Characteristic Galerkin Method 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.

N-Side Cell-Based Smoothed Finite Element Method for Incompressible Flow with Heat Transfer Problems

In this paper, an n-sided cell-based smoothed finite element method (nCS-FEM) is presented to solve the convective heat transfer in incompressible viscous fluid flow. The Characteristic-based-split (CBS) scheme is used to stabilize the spatial and pressure oscillations of CS-FEM that occur during the process of solving incompressible Navier-Stokes equations. Meanwhile, the characteristic-Galerkin method in CBS is adopted as the convective stabilization in the heat transfer equation. In order to verify the proposed method, typical numerical examples are used to test the computational accuracy of CS-FEM. Meanwhile, the investigation of the robustness of highly distorted element is conducted in a natural convection test case. The calculation results of numerical examples show that CS-FEM is qualified to solve the steady and unsteady convective heat transfer problems for incompressible laminar flow with high reliability. The flow across tube banks with complex geometry in the heat sinks is also simulated to prove the capabilities of the proposed method. Well agreed comparison with the results of CFD software, Fluent, proves that the present method can effectively simulate realistic and complex flow with heat transfer problems.

A semi-implicit characteristic-based polynomial pressure projection for FEM to solve incompressible flows

PurposeThe purpose of this paper is to investigate a novel stabilization scheme to handle convection and pressure oscillation in the process of solving incompressible laminar flows by finite element method (FEM).Design/methodology/approachThe semi-implicit stabilization scheme, characteristic-based polynomial pressure projection (CBP3) consists of the Characteristic-Galerkin method and polynomial pressure projection. Theoretically, the proposed scheme works for any type of element using equal-order approximation for velocity and pressure. In this work, linear 3-node triangular and 4-node tetrahedral elements are the focus, which are the simplest but most difficult elements for pressure stabilizations.FindingsThe present paper proposes a new scheme, which can stabilize FEM solution for flows of both low and relatively high Reynolds numbers. And the influence of stabilization parameters of the CBP3 scheme has also been investigated.Research limitations/implicationsThe research in this work is limited to the laminar incompressible flow.Practical implicationsThe verification and validation of the CBP3 scheme are conducted by several 2 D and 3 D numerical examples. The scheme could be used to deal with more practical fluid problems.Social implicationsThe application of scheme to study complex hemodynamics of patient-specific abdominal aortic aneurysm is also presented, which demonstrates its potential to solve bio-flows.Originality/valueThe paper simulated 2 D and 3 D numerical examples with superior results compared to existing results and experiments. The novel CBP3 scheme is verified to be very effective in handling convection and pressure oscillation.

Penalty and characteristic-based operator splitting with multistep scheme finite element method for unsteady incompressible viscous flows

The penalty and characteristic-based operator splitting with multistep scheme (penalty-MCBOS) has been developed for the unsteady incompressible Navier-Stokes N-S equations. At each time step, the N-S equations are split into the diffusive part, the convective part by adopting the operator splitting method and the pressure part by applying the penalty method with low penalty parameters. For the diffusive part, the temporal discretisation is based on backward difference method and is solved by preconditioned conjugate gradient (PCG) method. For the convective part, the temporal discretisation is performed by characteristic-Galerkin method. It is solved explicitly and the multistep technique is used. The pressure can be calculated from the pressure part without necessitating the solution of the pressure Poisson equation. The plane Poisseuille flow, the lid-driven cavity flow, the lid-driven triangular cavity flow and the backward-facing step flow are investigated to validate the present model. It is shown that the numerical results are in good agreement with the previously published data, and the present model has high efficiency. For the backward-facing step problem at Re = 3,000, the flow becomes periodical in time and dynamic evolution process of vortex can be simulated.

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.

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.

A Characteristic Galerkin Method for Eddy-Current Field Analysis in High-Speed Rotating Solid Conductors

Standard Galerkin finite element method (FEM) performs poorly when simulating convection dominated problems using the Eulerian formulation. A numerical method for analyzing transient eddy-current magnetic field in high-speed rotating solid conductors using characteristic Galerkin finite element method (CGFEM) is presented. The convection-diffusion equation is discretized implicitly in time domain via the method of characteristics and in space via piecewise quadratic finite elements. An example with extremely large Peclet number is solved by using the CGFEM to validate its effectiveness. TEAM workshop problem 30 A which is a practical case with high-speed rotating conductors is studied to showcase the efficiency and accuracy of the proposed method.

Characteristic-based operator-splitting finite element method for Navier-Stokes equations

A new finite element method, which is the characteristic-based operator-splitting (CBOS) algorithm, is developed to solve Navier-Stokes (N-S) equations. In each time step, the equations are split into the diffusive part and the convective part by adopting the operator-splitting algorithm. For the diffusive part, the temporal discretization is performed by the backward difference method which yields an implicit scheme and the spatial discretization is performed by the standard Galerkin method. The convective part can be discretized using the characteristic Galerkin method and solved explicitly. The driven square flow and backward-facing step flow are conducted to validate the model. It is shown that the numerical results agree well with the standard solutions or existing experimental data, and the present model has high accuracy and good stability. It provides a prospective research method for solving N-S equations.

A new algorithm for non-newtonian flow and its application in mould filling process

The simulation of injection molding process requires a stable algorithm to model the molten polymer with non-isothermal non-Newtonian property. In this paper, a staggered and iterative scheme is particularly designed to solve the velocity-pressure-temperature variables. In consideration of the polymer characteristic of high viscosity and low thermal conductivity, the non-Newtonian momentum-mass conservation equations are solved by the Crank-Nicolson method based split (CNBS) scheme, and the energy conservation equation with convective character is discretized by the characteristic Galerkin (CG) method. In addition, an arbitrary Lagrangian Eulerian (ALE) free surface tracking and mesh generation method is introduced to catch the front of the fluid flow. The efficiency of the proposed scheme is demonstrated by numerical experiments including a lid-driven cavity flow problem and an injection molding problem.

