Moment Scheme Research Articles

Self-Similar-Structure–Based All-Speed Genuinely Two-Dimensional Riemann Solver

In this study, the low-speed asymptotic analysis of the Multidimensional, Self-Similar, Strongly Interacting, Consistent with the Second-Order Moment (MuSIC2) scheme in the low-speed limit is conducted. Based on such analysis, a controlling function is employed to control the numerical dissipation in the momentum equations at low speeds, and the All-Speed MuSIC2 (AMuSIC2) scheme is proposed in curvilinear coordinates. Systematic numerical tests are illustrated. The one-dimensional cases show that the AMuSIC2 scheme is capable of accurately capturing one-dimensional shocks, expansion waves, and contact discontinuities. The two-dimensional odd–even decoupling case indicates that the AMuSIC2 scheme is robust against the unphysical shock anomaly in capturing strong shocks. The spherical blast wave case indicates that the AMuSIC2 scheme improves the traditional one-dimensional Riemann solvers’ mesh imprinting phenomenon as the MuSIC2 scheme. The two-dimensional inviscid flow over the NACA0012 airfoil case and the low-speed Gresho vortex case suggest that the AMuSIC2 scheme improves the MuSCI2 scheme’s accuracy at low speeds remarkably. The turbulent flow over the flat plate case and the turbulent flow past the NACA4412 airfoil case also suggest that the AMuSIC2 scheme has a much higher level of accuracy at low speeds than its counterpart.

SOLUTION OF PHYSICALLY NONLINEAR PROBLEMS OF DEFORMATION OF MASSIVE AND THIN-WALLED PRISMATIC BODIES

The results of a numerical study on test examples of the convergence of the developed version of the semi-analytical method of finite elementsare presented. A comparison of machine time and accuracy of solving elastic and plastic problems obtained on the basis of finite elements with variableand averaged mechanical parameters was performed.The convergence of the finite element method and the semianalytical finite element method in the calculation of prismatic bodies loaded with localized and distributed external influences was studied. An effective numerical approach to the study of arbitrarily loaded massive and thin-walled prismatic bodies of complex shape, the deformation of which can pass beyond theelastic limit of the material, is being developed based on the finite element moment scheme and the semianalytical variant of the finite element method. Due to the representation of displacements by polynomials and the use of iterative methods of solving systems of equations, this approach is developed in relation to the calculation of objects with arbitrary boundary conditions on the ends, which made it possible to expand the area of effective application of the semi-analytical method of finite elements to a new class ofproblems.A number of new complex problems of elastic and elastic-plastic deformation of massive and thinwalled prismatic bodies, which have an independentapplied value, have been solved.

FEATURES OF THE CONSTRUCTION OF THE STIFFNESS MATRIX OF THE SPATIAL HEXAGONAL FINITE ELEMENT FOR COMPOSITE MATERIAL WITH DISCRETE INCLUSIONS BASED ON THE MOMENT SCHEME

An approach to numerical modeling of the stress-strain state of composite structures with discrete inclusions is presented in the paper. The finite element method is used as the main method, namely its modification – the moment finite-element scheme. The moment scheme, in contrast to the classic scheme of finite elements, allows to avoid such negative properties as not taking in consideration the rigid rotation of the finite element and “false” shear. If both the material of the matrix and the material of the reinforcing inclusions are weakly compressible, then problems arise due to the fact that some elastic constants approach very large values. The Taylor series expansion of the components of the displacement vector, the components of the strain tensor, and the volume change function is used in order to eliminate the mentioned shortcomings, after that, according to the moment scheme, certain sums are removed from these expansions. Homogenization of the material with lamellar inclusions, a small proportion of spherical inclusions, and a large proportion of spherical inclusions is used for modeling the elastic properties of the composite. The chaotic nature of the location of inclusions after homogenization makes it possible to present a non-homogeneous composite material as a homogeneous quasi-isotropic one. The described approaches are used in the construction of the stiffness matrix of the spatial hexagonal finite element. The obtained expressions for the stiffness matrix are done in the software package for calculating structures from composite materials. The calculation of a thick-walled pipe under the action of internal pressure from a composite material with lamellar inclusions, a small proportion of spherical inclusions, and a large proportion of spherical inclusions was carried out using the software package. For different volume fractions of discrete inclusions, the numerical convergence of the results with different finite element meshes has been investigated, which shows great congruence with analytical solutions.

