Sparse deconvolution for the large-scale ill-posed inverse problem of impact force reconstruction

Most previous regularization methods for solving the inverse problem of force reconstruction are to minimize the l₂-norm of the desired force. However, these traditional regularization methods such as Tikhonov regularization and truncated singular value decomposition, commonly fail to solve the large-scale ill-posed inverse problem in moderate computational cost. In this paper, taking into account the sparse characteristic of impact force, the idea of sparse deconvolution is first introduced to the field of impact force reconstruction and a general sparse deconvolution model of impact force is constructed. Second, a novel impact force reconstruction method based on the primal-dual interior point method (PDIPM) is proposed to solve such a large-scale sparse deconvolution model, where minimizing the l₂-norm is replaced by minimizing the l₁-norm. Meanwhile, the preconditioned conjugate gradient algorithm is used to compute the search direction of PDIPM with high computational efficiency. Finally, two experiments including the small-scale or medium-scale single impact force reconstruction and the relatively large-scale consecutive impact force reconstruction are conducted on a composite wind turbine blade and a shell structure to illustrate the advantage of PDIPM. Compared with Tikhonov regularization, PDIPM is more efficient, accurate and robust whether in the single impact force reconstruction or in the consecutive impact force reconstruction.