Identification of Flexural Rigidity in a Kirchhoff Plates Model Using a Convex Objective and Continuous Newton Method

This work provides a detailed theoretical and numerical study of the inverse problem of identifying flexural rigidity in Kirchhoff plate models. From a mathematical standpoint, this inverse problem requires estimating a variable coefficient in a fourth-order boundary value problem. This inverse problem and related estimation problems associated with general plates and shell models have been investigated by numerous researchers through an optimization framework using the output least-squares (OLSs) formulation. OLS yields a nonconvex framework and hence it is suitable for investigating only the local behavior of the solution. In this work, we propose a new convex framework for the inverse problem of identifying a variable parameter in a fourth-order inverse problem. Existence results, optimality conditions, and discretization issues are discussed in detail. The discrete inverse problem is solved by using a continuous Newton method. Numerical results show the feasibility of the proposed framework.

CONTINUOUS NEWTON METHOD FOR NONLINEAR PARTIAL DIFFERENTIAL EQUATION

Newton method is a widely used iteration method in solving nonlinear algebraic equations. In this method, a linear algebraic equations need to be solved in every step. The coefficient matrix of the algebraic equations is so-called Jacobian matrix, which needs to be determined at every step. For a complex non-linear system, usually no explicit form of Jacobian matrix can be found. Several methods are introduced to obtain an approximated matrix, which are classified as Jacobian-free method. The finite difference method is used to approximate the derivatives in Jacobian matrix, and a small parameter is needed in this process. Some problems may arise because of the interaction of this parameter and round-off errors. In the present work, we show that this kind of Newton method may encounter difficulties in solving non-linear partial differential equation (PDE) on fine mesh. To avoid this problem, the continuous Newton method is presented, which is a modification of classical Newton method for non-linear PDE.