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.

Temperature Variation of Gilbert Damping Constant in 3d Transition Metals from First Principles

A theoretical investigation of temperature-dependent Gilbert damping is presented. We evaluate the Gilbert damping constant α (and the corresponding Landau frequency λ) at a finite temperature for hcp-Co, fcc-Ni, and Cu-doped fcc-Ni from first principles by combining the torque correlation model and the disordered local moment scheme, in which the finite-temperature effect is included as the transverse spin fluctuation. The calculated λ has, from the viewpoint of the material dependence, well reproduced the experimental results, but quantitatively, these values are more underestimated than the experimental values. A comparison with the lifetime approximation results reveals that a change in the electronic structure due to the spin fluctuation affects λ for Co and Ni but not for Fe. On the basis of the variation of the local magnetic moment of Co and Ni at finite temperature, we conclude that these differences in λ are due to the difference in the degree of itinerancy of spins.

Application of the finite element moment scheme to the investigation of thin elastic shells of inhomogeneous structure

The work is devoted to the problem of developing a universal method for studying the deformation, buckling, postbuckling behavior and vibrations of thin and medium-thickness shells of complex shape and structure under the action of mechanical and thermal loads. A wide class of shells is considered: of constant and smooth-variable thickness, with ribs and cover plates, channels and cavities, holes, sharp bends of the mid-surface, and with a multilayer structure of the material. The method is based on the positions of the 3D geometrically nonlinear theory of thermoelasticity without the use of simplifying hypotheses of the theory of shells. The development of the computational model is based on the use of a universal isoparametric spatial finite element with multilinear shape functions, which is the same for all sections of the shell with stepwise variable thickness. The governing equations are constructed in accordance with the requirements of the finite element moment scheme. The mid-surface of the shell’s casing is taken as a single reference surface. All matrices of governing equations for a universal spatial multilayer finite element are obtained in explicit form by analytical integration. This speeds up the execution of calculations in the algorithmic implementation of the method. Such an approach, based on a unified methodology, makes it possible to study the behavior of multilayer shells with different geometric features in terms of thickness under complex thermo-mechanical loading.

Implementation of a double moment cloud microphysics scheme in the UK met office regional numerical weather prediction model

AbstractCloud microphysics parametrizations control the transfer of water between phases and hydrometeor species in numerical weather prediction and climate models. As a fundamental component of weather modelling systems cloud microphysics can determine the intensity and timing of precipitation, the extent and longevity of cloud cover and its impact on radiative balance, and directly influence near surface weather metrics such as temperature and wind. In this paper we introduce and demonstrate the performance of a double moment cloud microphysical scheme (CASIM: Cloud AeroSol Interacting Microphysics) in both midlatitude and tropical settings using the same model configuration. Comparisons are made against a control configuration using the current operational single moment cloud microphysics, and CASIM configurations that use fixed in‐cloud droplet number or compute cloud droplet number concentration from the aerosol environment. We demonstrate that configuring CASIM as a single moment scheme results in precipitation rate histograms that match the operational single moment microphysics. In the midlatitude setting, results indicate that CASIM performs as well as the single moment microphysics configuration, but improves certain aspects of the surface precipitation field such as greater extent of light (1 mm hr) rain around frontal precipitation features. In the tropical setting, CASIM outperforms the single moment cloud microphysics as evident from improved comparison with radar derived precipitation rates.

