Arbitrary-Order Sensitivity Analysis in Wave Propagation Problems Using Hypercomplex Spectral Finite Element Method

Abstract

Many modern structural health monitoring (SHM) systems use piezoelectric transducers to induce and measure guided waves propagating in structures for structural damage detection. To increase the detection capabilities of SHM systems, gradient-based optimization of sensor placement is frequently necessary. However, available numerical differentiation methods for mechanical wave propagation problems suffer from truncation and subtraction errors and are difficult to extend to high-order sensitivities. This paper addresses these issues by introducing an approach to obtain highly accurate numerical sensitivities of arbitrary order in mechanical wave propagation problems. The hypercomplex time-domain spectral finite element method (ZSFEM) couples the hypercomplex Taylor series expansion method with the time-domain spectral finite element method. We show how ZSFEM can be implemented within the commercial finite element package ABAQUS/Explicit. For verification, we compared the numerical and analytical results of the displacement and its sensitivities with respect to mechanical parameters, geometry, and boundary conditions for a rod subjected to a sudden, distributed axial load. First- and second-order sensitivities were obtained with normalized root mean square deviations below 4×10−3. Mesh convergence analyses revealed that p-refinement offered better convergence rates than h-refinement for the outputs and their sensitivities. Also, the sensitivities obtained with ZSFEM were compared with finite differences showing higher accuracy and step-size independence (e.g., no iteration is needed to determine the step size that minimizes the error). For simplicity, ZSFEM was presented only for one-dimensional truss elements, but the method is general and can be applied to other elements.