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A numerical method is formulated for the solution of the advective Cahn-Hilliard (CH) equation with constant and degenerate mobility in three-dimensional porous media with non-vanishing velocity on the exterior boundary. The CH equation describes phase separation of an immiscible binary mixture at constant temperature in the presence of a mass cons...
Citations
... Numerical simulation of the phase field is an extremely important research topic, and it is widely used in reality, from the initial use for simulating phase transition [1][2][3] and microstructure evolution [4,5], to the recent development of simulating crystal growth [6][7][8], flows through porous media [9], image analysis [10][11][12][13][14], grain growth [15][16][17], planet formation [18], interface dynamics [19], foam material [20], malignant tumor growth [21,22], polymer separation [23,24], biological population competition and rejection and many other physical phenomena [25,26]. Various numerical methods have been developed and analyzed to solve the phase field model, such as finite element methods [27][28][29][30][31][32][33][34][35][36][37][38][39][40][41], finite difference methods [42][43][44][45], spectral methods [46][47][48], extended finite element methods [49,50], discontinuous Galerkin methods [12,[51][52][53][54][55], boundary finite element methods [56,57], finite volume methods [58][59][60][61][62][63][64][65], penalty projection methods [66], lattice Boltzmann methods [67,68], virtual element methods [69,70], and other methods [71][72][73][74][75][76][77][78][79]. As one of the most basic equations describing the phase field model, the Allen-Cahn equation was first proposed by Allen and Cahn in [80] to describe the motion of the antiphase boundary in the crystal. ...
In this paper, the discontinuous finite volume element method (DFVEM) is considered to solve the Allen-Cahn equation which contains strong nonlinearity. The method is based on the DFVEM in space and the backward Euler method in time. The energy stability and unique solvability of the proposed fully discrete scheme are derived. The error estimates for the semi-discrete and fully discrete scheme are also established. A series of numerical experiments verify the efficiency of the proposed numerical method. The results show that our method can not only accurately capture the dynamic information of the phase transition, but also ensure the stability of the system during long-term numerical simulations.
... We use P 2 scheme, e.g., discontinuous piecewise quadratic polynomials for space approximation, on cubic partitions of 3D domains. More details can be found in [13]. ...
For time-dependent PDEs, the numerical schemes can be rendered bound-preserving without losing conservation and accuracy, by a post processing procedure of solving a constrained minimization in each time step. Such a constrained optimization can be formulated as a nonsmooth convex minimization, which can be efficiently solved by first order optimization methods, if using the optimal algorithm parameters. By analyzing the asymptotic linear convergence rate of the generalized Douglas-Rachford splitting method, optimal algorithm parameters can be approximately expressed as a simple function of the number of out-of-bounds cells. We demonstrate the efficiency of this simple choice of algorithm parameters by applying such a limiter to cell averages of a discontinuous Galerkin scheme solving phase field equations for 3D demanding problems. Numerical tests on a sophisticated 3D Cahn-Hilliard-Navier-Stokes system indicate that the limiter is high order accurate, very efficient, and well-suited for large-scale simulations. For each time step, it takes at most $20$ iterations for the Douglas-Rachford splitting to enforce bounds and conservation up to the round-off error, for which the computational cost is at most $80N$ with $N$ being the total number of cells.
... and C is a stabilization matrix that is required only for lowest order = 0 to guarantee the full rank of A. We choose C as in [13], (3.84), in which case, C can be interpreted as a pressure-Laplacian discretized by cellcentered finite volumes [16]. For > 0, C is set to zero. ...
In recent years, convolutional neural networks (CNNs) have experienced an increasing interest in their ability to perform a fast approximation of effective hydrodynamic parameters in porous media research and applications. This paper presents a novel methodology for permeability prediction from micro-CT scans of geological rock samples. The training data set for CNNs dedicated to permeability prediction consists of permeability labels that are typically generated by classical lattice Boltzmann methods (LBM) that simulate the flow through the pore space of the segmented image data. We instead perform direct numerical simulation (DNS) by solving the stationary Stokes equation in an efficient and distributed-parallel manner. As such, we circumvent the convergence issues of LBM that frequently are observed on complex pore geometries, and therefore, improve the generality and accuracy of our training data set. Using the DNS-computed permeabilities, a physics-informed CNN (PhyCNN) is trained by additionally providing a tailored characteristic quantity of the pore space. More precisely, by exploiting the connection to flow problems on a graph representation of the pore space, additional information about confined structures is provided to the network in terms of the maximum flow value, which is the key innovative component of our workflow. The robustness of this approach is reflected by very high prediction accuracy, which is observed for a variety of sandstone samples from archetypal rock formations.
... where,¯ 0 = 7.63958172715414, which guarantees zero average pressure over Ω (up to machine precision). Here in above, the order parameter field is taken from [11]. The chemical potential is an intermediate variable, which value is derived by the order parameter. ...
