeneous model can be derived. Numerical convergence of a representative heterogeneous model to the effective macroscopic two-scale model for vanishing microstructure is studied in the case of the flow problem. This confirms the theoretical statement of the convergence behavior in the limit process. Furthermore, simulation results of a homogenized colloidal transport model which incorporates electrostatic interaction among and beween colloidal particles and charged surfaces on the microscale are presented. |