Solution Of Dynamic Problems Research Articles

Spectral integrated neural networks (SINNs) for solving forward and inverse dynamic problems

This study introduces an innovative neural network framework named spectral integrated neural networks (SINNs) to address both forward and inverse dynamic problems in three-dimensional space. In the SINNs, the spectral integration technique is utilized for temporal discretization, followed by the application of a fully connected neural network to solve the resulting partial differential equations in the spatial domain. Furthermore, the polynomial basis functions are employed to expand the unknown function, with the goal of improving the performance of SINNs in tackling inverse problems. The performance of the developed framework is evaluated through several dynamic benchmark examples encompassing linear and nonlinear heat conduction problems, linear and nonlinear wave propagation problems, inverse problem of heat conduction, and long-time heat conduction problem. The numerical results demonstrate that the SINNs can effectively and accurately solve forward and inverse problems involving heat conduction and wave propagation. Additionally, the SINNs provide precise and stable solutions for dynamic problems with extended time durations. Compared to commonly used physics-informed neural networks, the SINNs exhibit superior performance with enhanced convergence speed, computational accuracy, and efficiency.

Hyperbolic modeling of gradient damage and one-dimensional finite volume simulations

This study provides a new formulation of gradient damage model which allows an efficient explicit numerical solution of dynamics problems. The proposed methodology is based on an ”extended Lagrangian approach” developed by one of the authors for the nondissipative and dispersive shallow water equation. By using this strategy, the global minimization problem commonly derived for gradient damage models is recast as a purely local hyperbolic one with source terms that can be easily solved using finite volumes. The numerical solution of the governing system is then based on a fractional-step method consisting of a classical Godunov-type scheme and an implicit Ordinary Differential Equation solver for the local source terms. Numerical results are presented on one-dimensional multi-fragmentation and spalling tests for illustration purpose.

Precise solutions of dynamic problems in stratified transversely isotropic piezoelectric materials

Precise solutions of dynamic problems in stratified transversely isotropic piezoelectric materials

The u‐p approximation versus the exact dynamic equations for anisotropic fluid‐saturated solids. I. Hyperbolicity

AbstractThe numerical solution of dynamic problems for porous fluid‐saturated solids is often performed with the use of simplified equations known as the u‐p approximation. The simplification of the equations consists in neglecting some acceleration terms, which is justified for a certain class of problems related, in particular, to geomechanics and earthquake engineering. There exist two u‐p approximations depending on how many acceleration terms are neglected. All comparative studies of the exact and u‐p formulations are focused on the question of how well the u‐p solutions approximate those obtained with the exact equations. In this paper, the equations are compared from a different point of view, addressing the question of well‐posedness of boundary value problems. The exact equations must be hyperbolic and satisfy the corresponding hyperbolicity conditions for the boundary value problems to be well posed. The u‐p equations are not of the form to which the conventional definition of hyperbolicity applies. A slight extension of the approach makes it possible to derive hyperbolicity conditions as necessary conditions for well‐posedness for the u‐p approximations. The hyperbolicity conditions derived in this paper for the u‐p approximations are formulated in terms of the acoustic tensor of the skeleton. They differ essentially from the hyperbolicity conditions for the exact equations.

Solution of Dynamic Problems for Elliptical Cylindrical Shells Under Distributed Impulsive Loads*

Solution of Dynamic Problems for Elliptical Cylindrical Shells Under Distributed Impulsive Loads*

Dynamic optimization with side information

We develop a tractable and flexible data-driven approach for incorporating side information into multi-stage stochastic programming. The proposed framework uses predictive machine learning methods (such as k-nearest neighbors, kernel regression, and random forests) to weight the relative importance of various data-driven uncertainty sets in a robust optimization formulation. Through a novel measure concentration result for a class of supervised machine learning methods, we prove that the proposed approach is asymptotically optimal for multi-period stochastic programming with side information. We also describe a general-purpose approximation for these optimization problems, based on overlapping linear decision rules, which is computationally tractable and produces high-quality solutions for dynamic problems with many stages. Across a variety of multi-stage and single-stage examples in inventory management, finance, and shipment planning, our method achieves improvements of up to 15% over alternatives and requires less than one minute of computation time on problems with twelve stages.

