Time-marching Scheme Research Articles

Numerical simulation of micro gas turbine engine based on time-marching one-dimensional method

The advancement of gas turbine technology serves as a crucial indicator of high-tech and technological prowess, with widespread applications in aviation, marine vessels, power generation, and clean energy. The precision of the performance simulation model stands as a pivotal component in the development of gas turbines. The present study deals with the one-dimensional modeling and validation of gas turbines by utilizing a time-marching scheme, in order to establish the correlation between the engine components and full-engine performance. The flow through the turbomachinery was solved by the one-dimensional model. The combustor was modelled as a compressible actuator disk. The accuracy of the turbomachinery solver was validated using the KJ66 compressor and turbine, and the predicted results demonstrated consistency with the experimental data or the three-dimensional simulations. The characteristic interface method was developed based on Riemann invariants, in order to provide the flow properties on both sides of the combustor. The results indicated that compared to the component-by-component approach, the developed characteristic interface method significantly reduced the computational time. Finally, the developed one-dimensional method was employed to simulate the performance of the KJ66 micro gas turbine engine. The predicted performance for the KJ66 gas turbine engine was consistent with the trend in the experiment results. The novelty of the present study was its modeling approach, and its ability to eliminate the dependence on the general component map, and obtain meridional distributions of flow parameters, which cannot be achieved in traditional zero-dimensional thermal cycle calculations. These results can provide an accurate and powerful tool for the analysis and optimization of component and full-engine performance in the preliminary design stage.

Exponential time propagators for elastodynamics

We propose a computationally efficient and systematically convergent approach for elastodynamics simulations. We recast the second-order dynamical equation of elastodynamics into an equivalent first-order system of coupled equations, so as to express the solution in the form of a Magnus expansion. With any spatial discretization, it entails computing the exponential of a matrix acting upon a vector. We employ an adaptive Krylov subspace approach to inexpensively and accurately evaluate the action of the exponential matrix on a vector. In particular, we use an apriori error estimate to predict the optimal Krylov subspace size required for each time-step size. We show that the Magnus expansion truncated after its first term provides quadratic and superquadratic convergence in the time-step for nonlinear and linear elastodynamics, respectively. We demonstrate the accuracy and efficiency of the proposed method for one linear (linear cantilever beam) and three nonlinear (nonlinear cantilever beam, soft tissue elastomer, and hyperelastic rubber) benchmark systems. For a desired accuracy in energy, displacement, and velocity, our method allows for 10−100× larger time-steps than conventional time-marching schemes such as Newmark-β method. Computationally, it translates to a ∼1000× and ∼10−100× speed-up over conventional time-marching schemes for linear and nonlinear elastodynamics, respectively.

Energy-stable auxiliary variable viscosity splitting (AVVS) method for the incompressible Navier–Stokes equations and turbidity current system

In this work, we develop a novel energy-stable linear approach, which we name as auxiliary variable viscosity splitting (AVVS) method, to efficiently solve the incompressible fluid flows. Different from the projection-type methods with pressure correction, the AVVS method adopts the viscosity splitting strategy to split the original momentum equation into an intermediate momentum equation without divergence-free constraint and an advection-free momentum equation. A time-dependent auxiliary variable which has exact value 1 is introduced to construct a supplementary equation. The new model not only inherits the same dynamics of original incompressible Navier–Stokes equations, but also facilitates us to design linearly decoupled and energy-stable time-marching scheme. Comparing with the conventional projection-type schemes, the present method leads to an energy dissipation law with respect to kinetic energy instead of a modified energy including velocity and pressure gradient. In each time step, only two parabolic equations with constant coefficients and one Poisson equation need to be solved. Therefore, the numerical implementation is highly efficient. Moreover, the proposed AVVS method can be directly extended to construct linear, decoupled, and energy-stable scheme for the turbidity current system with slight modifications on the right-hand side of supplementary equation. Extensive numerical experiments are implemented to validate the accuracy, energy stability, and capability in complex fluid simulations.

