Abstract
A direct approach is developed using Streamline Upwind Petrov Galerkin (SUPG) concepts to determine the spatially varying property distribution in a nominally heterogeneous material. The approach is based on successful development of a SUPG-stabilized inverse finite element approach to solve the differential equations of equilibrium in terms of material properties, resulting in a matrix form [A] {E} = {R}, where [A] is a known function of measured axial strains (e.g., from StereoDIC) and axial positions, {R} is a known function of axial body forces, applied loads and reactions, and {E} is a vector of unknown material properties at discrete axial locations. Theoretical and computational developments for the SUPG-stabilized approach are described in detail for one-dimensional applications (e.g., heterogeneous tensile/compression specimens, tensile/compressive surfaces of beams). Property predictions using the SUPG method with analytic strains and additive Gaussian noise are shown to be in excellent agreement with known property values, whereas predictions using the classical Bubnov–Galerkin method exhibit large, spurious oscillations in the predicted material properties. To demonstrate the methodology using experimental measurements, a 3D-printed heterogeneous tensile specimen with independently measured material properties is tested and full-field strains measured at several load levels. Results confirm that SUPG finite element property predictions are in very good agreement with independently determined values at each load level along the specimen length, providing confidence that the SUPG FE analysis framework developed in this work is stable and extendable to multiple dimensions.
