Fast Barycentric-Based Evaluation Over Spectral/hp Elements

As the use of spectral/hp element methods, and high-order finite element methods in general, continues to spread, community efforts to create efficient, optimized algorithms associated with fundamental high-order operations have grown. Core tasks such as solution expansion evaluation at quadrature points, stiffness and mass matrix generation, and matrix assembly have received tremendous attention. With the expansion of the types of problems to which high-order methods are applied, and correspondingly the growth in types of numerical tasks accomplished through high-order methods, the number and types of these core operations broaden. This work focuses on solution expansion evaluation at arbitrary points within an element. This operation is core to many postprocessing applications such as evaluation of streamlines and pathlines, as well as to field projection techniques such as mortaring. We expand barycentric interpolation techniques developed on an interval to 2D (triangles and quadrilaterals) and 3D (tetrahedra, prisms, pyramids, and hexahedra) spectral/hp element methods. We provide efficient algorithms for their implementations, and demonstrate their effectiveness using the spectral/hp element library Nektar++ by running a series of baseline evaluations against the ‘standard’ Lagrangian method, where an interpolation matrix is generated and matrix-multiplication applied to evaluate a point at a given location. We present results from a rigorous series of benchmarking tests for a variety of element shapes, polynomial orders and dimensions. We show that when the point of interest is to be repeatedly evaluated, the barycentric method performs at worst $$50\%$$ 50 % slower, when compared to a cached matrix evaluation. However, when the point of interest changes repeatedly so that the interpolation matrix must be regenerated in the ‘standard’ approach, the barycentric method yields far greater performance, with a minimum speedup factor of $$7\times $$ 7 × . Furthermore, when derivatives of the solution evaluation are also required, the barycentric method in general slightly outperforms the cached interpolation matrix method across all elements and orders, with an up to $$30\%$$ 30 % speedup. Finally we investigate a real-world example of scalar transport using a non-conformal discontinuous Galerkin simulation, in which we observe around $$6\times $$ 6 × speedup in computational time for the barycentric method compared to the matrix-based approach. We also explore the complexity of both interpolation methods and show that the barycentric interpolation method requires $${\mathcal {O}}(k)$$ O ( k ) storage compared to a best case space complexity of $${\mathcal {O}}(k^2)$$ O ( k 2 ) for the Lagrangian interpolation matrix method.

RELATIVISTIC ENERGY APPROACH AND MANY-BODY PERTURBATION THEORY TO COMPUTING ELECTRON-COLLISION CROSS-SECTIONS OF COMPLEX MULTIELECTRON IONS

An advanced relativistic energy approach combined with a relativistic many-body perturbation theory with ab initio zeroth approximation is used to calculate the electron-collision excitation cross-sections for complex multielectron systems. The relativistic many-body perturbation theory is used alongside the gauge-invariant scheme to generate an optimal Dirac-Kohn-Sham- Debye-Hückel one-electron representation. The results of relativistic calculation (taking into account the exchange and correlation corrections) of the electron collision cross-sections of excitation for the neon-like ion of the krypton are presented and compared with alternative results calculation on the basis of the R-matrix method in the Breit-Pauli approximation, in the relativistic distorted wave approximation and R- matrix method in combination with Dirac-Fock approximation

Comparison of methods of nonlinear control systems design

The issue of designing nonlinear control systems is a complex problem. A lot of methods are known that allow us to find a suitable control for a given nonlinear object that provides asymptotic stability of the nonlinear system equilibrium and an acceptable quality of the transient process. Many of these methods are difficult to apply in practice. Thus, comparing some of the methods in terms of simplicity of use is of great interest. Two analytical methods for the synthesis of nonlinear control systems are considered. They are the algebraic polynomial-matrix method that uses a quasilinear model, and the feedback linearization method that uses the Brunovsky model in combination with special feedbacks. A comparative analysis of the algebraic polynomial-matrix method and the feedback linearization method is carried out. It is found out that the algebraic polynomial-matrix method (APM) is much simpler than the feedback linearization method (FLM). A numerical example of designing a system that is synthesized by these methods is considered. It is found out that the system synthesized by the APM method has a region of attraction of the equilibrium position twice as large as the region of attraction of the system synthesized by the FLM method. It is reasonable to use the algebraic polynomial-matrix method with the quasilinear models in case of synthesis of control systems of objects with differentiable nonlinearities.

Experimental substantiation of the composition of gelatin mass for obtaining capsules with essential oils