This paper is devoted to the analysis of an energy-stable discontinuous Galerkin algorithm for solving the Cahn-Hilliard-Navier-Stokes equations within a decoupled splitting framework. We show that the proposed scheme is uniquely solvable and mass conservative. The energy dissipation and the $L^\infty$ stability of the order parameter are obtained under a CFL condition. Optimal a priori error estimates in the broken gradient norm and in the $L^2$ norm are derived. The stability proofs and error analysis are based on induction arguments and do not require any regularization of the potential function.
... In recent years, an increasing number of advances has been published exploring the use of discontinuous Galerkin (DG) methods both for constant [2,24] and degenerate mobility [18,28,33,36]. Although DG methods lead in general to more complex algorithms and to bigger amounts of degrees of freedom, they exhibit some benefits of which one can take advantage also in CH equations, for instance doing mesh refining and adaptivity [2]. ...
... For instance, the work of [24] is focused on construction and convergence analysis of a DG method for the CCH equations with constant mobility, applying an interior penalty technique to the second-order terms and an upwind operator for discretization of the convection term. Authors of [18] consider CCH with degenerate mobility applying also an interior penalty to second-order terms in the mixed form (1) and a more elaborated upwinding technique, based on a sigmoid function, to the convective term. ...
... Then, we introduce our fully discrete scheme (29) providing a maximum principle for the discrete phase variable. Finally, in Section 4, we present several numerical tests, comparing our DG scheme with two different space discretizations found in the literature: classical continuous P 1 finite elements and the SIP+upwind sigmoid DG approximation proposed in [18]. We show error order tests and also we present qualitative comparisons where the maximum principle of our scheme is confirmed (and it is not conserved by the other ones). ...
The design of numerical approximations of the Cahn-Hilliard model preserving the maximum principle is a challenging problem, even more if considering additional transport terms. In this work, we present a new upwind discontinuous Galerkin scheme for the convective Cahn-Hilliard model with degenerate mobility which preserves the maximum principle and prevents non-physical spurious oscillations. Furthermore, we show some numerical experiments in agreement with the previous theoretical results. Finally, numerical comparisons with other schemes found in the literature are also carried out
... In recent years, an increasing number of advances has been published exploring the use of discontinuous Galerkin (DG) methods both for constant [20,2] and degenerate mobility [29,32,15,24]. Although DG methods lead in general to more complex algorithms and to bigger amounts of degrees of freedom, they exhibit some benefits of which one can take advantage also in CH equations, for instance doing mesh refining and adaptivity [2]. ...
... For instance, the work of Frank et al [20] is focused on construction and convergence analysis of a DG method for the CCH with constant mobility, applying an interior penalty technique to the second order terms and an upwind operator for discretization of the convection term. Authors of [15] consider CCH with degenerate mobility applying also an interior penalty to second order terms in the mixed form (1) and a more elaborated upwinding technique, based on a sigmoid function, to the convective term. ...
... Then we introduce our fully discrete scheme (28) providing a maximum principle for the discrete phase variable. Finally, in Section 4 we present several numerical test comparing our DG scheme with two different space discretizations found in the literature: classical continuous P 1 finite elements and the SIP+upwind sigmoid DG approximation proposed in [15]. We show error order tests and also we present qualitative comparisons where the maximum principle of our scheme is confirmed (and it is not conserved in the others). ...
The design of numerical approximations of the Cahn-Hilliard model preserving the maximum principle is a challenging problem, even more if considering additional transport terms. In this work we present a new upwind Discontinuous Galerkin scheme for the convective Cahn-Hilliard model with degenerate mobility which preserves the maximum principle and prevents non-physical spurious oscillations. Furthermore, we show some numerical experiments in agreement with the previous theoretical results. Finally, numerical comparisons with other schemes found in the literature are also carried out.
... We consider the instantaneous adsorption regime in [30], and simplify the model by assuming that the surfactant is only present in one of the fluid phases. Concerning numerical methods for similar type of models as discussed here, we refer to [35], where energy-stable schemes are proposed for a Cahn-Hiliard model for two-phase flow and surfactant transport, and to [36,37], where energy-stable methods based on discontinuous Galerkin discretization are analyzed. ...
... for i, j = 1, . . . , d, and where χ j and χ 0 solve the cell problems (36) and (37). ...
... Solve porescale problems (36) and (37) Compute B n and H n Solve the Darcyscale problem (42e) Solve the phase-field problem (43) Next time step (17), (37), (43), (46) and (48) are computed using the lowest order Raviart-Thomas elements (see [47]). For the pore-scale problems (16) and (36) we use the Crouzeix-Raviart elements (see [48,Section 8.6.2]). ...
