On the use of physical spline finite element method for acoustic scattering

This paper presents the comparison of physical spline finite element method (PSFEM), in which differential equations are incorporated into interpolations of basic elements, with least-squares finite element method (LSFEM) and mixed Galerkin finite element method (MGFEM) on the numerical solution of one dimensional Helmholtz equation applied to an acoustic scattering problem. Firstly, all three methods are explained in detail and then it is shown that PSFEM reaches higher precision in a shorter time with fewer nodes than the other methods. It is also observed that this method is well suited for high frequency acoustic problems. Consequently, the results of PSFEM point out better efficiency in terms of number of unknowns and accuracy level.     

A TRUST REGION METHOD WITH A CONIC MODEL FOR NONLINEARLY CONSTRAINED OPTIMIZATION

Trust region methods are powerful and effective optimization methods.The conic model method is a new type of method with more information available at each iteration than standard quadratic-based methods.The advantages of the above two methods can be combined to form a more powerful method for constrained optimization.The trust region subproblem of our method is to minimize a conic function subject to the linearized constraints and trust region bound.At the same time,the new algorithm still possesses robust global properties.The global convergence of the new algorithm under standard conditions is established.     

A superlinearly convergent norm-relaxed SQP method of strongly sub-feasible directions for constrained optimization without strict complementarity

In this paper, a kind of optimization problems with nonlinear inequality constraints is discussed. Combined the ideas of norm-relaxed SQP method and strongly sub-feasible direction method as well as a pivoting operation, a new fast algorithm with arbitrary initial point for the discussed problem is presented. At each iteration of the algorithm, an improved direction is obtained by solving only one direction finding subproblem which possesses small scale and always has an optimal solution, and to avoid the Maratos effect, another correction direction is yielded by a simple explicit formula. Since the line search technique can automatically combine the initialization and optimization processes, after finite iterations, the iteration points always get into the feasible set. The proposed algorithm is proved to be globally convergent and superlinearly convergent under mild conditions without the strict complementarity. Finally, some numerical tests are reported.     

A two‐grid stabilized mixed finite element method for Darcy‐Forchheimer model

A two‐grid stabilized mixed finite element method based on pressure projection stabilization is proposed for the two‐dimensional Darcy‐Forchheimer model. We use the derivative of a smooth function, , to approximate the derivative of in constructing the two‐grid algorithm. The two‐grid method consists of solving a small nonlinear system on the coarse mesh and then solving a linear system on the fine mesh. There are a substantial reduction in computational cost. We prove the existence and uniqueness of solution of the discrete schemes on the coarse grid and the fine grid and obtain error estimates for the two‐grid algorithm. Finally, some numerical experiments are carried out to verify the accuracy and efficiency of the method.     

A new stabilized finite element method for optimal control for a Ladyzhenskaya model for unsteady flows

This article considers the time‐dependent optimal control problem of tracking the velocity for the viscous incompressible flows which is governed by a Ladyzhenskaya equations with distributed control. The existence of the optimal solution is shown and the first‐order optimality condition is established. The semidiscrete‐in‐time approximation of the optimal control problem is also given. The spatial discretization of the optimal control problem is accomplished by using a new stabilized finite element method which does not need a stabilization parameter or calculation of high order derivatives. Finally a gradient algorithm for the fully discrete optimal control problem is effectively proposed and implemented with some numerical examples. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 263–287, 2012     

The finite volume method based on the Crouzeix–Raviart element for a fracture model

In this paper, we present and analyze a finite volume method based on the Crouzeix–Raviart element for the coupled fracture model, where the fluid flow is governed by Darcy's law in the one‐dimensional fracture and two‐dimensional surrounding matrix. In the numerical scheme, the pressure in the matrix and fracture is respectively approximated by the Crouzeix–Raviart elements and piecewise constant functions, and then the velocity is calculated by piecewise constant functions element by element. The existence and uniqueness of the numerical solution are discussed, and optimal order error estimates for both the pressure p and the velocity u are proved on general triangulations. We finally carry out numerical experiments, and results confirm our theoretical analysis.     

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      

A stabilized finite element method based on local polynomial pressure projection for the stationary Navier–Stokes equations

This article considers a stabilized finite element approximation for the branch of nonsingular solutions of the stationary Navier–Stokes equations based on local polynomial pressure projection by using the lowest equal-order elements. The proposed stabilized method has a number of attractive computational properties. Firstly, it is free from stabilization parameters. Secondly, it only requires the simple and efficient calculation of Gauss integral residual terms. Thirdly, it can be implemented at the element level. The optimal error estimate is obtained by the standard finite element technique. Finally, comparison with other methods, through a series of numerical experiments, shows that this method has better stability and accuracy.     