Efficiently linear and unconditionally energy-stable time-marching schemes with energy relaxation for the phase-field surfactant model

In this article, we consider consistent, efficient, and accurate numerical approximations for a recently developed Cahn–Hilliard type phase-field surfactant model enjoying the bounded-from-below condition of total energy. The phase-field surfactant model naturally satisfies the energy dissipation property in time. Based on this feature, we construct linearly decoupled and unconditionally energy-stable time-marching schemes based on an efficient variant of scalar auxiliary variable approach. To enhance the consistency of discrete modified energy and discrete original energy, a practical energy relaxation technique is adopted. We analytically proved the unique solvability and unconditional energy stability of the proposed schemes. In each time step, only two linear elliptic type equations with constants coefficients need to be solved and several algebraic operations need to be performed. Therefore, the proposed schemes are highly efficient and easy to implement. The space is discretized by the finite difference method and the linear multigrid algorithm is used to accelerate the convergence. The numerical results indicate that the proposed method not only has desired accuracy in time but also satisfies the energy dissipation law even if larger time steps are used. Furthermore, the surfactant laden spinodal decompositions and surfactant absorptions can be well simulated.

Modeling and numerical analysis for cylindrical flow-through catalytic membrane reactor with power-law reaction kinetics: Revealing dead-core phenomena

Cylindrical flow-through catalytic membrane reactors, employing porous membranes impregnated with catalysts, offer enhanced selectivity and yield in chemical reactions. This work focuses on the mathematical modeling and numerical analysis of a cylindrical flow-through catalytic membrane reactor. The proposed reactor model addresses a critical gap in research by incorporating realistic geometries of porous catalytic membranes. This model simulates a series of irreversible reactions following power-law kinetics under non-isothermal conditions within the reactor, formulated using a system of non-linear differential equations to account for mass and heat balances.The paper investigates the occurrence of dead zones within the membrane reactor as a result of rapid reactant depletion, a phenomenon that has not been extensively studied in prior literature for cylindrical membrane reactors. Problems with fractional reaction exponents require efficient numerical solvers since conventional iterative solvers encounter difficulties due to the fact that the power-law reaction term with fractional reaction exponent is not differentiable at the vanishing concentration. A novel time-marching scheme specifically designed for the cylindrical catalytic flow-through membrane reactor is developed and applied for simulations to get valuable insights into dead-core phenomena.The effects of dimensionless process parameters, including Thiele modulus, Peclet numbers, reaction exponent, Prater number, and a membrane geometrical parameter on the concentration and temperature profiles, as well as dead zone formation, are extensively investigated. The simulation results demonstrate that these parameters affect the occurrence of dead zones and their size. Finally, the effect of convective flow on the reactor performance indicators is presented.

Error estimate of a fully decoupled numerical scheme based on the Scalar Auxiliary Variable (SAV) method for the Boussinesq system

In this paper, we will investigate the unconditional stability and error estimate of the fully decoupled numerical scheme for the Boussinesq equations. The newly constructed numerical scheme is based on the pressure correction technique and the SAV method, in which all coupling terms and nonlinear terms are completely decoupled, that is, we only need to solve several linear constant-coefficient equations. We rigorously prove the unconditional stability and convergence of the time-marching scheme and discuss all the details of the algorithm implementation. Finally, we implement some numerical experiments to verify its stability and accuracy.

An efficient unconditional energy stable scheme for the simulation of droplet formation

We have developed an efficient and unconditionally energy-stable method for simulating droplet formation dynamics. Our approach involves a novel time-marching scheme based on the scalar auxiliary variable technique, specifically designed for solving the Cahn-Hilliard-Navier-Stokes phase field model with variable density and viscosity. We have successfully applied this method to simulate droplet formation in scenarios where a Newtonian fluid is injected through a vertical tube into another immiscible Newtonian fluid. To tackle the challenges posed by nonhomogeneous Dirichlet boundary conditions at the tube entrance, we have introduced additional nonlocal auxiliary variables and associated ordinary differential equations. These additions effectively eliminate the influence of boundary terms. Moreover, we have incorporated stabilization terms into the scheme to enhance its numerical effectiveness. Notably, our resulting scheme is fully decoupled, requiring the solution of only linear systems at each time step. We have also demonstrated the energy decaying property of the scheme, with suitable modifications. To assess the accuracy and stability of our algorithm, we have conducted extensive numerical simulations. Additionally, we have examined the dynamics of droplet formation and explored the impact of dimensionless parameters on the process. Overall, our work presents a refined method for simulating droplet formation dynamics, offering improved efficiency, energy stability, and accuracy.

