Implicit Time Scheme Research Articles

Computational gas dynamics in a wide range of Mach number on heterogeneous cluster architecture

The specific features of numerical method for gas dynamics problems are considered in this paper. The viscous flows are considered in a wide range of Mach number. The numerical method is based on conservation laws applied on unstructured heterogeneous mesh with further integration on complex cluster architectures. Application of unstructured mesh allows one to perform calculation of geometrically complex 3D objects with strict conservation properties. The developed numerical method includes multiple flux solvers (Godunov's, HLLC, Roe-Fix, AUSM, etc), first and second order spatial approximation, and explicit, semi-implicit or implicit first order time schemes. The method utilizes heterogeneous GPU HPC clusters. A wide range of problems is used for verification of the developed computational complex. All results on these problems had a good agreement with reference data.

Pressure boundary conditions for blood flows

Simulations of blood flows in arteries require numerical solutions of fluid-structure interactions involving Navier-Stokes equations coupled with large displacement visco-elasticity for the vessels. Among the various simplifications which have been proposed, the surface pressure model leads to a hierarchy of simpler models including one which involves only the pressure. The model exhibits fundamental frequencies which can be computed and compared with the pulse. Yet unconditionally stable time discretizations can be constructed by combining implicit time schemes with Galerkin-characteristic discretization of the convection terms in the Navier-Stokes equations. Such problems with prescribed pressure on the walls will be shown to be efficient and accurate as an approximation of the full fluid structure interaction problem.

Performance of the Alternating Direction Implicit Scheme With Recursive Sparsification for the Finite Element Time Domain Method

In this paper, we solve the $ \mathcal {E}$ and $ \mathcal {B}$ mixed finite element time domain formulation using the alternating-direction implicit (ADI) scheme in time. To save computational cost, the algorithm is improved using a recursive sparsification procedure that approximates the inverse matrix needed to calculate the electric field. Although the method is unconditionally stable, we show that the numerical performance of the algorithm is strongly related to the time step. Numerical results demonstrated that for the same accuracy our approach is much faster than the ADI with Cholesky factorization.

A multi-dimensional space method for dynamic cracks problems using implicit time scheme in the framework of the extended finite element method

The extended finite element method (XFEM) has been widely used to investigate the moving cracks without any remeshing. The degrees of freedom (DOFs) of nodes around crack surfaces and crack tips are added to represent the discontinuous fields and asymptotic fields. However, the XFEM encounters some challenges in dynamic problems because the total number of DOFs increases with the crack growth, which leads to difficulty in iteration algorithm of time integration. In this paper, based on the XFEM, a multi-dimensional space method is developed to simulate the crack growth under dynamic loads. Every node, not limited to the nodes around crack tips and crack surfaces, has 12 DOFs in the domain containing one crack (12n DOFs in the domain containing n cracks). The generalized boundary condition is proposed to modify the matrices of stiffness, mass and damping. The modified matrices of stiffness, mass and damping are highly sparse, the additional central processing unit time is little and the use of internal memory does not increase compared with the traditional XFEM. Moreover, the implicit Newmark time integration scheme is introduced to solve the dynamic problems of XFEM. Compared with the other time integration schemes, the time step of the present method is much longer. The convergence property of the present method is demonstrated in theoretical and numerical studies. In order to verify the robustness and the precision of the present method, three numerical cases are presented.

A posteriori analysis of the spectral element discretization of heat equation

Abstract In this paper, we present a posteriori analysis of the discretization of the heat equation by spectral element method. We apply Euler’s implicit scheme in time and spectral method in space. We propose two families of error indicators both of them are built from the residual of the equation and we prove that they satisfy some optimal estimates. We present some numerical results which are coherent with the theoretical ones.

