Article
Mathematics, Applied
Shuai Miao, Jiming Wu, Yanzhong Yao
Summary: We propose a novel cell-centered finite volume method for discretizing heterogeneous and anisotropic diffusion problems on polygonal meshes. The unknowns in the resulting linear scheme are the values at the cell centers without any auxiliary unknowns. The new scheme does not require the usual star-shaped assumption on the mesh and only needs each mesh cell to be simply-connected. It has a small stencil, a nine-point stencil on structured quadrilateral meshes, and a five-point stencil on rectangular meshes if the diffusion coefficient is a scalar function. More importantly, the new scheme can handle arbitrary discontinuities and maintains second order accuracy on arbitrary meshes. Numerical experiments demonstrate its robustness and efficiency, with optimal convergence rates for the solution and flux on general polygonal meshes in extreme cases.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Computer Science, Interdisciplinary Applications
Shuai Miao, Jiming Wu
Summary: In this paper, a nonlinear positivity-preserving finite volume scheme for the heterogeneous and anisotropic diffusion problems is proposed. The scheme includes novel interpolation algorithms and a nonlinear correction technique to improve numerical performance on distorted meshes. Numerical experiments show that the scheme is efficient and has approximately second order accuracy in extreme cases.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Cheng Dong, Tong Kang
Summary: In this work, a cell-centered finite volume scheme is derived using a linearity-preserving technique. The scheme introduces both cell-centered and vertex unknowns, and eliminates the vertex unknowns with a new vertex interpolation algorithm derived from the presented linearity-preserving technique. The proposed technique is flexible and can be used to design algorithms for 3D diffusion problems. Numerical experiments show nearly optimal accuracy, and the new vertex interpolation algorithm outperforms some commonly used linearity-preserving algorithms on Kershaw meshes and distorted meshes.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2022)
Article
Mathematics, Applied
Cheng Dong, Tong Kang
Summary: We propose a diamond scheme for solving 3D heterogeneous and anisotropic problems using piecewise linear approximation and least squares method. The scheme allows arbitrary diffusion tensors and achieves better performance than the least squares interpolation. The vertex interpolation algorithm used in this scheme does not require edge information, simplifying the programming and reducing topological searches.
APPLIED MATHEMATICS AND COMPUTATION
(2022)
Article
Computer Science, Interdisciplinary Applications
Ricardo J. M. de Lira Filho, Sidicley R. dos Santos, Tulio de M. Cavalcante, Fernando R. L. Contreras, Paulo R. M. Lyra, Darlan K. E. de Carvalho
Summary: The paper introduces a novel MPFA-D scheme for solving the 3-D steady state diffusion equation, which can accurately reproduce piecewise linear solutions on challenging heterogeneous and anisotropic media. The numerical validation demonstrates second order accuracy for the scalar unknown and at least first order accuracy for fluxes on unstructured tetrahedral meshes and arbitrarily anisotropic diffusion tensors. The performance of the new scheme is compared with existing methods in the literature, showing good results but also limitations in dealing with very distorted meshes and highly anisotropic diffusion tensors.
COMPUTERS & STRUCTURES
(2021)
Article
Mathematics, Applied
Longshan Luo, Cheng Dong
Summary: In this paper, a diamond scheme is proposed for solving heterogeneous and anisotropic diffusion problems on unstructured polyhedral meshes. The scheme features both cell-centered primary unknowns and cell-vertex auxiliary unknowns. The auxiliary unknowns are expressed as linear combinations with the surrounding cell-centered unknowns using the least square technique and a graph search algorithm. Numerical experiments show that the scheme is robust and linearity-preserving for the heterogeneous diffusion tensor on general unstructured grids.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Computer Science, Interdisciplinary Applications
Cheng Dong, Tong Kang
Summary: A new least squares based diamond scheme and vertex interpolation algorithm are proposed for anisotropic diffusion problems on polygonal meshes. These methods are applicable to diffusion problems with arbitrary diffusion tensors and maintain linearity under certain assumptions.
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS
(2021)
Article
Mathematics, Applied
Tarek Ghoudi, M. Shadi Mohamed, Mohammed Seaid
Summary: A new adaptive finite volume method is proposed for the simulation of wave problems in the time domain. The method discretizes the transient wave equations in time and space, and utilizes a vertex-centered finite volume method with both cell-centered and edge-midpoint. The method also includes a mesh adaptation procedure based on energy-norm error-estimates, allowing for multiple adaptations within a single error estimation.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Computer Science, Interdisciplinary Applications
Di Yang, Zhiming Gao, Guoxi Ni
Summary: Two kinds of nonlinear cell-centered positivity-preserving finite volume schemes are proposed for anisotropic diffusion problems on general three-dimensional polyhedral meshes. The schemes discretize the one-sided flux on cell-faces using a fixed stencil of all vertices, and then obtain a cell-centered discretization scheme using a nonlinear two-point flux approximation. In addition, a new explicit weighted second-order vertex interpolation algorithm for arbitrary polyhedral meshes is designed to eliminate the vertex auxiliary unknowns in the scheme. An improved Anderson acceleration algorithm is also adopted for nonlinear iteration.
