An appropriate mesh representation for tetrahedral finite volume computations

The amounts of information used for the solution of three-dimensional partial differential equations in engineering applications is formidable. Part of the problem is that in unstructured meshes (adequate for advanced numerical techniques) the information is local in nature. An alternate representation of an unstructured tetrahedral mesh is proposed. Besides having some storage advantages over representations commonly used for finite volume computations, we show that preprocessing complexity is lower than in the conventional representation. The resulting simplified structure may result in significant gains for applications that go through a number of mesh refinement cycles.     

The simpler GMRES method combined with finite volume method for simulating viscoelastic flows on triangular grid

An efficient solver integrating the restarted simpler generalized minimal residual method (SGMRES(m)) with finite volume method (FVM) on triangular grid is developed to simulate the viscoelastic fluid flows. In particular, the SGMRES(m) solver is used to solve the large-scale sparse linear systems, which arise from the course of FVM on triangular grid for modeling the Newtonian and the viscoelastic fluid flows. To examine the performance of the solver for the nonlinear flow equations of viscoelastic fluids, we consider two types of numerical tests: the Newtonian flow past a circular cylinder, and the Oldroyd-B fluid flow in a planar channel and past a circular cylinder. It is shown that the numerical results obtained by the SGMRES(m) are consistent with the analytical solutions or empirical values. By comparing CPU time of different solvers, we find our solver is a highly efficient one for solving the flow equations of viscoelastic fluids.     

Godunov-type upwind flux schemes of the two-dimensional finite volume discrete Boltzmann method

A simple unified Godunov-type upwind approach that does not need Riemann solvers for the flux calculation is developed for the finite volume discrete Boltzmann method (FVDBM) on an unstructured cell-centered triangular mesh. With piecewise-constant (PC), piecewise-linear (PL) and piecewise-parabolic (PP) reconstructions, three Godunov-type upwind flux schemes with different orders of accuracy are subsequently derived. After developing both a semi-implicit time marching scheme tailored for the developed flux schemes, and a versatile boundary treatment that is compatible with all of the flux schemes presented in this paper, numerical tests are conducted on spatial accuracy for several single-phase flow problems. Four major conclusions can be made. First, the Godunov-type schemes display higher spatial accuracy than the non-Godunov ones as the result of a more advanced treatment of the advection. Second, the PL and PP schemes are much more accurate than the PC scheme for velocity solutions. Third, there exists a threshold spatial resolution below which the PL scheme is better than the PP scheme and above which the PP scheme becomes more accurate. Fourth, besides increasing spatial resolution, increasing temporal resolution can also improve the accuracy in space for the PL and PP schemes.     

Quadratic finite volume element method for the air pollution model

In this article, we use the quadratic finite volume element method (FVEM) to solve the problem of the air pollution model, choose the trial function spaces as the Lagrange quadratic element function spaces and the test function spaces as the piecewise constant function spaces, then get the error estimates of L ² and H ¹. Finally, by numerical experiments, we analyse and compare the FVEM with the finite difference method, and the numerical results we get show that the FVEM is much better and more effective, so this article has some practical significance in the improvement and control of air pollution.     

Element residual error estimate for the finite volume method

Out of the wide range of a-posteriori error estimates for the finite element method (FEM), the group of estimates based on the element residual seems to be the most popular. One recent extension of the element residual method is the element residual error estimate (EREE) [Numerische Mathematik 65 (1993) 23], which includes the elements of the duality theory and consistently produces good results. In this paper, the EREE will be extended to allow its use in conjunction with the finite volume (FV) type of discretisation. The extension consists of three parts: an appropriate definition of the residual in the FV framework, a procedure for calculation of self-equilibrating fluxes based on the conservative properties of the FV solution and a simplified solution method for the Local Problem. The paper covers the extensions of the EREE to the convection-diffusion and the Navier-Stokes problem, following [Comp Meth Appl Mech Engng 101 (1992) 73] and [Comp Meth Appl Mech Engng 111 (1994) 185], respectively. The error estimate is tested on three test cases with analytical solutions, where its performance is shown to be similar to its FEM counterpart. Finally, the estimate is applied to a realistic laminar fluid flow problem.     

A finite volume method for solid mechanics incorporating rotational degrees of freedom

A novel finite volume (FV) based discretization method for determining displacement, strain and stress distributions in loaded two dimensional structures with complex geometries is presented. The method incorporates rotation variables in addition to the displacement degrees of freedom employed in earlier FV based structural analysis procedures and conventional displacement based finite element (FE) formulations. The method is used to predict the displacement fields in a number of test cases for which solutions are already available. The effect of mesh refinement upon the accuracy of the solutions predicted by the method is assessed. The results of this assessment indicate that the new method is more accurate than previous FV procedures incorporating displacement variables only, particularly in cases where bending is the predominant mode of deformation, and therefore the new method represents a significant advance in the development of this type of discretization procedure. Interestingly, the results of the accuracy assessment exercise also indicate that the new FV procedure is also more accurate than the equivalent FE formulation incorporating displacement and rotational degrees of freedom.     

