Impact Force Localization and Reconstruction via ADMM-based Sparse Regularization Method

Abstract In practice, simultaneous impact localization and time history reconstruction can hardly be achieved, due to the ill-posed and under-determined problems induced by the constrained and harsh measuring conditions. Although $$\ell_{1}$$ ℓ 1 regularization can be used to obtain sparse solutions, it tends to underestimate solution amplitudes as a biased estimator. To address this issue, a novel impact force identification method with $$\ell_{p}$$ ℓ p regularization is proposed in this paper, using the alternating direction method of multipliers (ADMM). By decomposing the complex primal problem into sub-problems solvable in parallel via proximal operators, ADMM can address the challenge effectively. To mitigate the sensitivity to regularization parameters, an adaptive regularization parameter is derived based on the K-sparsity strategy. Then, an ADMM-based sparse regularization method is developed, which is capable of handling $$\ell_{p}$$ ℓ p regularization with arbitrary p values using adaptively-updated parameters. The effectiveness and performance of the proposed method are validated on an aircraft skin-like composite structure. Additionally, an investigation into the optimal p value for achieving high-accuracy solutions via $$\ell_{p}$$ ℓ p regularization is conducted. It turns out that $$\ell_{0.6}$$ ℓ 0.6 regularization consistently yields sparser and more accurate solutions for impact force identification compared to the classic $$\ell_{1}$$ ℓ 1 regularization method. The impact force identification method proposed in this paper can simultaneously reconstruct impact time history with high accuracy and accurately localize the impact using an under-determined sensor configuration.