Two-dimensional approximations of three-dimensional eigenvalue problems in plate theory

The eigenvalues and eigenfunctions corresponding to the three-dimensional equations for the linear elastic equilibrium of a clamped plate of thickness 2?, are shown to converge (in a specific sense) to the eigenvalues and eigenfunctions of the well-known two-dimensional biharmonic operator of plate theory, as ? approaches zero. In the process, it is found in particular that the displacements and stresses are indeed of the specific forms usually assumed a priori in the literature. It is also shown that the limit eigenvalues and eigenfunctions can be equivalently characterized as the leading terms in an asymptotic expansion of the three-dimensional solutions, in terms of powers of ?. The method presented here applies equally well to the stationary problem of linear plate theory, as shown elsewhere by P. Destuynder.     

热传导方程的时间最优脉冲控制问题的有限元逼近

时间最优控制问题是一类典型的最优控制问题, 受到研究者的广泛关注. 脉冲控制是一种在工程控制中被广泛应用的控制方式. 偏微分方程描述系统的最优控制问题的数值逼近的收敛性为数值求解方法的可行性提供了定性依据. 本文研究热传导方程的时间最优脉冲控制问题的有限元逼近的收敛性. 通过利用投影算子的特性和系统状态的误差估计, 证明了逼近问题的最优时间收敛到原问题的最优时间. 由此进一步利用原问题最优控制的bangbang性证明了最优控制的收敛性.     

Finite element patch approximations and alternating-direction methods

Numerical results using two modified Crank-Nicolson time discretizations are presented for finite element approximations to the nonlinear parabolic equation in 1R^d, c(x,u) ? (a_ij (x,u)u_,j)_,i + b_i (x,u)u_,i = f(x,t,u), where the summation convection on repeated indices is assumed. Both procedures use a local approximation to the coefficients which is based on patches of finite elements. With the first method, the coefficients are updated at each time step; however, only one matrix decomposition is required per problem. This method can exploit efficient direct methods for solving the resulting matrix problem. The second method is an alternating-direction variation which is valid for certain nonrectangular regions. With the alternating-direction method the resulting matrix problem can be solved as a series of one-dimensional problems, which results in a significant savings of time and storage over traditional techniques.     

Solving nonlinear eigenvalue problems by algorithmic differentiation

The eigenvalues of a matrixA(λ), the elements of which are complex functions of a complex variable λ, can be found with a zero finding method applied to the determinant function detA(λ). It is proposed to evaluate the derivatives of det (λ) used in the zero finding method by algorithmic differentiation. This leads to a simple and lucid algorithm. Program listings and numerical examples are given.     

On the correction of finite difference eigenvalue approximations for Sturm-Liouville problems

The use of algebraic eigenvalues to approximate the eigenvalues of Sturm-Liouville operators is known to be satisfactory only when approximations to the fundamental and the first few harmonics are required. In this paper, we show how the asymptotic error associated with related but simpler Sturm-Liouville operators can be used to correct certain classes of algebraic eigenvalues to yield uniformly valid approximations.     

A posteriori error estimates for nonconforming approximations of Steklov eigenvalue problems

This paper deals with a posteriori error estimators for the non conforming Crouzeix-Raviart finite element approximations of the Steklov eigenvalue problem. First, we define an error estimator of the residual type which can be computed locally from the approximate eigenpair and we prove the equivalence between this estimator and the broken energy norm of the error with constants independent of the corresponding eigenvalue. Next, we prove that edge residuals dominate the volumetric part of the residual and that the volumetric part of the residual terms dominate the normal component of the jumps of the discrete fluxes across interior edges. Finally, based on these results, we introduce two simpler equivalent error estimators. The analysis shows that these a posteriori error estimates are optimal up to higher order terms and that may be used for the design of adaptive algorithms.     

On the numerical solution of nonlinear eigenvalue problems

We consider the numerical solution of the nonlinear eigenvalue problemA(λ)x=0, where the matrixA(λ) is dependent on the eigenvalue parameter λ nonlinearly. Some new methods (the BDS methods) are presented, together with the analysis of the condition of the methods. Numerical examples comparing the methods are given.     

Nonlinear optimal control as quantum mechanical eigenvalue problems

A novel approach for approximating the nonlinear optimal feedback control of a system with a terminal cost is proposed. To lessen the difficulty due to nonlinearity, we try to treat the system in a framework of linear theories. For this, we assume a quantum mechanical linear wave associated with the system. Since the control system is constrained by state equations, we handle the system according to quantum mechanics of constrained dynamics. A Hamiltonian is represented as a linear operator acting on a function that describes behavior of waves. Subsequently, nonlinear feedback is calculated without any time integration in the backward direction. Using eigenvalues and eigenfunctions of the linear Hamiltonian operator, an optimal feedback law is given as a combination of analytic functions of time and state variables. We take as an example a system described by two scalar variables for state and control input. Simulation studies on the system by the eigenvalue analysis show that the proposed method reduces calculation time to nearly a tenth that of a numerical calculation of a Hamilton-Jacobi equation by a finite difference method.     

