A space-time quasi-Trefftz DG method for the wave equation with piecewise-smooth coefficients

Trefftz methods are high-order Galerkin schemes in which all discrete functions are elementwise solution of the PDE to be approximated. They are viable only when the PDE is linear and its coefficients are piecewise constant. We introduce a 'quasi-Trefftz' discontinuous Galerkin method for the discretisation of the acoustic wave equation with piecewise-smooth wavespeed: the discrete functions are elementwise approximate PDE solutions. We show that the new discretisation enjoys the same excellent approximation properties as the classical Trefftz one, and prove stability and high-order convergence of the DG scheme. We introduce polynomial basis functions for the new discrete spaces and describe a simple algorithm to compute them. The technique we propose is inspired by the generalised plane waves previously developed for time-harmonic problems with variable coefficients; it turns out that in the case of the time-domain wave equation under consideration the quasi-Trefftz approach allows for polynomial basis functions.Comment: 25 pages, 9 figure

On polynomial Trefftz spaces for the linear time-dependent Schr\"odinger equation

We study the approximation properties of complex-valued polynomial Trefftz spaces for the $(d+1)$-dimensional linear time-dependent Schr\"odinger equation. More precisely, we prove that for the space-time Trefftz discontinuous Galerkin variational formulation proposed by G\'omez, Moiola (SIAM. J. Num. Anal. 60(2): 688-714, 2022), the same $h$-convergence rates as for polynomials of degree $p$ in $(d + 1)$ variables can be obtained in a mesh-dependent norm by using a space of Trefftz polynomials of anisotropic degree. For such a space, the dimension is equal to that of the space of polynomials of degree $2p$ in $d$ variables, and bases are easily constructed

Accuracy of 3D Tooth Movements in the Fabrication of Manual Setup Models for Aligner Therapy

Background: The clinical outcome of aligner therapy is closely related to the precision of its setup, which can be manually or digitally fabricated. The aim of the study is to investigate the suitability of manual setups made for aligner therapy in terms of the precision of tooth movements. Methods: Six dental technicians were instructed to adjust each of eleven duplicate plaster casts of a patient models as follows: a 1 mm pure vestibular translation of tooth 11 and a 15° pure mesial rotation of tooth 23. The processed setup models were 3D scanned and matched with the reference model. The one-sample Wilcoxon signed-rank test (p < 0.05) was used for evaluation. Results: The overall precision of the translational movement covers a wide range of values from 0.25 to 2.26 mm (median: 1.09 mm). The target value for the rotation of tooth 23 was achieved with a median rotation of 9.76° in the apical-occlusal direction. Unwanted movements in the other planes also accompanied the rotation. Conclusions: A manual setup can only be fabricated with limited precision. Besides the very high variability between technicians, additional unwanted movements in other spatial planes occurred. Manually fabricated setups should not be favored for aligner therapy due to limited precision

High-Resolution Optical Coherence Tomography in Healthy Individuals Provides Resolution at the Cellular and Subcellular Levels.

PURPOSE To assess the clinical resolution capacities of a novel high-resolution optical coherence tomography (High-Res OCT). METHODS Eight healthy volunteers were included in this observational study. Using the SPECTRALIS High-Res OCT device (Heidelberg Engineering, Heidelberg) macular b-scans were taken and compared with b-scans acquired with a SPECTRALIS HRA+OCT device (Heidelberg Engineering, Heidelberg). High-Res OCT scans were also compared with hematoxylin and eosin-stained sections from a human donor retina. RESULTS High-Res OCT allowed identification of several retinal structures at the cellular and subcellular levels, namely, cell nuclei of ganglion cells, displaced amacrine cells, cone photoreceptors and retinal pigment epithelial cells compared with the commercial device. Rod photoreceptor nuclei were partially detectable. Localization of cell type-specific nuclei were confirmed by histological sections of human donor retina. Additionally, all three plexus of the retinal vasculature could be visualized. CONCLUSIONS SPECTRALIS High-Res OCT device provides improved resolution compared with the conventional SPECTRALIS HRA+OCT device and allows to identify structures at the cellular level, similar to histological sections. TRANSLATIONAL RELEVANCE High-Res OCT shows improved visualization of retinal structures in healthy individuals and can be used to assess individual cells within the retina

Using task farming to optimise a street-scale resolution air quality model of the West Midlands (UK)

High resolution air quality models combining emissions, chemical processes, dispersion and dynamical treatments are necessary to develop effective policies for clean air in urban environments, but can have high computational demand. We demonstrate the application of task farming to reduce runtime for ADMS-Urban, a quasi-Gaussian plume air dispersion model. The model represents the full range of source types (point, road and grid sources) occurring in an urban area at high resolution. Here, we implement and evaluate the option to automatically split up a large model domain into smaller sub-regions, each of which can then be executed concurrently on multiple cores of a HPC or across a PC network, a technique known as task farming. The approach has been tested for a large model domain covering the West Midlands, UK (902 km2), as part of modelling work in the WM-Air (West Midlands Air Quality Improvement Programme) project. Compared to the measurement data, overall, the model performs well. Air quality maps for annual/subset averages and percentiles are generated. For this air quality modelling application of task farming, the optimisation process has reduced weeks of model execution time to approximately 35 h for a single model configuration of annual calculations