Article
Mathematics, Applied
Wietse M. Boon, Alessio Fumagalli
Summary: We propose a mixed finite element method for Stokes flow with one degree of freedom per element and facet of simplicial grids. The method is derived by considering the vorticity-velocity-pressure formulation and eliminating the vorticity locally through the use of a quadrature rule. The discrete solution is pointwise divergence-free and the method is pressure robust. The theoretically derived convergence rates are confirmed by numerical experiments.
APPLIED MATHEMATICS LETTERS
(2023)
Article
Mathematics, Applied
Wei Shi, Xin-Guang Yang, Lin Shen
Summary: This paper investigates the global well-posedness of a fluid-solid interaction model with vorticity, which is described as a hyperbolic-parabolic coupled system along the conjugate boundary. The fluid is governed by the incompressible Navier-Stokes equation with vorticity, while the elastic solid is modeled by the wave equation. The global existence of weak solution and uniqueness of the coupled system are proven using the energy method, delicate estimates, and the truncation-polishing technique to overcome the lack of smoothness in the variational form. (c) 2023 Elsevier B.V. All rights reserved.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2023)
Article
Mechanics
Ningyu Zhan, Rongqian Chen, Yancheng You
Summary: The meshfree method based on DGKS, named meshfree-DGKS, is proposed for simulating incompressible/compressible flows. Discretization is done using the least squares-based finite difference approach, with the concept of numerical flux introduced to handle compressible problems with discontinuities effectively. The method allows for capturing shock waves easily by reconstructing fluxes at mid-points based on the local solution of the Boltzmann equation.
Article
Engineering, Multidisciplinary
Sean Ingimarson, Leo G. Rebholz, Traian Iliescu
Summary: We investigate the consistency between nonlinear discretization in full order models (FOMs) and reduced order models (ROMs) for incompressible flows both theoretically and numerically. The study shows that consistent discretization yields more accurate results.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2022)
Article
Engineering, Multidisciplinary
Leo G. Rebholz, Duygu Vargun, Mengying Xiao
Summary: This paper enhances the classical IPP method for incompressible NavierStokes equations by using Anderson acceleration to improve convergence properties. By analyzing the fixed point operator associated with IPP iteration and applying a general theory for AA, it is shown that the linear convergence rate of IPP can be significantly improved with AA enhancement. Numerical tests demonstrate the effectiveness of IPP with penalty parameter 1 enhanced by AA.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2021)
Article
Computer Science, Interdisciplinary Applications
Mustafa E. Danis, Jue Yan
Summary: This study proposes a new formula for the nonlinear viscous numerical flux and extends it to the compressible Navier-Stokes equations using the direct discontinuous Galerkin method with interface correction (DDGIC). The new method simplifies the implementation and enables accurate calculation of physical quantities.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Bo Zheng, Yueqiang Shang
Summary: A parallel stabilized finite element variational multiscale method for the incompressible Navier-Stokes equations is proposed, utilizing a fully overlapping domain decomposition approach. The method computes a stabilized solution in a given subdomain using a locally refined global mesh, without the need for substantial recoding of the existing Navier-Stokes sequential solver. Error bounds for the approximate solutions are estimated using local a priori error estimates for the stabilized solution.
APPLIED NUMERICAL MATHEMATICS
(2021)
Article
Mathematics, Applied
Yuan Li, Rong An
Summary: Based on the equivalent form of variable density flows, a second order linearized finite element scheme is proposed and studied in this paper for approximating the three-dimensional incompressible Navier-Stokes equations with variable density. The discretization of the time derivative uses the two-step backward differentiation formula. It is proven that the proposed finite element scheme is unconditionally stable and achieves optimal second-order convergence rate in the L-2 norm through rigorous error analysis. Numerical results are provided to validate the theoretical analysis.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Computer Science, Interdisciplinary Applications
Pierre Lallemand, Li-Shi Luo, Manfred Krafczyk, Wen-An Yong
Summary: This review summarizes the rigorous mathematical theory behind the lattice Boltzmann equation (LBE), including the relevant properties of the Boltzmann equation, derivation of the LBE, and important LBE models. The focus is on the numerical analysis of the LBE as a solver for the nearly incompressible Navier-Stokes equations with appropriate boundary conditions, with several numerical results provided to demonstrate the efficacy of the lattice Boltzmann method.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Physics, Fluids & Plasmas
Ding Xu, Yisu Huang, Jinglei Xu
Summary: In this paper, a kinetic immersed boundary method (KIBM) based on particle distribution function discontinuity is proposed for solving the Boltzmann equation. It introduces a general immersed boundary force density that can handle various types of boundary conditions and solvers. Numerical experiments demonstrate the effectiveness of the proposed method.
