Numerical Integration Process Research Articles

Bending analysis of viscoelastic plates according to the Reissner’s theory using meshless local Petrov–Galerkin method and hereditary integral formulation

Purpose The purpose of this study is to apply the Meshless Local Petrov–Galerkin (MLPG) method to solve the bending problems of linear viscoelastic plates, considering Reissner’s theory.Design/methodology/approachThe weak formulation for the set of equations that govern Reissner’s plate theory is implemented in conjunction with the integral formulation applied to viscoelastic constitutive expressions. A meshless method based on the Moving Least Squares (MLS) approximation is considered in the numerical implementation. The final equation system is assembled by adopting simple and efficient schemes for numerical integration, considering a simplified formulation through centralization of the local interpolation domains and Gaussian quadrature at the same field point. The results obtained are compared with available solutions to demonstrate the efficiency of the proposed formulation.FindingsThe hereditary integral approach proved to be the most general way to analyze the viscoelastic problem, especially when applied together with the modified scheme for numerical integration. In addition, the variable changing technique is demonstrated to be an efficient formulation for solving shear-locking effects in thin plate problems.Originality/valueThe differential of the present study is related to the manner in which the properties of linear viscoelastic materials are considered in the formulation. Although most authors consider this point through the application of the correspondence principle, the present study works with a hereditary integral formulation. In addition, the variable changing technique is applied to solve the shear-locking effects, and an alternative approximation technique is considered to speed up the numerical integration process.

Friction tensor for an ellipsoid enclosed in a rarefied gas

This research renders calculations of rotranslational friction and diffusion tensors in a rarefied gas, considering both specular and diffuse reflection off ellipsoidal particles. Calculations were conducted in a spherical polar coordinate system, revealing the dynamic nature of these off-diagonal tensors. The tensor elements were determined analytically, intending to compute the surface integrals. All diagrams, and pictures of two-dimensional surfaces were obtained by the numerical integration process in MATLAB. Additionally, we performed calculations without using the time parameter and time derivatives in the relevant relations. Deviations from the state of spherical symmetry were observed in the results for an ellipsoid and show that the tensors are dynamic quantities. Therefore, it is necessary to provide average values. We established that the non-zero elements of the friction tensor for an ellipsoid represent the directions of linear momentum transfer. Also, the non-zero components of the diffusion tensor signify the extent of fluctuations in relevant momentum transfers in corresponding directions.

Equations of Motion of Mechanical Systems with Switchable Constraints

The article deals with a numerical and analytical approach to solving the equations of motion, which enables to treat the considered problems with the change of system structure or number of degrees of freedom without interrupting the numerical integration process. The described methodology allows effectively incorporate switchable constraints in the systems in accordance with their flexible structures. The crucial idea is based on the formulation of the resulting differentialalgebraic equations into a saddle point system, where the switchable constraints are represented by a sign matrix with variable rank. In connection with this property, a pseudoinversion is applied to eliminate algebraic variables and transform the problem to the first order system of ordinary differential equations. Moreover, the time independent case leads to linear autonomous systems with non-diagonalizable matrices, as is proved. The relevant numerical scheme is based on Runge-Kutta methods, that correspond to the power series of the resulting matrix exponential for time independent problems. The methodology presented is illustrated on the idealized two-mass oscillator with a switchable constraint. The numerical experiments performed range from initial stages, through simple transient cases to damped intentional control. The advanced applications can be found in robotics, active and controlled systems, and in the simulations of complex systems in biology and related areas. Moreover, the methodology can also be applied in the simulation of transport systems, especially in relation to vehicle technology, a quarter car suspension system, a vibration control mechanism, a torsion system with a clutch, and machine balancing and storage should to be highlighted.

Optimization of Weight Matrices for the Linear Quadratic Regulator Problem Using Algebraic Closed-Form Solutions

This work proposes an analytical gradient-based optimization approach to determine the optimal weight matrices that make the state and control input at the final time close to zero for the linear quadratic regulator problem. Most existing methodologies focused on regulating the diagonal elements using only bio-inspired approaches or analytical approaches. The method proposed, contrarily, optimizes both diagonal and off-diagonal matrix elements based on the gradient. Moreover, by introducing a new variable composed of the steady-state and time-varying terms for the Riccati matrix and using the coordinate transformation for the state, one develops algebraic equationsbased closed-form solutions to generate the required states and numerical partial derivatives for an optimization strategy that does not require the computationally intensive numerical integration process. The authors test the algorithm with one- and two-degrees-of-freedom linear plant models, and it yields the weight matrices that successfully satisfy the pre-defined requirement, which is the norm of the augmented states less than 10−5. The results suggest the broad applicability of the proposed algorithm in science and engineering.

