In isogeometric analysis, constructing bijective and low-distorted parameterizations is a fundamental task. Compared with the planar problem, the volumetric case is more challenging in both robustness and efficiency. In this paper, we present a robust and efficient volumetric parameterization method based on the idea of penalty functions and the Jacobian regularization technique. The proposed method does not require an already bijective initialization and thus avoids an extra foldover elimination step. The main contributions of this work lie in three aspects. First, a new objective function that characterizes the volume distortion is established using the Divergence Theorem. Second, we employ a novel penalty function for the Jacobian regularization. The full analytical gradient of the objective function is also deduced to enhance the numerical stability in gradient-based optimization. Third, we develop a reduced numerical integration strategy to accelerate the new algorithm. Several numerical examples demonstrate that our method significantly outperforms the current competitive approaches both in terms of robustness and efficiency.

Photoactive yellow proteins (PYPs) make up a diverse class of blue-light-absorbing bacterial photoreceptors. Electronic excitation of the p-coumaric acid chromophore covalently bound within PYP results in triphasic quenching kinetics; however, the molecular basis of this behavior remains unresolved. Here we explore this question by examining the excitation-wavelength dependence of the photodynamics of the PYP from Halorhodospira halophila via a combined experimental and computational approach. The fluorescence quantum yield, steady-state fluorescence emission maximum, and cryotrapping spectra are demonstrated to depend on excitation wavelength. We also compare the femtosecond photodynamics in PYP at two excitation wavelengths (435 and 475 nm) with a dual-excitation-wavelength-interleaved pump-probe technique. Multicompartment global analysis of these data demonstrates that the excited-state photochemistry of PYP depends subtly, but convincingly, on excitation wavelength with similar kinetics with distinctly different spectral features, including a shifted ground-state beach and altered stimulated emission oscillator strengths and peak positions. Three models involving multiple excited states, vibrationally enhanced barrier crossing, and inhomogeneity are proposed to interpret the observed excitation-wavelength dependence of the data. Conformational heterogeneity was identified as the most probable model, which was supported with molecular mechanics simulations that identified two levels of inhomogeneity involving the orientation of the R52 residue and different hydrogen bonding networks with the p-coumaric acid chromophore. Quantum calculations were used to confirm that these inhomogeneities track to altered spectral properties consistent with the experimental results.

RE³⁺ (RE = Pr, Sm, Tb, Er, Dy)-activated Y₄Si₂O₇N₂ samples were prepared by a solid-state reaction method at high temperature, and their photoluminescence properties were investigated. The absorption band located at about 250 nm is attributed to the host absorption. The 5d bands of Pr³⁺ and Tb³⁺ are at rather low energy in Y₄Si₂O₇N₂ compared to oxide. The direct Pr³⁺ 4f² → 4f¹5d¹ excitation at 275 nm leads to typical 4f² → 4f² line emissions (450–700 nm) and strong 4f¹5d¹ → 4f² broad band emission (300–450 nm), respectively. The charge transfer (N³⁻ → Sm³⁺) band of Sm³⁺ was observed at a somewhat lower energy of 4.68 eV compared to oxide, and Sm³⁺-activated sample shows a bright red emission originating from ⁴G_5/2 → ⁶H_J (J = 5/2, 7/2 and 9/2) transitions. For Tb³⁺-doped sample, the direct Tb³⁺ 4f⁸ → 4f⁷5d¹ excitation leads to ⁵D₃ → ⁷F_J (J = 6, 5, 4, 3) (blue) and ⁵D₄ → ⁷F_J (J = 6, 5, 4, 3) (green) line emissions, the cross-relaxation depended on Tb concentration has happened. The incorporation of Er³⁺ (or Dy³⁺) into Y₄Si₂O₇N₂ resulted in a typical Er³⁺ (or Dy³⁺) f-f line absorptions and emissions. Moreover, the energy transfer from the host lattice to the luminescent activators (Pr³⁺, Tb³⁺ and Sm³⁺) is observed. The energy level diagram containing the position of 4f and 5d energy levels of all Ln²⁺ and Ln³⁺ ions relative to the valence and conduction band of Y₄Si₂O₇N₂ has been established and studied based on the data presented in this work, and further provides a platform for studying the photoluminescence properties as well as the valence stability of the lanthanide ions.