Finite element approximation of nonlinear transient magnetic problems involving periodic potential drop excitations

This paper deals with the computation of nonlinear 2D transient magnetic fields when the data concerning the electric current sources involve potential drop excitations. In the first part, a mathematical model is stated, which is solved by an implicit time discretization scheme combined with a finite element method for space approximation. The second part focuses on developing a numerical method to compute periodic solutions by determining a suitable initial current which avoids large simulations to reach the steady state. This numerical method leads to solve a nonlinear system of equations which requires to approximate several nonlinear and linear magnetostatic problems. The proposed methods are first validated with an axisymmetric example and sinusoidal source having an analytical solution. Then, we show the saving in computational effort that this methodology offers to approximate practical problems specially with pulse-width modulation (PWM) voltage supply.     

Finite element analysis of nonlinear magnetic fields

The basic equations of electromagnetism are written in the form of a quasi-harmonic equation. The application of the weighted residual process leads to a non-linear system of algebraic equations which is solved by a full Newton–Raphson procedure. The iteration scheme is developed and applied to numerical examples.     

Finite element algorithms for contact problems

Summary The numerical treatment of contact problems involves the formulation of the geometry, the statement of interface laws, the variational formulation and the development of algorithms. In this paper we give an overview with regard to the different topics which are involved when contact problems have to be simulated. To be most general we will derive a geometrical model for contact which is valid for large deformations. Furthermore interface laws will be discussed for the normal and tangential stress components in the contact area. Different variational formulations can be applied to treat the variational inequalities due to contact. Several of these different techniques will be presented. Furthermore the discretization of a contact problem in time and space is of great importance and has to be chosen with regard to the nature of the contact problem. Thus the standard discretization schemes will be discussed as well as techiques to search for contact in case of large deformations.     

Finite element approximation of a nonlinear diffusion problem

This paper deals with the finite element approximation of the nonlinear diffusion problem: −div (\vbgrad u\vb^p−2 grad u) = f. Glowinski and Marrocco[3] have been shown that the rate of convergence decreases as p increases. In this paper we show that the rate of convergence is optimal and independent of p. This theoretical result agrees with the numerical experiments reported in the last section.     

Finite central difference/finite element approximations for parabolic integro-differential equations

We study the numerical solution of an initial-boundary value problem for parabolic integro-differential equation with a weakly singular kernel. The main purpose of this paper is to construct and analyze stable and high order scheme to efficiently solve the integro-differential equation. The equation is discretized in time by the finite central difference and in space by the finite element method. We prove that the full discretization is unconditionally stable and the numerical solution converges to the exact one with order O(Δt ² + h ^l ). A numerical example demonstrates the theoretical results.     

Finite element approximation of some indefinite elliptic problems

We consider the finite element approximation of some indefinite Neumann problems in a domain of IR^N. From the Fredholm Alternative this kind of problem admits a solution if and only if the right hand term has zero mean value with respect to a measure whose density m is the solution of a homogeneous adjoint problem. The first step consists in the construction of piecewise linear finite element approximations m_h of m, showing their optimal rate of convergence both in energy and L^p norms. The functions m_h are then shown to be crucial in testing admissible data for the Neumann problem and also in its numerical resolution (actually, the standard Galerkin approximation may not be solvable without suitable perturbations of the data).     

Finite element program for solution of geotechnical problems

A program, named RODSIM, based on the finite element displacement method has been developed, tested and applied to the solution of major geotechnical problems. It is specially suitable to analyze deformations due to sequential excavation of soil supported by diaphragm walls with anchors or struts in which case it produces the bending moments, shear forces, axial forces, horizontal and vertical pressures along the wall, after deformation. All element matrices are evaluated by exact integration considering the Young's modulus of the orthotropic, cross-anisotropic or isotropic material varying linearly within the six node triangular element. The node numbering system may be optimized automatically.     

Finite element methods for elliptic problems with singularities

It has been shown how singular isoparametric transformations defined on a triangular mesh can be used to provide finite element approximations that behave locally as r¹/m. The purpose of the present note is to show how this form of local approximation can be extended to quadrilateral meshes and how any approximations that behave as r^P/q can be obtained for arbitrary p and q.Examples are given of approximate solutions of elliptic partial differential equations using this form of finite element approximation. It is shown that accurate approximations to the coefficients of leading terms in the solution near the singularity are obtained as well as a good approximation to the solution itself.Finally, it is shown how curved elements with the desired singular behaviour can be constructed by means of a generalisation of the basic approach. Approximations based on transformations involving functions other than polynomials are also introduced.     

The adaptive finite element method based on multi-scale discretizations for eigenvalue problems

In this paper, adaptive finite element methods for differential operator eigenvalue problems are discussed. For multi-scale discretization schemes based on Rayleigh quotient iteration (see Scheme 3 in [Y. Yang, H. Bi, A two-grid discretization scheme based on shifted-inverse power method, SIAM J. Numer. Anal. 49 (2011) 1602–1624]), a reliable and efficient a posteriori error indicator is given, in addition, a new adaptive algorithm based on the multi-scale discretizations is proposed, and we apply the algorithm to the Schrödinger equation for hydrogen atoms. The algorithm is performed under the package of Chen, and satisfactory numerical results are obtained.     

Finite dimensional approximations for a class of infinite dimensional time optimal control problems

ABSTRACT

In this work, we study the numerical approximation of the solutions of a class of abstract parabolic time optimal control problems with unbounded control operator. Our main results assert that, provided that the target is a closed ball centered at the origin and of positive radius, the optimal time and the optimal controls of the approximate time optimal problems converge (in appropriate norms) to the optimal time and to the optimal controls of the original problem. In order to prove our main theorem, we provide a nonsmooth data error estimate for abstract parabolic systems.