Forward and inverse modeling of conducting lattices using lattice Green’s functions

Advances in additive manufacturing have created multiple opportunities for manufactured lattice structures (meta-materials) with interesting and useful properties. Analysis, design, and optimization of such lattices accentuates the need for modeling and computational methods that take advantage of lattice periodicity and discreteness. In this dissertation we develop discrete analogs of continuum integral operators. These include, the Newtonian potential, single-layer, double-layer, adjoint of the double-layer, and hyper-singular integral operators. This allows, via Green’s analysis framework for analyzing conducting lattices, construction of lattice analogs for both the singular and hyper-singular boundary integral equations, and the integral representation for the domain solution. Modeling techniques to analyze defective lattices using defect equations in pristine lattices are demonstrated. Existence of discrete analog of Calderón projector for forward problems is established. Application of these boundary algebraic equations are shown to be convenient for solution of inverse problems in conducting lattices. Defect detection problems are recast as inverse source problems using equivalency conditions relating defect’s conductance to the corresponding polarization dipole. Inverse modeling techniques in lattices are shown to be challenging for two reasons - first due to lack of high sensitivity of boundary data to the presence of internal defects and second due to the discrete nature of the defective region. Perturbed formulation is used to enhance sensitivity. Use of machine learning method of LASSO and iteratively weighted least squares methods are investigated to solve the discrete problem for both complete and incomplete Cauchy data.