Analytical-numerical Procedure Research Articles

Stability and convergence computational analysis of a new semi analytical-numerical method for fractional order linear inhomogeneous integro-partial-differential equations

Abstract The motive behind this research work, is to originate a semi-analytical-numerical approach for the solution of fractional order linear integro partial differential equations (FOLIPDEs), especially in-homogeneous FOLIPDEs of different types, like fractional version of Fredholm and Volterra type IEs etc. To attain the said purpose, we will study some fractional formulations of linear model integral equations from the existing literature. Then we will describe the proposed semi analytical-numerical procedure alongwith its stability analysis and convergence properties. We will then prove with the aid of some particular numerical examples that the said approach is not only lucid and efficient but also precise. The outcomes of the proposed technique will show that this scheme has notable prospectives for solving a large class of FOLIEs. At the end, the contribution of this work in the advancements of semi analytical-numerical approaches for
the solution of fractional order linear integro partial differential equations will be laid down and discussion for future research in this field.

A step-by-step method for time-dependent analysis of composite beams

Composite beams composed of reinforced concrete slab and steel section joined by mechanical connectors. The concrete slab provides compressive strength, fire resistance, and floor surface, whereas the steel section provides high tensile strength. Steel-concrete composite structures are widely used in frame buildings and bridges due to their economic and structural advantages. In the existing literature, a hybrid analytical-numerical procedure is available for service load analysis of steel-concrete composite beams considering cracking, creep, and shrinkage in concrete. The procedure uses an age-adjusted effective modulus method (AEMM) to model creep and shrinkage effects. The AEMM is computationally efficient however, it compromises accuracy owing to the use of the same aging coefficient for creep and shrinkage. Therefore, in this paper, a step-by-step method (SSM) has been proposed for modeling the creep and shrinkage effects. The method takes into account the progressive concrete cracking with time due to creep and shrinkage. The method is computationally efficient for large composite frame buildings compared to the finite element method since no discretization of the member, along the length and/or across the cross-section, is needed. The method has been validated against experimental results available in the literature. It is observed that the SSM has greater accuracy than the AEMM.

Study and analysis of the wiredrawing of CuZn37 wires via numerical simulations and slab method

Improving efficiency and productivity of industrial manufacturing activities is crucial in today’s global economy and the advanced engineering methods they have proven to be an effective solution to better understand the operation of the processes. The technological problem in wiredrawing is conditioned by multiple factors such as drawing die geometry, tribological system, process speed and the state of the metal to be drawn, among others. A wire rod of CuZn37 alloy brass with Ø8 mm is obtained by continuous casting process from electrolytic materials. The structural state of the initial wire rod and the draft sequence conditions the deformations and residual stresses accumulated in its cross section, affecting the sequential process and final quality. The present work proposes a complete analytical-numerical study aided to the better understanding of the problem for the design of a process proposal for an adequate continuous wiredrawing sequence for the production of Ø2 mm brass wire. An industrial sequence without intermediate annealing was evaluated by the analytical-numerical procedure allowing to better understand the technological problems involved. This methodology has allowed to check that it is not possible to achieve such a high degree of reduction ratio above of r=0.70÷0.75, without applying intermediate annealing heat treatment.

Biomechanical characterization of the passive response of the thoracic aorta in chronic hypoxic newborn lambs using an evolutionary strategy

The present study involves experiments and modelling aimed at characterizing the passive structural mechanical behavior of the chronic hypoxic lamb thoracic aorta, whose gestation, birth and postnatal period were carried at high altitude (3600 masl). To this end, the mechanical response was studied via tensile and pressurization tests. The tensile and pressurization tests measurements were used simultaneously to calibrate the material parameters of the Gasser–Holzapfel–Ogden (GHO) hyperelasctic anisotropic constitutive model through an analytical-numerical optimization procedure solved with an evolutionary strategy that guarantees a stable response of the model. The model and procedure of calibration adequately adjust to the material behavior in a wide deformation range with an appropriate physical description. The results of this study predict the mechanical response of the lamb thoracic aorta under generalized loading states like those that can occur in physiological conditions and/or in systemic arterial hypertension. Finally, the novel use of the evolutionary strategy, together with the set of experiments and tools used in this study, provide a robust alternative to validate biomechanical characterizations.

