Abstract: This paper presents a weak Galerkin (WG) finite element solver for Darcy flow and its implementation on the \texttt{deal.II} platform. The solver works for quadrilateral and hexahedral meshes in a unified way. It approximates pressure by $ Q $-type degree $ k (\ge 0) $ polynomials separately in element interiors and on edges/faces. Numerical velocity is obtained in the unmapped Raviart-Thomas space $ RT_{[k]} $ via postprocessing based on the novel concepts of discrete weak gradients. The solver is locally mass-conservative and produces continuous normal fluxes. It is implemented in \texttt{deal.II} in the dimension-independent paradigm and allows polynomial degrees up to $ 5 $. Numerical experiments show that our new WG solver performs better than the classical mixed finite element methods.

Abstract: We study a mathematical model and its finite element approximation for solving the coupled problem arising in the interaction between a free fluid and a fluid in a poroelastic material. The free fluid flow is governed by the Stokes equations, while the poroelastic material is modeled using the Biot system of poroelasticity. The model is based on a mixed stress-displacement-rotation elasticity formulation and mixed velocity-pressure Darcy and Stokes formulations. The mixed finite element approximation provides local mass and momentum conservation in the poroelastic media. We discuss stability, accuracy, and robustness of the method. Applications to flows in fractured poroelastic media and arterial flows are presented.

Abstract: The velocity, coupling term in the flow and transport problems, is important in the accurate numerical simulation or in the posteriori error analysis for adaptive mesh refinement. We consider Enhanced Velocity Mixed Finite Element Method for the incompressible Darcy flow. In this paper, our aim to study the improvement of velocity at interface to achieve the better approximation of velocity between subdomains. We propose the reconstruction of velocity at interface by using the post-processed pressure. Numerical results at the interface show improvement on convergence rate.

Abstract: In this work we propose a new combination of finite element methods to solve incompressible miscible displacements in heterogeneous media formed by the coupling of the free-fluid with the porous medium employing the stabilized hybrid mixed finite element method developed and analyzed by Igreja and Loula in \cite{Igreja:2018} and the classical Streamline Upwind Petrov--Galerkin (SUPG) method presented and analyzed by Brooks and Hughes in \cite{brooks-hughes:82}. The hydrodynamic problem is governed by the Stokes and Darcy systems coupled by Beavers-Joseph-Saffman interface conditions. To approximate the Stokes-Darcy coupled system we apply the stabilized hybrid mixed method, characterized by the introduction of the Lagrange multiplier associated with the velocity field in both domains. This choice naturally imposes the Beavers-Joseph-Saffman interface conditions on the interface between Stokes and Darcy domains. Thus, the global system is assembled involving only the degrees of freedom associated with the multipliers and the variables of interest can be solved at the element level. Considering the velocity fields given by the hybrid method we adopted the SUPG method combined with an implicit finite difference scheme to solve the transport equation associated with miscible displacements. Numerical studies are presented to illustrate the flexibility and robustness of the hybrid formulation. To verify the efficiency of the combination of hybrid and SUPG methods, computer simulations are also presented for the recovery hydrological flow problems in heterogeneous porous media, such as continuous injection.

Abstract: In this paper, we formulate a modeling theory and numerical algorithm for a multi-component two-phase fluid system together with the interface between phases and with gravity. We use a diffuse interface model based on Peng-Robinson equation of state (EOS) for the modeling of the fluid. We show that gravity has a significant influence on the phase equilibrium behavior, which is an expected phenomenon but has not been numerically studied in the literature regarding to Peng-Robinson fluid modeled by a diffuse interface model.

Abstract: In this paper, the effects of different numerical integration schemes on the distributed Lagrange multiplier/fictitious domain (DLM/FD) method with body-unfitted meshes are studied for solving different types of interface problems: elliptic-, Stokes- and Stokes/elliptic-interface problems, for which the corresponding mixed finite element approximations are developed. Commonly-used numerical integration schemes, compound type formulas and a specific subgrid integration scheme are presented and the comparison between them is illustrated in numerical experiments, showing that different numerical integration schemes have significant effects on approximation errors of the DLM/FD finite element method for different types of interface problems, especially for Stokes- and Stokes/elliptic-interface problems, and that the subgrid integration scheme always results in numerical solutions with the best accuracy.