Second-order accurate and unconditionally stable algorithm with unique solvability for a phase-field model of 3D volume reconstruction

Three-dimensional (3D) volume reconstruction is an important technique in the fields of 3D printing, artificial limb, and medical diagnostic. This work aims to develop an unconditionally stable algorithm with desired accuracy and unique solvability for a phase-field model of 3D volume reconstruction. Based on scattered points of a target object, a smooth narrow volume is reconstructed by solving a Allen–Cahn-type equation with control function. Since the non-negative property of control function, the evolution of governing equation can associate with an energy dissipation law (i.e., energy stability). By modifying the nonlinear term in governing equation into an appropriate truncated form and utilizing a stabilized Crank–Nicolson-type discretization in time, we develop a linear, second-order accurate, and unconditionally energy-stable time-marching scheme to update the solution. The space is discretized with the finite difference method and the fully discrete energy stability is analytically estimated. We also prove the unique solvability of our proposed time-marching scheme. The accuracy, stability, and capability of reconstructing various 3D volumes are verified via extensive numerical simulations. To facilitate the interested readers, we provide the open source codes of 3D volume reconstruction of a teapot on corresponding author's personal website: https://cfdyang521.github.io.

Complexity of direct and iterative solvers on space–time formulations and time-marching schemes for h-refined grids towards singularities

We study computational complexity aspects for Finite Element formulations considering hypercubic space–time full and time-marching discretization schemes for h-refined grids towards singularities. We perform a relatively comprehensive study comparing the computational time via time complexities of direct and iterative solvers. We focus on the space–time formulation with refined computational grids and the corresponding time slabs, namely, computational grids obtained by taking the “cross-sections” of the refined space–time mesh. We estimate the computational complexity of solving systems of linear equations for multidimensional meshes with arbitrary dimensional singularities encountered in space–time formulation and time-marching schemes. The choice between space–time formulation and time-marching schemes depends entirely on the problem’s nature and the properties that need to be addressed. Thus, the paper aims to discuss the computational complexities of both approaches rather than suggest a better formulation. Our considerations concern the computational complexity of sequental execution of the multi-frontal solvers, the iterative solvers, and the static condensation. Numerical experiments with Octave confirm our theoretical findings.

Numerical treatments of nonlinear Burgers–Fisher equation via a combined approximation technique

A combined spectral matrix collocation strategy is presented to solve the time-dependent nonlinear Burgers–Fisher equation pertaining to various important physical mechanisms such as advection, diffusion, and logistic reaction. The nonlinearity of the model is first tackled by a time-marching scheme based on the Taylor formula. Hence in each time step, we solve a linear initial boundary value problem (IBVP) by using the spectral collocation technique based on novel Boole polynomials. Various numerical computations are carried out to indicate the pertinent features and testify the applicability of the presented combined technique. Comparisons are made between our results and the exact analytical solutions and some available numerical outcomes in the literature to show the validity of the method.

A high-order explicit time-domain FEM using 15-node tetrahedral elements for room acoustics modeling: Basic performance

Time-domain room acoustic modeling using mass-lumped higher-order tetrahedral finite elements with an explicit time-marching scheme is highly attractive because of its excellent geometrical flexibility with unstructured meshing and applicability into massively parallel computing. However, the standard mass-lumped tetrahedral elements yield an unstable scheme, and only implicit time-marching schemes are available. This study proposes a novel wave-based room acoustics solver based on a high-order explicit time-domain FEM. The present solver uses mass-lumped 15-node tetrahedral elements for spatial discretization and a dissipation-free two-stage partitioned Runge-Kutta time integration for time discretization. The 15-node tetrahedral elements had been developed to apply mass-lumping methods to 10-node tetrahedral elements, which are the most common finite element in engineering applications, but its applicability in room acoustics problems is unknown. Higher accuracy and efficiency of the proposed method over the standard method using 10-node tetrahedral elements is presented through an eigenvalue analysis of a long duct model and acoustic simulations in a small room. Additionally, frequency-dependent sound absorbing boundary is implemented by revising an auxiliary differential equation method.

Numerical Approximations of Diblock Copolymer Model Using a Modified Leapfrog Time-Marching Scheme

An efficient modified leapfrog time-marching scheme for the diblock copolymer model is investigated in this paper. The proposed scheme offers three main advantages. Firstly, it is linear in time, requiring only a linear algebra system to be solved at each time-marching step. This leads to a significant reduction in computational cost compared to other methods. Secondly, the scheme ensures unconditional energy stability, allowing for a large time step to be used without compromising solution stability. Thirdly, the existence and uniqueness of the numerical solution at each time step is rigorously proven, ensuring the reliability and accuracy of the method. A numerical example is also included to demonstrate and validate the proposed algorithm, showing its accuracy and efficiency in practical applications.