Estimating Annual Temperate Rain Attenuation Distributions at 20/40 GHz With a High-Resolution Numerical Weather Prediction Model

Earth-space communications are to make use of higher frequencies, at and above 20 GHz, to offer new services and increased data rates. A consequence is a strong attenuation of the carrier waves by the troposphere, particularly during rain events. The statistical characterization of the rain attenuation on high-frequency channels is thus paramount to their link budget. Traditionally this knowledge is obtained via long-term ground measurements of spaceborne beacon signals. Here, simulated results based on numerical weather prediction models are considered as an alternative. A comparison is given based on two years of beacon data at 20 and 40 GHz in Toulouse (France). Four parametrizations of the weather research forecasting model's microphysics are tested: the best performances are achieved with the NSSL-2 moment scheme.

FEATURES OF THE USE OF PARALLEL CALCULATIONS IN THE PACKAGE OF APPLICATION PROGRAMS «MIRELA+»

The article examines approaches to the optimization of structural calculations, which require a comprehensive analysis of deformation processes under the action of operational loads. The finite element method is usually used to solve such problems. Modeling of structures taking into account stress concentrators requires the use of large-sized calculation grids. In real operating conditions, most structural elements are under the influence of mechanical and thermal loads. The package of application programs «MIRELA+» is intended for solving problems of thermoelasticity of structures made of weakly compressible elastomers and composites with an elastomeric matrix. One of the stages of solving problems of thermoelasticity is the formation of stiffness matrices taking into account weak compressibility and thermal conductivity of finite elements. When forming the stiffness and thermal conductivity matrices, it is necessary to perform integration over the area of ​​the finite element. For elements of the same type, calculations are used according to the same procedure, and the number of such calculations determines the time to solve the problem. In the traditional approach, these calculations are performed sequentially. In the case of large grid sizes, the number and time of calculations increase, which requires optimization of calculations using parallel calculation algorithms. Using the OpenMP library, a parallel algorithm for forming the matrix of solving equations was built. The calculation of terms according to the quadrature scheme is included in the parallel area of ​​the algorithm. Filling of the stiffness matrix of the finite element takes place in the sequential region. The execution time of parallel calculations is determined by the slowest subtask. Using the moment scheme of finite elements, displacement and deformation are approximated by the same polynomials, which simplifies the calculation of integrals. Parallel programming algorithms for constructing solving equations of the “MIRELA+” program package have been developed. Solutions for calculation grids of different sizes have been obtained. The effect of parallelization on the calculation time was studied.

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.

Vibrations of launch vehicle fairings with conical shape

Vibrations of launch vehicle conical fairings are investigated. Fairings are simulated using thin conic shells. The modal analysis of a thin elastic shell is based on the use of the developed finite element model of an inhomogeneous shell. In general, the technique makes it possible to investigate the geometrically nonlinear deformation, stability, and post-buckling behavior of a wide class of thin elastic shells. The modal analysis of the structure is implemented at each step of the static thermomechanical load. The subspace iteration method is used to determine the spectrum of the lowest vibration frequencies of shells of an inhomogeneous structure. The shell behavior analysis method is based on the relations of the three-dimensional theory of thermoelasticity and uses the finite element moment scheme. A thin elastic shell is simulated by a universal solid isoparametric finite element. The parameters of natural vibrations of conical shells of revolution with different thicknesses are investigated. Comparison of the calculation results obtained by the finite element moment scheme with the data of other authors shows a fairly good agreement between the solutions.