Discontinuous Galerkin method for multicomponent chemically reacting flows and combustion

This paper presents the development of a discontinuous Galerkin (DG) method for application to chemically reacting flows in subsonic and supersonic regimes under the consideration of variable thermo-viscous-diffusive transport properties, detailed and stiff reaction chemistry, and shock capturing. A hybrid-flux formulation is developed for treatment of the convective fluxes, combining a conservative Riemann-solver and an extended double-flux scheme. A computationally efficient splitting scheme is proposed, in which advection and diffusion operators are solved in the weak form, and the chemically stiff substep is advanced in the strong form using a time-implicit scheme. The discretization of the viscous-diffusive transport terms follows the second form of Bassi and Rebay, and the WENO-based limiter due to Zhong and Shu is extended to multicomponent systems. Boundary conditions are developed for subsonic and supersonic flow conditions, and the algorithm is coupled to thermochemical libraries to account for detailed reaction chemistry and complex transport. The resulting DG method is applied to a series of test cases of increasing physico-chemical complexity. Beginning with one- and two-dimensional multispecies advection and shock–fluid interaction problems, computational efficiency, convergence, and conservation properties are demonstrated. This study is followed by considering a series of detonation and supersonic combustion problems to investigate the convergence-rate and the shock-capturing capability in the presence of one- and multistep reaction chemistry. The DG algorithm is then applied to diffusion-controlled deflagration problems. By examining convergence properties for polynomial order and spatial resolution, and comparing these with second-order finite-volume solutions, it is shown that optimal convergence is achieved and that polynomial refinement provides advantages in better resolving the localized flame structure and complex flow-field features associated with multidimensional and hydrodynamic/thermo-diffusive instabilities in deflagration and detonation systems. Comparisons with standard third- and fifth-order WENO schemes are presented to illustrate the benefit of the DG scheme for application to detonation and multispecies flow/shock-interaction problems.

Numerical modeling of liquid-phase infiltration in the process of sintering ceramic composites

The paper presents a numerical model for infiltration of porous preforms of ceramic materials in the process of sintering. The mathematical model of infiltration is based on the Richards equation and Van Genuchten formulas describing permeability properties of the porous structure and the infiltrant. A finite-element method and stable implicit finite-difference time scheme are applied for discretization of the Richards equation. Based on the outcome of solving test problems, a comparison of the numerical results with the results obtained by other researchers and with the experimental data, the numerical model has been proved adequate. The paper provides some findings of computer modeling of infiltration of liquid silicon into porous silicon carbide preforms of various geometries and dimensions.

A Completely Discrete Heterogeneous Multiscale Finite Element Method for Multiscale Richards’ Equation of van Genuchten-Mualem Model

We propose a fully discrete method for the multiscale Richards’ equation of van Genuchten-Mualem model which describes the flow transport in unsaturated heterogenous porous media. Under the framework of heterogeneous multiscale method (HMM), a fully discrete scheme combined with a regularized procedure is proposed. Including the numerical integration, the discretization is given byC0piecewise finite element in space and an implicit scheme in time. Error estimates between the numerical solution and the solution of homogenized problem are derived under the assumption that the permeability is periodic. Numerical experiments with periodic and random permeability are carried out for the van Genuchten-Mualem model of Richards’ equation to show the efficiency and accuracy of the proposed method.

Fixed-Grid Modeling of Solid-Liquid Phase Change in Unstructured Meshes Using Explicit Time Schemes

Fixed-grid enthalpy models have been used extensively for solid-liquid phase-change computational fluid dynamics (CFD) simulations with implicit time schemes. In this work, this technique is implemented for explicit time schemes and collocated unstructured domain discretization, due to the interest in coupling phase-change formulations with turbulence models for liquid motion. Issues regarding the form of the energy equation, the treatment of the pressure equation, as well as the momentum source term coefficient introduced by the enthalpy-porosity method are described in detail. Numerical implementation is tested with different study cases, showing good agreement with other experimental and numerical results.

Finite volume method solution of pollutant transport in catchment sheet flow

A finite volume numerical method was employed in the solution of two-dimensional pollutant transport in catchment sheet flow. The full dynamic wave constituted the sheet flow while the advection–diffusion equation with sink/source terms was the pollutant transport model. It is assumed that the solute in the surface active layer is uniformly distributed and the exchange rate of the solute between the active layer and overland flow are proportional to the difference between the concentrations in soil and water volume. Decrease of the solute transfer rate in the active surface layer caused by the transfer of solutes from soil to the overlying runoff is assumed to follow an exponential law. The equations governing sheet flow and pollutant transport are discretized using the finite volume method in space and an implicit backward difference scheme in time. The model was subjected to several numerical tests involving varying microtopographic surface, roughness, and infiltration. The results revealed that spatially varying microtopography plays an important role unlike the roughness and infiltration with respect to the total pollutant rate from the outlet of a catchment.

