Characteristic Galerkin Research Articles

Explicit Numerical Manifold Characteristic Galerkin Method for Solving Burgers’ Equation

Explicit Numerical Manifold Characteristic Galerkin Method for Solving Burgers’ Equation

Fourth-Order Conservative Non-splitting Semi-Lagrangian Hermite WENO Schemes for Kinetic and Fluid Simulations

AbstractWe present fourth-order conservative non-splitting semi-Lagrangian (SL) Hermite essentially non-oscillatory (HWENO) schemes for linear transport equations with applications for nonlinear problems including the Vlasov–Poisson system, the guiding center Vlasov model, and the incompressible Euler equations in the vorticity-stream function formulation. The proposed SL HWENO schemes combine a weak formulation of the characteristic Galerkin method with two newly constructed HWENO reconstruction methods. The new HWENO reconstructions are meticulously designed to strike a delicate balance between curbing numerical oscillation and introducing excessive dissipation. Mass conservation naturally holds due to the weak formulation of the semi-Lagrangian discontinuous Galerkin method and the design of the HWENO reconstructions. We apply a positivity-preserving limiter to maintain the positivity of numerical solutions when needed. Abundant benchmark tests are performed to verify the effectiveness of the proposed SL HWENO schemes.

Study on Temperature Distribution Law of Tunnel Portal Section in Cold Region Considering Fluid–Structure Interaction

The design of tunnels in cold regions contributes greatly to the feasibility and sustainability of highways. Based on the heat transfer mechanism of the tunnel surrounding rock–lining–air, this paper uses FEPG software to carry out secondary excavation and development, then the air heat convection calculation model is established by using a three-dimensional extension of the characteristic-based operator-splitting (CBOS) finite-element method and the explicit characteristic–Galerkin method. By coupling with the heat conduction model of the tunnel lining and surrounding rock, the heat conduction-thermal convection fluid–structure interaction finite-element calculation model of tunnels in cold regions is established. Relying on the Qinghai Hekashan tunnel project, the temperature field of the tunnel portal section is calculated and studied by employing the fluid–structure interaction finite-element model and then compared with the field monitoring results. It is found that the calculated values are basically consistent with the measured values over time, which proves the reliability of the model. The calculation results are threefold: (1) The temperature of the air, lining, and surrounding rock in the tunnel changes sinusoidally with the ambient temperature. (2) The temperature of each layer gradually lags behind, and the temperature variation amplitude of the extreme value of the layer temperature gradually decreases with the increase in the radial distance of the lining. (3) In the vicinity of the tunnel entrance, the lining temperature of each layer remains unchanged, and the temperature gradually decreases or increases with the increase in the depth. The model can be used to study and analyze the temperature field distribution law of the lining and surrounding rock under different boundary conditions, and then provide a calculation model with both research and practical value for the study of the temperature distribution law of tunnels in cold regions in the future.

Application of vanishing diffusion stabilization in Oldroyd-B fluid flow simulations

Two different methods of artificial diffusion stabilization of the numerical simulations of steady Oldroyd-B fluids flows are presented. They are based on the idea of vanishing in time added stabilization terms which are only present during the initial stage of time-marching process towards the steady state solution. These extra terms naturally vanish and do not affect the final result. The numerical simulations are built on a simple steady 2D case of Oldroyd-B fluid flow in a symmetrical corrugated channel. Numerical solver uses finite element discretization in space and characteristic Galerkin method for pseudo-time discretization. Numerical results are presented in the form of isolines and graphs of selected flow variables, to assess the possible efficiency of the different stabilization techniques used.

Finite Element Model to Simulate Two-Phase Fluid Flow in Naturally Fractured Oil Reservoirs: Part I.