Бортовая система электроснабжения для электрических самолетов

The development of electric aircraft, the design of which will use, apart from doing away with pneumatic and hydraulic transmissions, electric propulsion, is one of promising lines in aviation technology. As an example, the project of the X-57 Maxwell electric aircraft developed by NASA, which is equipped with 14 propulsion electric motors, is given. In view of insufficient energy efficiency of existing storage batteries, a combined storage battery-and-generator power supply system for electric aircraft is considered. In the considered system, the aircraft propulsion is provided by electric motors, which are powered by storage batteries in combination with permanent magnet generators. The generator shaft is rotated by an auxiliary internal combustion engine, which operates in an economical mode. The power circuit of an active rectifier producing the +270 V DC bipolar output voltage is proposed, with which the permanent magnet generator operates with a sinusoidal phase current waveform and a close-to-unity power factor. The active rectifier control system uses an enhanced phase-locked loop, which eliminates the need to use generator rotor position sensors. For determining the stability domains and revealing undesirable hidden attractors in the considered storage battery-and-generator power supply system, it is proposed to use special analytical-numerical computation procedures based on modifications of Lyapunov's functions methods. The article will be of interest for developers of both aircraft and ground-based self-contained electric power supply systems with a bipolar 270 V DC voltage that must meet stringent requirements for improvement of their mass-and-dimensions and power performance characteristics.

Параметр термічної ректифікації у біматеріалі з міжфазною тріщинуватістю

The phenomenon of thermal rectification in bi-material with a periodic system of gas-filled interfacial cracks has been studied. The change of thermal resistance and pressure of the crack filler under the action of tensile forces and uniform heat flow is taken into account. The gas pressure is determined from the Clapeyron-Mendeleev equation. The thermal resistance of the gas is directly proportional to the opening of the cracks and inversely proportional to the thermal conductivity of the gas. The thermo-elastic problem is reduced to nonlinear systems of singular integro-differential equations for an opening cracks and a temperature jump between the cracks faces. An analytical-numerical iterative procedure is proposed to solve the system of singular integro-differential equations. The parameter of thermal rectification is determined and its dependence on the applied on the thermal and mechanical loads, thermal conductivity and mass of the gas and the interfacial cracking factor of the bi-material is analyzed. The nonlinear dependence of the thermal rectification parameter on the heat flow density and the interfacial cracking factor is established. The thermal rectification is manifested to a greater extent for higher values of heat flow and lower thermal conductivity of gas.

Determination of cop maximum of cold water loop of heat pump heating system by means of numerical-graphical optimization procedure

The paper presents the energy optimization of the cold water loop of the heat pump heating system using analytical-numerical procedure. The aim of the study is obtain the maximum COP of the heating system by optimum of the wall water mass-flow rate and well pump power. The objective function is the heating system?s COP. All components of the heating system: evaporator, condenser, compressor, circulation pump, and well pump are described by steady-state, lumped mathematical model. The model?s equations are coupled, non-linear, multivariable and algebraic, the solution is feasible using an iterative numerical method. The MATLAB?s program with Gauss elimination and Newton-linearization method is applied for solving the model. The obtained numerical data are presented in 3-D graphics. The optimum value of the cold-well water mass-flow rate is obtained from the graphics or by using a selection algorithm. The results of the study are the adequate mathematical model for energy optimization of the heating system, the numerical algorithm for solving the model and the ultimate goal to obtain the optimum of the power of well pump and compressor.