Comparison of different nonlinear solvers for 2D time-implicit stellar hydrodynamics

Time-implicit schemes are attractive since they allow numerical time steps that are much larger than those permitted by the Courant-Friedrich-Lewy criterion characterizing time-explicit methods. This advantage comes, however, at a cost: the solution of a system of nonlinear equations is required at each time step. In this work, the nonlinear system results from the discretization of the hydrodynamical equations with the Crank-Nicholson scheme. We compare the cost of different methods, based on Newton-Raphson iterations, to solve this nonlinear system, and benchmark their performances against time-explicit schemes. Since our general scientific objective is to model stellar interiors, we use as test cases two realistic models for the convective envelope of a red giant and a young Sun. Focusing on 2D simulations, we show that the best performances are obtained with the quasi-Newton method proposed by Broyden. Another important concern is the accuracy of implicit calculations. Based on the study of an idealized problem, namely the advection of a single vortex by a uniform flow, we show that there are two aspects: i) the nonlinear solver has to be accurate enough to resolve the truncation error of the numerical discretization; and ii) the time step has be small enough to resolve the advection of eddies. We show that with these two conditions fulfilled, our implicit methods exhibit similar accuracy to time-explicit schemes, which have lower values for the time step and higher computational costs. Finally, we discuss in the conclusion the applicability of these methods to fully implicit 3D calculations.

Numerical Simulation of Compressible Thermo-Buoyant Flow in a Partially Opened Enclosure with Localized Heater from Below

We present two numerical models for the study of unsteady two-dimensional flow of air subjected to a heat source in a cell divided into two areas of different size by a vertical lintel. The heat source placed on the floor of the enclosure produces hot gases for a finite time. First numerical model based on solving the equations of conservation of mass, momentum and energy, uses an implicit scheme in time and a finite difference hybrid space. The injection of hot gas in the two areas led to the abandonment of the Boussinesq hypothesis. This model also takes into account variations in the physical properties of the fluid. The second model is a finite element approach which gave more detailed results, thanks to the high density of the mesh node and the incorporation of the boundary conditions in the general equations of the problem. Dynamically, the results are very significant in terms of the flow direction, circulation areas and creation of vortices. The results for two simple cases of a cell partitioned allowed us to know the characteristics of the flow adjacent to the heat source and through the two compartments of the enclosure.

Modeling pollutant transport in overland flow over non-planar and non-homogenous infiltrating surfaces

Pollutant transport in overland flow over surfaces with spatially varying microtopography, roughness, and infiltration was investigated using the diffusion wave equation and transport rate-based equation. The finite volume method in space and an implicit backward difference scheme in time were employed in the numerical solution of the 2D governing equations. The developed model was first tested against an analytical solution and an experimental study involving overland flow and the associated pollutant transport, subsequently a series of numerical tests were carried out. Non-point source pollution was investigated under spatially varying microtopography, roughness, and infiltration. The simulation results showed that microtopography and roughness were the dominant factors causing significant spatial variations in solute concentration. When the spatially varying microtopography was replaced by a smooth surface, the result was an overestimation of the solute rate at the outlet of the upland. On the other hand, when the spatially varying roughness was replaced by the average roughness and spatially varying infiltration rate by the average infiltration rate, the pollutant discharge at the outlet of the upland was not significantly affected. The numerical results further showed that one cannot ignore the spatial variations of slope and roughness when investigating the local pollutant concentration distribution.

3-D Carotid Multi-Region MRI Segmentation by Globally Optimal Evolution of Coupled Surfaces

In this paper, we propose a novel global optimization based 3-D multi-region segmentation algorithm for T1-weighted black-blood carotid magnetic resonance (MR) images. The proposed algorithm partitions a 3-D carotid MR image into three regions: wall, lumen, and background. The algorithm performs such partitioning by simultaneously evolving two coupled 3-D surfaces of carotid artery adventitia boundary (AB) and lumen-intima boundary (LIB) while preserving their anatomical inter-surface consistency such that the LIB is always located within the AB. In particular, we show that the proposed algorithm results in a fully time implicit scheme that propagates the two linearly ordered surfaces of the AB and LIB to their globally optimal positions during each discrete time frame by convex relaxation. In this regard, we introduce the continuous max-flow model and prove its duality/equivalence to the convex relaxed optimization problem with respect to each evolution step. We then propose a fully parallelized continuous max-flow-based algorithm, which can be readily implemented on a GPU to achieve high computational efficiency. Extensive experiments, with four users using 12 3T MR and 26 1.5T MR images, demonstrate that the proposed algorithm yields high accuracy and low operator variability in computing vessel wall volume. In addition, we show the algorithm outperforms previous methods in terms of high computational efficiency and robustness with fewer user interactions.