Article
Mathematics, Applied
Alejandro Allendes, Gabriel R. Barrenechea, Julia Novo
Summary: This work focuses on the finite element discretization of the incompressible Navier-Stokes equations, using a low order stabilized finite element method with piecewise linear continuous discrete velocities and piecewise constant pressures. The modified continuity equation involves a stabilizing bilinear form based on the jumps of the pressure, resulting in a divergence-free velocity field. The stability of the discrete problem is proven without needing to rewrite the convective field in its skew-symmetric way, and error estimates with constants independent of viscosity are established and validated through numerous numerical experiments.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2021)
Article
Multidisciplinary Sciences
M. C. Lopes Filho, H. J. Nussenzveig Lopes
Summary: In this work, the authors proved that physically realizable weak solutions of the incompressible two-dimensional Euler equations on a torus conserve kinetic energy. Physically realizable weak solutions refer to solutions that can be obtained as limits of vanishing viscosity. The authors extended the previous research by adding forcing to the flow.
PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES
(2022)
Article
Computer Science, Interdisciplinary Applications
Kaibo Hu, Young-Ju Lee, Jinchao Xu
Summary: The study introduces finite element methods for the incompressible magnetohydrodynamics (MHD) system that accurately preserve the magnetic and cross helicity, the energy law and the magnetic Gauss law at the discrete level. Numerical tests are presented to demonstrate the algorithm's performance.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
Yanghai Yu, Jinlu Li, Zhaoyang Yin
Summary: In this paper, a new smallness hypothesis of initial data for the three-dimensional incompressible Navier-Stokes equations is derived and used to prove the existence of a unique global solution. Additionally, two examples of initial data satisfying the smallness condition are constructed, demonstrating that the norm of the initial data can be arbitrarily large.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Mathematics, Applied
Mofdi El-Amrani, Abdelouahed Ouardghi, Mohammed Seaid
Summary: An adaptive enriched Galerkin-characteristics finite element method is proposed for efficient numerical solution of incompressible Navier-Stokes equations, combining various techniques for improved efficiency and accuracy. The method adapts enrichments based on the gradient of the velocity field as an error indicator and is capable of resolving complex flow features in irregular geometries.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Maxim A. Olshanskii, Arnold Reusken, Alexander Zhiliakov
Summary: The paper introduces a trace finite element method for solving the Stokes system on a closed smooth surface. It achieves inf-sup stability through volume normal derivative stabilization and demonstrates optimal convergence and interpolation properties. Numerical examples illustrate the method's effectiveness.
MATHEMATICS OF COMPUTATION
(2021)
Article
Mathematics, Applied
Thomas Jankuhn, Maxim A. Olshanskii, Arnold Reusken, Alexander Zhiliakov
Summary: The paper introduces a higher order unfitted finite element method for the Stokes system on a surface in R-3, utilizing parametric P-k-Pk-1 finite element pairs on a tetrahedral bulk mesh. It proves stability and optimal order convergence results, including a quantification of geometric errors from approximate parametric representation of the surface. Numerical experiments demonstrate the method's effectiveness.
JOURNAL OF NUMERICAL MATHEMATICS
(2021)
Article
Biochemistry & Molecular Biology
A. Zhiliakov, Y. Wang, A. Quaini, M. Olshanskii, S. Majd
Summary: Membrane phase-separation is a mechanism used by biological membranes to concentrate specific lipid species for organizing membrane processes. The computational approach based on the surface Cahn-Hilliard phase-field model complements experimental investigations in designing patchy liposomes by providing both qualitative and accurate quantitative information about membrane organization dynamics. The computational model informed by experiments has the potential to assist in designing liposomes with spatially organized surfaces, reducing cost and time in the design process.
