Application of fuzzy decision making to the fuzzy finite element method

In this paper we present a new approach to handle uncertainty in the Finite Element Method. As this technique is widely used to tackle real-life design problems, it is also very prone to parameter-uncertainty. It is hard to make a good decision regarding design optimization if no claim can be made with respect to the outcome of the simulation. We propose an approach that combines several techniques in order to offer a total order on the possible design choices, taking the inherent fuzziness into account. Additionally we propose a more efficient ordering procedure to build a total order on fuzzy numbers.     

Convergence analysis of a finite volume method via a new nonconforming finite element method

The article is devoted to the study of convergence properties of a Finite Volume Method (FVM) using Voronoi boxes for discretization. The approach is based on the construction of a new nonconforming Finite Element Method (FEM), such that the system of linear equations coincides completely with that for the FVM. Thus, by proving convergence properties of the FEM, we obtain similar ones of the FVM. In this article, the investigations are restricted to the Poisson equation. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14:213–231, 1998     

Fuzzy temporal fault tree analysis of dynamic systems

Fault tree analysis (FTA) is a powerful technique that is widely used for evaluating system safety and reliability. It can be used to assess the effects of combinations of failures on system behaviour but is unable to capture sequence dependent dynamic behaviour. A number of extensions to fault trees have been proposed to overcome this limitation. Pandora, one such extension, introduces temporal gates and temporal laws to allow dynamic analysis of temporal fault trees (TFTs). It can be easily integrated in model-based design and analysis techniques. The quantitative evaluation of failure probability in Pandora TFTs is performed using exact probabilistic data about component failures. However, exact data can often be difficult to obtain. In this paper, we propose a method that combines expert elicitation and fuzzy set theory with Pandora TFTs to enable dynamic analysis of complex systems with limited or absent exact quantitative data. This gives Pandora the ability to perform quantitative analysis under uncertainty, which increases further its potential utility in the emerging field of model-based design and dependability analysis. The method has been demonstrated by applying it to a fault tolerant fuel distribution system of a ship, and the results are compared with the results obtained by other existing techniques.     

Analysis of newton multilevel stabilized finite volume method for the three‐dimensional stationary Navier‐Stokes equations

This article proposes and analyzes a multilevel stabilized finite volume method(FVM) for the three‐dimensional stationary Navier–Stokes equations approximated by the lowest equal‐order finite element pairs. The method combines the new stabilized FVM with the multilevel discretization under the assumption of the uniqueness condition. The multilevel stabilized FVM consists of solving the nonlinear problem on the coarsest mesh and then performs one Newton correction step on each subsequent mesh thus only solving one large linear systems. The error analysis shows that the multilevel‐stabilized FVM provides an approximate solution with the convergence rate of the same order as the usual stabilized finite element solution solving the stationary Navier–Stokes equations on a fine mesh for an appropriate choice of mesh widths: h_j ～ h_j‐1², j = 1,…,J. Therefore, the multilevel stabilized FVM is more efficient than the standard one‐level‐stabilized FVM. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Stochastic optimal control of ultradiffusion processes with application to dynamic portfolio management

We consider theoretical and approximation aspects of the stochastic optimal control of ultradiffusion processes in the context of a prototype model for the selling price of a European call option. Within a continuous-time framework, the dynamic management of a portfolio of assets is effected through continuous or point control, activation costs, and phase delay. The performance index is derived from the unique weak variational solution to the ultraparabolic Hamilton–Jacobi equation; the value function is the optimal realization of the performance index relative to all feasible portfolios. An approximation procedure based upon a temporal box scheme/finite element method is analyzed; numerical examples are presented in order to demonstrate the viability of the approach.     

A posteriori error analysis of a quadratic finite volume method for nonlinear elliptic problems

In this article, we construct and analyze a residual-based a posteriori error estimator for a quadratic finite volume method (FVM) for solving nonlinear elliptic partial differential equations with homogeneous Dirichlet boundary conditions. We shall prove that the a posteriori error estimator yields the global upper and local lower bounds for the norm error of the FVM. So that the a posteriori error estimator is equivalent to the true error in a certain sense. Numerical experiments are performed to illustrate the theoretical results.     

A posteriori error analysis of a cell-centered finite volume method for semilinear elliptic problems

In this paper, we conduct a goal-oriented a posteriori analysis for the error in a quantity of interest computed from a cell-centered finite volume scheme for a semilinear elliptic problem. The a posteriori error analysis is based on variational analysis, residual errors and the adjoint problem. To carry out the analysis, we use an equivalence between the cell-centered finite volume scheme and a mixed finite element method with special choice of quadrature.     

