Numerical Integration Procedure Research Articles

Геодезическая нутация Юпитера и его галилеевых спутников

In the article, the relativistic effect of geodetic nutation in the rotation of Jupiter and its Galilean moons (Io, Europa, Ganymede and Callisto) was investigated in two ways over an 800-year time interval. As a result, for the first time, the most significant periodic terms of the geodetic rotation of these celestial bodies relative to: a) the barycenter of the Solar system and the plane of the mean orbit of Jupiter of the epoch J2000.0 in the Euler angles, in the disturbing terms of the physical libration and in the absolute magnitude of the angular rotation vector of the geodetic rotation of the body under study; b) (with the exception of Jupiter) the barycenter of the Jupiter satellite system and the mean orbit of the studied satellite of the epoch J2000.0 in the disturbing terms of the physical libration and in the absolute magnitude of the angular rotation vector of the geodetic rotation of the body under study. It is shown that the use of Way II in this study, which uses the numerical integration procedure, is more preferable, and the values of the geodetic nutation terms obtained by this way are more accurate. The accuracy level of calculating the parameters (obtained by Way II) for the geodetic nutation values of the studied celestial bodies was 0.1 microarcseconds. The obtained analytical values of geodetic nutation of the studied celestial bodies can be used for numerical investigation of the rotation of these bodies in the relativistic approximation.

Determination of internal efforts in the base finite elements of SAFEM

The principles of calculating the internal efforts of a circular finite element in the semi-analytical finite element method (FEM) based on the obtained components of the stress tensor and the peculiarities of the approach associated with the use of the moment scheme of the finite element (FEM) are considered. Formulas for determining longitudinal, shear forces, bending and torque moments have been obtained.
 A special place, among the variety of objects considered with the help of analytical and numerical methods, is occupied by bodies of revolution of complex shape and cross-sectional structure, formed by the movement of some creative surface along a closed or opened line without breaks. The selected geometric class is used as natural structures of nodes and details in construction of mechanical engineering. The sufficiently wide distribution of the specified forms in the construction and machine-building industries, on the one hand, and the possibility of significantly simplifying the solving relationships by taking into account their geometric features, on the other hand, provide a basis for the development and use of various modifications of the finite element method (FEM). The semi-analytical finite element method (SAFEM) is one such approach that has gained widespread use for solving problems whose objects are prismatic bodies and bodies of revolution of complex shape and cross-sectional structure. Due to the introduction of additional hypotheses that do not reduce the accuracy of the approximation, the representation of deformations and stresses in physical terms and in accordance with the moment scheme of the finite element (MSFE), on the one hand, it is possible to avoid the time-consuming procedure of numerical integration over the cross-sectional area of the finite element (FE), on the other hand - maintain the high efficiency of 3D discretization.
 Despite the large number of publications devoted to the semi-analytical method of finite elements, the question of determining internal forces, which are often component factors of the strength criteria laid down in state building codes, is inappropriately neglected. The use of SAFEM in combination with МSFЕ creates some mathematical features of calculating internal longitudinal, shearing forces and moments.

About the Differential Sweep Method for Measuring of Ferrocolloids Magnetization Curves

Purpose. Justification and description of a laboratory method for measuring static magnetization curves specialized for ferrocolloids.Methods. The measurement method is based on the paramagnetism of magnetic colloids and the quasi-linear response of their magnetisation to small perturbations of the external magnetic field. To obtain the magnetisation curve, the studying ferrocolloid sample is placed in a constant homogeneous field of a laboratory electromagnet with an iron core. By low-frequency modulation of the current in the coils of the electromagnet, a co-directional perturbation is applied to the constant field. Information about the response of the sample to the external field perturbation - the differential magnetic susceptibility of ferrocolloid - is extracted by electrical measurements. These measurements are carried out using a classical compensation device of two counter-connected wire coils, one of which contains the investigated sample. Conducting (sweeping) the measurements in a wide range of applied fields allows to collect a sequence of experimental values of differential susceptibility from which the desired magnetisation curve is reconstructed by numerical integration.Results. The experimental setup for measuring the magnetisation curves of ferrocolloids was assembled. A theoretical description of the compensating electrical measuring device of the setup was proposed. The adjustment of the electrical scheme was carried out within several series of calibration experiments aimed at establishing the material parameters of the setup that were unknown from the theory. On the example of ferrocolloid of the type “magnetite - kerosene – oleic acid” both the process of obtaining primary experimental data and their subsequent processing, including the procedure of numerical integration, were demonstrated. It is established that the use of integration methods of a higher accuracy allows reducing the number of required experimental points and accelerating the measurement process without reducing the quality of the obtained curves.Conclusion: The method applicable for measuring the magnetisation curves of ferrocolloids by differential sweeping is described, substantiated and implemented using laboratory equipment.