Naturally fractured reservoirs host more than 20% of the world’s total oil and gas reserves. To produce from such reservoirs efficiently, a good understanding of the reservoir behavior at various conditions is essential. This allows us to predict the reservoir performance in advance and assess its economic feasibility. However, the production from such reservoirs is challenging due to (a) uncertainty associated with the fracture map, (b) complex physics phenomena of fluid and rock interaction, and (c) lack of comprehensive knowledge of the extent, orientation, and permeability sensitivity of the fracture network. This paper addresses the abovementioned challenges by presenting a three-dimenisonal (3-D) two-phase fluid flow model in a poroelastic environment. The model is based on a hybrid methodology by combining single continuum and discrete fracture network approaches. Also, the capillary pressure effect, saturation, and relative permeability variations are considered. A mathematical formulation for three-dimensional, two immiscible fluid flows including rock deformation for the fracture network and the rock matrix is presented. A standard Galerkin-based finite element method is applied to discretize the poroelastic governing equations in space and time. The characteristic Galerkin discretization method is used to stabilize the solution of the convection equation in a finite element approach. The 3-D model is validated against IMEX commercial software and finite element package to test its robustness. The results show that the developed model has the ability to predict the two-phase flow behavior precisely, which can be used to assess the production performance of naturally fractured reservoirs. Numerical results of fluid flow profiles for single and multiple fractures with different orientations that match experimental oil drainage tests and field case studies are presented that proves the reliability of the developed multiphase flow model. Results of simulated well production data show an excellent match with the field production data.

Investigation of Relaxation Time on Viscoelastic Two‐Dimensional Flow Characteristics Using FreeFem++

Viscoelastic fluid flow models have shown promising scope in modeling the behavior of many industrial materials such as polymeric materials, microfluidics, biological liquids, gels, plastic melts, and geomaterials. The relaxation time in these models is of great physical significance. In this article, we study the impact of relaxation time on the viscoelastic flow characteristics in a two‐dimensional baffled cavity. To the best of authors’ knowledge, relaxation time impact on the chosen flow characteristics in the present context has not been studied and presented in the literature before. The constitutive theory of upper convected Maxwell viscoelastic flow incorporating the viscosity ad relaxation time is taken into consideration. To this account, the flow governing PDEs are derived, and an unsteady variational numerical approach based on classical variational setting is presented. A numerical algorithm based on characteristic Galerkin finite elements method is designed and implemented using the programming language FreeFem++. Computations are carried out and drag and lift forces along with other parameters of interests are calculated. Impact of relaxation time on these flow characteristics are studied and analyzed. The relaxation time Rf is assumed to be in the range of 0 ≤ Rf ≤ 1 × 10 − 3. The flow simulations are carried out for large Reynolds number in the range of 200 ≤ Re ≤ 5000. In addition to the application of FreeFem++, some new and interesting features of the flow characteristics are presented and discussed.

A stabilized finite element method based on characteristic‐based polynomial pressure projection scheme for incompressible flows

AbstractIn this paper, a characteristic‐based polynomial pressure projection (CBP3) scheme is proposed to stabilize finite element method for solving incompressible laminar flow. The characteristic‐Galerkin (CG) method is adopted as the stabilization for convection caused oscillation in CBP3 scheme. The pressure oscillation caused by incompressible constraint is stabilized by the polynomial pressure projection (P3) technique. Proposed scheme is suitable for any element using the equal‐order approximation for velocity and pressure. In this paper, the linear triangular and bilinear quadrilateral elements are adopted. The constant pressure projection is used for triangular elements. The CBP3 formulations for quadrilateral element are derived using both constant and linear pressure projections. Besides, the quasi‐implicit second‐order time stepping is adopted. The verification of CBP3 scheme implementation and validation of CBP3 scheme are accomplished by calculating several benchmarks. The results of CBP3 scheme reveal that the linear pressure projection for quadrilateral element is not appropriate due to its severe pressure oscillation. The well‐agreed results of CBP3 scheme using constant pressure projection demonstrate its good stabilization for FEM to solve both low and relatively high Reynolds number flows.

