Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations

In this paper, we extend to the time-harmonic Maxwell equations the p-version analysis technique developed in [R. Hiptmair, A. Moiola and I. Perugia, Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the p-version, SIAM J. Numer. Anal., 49 (2011), 264-284] for Trefftz-discontinuous Galerkin approximations of the Helmholtz problem. While error estimates in a mesh-skeleton norm are derived parallel to the Helmholtz case, the derivation of estimates in a mesh-independent norm requires new twists in the duality argument. The particular case where the local Trefftz approximation spaces are built of vector-valued plane wave functions is considered, and convergence rates are derived.

Frying of potato chips in a blend of canola oil and palm olein: changes in levels of individual fatty acids and tocols

The changes occurring in the levels of nutritionally relevant oil components were assessed during repeated frying of potato chips in a blend of palm olein and canola oil (1:1 w/w). The blend suffered minimal reductions in omega-3 and omega-6 polyunsaturated fatty acids. There was no significant difference between the fatty acid composition of the oil extracted from the product and that of the frying medium, in all three cases. The blend also contained a significant amount of tocols which add a nutritional value to the oil. The concentration of the tocols was satisfactorily retained over the period of oil usage, in contrast to the significant loses observed in the case of the individual oils. The blend also performed well when assessed by changes in total polar compounds, free fatty acids, p-anisidine value. When fried in used oil, the product oil content increased progressively with oil usage time. This study shows that blended frying oils can combine good stability and nutritional quality

Post-frying oil drainage from potato chips and french fries: a comparative study of atmospheric and vacuum drainage

This paper represents a study of the transient changes occurring in temperature, and moisture and oil contents during the so called “post-frying drainage”—which is the duration for which a product is held in the head space of the fryer after it is removed from the oil. Since most of the oil adhering to the product penetrates into the structure during this period, this paper examines the effects of applying vacuum during drainage (1.33 kPa) to maintain the product temperature consistently above the water saturation temperature corresponding to the prevailing pressure (11 °C), which potentially eliminates water condensation and prevents the occluded surface oil from penetrating into the product structure. Draining under vacuum significantly lowers the oil content of potato chips by 38% compared to atmospheric drainage. This phenomenon can be further confirmed by confocal laser scanning microscopy (CLSM) images, which show that the boundary between the core and the crust regions is clearly visible in the case of vacuum drainage, whereas in the case of atmospheric drainage, the oil is distributed throughout the structure. Unfortunately, the same approach did not reduce the oil content of French fries—the oil content of vacuum-drained product was found similar to the product obtained by draining under atmospheric pressure. This is because the reduction in oil content only occurs when there is net moisture evaporation from the product and the evaporation rate is sufficient to force out the oil from the product; this was clearly not the case with French fries. The CLSM images show that the oil distribution in the products drained under atmospheric pressure and vacuum was similar.

A kinetic model for the formation of acrylamide during the finish-frying of commercial French fries

Acrylamide is formed from reducing sugars and asparagine during the preparation of French fries. The commercial preparation of French fries is a multi-stage process involving the preparation of frozen, par-fried potato strips for distribution to catering outlets where they are finish fried. The initial blanching, treatment in glucose solution and par-frying steps are crucial since they determine the levels of precursors present at the beginning of the finish frying process. In order to minimize the quantities of acrylamide in cooked fries, it is important to understand the impact of each stage on the formation of acrylamide. Acrylamide, amino acids, sugars, moisture, fat and color were monitored at time intervals during the frying of potato strips which had been dipped in varying concentrations of glucose and fructose during a typical pretreatment. A mathematical model of the finish-frying was developed based on the fundamental chemical reaction pathways, incorporating moisture and temperature gradients in the fries. This showed the contribution of both glucose and fructose to the generation of acrylamide, and accurately predicted the acrylamide content of the final fries.

Combination of moderate vacuum frying with high vacuum drainage: relationship between process conditions and oil uptake

This paper explores the possibility of combining moderate vacuum frying followed by post-frying high vacuum application during the oil drainage stage, with the aim to reduce oil content in potato chips. Potato slices were initially vacuum fried under two operating conditions (140 °C, 20 kPa and 162 °C, 50.67 kPa) until the moisture content reached 10 and 15 % (wet basis), prior to holding the samples in the head space under high vacuum level (1.33 kPa). This two-stage process was found to lower significantly the amount of oil taken up by potato chips by an amount as high as 48 %, compared to drainage at the same pressure as the frying pressure. Reducing the pressure value to 1.33 kPa reduced the water saturation temperature (11 °C), causing the product to continuously lose moisture during the course of drainage. Continuous release of water vapour prevented the occluded surface oil from penetrating into the product structure and released it from the surface of the product. When frying and drainage occurred at the same pressure, the temperature of the product fell below the water saturation temperature soon after it was lifted out of the oil, which resulted in the oil getting sucked into the product. Thus, lowering the pressure after frying to a value well below the frying pressure is a promising method to lower oil uptake by the product.