A posteriori error estimates for the coupling equations of scalar conservation laws

In this paper we prove a posteriori L 2(L 2) and L ∞(H −1) residual based error estimates for a finite element method for the one-dimensional time dependent coupling equations of two scalar conservation laws. The underlying discretization scheme is Characteristic Galerkin method which is the particular variant of the Streamline diffusion finite element method for δ=0. Our estimate contains certain strong stability factors related to the solution of an associated linearized dual problem combined with the Galerkin orthogonality of the finite element method. The stability factor measures the stability properties of the linearized dual problem. We compute the stability factors for some examples by solving the dual problem numerically.

An accelerated algorithm for Navier–Stokes equations

In this paper, the numerical solution of the Navier–Stokes equations by the Characteristic-Based-Split (CBS) scheme is accelerated with the Minimum Polynomial Extrapolation (MPE) method to obtain the steady state solution for evolution incompressible and compressible problems. The CBS is essentially a fractional time-stepping algorithm based on an original finite difference velocity-projection scheme where the convective terms are treated using the idea of the Characteristic-Galerkin method. In this work, the semi-implicit version of the CBS with global time-stepping is used for incompressible problems whereas the fully-explicit version is used for compressible flows. At the other end, the MPE is a vector extrapolation method that transforms the original sequence into another sequence converging to the same limit faster then the original one without the explicit knowledge of the sequence generator. The developed algorithm, tested on two-dimensional benchmark problems, demonstrates the new computational features arising from the introduction of the extrapolation procedure to the CBS scheme. In particular, the results show a remarkable reduction of the computational cost of the simulation.

303 特性有限要素法による自由界面問題の表面張力計算に関する検討(OS3.CIP法とその関連手法(1),オーガナイズドセッション)

In this paper we propose a new approximation of upwind streamlines in a framework of the Hermitian characteristic Galerkin (HCG) method for advection diffusion problems. The HCG method applies the Hermitian type element which can improve accuracy of the characteristic Galerkin method. We present a new high order method using the Newmark scheme for the approximation of upwind streamlines, which is often used for the dynamic structural analysis. We show the effectiveness of the present method by evaluating the computational accuracy.

A Family of Characteristic Discontinuous Galerkin Methods for Transient Advection-Diffusion Equations and Their Optimal-Order L^2 Error Estimates

We develop a family of characteristic discontinuous Galerkin methods for transient advection-diffusion equations, including the characteristic NIPG, OBB, IIPG, and SIPG schemes. The derived schemes possess combined advantages of Eulerian- Lagrangian methods and discontinuous Galerkin methods. An optimal-order error estimate in the L 2 norm and a superconvergence estimate in a weighted energy norm are proved for the characteristic NIPG, IIPG, and SIPG scheme. Numerical experi- ments are presented to confirm the optimal-order spatial and temporal convergence rates of these schemes as proved in the theorems and to show that these schemes com- pare favorably to the standard NIPG, OBB, IIPG, and SIPG schemes in the context of advection-diffusion equations. AMS subject classifications: 35R32, 37N30, 65M12, 76N15

Element-free characteristic Galerkin method for Burgers’ equation

A new meshfree method named element-free characteristic Galerkin method (EFCGM) is proposed for solving Burgers’ equation with various values of viscosity. Based on the characteristic method, the convection terms of Burgers’ equation disappear and this process makes Burgers’ equation self-adjoint, which ensures that the spatial discretization by the Galerkin method can be optimal. After the temporal discretization, element-free Galerkin method is then applied to solve the semi-discrete equation in space. Moreover, the process is fully explicit at each time step. In order to show the efficiency of the presented method, one-dimensional and two-dimensional Burgers’ equations are considered. The numerical solutions obtained with different values of viscosity are compared with the analytical solutions as well as the results by other numerical schemes. It can be easily seen that the proposed method is efficient, robust and reliable for solving Burgers’ equation, even involving high Reynolds number for which the analytical solution fails.

An optimal shape problem related to the realistic design of river fishways

A river fishway is a hydraulic structure that facilitates fish in overcoming obstacles (dams, waterfalls, etc.) to their spawning and other migrations in rivers. In this work we present a mathematical formulation of an optimal design problem for a vertical slot fishway, where the state system is given by the 2D shallow water equations fixing the height and velocity of water, the design variables are the geometry of the slots, and the objective function is determined by the existence of rest areas for fish and of a water velocity suitable for fish swimming capability. We also derive an expression for the gradient of the objective function via the adjoint system. From the numerical point of view, we present a characteristic-Galerkin method for solving the shallow water equations, and a direct search algorithm for the computation of the optimal design variables. Finally, we give numerical results obtained for a standard ten pools channel.

610 特性有限要素法における時間離散および上流点近似に関する考察(OS2.CIP法とその関連手法(3),オーガナイズドセッション)

In this paper we propose a new approximation of upwind streamlines in a framework of the Hermitian characteristic Galerkin (HCG) method for advection diffusion problems. The HCG method applies the Hermitian type element which can improve accuracy of the characteristic Galerkin method. We present a new high order method using the Newmark scheme for the approximation of upwind streamlines, which is often used for the dynamic structural analysis. We show the effectiveness of the present method by evaluating the computational accuracy.