Convergence analysis of an upwind finite volume element method with Crouzeix-Raviart element for non-selfadjoint and indefinite problems

In this paper we construct an upwind finite volume element scheme based on the Crouzeix-Raviart nonconforming element for non-selfadjoint elliptic problems. These problems often appear in dealing with flow in porous media. We establish the optimal order H ¹-norm error estimate. We also give the uniform convergence under minimal elliptic regularity assumption      

Superconvergence analysis of a nonconforming finite element method for monotone semilinear elliptic optimal control problems

A nonconforming finite element method (FEM) is proposed for optimal control problems (OCPs) governed by monotone semilinear elliptic equations. The state and adjoint state are approximated by the nonconforming elements, and the control is approximated by the orthogonal projection of the adjoint state, respectively. Some global supercloseness and superconvergence estimates are achieved by making full use of the distinguish characters of this element, such as the interpolation equals to its Ritz projection, and the consistency error is 1 − ε ( is small enough) order higher than its interpolation error in the broken energy norm when the exact solution belongs to H^{3 − ε}(Ω). Finally, some numerical results are presented to verify the theoretical analysis.     

Convergence analysis of an approximation to miscible fluid flows in porous media by combining mixed finite element and finite volume methods

This article deals with development and analysis of a numerical method for a coupled system describing miscible displacement of one incompressible fluid by another through heterogeneous porous media. A mixed finite element (MFE) method is employed to discretize the Darcy flow equation combined with a conservative finite volume (FV) method on unstructured grids for the concentration equation. It is shown that the FV scheme satisfies a discrete maximum principle. We derive L^∞ and BV estimates under an appropriate CFL condition. Then we prove convergence of the approximate solutions to a weak solution of the coupled system. Numerical results are presented to see the performance of the method in two space dimensions. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Error analysis of discontinuous Galerkin finite element method for optimal control problem governed by the transport equation

This article discusses a priori and a posteriori error estimates of discontinuous Galerkin finite element method for optimal control problem governed by the transport equation. We use variational discretization concept to discretize the control variable and discontinuous piecewise linear finite elements to approximate the state and costate variable. Based on the error estimates of discontinuous Galerkin finite element method for the transport equation, we get a priori and a posteriori error estimates for the transport equation optimal control problem. Finally, two numerical experiments are carried out to confirm the theoretical analysis.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1493–1512, 2017     

The modified characteristic finite difference fractional steps method for the coupled system of fluid dynamics in porous media and its analysis

For the coupled system of multilayer fluid dynamics in porous media, the modified characteristic finite difference fractional steps method applicable to parallel arithmetic is put forward and two‐dimensional and three‐dimensional schemes are used to form a complete set. Some techniques, such as calculus of variations, energy method, piecewise biquadratic interpolation, multiplicative commutation rule of difference operators, decomposition of high order difference operators and prior estimates are adopted. Optimal order estimates in L² norm are derived to determine the error in the approximate solution. This method has already been applied to the numerical simulation of multilayer fluid dynamics in porous media. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 665–681, 2003.     

A local boundary condition coupled to a finite element method to compute guided modes of optical fibres under the weak guidance assumptions

An efficient method coupling a local boundary condition to a finite element technique is proposed to compute guided modes of optical fibres under the weak guidance assumptions. This method is designed to provide accurate solutions for optical fibres with no restriction on their shape as well as on their refractive index profile. Several numerical experiments are performed to demonstrate this point. Copyright © 2000 John Wiley & Sons, Ltd.     

Numerical analysis of a Picard multilevel stabilization of mixed finite volume method for the 2D/3D incompressible flow with large data

In this article, we develop a branch of nonsingular solutions of a Picard multilevel stabilization of mixed finite volume method for the 2D/3D stationary Navier‐Stokes equations without relying on the unique solution condition. The method presented consists of capturing almost all information of initial problem (the nonlinear problems) on the coarsest mesh and then performs one Picard defect correction (the linear problems) on each subsequent mesh based on previous information thus only solving one large linear systems. What is more, the method presented can results in a better coefficient matrix in the model presented with small viscosity. Theoretical results show that the method presented is derived with the convergence rate of the same order as the corresponding finite volume method/finite element method solving the stationary Navier‐Stokes equations on a fine mesh. Therefore, the method presented is definitely more efficient than the standard finite volume method/finite element method. Finally, numerical experiments clearly show the efficiency of the method presented for solving the stationary Navier‐Stokes equations.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 34: 30–50, 2018     

Superconvergence analysis and a posteriori error estimation of a finite element method for an optimal control problem governed by integral equations

In this paper, we discuss the numerical simulation for a class of constrained optimal control problems governed by integral equations. The Galerkin method is used for the approximation of the problem. A priori error estimates and a superconvergence analysis for the approximation scheme are presented. Based on the results of the superconvergence analysis, a recovery type a posteriori error estimator is provided, which can be used for adaptive mesh refinement. The research project is supported by the National Basic Research Program under the Grant 2005CB321701 and the National Natural Science Foundation of China under the Grant 10771211.     