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS
(2022)
Article
Computer Science, Interdisciplinary Applications
Di Yang, Zhiming Gao, Guoxi Ni
Summary: Two new nonlinear cell-centered positivity-preserving finite volume schemes are proposed for anisotropic diffusion problems on general three-dimensional polyhedral meshes. The schemes involve one-sided flux discretization, two-point flux approximation, and explicit weighted second-order vertex interpolation algorithm.
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS
(2022)
Article
Geochemistry & Geophysics
Qingyu Zhang, Xiao Ma, Yufeng Nie
Summary: A new iterative fast sweeping method on unstructured triangular meshes is proposed for solving the eikonal equation in models with irregular topography and subsurface interfaces, demonstrating validity and accuracy for rough topography and subsurface interface models.
Article
Mathematics, Applied
Yanhui Zhou, Yanlong Zhang, Jiming Wu
Summary: In this paper, a polygonal finite volume element method (PFVEM) is proposed and analyzed for solving the anisotropic diffusion equation on convex polygonal meshes, based on the Wachspress generalized barycentric coordinates. The PFVEM reduces to the classical P-1-FVEM on triangular meshes but is not identical to the classical Q(1)-FVEM on quadrilateral meshes. The paper provides a new proof for Proposition 8 in [19], a crucial result for the derivation of interpolation error estimates. Furthermore, the H-2 error estimate of the Wachspress interpolation is proven for the error analysis of the PFVEM, and the optimal H-1 error estimate for the finite volume element solution is obtained under the coercivity assumption. Several numerical examples are presented to demonstrate the efficiency and robustness of the proposed method.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Yanhui Zhou, Jiming Wu
Summary: The paper presents a family of quadratic finite volume element schemes for anisotropic diffusion problems on triangular meshes, with the introduction of special parameters to split the element stiffness matrix and ensure the existence, uniqueness, and coercivity of the finite volume element solution. The optimal H-1 error estimate is obtained, and theoretical results are verified through numerical experiments.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Pablo Castrillo, Alfredo Canelas, Eugenio Schillaci, Joaquim Rigola, Asensio Oliva
Summary: This paper presents a high-order finite volume method using Moving Least Squares (MLS) and Local Regression Estimators (LRE) for solving linear elasticity problems on two-dimensional unstructured meshes. The method effectively solves structural problems affected by shear locking and demonstrates accuracy and flexibility through canonical tests and analytical examples.
COMPUTERS & STRUCTURES
(2022)
Article
Computer Science, Interdisciplinary Applications
Gang Peng, Zhiming Gao, Wenjing Yan, Xinlong Feng
Summary: This paper introduces a new cell-centered positivity-preserving finite volume scheme for solving 3D anisotropic diffusion problems on distorted meshes. The scheme utilizes primary and auxiliary unknowns, with discretization, interpolation, and acceleration methods to improve computational efficiency and numerical accuracy.
COMPUTER PHYSICS COMMUNICATIONS
(2021)
Article
Engineering, Multidisciplinary
G. Galindez-Ramirez, F. R. L. Contreras, D. K. E. Carvalho, P. R. M. Lyra
Summary: In this paper, a new high-order numerical methodology based on unstructured quadrilateral meshes is proposed for modeling oil-water displacements in highly heterogeneous and anisotropic petroleum reservoirs. The method achieves comparable accuracy to lower-order counterparts while reducing computational cost, and its accuracy, efficiency, and robustness are demonstrated through representative examples.
APPLIED MATHEMATICAL MODELLING
(2022)
Article
Engineering, Multidisciplinary
Artur Castiel Reis de Souza, Darlan Karlo Elisiario de Carvalho, Jose Cicero Araujo dos Santos, Ramiro Brito Willmersdorf, Paulo Roberto Maciel Lyra, Michael G. Edwards
Summary: This paper introduces a new multiscale finite volume framework for simulating multi-phase flow in heterogeneous and anisotropic porous media. The framework allows the use of geophysical grid defined properties on high-definition grids, addressing the issues of basis function leakage and mass conservation. The accuracy of the framework is validated through comparisons with direct simulations on fine-scale, and the results demonstrate its ability to produce well-resolved solutions for complex geological formations found in petroleum reservoir problems.
APPLIED MATHEMATICAL MODELLING
(2022)
Article
Engineering, Civil
Rodolfo M. S. Cabral, Adriano D. M. Ferreira, Julio T. Pimentel, Marco A. F. da Silva Cabral, Paulo R. M. Lyra, Silvana M. B. Afonso, Ramiro B. Willmersdorf
Summary: This paper presents a new methodology for predicting pipelines failure pressure using axisymmetric modeling and the River Bottom Profile (RBP) of corrosion defects. This practical tool allows pipeline engineers in the oil and gas industry to quickly and accurately assess pipeline integrity.
