A novel inverse super element method for structural shape and stress sensing

The inverse finite element method (iFEM) has been widely adopted in shape sensing, as it enables full-field reconstruction of structural displacements and stresses by minimizing a weighted least–squares functional of the discrepancy between theoretical and measured strains. However, when the measured strain data are sparse, the equivalent equilibrium equations derived from iFEM become severely ill–conditioned, which tends to underestimate the reconstruction results. To overcome this limitation, this paper presents a novel inverse super–element method (iSEM). Unlike iFEM, where the error functional is formulated at the element level, iSEM constructs the strain error functional directly over the entire structural domain. Through static condensation of degrees of freedom, the equivalent equilibrium equations are reduced to a set of user–selected Primary degrees of freedoms. Together with an energy–based regularization (EBR) strategy grounded in the principle of minimum energy, the proposed method markedly alleviates the ill–posedness of the equilibrium equations. Notably, iSEM does not rely on any concept of master and slave elements; so strain measurements can be taken with greater flexibility in sensor placement. The performance of iSEM is validated through both finite–element simulations and physical model experiments. The results indicate that under sparse uniaxial strain measurements, iSEM reconstructs the structural displacement and strain fields with high fidelity, achieving significantly lower relative errors than iFEM while exhibiting reduced sensitivity to the EBR coefficient. Overall, iSEM provides a novel theoretical framework for high–accuracy shape and stress sensing in scenarios involving sparse strain measurements.