Automated finite element model updating of full-scale structures with PARameter Identification System (PARIS)

This paper presents a software framework, PARIS (PARameter Identification System), developed for automated finite element model updating for structural health monitoring. With advances in Application Programming Interfaces (API) for modern computing, the traditional boundaries between different standalone software packages hardly exist. Now complex problems can be distributed between different software platforms with advanced and specialized capabilities. PARIS takes advantage of the advancements in the computing environment and interfacing capabilities provided by commercial software to systematically distribute the structural parameter estimation problem into an iterative optimization and finite element analysis problem across different computing platforms. Three validation examples using simulated nondestructive test data for updating full-scale structural models under typically encountered damage scenarios are included. The results of model updating process for realistic structural models and their systematic treatment provide enhanced understanding of the aforementioned parameter estimation process and an encouraging path towards its feasible field application for structural health monitoring and structural condition assessment.     

Physical-bound-preserving finite volume methods for the Nagumo equation on distorted meshes

In this paper, we present a boundedness preserving finite volume scheme for the Nagumo equation. In this method, we use the implicit Euler method for the time discretization, and construct a maximum-principle-preserving discrete normal flux for the diffusion term. For the nonlinear reaction term, we design a type of Picard iteration to ensure that at each iterative step it keeps physical boundedness. Moreover we prove that the numerical solution of the resulting scheme can preserve the bound of the solution for the Nagumo equation on distorted meshes. Some numerical results are presented to verify the theoretical analysis.     

Iterative inflow-implicit outflow-explicit finite volume scheme for level-set equations on polyhedron meshes

In this paper, we propose a cell-centered finite volume method for advective and normal flows on polyhedron meshes which is second-order accurate in space and time for smooth solutions. In order to overcome a time restriction caused by CFL condition, an implicit time discretization of inflow fluxes and an explicit time discretization of outflow fluxes are used in an iterative procedure. For an efficient computation, an 1-ring face neighborhood structure is introduced. Since it is limited to access unknown variables in an 1-ring face neighborhood structure, an iterative procedure is proposed to resolve the limitation of assembled linear system. Two types of gradient approximations, an inflow-based gradient and an average-based gradient, are studied and compared from the point of numerical accuracy. Numerical schemes are tested for an advective and a normal flow of level-set functions illustrating a behavior of the proposed method for an implicit tracking of a smooth and a piecewise smooth interface.     

A finite volume method for Stokes problems on quadrilateral meshes

We present a finite volume method for Stokes problems using the isoparametric $Q_1$–$Q_0$ element pair on quadrilateral meshes. To offset the lack of the $inf$–$sup$ condition, a jump term of discrete pressure (stabilizing term) is added to the continuity approximation equation. Thus, we establish a stabilized finite volume scheme on quadrilateral meshes. Then, based on some superclose estimates, we derive the optimal error estimates in the $H^1$- and $L_2$-norms for velocity and in the $L_2$-norm for pressure, respectively. Numerical examples are provided to illustrate our theoretical analysis. We emphasize that our work is the first time to propose and analyze a finite volume method for Stoke problems using isoparametric elements on quadrilateral meshes.     

基于非线性有限元的可变形模型的变形方法

三维模型的变形是计算机图形学与其他学科交叉研究的热点问题之一。针对采用线性应变对模型变形时,容易出现不符合物体在客观实际情况下的形变。给出了一个有效的模型变形方法。首先,通过采用几何删除方法对原始模型进行简化,使简化后的模型尽可能地保持原模型的拓扑形态; 在此基础之上对其进行四面体剖分,进而采用格林应变对剖分后的模型进行变形。实验结果表明,在采用非线性有限元方法对模型进行变形时,先对模型进行简化,然后再四面体剖分,可以获得变形速率上的提升。     

Numerical analysis of a finite volume scheme for two incompressible phase flow with dynamic capillary pressure

In this paper, we propose and analyze the convergence of a TPFA (Two Points Flux Approximation) finite volume scheme to approximate the two incompressible phase flow with dynamic capillary pressure in porous media. The fully implicit scheme is based on nonstandard approximation on mobilities and capillary pressure on the dual mesh. We derive a discrete variational formulation and we present a new result of convergence in a two or three dimensional porous medium. In comparison with static capillary pressure, the non-equilibrium capillary model requires more powerful techniques; especially the discrete energy estimates are not standard.     