Development of an efficient calculation technique for dynamics of mooring lines by using discrete forms of rotation

Development of efficient calculation techniques is of key importance for design of deepwater moored floating structures consisting of many subsystems, such as a floating vessel, risers and mooring lines. In particular, this study focuses on a method for simulating dynamic behavior of the mooring line, since it includes a lot of degrees of freedom and geometric nonlinear effects. A modified Morison equation based on the linear potential flow theory is employed to derive the hydrodynamic forces acting on the lines. The present approach describes the mooring lines (chain) as multi-rigid bodies based on the multibody dynamics approach. In addition, a discrete form of rotation called “the rotation update equation” is introduced. This technique approximates the rotation by using incremental components of rotation in terms of the angular velocity. This approach can bypass the parameterization of rotation in the equations of motion. Clearly, it contributes to reduction of computational costs. Furthermore, a preconditioning method is applied to the equations of motions in order to prevent the ill-conditioning which affects convergence of the numerical integration procedures for the set of equations with configuration level constraints. Finally, a comparison of results by the present method and the conventional method are presented in order to show the validity of the method.

Optimization on Elastoplasticity of Functionally Graded Shells of Revolution under Axisymmetric Loading

Depending on the load level, structures can experience a material nonlinearity known as elastoplasticity, which has an important role in the behaviour of structures. In order to avoid the elastoplastic behaviour, it is necessary to find the optimal thickness distribution, which corresponds to the minimum mass that provides an elastic behaviour for a certain load level. The elastoplasticity analysis of functionally graded axisymmetric shells under axisymmetric mechanical loading, and the subsequent optimization, was performed by using a simple conical frustum finite element model with two nodal circles; three degrees of freedom per node, which was based on Kirchhoff’s theory allowing for shear deformation; and using a reduced numerical integration procedure that is essential for its success when applied to thin shells. The formulation accounts for the calculation of the displacements and through-thickness stress distribution, including the effective stress. In this work, the thickness was the design variable in the optimization procedure and the mass was the objective function that needed to be minimized subject to a constraint imposed on the effective stress. The optimization solutions were obtained by using a feasible arc interior point gradient-based algorithm. Some illustrative examples were performed, and the corresponding results are presented and discussed.

Approximating piecewise nonlinearities in dynamic systems with sigmoid functions: advantages and limitations

In the industry field, the increasingly stringent requirements of lightweight structures are exposing the ultimately nonlinear nature of mechanical systems. This is extremely true for systems with moving parts and loose fixtures which show piecewise stiffness behaviours. Nevertheless, the numerical solution of systems with ideal piecewise mathematical characteristics is associated with time-consuming procedures and a high computational burden. Smoothing functions can conveniently simplify the mathematical form of such systems, but little research has been carried out to evaluate their effect on the mechanical response of multi-degree-of-freedom systems. To investigate this problem, a slightly damped mechanical two-degree-of-freedom system with soft piecewise constraints is studied via numerical continuation and numerical integration procedures. Sigmoid functions are adopted to approximate the constraints, and the effect of such approximation is explored by comparing the results of the approximate system with the ones of the ideal piecewise counter-part. The numerical results show that the sigmoid functions can correctly catch the very complex dynamics of the proposed system when both the above-mentioned techniques are adopted. Moreover, a reduction in the computational burden, as well as an increase in numerical robustness, is observed in the approximate case.