A SIMPLE based discontinuous Galerkin solver for steady incompressible flows

In this paper we present how the well-known SIMPLE algorithm can be extended to solve the steady incompressible Navier–Stokes equations discretized by the discontinuous Galerkin method. The convective part is discretized by the local Lax-Friedrichs fluxes and the viscous part by the symmetric interior penalty method. Within the SIMPLE algorithm, the equations are solved in an iterative process. The discretized equations are linearized and an equation for the pressure is derived on the discrete level. The equations obtained for each velocity component and the pressure are decoupled and therefore can be solved sequentially, leading to an efficient solution procedure. The extension of the proposed scheme to the unsteady case is straightforward, where fully implicit time schemes can be used.Various test cases are carried out: the Poiseuille flow, the channel flow with constant transpiration, the Kovasznay flow, the flow into a corner and the backward-facing step flow. Using a mixed-order formulation, i.e. order k for the velocity and order k-1 for the pressure, the scheme is numerically stable for all test cases. Convergence rates of k+1 and k in the L2-norm are observed for velocity and pressure, respectively. A study of the convergence behavior of the SIMPLE algorithm shows that no under-relaxation for the pressure is needed, which is in strong contrast to the application of the SIMPLE algorithm in the context of the finite volume method or the continuous finite element method. We conclude that the proposed scheme is efficient to solve the steady incompressible Navier–Stokes equations in the context of the discontinuous Galerkin method comprising hp-accuracy.

Plasma Control for a Maneuvering Low-Aspect-Ratio Wing at Low Reynolds Number

Abstract Plasma-based flow control was explored as a means of enhancing the performance of a maneuvering flat-plate wing. For this purpose, a numerical investigation was conducted via large-eddy simulation (LES). The wing has a rectangular planform, a thickness to chord ratio of 0.016, and an aspect ratio of 2.0. Computations were carried out at a chord-based Reynolds number of 20,000, such that the configuration and flow conditions are typical of those commonly utilized in a small unmanned air system (UAS). Solutions were obtained to the Navier–Stokes equations, that were augmented by source terms used to represent body forces imparted by plasma actuators on the fluid. A simple phenomenological model provided these body forces resulting from the electric field generated by the plasma. The numerical method is based upon a high-fidelity time-implicit scheme and an implicit LES approach, which were applied to obtain solutions on an overset mesh system. Specific maneuvers considered in the investigation all began at 0 deg angle of attack, and consisted of a pitch-up and return, a pitch-up and hold, and a pitch-up to 60 deg. The maximum angle of attack for the first two maneuvers was 35 deg, which is well above that for static stall. Two different pitch rates were imposed for each of the specified motions. In control situations, a plasma actuator was distributed in the spanwise direction along the wing leading edge, or extended in the chordwise direction along the wing tip. Control solutions were compared with baseline results without actuation in order to assess the benefits of flow control and to determine its effectiveness. In all cases, it was found that plasma control can appreciably improve the time integrated lift over the duration of the maneuvers. The wing-tip actuator could achieve up to a 40% increase in the integrated lift, above that of the baseline value.