Исследование амплитудно-частотных характеристик вязкоупругого тела типа плотины при установившихся вынужденных гармонических колебаниях

The article discusses the amplitude-frequency response of a viscoelastic body of the dam type, under steady-state forced harmonic vibrations. An important factor is the determination of the number of frequencies and resonance peaks that arise in the process of harmonic effect of the water body on the dam. The use of the finite element method (FEM) for the numerical solution of dynamic problems allows, by expanding the solution in terms of eigenmodes and frequencies, to reduce the original problem to a system of separated linear integro-differential equations with respect to the sought parameters of generalized functions. The process of the influence of viscoelastic properties of the dam material on the resonance curves that arise under harmonic loads of different frequencies at different dimensions of the foot of the dam is investigated. The analysis of the curves of the amplitude-frequency responses of a dam-type viscoelastic body, under steady-state forced harmonic oscillations, showed that the occurrence of resonance peaks depends on the viscoelastic properties of the dam body and the dimensions of the foot of the dam. The main resonance peaks occur at frequencies less than the sixth eigenfrequency, as a result of which a further increase in the number of eigenmodes in the expansion does not introduce any significant changes in the amplitude of the distribution of resonance curves of the amplitude-frequency response of the dam.

МЕТОД АГРЕГИРОВАНИЯ ПРИ РЕШЕНИИ ДИНАМИЧЕСКИХ ЗАДАЧ БОЛЬШОЙ РАЗМЕРНОСТИ В СИСТЕМНОМ АНАЛИЗЕ

Software development, which includes models of complex systems, implies the emergence of tasks with a large number of desired quantities and relationships between quantities. The solution of dynamic problems of large dimension is always complicated by the limited RAM of an electronic computing device or too much time spent on finding a solution. Therefore, for dynamic problems of large dimension, solution methods are always used to simplify this process. In some situations, dividing the initial task into several tasks of smaller dimensions is sufficient, but this is not always possible. Then the solution may be approximate, but the question of the accuracy of such a solution remains open. Today there are many algorithms that help to lower the dimension in a multidimensional dynamic problem using differential equations. Dynamic tasks of large dimension are specific, so there are many approaches, but the most convenient and affordable today is aggregation.

A local meshless analysis of dynamics problems / Uma análise local desordenada dos problemas dinâmicos

This paper is concerned with new formulations of local meshfree numerical method, for the solution of dynamic problems in linear elasticity, Integrated Local Mesh Free (ILMF) method. The key attribute of local numerical methods is the use of a modeling paradigm based on a node-by-node calculation, to generate the rows of the global system of equations of the body discretization. In the local domain, assigned to each node of a discretization, the work theorem is kinematically formulated, leading thus to an equation of mechanical equilibrium of the local node, that is used by local meshfree method as the starting point of the formulation. The main feature of this paper is the use of a linearly integrated local form of the work theorem. The linear reduced integration plays a key role in the behavior of local numerical methods, since it implies a reduction of the nodal stiffness which, in turn, leads to an increase of the solution accuracy. As a consequence, the derived meshfree and finite element numerical methods become fast and accurate, which is a feature of paramount importance, as far as computational efficiency of numerical methods is concerned. The cantilever beam was analyzed with this technique, in order to assess the accuracy and efficiency of the new local numerical method for dynamic problems with regular and irregular nodal configuration. The results obtained in this work are in perfect agreement with Mesh-Free Local Petrov-Galerkin (MLPG) and the Finite Element Method (FEM) solutions.