EFFECTS OF SLIP VELOCITY AND VISCOUS DISSIPATION ON MHD BOUNDARY LAYER FLOW OF A MICROPOLAR-NANOFLUID OVER A WEDGE WITH INTERNAL HEAT GENERATION

A numerical model is developed to examine the boundary layer flow of an electrically conducting, viscous incompressible micropolar nanofluid (Al2O3/water) over a wedge in the presence of a transverse magnetic field and viscous dissipation with internal heat generation/absorption.
 The combined effect of both cases constant fluid suction and injection is considered, also the velocity slip’s effect is also taken into account. The governing equations have been solved using the Runge-Kutta numerical integration procedure after reducing them to boundary layer equations. Various effects of parameters that govern the flow like velocity, micro-rotation, temperature as well as for local skin friction coefficient, local Nusselt number and local wall couple stress have been illustrated graphically.

Dynamic responses of an axisymmetric thermo-poroelastic half-space subjected to thermo-mechanical loads

The dynamic responses of a thermo-poroelastic half-space subjected to circular surface thermal or/and mechanical loads are studied semi-analytically. With the consideration of water seepage and inertia of both the soil particles and pore water, the governing equations of the thermo-poroelastic medium are established based on Biot’s dynamic poroelastic theory. The solutions of the governing equations are first derived in the transformed domain using Laplace-Hankel transform, following which the solutions in the time domain are obtained utilizing a numerical integration procedure. Finally, the dynamic responses of the thermo-poroelastic half-space under a rectangular thermal pulse load and/or a half-sinusoidal impact mechanical load are graphically presented. It is found that both the thermal loads and mechanical loads can induce time-dependent responses in the vertical displacements, vertical stresses, and pore water pressures within the saturated poroelastic medium. The resulting coupled dynamic responses of the thermo-poroelastic half-space due to the simultaneous application of the thermal and mechanical loads can be obtained as the superposition of the individual thermal and mechanical responses.

Algorithms analysis for solving geometrically nonlinear mechanics problems in the scheme of the semi-analytical finite element method

The effectiveness of using the semi-analytical finite element method (SAFEM) to research geometrically nonlinear construction mechanics problems for 3D bodies of revolution under the arbitrary loads based on a basic ring finite element is considered. An estimate of the rational application parameters of algorithms for taking into account the geometric nonlinearity of a defined above structures class, which has been obtained.
 In the process of numerically solving spatial problems of the theory of elasticity and plasticity with finite displacements, the choice of rational algorithms for solving systems of nonlinear equations is of fundamental importance. It is conditioned by the need of determining the coordinates of the discrete model in the actual configuration and changing the metric characteristics of the finite elements, which, in its turn, leads to the necessity for multiple solutions of systems of nonlinear equations of high order. Due to the introduction of additional hypotheses that do not reduce the accuracy of the approximation: the representation of deformations and stresses in physical terms and in accordance with the moment scheme of the finite element (MSFEM), it is possible, on the one hand, to avoid the time-consuming procedure of numerical integration over the cross-sectional area of the finite element, on the other hand- to maintain a high efficiency of spatial discretization. An important stage in the implementation of computer systems for solving spatial problems is the selection of optimal, from the point of the solution convergence speed and algorithms for solving equilibrium equations. The specificity of the algebraic equations of the SAFEM is conditioned by violation of the trigonometric function orthogonality in the space of the elasticity operator for bodies with variable stiffness and mass parameters along the guide. The clearly defined block structure of the stiffness matrix became the basis for using algorithms combining direct and iterative methods of solving. The reliability of the obtained results and the effectiveness of the approach are confirmed by the solution of a wide range of control examples under various boundary conditions and external loads.

Analytical ray transfer matrix for the crystalline lens.

We present the formulation of a paraxial ray transfer or ABCD matrix for onion-type GRIN lenses. In GRIN lenses, each iso-indicial surface (IIS) can be considered a refracting optical surface. If each IIS is a shell or layer, the ABCD matrix of a GRIN lens is computed by multiplying a typically high number of translation and refraction matrices corresponding to the K layers inside the lens. Using a differential approximation for the layer thickness, this matrix product becomes a sum. The elements A, B, C, and D of the approximated GRIN ray transfer matrix can be calculated by integrating the elements of a single-layer matrix. This ABCD matrix differs from a homogeneous lens matrix in only one integration term in element C, corresponding to the GRIN contribution to the lens power. Thus the total GRIN lens power is the sum of the homogeneous lens power and the GRIN contribution, which offers a compact and simple expression for the ABDC matrix. We then apply this formulation to the crystalline lens and implement both numerical and analytical integration procedures to obtain the GRIN lens power. The analytical approximation provides an accurate solution in terms of Gaussian hypergeometric functions. Last, we compare our numerical and analytical procedures with published ABCD matrix methods in the literature, and analyze the effect of the iso-indicial surface's conic constant (Q) and inner curvature gradient (G) on the lens power for different lens models.