Physicochemical changes occuring in oil when atmospheric frying is combined with post-frying vacuum application

The aim of this study was to investigate the effect of atmospheric frying followed by drainage under vacuum on the stability of oil, compared to similar frying with drainage at atmospheric pressure. Changes in the oil were assessed by the free fatty acid (FFA) content, p-anisidine value (AnV), colour, viscosity, fatty acid profile and concentration of tocols. The rate of FFA formation in the case of vacuum drainage was found to be about half that of atmospheric drainage. Oil deterioration by oxidation and polymerisation was also reduced by the use of vacuum drainage. The AnV of the oil after vacuum drainage was lower by about 12%, the total colour difference was improved by 14% and viscosity was slightly reduced after 5 days of frying, compared to the values for oil that had been drained at atmospheric pressure. There was a reduction in the loss of polyunsaturated fatty acids in the case of vacuum drainage after 5 days of frying but differences in retention of tocols were only evident in the first two days of frying.

Kinetic model for the formation of acrylamide during the finish-frying of commercial French fries

Acrylamide is formed from reducing sugars and asparagine during the preparation of French fries. The commercial preparation of French fries is a multistage process involving the preparation of frozen, par-fried potato strips for distribution to catering outlets, where they are finish-fried. The initial blanching, treatment in glucose solution, and par-frying steps are crucial because they determine the levels of precursors present at the beginning of the finish-frying process. To minimize the quantities of acrylamide in cooked fries, it is important to understand the impact of each stage on the formation of acrylamide. Acrylamide, amino acids, sugars, moisture, fat, and color were monitored at time intervals during the frying of potato strips that had been dipped in various concentrations of glucose and fructose during a typical pretreatment. A mathematical model based on the fundamental chemical reaction pathways of the finish-frying was developed, incorporating moisture and temperature gradients in the fries. This showed the contribution of both glucose and fructose to the generation of acrylamide and accurately predicted the acrylamide content of the final fries.

Trefftz discontinuous Galerkin methods for acoustic scattering on locally refined meshes

We extend the a priori error analysis of Trefftz-discontinuous Galerkin methods for time-harmonic wave propagation problems developed in previous papers to acoustic scattering problems and locally refined meshes. To this aim, we prove refined regularity and stability results with explicit dependence of the stability constant on the wave number for non convex domains with non connected boundaries. Moreover, we devise a new choice of numerical flux parameters for which we can prove L2-error estimates in the case of locally refined meshes near the scatterer. This is the setting needed to develop a complete hp-convergence analysis.

Validation of Canopy Height Profile methodology for small-footprint full-waveform airborne LiDAR data in a discontinuous canopy environment

A Canopy Height Profile (CHP) procedure presented in Harding et al. (2001) for large footprint LiDAR data was tested in a closed canopy environment as a way of extracting vertical foliage profiles from LiDAR raw-waveform. In this study, an adaptation of this method to small-footprint data has been shown, tested and validated in an Australian sparse canopy forest at plot- and site-level. Further, the methodology itself has been enhanced by implementing a dataset-adjusted reflectance ratio calculation according to Armston et al. (2013) in the processing chain, and tested against a fixed ratio of 0.5 estimated for the laser wavelength of 1550nm. As a by-product of the methodology, effective leaf area index (LAIe) estimates were derived and compared to hemispherical photography-derived values. To assess the influence of LiDAR aggregation area size on the estimates in a sparse canopy environment, LiDAR CHPs and LAIes were generated by aggregating waveforms to plot- and site-level footprints (plot/site-aggregated) as well as in 5m grids (grid-processed). LiDAR profiles were then compared to leaf biomass field profiles generated based on field tree measurements. The correlation between field and LiDAR profiles was very high, with a mean R2 of 0.75 at plot-level and 0.86 at site-level for 55 plots and the corresponding 11 sites. Gridding had almost no impact on the correlation between LiDAR and field profiles (only marginally improvement), nor did the dataset-adjusted reflectance ratio. However, gridding and the dataset-adjusted reflectance ratio were found to improve the correlation between raw-waveform LiDAR and hemispherical photography LAIe estimates, yielding the highest correlations of 0.61 at plot-level and of 0.83 at site-level. This proved the validity of the approach and superiority of dataset-adjusted reflectance ratio of Armston et al. (2013) over a fixed ratio of 0.5 for LAIe estimation, as well as showed the adequacy of small-footprint LiDAR data for LAIe estimation in discontinuous canopy forests.