An integrated design approach for simultaneous shape and topology optimization of shell structures

In this work, a novel design approach is developed to carry out shape and topology optimization of shell structures simultaneously. This approach effectively integrates the IsoGeometric Analysis (IGA) method, the Adaptive Bubble Method (ABM) and the Finite Cell Method (FCM) to make full use of their respective advantages in shell structure optimization. To be specific, the IGA method using Non-Uniform Rational B-Splines (NURBS) elements is adopted to analyze the shell structure and realize shape optimization with merits of exact shell surface representation geometrically and rotation-free shell formulation physically; the ABM is utilized to insert deformable holes adaptively in the parametric domain of the NURBS surface for topology optimization of the shell structure; the FCM is employed to facilitate the optimization process by simplifying significantly the discrimination and the numerical integration processes of trimmed elements involved in fixed-mesh analysis of the holed shell structure. Based on this integrated design approach of IGA/ABM/FCM, shape optimization can be implemented by directly optimizing the control-point coordinates of the NURBS surface in the physical space, while topology optimization could be carried out simultaneously in the parametric domain as easily as for 2D planar structures. Finally, representative examples are investigated to demonstrate the effectiveness and advantages of the proposed design approach.

Practical Approach to Radar Detection Performance for NCI System

In noncoherent integration (NCI) radar systems, analyzing detection probability is a basis for designing and evaluating the system performance. Most previous studies have approached it with a complex integrand: a joint probability distribution function of signal components. In calculating the detection probability of NCI systems, some numerical computation techniques are essential in the numerical integration process. The calculation becomes even more intricate when the system considers complex clutter or propagation environment, as well as additive white Gaussian noise. This letter discusses a practical approach to estimate radar detection performance without using complicated numerical methods. This is based on the ideas of how the mixed signal sample follows sample mean properties and central limit theorem under the NCI system. Some comparisons are given against the Swerling II case of target fluctuation and compound sea clutter. The result shows how much this approach is practically applicable in design and analysis of the NCI system.

Research on Algorithm and Coupling Damage Model of Rock Under High Temperature and Loading Based on Mohr–Coulomb Criterion

For the coupled damage problem caused by the surrounding rock of tunnel under high temperature and external load under fire conditions, an elastoplastic damage model is established based on the Mohr–Coulomb (M-C) criterion and the theory of continuous damage mechanics by introducing high temperature and load coupling damage factors. In order to solve the “singularity” problem of the model in the numerical integration process, the fully implicit return mapping algorithm in the principal stress space is deduced by region and including three steps of elastic prediction, plastic correction and damage correction. The finite element solution program of the model is written in C++ language, and uniaxial compression test and tunnel cavern calculation examples are used to verify and analyze the program calculation results. The results show that the difference between the peak intensity calculated by the model and the peak intensity data of the uniaxial test under each temperature is within 5% and the stress–strain curve calculated by the model is consistent with the overall trend of the test curve. The conclusion that damage around the cavity increases with increasing temperature is obtained by the simulation calculation of the cavern, which verifies the accuracy and feasibility of the model and calculation program. The safety and stability evaluation of the surrounding rock of tunnel under the action of fire is provided with a certain theoretical basis by the research content of this paper.

Energy-based monitoring and correction to enhance the accuracy and stability of explicit co-simulation

The simulation of complex engineering applications often requires the consideration of component-level dynamics whose nature and time-scale differ across the elements of which the system is composed. Co-simulation offers an effective approach to deal with the modelling and numerical integration of such assemblies by assigning adequate description and solution methods to each component. Explicit co-simulation, in particular, is frequently used when efficient code execution is a requirement, for instance in real-time setups. Using explicit schemes, however, can lead to the introduction of energy artifacts at the discrete-time interface between subsystems. The resulting energy errors deteriorate the accuracy of the co-simulation results and may in some cases develop into the instability of the numerical integration process. This paper discusses the factors that influence the severity of the energy errors generated at the interface in explicit co-simulation applications, and presents a monitoring and correction methodology to detect and remove them. The method uses only the information carried by the variables exchanged between the subsystems and the co-simulation manager. The performance of this energy-correction technique was evaluated in multi-rate co-simulation of mechanical and multiphysics benchmark examples.

