Spline-based Rayleigh-Ritz methods for the approximation of the natural modes of vibration for flexible beams with tip bodies

Rayleigh-Ritz methods for the approximation of the natural modes for a class of vibration problems involving flexible beams with tip bodies using subspaces of piecewise polynomial spline functions are developed. An abstract operator theoretic formulation of the eigenvalue problem is derived and spectral properties investigated. The existing theory for spline-based Rayleigh-Ritz methods applied to elliptic differential operators and the approximation properties of interpolatory splines are useed to argue convergence and establish rates of convergence. An example and numerical results are discussed

Radially symmetric thin plate splines interpolating a circular contour map

Profiles of radially symmetric thin plate spline surfaces minimizing the Beppo Levi energy over a compact annulus $R_{1}\leq r\leq R_{2}$ have been studied by Rabut via reproducing kernel methods. Motivated by our recent construction of Beppo Levi polyspline surfaces, we focus here on minimizing the radial energy over the full semi-axis $0<r<\infty$. Using a $L$-spline approach, we find two types of minimizing profiles: one is the limit of Rabut's solution as $R_{1}\rightarrow0$ and $R_{2}\rightarrow\infty$ (identified as a `non-singular' $L$-spline), the other has a second-derivative singularity and matches an extra data value at $0$. For both profiles and $p\in\left[ 2,\infty\right]$, we establish the $L^{p}$-approximation order $3/2+1/p$ in the radial energy space. We also include numerical examples and obtain a novel representation of the minimizers in terms of dilates of a basis function.Comment: new figures and sub-sections; new Proposition 1 replacing old Corollary 1; shorter proof of Theorem 4; one new referenc

Optimal discrete-time LQR problems for parabolic systems with unbounded input: Approximation and convergence

An abstract approximation and convergence theory for the closed-loop solution of discrete-time linear-quadratic regulator problems for parabolic systems with unbounded input is developed. Under relatively mild stabilizability and detectability assumptions, functional analytic, operator techniques are used to demonstrate the norm convergence of Galerkin-based approximations to the optimal feedback control gains. The application of the general theory to a class of abstract boundary control systems is considered. Two examples, one involving the Neumann boundary control of a one-dimensional heat equation, and the other, the vibration control of a cantilevered viscoelastic beam via shear input at the free end, are discussed

Computational methods for the identification of spatially varying stiffness and damping in beams

A numerical approximation scheme for the estimation of functional parameters in Euler-Bernoulli models for the transverse vibration of flexible beams with tip bodies is developed. The method permits the identification of spatially varying flexural stiffness and Voigt-Kelvin viscoelastic damping coefficients which appear in the hybrid system of ordinary and partial differential equations and boundary conditions describing the dynamics of such structures. An inverse problem is formulated as a least squares fit to data subject to constraints in the form of a vector system of abstract first order evolution equations. Spline-based finite element approximations are used to finite dimensionalize the problem. Theoretical convergence results are given and numerical studies carried out on both conventional (serial) and vector computers are discussed

The INTERNODES method for the treatment of non-conforming multipatch geometries in Isogeometric Analysis

In this paper we apply the INTERNODES method to solve second order elliptic problems discretized by Isogeometric Analysis methods on non-conforming multiple patches in 2D and 3D geometries. INTERNODES is an interpolation-based method that, on each interface of the configuration, exploits two independent interpolation operators to enforce the continuity of the traces and of the normal derivatives. INTERNODES supports non-conformity on NURBS spaces as well as on geometries. We specify how to set up the interpolation matrices on non-conforming interfaces, how to enforce the continuity of the normal derivatives and we give special attention to implementation aspects. The numerical results show that INTERNODES exhibits optimal convergence rate with respect to the mesh size of the NURBS spaces an that it is robust with respect to jumping coefficients.Comment: Accepted for publication in Computer Methods in Applied Mechanics and Engineerin

Wavelet and its Applications

Ph.DDOCTOR OF PHILOSOPH

On Hilbert-Schmidt norm convergence of Galerkin approximation for operator Riccati equations

An abstract approximation framework for the solution of operator algebraic Riccati equations is developed. The approach taken is based on a formulation of the Riccati equation as an abstract nonlinear operator equation on the space of Hilbert-Schmidt operators. Hilbert-Schmidt norm convergence of solutions to generic finite dimensional Galerkin approximations to the Riccati equation to the solution of the original infinite dimensional problem is argued. The application of the general theory is illustrated via an operator Riccati equation arising in the linear-quadratic design of an optimal feedback control law for a 1-D heat/diffusion equation. Numerical results demonstrating the convergence of the associated Hilbert-Schmidt kernels are included

Inner product quadrature formulas

We investigate an approach to approximating the integral (0.1) ⨍[sub]R w(x)f(x)g(x)dx ≡ I (f;g), where R is a region in one-dimensional Euclidean space, and w a weight function. Since (0.1) may be regarded as a continuous bi-linear functional in f and g we approximate it by a discrete bi-linear functional, which we term an Inner Product Quadrature Formula (I.P.Q.F.). (0.2) Q(f;g) ≡ f̲ᵀAg̲, Where f̲ᵀ = (Sₒ(f), . . . , s[sub]m(f))ᵀ g̲ᵀ = (Tₒ(g), . . . , T[sub]n(g)) ᵀ A = (aᵢ[sub]j)ᵐi=o,ⁿj=0, And a[sub]i[sub]j are real numbers, ᵐi=0 ⁿj =0 |aᵢ [sub]j | > 0 The so-called elementary functionals {Sᵢ}ᵐi=0 and {T[sub]j}ⁿj=0 are two sets of linearly independent linear functionals, acting f and g respectively, defined over a certain subspace of functions to which f and g belong. The simplest example of these functionals is function evaluation at a given point. The matrix A is determined by requiring (0.2) to be exact for certain classes of functions f and g, say F ≡ {₀, . . . , ᵧ}, ≥0 G ≡ {₀, . . . , [sub] } ≥0 In Chapter 1 we introduce the concept of I.P.Q.F. in more detail and make some general comments about approaches available when examining numerical integration. After explaining in some detail why we feel I.P.Q.F. are a useful tool in §2.1, we proceed in the remainder of Chapter 2 to investigate various conditions which may be placed on ᵞ, [super] {Sᵢ}ᵐi=0 and {T[sub]j}ⁿj=0 in order to guarantee the existence of I.P.Q.F. exact when F and G. In particular we investigate the question of maximizing + . In the case where ᵢ and [sub]j are the standard monomials of degree i and j respectively, some results have already been published in B.I.T. (1977) p. 392-408. We investigate various choices of ᵢ and [sub]j : (a) {ᵢ}ᵐ⁺¹ I = 0 (i.e. = m+1) and {[sub]j}ᵐ[sub]j = 0 (i.e. = m) being Tchebychev sets (§2.7), (b) {ᵢ}²ᵐ⁺¹ I = 0 (i.e. = 2m+1) being a Tchebychev set and [super] contains only one function (i.e. = 0) (§2.6) (c) ᵢ ≡ ([sub]l)ⁱ, i=0,1, . . . and ᵢ = ᵢ, i= 0, 1, … (§2.8). In Chapter 3 we consider the question of compounding I.P.Q.F. both in the classical sense, and, briefly, by examining spline functions, regarding them as providing a link between an I.P.Q.F on one hand and a compounded I.P.Q.F. on the other. Various methods of theoretically estimating the errors involved are considered in Chapter M-. In the fifth Chapter we examine various ways in which the concept of I.P.Q.F. might (or might not) be extended. Finally, we make some brief comments about the possible applications of I.P.Q.F., and give a few examples