Convergence analysis of some high–order time accurate schemes for a finite volume method for second order hyperbolic equations on general nonconforming multidimensional spatial meshes

The present work is an extension of our previous work (Bradji, Numer Methods Partial Differ Equations, to appear) which dealt with error analysis of a finite volume scheme of a first convergence order (both in time and space) for second‐order hyperbolic equations on general nonconforming multidimensional spatial meshes introduced recently in (Eymard et al. IMAJ Numer Anal 30(2010), 1009–1043). We aim in this article to get some higher‐order time accurate schemes for a finite volume method for second‐order hyperbolic equations using the same class of spatial generic meshes stated above. We derive a family of finite volume schemes approximating the wave equation, as a model for second‐order hyperbolic equations, in which the discretization in time is performed using a one‐parameter scheme of the Newmark's method. We prove that the error estimate of these finite volume schemes is of order two (or four) in time and it is of optimal order in space. These error estimates are analyzed in several norms which allow us to derive approximations for the exact solution and its first derivatives whose the convergence order is two (or four) in time and it is optimal in space. We prove in particular, when the discrete flux is calculated using a stabilized discrete gradient, that the convergence order is \begin{align*}k^2+h_\mathcal{D}\end{align*} or \begin{align*}k^4+h_\mathcal{D}\end{align*}, where \begin{align*}h_\mathcal{D}\end{align*} (resp. k) is the mesh size of the spatial (resp. time) discretization. These estimates are valid under the regularity assumption \begin{align*}u\in C^4(\lbrack 0,T\rbrack;C^2(\overline{\Omega}))\end{align*}, when the schemes are second‐order accurate in time, and \begin{align*}u\in C^6(\lbrack 0,T\rbrack;C^2(\overline{\Omega}))\end{align*}, when the schemes are four‐order accurate in time for the exact solution u. The proof of these error estimates is based essentially on a comparison between the finite volume approximate solution and an auxiliary finite volume approximation. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A recursive method to the optimal control of an M/G/1 queueing system with finite capacity and infinite capacity

We study a single removable server in an infinite and a finite queueing systems with Poisson arrivals and general distribution service times. The server may be turned on at arrival epochs or off at service completion epochs. We present a recursive method, using the supplementary variable technique and treating the supplementary variable as the remaining service time, to obtain the steady state probability distribution of the number of customers in a finite system. The method is illustrated analytically for three different service time distributions: exponential, 3-stage Erlang, and deterministic. Cost models for infinite and finite queueing systems are respectively developed to determine the optimal operating policy at minimum cost.     

A method for analysis of linear dynamic systems driven by stationary non-Gaussian noise with applications to turbulence-induced random vibration

A method is developed for approximating the properties of the state of a linear dynamic system driven by a broad class of non-Gaussian noise, namely, by polynomials of filtered Gaussian processes. The method involves four steps. First, the mean and correlation functions of the state of the system are calculated from those of the input noise. Second, higher order moments of the state are calculated based on Itô’s formula for continuous semimartingales. It is shown that equations governing these moments are closed, so that moment of any order of the state can be calculated exactly. Third, a conceptually simple technique, which resembles the Galerkin method for solving differential equations, is proposed for constructing approximations for the marginal distribution of the state from its moments. Fourth, translation models are calibrated to representations of the marginal distributions of the state as well as its second moment properties. The resulting models can then be utilized to estimate properties of the state, such as the mean rate at which the state exits a safe set. The implementation of the proposed method is demonstrated by numerous examples, including the turbulence-induced random vibration of a flexible plate.     

Application of a hybrid method to the nonlinear dynamic analysis of a flexible rotor supported by a spherical gas-lubricated bearing system

This paper employs a hybrid numerical method combining the differential transformation method and the finite difference method to study the nonlinear dynamic behavior of a flexible rotor supported by a spherical gas-lubricated bearing system. The analytical results reveal a complex dynamic behavior comprising periodic, sub-harmonic, and quasi-periodic responses of the rotor center and the journal center. Furthermore, the results reveal the changes which take place in the dynamic behavior of the bearing system as the rotor mass and bearing number are increased. The current analytical results are found to be in good agreement with those from other numerical methods. Therefore, the proposed method provides an effective means of gaining insights into the nonlinear dynamics of spherical gas film rotor–bearing systems.