A new analytical-numerical solution to analyze a circular tunnel using 3D Hoek-Brown failure criterion

In this study, a new analytical-numerical procedure is developed to give the stresses and strains around a circular tunnel in rock masses exhibiting different stress-strain behavior. The calculation starts from the tunnel wall and continues toward the unknown elastic-plastic boundary by a finite difference method in the annular discretized plastic zone. From the known stresses in the tunnel boundary, the strains are calculated using the elastic-plastic stiffness matrix in which three dimensional Hoek-Brown failure criterion (Jiang and Zhao 2015) and Mohr-Coulomb potential function with proper dilation angle (i.e., non-associated flow rule) are employed in terms of stress invariants. The illustrative examples give ground response curve and show correctness of the proposed approach. Finally, from the results of a great number of analyses, a simple relationship is presented to find out the closure of circular tunnel in terms of rock mass strength and tunnel depth. It can be valuable for the preliminary decision of tunnel support and for prediction of tunnel problems.

Establishment of analytical-numerical method to determine the water level in the railway ballast layer contaminated with sandy material

Contamination of railway ballast can influence the permeability of ballast course and consequently the drainage potential of the track substructure. The present study elaborates the water level of the contaminated ballast with sandy material by establishment of laboratory test as well as analytical-numerical method. For this goal, the large-scale permeability tester is utilised to determine the hydraulic conductivity of ballast aggregate fouled with disparate ratios of sandy material. Then, the developed analytical-numerical procedure is applied to predict the elevation of water in the ballast course for considered conditions of boundary of ballast layer as well as specified values of rainfall rate. Based on the obtained results, the permeability of extremely contaminated ballast with sand is nevertheless appropriate. However, the estimated water level of the ballast course is drastically influenced by further contamination of railway ballast layer especially under higher rainfall rates. Meanwhile, the increment of water level on two sides of the ballast course eventuates in more uniform water table along the cross-section of the course which results in more saturation of track substructure.

Enhanced variational approach for damage analysis of laminated composite

The main objective of this research is to introduce a new coupled analytical–numerical procedure to calculate stiffness reduction due to matrix cracks in composite laminates subjected to general loading conditions. A novel micromodel is developed for transverse cracking of general symmetric laminates based on a variational approach. The proposed method overcomes the restrictions of previous similar approaches regarding damage in laminates with general symmetric stacking sequences. A polynomial series is used for calculating the stress perturbation due to intralaminar cracks in conjunction with the principle of minimum complementary energy. The numerical part deals with well-known Newton–Raphson procedure to solve the governing equations for the unknown parameters for the stress functions. Furthermore, the analytical part can be solved just one time by a symbolic calculation for unknown geometry parameters, material properties, and loading conditions, which has the benefit to be ready for Finite Element software implementations. Comparison between the existing experimental and analytical results and the result of the present study are in good agreement.

Controlling the One-Dimensional Motion of Hybrid Vibrational Rod Systems

A mathematical model of the one-dimensional controlled motion of a hybrid vibrational system is proposed. The plant includes an inhomogeneous elastic rod with loads concentrated at its endpoints. An analytical-numerical procedure for finding the eigenvalues (eigenfrequencies) and eigenfunctions is developed. A novel procedure to take into account the nonuniform distributed forces and forces concentrated at the endpoints using a modification of Grinberg’s approach is presented. Efficient statements and constructive approximate solutions of the problems for controlling the motion of a countably dimensional vibrational system with a complex distribution of eigenfrequencies are proposed.

Neural network based approach for rapid prediction of deflections in RC beams considering cracking

Maximum deflection in a beam is a serviceability design criterion and occurs generally at or close to the mid-span. This paper presents a methodology using neural networks for rapid prediction of mid-span deflections in reinforced concrete beams subjected to service load. The closed form expressions are further obtained from the trained neural networks. The closed form expressions take into account cracking in concrete at in-span and at near the interior supports and tension stiffening effect. The expressions predict the inelastic deflections (incorporating the concrete cracking) from the elastic moments and the elastic deflections (neglecting the concrete cracking). Five separate neural networks are trained since these have been postulated to represent all beams having any number of spans. The training, validating, and testing data sets for the neural networks are generated using an analytical-numerical procedure of analysis. The proposed expressions have been verified by comparison with the experimental results reported elsewhere and also by comparison with the finite element method (FEM). The proposed expressions, at minimal input data and minimal computation effort, yield results that are close to FEM results. The expressions can be used in every day design since the errors are found to be small.

