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Elliptic obstacle problems are formulated to find either superharmonic solutions or minimal surfaces that lie on or over the obstacles, by incorporating inequality constraints. In order to solve such problems effectively using finite difference (FD) methods, the article investigates simple iterative algorithms based on the successive over-relaxation (SOR) method. It introduces subgrid FD methods to reduce the accuracy deterioration occurring near the free boundary when the mesh grid does not match with the free boundary. For nonlinear obstacle problems, a method of gradient-weighting is introduced to solve the problem more conveniently and efficiently. The iterative algorithm is analyzed for convergence for both linear and nonlinear obstacle problems. An effective strategy is also suggested to find the optimal relaxation parameter. It has been numerically verified that the resulting obstacle SOR iteration with the optimal parameter converges about one order faster than state-of-the-art methods and the subgrid FD methods reduce numerical errors by one order of magnitude, for most cases. Various numerical examples are given to verify the claim.

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Background Toxicokinetics of nanomaterials, including studies on the absorption, distribution, metabolism, and elimination of nanomaterials, are essential in assessing their potential health effects. The fate of nanomaterials after inhalation exposure to multiple nanomaterials is not clearly understood. Methods Male Sprague–Dawley rats were exposed to similar sizes of silver nanoparticles (AgNPs, 10.86 nm) and gold nanoparticles (AuNPs, 10.82 nm) for 28 days (6-h/day, 5-days/week for four weeks) either with separate NP inhalation exposures or with combined co-exposure in a nose-only inhalation system. Mass concentrations sampled from the breathing zone were AuNP 19.34 ± 2.55 μg/m3 and AgNP 17.38 ± 1.88 μg/m3 for separate exposure and AuNP 8.20 μg/m3 and AgNP 8.99 μg/m3 for co-exposure. Lung retention and clearance were previously determined on day 1 (6-h) of exposure (E-1) and on post-exposure days 1, 7, and 28 (PEO-1, PEO-7, and PEO-28, respectively). In addition, the fate of nanoparticles, including translocation and elimination from the lung to the major organs, were determined during the post-exposure observation period. Results AuNP was translocated to the extrapulmonary organs, including the liver, kidney, spleen, testis, epididymis, olfactory bulb, hilar and brachial lymph nodes, and brain after subacute inhalation and showed biopersistence regardless of AuNP single exposure or AuNP + AgNP co-exposure, showing similar elimination half-time. In contrast, Ag was translocated to the tissues and rapidly eliminated from the tissues regardless of AuNP co-exposure. Ag was continually accumulated in the olfactory bulb and brain and persistent until PEO-28. Conclusion Our co-exposure study of AuNP and AgNP indicated that soluble AgNP and insoluble AuNP translocated differently, showing soluble AgNP could be dissolved into Ag ion to translocate to the extrapulmonary organs and rapidly removed from most organs except the brain and olfactory bulb. Insoluble AuNPs were continually translocated to the extrapulmonary organs, and they were not eliminated rapidly.

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