Cs-smooth isogeometric spline spaces over planar bilinear multi-patch parameterizations

The design of globally Cs-smooth (s ≥ 1) isogeometric spline spaces over multi-patch geometries with possibly extraordinary vertices, i.e. vertices with valencies different from four, is a current and challenging topic of research in the framework of isogeometric analysis. In this work, we extend the recent methods Kapl et al. Comput. Aided Geom. Des. 52–53:75–89, 2017, Kapl et al. Comput. Aided Geom. Des. 69:55–75, 2019 and Kapl and Vitrih J. Comput. Appl. Math. 335:289–311, 2018, Kapl and Vitrih J. Comput. Appl. Math. 358:385–404, 2019 and Kapl and Vitrih Comput. Methods Appl. Mech. Engrg. 360:112684, 2020 for the construction of C1-smooth and C2-smooth isogeometric spline spaces over particular planar multi-patch geometries to the case of Cs-smooth isogeometric multi-patch spline spaces of degree p, inner regularity r and of a smoothness s ≥ 1, with p ≥ 2s + 1 and s ≤ r ≤ p − s − 1. More precisely, we study for s ≥ 1 the space of Cs-smooth isogeometric spline functions defined on planar, bilinearly parameterized multi-patch domains, and generate a particular Cs-smooth subspace of the entire Cs-smooth isogeometric multi-patch spline space. We further present the construction of a basis for this Cs-smooth subspace, which consists of simple and locally supported functions. Moreover, we use the Cs-smooth spline functions to perform L2 approximation on bilinearly parameterized multi-patch domains, where the obtained numerical results indicate an optimal approximation power of the constructed Cs-smooth subspace.

Performance Evaluation of the Serological Test and Dynamic Changes of Anti-2019-nCoV Antibodies Among COVID‐19 Patients

BackgroundSerological tests for anti-2019-nCoV antibodies have been developed, however, validation with clinical samples was insufficient and mixed. MethodsA total of 197 patients with COVID-19 hospitalized in the Wuhan Pulmonary Hospital and 114 healthy people were enrolled in this study. The IgM/IgG was measured by chemiluminescence detection kit and the performance of IgM/IgG was assessed through diagnostic test. A smooth spline function was used to fit a possible dynamic changes of antibody content.ResultsThe sensitivity and specificity to diagnose COVID-19 were 96.95% and 92.98% for IgG, and 65.99% and 98.25% for IgM, respectively. The period with the highest IgM positive rate (93.75%) was 11 to 13 days after the onset, while the IgG positive rate was almost 100% in the patients whose serum was collected 7 days after onset. Small differences were found in IgM content among mild/regular, severe and critical patients. IgG content in critical patients were highest in the 2-week post-symptom onset group, while the IgG in the severe patients were highest in the 15-28 days and more than 28 days post-symptom onset.ConclusionsSerological testing performed well in the diagnosis for COVID-19, and the positive rate and variance of IgG are higher than those of IgM.