Appropriate stabilized Galerkin approaches for solving two-dimensional coupled Burgers’ equations at high Reynolds numbers

This paper aims to seek proper stabilized Galerkin methods for solving the two-dimensional coupled Burgers’ equations at high Reynolds numbers. The stabilization techniques employed here include the streamline upwind/Petrov–Galerkin (SUPG) method, the spurious oscillations at layers diminishing (SOLD) method and the characteristic Galerkin (CG) method. The first two methods are combined with the Crank–Nicolson scheme for time discretization and the last one is applied in its semi-implicit version. Different from most of the studies on the equations which are usually devoted to improving the accuracy of computed solution in the case of low Reynolds numbers, this paper mainly focuses on keeping the stability of the solution at high Reynolds numbers, which is significant in practical applications and also challenging in numerical computation. We study two problems, equipped with mixed boundary conditions and only Dirichlet boundary conditions, respectively. Numerical experiments reveal that the SUPG method is optimal for the former problem, and the SOLD method is more appropriate for the latter one. In addition, the performances of these methods demonstrate the difference between the two problems, which is seldom mentioned previously and might be helpful to other conventional methods intending to solve the problems at high Reynolds numbers. And last, since SOLD methods have rarely been utilized to solve nonlinear unsteady problems before, this study also indicates the potential of this class of methods to solve nonlinear unsteady convection-dominated problems.

Finite element analysis of nitric oxide (NO) transport in system of permeable capillary and tissue

The endothelial dysfunction and the unbalanced secretion of vasoactive substances are the main causes of the macro and micro vascular complications for patients with diabetic mellitus (DM). This paper investigates the blood flow and the nitric oxide (NO) transport in a permeable capillary by using a finite element method. The computational domain consists of a permeable straight or bifurcated capillary and the surrounding tissue with different endothelial hydraulic permeability. The blood flow in the lumen of a capillary is assumed to be governed by the Stokes equation. The fluid flow in the surrounding tissue is simplified as the Darcy flow. The advection–diffusion reaction equation is employed to investigate the NO distribution in a lumen-tissue model, which is originated from the endothelial cells inside the capillary wall. The characteristic Galerkin method is employed for the discretization of the advection-diffusion reaction equation. The simulated results show that the NO transport is effectively affected by different hydraulic permeabilities due to the changes of the blood velocity. When the hydraulic permeability increases considerably, the NO concentration ([NO]) in the whole domain decreases accordingly. Moreover, the NO concentration increases in the area after the bifurcation of the capillary owing to the convective effect. It is also shown that even if the NO production by the endothelial cells is enhanced, the increase of the convection inside the vessel may reduce the endothelial NO concentration. As a signal transduction molecule, the spatial location with discriminating NO distribution may be useful for determining the position of the microangiopathy in the DM.

Finite element lattice Boltzmann method for fluid flow through complex fractured media with permeable matrix

Abstract This study proposed an improved finite element based generalized lattice Boltzmann model (FE-GLBM) to simulate the fluid flow in extremely heterogeneous porous media, in which the generalized lattice Boltzmann equations proposed by Guo and Zhao (2002) are solved using the characteristic Galerkin finite element method (GFEM), and the FEM is applied to improve the flexibility of GLBM and to optimize the flow simulation in complex geometries. An image-based automatic meshing method is adopted to generate advanced unstructured mesh of the structure. The proposed numerical model is validated with analytical results, finite-difference results, and previously published data in three cases, including the generalized two-dimensional Poiseuille flow, the lid-driven cavity flow, and the circular cylinder flow, respectively. The sensitivity study of the fluid flow in three-dimensional fractured porous media with a disordered fracture network and permeable matrix is carried out as a practical application example, and the results indicate that the matrix permeability plays an important role in controlling the flow dynamics in fractured porous media.

Lattice Boltzmann Simulation of Shallow Water Equations in Finite Element Framework

Lee, H. and Lee, N., 2018. Lattice Boltzmann Simulation of Shallow Water Equations in Finite Element Framework. In: Shim, J.-S.; Chun, I., and Lim, H.S. (eds.), Proceedings from the International Coastal Symposium (ICS) 2018 (Busan, Republic of Korea). Journal of Coastal Research, Special Issue No. 85, pp. 106–110. Coconut Creek (Florida), ISSN 0749-0208.The characteristic Galerkin finite element model for the discrete Boltzmann equation is developed for the depth-integrated shallow water equations. The lattice Boltzmann method (LBM) simulates microscopic collision and streaming of fluid particles and, then, evaluates the macroscopic velocity, pressure, etc. from the computed particle distribution. Due to the inherent geometric flexibility of the unstructured mesh in finite element method, the mesh clustering near large gradients and the numerical treatments of complex geometries has become possible. Some benchmark problems of practical importance (subcritical flow over hump, flow around cylinder, curved channel, etc.) are tested and the numerical results are in good agreement with previous experimental or analytic solutions.

