Article
Mechanics
Li-Wei Chen, Berkay A. Cakal, Xiangyu Hu, Nils Thuerey
Summary: In this study, deep learning methods were used to efficiently predict flow fields and loads for aerodynamic shape optimization. The trained U-net-based deep neural network models successfully inferred flow fields and calculated gradient flows for optimizing shapes, showing great promise for general aerodynamic design problems. The results demonstrate that the DNN models are capable of accurately predicting flow fields and generating satisfactory aerodynamic forces, even without specific training for aerodynamic forces.
JOURNAL OF FLUID MECHANICS
(2021)
Article
Mathematics, Applied
L. Rebholz, F. Tone
Summary: This paper studies the H1-stability of the BDF2 scheme for the 2D Navier-Stokes equations for all positive time. Specifically, we discretize in time using the backward differentiation formula (BDF2) and prove the stability of the numerical scheme with the help of the discrete Gronwall lemma and the discrete uniform Gronwall lemma.
APPLIED MATHEMATICS LETTERS
(2023)
Article
Mechanics
T. H. B. Demont, S. K. F. Stoter, E. H. van Brummelen
Summary: In this article, the behavior of the Abels-Garcke-Grun Navier-Stokes-Cahn-Hilliard diffuse-interface model for binary-fluid flows as the diffuse-interface thickness approaches zero is studied. The optimal order of the m-epsilon scaling relation and its impact on the convergence rate of the diffuse-interface solution to the sharp-interface solution are elucidated. The case of an oscillating droplet is investigated, and new analytical expressions for small-amplitude oscillations are derived. The sharp-interface limit of the Navier-Stokes-Cahn-Hilliard equations is probed using an adaptive finite-element method.
JOURNAL OF FLUID MECHANICS
(2023)
Article
Physics, Mathematical
Masahiro Suzuki, Katherine Zhiyuan Zhang
Summary: In this paper, we investigate the compressible Navier-Stokes equation in a perturbed half-space with an outflow boundary condition and the supersonic condition. We demonstrate the unique existence of stationary solutions for the perturbed half-space, which exhibit multidirectional flow and are independent of the tangential directions. Additionally, we prove the asymptotic stability of these stationary solutions.
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2021)
Article
Mathematics, Applied
Hui Zhang, Fanhai Zeng, Xiaoyun Jiang, George Em Karniadakis
Summary: This paper introduces the application of a generalized discrete Gronwall inequality in the numerical methods for time-fractional evolution equations, proving their convergence and unifying the convergence analysis of several existing time-stepping schemes.
FRACTIONAL CALCULUS AND APPLIED ANALYSIS
(2022)
Article
Mechanics
Chuong V. Tran, Xinwei Yu, David G. Dritschel
Summary: Incompressible fluid flows are characterized by high correlations between velocity and pressure, as well as between vorticity and pressure. This correlation plays a significant role in maintaining regularity in Navier-Stokes flows. The study suggests that as long as global pressure minimum (or minima) and velocity maximum (or maxima) are mutually exclusive, regularity is likely to persist.
JOURNAL OF FLUID MECHANICS
(2021)
Article
Computer Science, Interdisciplinary Applications
Xinjie Ji, James Gabbard, Wim M. van Rees
Summary: This paper introduces a sharp-interface approach based on the immersed interface method for handling the one- and two-way coupling between an incompressible flow and rigid bodies using the vorticity-velocity Navier-Stokes equations. The authors develop a moving boundary treatment and a two-way coupling methodology that do not require the pressure field. Extensive testing shows that the resulting solver achieves second-order accuracy and provides efficiency benefits compared to a representative first-order approach.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Mathematics, Applied
Zhenzhen Lou, Qixiang Yang, Jianxun He, Kaili He
Summary: This paper presents the existence of the uniform analytic solution of the Cauchy problem for fractional incompressible Navier-Stokes Equations in critical Fourier-Herz spaces. The main strategy is to prove that the existence of the uniform analytic solution is equivalent to the boundedness of convolution inequality on Herz space.
APPLIED MATHEMATICS LETTERS
(2021)
Article
Mathematics, Applied
Sameh Abidi, Jamil Satouri
Summary: The article proposes a new numerical method to solve the problem of optimal control of the steady-state Navier-Stokes equations through the velocity-pressure formulation. A system of equations is derived using a new technique for calculating the solution. The spectral method is used to discretize the problem, and an extended relaxation method is proposed to ensure proper convergence of the system. Numerical results are provided to confirm the effectiveness of this approach.