Abstract: This paper discusses the application of the double potential method for modeling flow of incompressible fluid. The algorithm allows us to avoid a numeric calculation of pressure. This procedure is not easy for case of an incompressible fluid flow. It may lead to solution instability with approximation by cell center grid methods. Also, the double potential method overcomes a problem of a complex boundary conditions which arises in case of modelling with using the Navier-Stokes equations in the vector potential-vortex formulation. The resulting system of equations is approximated by using the finite volume method and the exponential transformation. As a verification problem, the problem of establishing the Poiseuille flow on three-dimensional cylindrical geometry was applied

Abstract: The formation of the bubbles in the liquid was examined numerically and obtained results were successfully compared with the results provided by experiments. The study covered two different patterns defined by different Morton numbers or gas flow rates. The unsteady three dimensional calculations were carried out in code OpenFoam with the volume of fluid approach. Found numerical results were in a good math to the experiments in respect to bubble shapes, diameters and Reynolds numbers. More accurate comparison was found for lower gas flow rate then for the higher one. The main reason can be that under higher gas flow rate, a complex flow behavior between gas bubbles and surrounding liquid flow is created which after that worsen the accuracy of calculations. The main important output of the study was a comparison of the bubble diameters in time. Especially for higher gas flow rates, bubbles are growing rapidly during its climbing. Nevertheless a satisfactory agreement was found between numerics and experiments.

Abstract: Multiphase fluid flow with complex compositions is an increasingly attractive research topic with more and more attentions paid on related engineering problems, including global warming and green house effect, oil recovery enhancement and subsurface water pollution treatment. Prior to study the flow behaviors and phase transitions in multi-component multiphase flow, the first effort should be focused on the accurate prediction of the total phase numbers existing in the fluid mixture, and then the phase equilibrium status can be determined. In this paper, a novel and fast prediction technique is proposed based on deep learning method. The training data is generated using a selected VT dynamic flash calculation scheme and the network constructions are deeply optimized on the activation functions. Compared to previous machine learning techniques proposed in literatures to accelerate vapor liquid phase equilibrium calculation, the total number of phases existing in the mixture is determined first and other phase equilibrium properties will be estimated then, so that we do not need to ensure that the mixture is in two phase conditions any more. Our method could handle fluid mixtures with complex compositions, with 8 different components in our example and the original data is in a large amount. The analysis on prediction performance of different deep learning models with various neural networks using different activation functions can help future researches selecting the features to construct the neural network for similar engineering problems. Some conclusions and remarks are presented at the end to help readers catch our main contribution and insight the future related research.

Abstract: Modeling and simulation of fluid flow and heat transfer processes occurring in heterogeneous irregular regions have received extensive attention in recent years. The presence of heterogeneous properties would exert crucial impacts on the overall performance of fluid flow and heat transfer simulations. For example, the high heterogeneous properties always worsen the model coefficient matrix and complicate the simulation difficulty. Therefore, the need to develop high-efficient and accurate numerical methods for general fluid flow and heat transfer occurring in heterogeneous irregular regions which could significantly reduce the computational efforts at the same time conserve the main physical properties, is highly addressed among engineering and academic communities. In this study, we present a highly efficient solver, geometrical multi-grid (GMG), for the fast simulation of fluid flow and heat transfer problems occurring in heterogeneous irregular regions in the framework of body-fitted coordinate (BFC) system. The key point of the proposed multigrid solver lies in the calculation of heterogeneous properties on coarse grid levels within the original physical domain, in which the up-scaling method is widely used. However, different upscaling methods would yield the effective properties with different numerical accuracy and computational efficiency. To explore the influence of the upscaling approaches on overall performances of the proposed multigrid solver, in this study we adopt the general statistical averages (e.g. harmonic average, arithmetic average, geometric average, harmonic-arithmetic average, etc.) and flow-based methods (e.g. sealed-side boundary condition, open-side boundary conditions, etc. for fluid flows) to compute the unscaled effective properties on corresponding coarse grid levels. The numerical accuracy of the quantity of interest on different grid levels and the computational speed-up of the proposed multigrid solver for flow and heat transfer problems occurring in heterogeneous irregular regions are validated by several examples to assess the influence of different upscaling approaches on computations. The proposed multigrid solver for fluid flow and heat transfer problems in heterogeneous irregular regions can not only markedly improve the computational efficiency of the fine grid solution, but also can provide the computation byproduct - solution on coarse grid levels for specific applications, for example in which the coarse grid solution can be used for sample recycling in multigrid multilevel Monte Carlo method to avoid the repeated realization of sampling on coarse grid levels.

Abstract: Enhanced geothermal systems (EGS) are the major way of the hot dry rock (HDR) exploitation. At present, the finite element method (FEM) is often used to simulate the thermal energy extraction process of the EGS. Satisfactory results can be obtained by this method to a certain extent. However, when many discrete fractures exist in the computational domain, a large number of unstructured grids must be used, which seriously affects the computational efficiency. To solve this challenge, based on the embedded discrete fracture model (EDFM), two sets of seepage and energy conservation equations are respectively used to describe the flow and heat transfer processes of the matrix media and the fracture media. The main advantage of the proposed model is that the structured grid can be used to mesh the matrix, and there is no need to refine the mesh near the fracture. Comparing with commercial software, COMSOL Multiphysics, the accuracy of the proposed model is verified. Subsequently, a specific example of geothermal exploitation is designed, and the spatial-temporal evolutions of pressure and temperature fields are analyzed.