A Novel Spacetime Boundary-Type Meshless Method for Estimating Aquifer Hydraulic Properties Using Pumping Tests

This article introduces a new boundary-type meshless method designed for solving axisymmetric transient groundwater flow problems, specifically for aquifer tests and estimating hydraulic properties. The method approximates solutions for axisymmetric transient groundwater flow using basis functions that satisfy the governing equation by solving the inverse boundary value problem in the spacetime domain. The effectiveness of this method was demonstrated through validation with the Theis solution, which involves transient flow to a well in an infinite confined aquifer. The study included numerical examples that predicted drawdown at various radial distances and times near pumping wells. Additionally, an iterative scheme, namely, the fictitious time integration method, was employed to iteratively determine the hydraulic properties during the pumping test. The results indicate that this approach yielded highly accurate solutions without relying on the conventional time-marching scheme. Due to its temporal and spatial discretization within the spacetime domain, this method was found to be advantageous for estimating crucial hydraulic properties, such as the transmissivity and storativity of an aquifer.

Optimization Problems for PDEs in Weak Space-Time Form

Optimization problems constrained by time-dependent Partial Differential Equations (PDEs) are challenging from a computational point of view: even in the simplest case, one needs to solve a system of PDEs coupled globally in time and space for the unknown solutions (the state, the costate and the control of the system). Typical and practically relevant examples are the control of nonlinear heat equations as they appear in laser hardening or the thermic control of flow problems (Boussinesq equations). Specifically for PDEs with a long time horizon, conventional time-stepping methods require an enormous amount of computer memory allocations for the respective other variables. In contrast, adaptive-in-time-and-space methods aim at distributing the available degrees of freedom in an a-posteriori fashion to capture singularities and are, therefore, most promising. Recently, well-posed weak variational formulations have been introduced for time-dependent PDEs such as the heat equation, linear transport and the wave equation. Those formulations also allow for a sharp relation between the approximation error and the residual, which is particularly relevant for model reduction. Moreover, for those tensor-basis formulations, advanced algebraic solvers designed to take into account these multiarray (tensorial) formulations appear to be particularly competitive with respect to time-marching schemes, especially in higher dimensions. We plan to discuss whether these techniques can be extended to nonlinear PDEs like Hamilton-Jacobi-Bellman equations, or stochastic PDEs and variational inequalities. Another topic will be adaptive schemes which, when properly designed, inherit the stability of the continuous formulation. The central goals of the workshop are the analysis, fast solvers and model reduction for PDE-constrained control and optimization problems based on weak formulations of the underlying PDE(s).

Interpolation operators for parabolic problems

We introduce interpolation operators with approximation and stability properties suited for parabolic problems in primal and mixed formulations. We derive localized error estimates for tensor product meshes (occurring in classical time-marching schemes) as well as locally in space-time refined meshes.

Phase-field simulation of multiple fluid vesicles with a consistently energy-stable implicit–explicit method

When certain polymer amphiphiles and phospholipids are dispersed in a liquid, these molecules combine to form various closed bilayer structures known as multiple vesicles. The specific structures of these vesicles play fundamental roles in cytobiology and drug transportation. To model the multiple lipid vesicles in a fluid environment, we use the multi-phase conservative Allen–Cahn-type equations with approximately area-preserving penalty terms to capture the vesicle interfaces and the incompressible Navier–Stokes equations to describe the evolution of velocity and pressure. To efficiently simulate this complex coupled fluid system, we introduce several time-dependent variables to transform the total energy functional and the governing equations into equivalent forms. Based on these equivalent models, we develop a totally decoupled, energy-stable, and temporally second-order accurate time-marching scheme. A consistency-enhanced technique is utilized to correct the computed energy so that a more accurate solution is obtained. The time-discretized energy stability can be analytically estimated and the solution algorithm in each time step is easy to implement. Various numerical experiments, such as multi-component vesicles with different initial shapes, multiple vesicles in shear flow, and multiple vesicles in complex domains, are performed to verify the accuracy, stability, and capability of our proposed method.

Modular Framework for the Solution of Boundary-coupled Multiphysics Problems

This paper presents a modular multiphysics framework developed for OpenFOAM. The framework is built around an iterative implicit coupling scheme based on a multi-region partitioned approach. This scheme allows the implementation of formal implicit time-marching schemes, which improves the stability of strongly interacting coupled problems. This methodology allows physical interactions to be handled through specifically designed interface boundary conditions. It also allows region-specific solvers to be implemented as modular class solvers. The coupling methodology is handled with a main program that manages solver-specific actions. The aim of this framework is to facilitate the implementation and testing of new multiphysics coupling problems in an integrated code structure. To show the capabilities of the framework to integrate new physics, solvers and boundary conditions requirements are discussed. Also, three validated examples involving fluid-structure interactions, conjugate heat transfer, and fluid-structure-thermal interactions are presented. Although all these problems are boundary-coupled multiphysics problems, the framework is conceptually not limited to this kind of problems. The benefit of this work to the OpenFOAM community is a general and modular framework that facilitates the setup and solution of diversified multiphysics problems, and that illustrates the implementation of modular interface boundary conditions between physics regions.