Plasma flow control simulations of a low-Reynolds number low-aspect-ratio wing

Large-eddy simulation (LES) is employed to investigate use of plasma-based control in order to improve aerodynamic performance of a flat-plate wing. The wing has a rectangular planform, a thickness to chord ratio of 0.016, and an aspect ratio of 2.0. Computations are carried out at a chord-based Reynolds number of 20,000, such that the configuration and flow conditions are typical of those commonly utilized in micro air vehicle (MAV) applications. Solutions are obtained to the Navier–Stokes equations, that were augmented by source terms used to represent body forces imparted by plasma actuators on the fluid. A simple phenomenological model provided these body forces resulting from the electric field generated by the plasma. The numerical method is based upon a high-fidelity time-implicit scheme and an implicit LES approach, which are applied to obtain solutions on an overset mesh system. The control strategy explores plasma actuators that are distributed in the spanwise direction along the wing leading edge, and actuators extending in the chordwise direction along the wing tip. The investigation considers plasma-induced forces that are imposed in either the co-flowing or counter-flow directions for the former arrangement, and directed inboard or outboard for the latter. Both continuous and pulsed operation of actuators are simulated, and the magnitude of the plasma force is varied in order to establish the most efficient means of control. Computations are carried out at angles of attack that are below (8.0deg) and above (25.0deg) the stall value. Control solutions are compared with baseline results without actuation to determine the most beneficial control strategies. It was found that actuation is effective only for sufficiently large values of the plasma force. A favorable comparison is shown with the limited available experimental data.

A modular numerical method for implicit 0D/3D coupling in cardiovascular finite element simulations

Implementation of boundary conditions in cardiovascular simulations poses numerical challenges due to the complex dynamic behavior of the circulatory system. The use of elaborate closed-loop lumped parameter network (LPN) models of the heart and the circulatory system as boundary conditions for computational fluid dynamics (CFD) simulations can provide valuable global dynamic information, particularly for patient specific simulations. In this paper, the necessary formulation for coupling an arbitrary LPN to a finite element Navier–Stokes solver is presented. A circuit analogy closed-loop LPN is solved numerically, and pressure and flow information is iteratively passed between the 0D and 3D domains at interface boundaries, resulting in a time-implicit scheme. For Neumann boundaries, an implicit method, regardless of the LPN, is presented to achieve the desired stability and convergence properties. Numerical procedures for passing flow and pressure information between the 0D and 3D domains are described, and implicit, semi-implicit, and explicit quasi-Newton formulations are compared. The issue of divergence in the presence of backflow is addressed via a stabilized boundary formulation. The requirements for coupling Dirichlet boundary conditions are also discussed and this approach is compared in detail to that of the Neumann coupled boundaries. Having the option to select between Dirichlet and Neumann coupled boundary conditions increases the flexibility of current framework by allowing a wide range of components to be used at the 3D–0D interface.

An Adaptive Time-Stepping Strategy for the Cahn-Hilliard Equation

AbstractThis paper studies the numerical simulations for the Cahn-Hilliard equation which describes a phase separation phenomenon. The numerical simulation of the Cahn-Hilliard model needs very long time to reach the steady state, and therefore large time-stepping methods become useful. The main objective of this work is to construct the unconditionally energy stable finite difference scheme so that the large time steps can be used in the numerical simulations. The equation is discretized by the central difference scheme in space and fully implicit second-order scheme in time. The proposed scheme is proved to be unconditionally energy stable and mass-conservative. An error estimate for the numerical solution is also obtained with second order in both space and time. By using this energy stable scheme, an adaptive time-stepping strategy is proposed, which selects time steps adaptively based on the variation of the free energy against time. The numerical experiments are presented to demonstrate the effectiveness of the adaptive time-stepping approach.

Performance evaluation of a direct drive wave energy converter using CFD

This paper presents a numerical study of the savonius type direct drive turbine incorporated in the rear bottom of typical chamber geometry of an oscillating water column chamber (OWC) for wave energy conversion. The paper deals with a numerical modelling devoted to investigate the effect of wave on the performance and internal flow of the savonius turbine in the components of an oscillating water column (OWC) system used for the wave energy capture. In the present paper, the flow behaviour is modelled by using the commercial code ANSYS CFX. The numerical flow models have been elaborated and simulated independently with different inlet geometries of a water chamber. Constant periodic wave flow calculations were performed to investigate the flow distribution at the turbines inlet section, as well as the properties of the savonius type turbine. The flow is assumed to be two-dimensional (2D), viscous, turbulent and unsteady. The commercial CFD code is used with a solver of the coupled conservation equations of mass, momentum and energy, with an implicit time scheme. Turbulence is modelled with the k–e model. Simulation result shows that the water velocity, differential pressure, and the best efficiency vary considerably depending on the savonius rotor angle. The obtained results also indicate that the developed models are well suitable to analyze the water flows both in the chamber and around the savonius rotor.