Abstract: Modelling the fusion and heat affected microstructure of an Additive Manufacturing (AM) process bridges many length and time scales and requires more than intelligent meshing schemes to make simulations feasible. The aim of this research was to develop an efficient and simple, yet significantly accurate high quality and high precision thermal model in wire + arc additive manufacturing process. To describe the influence of the process parameters and materials on the entire welding process, a 3D transient non-linear finite element model to simulate multi-layer deposition of cast IN-738LC alloy onto SAE-AISI 1524 Carbon Steel Substrates was developed. Temperature-dependent material properties and the effect of forced convection were included in the model. The heat source represented by a moving Gaussian power density distribution was applied over the top surface of the specimen during a period of time that depends on the welding speed. The effect of multi-layer deposition on the prediction and validation of melting pool shape and thermal cycles was also investigated. The effect of convection and radiation heat loss from the weldment (layers) surfaces were included into the finite element analysis. As the AM layers itself act as extended surfaces (fins), it was found that the heat extraction is quite significant. It is encouraging to note that the thermal model is sufficiently accurate to predict thermal cycles, FZ and HAZ weld profiles. A firm foundation for modelling thermal transport in wire + arc additive manufacturing process it was established.

Abstract: With rapid advancement in exploration and production of shale gas reservoirs over the world, the fast simulation of shale gas flows that is required especially in engineering applications has attracted extensive attentions from the engineering and academic communities. In this study, we apply a popular global model reduction method, proper orthogonal decomposition (POD), to speed up the simulation of shale gas reservoirs. However, different from incompressible fluid flows, the compressibility of shale gas induces additional challenges to construct accurate and efficient POD reduced-order model (ROM). First, the compressibility of shale gas increases the nonlinearity of the flow system, the dimension of the projected Darcy-type pressure equation in low-dimensional space still depends on the dimension of the original system, which complicates the computation and worsens the acceleration of the POD-ROM substantially. Second, due to the POD projection term containing the compressibility of shale gas, the additional computational cost is needed to solve the equation of state (EOS) of shale gas to obtain the compressibility factor. To handle these problems, we adopt another model reduction approach, discrete empirical interpolation method (DEIM), to approximate the nonlinearity in pressure equation by only using few selected representative interpolation points over the domain, thus the nonlinearity of the variables and the computation of EOS can be greatly reduced. Combined the POD-ROM with DEIM, a POD-DEIM-ROM for fast simulation of shale gas flows with different gas states in single-continuum porous media is developed. The performances of the proposed model are validated by comparing to the traditional method without acceleration techniques through different numerical cases, the computational speed and numerical accuracy are analyzed in detail. Results show that the proposed model can achieve great acceleration of the simulation (two orders of magnitude) without sacrificing the numerical accuracy obviously. Especially, the influence of the type of EOS (such as ideal gas EOS, Van der Waals EOS, and Peng–Robinson EOS, etc.), the number of POD modes and interpolation points, as well as the permeability filed distribution on the overall performance of the proposed model are investigated.

Abstract: A direct numerical simulation (DNS) code was developed for solving incompressible homogeneous isotropic turbulence with high Reynolds numbers in a periodic box using the Fourier spectral method. The code was parallelized using the Message Passing Interface and OpenMP with two-directional domain decomposition and optimized on the K computer. High resolution DNSs with up to $12288^3$ grid points were performed on the K computer using the code. Efficiencies of 3.84\%, 3.14\%, and 2.24\% peak performance were obtained in double precision DNSs with $6144^3$, $8192^3$, and $12288^3$ grid points, respectively. In addition, a two-path alias-free procedure is proposed and clarified its effectiveness for some number of parallel processes.

Abstract: This work presents Direct Numerical Simulation of mass transfer from buoyancy-driven bubbles rising in a wall-confined vertical channel, by means of a multiple marker level-set method. The Navier-Stokes equations and mass transfer equations are discretized using a finite-volume method on a collocated unstructured mesh, whereas a multiple marker approach is used to avoid the numerical coalescence of bubbles. This approach is implemented in the framework of a mass conservative level-set method, whereas unstructured flux-limiter schemes are used to discretize the convective term of momentum equation, level-set equations, and mass transfer equation, in order to improve the stability of the solver in bubbly flows with high Reynolds number and high-density ratio. The capabilities of this model are proved in the buoyancy-driven motion of single bubbles and bubble swarms in a vertical channel of circular cross-section.

Abstract: A hybrid particle/mesh Vortex Method, called remeshed vortex method, is proposed in this work to simulate three-dimensional incompressible flows. After a validation study of the present method in the context of Direct Numerical Simulations, an anisotropic artificial viscosity model is proposed in this paper in order to handle bi-level simulations. The bi-level approach implies two different mesh sizes for the discretization of the two coupled variables of the problem, namely the vorticity field and the velocity field: the vorticity field is computed on a fine grid while the velocity field is solved on a coarse grid. This approch is proposed with the objective of performing efficient computations based on hybrid GPU-CPU architectures.