Parallel characteristic mixed element method for saltwater intrusion problem

Numerical simulation of saltwater intrusion problem has attracted more and more attention in the past decades. Based on parallel subspace correction method proposed by Xu (1992), a new overlapping domain decomposition parallel algorithm is constructed to solve saltwater intrusion problem. In this algorithm, the splitting positive definite mixed element method is used to discrete head water equation, while the characteristic Galerkin finite element method is used to approximate concentration equation. The relation between the convergence rate and discretization parameters, including the overlapping degree of the subspaces, is considered. Furthermore, the convergence of this algorithm is analyzed and the corresponding error estimate is given. The convergence result suggests that only two iterations are needed to reach the given accuracy at each time level.

A level set method for shape optimization in semilinear elliptic problems

We develop a finite-element based level set method for numerically solve shape optimization problems constrained by semilinear elliptic problems. By combining the shape sensitivity analysis and level set method, a gradient descent algorithm is proposed to solve the model problem. Different from solving the nonlinear Hamilton–Jacobi equations with finite differences in traditional level set methods, we solve the linear convection equation and reinitialization equation using the characteristic Galerkin finite element method. The methodology can handle topology as well as shape changes in both regular and irregular design regions. Numerical results are presented to demonstrate the effectiveness of our algorithm as well as to verify symmetry preserving and breaking properties of optimal subdomains.

A High Order Conservative Semi-Lagrangian Discontinuous Galerkin Method for Two-Dimensional Transport Simulations

In this paper, we develop a class of high order conservative semi-Lagrangian (SL) discontinuous Galerkin methods for solving multi-dimensional linear transport equations. The methods rely on a characteristic Galerkin weak formulation, leading to $$L^2$$ stable discretizations for linear problems. Unlike many existing SL methods, the high order accuracy and mass conservation of the proposed methods are realized in a non-splitting manner. Thus, the detrimental splitting error, which is known to significantly contaminate long term transport simulations, will be not incurred. One key ingredient in the scheme formulation, borrowed from CSLAM (Lauritzen et al. in J Comput Phys 229(5):1401–1424, 2010), is the use of Green’s theorem which allows us to convert volume integrals into a set of line integrals. The resulting line integrals are much easier to approximate with high order accuracy, hence facilitating the implementation. Another novel ingredient is the construction of quadratic curves in approximating sides of upstream cell, leading to quadratic-curved quadrilateral upstream cells. Formal third order accuracy is obtained by such a construction. The desired positivity-preserving property is further attained by incorporating a high order bound-preserving filter. To assess the performance of the proposed methods, we test and compare the numerical schemes with a variety of configurations for solving several benchmark transport problems with both smooth and nonsmooth solutions. The efficiency and efficacy are numerically verified.

Analysis of scattering from multi-scale and multiple targets using characteristic basis function (CBFM) and integral equation discontinuous Galerkin (IEDG) methods

In this paper, the characteristic basis function method is adapted to analyze the problem of scattering from multiple and multi-scale impenetrable targets in conjunction with the discontinuous Galerkin method, and the monopolar RWG functions are chosen as the basis functions. This method enables us to analyze multiple and multi-scale targets using nonconforming discretizations. The use of the CBFs helps reduce the size of the impedance matrix of the associated method of moment significantly, enabling us to employ direct solvers as opposed to iterative solvers. The process of generating the reduced matrix is naturally parallel, and the reduced matrix is well conditioned, which obviates the need for pre-conditioning. In addition, the adaptive cross-approximation algorithm is implemented to reduce the complexity of the computation. Numerical results are included to demonstrate the accuracy and efficiency of the present approach when analyzing scattering from multiple and multi-scale targets using nonconforming discretizations.