On the Construction of Growth Models via Symmetric Copulas and Stochastic Differential Equations

By nature, growth regulatory networks in biology are dynamic and stochastic, and feedback regulates their growth function at different ages. In this study, we carried out a stochastic modeling of growth networks and demonstrated this method using three mixed effect four-parameter Gompertz-type diffusion processes and a combination thereof using the conditional normal copula function. Using the conditional normal copula, newly derived univariate distributions can be combined into trivariate and bivariate distributions, and their corresponding conditional bivariate and univariate distributions. The link between the predictor variable and the remaining one or two explanatory variables can be formalized using copula-type densities and a numerical integration procedure. In this study, for parameter estimation, we used a semiparametric maximum pseudo-likelihood estimator procedure, which was characterized by a two-step technique, namely, separately estimating the parameters of the marginal distributions and the parameters of the copula. The results were illustrated using two observed longitudinal datasets, the first of which included the age, diameter, and potentially available area of 39,437 trees (48 stands), while the second included the age, diameter, potentially available area, and height of 8604 trees (47 stands) covering uneven mixed-species (pine, spruce, and birch) stands. All results were implemented using the MAPLE symbolic algebra system.

Dynamic fracture analysis in quasi-brittle materials via a finite element approach based on the combination of the ALE formulation and M−integral method

This work proposes an efficient FE-based approach for reproducing dynamic crack propagation phenomena in quasi-brittle materials. The proposed model uses a Moving Mesh technique based on the Arbitrary Lagrangian-Eulerian formulation (ALE) to adapt the computational mesh consistently to the geometry variations caused by dynamically growing cracks. Specifically, the motion of mesh nodes occurs according to Fracture Mechanics criteria, which provide suitable conditions regarding the direction and velocity of a growing crack tip. These conditions usually depend on the Dynamic Stress Intensity Factors (DSIFs) at the crack front. For extracting the DIFSs at a moving crack tip, this work introduces the ALE formulation of the dynamic M−integral as a key novelty. This strategy offers the key advantage of performing numerical integration procedures on deforming finite elements without losing accuracy. The validity of the proposed method has been assessed through comparisons with experimental and numerical data reported in the literature.

Transport coefficients for modeling fission dynamics

Transport coefficients are important ingredients in dynamical investigations of a nuclear fission process, which determine the dynamical evolution of the fissioning system. The inertia (mass) and friction (dissipation) tensors influence substantially on different observables. The proper calculation of these tensors is a necessary condition for correct modeling of the fission process. The calculation of the inertia and friction tensors is presented for a large variety of nuclear shapes generated by a {c,h,α} parametrization. The inertia tensor is calculated within the Werner-Wheeler approximation for incompressible irrotational flow. The friction tensor is calculated within the one-body and two-body mechanisms of nuclear dissipation. A numerical code is made available, which calculates all these quantities. Program summaryProgram Title: Mass_DissCPC Library link to program files:https://doi.org/10.17632/rcndf6sbns.1Licensing provisions: CC BY NC 3.0Programming language: C++Nature of problem: The modeling of the nuclear fission within dynamical consideration requires calculation of the inertia and friction coefficients. This calculation should be performed for a large variety of the nuclear shapes, which describe the evolution of the shape of compound nucleus from the ground state till the separated fission fragments.Solution method: The inertia tensor is calculated within the Werner-Wheeler approximation for incompressible irrotational flow. The friction tensor is calculated within the one-body and two-body mechanisms of the nuclear dissipation. The shape of the fissioning nucleus is described by the {c,h,α} parametrization. This parametrization determines the axially symmetric shapes in cylindrical coordinates. The inertia and friction tensors are calculated using numerical integration procedure as a function of the three shape parameters. All subroutines, which define the nuclear shape, are separated from the code, which calculates the inertia and friction tensors. Thus, the program could be adopted for any other shape parameterization in case if it is given in cylindrical coordinates.