Experimental and numerical investigation on the flexural behavior of a large-scale prestressed UHPC T-Shaped girder

A large scale (10 m long) prestressed ultra-high-performance-concrete (UHPC) T-shaped girder was experimentally investigated in this paper to study its flexural behavior, cracking moment, and ultimate flexural capacity through the bending test. By considering contribution of the tensile strength of UHPC, this paper suggested a modified tensile constitutive model of UHPC to achieve a reliable cracking moment and ultimate flexural design scheme for the prestressed UHPC girders. Effectiveness of this design scheme was demonstrated through the comparative studies on the developed load–deflection curves and ultimate flexural capacity of the girder obtained from the experimental results and that from the suggested design scheme. In addition, this study developed a nonlinear numerical analysis program to investigate the nonlinear flexural performance of the girder, and its flexural capacity calculated by using this program agreed well with the experimental results. Moreover, parametric studies regarding the influences of reinforcement ratios, prestresses of the prestressing tendons, and girder depths on the girder’s flexural resistance were also conducted. Finally, it is concluded that (1) UHPC girder exhibited good ductility and excellent flexural deformation capacity and cracking performance; (2) the tensile strength of UHPC has limited contribution to the ultimate flexural capacity of UHPC girders; and (3) appropriately increasing reinforcement ratio of the prestressing tendons, ultrahigh compressive strength of UHPC could be fully utilized, and thus the cracking moment and ultimate flexural capacity of UHPC girders could be effectively improved.

Performance of Regional Climate Model (WRF 4.3) in Medium Range Rainfall Forecast (MRRF) for Tamil Nadu, India

The weather events are highly dynamic and fluctuating for the next few days due to enormous processes carried out by nature and physics and it is even more highly variable in tropics. The Medium Range Weather Forecast is incredibly helpful and trustworthy for agricultural purposes and rainfall is one of the most imminent events determining productivity. The Medium Range Rainfall Forecast (MRRF) given by Weather Research and Forecast model (WRF v 4.3) is verified using forecast verification scores including Ratio of Root Mean Square Error (RMSE) to the standard deviation of the observations (RSR), Nash-Sutcliffe Efficiency (NSE), Percent Bias (PBIAS), Kling-Gupta Efficiency (KGE), and Root Mean Square Error (RMSE). Scores were computed by comparing forecast generated using two microphysics options viz., WRF Single Moment scheme (WSM-3) and Kessler scheme during South West Monsoon (SWM) and North East Monsoon (NEM) of the year 2021 for five different physiographic regions of Tamil Nadu. WSM-3 microphysics scheme outperformed in predicting MRRF for all the five regions and during both the monsoons.

Entropy bounds for the space–time discontinuous Galerkin finite element moment method applied to the BGK–Boltzmann equation

This paper presents a numerical analysis for the time-implicit numerical approximation of the Boltzmann equation based on a moment system approximation in velocity dependence and a discontinuous Galerkin finite-element (DGFE) approximation in time and position dependence. The implicit nature of the DGFE moment method in position and time dependence provides a robust numerical algorithm for the approximation of solutions of the Boltzmann equation. The closure relation for the moment systems derives from minimization of a suitable φ-divergence. We present sufficient conditions such that this divergence-based closure yields a hierarchy of tractable symmetric hyperbolic moment systems that retain the fundamental structural properties of the Boltzmann equation. The resulting combined space-time DGFE moment method corresponds to a Galerkin approximation of the Boltzmann equation in renormalized form. We propose a renormalization map that facilitates the approximation of multidimensional problems in an implicit manner. Moreover, upper and lower entropy bounds are derived for the proposed DGFE moment scheme. Numerical results for benchmark problems governed by the BGK-Boltzmann equation are presented to illustrate the approximation properties of the new DGFE moment method, and it is shown that the proposed velocity-space-time DGFE moment method is entropy bounded.