Analytical Calculation of the RMS Value and the Spectrum of the DC-Link Current of a Dual-Inverter

This article is focused on the analytic definition of the dc-link current of a dual-inverter. Therefore, a two-level dual-inverter for driving two independent three-phase electrical motors is considered. Space-vector modulation is taken into account as the modulation method of the inverter. Based on that, an analytic description is derived for the frequency spectrum as well as for the rms value of the dc-link current of a dual-inverter. Contrary to the current definitions known from previous publications, the expressions presented below are defined in solved form. Therefore, the use of integral or differential equation expressions is not necessary. In this way, a basis is created for calculating the dc-link current without computationally intensive numerical integration processes or SPICE simulations. Furthermore, this article includes the metrological verification of the novel expressions. The results discussed in this publication provide the basis for stabilizing the HV-dc system and for minimizing the dc-link capacitor load. Thus, they enable the dc-link capacitor of a dual-inverter to be downsized. Based on the presented definition of the dc-link current in solved form, the possibility of real-time optimization is created. Furthermore, the results outlined in this article form the foundation to derive the control optimization for a multiphase inverter.

Robust Trajectory Planning for Hypersonic Glide Vehicle with Parametric Uncertainties

A hybrid double-loop optimization algorithm combing particle swarm optimization (PSO) and nonintrusive polynomial chaos (NIPC) is proposed for solving the robust trajectory optimization of hypersonic glide vehicle (HGV) under uncertainties. In the outer loop, the PSO method searches globally for the robust optimal control law according to a penalized fitness function that contains the system robustness considerations. In the inner loop, uncertainty propagation of the stochastic system is performed using the NIPC method, to provide statistical moments for the iterative scheme of the PSO method in the outer loop. Only control variables are discretized, and the state constraints are satisfied implicitly through the numerical integration process, which reduces the number of decision variables as well as the huge amount of computation increased by NIPC. In the end, the robust optimal control law is achieved conveniently. Numerical simulations are carried out considering a classical time-optimal trajectory optimization problem of HGV with uncertainties in both initial states and aerodynamic coefficients. The results demonstrate the feasibility and effectiveness of the proposed method.

CONSTRUCTION OF EXPLICIT RUNGE-KUTTA METHODS FOR MODELING OF DYNAMIC SYSTEMS WITH DELAY

A system of differential equations with delay is considered, which is a mathematical model of many technical processes with a time delay. Numerical Runge-Kutta methods and the method of expansion onTaylorseries on time delay are most often used to model such systems.When the value of the delay is greater than the step of numerical integration, the numerical solution of these systems is not difficult. For this, interpolation of the prehistory of the model and numerical methods for conventional systems of differential equations are used. For example, explicit Runge-Kutta methods. Using the method of steps, a numerical solution is obtained for the required period of time. But, if the time interval is large enough in comparison with the delay, then the number of steps turns out to be large, which slows down the process of numerical integration and leads to the accumulation of errors.For small delays,Taylortime delay expansions are used and the further solution of the usual system of differential equations by a numerical method, for example, by the Runge-Kutta method. This approach has limitations on the amount of delay and is not applicable for many models.Thus, it is often necessary to apply a step of numerical integration that is larger than the delay value and continuous implicit Runge-Kutta methods, which leads to a complication of the numerical algorithm, since at each step of numerical integration it is necessary to solve systems of nonlinear equations.In this paper, based on the construction of Newton's and Taylor's polynomials, an algorithm has been developed that allows using explicit Runge-Kutta methods to solve systems with delay and the step size of numerical integration is more than the delay value. For systems with delay, an explicit Runge-Kutta method of the fifth order of approximation is constructed on the basis of the most frequently used explicit Runge-Kutta methods of the fifth order of approximation. These methods are convenient in programming, have a higher speed of calculation than implicit methods, and are applicable for steps of numerical integration that are large in comparison with the lag.

Iterative refinement algorithm for efficient velocities and accelerations solutions in closed-loop multibody dynamics

To simulate closed-loop multibody systems in effective and accurate way, the semi-recursive formulations can be used together with optimal numerical integrators. Although numerous contributions have been reported in this field, there still exists a demand to improve the computation efficiency for the faster-than-real-time simulation and hardware-in-the-loop control applications. This paper presents an iterative refinement algorithm to accelerate the numerical integration process of a semi-recursive multibody formulation. By reusing the constraint Jacobian matrix factorization and generalized mass matrix factorization, respectively, the presented algorithm is introduced to enhance the solution of dependent relative velocities and independent relative accelerations. The introduced procedure consists of three steps; initial guess determination, iterative refinement process and termination criteria. The iterative refinement process is designed to find an accurate numerical solution, which contains residual computing, solution increment computing and solution updating. For higher efficiency, the initial guess can be re-determined to repeat the iterative refinement process. Three closed-loop multibody systems with increasing complexity are taken as numerical examples to verify the accuracy and efficiency of the presented iterative refinement algorithm. The results highlight more than 15% efficiency gain. The algorithm also can be implemented in other numerical integrators and multibody formulations.