Analysis and Design of the Energy Storage Requirement of Hybrid Modular Multilevel Converters Using Numerical Integration and Iterative Solution

Increasing the modulation index by utilizing the negative voltage states of full-bridge submodules (FBSMs) can greatly reduce capacitor usage of modular multilevel converters (MMCs), thereby optimizing the cost and volume. The hybrid MMC is composed of half-bridge submodules (HBSMs) and FBSMs, and the capacitor voltages of the two types of submodules (SMs) have different shapes as long as negative voltage states exist. This condition greatly complicates the analysis and design of the energy storage requirement of the hybrid MMC, which utilizes the negative voltage states of FBSMs to boost the AC voltage. A numerical calculation method for solving the capacitor voltages and designing the capacitances of FBSMs and HBSMs is proposed in order to accurately determine the minimum energy storage requirement considering the difference between the energy variations in FBSMs and HBSMs. In the numerical calculation, the energy storage and voltage of the arm are decomposed into FBSM and HBSM parts. According to the physical switching process, the output voltages of FBSM and HBSM parts are determined separately. The one-cycle waveforms of the capacitor voltages are then obtained by numerical integration of the power flows in FBSM and HBSM parts. An iterative solution procedure and the termination criterion that can ensure the accuracy of the obtained one-cycle waveforms are also proposed. Using the numerical integration and iterative solution procedure as the kernel algorithm, the proposed method can accurately analyze the capacitor voltages of the FBSMs and HBSMs and determine the minimum energy storage requirement of the hybrid MMC. Furthermore, the proposed method is applicable for various operating working conditions and various proportions of FBSMs. The simulation results verify the feasibility and accuracy of the analysis and design method.

The Numerical simulation of composite sandwich cylindrical shells reinforced with circular frames under local loadings

The paper considers a woven fabric (based on fiberglass) composite structure in the form of a foam core sandwich cylindrical shell reinforced with circular frames, under the action of local loads applied to the frames. A technique is described for obtaining a numerical solution (with the confirmed reliability) to the problem of the stress-strain state of this type of structure using two alternative computational models, one of which is based on the numerical integration method, and the other one is based on the finite element method. The first model is developed by adopting a scheme, in which the frames are considered as short cylindrical shells obeying the single normal hypothesis. The foam core sandwich sections connected to them are considered within the framework of the zig-zag theory of soft core sandwich shells based on the assumption of incompressibility of the core along thickness. In this case, the corresponding calculation problem is formulated in the form of systems of algebraic and differential (in partial derivatives) equations for each of the shell sections being considered, which are supplemented by kinematic and force conditions at their joints, as well as boundary conditions at the left and right ends of the structure. The problem solving for each harmonic number is reduced to solving a set of boundary value problems for systems of 8 and 12 linear ordinary differential equations of the first order, related by conditions at the specified joints and using the procedure for expanding the parameters of the stress-strain state and applied loads into Fourier series in the circumferential direction. The solution algorithm is constructed using the numerical integration procedure in the orthogonal sweep version combined with the displacement method procedure (to satisfy the conditions at the joints). The declared finite element model is built within the ABAQUS software package using S4 shell elements (for composite layers) and C3D20 volume elements (for frames and core). The formed model, when setting deliberately overestimated values of the corresponding elastic modules, can implement a situation close to fulfilling the set of hypotheses adopted when constructing the first model. Having fixed in such a situation the consistency of the calculation results on the basis of the alternative computational models constructed in this way and thereby confirming the reliability of the obtained numerical solution, we carry out the transition to the calculation using the real values of the mentioned modules and the analysis of their influence on the stress-strain state of the investigated sandwich shell. The presented example of the calculation of a composite sandwich cylindrical structure, one of the frames of which is under the action of two local axial loads, demonstrates the possibilities of the adopted modeling method.