Comparative Analysis of Nonlinear Deformation and Buckling of Thin Elastic Shells of Step-Variable Thickness

A comparative analysis of finite element models and methods for solving complex problems of geometrically nonlinear deformation, buckling and post-buckling behavior of thin shells of stepwise variable thickness is carried out. An approach based on the use of the moment scheme of finite elements is considered. The features of using the software suite LIRA and integrated software system SCAD for solving the assigned problems are also provided. Thin and medium thickness shells are considered. They can have different geometric features in thickness and be under the action of static thermomechanical loads. A technique for solving these problems with the help of an efficient refined approach is presented. The technique is based on the general methodological positions of the three-dimensional theory of thermoelasticity and the use of the finite element moment scheme. With this approach, the approximation through the shell thickness is carried out by a single universal spatial finite element. The element can be modified in different portions of the shell with a step-variable thickness. It can be located eccentrically relative to the middle surface of the casing and can change its dimensions in the direction of the shell thickness. Such a unified approach made it possible to create a unified designed finite element model of a shell of an inhomogeneous geometric structure under the combined action of a thermomechanical load. A comparative analysis of the application of three finite element approaches for problems of geometrically nonlinear deformation and buckling of shells of stepwise variable thickness is carried out.

A Sensitivity Study on High Resolution NWP ICON—LAM Model over Italy

In this work, the results of a sensitivity study based on ICON-LAM simulations at 2.5 km of spatial resolution over the Italian area, driven by ECMWF IFS global data, are presented. The main aim is to provide a contribution to the selection of suitable parameterization schemes that result in more effective for a proper representation of the Italian climate features. Model evaluation was conducted in terms of the air temperature and precipitation for three subregions, comparing a set of 13 simulations against SCIA and E-OBS standard datasets. In addition, evaluation was also conducted against selected data stations scattered over the Italian area. We found that the ICON-LAM model was able to provide a good representation of the temperature over Italy, whereas non-negligible biases were observed for precipitation in certain regions. The model proved to be sensitive to changes in physical parameterization schemes. In particular, we found that the explicit treatment of deep convection and the “clouds as in turbulence” scheme for cloud cover allowed for a better representation of precipitation in the summer over the Alpine region. The single moment scheme is currently the best option for cloud microphysics.

A new moment-based general-relativistic neutrino-radiation transport code: Methods and first applications to neutron star mergers

ABSTRACTWe present a new moment-based energy-integrated neutrino transport code for neutron star merger simulations in general relativity. In the merger context, ours is the first code to include Doppler effects at all orders in υ/c, retaining all non-linear neutrino–matter coupling terms. The code is validated with a stringent series of tests. We show that the inclusion of full neutrino–matter coupling terms is necessary to correctly capture the trapping of neutrinos in relativistically moving media, such as in differentially rotating merger remnants. We perform preliminary simulations proving the robustness of the scheme in simulating ab-initio mergers to black hole collapse and long-term neutron star remnants up to ${\sim }70\,$ ms. The latter is the longest dynamical space-time, 3D, general relativistic simulations with full neutrino transport to date. We compare results obtained at different resolutions and using two different closures for the moment scheme. We do not find evidences of significant out-of-thermodynamic equilibrium effects, such as bulk viscosity, on the post-merger dynamics or gravitational wave emission. Neutrino luminosities and average energies are in good agreement with theory expectations and previous simulations by other groups using similar schemes. We compare dynamical and early wind ejecta properties obtained with M1 and with our older neutrino treatment. We find that the M1 results have systematically larger proton fractions. However, the differences in the nucleosynthesis yields are modest. This work sets the basis for future detailed studies spanning a wider set of neutrino reactions, binaries, and equations of state.

