• Coupled systems mechanics
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• 제9권1호
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• pp.5-28
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• 2020

Despite a widely-held belief that the finite element method is the method for the solution of solid mechanics problems, which has for 30 years dissuaded solid mechanics scientists from paying any attention to the finite volume method, it is argued that finite volume methods can be a viable alternative. It is shown that it is simple to understand and implement, strongly conservative, memory efficient, and directly applicable to nonlinear problems. A number of examples are presented and, when available, comparison with finite element methods is made, showing that finite volume methods can be not only equal to, but outperform finite element methods for many applications.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제5권1호
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• pp.55-78
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• 2001

Given the anisotropic Poisson equation $-{\nabla}{\cdot}{\mathcal{K}}{\nabla}p=f$, one can convert it into a system of two first order PDEs: the Darcy law for the flux $u=-{\mathcal{K}{\nabla}p$ and conservation of mass ${\nabla}{\cdot}u=f$. A very natural mixed finite volume method for this system is to seek the pressure in the nonconforming P1 space and the Darcy velocity in the lowest order Raviart-Thomas space. The equations for these variables are obtained by integrating the two first order systems over the triangular volumes. In this paper we show that such a method is really a standard finite element method with local recovery of the flux in disguise. As a consequence, we compare two approaches in analyzing finite volume methods (FVM) and shed light on the proper way of analyzing non co-volume type of FVM. Numerical results for Dirichlet and Neumann problems are included.

• Journal of applied mathematics & informatics
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• 제41권2호
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• pp.295-309
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• 2023

The Richards' equation attracts the attention of several scientific researchers due to its importance in the hydrogeology field especially porous soil. This work presents a numerical method to solve the two dimensional Richards' equation. The pressure form and the mixed form of Richards' equation are solved numerically using a bivariate diamond finite volumes scheme. Euler explicit scheme is used for the time discretization. Different test cases are done to validate the accuracy and the efficiency of our numerical model and to compare the possible numerical strategies. We started with a first simple test case of Richards' pressure form where the hydraulic capacity and the hydraulic conductivity are taken constant and then a second test case where the hydrodynamics parameters are linear variables. Finally, a third test case where the soil parameters are taken according the Van Gunchten empirical model is presented.

• 대한수학회지
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• 제56권4호
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• pp.1083-1112
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• 2019

In this paper, we present and analyze a cell-centered collocated finite volume scheme for incompressible flows to compute solutions simultaneous to Stokes and Darcy equations by applying a pressure jump stabilization term to avoid locking. We prove that the new stabilized FV formulation satisfies a discrete inf-sup condition and error estimates for both problems. Finally, we present some numerical examples confirming this analysis.

• Advances in Computational Design
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• 제5권3호
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• pp.277-289
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• 2020

This article describes the technology for constructing of a multiply-connected three-dimensional area's finite element representation. Representation of finite-element configuration of an area is described by a discrete set that consist of the number of nodes and elements of the finite-element grid, that are orderly set of nodes' coordinates and numbers of finite elements. Corresponding theorems are given, to prove the correctness of the solution method. The adequacy of multiply-connected area topology's finite element model is shown. The merging of subareas is based on the criterion of boundary nodes' coincidence by establishing a simple hierarchy of volumes, surfaces, lines and points. Renumbering nodes is carried out by the frontal method, where nodes located on the outer edges of the structure are used as the initial front.

• 한국해양공학회지
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• 제30권3호
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• pp.169-176
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• 2016

Considering the artificial reef (AR) canyon intervals facilitated by flatly distributed placement models, the wake volumes of 25 AR sets were characterized through the following works. First, twenty-five different canyon intervals were established to investigate how the intervals affect the wake volumes of the AR placement models, each with nine cube-type ARs. Second, the element-based finite-volume method was used to facilitate flow analyses. Third, the so-called wake volume concept was adopted, and finally a reasonable placement interval was found based on the size of the wake volumes and the associated unit propagation indices. From the analysis results, it was found that a maximum wake volume of 25.18 m³ was generated when the longitudinal and transverse intervals were fixed at 6 m and 0 m, respectively. Thus, to magnify the wake volume, it is recommended that artificial reefs be placed at intervals of 6 m (3 times the reef length) in the flow direction, with no intervals in the normal direction, implicitly indicating that an intensively stacked placement model is a better option to efficiently secure a larger wake volume for the cube-type ARs.