Projection- and characteristic-based operator-splitting simulation of mixed convection flow coupling heat transfer and fluid flow in a lid-driven square cavity

ABSTRACTMixed convection flow in a 2D rectangular cavity is simulated by a novel finite element method, namely the projection- and characteristic-based operator-splitting algorithm. In each time step, the Navier–Stokes equations are split as follows: the diffusion part, the convection part by applying operator-splitting method, and the Poisson’s equation by adopting projection method. The implicit diffusion part is solved by the preconditioned conjugate gradient (PCG) method, whereas characteristic method is applied for the convection part in a multistep explicit scheme. The characteristic Galerkin approach is used to solve the energy equation. To validate the model, lid-driven cavity flow and natural convection flow are simulated.

확장 유한 요소 법을 적용한 RTM 공정 해석

ABSTRACT: Numerical simulation for Resin Transfer Molding manufacturing process is attempted by using theeXtended Finite Element Method (XFEM) combined with the level set method. XFEM allows to obtaining a goodnumerical precision of the pressure near the resin flow front, where its gradient is discontinuous. The enriched shapefunctions of XFEM are derived by using the level set values so as to correctly describe the interpolation with the resinflow front. In addition, the level set method is used to transport the resin flow front at each time step during themold filling. The level set values are calculated by an implicit characteristic Galerkin FEM. The multi-frontal solver ofIPSAP is adopted to solve the system. This work is validated by comparing the obtained results with analyticsolutions. Moreover, a localization method of XFEM and level set method is proposed to increase the computingefficiency. The computation domain is reduced to the small region near the resin flow front. Therefore, the totalcomputing time is strongly reduced by it. The efficiency test is made with a simple channel flow model. Severalapplication examples are analyzed to demonstrate ability of this method.초록

Effects of Ion Advection and Thermal Convection on Pore Pressure Changes in High Permeable Chemically Active Shale Formations

Wellbore instability problems are often encountered while drilling in water active shales due to changes in pore pressure. The change in pore pressure is caused by hydraulic, thermal, chemical, and electrical potential gradients. In all previous studies it has been found that the effects of ion advection and thermal convection have a negligible effect on changes in pore pressure for a range of very low permeable shale formations (>10−5 md). This is an appropriate assumption for very low permeable shale formations. For high permeable shale formations (e.g., shale with a disseminated microfissure network), however, thermal convection and ion advection can play a significant role. The authors present a hydro-chemo-thermo-electrical model based on finite element method to investigate the effect of advection on ion transfer and thermal convection on temperature and their combined effect on pore pressure in shale formations. All equations are based on the thermodynamics of irreversible processes in a discontinuous system. The characteristic Galerkin discretization method is used to stabilize the solution of advection and convection equations in the finite element approach. Results of this study revealed that ion and heat transfer are controlled primarily by permeability of the shale formations. Movement of fluid into or out of the formation is due to a combination of hydraulic, chemical, electrical, and thermal osmotic flow. Results have also shown that in high permeable shale formations the chemical potential gradient between the pore fluid and drilling fluid reaches equilibrium faster than in low permeable shale formations. This is mainly due to the advection of ion from drilling fluid to the shale formation.

Numerical simulation of RTM process using the extended finite element method combined with the level set method

Numerical simulation for resin transfer moulding manufacturing process is attempted using the extended finite element method combined with the level set method. The level set method is used to transport the resin flow front at each time step during the mould filling. Extended finite element method allows to obtaining a good numerical precision of the resin pressure near the resin flow front, where the gradient of the pressure could be discontinuous. The enriched shape functions of extended finite element method are derived using the level set values so as to correctly describe these shape functions with the resin flow front. In addition, the position of the resin flow front at each time step is calculated by an implicit characteristic Galerkin finite element method. Some examples are presented to validate our study in comparison with analytical or experimental results and to illustrate its industrial applications.

305 スプライン基底を用いた特性ガラーキン法に対するNitcheの方法に基づく境界拘束の適用

305 スプライン基底を用いた特性ガラーキン法に対するNitcheの方法に基づく境界拘束の適用