ENGINEERING STRUCTURES
(2022)
Letter
Computer Science, Interdisciplinary Applications
Jose Cicero Araujo dos Santos, Paulo Roberto Maciel Lyra, Joao Paulo Rodrigues de Andrade, Artur Castiel Reis de Souza, Ricardo Jorge Morais de Lira Filho, Darlan Karlo Elisiario de Carvalho
Summary: This paper proposes an adaptive flow-based dual volume agglomeration strategy for correcting non-physical terms in the coarse transmissibility matrix of classical multiscale finite volume (MsFV) methods. It also presents a framework to handle non-uniform levels at each coarse control volume, in order to reduce the size of coarse scale matrices. The proposed methodologies are applied to approximate pressure solutions in an Implicit Pressure Explicit Saturation (IMPES) strategy, and their accuracy and efficiency are demonstrated through testing with challenging benchmark problems.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Computer Science, Interdisciplinary Applications
J. C. Teixeira, D. K. E. Carvalho, T. M. Cavalcante, K. C. L. Angelim, P. R. M. Lyra
Summary: This paper presents a numerical formulation for simulating two-phase flow in naturally fractured oil reservoirs using unstructured quadrilateral meshes and Hybrid-Grid MPFA-Streamline method. It can handle irregular polygonal grids and decouples transport equations using streamline-based method to solve advective saturation problems.
COMPUTERS AND GEOTECHNICS
(2022)
Article
Mathematics, Applied
T. M. Cavalcante, R. J. M. Lira Filho, A. C. R. Souza, D. K. E. Carvalho, P. R. M. Lyra
Summary: In this paper, we solve the steady state diffusion equation in 3D domains using the MPFA-DNL method, which guarantees the Discrete Maximum Principle by introducing a non-linear defect correction strategy. The method is locally conservative and capable of handling arbitrary anisotropic diffusion tensors and unstructured meshes while maintaining second order convergence rates.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Review
Water Resources
Fernando R. L. Contreras, Uewerton A. O. Vaz, Gustavo L. S. S. Pacheco, Alessandro R. E. Antunes, Paulo R. M. Lyra, Darlan K. E. Carvalho
Summary: This paper proposes a novel full finite volume method to solve the advection-dispersion transport equation, combining various numerical methods to ensure accuracy and robustness of the numerical solution. Numerical experiments show that the method can provide accurate solutions when simulating groundwater processes with complex physical and geological properties.
ADVANCES IN WATER RESOURCES
(2023)
Article
Energy & Fuels
M. E. S. Galindo, I. V. Lacerda, G. Galindez-Ramirez, P. R. M. Lyra, D. K. E. Carvalho
Summary: Compositional reservoir simulation is a crucial tool for modeling fluid flow in complex petroleum reservoirs, especially for volatile reservoir fluids or those involving enhanced oil recovery. Simple black-oil models are inadequate in these cases. The compositional model involves solving a large system of partial differential equations that describe mass conservation, Darcy's law, and fugacity constraints. However, the complexity and computational demands of the compositional problems are high due to the large number of equations and constraints. To improve accuracy and reduce computational costs, higher-order methods can be used to approximate the advective flux terms in the reservoir's multicomponent transport.
GEOENERGY SCIENCE AND ENGINEERING
(2023)
Article
Computer Science, Interdisciplinary Applications
Ashish Bhole, Herve Guillard, Boniface Nkonga, Francesca Rapetti
Summary: Finite elements of class C-1 are used for computing magnetohydrodynamics instabilities in tokamak plasmas, and isoparametric approximations are employed to align the mesh with the magnetic field line. This numerical framework helps in understanding the operation of existing devices and predicting optimal strategies for the international ITER tokamak. However, a mesh-aligned isoparametric representation encounters issues near critical points of the magnetic field, which can be addressed by combining aligned and unaligned meshes.
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS
(2024)
Article
Computer Science, Interdisciplinary Applications
Federico Vismara, Tommaso Benacchio
Summary: This paper introduces a method for solving hyperbolic-parabolic problems on multidimensional semi-infinite domains. By dividing the computational domain into bounded and unbounded subdomains and coupling them using numerical fluxes at the interface, accurate numerical solutions are obtained. In addition, computational cost can be reduced by tuning the parameters of the basis functions.
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS
(2024)
Article
Computer Science, Interdisciplinary Applications
Keigo Enomoto, Takato Ishida, Yuya Doi, Takashi Uneyama, Yuichi Masubuchi
Summary: We have developed a novel Moving Particle Simulation (MPS) method to accurately reproduce the motion of fibers in sheared liquids. By introducing the micropolar fluid model, we address the issue of fibers being aligned with the flow direction in conventional MPS simulations. Our method is capable of accurately reproducing the fiber motion predicted by Jeffery's theory.
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS
(2024)