Article
Computer Science, Interdisciplinary Applications
Ning Li, Jilian Wu, Xinlong Feng
Summary: In this paper, a filtered time-stepping method for incompressible Navier-Stokes equations with variable density is presented. By using a time filter, the time accuracy of the method is increased from first order to second order with only one backward Euler solve at each time step. The added time filter is expressed by linear combinations of the solutions at previous time levels without additional complexity. The stability of density and velocity for both the fully implicit backward Euler algorithm and the backward Euler plus time filter algorithm is proved. Furthermore, the approach is extended to a variable time stepsize BETF algorithm, and new adaptive BE algorithm and variable stepsize variable order algorithm with low cost error estimators are constructed. Experimental results demonstrate the stability and efficiency of the proposed methods.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Mathematics, Applied
Yanqing Wang, Yulin Ye
Summary: In this paper, an energy conservation criterion is derived for weak solutions of both the incompressible and compressible Navier-Stokes equations. The criterion is based on a combination of velocity and its gradient. For the incompressible case, it extends known results on periodic domain, including the famous Lions' energy conservation criterion. For the compressible case, it improves recent results and extends criteria for energy conservation from incompressible to compressible flow.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
Xiaocui Li, Xu You
Summary: This paper presents a detailed numerical analysis of the mixed finite element method for fractional Navier-Stokes equations, including stability analysis and convergence analysis, with numerical examples demonstrating the effectiveness of the proposed method.
JOURNAL OF COMPUTATIONAL MATHEMATICS
(2021)
Article
Mechanics
Lukas Unglehrt, Michael Manhart
Summary: This study investigates the flow behavior of a hexagonal close-packed arrangement of spheres under steady and oscillatory conditions. The friction and pressure drag contributions to the momentum budget are quantified and analyzed. The findings show that for steady flow, the friction and viscous pressure drag scale with the Reynolds number, while the convective pressure drag exhibits different scaling behaviors. Under oscillatory flow conditions, the amplitudes of the drag components are similar to the steady cases at low and medium Womersley numbers, but behave differently at high Womersley numbers, indicating the need for new models beyond the current quasisteady approaches.
JOURNAL OF FLUID MECHANICS
(2023)
Article
Mechanics
Niklas Fehn, Martin Kronbichler, Peter Munch, Wolfgang A. Wall
Summary: This study contributes to the investigation of the well-known energy dissipation anomaly in inviscid limit by conducting high-resolution numerical simulations of the three-dimensional Taylor-Green vortex problem. The interesting observation is made that the kinetic energy evolution does not tend towards exact energy conservation as the spatial resolution of numerical scheme increases. This raises the question of whether the results obtained can be seen as a numerical confirmation of the famous energy dissipation anomaly and elaborates on an indirect approach for the identification of finite-time singularities based on energy arguments.
JOURNAL OF FLUID MECHANICS
(2021)
Article
Mathematics, Applied
Dominic Breit, Andreas Prohl
Summary: In this paper, we propose and study a temporal and a spatio-temporal discretisation method for the two-dimensional stochastic Navier-Stokes equations in bounded domains with no-slip boundary conditions. By considering additive noise, we construct the discretisation based on the solution of the related nonlinear random partial differential equation, which is solved using a transform of the solution of the stochastic Navier-Stokes equations. We show a strong rate (up to) 1 in probability for the corresponding discretisation in space and time (and space-time).
IMA JOURNAL OF NUMERICAL ANALYSIS
(2023)
Article
Physics, Multidisciplinary
Yaxin Wei, Pengzhan Huang
Summary: This paper investigates the stationary double-diffusive natural convection model and proves the existence and uniqueness of the model. Three finite element iterative methods are designed to solve the problem, and their stability under different viscosity conditions is analyzed.
Article
Mathematics, Applied
Yanan Yang, Pengzhan Huang
Summary: This paper presents a defect-deferred correction method to solve the non-stationary coupled Stokes/Darcy model, and provides theoretical proof and numerical experiments to verify its second order accuracy and stability in time. Compared with the standard finite element method, this method has advantages in calculating small viscosity and hydraulic conductivity coefficients.
Article
Computer Science, Interdisciplinary Applications
Yunhua Zeng, Pengzhan Huang, Yinnian He
Summary: This paper proposes a time filter method to solve the double-diffusive natural convection model. The method improves the temporal accuracy of the backward Euler scheme from the first order to the second order by adding a simple time filter. The stability analysis and error estimate of the method are also proved.
COMPUTERS & FLUIDS
(2022)
Article
Physics, Multidisciplinary
Aytura Keram, Pengzhan Huang
Summary: In this study, a Uzawa-type iterative algorithm is presented for solving the stationary natural convection model, which produces weakly divergence-free velocity approximation. The convergence of the algorithm is provided and supported by numerical tests.
Article
Mathematics, Applied
Pengzhan Huang, Yinnian He, Ting Li
Summary: This paper presents a finite element algorithm for the time-dependent nematic liquid crystal flow based on the Gauge-Uzawa method. The algorithm combines the Gauge and Uzawa methods within a finite element variational formulation, which is a fully discrete projection type algorithm. Error estimates for velocity and molecular orientation of the nematic liquid crystal flow are also shown. Numerical results confirm the reliability of the presented algorithm and validate the theoretical analysis.
