Article
Mathematics, Applied
Gwanghyun Jo, Do Y. Kwak, Young-Ju Lee
Summary: This paper introduces a locally conservative enriched immersed finite element method (EIFEM) for elliptic problems with interfaces, addressing the lack of conservative flux conservation in current methods under the IFEM framework. By introducing a local piecewise constant enrichment, locally conservative flux is provided, and an auxiliary space preconditioner based on algebraic multigrid method is constructed and analyzed for the resulting system. A new observation is made that imposing strong Dirichlet boundary conditions can remove zero eigen-modes of the EIFEM system while still weakly applying Dirichlet boundary conditions to the piecewise constant enrichment. Various issues related to the piecewise constant enrichment for mesh unfit to the interface are also discussed and clarified, with numerical tests confirming the theoretical developments.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Saihua Wang, Feng Wang, Xuejun Xu
Summary: This paper proposes a robust multigrid method for 1D immersed finite element method (IFEM) and proves its optimality, with convergence rate independent of mesh size and level, as well as jump of discontinuous coefficients.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Peipei Lu, Andreas Rupp, Guido Kanschat
Summary: In this paper, two-level domain decomposition methods are proposed for hybridizable discontinuous Galerkin discretizations, including hybridized local discontinuous Galerkin, Raviart-Thomas, and Brezzi-Douglas-Marini finite elements for Poisson's equation. The additive Schwarz method is studied as a preconditioner and the multiplicative method as an iterative solver. The algorithm employs the same discretization scheme on the coarse mesh. The injection operator developed in [13] is used and it is proven that the condition number of the preconditioned system depends only on the fraction between coarse and fine mesh sizes and the overlap width. Numerical experiments validate the analytical findings.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Gwanghyun Jo, Do Y. Kwak
Summary: In this study, two IFEMs are developed for convection-diffusion equations with interfaces. The first method introduces judiciously defined convection-related line integrals to define bilinear forms. Optimal error estimates in both L2 and H1-norms are proved by establishing Garding's inequality. The second method focuses on the convection-dominated case and uses piecewise constant functions on vertex-associated control volumes. With the use of upwinding concepts, the control-volume based IFEM is made robust to the magnitude of convection terms. The H1 optimal error estimate is proved for the control-volume based IFEM. Numerical experiments are presented to confirm the analysis.
Article
Engineering, Multidisciplinary
Cuiyu He, Shun Zhang, Xu Zhang
Summary: This paper presents a new and stable Petrov-Galerkin (PG) immersed finite element method (IFEM) for second-order elliptic interface problems. The method introduces stabilization terms to overcome the lack of local positivity in the classical PG-IFEM. The a priori and a posteriori error estimates of the method are analyzed, and the proposed a posteriori error estimator is proven to be reliable and efficient. Extensive numerical results demonstrate the optimal convergence and robustness of the numerical scheme.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2023)
Article
Mathematics, Applied
Yuan Chen, Songming Hou, Xu Zhang
Summary: In this paper, an immersed finite element (IFE) method is proposed for solving elastodynamics interface problems on interface-unfitted meshes. Vector-valued P1 and Q1 IFE spaces are used for spatial discretization. Important properties of these IFE spaces, such as inverse inequalities, are established, which are crucial in the error analysis. Both semi-discrete and fully discrete schemes are derived for temporal discretization. The proposed schemes are proved to be unconditionally stable and have optimal convergence rates in the energy, H2, and semi-H1 norms. Numerical examples are provided to verify the theoretical analysis and demonstrate the stability and robustness of the schemes.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mechanics
S. Parameswaran, J. C. Mandal
Summary: This paper presents a novel reinitialization approach for the conservative level set method, addressing the drawbacks of existing methods and reducing the numerical computations. The effectiveness of the proposed approach is demonstrated through various test problems, and its ability to deal with complex mesh types is also shown.
EUROPEAN JOURNAL OF MECHANICS B-FLUIDS
(2023)
Article
Computer Science, Interdisciplinary Applications
Vladimir Bogdanov, Felix S. Schranner, Josef M. Winter, Stefan Adami, Nikolaus A. Adams
Summary: This work presents a method for modeling multi-phase flows with moving contact lines at no-slip walls using a level-set-based sharp-interface method. The method includes a dynamic-contact-angle level-set boundary condition and an extrapolation equation for interface velocities. It reduces spurious currents and has been verified for its convergence and stability.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Derrick Jones, Xu Zhang
Summary: This paper introduces a class of lowest-order nonconforming immersed finite element methods for solving two-dimensional Stokes interface problems, which do not require the solution mesh to align with the fluid interface and can use triangular or rectangular meshes. The new vector-valued IFE functions are constructed to approximate the interface jump conditions, and the approximation capabilities of these new IFE spaces for the Stokes interface problems are examined through numerical examples, showing optimal convergence rates.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Computer Science, Interdisciplinary Applications
Mehrdad Yousefzadeh, Yinuo Yao, Ilenia Battiato
Summary: A simulation framework based on the level-set and the immersed boundary methods (LS-IBM) is developed for accurately modeling reactive transport problems with moving solid-fluid interface in porous media. The level-set method tracks interface movement, while the immersed boundary method captures momentum and mass transport at the interface. The proposed method guarantees second order accuracy in space and includes an interface velocity propagation method.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Mathematics, Applied
Huili Zhang, Xinlong Feng, Kun Wang
Summary: This article focuses on anisotropic immersed finite element (AIFE) methods for the elliptic interface system. The existence and uniqueness of the weak solution are established, and a linear AIFE method is constructed to solve the interface system. Unlike the scalar problem, the jump conditions for each unknown function are coupled together in the anisotropic problem considered here. By defining suitable interpolations, error estimates in L2-$$ {L}<^>2\hbox{-} $$ and H1-norms$$ {H}<^>1\hbox{-} \mathrm{norms} $$ are derived for the proposed AIFE method. Furthermore, numerical experiments are conducted to verify the theoretical predictions.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Cheng Wang, Wanli Wang, Shucheng Pan, Fuyu Zhao
Summary: This work presents an adaptive particle reseeding method for the particle level set method based on local curvature, achieving higher efficiency by placing more particles near interfaces with larger curvature. The method also includes a correction procedure with grid independence. Numerical tests demonstrate improved area/volume conservation and computational efficiency compared to the original method, with a CPU time savings of 50-75% without sacrificing accuracy.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Shuhao Cao, Long Chen, Ruchi Guo
Summary: In this work, a novel immersed virtual element method is proposed to solve three-dimensional H(curl) interface problems. The method combines the conformity of virtual element spaces and the robust approximation capabilities of immersed finite element spaces, achieving optimal convergence.
MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES
(2023)
Article
Computer Science, Interdisciplinary Applications
Seiji Kubo, Atsushi Koguchi, Kentaro Yaji, Takayuki Yamada, Kazuhiro Izui, Shinji Nishiwaki
Summary: This study presents a topology optimization method for two dimensional turbulent flow based on RANS equations, utilizing the level set method and IBM. By imposing boundary conditions explicitly and estimating interpolated velocity and pressure values using standard wall functions, a topology optimization method for two dimensional turbulent flow is constructed.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
Shuhao Cao, Long Chen, Ruchi Guo, Frank Lin
Summary: This article presents an immersed virtual element method for solving interface problems, which combines the advantages of both body-fitted mesh methods and unfitted mesh methods. The method is capable of handling more complicated interface element configuration and provides better performance than the conventional penalty-type IFE method for the H(curl)-interface problem arising from Maxwell equations.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
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)