Modal truncation damping in reduced modal analysis of piecewise linear continuum structures

Modal analysis is an often-used method for the solution of dynamical problems in engineering practice. In the case of continuous systems and discretization methods (e.g., FEM) with a large number of degrees of freedom, the reduced modal analysis is used, where only a selected number of modes are applied, while the higher modes are neglected. A piecewise linear elastic structure has state-dependent stiffness and as such, state-dependent vibration modes. During the dynamic modal analysis of such systems, one has to switch between the modes. During the switch a part of the kinetic and potential energy is transferring into modes, which are neglected in the reduced modal analysis. Thus, using the reduced modal analysis for the numerical calculation of the dynamic behavior of piecewise linear elastic structures introduces the modal truncation damping. In this paper, we present on a continuous system, how this modal truncation damping can be characterized, what are the upper and lower limits of that damping and how the characteristics of the free vibration of a piecewise linear elastic structure can be predicted.

Термическая реакция при тепловом ударе массивного тела с внутренней цилиндрической полостью

The paper examines mathematical models of thermal shock in terms of dynamic thermoelasticity and their application to specific cases during intense heating of a solid boundary. We introduce a stress compatibility equation for dynamical problems, generalizing the well-known Beltrami --- Mitchell relation for quasi-static cases. It is convenient to use this relation when considering numerous special cases in the theory of heat shock in Cartesian coordinates both for bounded bodies of a canonical form, i.e., an infinite plate, and for partially bounded bodies, i.e., space bounded from the inside by a flat surface. In the latter case, the obtained analytical solutions of dynamic problems of thermoelasticity lead to visual and convenient for physical analysis functional structures describing the kinetics of thermal stresses. For cylindrical and spherical coordinate systems, we propose a compatibility equation in displacements, which is convenient for studying the problem of thermal shock in bodies with a radial heat flux and under conditions of central symmetry. In the study, we singled out a class of problems in which the consideration of the geometric dimensions of a structure investigated for a thermomechanical reaction under conditions of intense heating concerns mainly the near-surface layers. According to the experimental results, it is these layers that absorb the main amount of heat during a time close to the beginning of heating, which corresponds to the time of the microsecond duration of the inertial effects. We investigated the thermal reaction of a massive body with an inner cylindrical cavity within the framework of dynamic thermoelasticity under various modes of intense heating of the cavity surface. Finally, we carried out numerical experiments and described the wave character of thermal stresses with the corresponding quasi-static values, and established the role of inertial effects in mathematical models of the theory of thermal shock

New Classes of Solutions of Dynamical Problems of Plasticity

Dynamical problems of the theory of plasticity have not been adequately studied. Dynamical problems arise in various fields of science and engineering but the complexity of original differential equations does not allow one to construct new exact solutions and to solve boundary value problems correctly. One-dimensional dynamical problems are studied rather well but two-dimensional problems cause major difficulties associated with nonlinearity of the main equations. Application of symmetries to the equations of plasticity allow one to construct some exact solutions. The best known exact solution is the solution obtained by B.D. Annin. It describes non-steady compression of a plastic layer by two rigid plates. This solution is a linear one in spatial variables but includes various functions of time. Symmetries are also considered in this paper. These symmetries allow transforming exact solutions of steady equations into solutions of non-steady equations. The obtained solution contains 5 arbitrary functions

Research on “concentric drum” project based on discrete model

The “concentric drum” project tests team members’ ability of cooperation, and the team members can hit the ball more times by working together. It is a typical dynamic problem. In this paper, the complete elastic collision model and the drum surface segmentation model are established to complete the discrete model. Through the optimization of the system, the good solution results of the problem are obtained gradually. The “concentric drum” strategy based on the discrete model has a certain reference for the solution of dynamic problems.

Dynamic Models of Environmental Data Envelopment Analysis with Stock and Flow Variables

The paper devoted to the elaboration of methodological approach to the solution of dynamic problems of Environmental Data Envelopment Analysis for production facilities, whose activities are characterized by a set of variables of two different types—stock variables and flow variables. The limitations that are additionally imposed on the Production Possibility Set are studied. A task is set for assessing the comparative effectiveness of regional systems of environmental management that operate at a certain time interval. A computational example is given for assessing the comparative effectiveness of environmental management systems in the regions of the Central Federal District in the period from 2010 to 2014. The possibilities of using the developed method in practice are discussed.