Plane wave discontinuous Galerkin methods: exponential convergence of the hp-version

We consider the two-dimensional Helmholtz equation with constant coefficients on a domain with piecewise analytic boundary, modelling the scattering of acoustic waves at a sound-soft obstacle. Our discretisation relies on the Trefftz-discontinuous Galerkin approach with plane wave basis functions on meshes with very general element shapes, geometrically graded towards domain corners. We prove exponential convergence of the discrete solution in terms of number of unknowns.

Energy consistent discontinuous Galerkin methods for the Navier–Stokes–Korteweg system

We design consistent discontinuous Galerkin finite element schemes for the approximation of the Euler-Korteweg and the Navier-Stokes-Korteweg systems. We show that the scheme for the Euler-Korteweg system is energy and mass conservative and that the scheme for the Navier-Stokes-Korteweg system is mass conservative and monotonically energy dissipative. In this case the dissipation is isolated to viscous effects, that is, there is no numerical dissipation. In this sense the methods are consistent with the energy dissipation of the continuous PDE systems. - See more at: http://www.ams.org/journals/mcom/2014-83-289/S0025-5718-2014-02792-0/home.html#sthash.rwTIhNWi.dpuf

A Posteriori analysis of discontinuous Galerkin schemes for systems of hyperbolic conservation laws

In this work we construct reliable a posteriori estimates for some semi- (spatially) discrete discontinuous Galerkin schemes applied to nonlinear systems of hyperbolic conservation laws. We make use of appropriate reconstructions of the discrete solution together with the relative entropy stability framework, which leads to error control in the case of smooth solutions. The methodology we use is quite general and allows for a posteriori control of discontinuous Galerkin schemes with standard flux choices which appear in the approximation of conservation laws. In addition to the analysis, we conduct some numerical benchmarking to test the robustness of the resultant estimator.

Energy consistent discontinuous Galerkin methods for a quasi-incompressible diffuse two phase flow model

We design consistent discontinuous Galerkin finite element schemes for the approximation of a quasi-incompressible two phase flow model of Allen–Cahn/Cahn–Hilliard/Navier–Stokes–Korteweg type which allows for phase transitions. We show that the scheme is mass conservative and monotonically energy dissipative. In this case the dissipation is isolated to discrete equivalents of those effects already causing dissipation on the continuous level, that is, there is no artificial numerical dissipation added into the scheme. In this sense the methods are consistent with the energy dissipation of the continuous PDE system.

Discontinuous Galerkin methods for the p-biharmonic equation from a discrete variational perspective

We study discontinuous Galerkin approximations of the p-biharmonic equation for p∈(1,∞) from a variational perspective. We propose a discrete variational formulation of the problem based on an appropriate definition of a finite element Hessian and study convergence of the method (without rates) using a semicontinuity argument. We also present numerical experiments aimed at testing the robustness of the method.

On the possibility of non-fat frying using molten glucose

Fried products impose a health concerns due to considerable amount of oil they contain. Production of snack foods with minimal oil content and good management of oil during frying to minimise the production of toxic compounds continue to be challenging aims. This paper aims to investigate the possibility of producing a fat-free food snack by replacing frying oil with a non-fat medium. Glucose was melted and its temperature was then brought to 185°C and used to fry potato strips, to obtain a product referred here as glucose fries. The resulting product was compared with French fries prepared conventionally under conditions that resulted in similar final moisture content. The resulting products were also examined for crust formation, texture parameters, colour development and glucose content. Stereo microscope images showed that similar crusts were formed in the glucose fries and French fries. Texture parameters were found to be similar for both products at 5mm and 2 mm penetration depth. The maximum hardness at 2mm penetration depth was also similar for both products, but different from cooked potato. The colour development which characterised French fries was also observed in glucose fries. The glucose content in glucose fries was found to be twice the content of French fries, which is to be expected since glucose absorbed or adhered to the surface. In conclusion, glucose fries, with similar texture and colour characteristics to that of French fries, can be prepared by using a non-fat frying medium.