Unbounded Term Research Articles

NON-SINGULAR SOMIGLIANA STRESS IDENTITIES IN ELASTICITY

The paper presents two fully equivalent and regular forms of the hypersingular Somigliana stress identity in elasticity that are appropriate for problems in which the displacement field (and resulting stresses) is C1,α continuous. Each form is found as the result of a single decomposition process on the kernels of the Somigliana stress identity in three dimensions. The results show that the use of a simple stress state for regularization arises in a direct manner from the Somigliana stress identity, just as the use of a constant displacement state regularization arose naturally for the Somigliana displacement identity. The results also show that the same construction leads naturally to a finite part form of the same identity. While various indirect constructions of the equivalents to these findings are published, none of the earlier forms address the fundamental issue of the usual discontinuities of boundary data in the hypersingular Somigliana stress identity that arise at corners and edges. These new findings specifically focus on the corner problem and establish that the previous requirements for continuity on the densities in the hypersingular Somigliana stress identity are replaced by a sole requirement on displacement field continuity. The resulting regularized and finite part forms of the Somigliana stress identity leads to a regularized form of the stress boundary integral equation (stress-BIE). The regularized stress-BIE is shown to properly allow piecewise discontinuity of the boundary data subject only to C1,α continuity of the underlying displacement field. The importance of the findings is in their application to boundary element modeling of the hypersingular problem. The piecewise discontinuity derivation for corners is found to provide a rigorous and non-singular basis for collocation of the discontinuous boundary data for both the regularized and finite part forms of the stress-BIE. The boundary stress solution for both forms is found to be an average of the computed stresses at collocation points at the vertices of boundary element meshes. Collocation at these points is shown to be without any unbounded terms in the formulation thereby eliminating the use of non-conforming elements for the hypersingular equations. The analytical findings in this paper confirm the correct use of both regularized and finite part forms of the stress-BIE that have been the basis of boundary element analysis previously published by the first author of the current paper.

An H/sup ∞//MinMax periodic control in a two-dimensional structural acoustic model with piezoceramic actuators

A feedback control computational methodology for reducing acoustic sound pressure levels in a two-dimensional cavity with a flexible boundary (a beam) is investigated. The control is implemented in this model through voltages to piezoceramic patches on the beam which are excited in a manner (out-of-phase) so as to produce pure bending moments. The incorporation of the output feedback control in this system leads to a problem with unbounded input and output terms. By writing the resulting system as an abstract Cauchy equation, the problem of reducing interior pressure levels can be posed in the context of an H/sub /spl infin///MinMax time-domain state-space formulation. A summary of extensive computational efforts comparing output feedback to full state feedback and investigating the effect of the number and location of sensors on performance of the control is presented.

Well posedness for damped second-order systems with unbounded input operators

We consider damped second order in time systems such as those arising in structures with piezoceramic actuators and sensors. These systems are naturally formulated as abstract second order systems with unbounded nonhomogeneous term. Existence, uniqueness and continuous dependence of solutions in a weak or variational setting are given. A semigroup formulation is presented and conditions under which the variational solutions and semigroup solutions are the same are discussed.

Hypersingular formulation for boundary stress evaluation

Instead of using shape function derivatives and Hooke's law, the full stress tensor is evaluated at boundary points by direct application of boundary integral identities for the displacement derivatives. It is first shown that integral equations with singular or hypersingular kernels do not give rise to unbounded terms, even when the source point is on the boundary. A general method for performing the integration is also described. Numerical results are quite interesting, since the stress components evaluated through the hypersingular integral equation method show very good accuracy even on coarse meshes.

The Vector Harmonic Transform Method for Solving Partial Differential Equations in Spherical Geometry

Abstract The development of computational methods for solving partial differential equations in spherical geometry is complicated by problems induced by the spherical coordinate system itself. Even though the solution is smooth in Cartesian coordinates, in spherical coordinates the components of vector fields such as the wind are multivalued at the poles and the differential equations have unbounded terms. For example, the total derivative of the velocity is unbounded at the poles. Here, the vector harmonic transform method for the effective treatment of these problems is presented. Vector fields such as the wind are expanded in terms of vector harmonics, and scalar fields such as pressure and temperature are expanded in terms of scalar harmonies. Unbounded terms in the differential equation are grouped into bounded expressions that are evaluated by their formal application to the spectral expansions. The method can be applied to any differential equation without introducing scalar-dependent variables, su...

Approximation Methods for Control of Structural Acoustics Models with Piezoceramic Actuators

The active control of acoustic pressure in a 2-D cavity with a flexible boundary (a beam) is considered. Specifically, this control is implemented via piezoceramic patches on the beam which produce pure bending moments. The incorporation of the feedback control in this manner leads to a system with an unbounded input term. Approximation methods in the context of an LQR state space formulation are discussed and numerical results demonstrating the effectiveness of this approach in computing feedback controls for noise reduction are presented.

Viscosity solutions for a class of Hamilton-Jacobi equations in Hilbert spaces

We study Hamilton-Jacobi equations with an unbounded term in Hilbert spaces. We introduce a new variant of the notion of viscosity solution for a class of Hamilton-Jacobi equations, and obtain comparison and existence results for viscosity solutions. We also examine the usefulness of the notion of viscosity solution in optimal control.

Asymptotic perturbation theory applied to diffractive atomic scattering

We use the multiple-time-scale perturbation technique to treat the long interaction time limit of atomic scattering from a standing light wave, i.e., the atomic Kapitza–Dirac effect. The use of the method is restricted to an exactly resonant interaction and an oblique angle of incidence. We develop the theory in some detail in view of its general applicability to a large class of physical phenomena. The accuracy of the ensuing analytical expressions is tested against exact numerical computations of the effect, and the allowed ranges of the parameters are given with reference to the actual atomic experiments. For moderately large values of the parameters that characterize the field strength and the photon recoil, the method is found to give excellent approximations for the long-time behavior, whereas straightforward perturbation expansions diverge because of unbounded secular terms.