An alternative alpha finite element method with discrete shear gap technique for analysis of laminated composite plates

This paper presents an alternative alpha finite element method using triangular meshes (AαFEM) for static, free vibration and buckling analyses of laminated composite plates. In the AαFEM, an assumed strain field is carefully constructed by combining compatible strains and additional strains with an adjustable parameter α which can produce an effectively softer stiffness formulation compared to the linear triangular element. The stiffness matrices are obtained based on the strain smoothing technique over the smoothing domains and the constant strains on triangular sub-domains associated with the nodes of the elements. The discrete shear gap (DSG) method is incorporated into the AαFEM to eliminate transverse shear locking and an improved triangular element termed as AαDSG3 is proposed. Several numerical examples are then given to demonstrate the effectiveness of the AαDSG3.     

A successive quadratic programming method for a class of constrained nonsmooth optimization problems

In this paper we present an algorithm for solving nonlinear programming problems where the objective function contains a possibly nonsmooth convex term. The algorithm successively solves direction finding subproblems which are quadratic programming problems constructed by exploiting the special feature of the objective function. An exact penalty function is used to determine a step-size, once a search direction thus obtained is judged to yield a sufficient reduction in the penalty function value. The penalty parameter is adjusted to a suitable value automatically. Under appropriate assumptions, the algorithm is shown to produce an approximate optimal solution to the problem with any desirable accuracy in a finite number of iterations.     

Analysis of a fully discrete finite element method for the phase field model and approximation of its sharp interface limits

We propose and analyze a fully discrete finite element scheme for the phase field model describing the solidification process in materials science. The primary goal of this paper is to establish some useful a priori error estimates for the proposed numerical method, in particular, by focusing on the dependence of the error bounds on the parameter , known as the measure of the interface thickness. Optimal order error bounds are shown for the fully discrete scheme under some reasonable constraints on the mesh size and the time step size . In particular, it is shown that all error bounds depend on only in some lower polynomial order for small . The cruxes of the analysis are to establish stability estimates for the discrete solutions, to use a spectrum estimate result of Chen, and to establish a discrete counterpart of it for a linearized phase field operator to handle the nonlinear effect. Finally, as a nontrivial byproduct, the error estimates are used to establish convergence of the solution of the fully discrete scheme to solutions of the sharp interface limits of the phase field model under different scaling in its coefficients. The sharp interface limits include the classical Stefan problem, the generalized Stefan problems with surface tension and surface kinetics, the motion by mean curvature flow, and the Hele-Shaw model.

     

A three‐dimensional finite element model for the control of certain non‐linear bioreactors

This paper deals with the dynamics of non‐linear distributed parameter fixed‐bed bioreactors. The model consists of a pair of non‐linear partial differential (evolution) equations. The true spatially three‐dimensional situation is considered instead of the usual one‐dimensional approximation. This enables one to take into account the effects of flow profiles and the true location of the measurement transducer. The (output) evolution of the corresponding open‐loop control system is simulated. Furthermore, the associated closed‐loop system with respect to the relevant output function is considered. Especially, the asymptotic output tracking is found to be successful by applying the usual process based on the state feedback linearization. Copyright © 2000 John Wiley & Sons, Ltd.     

非饱和土壤水流问题基于POD 方法的降阶有限体积元格式及外推算法实现

利用特征投影分解(proper orthogonal decomposition, 简记为POD) 方法对非饱和土壤水流问题的经典有限体积元格式做降阶处理, 建立一种具有足够高精度维数较低的降阶有限体积元格式, 并给出这种降阶有限体积元解的误差估计和外推算法的实现, 最后用数值例子说明数值结果与理论结果是相吻合的. 进一步表明了基于POD 方法的降阶有限体积元格式对求解非饱和土壤水流问题数值解是可靠和有效的.     

A trust region method based on interior point techniques for nonlinear programming

An algorithm for minimizing a nonlinear function subject to nonlinear inequality constraints is described. It applies sequential quadratic programming techniques to a sequence of barrier problems, and uses trust regions to ensure the robustness of the iteration and to allow the direct use of second order derivatives. This framework permits primal and primal-dual steps, but the paper focuses on the primal version of the new algorithm. An analysis of the convergence properties of this method is presented. Received: May 1996 / Accepted: August 18, 2000?Published online October 18, 2000