Closed-form expressions for long-term deflections in high-rise composite frames

This paper presents closed-form expressions for rapid prediction of long-term deflections in high-rise steel concrete composite frames subjected to service load. The closed-form expressions predict the inelastic mid-span deflections in beams of frames (typically for 20 years, considering cracking, and time effects, i.e., creep and shrinkage in concrete) from the elastic moments and elastic mid-span deflections (neglecting cracking, and time effects). The expressions also take into account the sagging moments developed in beams due to the substantial differential shortening of adjacent columns in high-rise frames. The expressions can be used for frames with any number of bays and storeys. The expressions have been obtained from trained neural networks. The training, validating, and testing data sets for the neural networks are generated using a hybrid analytical-numerical procedure of analysis. The proposed expressions are verified for example frames of different number of spans and storeys and the errors are shown to be small. The expressions can be used in every day design as they enable a rapid prediction of inelastic deflections with reasonable accuracy for practical purposes without detailed complex analysis and require computational effort that is a fraction of that required for the available methods.

Zeroes of functions of Fresnel complementary integral type

Theoretical upper and lower bounds are established for zeroes of a parametric family of functions which are defined by integrals of the same type as the Fresnel complementary integral. Asymptotic properties for these bounds are obtained as well as monotony properties of the localization intervals. Given the value of the parameter an analytical-numerical procedure is deduced to enclose all zeros of a given function with an a priori error.

Explicit expressions for inelastic design quantities in composite frames considering effects of nearby columns and floors

Explicit expressions for rapid prediction of inelastic design quantities (considering cracking of concrete) from corresponding elastic quantities, are presented for multi-storey composite frames (with steel columns and steel-concrete composite beams) subjected to service load. These expressions have been developed from weights and biases of the trained neural networks considering concrete stress, relative stiffness of beams and columns including effects of cracking in the floors below and above. Large amount of data sets required for training of neural networks have been generated using an analyticalnumerical procedure developed by the authors. The neural networks have been developed for moments and deflections, for first floor, intermediate floors (second floor to ante-penultimate floor), penultimate floor and topmost floor. In the case of moments, expressions have been proposed for exterior end of exterior beam, interior end of exterior beam and both interior ends of interior beams, for each type of floor with a total of twelve expressions. Similarly, in the case of deflections, expressions have been proposed for exterior beam and interior beam of each type of floor with a total of eight expressions. The proposed expressions have been verified by comparison of the results with those obtained from the analytical-numerical procedure. This methodology helps to obtain the inelastic design quantities from the elastic quantities with simple calculations and thus would be very useful in preliminary design.

An automated computationally efficient two-stage procedure for service load analysis of RC flexural members considering concrete cracking

An automated computationally efficient two-stage procedure has been proposed for service load analysis of reinforced concrete (RC) flexural members considering concrete cracking and tension stiffening. The proposed procedure yields cracked lengths, redistributed bending moments and inelastic deflections. The computation of final state, including cracked lengths and interpolation coefficients for tension stiffening, is automated by the initialization of second stage from the results of first stage. The procedure combines the analytical–numerical procedure developed by authors and the neural networks methodology. The cracked lengths and corresponding interpolation coefficients are rapidly estimated in the first stage using the closed-form expressions which are obtained from the trained neural networks. Eight separate neural networks are trained for the estimation of final cracked lengths and interpolation coefficients at in-span locations and supports. The use of estimated cracked lengths and interpolation coefficients of the first stage, in the beginning of second stage, significantly reduces number of iterations in the second stage. The deflections and bending moments obtained from the two-stage procedure are compared with those from the analytical–numerical procedure for a number of beams. The analytical–numerical procedure requires around six analyses to yield results with sufficient accuracy for design purpose (within 2–3%); whereas, in the two-stage procedure, only two analyses, one in each stage, are required to yield results with similar accuracy. The developed two-stage procedure requires a significantly small computational effort as compared to similarly accurate methods available in literature.