BIOCHIMICA ET BIOPHYSICA ACTA-BIOMEMBRANES
(2021)
Article
Mathematics, Applied
Maxim Olshanskii, Xianmin Xu, Vladimir Yushutin
Summary: The paper studies an Allen-Cahn-type equation defined on a time-dependent surface as a model of phase separation with order-disorder transition in a thin material layer. It shows that the limiting behavior of the solution is a geodesic mean curvature type flow in reference coordinates through formal inner-outer expansion. A geometrically unfitted finite element method, known as trace FEM, is considered for numerical solution with full stability and convergence analysis accounting for interpolation errors and approximate geometry recovery.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Engineering, Multidisciplinary
Yerbol Palzhanov, Alexander Zhiliakov, Annalisa Quaini, Maxim Olshanskii
Summary: This paper presents a thermodynamically consistent phase-field model of two-phase flow of incompressible viscous fluids which allows for a non-linear dependence of the fluid density on the phase-field order parameter. An unfitted finite element method is applied to discretize the system, and a fully discrete time-stepping scheme is introduced to ensure the stability of the numerical solution. Numerical examples demonstrate the stability, accuracy, and overall efficiency of the approach, revealing interesting dependencies of flow statistics on geometry in two-phase surface flows.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2021)
Article
Mathematics, Applied
Maxim Olshanskii, Annalisa Quaini, Qi Sun
Summary: We propose an isoparametric unfitted finite element approach for simulating two-phase Stokes problems with slip, demonstrating stability and optimal error estimates independent of various factors. Numerical results in two and three dimensions support the theoretical findings and showcase the robustness of the approach.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Maxim A. Olshanskii, Alexander Zhiliakov
Summary: The paper discusses the reuse of matrix factorization as a building block in augmented Lagrangian and modified AL preconditioners for nonsymmetric saddle point linear algebraic systems. The strategy is applied to efficiently solve two-dimensional incompressible fluid problems independent of Reynolds number, and is tested on simulating surface fluid motion motivated by lateral fluidity of inextensible viscous membranes. New eigenvalue estimates for the AL preconditioner are derived from numerical examples including Kelvin-Helmholtz instability problems on the sphere and torus.
NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS
(2022)
Article
Mathematics
Maxim Olshanskii, Yerbol Palzhanov, Annalisa Quaini
Summary: This paper investigates phase-field models for the numerical simulation of lateral phase separation and coarsening in lipid membranes, with and without lateral flow. An unfitted finite element method is used for the numerical solution of these models, allowing for complex and evolving shapes without explicit surface parametrization. The effect of lateral flow on phase evolution is examined through several numerical tests, focusing on the impact of variable line tension, viscosity, membrane composition, and surface shape on pattern formation.
VIETNAM JOURNAL OF MATHEMATICS
(2022)
Article
Biochemistry & Molecular Biology
Y. Wang, Y. Palzhanov, A. Quaini, M. Olshanskii, S. Majd
Summary: Researchers have proposed a computational platform based on the principles of continuum mechanics and thermodynamics to model the membrane coarsening dynamics of liposomes. The platform has been quantitatively validated and shown to be a valuable tool in experimental practice.
BIOCHIMICA ET BIOPHYSICA ACTA-BIOMEMBRANES
(2022)
Article
Engineering, Multidisciplinary
Alexander V. Mamonov, Maxim A. Olshanskii
Summary: This paper presents a reduced order model (ROM) for numerical integration of a dynamical system with multiple parameters. The ROM utilizes compressed tensor formats to find a low rank representation for high-fidelity snapshots of the system state. The computational cost of the online phase depends only on tensor compression ranks, making it efficient and accurate.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2022)
Article
Mathematics, Applied
Alexander Lozovskiy, Maxim A. Olshanskii, Yuri V. Vassilevski
Summary: This paper investigates a finite element method for a fluid-porous structure interaction problem, with a corrected balance of stresses on the fluid-structure interface. The deformations of the elastic medium are not necessarily small and are modeled using the Saint Venant-Kirchhoff (SVK) constitutive relation. The stability of the method is proved through an energy bound for the finite element solution.