High resolution finite volume computations on unstructured grids using solution dependent weighted least squares gradients

An improved high resolution finite volume method based on linear and quadratic variable reconstructions using solution dependent weighted least squares (SDWLS) gradients has been presented here. An extended stencil consisting of vertex-based neighbours of a cell is used in the higher order reconstructions for inviscid flux computations. A QR algorithm with Householder transformation is used to solve the weighted least squares problem. In case of Navier–Stokes equations, viscous fluxes are discretized in a central differencing manner based on the Coirier’s diamond path. A few inviscid and viscous test cases are solved in order to demonstrate the efficacy of the present method. Progressive improvements in solution accuracy are observed with the increase in the order of variable reconstructions. In most cases, results of quadratic reconstruction show significant improvements over that of linear reconstruction.     

Analysis and construction of cell-centered finite volume scheme for diffusion equations on distorted meshes

A finite volume scheme solving diffusion equation on non-rectangular meshes is introduced by Li [Deyuan Li, Hongshou Shui, Minjun Tang, On the finite difference scheme of two-dimensional parabolic equation in a non-rectangular mesh, J. Numer. Meth. Comput. Appl. 4 (1980) 217 (in Chinese), D.Y. Li, G.N. Chen, An Introduction to the Difference Methods for Parabolic Equation, Science Press, Beijing, 1995 (in Chinese)], which is the so-called nine-point scheme on arbitrary quadrangles. The vertex unknowns can be represented as some weighted combination of the cell-centered unknowns, but it is difficult to choose the suitable combination coefficients for the multimaterial computation on highly distorted meshes. We present a nine-point scheme for discretizing diffusion operators on distorted quadrilateral meshes, and derive a new expression for vertex unknowns. The stability and convergence of the resulting scheme are proved. We give numerical results for various test cases which exhibit the good behavior of our scheme.     

The finite volume spectral element method to solve Turing models in the biological pattern formation

It is well known that reaction-diffusion systems describing Turing models can display very rich pattern formation behavior. Turing systems have been proposed for pattern formation in various biological systems, e.g. patterns in fish, butterflies, lady bugs and etc. A Turing model expresses temporal behavior of the concentrations of two reacting and diffusing chemicals which is represented by coupled reaction-diffusion equations. Since the base of these reaction-diffusion equations arises from the conservation laws, we develop a hybrid finite volume spectral element method for the numerical solution of them and apply the proposed method to Turing system generated by the Schnakenberg model. Also, as numerical simulations, we study the variety of spatio-temporal patterns for various values of diffusion rates in the problem.     

High-order IMEX-WENO finite volume approximation for nonlinear age-structured population model

ABSTRACT

In this article, we design a high-order implicit–explicit weighted non-oscillatory (IMEX-WENO) scheme for the solution of the population age density model with the nonlinear mortality rate as well as fertility rate. The mortality rate is the combination of natural mortality that comes to be unbounded at maximum age and bounded external mortality rate that includes external resources as well as seasonality. The existence of global terms in the mortality function as well as in the boundary condition is the main technical complication which provides high nonlinearity for the model equation. We carefully construct a numerical scheme in such way, that the high-order accuracy is maintained in the global terms and boundary condition. The performance of the designed scheme is shown by comparing with the exact solution of the examples considered.     

An efficient unstructured MUSCL scheme for solving the 2D shallow water equations

The aim of this paper is to present a novel monotone upstream scheme for conservation law (MUSCL) on unstructured grids. The novel edge-based MUSCL scheme is devised to construct the required values at the midpoint of cell edges in a more straightforward and effective way compared to other conventional approaches, by making better use of the geometrical property of the triangular grids. The scheme is incorporated into a two-dimensional (2D) cell-centered Godunov-type finite volume model as proposed in Hou et al. (2013a,c) to solve the shallow water equations (SWEs). The MUSCL scheme renders the model to preserve the well-balanced property and achieve high accuracy and efficiency for shallow flow simulations over uneven terrains. Furthermore, the scheme is directly applicable to all triangular grids. Application to several numerical experiments verifies the efficiency and robustness of the current new MUSCL scheme.     

A posteriori error estimates of two-grid finite volume element methods for nonlinear elliptic problems

In this paper, we study the a posteriori error estimates of two-grid finite volume element method for second-order nonlinear elliptic equations. We derive the residual-based a posteriori error estimator and prove the computable upper and lower bounds on the error in $H^1$-norm. The a posteriori error estimator can be used to assess the accuracy of the two-grid finite volume element solutions in practical applications. Numerical examples are provided to illustrate the performance of the proposed estimator.     

A finite volume method parallelization for the simulation of free surface shallow water flows

We construct a parallel algorithm, suitable for distributed memory architectures, of an explicit shock-capturing finite volume method for solving the two-dimensional shallow water equations. The finite volume method is based on the very popular approximate Riemann solver of Roe and is extended to second order spatial accuracy by an appropriate TVD technique. The parallel code is applied to distributed memory architectures using domain decomposition techniques and we investigate its performance on a grid computer and on a Distributed Shared Memory supercomputer. The effectiveness of the parallel algorithm is considered for specific benchmark test cases. The performance of the realization measured in terms of execution time and speedup factors reveals the efficiency of the implementation.