Service load analysis of composite frames using cracked span length frame element

A cracked span length frame element has been proposed for service load analysis of composite frames consisting of steel columns and steel-concrete composite beams. The element comprises a maximum of three regions (cracked or uncracked) with simplified formulation and is applicable for all types of loading. This element has been used in an analytical-numerical procedure for the inelastic analysis of such composite frames, subjected to service loads. Analytical expressions have been derived for flexibility and stiffness coefficients, end displacements, load vector and mid span deflections of the cracked span length frame element. Average values over the cracked regions have been used for the tension stiffening characteristics, so as to retain the analytical nature of the procedure at the element level. The procedure uses an iterative technique for establishing the cracked region lengths and the distribution coefficients (for tension stiffening), and yields the inelastic deflections and redistributed moments. The procedure using the proposed element has been validated by comparing the results with experimental results available in literature and also with the results obtained from finite element analysis. The procedure requires a fraction of the computational effort that is required for the numerical methods available in literature and gives sufficiently accurate results. Therefore, the reduction in the computational efforts in the case of high-rise composite frames would be considerable.

Rapid prediction of inelastic bending moments in RC beamsconsidering cracking

A methodology using neural networks has been proposed for rapid prediction of inelastic bending moments in reinforced concrete continuous beams subjected to service load. The closed form expressions obtained from the trained neural networks take into account cracking in concrete at in-span and at near the internal supports and tension stiffening effect. The expressions predict the inelastic moments (considering the concrete cracking) from the elastic moments (neglecting the concrete cracking) at supports. Three separate neural networks are trained since these have been postulated to represent all the beams having any number of spans. The training, validating, and testing data sets for the neural networks are generated using an analytical-numerical procedure of analysis. The proposed expressions are verified for example beams of different number of spans and cross-section properties and the errors are found to be small. The proposed expressions, at minimal input data and computation effort, yield results that are close to FEM results. The expressions can be used in preliminary every day design as they enable a rapid prediction of inelastic moments and require a computational effort that is a fraction of that required for the available methods in literature.

An element incorporating cracking for reinforced concrete skeletal structures at service load

An element has been proposed to take into account cracking in the reinforced concrete skeletal structures subjected to a service load. A typical skeletal member is modeled as a single element and is visualized to consist of at most five zones (cracked or uncracked). Closed-form expressions for the flexibility and stiffness coefficients and end displacements have been obtained. Furthermore, for use in everyday design, a hybrid analytical–numerical procedure has been developed using the proposed element. The procedure is analytical at the element level and numerical at the structural level. To keep the procedure analytical at the element level, the average tension stiffening characteristics are arrived at for the cracked zones. The developed procedure has been validated in limiting cases by comparison with the experimental results reported elsewhere and by comparison with the finite element method results. The proposed element would lead to a drastic reduction in computational time for large reinforced concrete structures, for example, tall reinforced concrete building frames.

Analytical-Numerical Determination of Stress Distribution around a Tip of Polygon-Like Inclusion

A stress distribution in vicinity of a tip of polygon-like inclusion exhibits a singular stress behaviour. Singular stresses at the tip can be a reason for a crack initiation in composite materials. Knowledge of stress field is necessary condition for reliable assessment of such composites. A stress field near the general singular stress concentrator can be analytically described by means of Muskhelishvili plane elasticity based on complex variable functions. Parameters necessary for the description are the exponents of singularity and Generalized Stress Intensity Factors (GSIFs). The stress field in the closest vicinity of the SMI tip is thus characterized by 1 or 2 singular exponents (1 - λ) where, 0<Re (λ)<1, and corresponding GSIFs that follow from numerical solution. In order to describe stress filed further away from the SMI tip, the non-singular exponents for which 1<Re (λ), and factors corresponding to these non-singular exponents have to be taken into account. Analytical-numerical procedure of determination of stress distribution around a tip of sharp material inclusion is presented. Parameters entering to the procedure are varied and tuned. Thus recommendations are stated in order to gain reliable values of stresses and displacements.