• 한국항만학회지
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• 제13권1호
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• pp.167-174
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• 1999

Numerical analysis of two-fluid flows for both water and air is carried out. Free-Surface flows with an arbitrary deformation have been simulated around two dimensional submerged hydrofoil. The computation is performed using a finite volume method with unstructured meshes and an interface capturing scheme to determine the shape of the free surface. The method uses control volumes with an arbitrary number of faces and allows cell-wise local mesh refinement. the integration in space is of second order based on midpoint rule integration and linear interpolation. The method is fully implicit and uses quadratic interpolation in time through three time levels The linear equation systems are solved by conjugate gradient type solvers and the non-linearity of equations is accounted for through picard iterations. The solution method is of pressure-correction type and solves sequentially the linearized momentum equations the continuity equation the conservation equation of one species and the equations or two turbulence quantities.

• Journal of Mechanical Science and Technology
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• 제14권7호
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• pp.789-795
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• 2000

Free-surface flows with an arbitrary deformation, induced by a submerged hydrofoil, are simulated numerically, considering two-fluid flows of both water and air. The computation is performed by a finite volume method using unstructured meshes and an interface capturing scheme to determine the shape of the free surface. The method uses control volumes with an arbitrary number of faces and allows cell wise local mesh refinement. The integration in space is of second order, based on midpoint rule integration and linear interpolation. The method is fully implicit and uses quadratic interpolation in time through three time levels. The linear equations are solved by conjugate gradient type solvers, and the non-linearity of equations is accounted for through Picard iterations. The solution method is of pressure-correction type and solves sequentially the linearized momentum equations, the continuity equation, the conservation equation of one species, and the equations for two turbulence quantities. Finally, a comparison is quantitatively made at the same speed between the computation and experiment in which the grid sensitivity is numerically checked.

• 소성∙가공
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• 제4권3호
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• pp.214-221
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• 1995

In the paper, we have proposed a new method to determine the initial billet for the forged products using a function approximation in the neural network. The architecture of neural network is a three-layer neural network and the back propagation algorithm is employed to train the network. By utilizing the ability of function approximation of a neural network, an optimal billet is determined by applying the nonlinear mathematical relationship between the aspect ratios in the initial billet and the final products. The amount of incomplete filling in the die is measured by the rigid-plastic finite element method. The neural network is trained with the initial billet aspect ratios and those of the unfilled volumes. After learning, the system is able to predict the filling regions which are exactly the same or slightly different to the results of finite element simulation. This new method is applied to find the optimal billet size for the plane strain rib-web product in cold forging. This would reduce the number of finite element simulation for determining the optimal billet size of forging product, further it is usefully adapted to physical modeling for the forging design.

• 대한수학회지
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• 제59권3호
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• pp.519-548
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• 2022

In this paper, a stabilized-penalized collocated finite volume (SPCFV) scheme is developed and studied for the stationary generalized Navier-Stokes equations with mixed Dirichlet-traction boundary conditions modelling an incompressible biological fluid flow. This method is based on the lowest order approximation (piecewise constants) for both velocity and pressure unknowns. The stabilization-penalization is performed by adding discrete pressure terms to the approximate formulation. These simultaneously involve discrete jump pressures through the interior volume-boundaries and discrete pressures of volumes on the domain boundary. Stability, existence and uniqueness of discrete solutions are established. Moreover, a convergence analysis of the nonlinear solver is also provided. Numerical results from model tests are performed to demonstrate the stability, optimal convergence in the usual L² and discrete H¹ norms as well as robustness of the proposed scheme with respect to the choice of the given traction vector.