Article
Engineering, Multidisciplinary
Robert E. Bird, Charles E. Augarde, William M. Coombs, Ravindra Duddu, Stefano Giani, Phuc T. Huynh, Bradley Sims
Summary: This paper presents a 2D hp-adaptive discontinuous Galerkin finite element method for phase field fracture that can reliably and efficiently solve phase field fracture problems with arbitrary initial meshes. The method uses a posteriori error estimators to drive mesh adaptivity based on both elasticity and phase field errors, and it is validated on several example problems.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2023)
Article
Computer Science, Interdisciplinary Applications
Thomas Wiltshire, Robert E. Bird, William M. Coombs, Stefano Giani
Summary: Open source codes are crucial for enhancing research integrity and accountability in computational science and engineering. However, many existing open source codes lack consideration for the ease of modifying the base code. This paper presents an open source finite element code written in MATLAB, which is designed to facilitate user understanding and implementation of new ideas within the core code.
ADVANCES IN ENGINEERING SOFTWARE
(2022)
Review
Mathematics, Applied
Pascal Mossier, Andrea Beck, Claus-Dieter Munz
Summary: This study introduces a novel hybrid Discontinuous Galerkin scheme with hp-adaptivity capabilities for the compressible Euler equations. The scheme achieves efficient and accurate discretization through local p-adaptation in smooth regions, while ensuring robustness through a finite volume scheme on an h-refined element-local subgrid at strong discontinuities and shocks. The method utilizes a single a priori indicator based on the modal decay of the local polynomial solution representation to control p-refinement and distinguishes between discontinuous and smooth regions. The findings demonstrate the efficiency of the adaptive scheme and its ability to produce comparable or even better results with significantly reduced computational costs.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Computer Science, Interdisciplinary Applications
Shukai Du, Samuel N. Stechmann
Summary: This article presents a numerical method for efficient and low-memory calculations of the radiative transfer equation. The method reduces the number of spatial degrees of freedom and utilizes a suitable preconditioner to ensure a computational cost of O(N). Numerical examples demonstrate the effectiveness of the method in reducing memory requirements and computational cost.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Mathematics, Applied
Jens Markus Melenk, Alexander Rieder
Summary: The study involves a time-dependent problem generated by a nonlocal operator in space. The approach includes spatial discretization using hp-finite elements and a Caffarelli-Silvestre extension, and time discretization using hp-discontinuous Galerkin based time stepping. Exponential convergence is proven in an abstract framework for the spatial domain Omega.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2021)
Article
Mathematics, Applied
Yunqing Huang, Jichun Li, Chanjie Li, Kai Qu
Summary: We investigate the reformulated two-dimensional (2-D) perfectly matched layer (PML) models based on the original 3-D PML model developed by Cohen and Monk in 1999. We propose the discontinuous Galerkin methods for solving both 2-D TMz and TEz models, and establish the proofs of stability and error estimate. The numerical results demonstrate the accuracy and performance of our method.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
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
Mathematics, Applied
R. A. M. van Gestel, M. J. H. Anthonissen, J. H. M. ten Thije Boonkkamp, W. L. IJzerman
Summary: Liouville's equation describes the evolution of energy distribution in optical systems, with the discontinuous Galerkin spectral element method being suitable for this equation. Optical interfaces in phase space may lead to non-local boundary conditions which must also adhere to energy conservation. The numerical experiments show that the discontinuous Galerkin spectral element method outperforms ray tracing in terms of computational efficiency for low-error scenarios.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Zhaonan Dong, Lorenzo Mascotto, Oliver J. Sutton
Summary: The novel residual-based a posteriori error estimator for the biharmonic problem in two and three dimensions gives an upper bound and a local lower bound on the error, with the lower bound being robust to local mesh size but not to local polynomial degree. The analysis is based on elliptic reconstruction and Helmholtz decomposition, showing explicit suboptimality in terms of polynomial degree that grows at most algebraically.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2021)
Article
Mathematics, Applied
Yunqing Huang, Jichun Li, Xin Liu
Summary: In this paper, a local discontinuous Galerkin (LDG) method is proposed to simulate wave propagation in an electromagnetic concentrator. The concentrator model consists of a coupled system of four partial differential equations and one ordinary differential equation. Discrete stability and error estimate are proven for both semi-discrete and full-discrete LDG schemes. Numerical results justify the theoretical analysis and demonstrate the interesting wave concentration property of the electromagnetic concentrator.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Computer Science, Interdisciplinary Applications
Wei Guo, Juntao Huang, Zhanjing Tao, Yingda Cheng
Summary: In this paper, an adaptive sparse grid local discontinuous Galerkin method is proposed to solve Hamilton-Jacobi equations in high dimensions, using multiwavelets for multiresolution. Numerical tests show the method performs well in up to four dimensions.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Engineering, Multidisciplinary
Robert Bird, William M. Coombs, Stefano Giani
Summary: This article presents a hpr-adaptive crack propagation method for highly accurate 2D crack propagation paths which requires no a priori knowledge of the tip solution. The method is simple to implement and enables users to obtain high fidelity crack path predictions for domains containing multiple cracks propagating at different rates. The method also includes a crack path derefinement scheme to capture the fidelity of the crack path.
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING
(2022)
Article
Mathematics, Applied
Yanping Chen, Lina Wang, Lijun Yi
Summary: We propose an hp-discontinuous Galerkin method for solving nonlinear fractional differential equations. The method first transforms the fractional differential equations into equivalent integral equations, and then uses the hp-discontinuous Galerkin method to solve these integral equations. We prove that under certain conditions, the method can achieve exponential convergence, thus effectively solving problems with endpoint singularities.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Engineering, Multidisciplinary
Sergio Rojas, David Pardo, Pouria Behnoudfar, Victor M. Calo
Summary: This study presents a goal-oriented mesh-adaptive algorithm for a finite element method stabilized via residual minimization, which provides stable solutions on each mesh instance and minimizes errors by automatic mesh refinement.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2021)
Article
Mathematics, Applied
Ruo Li, Qicheng Liu, Fanyi Yang
Summary: The proposed discontinuous least squares finite element method efficiently solves the indefinite time-harmonic Maxwell equations by minimizing the functional over piecewise polynomial spaces. This method is stable without any constraint on mesh size and demonstrates optimal convergence rate under the energy norm and sub-optimal convergence rate under the L-2 norm. Numerical results in two and three dimensions confirm the accuracy of error estimates.
IMA JOURNAL OF NUMERICAL ANALYSIS
(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)