Modeling of the Maximum Deflection of a Stepped Rod Having an Initial Curvature Upon Impact Against a Rigid Barrier

The solution of dynamic problems of the longitudinal impact of a homogeneous rod in a nonlinear formulation presents, notoriously, significant mathematical difficulties. Existing approaches have a rather limited application mainly due to the significant approximation of the solution of the problem. In addition, such approaches affect an extensive mathematical apparatus, which complicates their use in engineering calculations. The task of solving the problem of the longitudinal impact of a geometrically heterogeneous (stepped) rod remains far from the final solution to this day. The solution of this problem is complicated by the randomness of the interference pattern of longitudinal waves during their transition through the boundaries of homogeneous sections of a stepped rod, and therefore the deformations and longitudinal forces along the length of the rod quickly change in time. In this work, using the initial parameter method and the wave model of longitudinal impact, an attempt is made to develop a methodology for calculating the dynamic deflection of a stepped rod that performs transverse vibrations during a longitudinal impact on a rigid barrier. The value of the maximum deflection of a stepped rod with the initial curvature of one of its sections is modeled. The results of modeling the maximum deflection for various values of the initial curvature and the pre-impact speed of the rod are obtained. During the processing of simulation results, the so-called “Zone of maximum deflections” is the interval of variation of the pre-shock velocity, at which the maximum amplitude of the transverse vibrations of the curved section of the stepped rod is observed. The possibility of calculating the amplitudes of the transverse vibrations of the rods and rod elements under the most various fixing schemes and the various nature of the pre-shock state is noted. The relevance of applying this technique in dynamic calculations of various impact mechanisms, as well as in engineering calculations of rod systems for various purposes, is emphasized.

Transient dynamic analysis of bars and trusses by the adaptive generalized finite element method

PurposeThis paper aims to present an adaptive approach of the generalized finite element method (GFEM) for transient dynamic analysis of bars and trusses. The adaptive GFEM, previously proposed for free vibration analysis, is used with the modal superposition method to obtain precise time-history responses.Design/methodology/approachThe adaptive GFEM is applied to the transient analysis of bars and trusses. To increase the precision of the results and computational efficiency, the modal matrix is responsible for the decoupling of the dynamic equilibrium equations in the modal superposition method, which is used with only the presence of the problem’s most preponderant modes of vibration. These modes of vibration are identified by a proposed coefficient capable of indicating the influence of each mode on the transient response.FindingsThe proposed approach leads to more accurate results of displacement, velocity and acceleration when compared to the traditional finite element method.Originality/valueIn this paper, the application of the adaptive GFEM to the transient analysis of bars and trusses is presented for the first time. A methodology of identification of the preponderant modes to be retained in the modal matrix is proposed to improve the quality of the solution. The examples showed that the method has a strong potential to solve dynamic analysis problems, as the approach had already proved to be efficient in the modal analysis of different framed structures. A simple way to performh-refinement of truss elements to obtain reference solutions for dynamic problems is also proposed.

Two-criteria technical and economic optimization of forest transportation problem

The systematic analysis of optimizing the producer-consumer structure transportation problem is carried out in paper. It is in demand for the timber industry complex. The well-known minimum transportation cost objective is solved as a static problem using the linear programming method. Due to complicated market development conditions, it is necessary to set up and solve the problem of minimizing transportation time (quick performance) as a dynamic problem. Solving it in general case, the transportation cost may be different from the minimum cost determined by the static problem solution. The dynamic problem solution is based on the carrier’s vehicle performance information used for determining the functional time spent per transported product unit. The rational decision making under market conditions is only possible by adopting a compromise obtained on the basis of the multi-criteria optimization theory. The task of determining the compromise area of two-criteria optimization (cost-time minimum ratio) is decided by experts. It is solved using the linear programming method for static and dynamic problems and their linear approximation. To develop a compromise transportation plan becomes the subject of solving the inverse transportation problem. The systemic “producer – transport – consumer” approach is necessary for sustainable timber industry complex development, when the producer-transport system makes up a single production process. Its parameters are determined on the basis of systems theory.