Model reference control of nonlinear plants with uncertain and unbounded terms

In the paper, a few new results are presented relating to model reference control (MRC) and model reference adaptive control (MRAC) for nonlinear uncertain plants of a very general nature. No a priori knowledge about the structure of uncertainties (plant terms and coefficients) is assumed, nor is any constant bound on their values. Such plant models result from many applications. In particular, if uncertainties represent some unmodelled dynamic effects, they will depend on the system states and so be unbounded. The Lyapunov type of stability is considered. A new MRAC law has been developed, which provides stable tracking to within a certain comput able tolerance. The paper includes a few examples of the application of the developed adaptive con troller to nonlinear plants with potentially unbounded terms. The tracking control of robotic manipulators has been especially elaborated.

Viscosity solutions of Hamilton-Jacobi equations in infinite dimensions. IV. Hamiltonians with unbounded linear terms

In this work we show how to interpret (S) and (E) in the viscosity sense when the solutions may be expected to be weakly sequentially continuous ― in particular, we introduce notions which allow us to handle the terms and for which we can obtain existence and uniqueness results in a variety of circumstances

Some results on non-linear optimal control problems and Hamilton-Jacobi equations in infinite dimensions

The optimal control of a distributed parameter system is connected to the solution of the corresponding Hamilton-Jacobi equation. This is a first-order equation in infinite dimensions with discontinuous coefficients. We study the Hamilton-Jacobi equation of a system governed either by a “semilinear” or by a “monotone” dynamics, replacing the unbounded terms of this equation by their Yosida approximations. We prove convergence of the approximate solutions to the value function of the original problem, by using the uniqueness result for viscosity solutions. Examples and applications are included.

Asymptotic expansions for the variance of stopping times in nonlinear renewal theory

We treat the problem of finding asymptotic expansions for the variance of stopping times for Wiener processes with positive drift (continuous time case) as well as sums of i.i.d. random variables with positive mean (discrete time case). Carrying over the setting of nonlinear renewal theory to Wiener processes, we obtain an asymptotic expansion up to vanishing terms in the continuous time case. Applying the same methods to sums of i.i.d. random variables, we also provide an expansion in the discrete time case up to terms of order o( b 1/2) where the leading term is of order O( b), as b → ∞. The possibly unbounded term is the covariance of nonlinear excess and stopping time.

The Differentiability with Respect to a Parameter of the Solution of a Linear Abstract Cauchy Problem

The differentiability with repect to a parameter of the solution of a linear inhomogeneous abstract Cauchy problem is considered by employing the theory of strongly continuous semigroups. Criteria for differentiability are developed which allow the parameter to appear in unbounded terms of the generator. Sufficient conditions are given for the solvability of an inhomogeneous Cauchy problem with zero initial condition by the variation of constants formula. These conditions are used to show that the derivatives with respect to a parameter satisfy certain sensitivity equations in time. The results are applied to the differentiability of the solution of a linear delay differential equation with respect to its delays.

The Approximation of Vector Functions and Their Derivatives on the Sphere

A vector basis is defined for vector functions that are given on the sphere. It is shown that the basis is orthogonal, that it is complete and that the convergence properties of the spectral representation are determined by the smoothness of the vector function in Cartesian coordinates. The method for analyzing a vector function in terms of this basis uses the methods for analyzing scalar functions in terms of surface harmonics. A vector function that is smooth in Cartesian coordinates and nonzero at the pole will be discontinuous in spherical coordinates. This is a result of the discontinuous spherical coordinate system rather than the vector function itself. This in turn induces singularities in certain surface derivatives on the sphere, even though the function is bounded and differentiable in Cartesian coordinates. Many of the individual terms in a partial differential equation on the sphere are unbounded at the poles; however, the cancellation between these terms is sufficient to yield a bounded result. This does not create a problem from a theoretical point of view; however, it becomes a significant problem from a computational point of view. Indeed, the finer the grid the more pronounced the difficulty becomes near the pole where the computation fails completely. In this paper, we show how to identify the unbounded terms and group them into bounded expressions. We also show how to compute these bounded expressions in a stable way that avoids determining their individual unbounded terms when the function is given in terms of the vector basis.

Asymptotic stability for some nonlinear evolution equations of second order with unbounded dissipative terms

Let H be a real Hilbert space and let V, W be real Banach spaces with V C WC H. We assume that V is dense in W and H, and that the natural injections from V into Wand from W into H are both continuous. We identify H with its dual H* (i.e., V C WC NC W* C V*). Pairing between

Semigroups on Product Spaces with Application to Initial Value Problems with Non-Local Boundary Conditions

Interest has recently been shown in conditions that a linear operator generates a C0 semigroup on a product space. Such semigroups arise in the study of initial value problems with nonlocal boundary conditions as well as linear functional differential equations. Introduction of semigroups on product spaces also permits study of a wide class of distributed systems with boundary input by means of evolution equations with bounded input operators. New conditions are given which permit unbounded coupling terms in the infinitesimal generator. Their application is examined in a number of examples.

De Giorgi techniques applied to Hamilton–Jacobi equations with unbounded right-hand side

In this article we obtain Holder estimates for solutions to second-order Hamilton-Jacobi equations with super-quadratic growth in the gradient and unbounded source term. The estimates are uniform with respect to the smallness of the diffusion and the smoothness of the Hamiltonian. Our work is in the spirit of a result by P. Cardaliaguet and L. Silvestre [5]. We utilize De Giorgi's method, which was introduced to this class of equations in [6].