Purification of gas mixtures based on the removal of small impurities from a counterflow membrane module: engineering calculation method

One of the tasks of the membrane gas separation process is the extraction of low-content highly penetrating impurities from gas mixtures.
 The gas separation process is efficiently organized using hollow fiber-type membrane modules.
 In this study, an attempt was made to develop an engineering method for calculating the purification of gas mixtures based on the removal of small impurities from a countercurrent membrane module, which would enable the calculations related to the separation mode using analytical dependencies, without the use of numerical integration procedures and iterations.
 A mathematical model representing the gas separation process from binary mixtures having low content of a highly penetrating component was analyzed to address the challenges related to cleaning, purifying, and drying gas mixtures. Parametric studies related to the process were carried out and an engineering calculation method that enabled calculating separation in the module without using complex, resource-intensive algorithms having analytical formulas was developed herein. During the parametric study, the separation of a binary gas mixture was considered a limiting case. The proposed method was tested with respect to the example of air drying in a module comprising a (polyvinyltrimethylsilane) PVTMS membrane, which was an approximation of a binary mixture. The numerical and analytical results of the calculation are consistent with each other, and the calculation herein based on the developed method requires several orders of magnitude fewer calculations, that can even be performed manually.

Γ-profilometry: a new paradigm for precise optical metrology.

We show that the shape of a surface can be unambiguously determined from investigating the coherence function of a wave-field reflected by the surface and without the requirement of a reference wave. Spatio-temporal sampling facilitates the identification of temporal shifts of the coherence function that correspond to finite height differences of the surface. Evaluating these finite differences allows for the reconstruction of the surface using a numerical integration procedure. Spatial sampling of the coherence function is provided by a shear interferometer whereas temporal sampling is achieved by means of a Soleil-Babinet compensator. This low coherence profiling method allows to determine the shape of an object with sub-micrometer resolution and over a large unambiguity range, although it does not require any isolation against mechanical vibration. The approach therefore opens up a new avenue for precise, rugged optical metrology suitable for industrial in-line applications.

Elastic and absorption electron collisions with acetaldehyde

We report a theoretical–experimental investigation on elastic electron scattering by acetaldehyde (CH $$_3$$ CHO) in the 0.5–800 eV energy range. Theoretically, the Pade approximation and the single-center partial-wave expansion method were applied to solve the Lippmann–Schwinger scattering equation, whereas a molecular complex optical potential was used to describe the electron–molecule interaction. Calculations in the independent-atom framework were also performed. Angular distributions of the elastically scattered electrons were measured in the 10 $$^{\circ }$$ –130 $$^{\circ }$$ and 30–800 eV ranges using a crossed beams geometry. Absolute values of differential cross sections were derived by the relative flow technique. In addition, integral and momentum transfer cross sections were obtained from the experimental and theoretical differential cross sections via a numerical integration procedure. Comparison between our results and previous experimental and theoretical calculations is made.

The effect of structural aspect for planar systems with 2DOF upon the stability of motion

The paper analyses all planar chains with 2DOF from the point of view of stability of motion. For the rotation-prismatic structural solution first there are obtained the equations of motion and then, the numerical integration procedure is applied. A strong instability of the system can be noticed. The same dynamical system is modelled using dynamical analysis software and the instability is confirmed.

Nonlinear time-history earthquake analysis for steel frames

This study presents a simple, efficient, accurate method for nonlinear time-history earthquake analysis of spatial steel frames. The proposed new fiber plastic hinge method simulating only one element for each member captures the time-history dynamic behavior of steel frames as accurately as sophisticated plastic zone methods. Stability functions and the geometric stiffness matrix are employed for predicting the second-order effects that aim to minimize computational time and computer resources. Residual stresses are considered through two fiber plastic hinges by assigning initial stress values. A numerical integration procedure using Newmark integration combined with the Newton-Raphson balanced iteration algorithm is developed to find a solution to nonlinear dynamic equilibrium equations of the structural system. Shear deformation effects are also considered in the dynamic analysis. The proposed software, named Direct Advanced Analysis and Design (DAAD), is proved to be accurate and reliable as compared with the analysis results generated by expensive commercial Finite Element Analysis (FEA) software packages. The proposed DAAD program can be improved for performance-based direct design and analysis.