In this paper we present a proof about the convergence of the 3D Nodal-LTSN Method in order to solve the transport problem in a parallelepiped domain. For that, we define functions associated to the errors, one in the approximated flux, another in the quadrature formula and establish a relation between them. We present a Nodal-LTSN method to generate an analytical solution for discrete ordinates problems in three-dimensional cartesian geometry. We first transverse integrate the SN equations and then we apply the Laplace transform. The essence of this method is the diagonalization of the LTSN transport matrices and the spectral analysis garantees this. The transverse leakage terms that appear in the transverse integrated SN equations are represented by exponential functions with decay constants that depend on the characteristics of the material of the medium the particles leave behind. We present numerical results generated by the offered method applied to typical shielding model problems.
We describe a Laplace transform exponential method applied to x-y-z geometry heterogeneous neutron transport problems in the discrete ordinates (S N ) formulation that we refer to as the LTS N -Exp method. This numerical method uses the space Laplace transform technique to solve the one-dimensional transverse-integrated S N exponential equations within one of the homogeneous regions of the domain of solution. Based on the physics of shielding problems where the neutron flux attenuates exponentially with increasing distance from the source, we approximate the transverse leakage terms by exponential functions. We show in two numerical experiments that the LTS N -Exp method generates very accurate results in highly absorbing media.
Objectives: To investigate the quality of newly formed bone in sheep mandibles submitted to distraction osteogenesis and low-level laser therapy (LLLT), based on hardness and modulus of elasticity values. The ideal moment for laser application (during the latency/activation period vs. during the bone consolidation period) was also evaluated. Computed tomography imaging was used to assess relapse as a result of early device removal. Study design: Extraoral distraction devices were placed in five sheep so as to achieve 1.5 cm of lengthened bone in 60 days. Distraction devices were removed 50, 40, and 33 days after surgery. Four animals were treated with LLLT, at different times, and one was used as control (no LLLT). Results: When applied during the bone consolidation period, LLLT caused an increase in hardness and modulus of elasticity values. On the other hand, animals irradiated with LLLT during the latency/activation period presented a delay in bone healing. A period of consolidation of 13 days (early device removal) was associated with relapse. Conclusions: Nanoindentation tests were able to detect slight abnormalities in bone metabolism and proved to be important tools for the assessment of bone quality following distraction osteogenesis. LLLT provided increased benefits when applied during the bone consolidation period, once it promoted an increase in hardness and modulus of elasticity values. According to our results, the bone consolidation period should be of at least 3 weeks, so as to prevent relapse.
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