# Browsing by Subject "Boundary algebraic equations"

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Item Forward and inverse modeling of conducting lattices using lattice Green’s functions(2021-12-01) Bhamidipati, Vikram; Rodin, G. J. (Gregory J.); Kallivokas, Loukas F.; Demkowicz, Leszek F; Landis, Chad M; Martinsson, Per-Gunnar JShow more 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.Show more Item Modeling and computing based on lattices(2010-12) Zhao, Haifeng, 1980-; Rodin, G. J. (Gregory J.); Mear, Mark E.; Ravi-Chandar, K.; Makarov, Dmitrii E.; Kovar, DesiderioShow more This dissertation presents three studies addressing various modeling and computational aspects of lattice structures. The first study is concerned with characterization of the threshold behavior for very slow (subcritical) crack growth. First, it is shown that this behavior requires the presence of a healing mechanism. Then thermodynamic analysis of brittle fracture specimens near the threshold developed by Rice (1978) is extended to specimens undergoing microstructural changes. This extension gives rise to a generalization of the threshold concept that mirrors the way the resistance R-curve generalizes the fracture toughness. In the absence of experimental data, the resistance curve near the threshold is constructed using a lattice model that includes healing and rupture mechanisms. The second study is concerned with transmission of various boundary conditions through irregular lattices. The boundary conditions are parameterized using trigonometric Fourier series, and it is shown that, under certain conditions, transmission through irregular lattices can be well approximated by that through classical continuum. It is determined that such transmission must involve the wavelength of at least 12 lattice spacings; for smaller wavelength classical continuum approximations become increasingly inaccurate. Also it is shown that this restriction is much more severe than that associated with identifying the minimum size for representative volume elements. The third study is concerned with extending the use of boundary algebraic equations to problems involving irregular rather than regular lattices. Such an extension would be indispensable for solving multiscale problems defined on irregular lattices, as boundary algebraic equations provide seamless bridging between discrete and continuum models. It is shown that, in contrast to regular lattices, boundary algebraic equations for irregular lattices require a statistical rather than deterministic treatment. Furthermore, boundary algebraic equations for irregular lattices contain certain terms that require the same amount of computational effort as the original problem.Show more