АЛГОРИТМ КОНСЕРВАТИВНОГО СГЛАЖИВАНИЯ РАЗРЫВНЫХ ВОЛН НАПРЯЖЕНИЙ В МКЭ

Three-dimensional geometrically and physically nonlinear problems of nonstationary deformation of structures are considered. The defining system of equations is formulated in Lagrange variables. The equation of motion is derived from the balance of virtual work capacities. The elastic-plastic deformation of structural materials is described by the relations of the flow theory with isotropic hardening. The solution of the problem is based on the moment scheme of the finite element method. The discretization of the problem by spatial variables is carried out by eight nodal finite elements with multilinear functions of the approximation form of the displacement velocity. Time integration is performed according to an explicit finite-difference scheme of the “cross” type, which does not have the monotonicity property. Due to the dispersion in the vicinity of the gap, it generates non-physical oscillations, which significantly limits the scope of its applicability. In this paper, to suppress high-frequency oscillations, it is proposed to use an algorithm for conservative smoothing of a numerical solution with a space-time monotony analyzer. Based on it, software modules have been developed for the “Dynamics-3” computing complex. Verification of the developed technique and its software implementation was carried out by solving a one-dimensional problem in a three-dimensional formulation of the compression wave passing through a fixed elastic layer. For comparison, the results of numerical solution of the problem according to the “cross” scheme without smoothing, with linear viscosity, with conservative smoothing using spatial and spatiotemporal monotonicity analyzers are obtained. The problem of penetration of an elastic cylinder into a round steel plate is solved in a three-dimensional formulation. It is shown that the developed technique not only suppresses high-frequency oscillations, but also prevents the occurrence of zero-energy modes. So, in the second problem, without using the procedure of conservative smoothing of the numerical solution, the final elements of the plate in the collision zone are significantly distorted, which leads to an early interruption of the calculation.

Rainfall types in the lifecycle of a stationary cloud cluster during the Indian Summer Monsoon: An investigation with numerical simulations and radar observation

The performance of three microphysics schemes (WRF Double Moment 6-class (WDM6), Morrison double moment (MORR), and spectral bin microphysics (Fast version) (SBMF)), with increasing complexities, in simulating various rainfall-types during the lifecycle of a tropical continental mesoscale cloud cluster (CC) are presented in this study. The CC spent its entire lifetime over the surveillance area of a polarimetric C-band radar, over the peninsular Indian region in a monsoon environment. Observed and simulated rainfall properties throughout the lifecycle of the CC are intercompared. A rainfall-type categorization algorithm has been applied to the observed and model simulated reflectivity fields. Microphysics scheme with the least complexities produced the highest convective rainfall-type, lowest stratiform rainfall-type, and a remarkable early cessation of the CC. With increasing complexities, the microphysics schemes are found to represent the rainfall-types more close to radar observations. Major differences are noted during the mature and dissipation phases, especially in the stratiform rainfall-type. The rainfall-types derived from radar, as well as three simulations with WDM6, MORR, and SBMF schemes, showed ~18%, ~54%, ~41%, and ~29% of convective rainfall fraction (including isolated convective categories), ~14%, ~33%, ~18%, and ~20% of mixed rainfall fraction and ~68%, ~13%, ~42%, and ~50% of stratiform rainfall fraction respectively during the lifecycle of the CC. The accumulated rainfall derived from radar observations, using a blended rainfall estimation algorithm, indicated substantial differences in the observed and simulated rainfall pattern. The lower moment scheme has a tendency to produce more lighter as well as heavy rainfall compared to the higher moment schemes.

CONSTRUCTION OF SOLVING EQUATIONS OF SEMI-ANALYTICAL METHOD OF FINISHED ELEMENTS FOR PRISMATIC BODIES OF COMPLEX SHAPE

The article presents an effective numerical approach to the study of arbitrarily loaded massive and thin-walled prismatic bodies of complex shape, the deformation of which can take place beyond the elasticity of the material. The equations of the semi-analytical finite element method (SAFEM) when used to decompose the displacements of Fourier series. The main relations between the spatial problem of the theory of elasticity in a curvilinear coordinate system and the theory of plastic flow for an isotropically reinforcing material under the Mises fluidity condition are presented. In accordance with the method of the moment scheme of finite elements (MSFE), the expressions of deformations of the prismatic finite element due to the nodal values of amplitude displacements are obtained. Formulas for calculating the stiffness matrix coefficients of a finite element (FE) with variable and averaged in the cross-sectional plane mechanical and geometric parameters are derived.