Solving nonlinear boundary value problems by a boundary shape function method and a splitting and linearizing method

In the paper, we develop two novel iterative methods to determine the solution of a second-order nonlinear boundary value problem (BVP), which precisely satisfies the specified non-separable boundary conditions by taking advantage of the property of the corresponding boundary shape function (BSF). The first method based on the BSF can exactly transform the BVP to an initial value problem for the new variable with two given initial values, while two unknown terminal values are determined iteratively. By using the BSF in the second method, we derive the fractional powers exponential functions as the bases, which automatically satisfy the boundary conditions. A new splitting and linearizing technique is used to transform the nonlinear BVP into linear equations at each iteration step, which are solved to determine the expansion coefficients and then the solution is available. Upon adopting those two novel methods very accurate solution for the nonlinear BVP with non-separable boundary conditions can be found quickly. Several numerical examples are solved to assess the efficiency and accuracy of the proposed iterative algorithms, which are compared to the shooting method.

A machine learning pipeline for autonomous numerical analytic continuation of Dyson-Schwinger equations

Dyson-Schwinger equations (DSEs) are a non-perturbative way to express n-point functions in quantum field theory. Working in Euclidean space and in Landau gauge, for example, one can study the quark propagator Dyson-Schwinger equation in the real and complex domain, given that a suitable and tractable truncation has been found. When aiming for solving these equations in the complex domain, that is, for complex external momenta, one has to deform the integration contour of the radial component in the complex plane of the loop momentum expressed in hyper-spherical coordinates. This has to be done in order to avoid poles and branch cuts in the integrand of the self-energy loop. Since the nature of Dyson-Schwinger equations is such, that they have to be solved in a self-consistent way, one cannot analyze the analytic properties of the integrand after every iteration step, as this would not be feasible. In these proceedings, we suggest a machine learning pipeline based on deep learning (DL) approaches to computer vision (CV), as well as deep reinforcement learning (DRL), that could solve this problem autonomously by detecting poles and branch cuts in the numerical integrand after every iteration step and by suggesting suitable integration contour deformations that avoid these obstructions. We sketch out a proof of principle for both of these tasks, that is, the pole and branch cut detection, as well as the contour deformation.

A regularization method for solving dynamic problems with singular configuration

In the simulation process for multi-body systems, the generated redundant constraints will result in ill-conditioned dynamic equations, which are not good for stable simulations when the system motion proceeds near a singular configuration. In order to overcome the singularity problems, the paper presents a regularization method with an explicit expression based on Gauss principle, which does not need to eliminate the constraint violation after each iteration step compared with the traditional methods. Then the effectiveness and stability are demonstrated through two numerical examples, a slider-crank mechanism and a planar four-bar linkage. Simulation results obtained with the proposed method are analyzed and compared with augmented Lagrangian formulation and the null space formulation in terms of constraints violation, drift mechanical energy and computational efficiency, which shows that the proposed method is suitable to perform efficient and stable dynamic simulations for multi-body systems with singular configurations.

Simulation of heat transfer with considering permafrost thawing in 3D media

An approach to mathematical modeling of heat transfer with a permafrost algorithm in 3D media based on the idea of localizing the phase transition area is considered. The paper presents a problem statement for a non-stationary heat transfer and a description of a numerical method based on a predictor-corrector scheme. For a better understanding of the proposed splitting method, the accuracy order of approximation considering inhomogeneous right-hand side was studied. The phase changes in the numerical implementation of permafrost thawing is considered in the temperature range and requires recalculation of coefficients values of the heat equation at each iteration step with respect to time. A brief description of the parallel algorithm based on a 3D decomposition method and the parallel sweep method is presented. A study of the parallel algorithm implementations using a high-performance computing system of the Siberian Supercomputer Center of the SB RAS was performed. The results of the permafrost algorithm on models with wellbores are also presented.

Improved trilateration for indoor localization: Neural network and centroid-based approach

Location awareness is the key to success to many location-based services applications such as indoor navigation, elderly tracking, emergency management, and so on. Trilateration-based localization using received signal strength measurements is widely used in wireless sensor network–based localization and tracking systems due to its simplicity and low computational cost. However, localization accuracy obtained with the trilateration technique is generally very poor because of fluctuating nature of received signal strength measurements. The reason behind such notorious behavior of received signal strength is dynamicity in target motion and surrounding environment. In addition, the significant localization error is induced during each iteration step during trilateration, which gets propagated in the next iterations. To address this problem, this article presents an improved trilateration-based architecture named Trilateration Centroid Generalized Regression Neural Network. The proposed Trilateration Centroid Generalized Regression Neural Network–based localization algorithm inherits the simplicity and efficiency of three concepts namely trilateration, centroid, and Generalized Regression Neural Network. The extensive simulation results indicate that the proposed Trilateration Centroid Generalized Regression Neural Network algorithm demonstrates superior localization performance as compared to trilateration, and Generalized Regression Neural Network algorithm.