JOURNAL OF COMPUTATIONAL MATHEMATICS
(2022)
Article
Mathematics, Applied
Aytura Keram, Pengzhan Huang
Summary: In this paper, the Arrow-Hurwicz iterative finite element method is proposed to solve the stationary thermally coupled incompressible magnetohydrodynamics system. It is proven that the iterative solution obtained by this method is convergent under certain conditions, and the effectiveness is illustrated with numerical examples.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Pengzhan Huang
Summary: This paper studies a finite element algorithm for solving the nonstationary incompressible magnetohydrodynamic equations using a correction method. The algorithm consists of two steps, one of which is a post-processing step. By using the correction method, the accuracy of the approximation solution in time can be improved, and the stability and error estimates of the solution are also proved.
MEDITERRANEAN JOURNAL OF MATHEMATICS
(2022)
Article
Mathematics, Applied
Huimin Ma, Pengzhan Huang
Summary: This paper studies a fully discrete vector penalty-projection method for time-dependent incompressible magnetohydrodynamics flows. The method combines mixed finite element approximation for spatial discretization and first-order backward Euler for temporal discretization. Moreover, it establishes unconditional energy stability and derives error estimates for the fully discrete scheme. Finally, the theoretical results are verified and the accuracy and efficiency of the proposed scheme are illustrated through numerical experiments.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Mathematics, Applied
Shuaijun Liu, Pengzhan Huang
Summary: This paper proposes a grad-div stabilization with the Jacobi iteration to the thermally coupled incompressible magnetohydrodynamic system, which avoids breakdown of solver and increase of computational time with increasing value of grad-div stabilized parameter. The proposed method includes two steps: a fully discrete first-order Euler semi-implicit scheme based on the mixed finite element method is implemented for the considered problem in Step 1. Then, in Step 2, the Jacobi approximation is applied to the grad-div operator whose each component will be decoupled. Moreover, theoretical analysis results of unconditional stability and convergence analysis are obtained for the presented method. Finally, numerical experiments are conducted to demonstrate the accuracy and effectiveness of the proposed method.
ZAMM-ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND MECHANIK
(2023)
Article
Mathematics, Applied
Shuaijun Liu, Pengzhan Huang
Summary: In this paper, a sparse grad-div stabilized algorithm is proposed to penalize for lack of divergence-free solution in the incompressible magnetohydrodynamics equations. This algorithm adds a minimally intrusive module that implements grad-div stabilization with a sparse block structure matrix. The unconditional stability and error estimates of the proposed algorithm are provided, and numerical tests are conducted. Compared to other grad-div stabilizations, the sparse grad-div stabilized algorithm is more efficient with some large values of grad-div parameters.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Huimin Ma, Pengzhan Huang
Summary: In this work, a fully decoupled, unconditionally stable finite element scheme is proposed to solve a nonlinear and coupled multi-physics system of a thermally coupled incompressible magnetohydrodynamic problem. The scheme combines the scalar auxiliary variable method and vector penalty projection approach, and has the features of unconditional stability, divergence-free approximation, and decoupling of unknown physical quantities. The scheme is provably unconditionally stable and error estimates for the velocity field, magnetic field, and temperature are established. Numerical simulations are provided to verify the proposed scheme's features.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Aytura Keram, Pengzhan Huang
Summary: We propose an iterative method based on a linearization approach for solving the thermally coupled stationary incompressible magnetohydrodynamics equations at high physical parameters. The method not only improves computational capability, but also reduces computational time and iterative steps. Stability of the method is proven and iterative error estimates are derived. Numerical tests are conducted to demonstrate the efficiency of the proposed method.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
Shuaijun Liu, Pengzhan Huang, Yinnian He
Summary: In this paper, a fully discrete second-order-in-time scheme for the incompressible MHD equations is proposed, which utilizes blended BDF and extrapolation treatments for nonlinear terms. The scheme is shown to be more accurate than the traditional two-step BDF scheme, while maintaining A-stability. The unconditional stability, long-time stability, and optimal convergence rate of the scheme are discussed, and numerical experiments with comparison to the two-step BDF scheme are conducted to verify the findings.
ADVANCES IN COMPUTATIONAL MATHEMATICS
(2023)
Article
Mathematics, Applied
Yaxin Wei, Pengzhan Huang
Summary: This paper analyzes several iterative finite element methods for a steady-state double-diffusive natural convection model, which includes the diffusion of temperature and concentration and can simulate heat and mass transfer phenomena. These iterations include the Stokes-type, Newton-type, and Oseen-type iterative methods. Then, based on the uniqueness condition, stability and convergence of these iterative methods are proved for the considered model. Finally, some numerical examples are presented to validate the correctness of the theoretical analysis.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
Moldir Serik, Rena Eskar, Pengzhan Huang
Summary: In this paper, a high-accuracy difference scheme for the time-fractional Schrodinger equation is established, and the scheme's unconditional stability and convergence in the maximum norm are proved.