RUSSIAN JOURNAL OF NUMERICAL ANALYSIS AND MATHEMATICAL MODELLING
(2022)
Article
Mathematics, Applied
Maxim A. Olshanskii, Arnold Reusken, Alexander Zhiliakov
Summary: This paper studies the lateral flow of a Boussinesq-Scriven fluid on a passively evolving surface embedded in Double-struck capital R-3, and introduces a well-posed weak formulation and a numerical solution method.
MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES
(2022)
Article
Mechanics
Maxim A. A. Olshanskii
Summary: The paper explores equilibrium configurations of inextensible elastic membranes that exhibit lateral fluidity. The mechanical equilibrium is described by differential equations derived from a continuum description of membrane motions using the surface Navier-Stokes equations with bending forces. The equilibrium conditions are independent of lateral viscosity and relate tension, pressure, and tangential velocity of the fluid. Only surfaces with Killing vector fields, such as axisymmetric shapes, can support non-zero stationary mass flow.
Article
Mathematics, Applied
Haoran Liu, Michael Neilan, Maxim Olshanskii
Summary: We propose a CutFEM discretization for the Stokes problem based on the Scott-Vogelius pair, where discrete piecewise polynomial spaces are defined on non-fitted macro-element triangulations. Boundary conditions are imposed through Nitsche-type discretization, and stability is ensured by adding local ghost penalty stabilization terms. The scheme exhibits stability and a divergence-free property of the discrete velocity outside an O(h) neighborhood of the boundary. Additionally, a local grad-div stabilization is introduced to mitigate the error caused by violation of the divergence-free condition. Error analysis shows optimal order error estimates with a grad-div parameter scaling like O(h(-1)).
ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS
(2023)
Article
Mechanics
Maxim A. Olshanskii
Article
Computer Science, Interdisciplinary Applications
Tian Liang, Lin Fu
Summary: In this work, a new shock-capturing framework is proposed based on a new candidate stencil arrangement and the combination of infinitely differentiable non-polynomial RBF-based reconstruction in smooth regions with jump-like non-polynomial interpolation for genuine discontinuities. The resulting scheme achieves high order accuracy and resolves genuine discontinuities with sub-cell resolution.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Lukas Lundgren, Murtazo Nazarov
Summary: In this paper, a high-order accurate finite element method for incompressible variable density flow is introduced. The method addresses the issues of saddle point system and stability problem through Schur complement preconditioning and artificial compressibility approaches, and it is validated to have high-order accuracy for smooth problems and accurately resolve discontinuities.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Gabriele Ciaramella, Laurence Halpern, Luca Mechelli
Summary: This paper presents a novel convergence analysis of the optimized Schwarz waveform relaxation method for solving optimal control problems governed by periodic parabolic PDEs. The analysis is based on a Fourier-type technique applied to a semidiscrete-in-time form of the optimality condition, which enables a precise characterization of the convergence factor at the semidiscrete level. The behavior of the optimal transmission condition parameter is also analyzed in detail as the time discretization approaches zero.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Jonas A. Actor, Xiaozhe Hu, Andy Huang, Scott A. Roberts, Nathaniel Trask
Summary: This article introduces a scientific machine learning framework that uses a partition of unity architecture to model physics through control volume analysis. The framework can extract reduced models from full field data while preserving the physics. It is applicable to manifolds in arbitrary dimension and has been demonstrated effective in specific problems.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Nozomi Magome, Naoki Morita, Shigeki Kaneko, Naoto Mitsume
Summary: This paper proposes a novel strategy called B-spline based SFEM to fundamentally solve the problems of the conventional SFEM. It uses different basis functions and cubic B-spline basis functions with C-2-continuity to improve the accuracy of numerical integration and avoid matrix singularity. Numerical results show that the proposed method is superior to conventional methods in terms of accuracy and convergence.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Timothy R. Law, Philip T. Barton
Summary: This paper presents a practical cell-centred volume-of-fluid method for simulating compressible solid-fluid problems within a pure Eulerian setting. The method incorporates a mixed-cell update to maintain sharp interfaces, and can be easily extended to include other coupled physics. Various challenging test problems are used to validate the method, and its robustness and application in a multi-physics context are demonstrated.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Xing Ji, Fengxiang Zhao, Wei Shyy, Kun Xu