A computational strategy for nonlinear coupled constitutive relations of rarefied nonequilibrium flows on unstructured grids

The Navier-Stokes-Fourier (NSF) equations are no longer valid in high-Knudsen and high-Mach flows, where the local thermodynamically equilibrium assumption fails, while the nonlinear coupled constitutive relations (NCCR) have been proven to be an efficient approach for rarefied non-equilibrium flows under the structured finite-volume-method computational fluid dynamic (FVM-CFD) solver. In this paper, by adding the time derivatives of non-conserved variables to the NCCR model, a computational strategy for NCCR model under the unstructured FVM-CFD framework is first proposed for further engineering applications. An upwind flux-splitting scheme with the Lower-Upper Symmetric Gauss-Seidel (LU-SGS) implicit time-marching scheme is employed. The linear stability of one-dimensional modified NCCR model in time and space is analyzed subsequently, which indicates the modified NCCR model is unconditionally stable. The accuracy and computational performance of the proposed NCCR solution are assessed by a feat of the numerical tests for hypersonic flow over a blunt cylinder without a wake, supersonic flow past a circular cylinder with a wake, and hypersonic flow around a planetary probe. Computational results show that the newly-developed NCCR solution can get solutions in better agreement with the direct simulation Monte Carlo (DSMC) than NSF results. Moreover, the newly-developed NCCR solution has a better property of convergence than the undecomposed NCCR algorithm under the unstructured FVM-CFD framework. These superior advantages of the present computational strategy are expected to make the NCCR method a promising engineering tool for modeling rarefied non-equilibrium flows.

Phase-field modeling and consistent energy-stable simulation of binary creeping flows in contact with solid

In this work, we present an efficient and practical model for describing two-phase creeping fluid flows in contact with a solid substrate. In the framework of phase-field method, a ternary Cahn–Hilliard model is modified by adding a term reflecting the wetting condition of liquid phase on the liquid–solid interface. The contact angle dynamics can be implicitly achieved by solving the phase-field equations and the explicit treatment on liquid–solid boundary is absent. Therefore, various discretization methods in space can be naturally adopted. To update the creeping flows in arbitrary domains, we herein consider the incompressible Darcy equations with a penalty term. The coupled binary fluid system theoretically satisfies the energy dissipation law with respect to a total energy functional. An energy dissipation-preserving time-marching scheme is constructed based on the auxiliary variable approach. Furthermore, a simple correction technique is utilized to improve the consistency between original and numerical values. We analytically prove that the proposed scheme still satisfies the energy law in its discrete version. Extensive numerical experiments are performed to validate the accuracy, consistent stability, and capability of our method.

An optimal spatial-filtering method derived from eigenvalue perturbation for extending the Courant-Friedrichs-Lewy stability limit

Explicit time-marching schemes are widely used in numerical simulations. However, their maximum time step is constrained by the Courant-Friedrichs-Lewy (CFL) stability limit. Two methods have been developed to extend this limit: (1) the eigenvalue perturbation method, which exhibits high numerical accuracy but incurs unaffordable memory demand and computational costs, even for middle-scale models, and (2) the spatial-filtering method, which can be implemented easily but results in significant numerical errors under large time steps. However, the intrinsic relationship between these two methods remains unknown. We reveal the intrinsic relationship between these two methods by considering the eigenvalue perturbation method for a homogeneous model. Using this relationship, we derive an analytical spatial filter for the homogeneous model and develop an optimal spatial filter for heterogeneous models. Compared to the classical spatial-filtering method, which removes all high-wavenumber components, our method retains all high-wavenumber components that contribute to wave propagation while eliminating other high-wavenumber components that cause instability. For the same numerical accuracy, the maximum time step allowed by the proposed method is approximately twice that of the classical spatial-filtering method. Compared to the eigenvalue perturbation method, our method can be used in large-scale models without additional memory consumption and computational cost. This can significantly accelerate the pseudospectral method with significantly larger time steps beyond the CFL stability limit, which is particularly promising for large-scale models with a fine grid.