... However, the combination of using constant mobility and discontinuous piecewise polynomials of degree greater than or equal to one, produces numerical solutions that do not automatically satisfy a maximum principle in general. The phenomena of bulk shift, overshoot and undershoot have been observed for the advective Cahn-Hilliard equations [20,21]; they can be reduced by carefully selecting mesh sizes, time step values, and penalty values in the DG discrete forms. Thus, in this work, we propose two post-processing techniques to eliminate bulk shift, overshoot and undershoot with respect to the order parameter. ...
... The order parameter takes the value c * −1 in one bulk phase and the value c * +1 in the other bulk phase. It is well known that the numerical approximation of the order parameter may exhibit a bulk shift in some parts of the domain, i. e., all the bulk values are either shifted up or down by a small amount [21,20]. The amount of bulk shift depends on the curvature of the interface, and it is reduced with decreasing mesh size. ...
... For linear polynomials, the extrema will occur at the vertices and after application of the slope limiter, these extrema will belong to [−1, 1]. Clearly this implies (21). For polynomials of degree greater than one, the approximation on a given troubled cell is reduced to a linear polynomial. ...
In this paper, we present an efficient numerical algorithm for solving the time-dependent Cahn–Hilliard–Navier–Stokes equations that model the flow of two phases with different densities. The pressure-correction step in the projection method consists of a Poisson problem with a modified right-hand side. Spatial discretization is based on discontinuous Galerkin methods with piecewise linear or piecewise quadratic polynomials. One important contribution of this work is the formulation of flux and slope limiting techniques that successfully eliminate the bulk shift, overshoot and undershoot in the order parameter. The bound-preserving property of the discrete order parameter is proved. Several numerical results demonstrate that the proposed numerical algorithm is effective and robust for modeling two-component immiscible flows in porous structures and digital rocks.
... In a recent series of works [32][33][34], a diffusive-interface framework was considered for an immiscible two-phase flows at the pore-scale in rock samples. The location of the two-phases in the pore space of the rock is expressed in terms of an auxiliary phase-field acting as an order parameter [35]. ...
... The system is mathematically modeled by the Cahn-Hilliard equations coupled with the incompressible Navier-Stokes equations. An interior penalty discontinuous Galerkin (IPDG) scheme was proposed to solve the system, while a temporal semi-implicit convex-concave splitting ensured the scheme to be unconditionally energy stable [32]. The coupled Cahn-Hilliard-Navier-Stokes problem has received much attention recently and several numerical methods have been employed to solve this problem, namely finite element methods and mixed element methods in [36][37][38], finite volume methods [39] and discontinuous Galerkin methods [34,40]. ...
... The spatial discretization of (9) is performed using IPDG. We follow closely the formulation considered for the advective pure Cahn-Hilliard system considered in [32]. Define P q (E) to be set of all polynomials on E of degree at most q and define the broken polynomial space ...
A numerical method using discontinuous polynomial approximations is formulated for solving a phase-field model of two immiscible fluids with a soluble surfactant. The proposed scheme is shown to decay the total free Helmholtz energy at the discrete level, which is consistent with the continuous model dynamics. The scheme recovers the Langmuir adsorption isotherms at equilibrium. Simulations of spinodal decomposition, flow through a cylinder and flow through a sequence of pore throats show the dynamics of the flow with and without surfactant. Finally the numerical method is used to simulate fluid flows in the pore space of Berea sandstone obtained by micro-CT imaging.
... and C is a stabilization matrix that is required only for lowest order = 0 to guarantee the full rank of A. We choose C as in [31], (3.84), in which case, C can be interpreted as a pressure-Laplacian discretized by cell-centered finite volumes [33]. For > 0, C is set to zero. ...
In recent years, convolutional neural networks (CNNs) have experienced an increasing interest for their ability to perform fast approximation of effective hydrodynamic parameters in porous media research and applications. This paper presents a novel methodology for permeability prediction from micro-CT scans of geological rock samples. The training data set for CNNs dedicated to permeability prediction consists of permeability labels that are typically generated by classical lattice Boltzmann methods (LBM) that simulate the flow through the pore space of the segmented image data. We instead perform direct numerical simulation (DNS) by solving the stationary Stokes equation in an efficient and distributed-parallel manner. As such, we circumvent the convergence issues of LBM that frequently are observed on complex pore geometries, and therefore, improve on the generality and accuracy of our training data set. Using the DNS-computed permeabilities, a physics-informed CNN PhyCNN) is trained by additionally providing a tailored characteristic quantity of the pore space. More precisely, by exploiting the connection to flow problems on a graph representation of the pore space, additional information about confined structures is provided to the network in terms of the maximum flow value, which is the key innovative component of our workflow. As a result, unprecedented prediction accuracy and robustness are observed for a variety of sandstone samples from archetypal rock formations.
