A Direct Solution to the Self-Sensing Inverse Problem Via the Primal-Dual Interior Point Method

Abstract Materials that are self-sensing via the piezoresistive effect have been widely explored for embedded sensing in aerospace structural composites, pressure sensing in touch pads, diagnostics in biomedical implant technology, and many other applications. In this approach, changes in electrical transport of the material are used as an indicator of stress/strain, pressure, or damage. However, engineers and other users of self-sensing materials are typically not directly interested in the electrical state of the material. Rather, they want to know the underlying mechanical state of the material that gives rise to an observed electrical change. Recovering material condition from electrical observations is referred to as the self-sensing inverse problem (SSIP). Prior work by the author has shown that the SSIP can be solved using an electrical impedance tomography (EIT)-generated conductivity map as an input, but this is undesirable because it requires solving a second inverse problem (i.e., the SSIP) on top of the EIT inverse problem. To that end, a direct formulation for the SSIP is herein presented. In this approach, voltage-current data is directly inverted to find the displacement field without using EIT as an intermediate step. Additionally, the direct SSIP is solved within the primal-dual interior point (PDIPM) framework such that an ℓ1-norm can be used on the regularization term, which promotes sparsity in the solution space. This approach is applied to a representative piezoresistive nanocomposite using the Laplace matrix as regularization to promote a spatially smooth solution. From the displacement field, strains and stresses are calculated and compared to a commercial finite element solution with good accuracy.