Summary: This paper presents the development of a third-order compact gas-kinetic scheme for compressible Euler and Navier-Stokes solutions, constructed particularly for an unstructured tetrahedral mesh. The scheme demonstrates robustness in high-speed flow computation and exhibits excellent adaptability to meshes with complex geometrical configurations.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Alsadig Ali, Abdullah Al-Mamun, Felipe Pereira, Arunasalam Rahunanthan
Summary: This paper presents a novel Bayesian statistical framework for the characterization of natural subsurface formations, and introduces the concept of multiscale sampling to localize the search in the stochastic space. The results show that the proposed framework performs well in solving inverse problems related to porous media flows.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Jacob Rains, Yi Wang, Alec House, Andrew L. Kaminsky, Nathan A. Tison, Vamshi M. Korivi
Summary: This paper presents a novel method called constrained optimized DMD with Control (cOptDMDc), which extends the optimized DMD method to systems with exogenous inputs and can enforce the stability of the resulting reduced order model (ROM). The proposed method optimally places eigenvalues within the stable region, thus mitigating spurious eigenvalue issues. Comparative studies show that cOptDMDc achieves high accuracy and robustness.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Andrea La Spina, Jacob Fish
Summary: This work introduces a hybridizable discontinuous Galerkin formulation for simulating ideal plasmas. The proposed method couples the fluid and electromagnetic subproblems monolithically based on source and employs a fully implicit time integration scheme. The approach also utilizes a projection-based divergence correction method to enforce the Gauss laws in challenging scenarios. Numerical examples demonstrate the high-order accuracy, efficiency, and robustness of the proposed formulation.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Junhong Yue, Peijun Li
Summary: This paper proposes two numerical methods (IP-FEM and BP-FEM) to study the flexural wave scattering problem of an arbitrary-shaped cavity on an infinite thin plate. These methods successfully decompose the fourth-order plate wave equation into the Helmholtz and modified Helmholtz equations with coupled conditions on the cavity boundary, providing an effective solution to this challenging problem.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
William Anderson, Mohammad Farazmand
Summary: We develop fast and scalable methods, called RONS, for computing reduced-order nonlinear solutions. These methods have been proven to be highly effective in tackling challenging problems, but become computationally prohibitive as the number of parameters grows. To address this issue, three separate methods are proposed and their efficacy is demonstrated through examples. The application of RONS to neural networks is also discussed.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Marco Caliari, Fabio Cassini
Summary: In this paper, a second order exponential scheme for stiff evolutionary advection-diffusion-reaction equations is proposed. The scheme is based on a directional splitting approach and uses computation of small sized exponential-like functions and tensor-matrix products for efficient implementation. Numerical examples demonstrate the advantage of the proposed approach over state-of-the-art techniques.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Sebastiano Boscarino, Seung Yeon Cho, Giovanni Russo
Summary: This work proposes a high order conservative semi-Lagrangian method for the inhomogeneous Boltzmann equation of rarefied gas dynamics. The method combines a semi-Lagrangian scheme for the convection term, a fast spectral method for computation of the collision operator, and a high order conservative reconstruction and a weighted optimization technique to preserve conservative quantities. Numerical tests demonstrate the accuracy and efficiency of the proposed method.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Jialei Li, Xiaodong Liu, Qingxiang Shi
Summary: This study shows that the number, centers, scattering strengths, inner and outer diameters of spherical shell-structured sources can be uniquely determined from the far field patterns. A numerical scheme is proposed for reconstructing the spherical shell-structured sources, which includes a migration series method for locating the centers and an iterative method for computing the inner and outer diameters without computing derivatives.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