Introduction. Soft gelatin capsules are a promising dosage form comprising essential oils as active agents. Joint research of the staff of the Perm State Pharmaceutical Academy, the Research Institute of Nutrition of the Ministry of Energy and Industry of the Republic of Tajikistan have proposed the composition of gelatin mass for encapsulation by the rotary matrix method. The mechanical and physical-technological parameters required to preserve the strength and elasticity of the capsule shell during the production process and storage are determined.Aim. Study of the rheological properties of gelatin masses based on gelatin of different grades, as well as the migration of essential oils through capsule shells.Materials and methods. Pharmaceutical active substances "Lipovitol" and "Limoneol" obtained in the Republic of Tajikistan were used as active substances introduced into the composition of soft gelatin capsules. Sunflower oil was used as the solvent. Gelatin samples were used to obtain gelatin masses: 1 – Foodchem (China), 2 – Brodnickie Zaklady Zelatyny Sp. zo.o. (Poland), 3 - Italgelatine s.p.a. (Italy), 4 – Ewald-Gelatine GmbH (Germany), 5 – Weishardt International (France); glycerol; sunflower oil. Gelatin mass for manufacturing soft capsules was prepared in a closed reactor. Capsules were prepared on an automatic encapsulation line RJWJ – 115 Soft Gelatin Encapsulator Machine (China). The structural and mechanical properties of soft gelatin masses were determined on a rotary viscometer RV type "Reotest 2" (Germany). The dynamics of the process of migration of essential oils and its components were studied by changing their amount in a capsule by chromato-mass spectrometry method on a chromatograph Varian CP 3800 with a quadrupole mass spectrometer 4000 MS as a detector (USA).Results and discussion. When studying the rheological properties of model compositions, it was found that for all samples of gelatin masses there is a decrease in values of effective viscosity when the shear rate increases, which characterizes the tested samples as a structured dispersion system. Additional studies have shown that the gelatin masses have thixotropic properties. Samples of gelatin masses 3–5 had narrower hysteresis loops, while sample 5 the narrowest, restoration of the structure took place quite quickly. Capsules obtained from gelatin mass 3 and 4 samples had a strong seam and were well cut out of the tape. From the mass of sample 5, high strength ribbons were obtained, a high temperature was required to seal the capsules, in some capsules the seam was not glued on one side, as a result, the capsules were rigid and brittle. As a result of the study, the rheological optimum of the gelatin mass suitable for preparing capsules by a rotary matrix method was determined, which has boundaries in the ranges of shear rates of 0.556–243 s-1 and viscosity ranges of 11.46-5028.76 Pa ⋅ s and shear stress of 2788–2808 Pa developing at these rates. When studying the migration of active substances through the capsule shell, it was found that over three years of storage of capsules in a closed polymer can, the content of essential oil in Lipovitol capsules decreased by 4.88 %, in Limoneol capsules by 5 %, which indicates partial migration of oil through the gelatin shell. The content remained within the permissible deviations (±10 %). The content of essential essential oil components also remained within acceptable deviations throughout the shelf life.Conclusion. The optimal composition of the shell for producing soft capsules by a rotary matrix method is justified. It was found that the rheological optimum of gelatin mass is characterized by viscosity ranges of 11.46-5028.76 Pa ⋅ s and shear stress of 2788–2808 Pa. According to the results of the study of the migration of essential oils through the shell, has been established the shelf life of soft gelatin capsules in glass jars made of dark glass and a temperature of 15 to 25 °C – 3 years.

Analytic Matrix Method for Frequency Response Techniques Applied to Nonlinear Dynamical Systems I: Small and Medium Amplitude Oscillations

This is the first on a series of articles that deal with nonlinear dynamical systems under oscillatory input that may exhibit harmonic and non-harmonic frequencies and possibly complex behavior in the form of chaos. Frequency response techniques of nonlinear dynamical systems are usually analyzed with numerical methods because, most of the time, analytical solutions turn out to be difficult, if not impossible, since they are based on infinite series of trigonometric functions. The analytic matrix method reported here is a direct one that speeds up the solution processing compared to traditional series solution methods. In this method, we work with the invariant submanifold of the problem, and we propose a series solution that is equivalent to the harmonic balance series solution. However, the recursive relation obtained for the coefficients in our analytical method simplifies traditional approaches to obtain the solution with the harmonic balance series method. This method can be applied to nonlinear dynamic systems under oscillatory input to find the analog of a usual Bode plot where regions of small and medium amplitude oscillatory input are well described. We found that the identification of such regions requires both the amplitude as well as the frequency to be properly specified. In the second paper of the series, the method to solve problems in the field of large amplitudes will be addressed.

Multivariate analytical methods for simultaneous estimation of Atenolol and Hydrochlorothiazide in bulk and tablet dosage form

The present study aimed to develop simple, accurate and precise FTIR and UV spectrophotometric methods for the quantification of Atenolol and Hydrochlorothiazide in bulk and tablet dosage forms. FT-IR method like classical least squares (CLS) was developed within the range of 2366.69-3433.44; 1564.40-1673.30 cm- UV methods like Cramer’s matrix method (method-I) and linear regression analysis (Method II) were developed and they are based upon constructing the matrix set by using molar absorptivity values at 275.60 nm and 270.40 nm. The assay values for FTIR- CLS method were 102% and 108 % for Atenololand Hydrochlorothiazide respectively. Cramer’s matrix method results were found to be 95.15% and 104% for Atenolol and Hydrochlorothiazide respectively and for linear regression method they were found to be 98.50% and 106% (w/w).

Fluid-Structure Interaction Analysis of Aircraft Hydraulic Pipe with Complex Constraints Based on Discrete Time Transfer Matrix Method

Fluid-structure interaction (FSI) is prevalent in aircraft hydraulic pipes due to high-pressure fluid pulsation, complex pipe path routing and boundary constraints, which pose a serious threat to the safety and reliability of the aircraft hydraulic system. This paper focuses on the FSI response of aircraft hydraulic pipes with complex constraints. A comprehensive fourteen-equation model for describing the FSI of pipe conveying fluid with wide pressure and Reynolds number range is proposed. The excitation models and complex boundary constraints of liquid-filled pipes are established. Moreover, based on the transfer matrix method (TMM), combined with the time discreteness and analytical integral method, a discrete time transfer matrix method (DTTMM) for solving the FSI fourteen-equation model in time domain is presented. Then, the numerical solution and experiment of an ARJ21-700 aircraft hydraulic pipe with complex constraints is carried out with four working conditions. The obtained results verify the correctness of the proposed model and solution method, and reveal the universal laws of the FSI response about aircraft hydraulic pipes, which can also provide theoretical and experimental references for modeling, solutions and verification in the FSI analysis of pipe conveying fluid.