Analysis Of The Effect Of Bending Strength On Scaffolding System With Direct Analysis Method

Steel scaffolding is a very important component in formwork work to support further work. The purpose of this analysis is to review the maximum compressive strength that occurs in 3-story scaffolding before buckling occurs using the direct analysis method (DAM). The design of steel structures, which are generally slender, requires stability analysis. The result is influenced by imperfections (non-linear geometry) and inelastic conditions (non-linear material). In this final project, we use second-order inelastic analysis based on direct analysis method. The 3-level scaffolding model was analyzed using beam elements in the SAP2000 program with 6 variations of notional loads applied to the weak axis direction of the scaffolding pipe.The lowest compressive strength on 3-story scaffolding before buckling occurs is 18.24 kN with horizontal notional loads to the right on the first level scaffolding, left on the second level scaffolding and to the right on the 3rd level scaffolding. The results of the analysis show that the maximum compressive strength obtained results in a large displacement drastically in the iteration step. By using the analysis on the DAM method, the results obtained are more effective.

Feature Extraction of Music Signal Based on Adaptive Wave Equation Inversion

The digitization, analysis, and processing technology of music signals are the core of digital music technology. There is generally a preprocessing process before the music signal processing. The preprocessing process usually includes antialiasing filtering, digitization, preemphasis, windowing, and framing. Songs in the popular wav format and MP3 format on the Internet are all songs that have been processed by digital technology and do not need to be digitalized. Preprocessing can affect the effectiveness and reliability of the feature parameter extraction of music signals. Since the music signal is a kind of voice signal, the processing of the voice is also applicable to the music signal. In the study of adaptive wave equation inversion, the traditional full-wave equation inversion uses the minimum mean square error between real data and simulated data as the objective function. The gradient direction is determined by the cross-correlation of the back propagation residual wave field and the forward simulation wave field with respect to the second derivative of time. When there is a big gap between the initial model and the formal model, the phenomenon of cycle jumping will inevitably appear. In this paper, adaptive wave equation inversion is used. This method adopts the idea of penalty function and introduces the Wiener filter to establish a dual objective function for the phase difference that appears in the inversion. This article discusses the calculation formulas of the accompanying source, gradient, and iteration step length and uses the conjugate gradient method to iteratively reduce the phase difference. In the test function group and the recorded music signal library, a large number of simulation experiments and comparative analysis of the music signal recognition experiment were performed on the extracted features, which verified the time-frequency analysis performance of the wave equation inversion and the improvement of the decomposition algorithm. The features extracted by the wave equation inversion have a higher recognition rate than the features extracted based on the standard decomposition algorithm, which verifies that the wave equation inversion has a better decomposition ability.

FRACTAL MODEL OF ESTIMATING QUALITY OF COLD WORKED FUEL CLADDING TUBES

A possibility was considered concerning estimation of grain anisomery in the structure of fuel cladding tubes of corrosion-resistant 026Cr16Ni15Mo3Nb steel of austenitic class rolled according to two flow charts: regular and intensive technologies using fractal formalism. Role of grain boundary hardening during cold plastic deformation was analyzed by studying the effect of the fractal dimension of grains D and their boundaries Dg on 0.2, w, and 5. The best correlation among those that were considered was observed between relative elongation and fractal dimensions of the grain structure (R2 = 0.90). The smallest correlation was observed with the yield stress (R2 = 0.64). It is because of variation of plastic flow processes towards a decrease in the degree of hardening in the material rolled according to the intensive technology. Cold deformation results in refining of the average grain size from 15.50 to 15.42 µm. In this case, extent of the grain boundary length L increased by 17.62% at an iteration step  commensurate with the average grain size which is indicated by a change in the fractal dimension according to L ~ δ1-D. Degree of the grain structure inhomogeneity was estimated using ratios of self-similarity of regions of fractal dimensions of the structure. The obtained results on the level of mechanical properties of fuel cladding tubes made of austenitic steel indicate advantage of the intensive technology over regular one that was confirmed by results of fractal modeling.

Application of Magnetic Nano-Immobilized Enzyme in Soybean Oil Degumming: Numerical Simulation in a Liquid-Solid MFB

Using crude soybean oil (CSO) as fluid and nanomagnetic immobilized phospholipase C (PLC) as fluidizing particles, the Eulerian–Lagrangian fluid-particle two-phase flow model was used to numerically simulate the law of motion of fluidizing particles in the magnetic fluidized bed (MFB). The main parameters were obtained by numerical simulation based on the discrete element method (DEM). The nanomagnetic PLC in the MFB was optimal to the enzymatic reaction by limiting the iteration step size to 3 × 10−5, the boundary condition to 20 × 300 mm, the opening rate to 37.5%, the condition of CSO flow rate to 0.01 m/s, and the magnetic field strength to 0.02T. After 2.0 h of reaction, the amount of residual phosphorus in the oil was 55.73 mg/kg, the content of 1, 2-DAG was 1.42%, and the nanomagnetic enzyme still had 97% relative activity. Hence, these optimal conditions can improve the efficiency and the stability of the nanomagnetic enzymatic reaction.

The Proof of a Conjecture Relating Catalan Numbers to an Averaged Mandelbrot-Möbius Iterated Function

In 2021, Mork and Ulness studied the Mandelbrot and Julia sets for a generalization of the well-explored function ηλ(z)=z2+λ. Their generalization was based on the composition of ηλ with the Möbius transformation μ(z)=1z at each iteration step. Furthermore, they posed a conjecture providing a relation between the coefficients of (each order) iterated series of μ(ηλ(z)) (at z=0) and the Catalan numbers. In this paper, in particular, we prove this conjecture in a more precise (quantitative) formulation.