Mathematical Modeling of Stress-Strain Evolution in the Rock Mass around a Mine Opening. Evaluation of the Steps of First Roof Caving at Different Thicknesses of the Main Roof

Intensive mining activity has highlighted the need to study the dynamics of rock mass elements and to assess the degree of their nonequilibrium. The roof, floor, pillars, gateways, and so on are in a highly nonequilibrium state due to difficult mining conditions and high face advance rates. A good tool for predicting the mechanical behavior of the rock mass during mining is mathematical modeling, which is based on the solution of dynamic problems taking into account the nonequilibrium and nonstationary deformation and failure of rocks. In this paper, dynamic modeling has been performed within an evolutionary framework to investigate the steps of first caving in a model rock mass during face advance. All other things being equal, the thickness of the main coal seam roof is varied in the calculations. The steps of first caving are evaluated in the conditions of highly nonstationary deformation of the rock mass. The fluctuation statistics of the stress-strain parameters of the rock mass is analyzed. It is shown that the first roof caving is preceded by a fall in the slope of the amplitude-frequency curve of stress fluctuation.

Novel frictional algorithm implementing angular increment for two-dimensional frictional systems

This study proposes a new frictional algorithm that implements angular increment. The proposed algorithm is used to solve the numerical solutions of dynamic problems in two-dimensional frictional systems. It can accurately obtain the motion responses of a lumped mass under time-varying external forces, and it can compensate for the shortcomings of the numerical frictional algorithm that implements a time step. Specifically, the proposed algorithm 1) overcomes the difficulties encountered when the angles between resultant tangential forces and slip motion are infinitely close, 2) provides accurate solutions for two-dimensional systems under fierce planar motions, and 3) calculates the responses of the mass within a reasonable period. We compare the computation accuracy, efficiency, and robustness of the proposed frictional algorithm and the previous frictional algorithm [1] through several representative scenarios. We reveal that the proposed algorithm has superior computation accuracy, efficiency, and robustness for two-dimensional frictional problems involving slip/stick transitions and sharp bending.

A strain-gradient isotropic elastoplastic damage model with J3 dependence

A “plastic-strain-gradient” version of an isotropic elastoplastic damage model that depends on the third invariant J3 of the stress deviator is developed. The model is based on the “non-local” equivalent plastic strain ep defined by Peerlings et al. (2001) and Engelen et al. (2003) and introduces a “material length” ℓ to the constitutive equations. It is shown that the non-local equivalent plastic strain ep at a material point P can be identified with the average value of the local von Mises equivalent plastic strain ε¯p over a sphere centered at P and of radius approximately equal to 3 ℓ. A methodology for the numerical integration of the constitutive equations is presented. The algorithm is appropriate for rate-independent as well as rate-dependent (viscoplastic) models. The model is implemented in the ABAQUS general-purpose finite element program and both quasi-static and dynamic problems are solved. Two possible ABAQUS implementations are discussed. First,“user elements” are developed, which can be used for the solution of both quasi-static and dynamic problems. Reduced 1-point Gauss integration is discussed in 8-node hexahedral elements and the “physical stabilization” method of Puso (2000) is used to remove the resulting numerical singularities (hourglass control). Second, the implementation of the model via “user material” subroutines is discussed. Quasi-static problems can be solved with ABAQUS/Standard using a *COUPLED TEMPERATURE-DISPLACEMENT, STEADY STATE analysis together with user subroutine UMAT, in which temperature is identified with the non-local equivalent plastic strain ep; the solution of dynamic problems requires use of ABAQUS/Explicit together with a *DYNAMIC TEMPERATURE-DISPLACEMENT analysis option and user subroutines VUMAT and DFLUX. Several example problems are solved.