Kenji Shirota

A variational approach for an inverse dynamical problem for composite beams

This paper deals with a problem of nondestructive testing for a composite system formed by the connection of a steel beam and a reinforced concrete beam. The small vibrations of the composite beam are described by a differential system where a coupling takes place between longitudinal and bending motions. The motion is governed in space by two second order and two fourth order differential operators, which are coupled in the lower order terms by the shearing, k, and axial, µ, stiffness coefficients of the connection. The coefficientsk and µ define the mechanical model of the connection between the steel beam and the concrete beam and contain direct information on the integrity of the system. In this paper we study the inverse problem of determining k and µ by mixed data. The inverse problem is transformed to a variational problem for a cost function which includes boundary measurements of Neumann data and also some interior measurements. By computing the Gateaux derivatives of the functional, an algorithm based on the projected gradient method is proposed for identifying the unknown coefficients. The results of some numerical simulations on real steel-concrete beams are presented and discussed.

A non-destructive method for damage detection in steel-concrete structures based on finite eigendata

This article proposes a non-destructive method for damage detection in steel-concrete beams based on finite spectral data associated with a given set of boundary conditions. The inverse problem consists of determining two stiffness coefficients of the connection between the steel beam and the concrete beam. The inverse problem is transformed to a variational problem for a cost function which includes eigenvalue data and transversal displacements of eigenfunctions. A projected gradient method which uses the analytical expressions of the first partial derivatives of the eigenvalues and eigenfunctions is proposed for identifying the unknown coefficients. The results of an extended series of numerical simulations on real steel-concrete beams are presented and discussed.

Combination of the finite and the boundary element methods for an external boundary value problem of the Poisson equation

Abstract A numerical algorithm for an external Dirichlet problem of the Poisson equation is considered. The domain O extending to infinity is divided into a bounded subdomain O0 and the unbounded subdomain O1. The finite and the boundary element methods are applied to the boundary value problems in the bounded and the unbounded subdomains, respectively. An iterative scheme using the Dirichlet‐Neumann map on the interface ?O1 is presented. The convergence of the scheme is mathematically guaranteed. A simple numerical example shows the effectiveness of our scheme.

Numerical method for an inverse dynamical problem for composite beams

In this paper we present a numerical method for an inverse problem of nondestructive testing for a composite system formed by the connection of a steel beam and a reinforced concrete beam. The small vibrations of the composite beam are governed in space by two second order and two fourth order differential operators, which are coupled in the lower order terms by two coefficients which express the shearing and axial stiffness of the connection. Our inverse problem is to determine these stiffness coefficients by using Neumann type boundary data measured at one end of the beam and transversal displacements given in an interior portion of the beam axis. We recast the inverse problem as a constrained variational issue and an iterated projected gradient method is proposed for the numerical solution of the minimizing problem. Suitable clip-off and mollifier operators are introduced in order to describe the constrained conditions. The effectiveness of method and the sensitivity of the results to errors in the measured data are tested on the basis of an extensive series of numerical experiments.

Adjoint Method for the Problem of Coefficient Identification in Linear Elastic wave Equation

In this study, we consider the numerical method for the problem of coefficient identification in linear elastic wave equation in two dimensions in space. The elastic body is assumed to be a state of plane strain. Our problem is to determine unknown Young’s modulus from the knowledge of the plural sets of surface displacements and tractions. We suppose that the density and Poisson’s ration are known. To identify unknown Young’s modulus numerically, we adopt the adjoint method. We introduce an object functional, then our problem is recast to minimizing problem with constraint. To find the minimum of the functional, the algorithm based on the projected gradient method is proposed. By numerical experiments, we confirm the effectiveness of our algorithm.

Coefficient Identification of the Wave Equation Using the Alternating Directions Method

The purpose of this paper is to present an algorithm for the numerical resolution of the problem of coefficient identification for the wave equation: Determine the unknown coefficient function by means of measurement of Dirichlet and Neumann boundary values. We introduce a functional to be minimized and recast our problem to a variational problem. To find the minimizing function, we use the alternating directions method. The numerical experiments support that further investigation is required to confirm the effectiveness of our algorithm.

A Variational Approach for Finding the Source Function of the Wave Equation

ABSTRACT The purpose of this paper is to present an algorithm for numerical identification of source function. The algorithm is based on the direct variational method, and a functional is minimized by the gradient method. A direct method of solution using the Newmark β-method is presented. We confirm the effectiveness of our algorithm by numerical experiments.

Variational approach for the problem of coefficient identification of the wave equation

Publisher Summary This chapter provides an overview of a new numerical method for the inverse problem of coefficient identification of the wave equation. It also determines the unknown coefficient function with the knowledge of the Dirichlet boundary value and the Neumann boundary value. In order to find the minimizing function, projected gradient method is used. The first variation of the functional is given by solving two initial-boundary value problems. Furthermore, an algorithm for the numerical resolution of inverse problem is also examined. The chapter introduces an object functional to minimize the problem and it is recast to a variational problem.