Inverse problems solution using spectral finite element methods

Abstract

Inverse problems are very challenging as these problems involve, finding the cause by analyzing the effects. In structural dynamics problems, the effects are normally measured in terms of dynamic responses in structures. These responses which are used to find the cause generally have partial data, embedded with measurement noise, and are truncated. Due to these problems, inverse problems are generally ill-posed in most cases as against forward problems. In this dissertation, we solve five different types of inverse problems involving high-frequency transient loads. All these problems are solved using the time-domain spectral element method (TSFEM) along with experimental or numerically simulated responses. The dissertation starts with the formulation of the forward problem, which is obtaining the responses from known input forces. The general formulation of TSFEM of composite Timoshenko beam is derived. The isotropic beam formulation is shown as a special case in this formulation. Five different inverse problems solved in the thesis are: 1. Force identification problem: A new algorithm is developed using a 1-D waveguide, involving an eight noded spectral finite element. The force identification is carried out, using a few measured responses on the structure, and using TSFEM we reconstruct the input force. This is followed by a portal frame example to demonstrate the wave reflection complexities. New procedures are developed to use various types of response data like displacement, velocity, acceleration, and strain to identify the force. 2. Material identification problem: A new procedure making use of the developed TSFEM, few responses, and nonlinear least square techniques are used to determine the material properties. Also, we show the case, in which we derive the material properties without force input consideration. 3. Crack location detection problem: A new procedure is developed using TSFEM and mechanics of crack. Three methods are described, in which the first method uses only responses and wave speeds to determine the location of the crack. In the second method, force reconstruction using the measured responses is carried out and this, in turn, is used to determine the location of the crack. The third method uses the residues of the actual force and the reconstructed forces using the healthy beam matrices and cracked beam responses. A new procedure to identify the crack location using a general force input pulse having many frequency components is also developed. 4. Material defect identification: Material defects like voids or density changes are identified using TSFEM. Location and magnitude of defect are identified using response computation and using the method of residues. 5. Porous location and identification in a composite material: TSFEM is used to construct a porous element and this is used along with a healthy beam to generate the responses. A force reconstruction algorithm is used to identify the location of the porous element. The Force residue method to identify the location of the defect is also demonstrated. Further, we make use of the material identification algorithm with a few modifications to evaluate all the parameters for the porous element.