An efficient time-space formulation for dynamic transient analyses: Application to the beam assemblies subjected to moving loads and masses

In this study, the formulation of a time marching method has been developed for the dynamic analysis of beam-type structures in the presence of moving loads/masses. By considering each flexural member of the system as a beam element, the introduced idea of the time-weighted residual method in our recent study “Int. J. Numer. Methods Eng., Vol. 114(7); pp. 719–748″, has been evoked to express the beam response in terms of a series of exponential basis functions. The solution advances in time by modifying the series coefficients through employing a pre-integration process as well as the governing equilibrium equation. The presented method needs neither discretization of the members nor any numerical integration processes so the solution can be obtained by the minimum numbers of required degrees of freedom. Six sample problems have been provided to show the capabilities of the proposed method in the solution of a variety of single and multi-spans beam under constant and accelerated moving loads/masses.

A Bisection Algorithm for Time-Optimal Trajectory Planning Along Fully Specified Paths

The time-optimal trajectory planning problem involves minimizing the time required to follow a path defined in space, subject to kinematic and dynamic constraints. Here, we introduce a novel technique, called the bisection algorithm (BA), which is fully implemented in C++ and extends dynamic programming approaches to the problem. These approaches, which rely on dividing the global problem into a series of simpler subproblems, become increasingly advantageous compared to direct transcription methods as the number of problem constraints increases. In contrast to nearly all other dynamic programming approaches, BA does not rely on finding a maximum-velocity curve or explicitly finding acceleration switching points during the trajectory planning process. Additionally, only one forward and one backward integration are used, during which all constraints are imposed. This approach is made feasible through careful control of the numerical integration process and the use of a bisection algorithm to resolve constraint violations during integration. BA is shown to be significantly simpler, faster, and more robust than recently proposed algorithms: a direct comparison is made for a series of paths to be followed by a serial manipulator, subject to kinematic constraints. The wide applicability of BA is then established by solving the time-optimal problem for a parallel manipulator following a complex path, subject to both kinematic and dynamic constraints.

Influence of friction and asymmetric freeplay on the limit cycle oscillation in aeroelastic system: An extended Hénon’s technique to temporal integration

Nonlinear effects such as friction and freeplay on the control surfaces can affect aeroelastic dynamics during flight. In particular, these nonlinearities can induce limit cycle oscillations (LCO), changing the system stability, and because of this it is essential to employ computational methods to predict this type of motion during the aircraft development cycle. In this context, the present article presents a matrix notation for describing the Hénon’s method used to reduce errors when considering piecewise linear nonlinearities in the numerical integration process. In addition, a new coordinate system is used to write the aeroelastic system of equations. The proposal defines a displacement vector with generalized and physical variables to simplify the computational implementation of the Hénon’s technique. Additionally, the article discusses the influence of asymmetric freeplay and friction on the LCO of an airfoil with control surface. The results show that the extended Hénon’s technique provides more accurate LCO predictions, that friction can change the frequency and amplitude of these motions, and the asymmetry of freeplay is important to determine the LCO behavior.

Analog Accelerator for Simulation and Diagnostics

We propose a new method for solving Initial Value Problems (IVPs). Our method is based on analog computing and has the potential to almost eliminate traditional switching time in digital computing. The approach can be used to simulate large systems longer, faster, and with higher accuracy. Many algorithms for Model-Based Diagnosis use numerical integration to simulate physical systems. The numerical integration process is often either computationally expensive or imprecise. We propose a new method, based on Field-Programmable Analog Arrays (FPAAs) that has the potential to overcome many practical problems. We envision a software/hardware framework for solving systems of simultaneous Ordinary Differential Equations (ODEs) in fraction of the time of traditional numerical algorithms. In this paper we describe the solving of an IVP with the help of an Analog Computing Unit (ACU). To do this we build a special calculus based on operational amplifiers (op-amps) with local feedback. We discuss the implementation of the ACU on an Integrated Circuit (IC). We analyze the working if the IC and simulate the dynamic Lotka-Volterra system with the de-facto standard tool for electrical simulation: Spice.

A new model for analyzing the vibration behaviors of rotor-bearing system

Considering the strong non-linear properties of rolling element bearings (REBs), a new model for the dynamic analysis of rotor-bearing system, especially for the heavy-load and high-speed condition, is proposed in this work. Two types of bearing stiffness are classified and analyzed with physical meanings based on the load-deflection curves, which can full preserve the geometric and kinematic nature of REBs for the vibration model. The improved numerical integration process based on 4th order Runge-Kutta method is described in details. One type of cylindrical bearing with internal clearance is brought in for case study. Complicated non-linear vibration behaviors caused by REBs instinct properties are observed. The effects from speed fluctuation, unbalance forces, external loads on the vibration responses of the system are investigated. This work may provide some references for the structure design, vibration prognosis and other engineering applications of the rotor-bearing system.

Nonlinear dynamics of a spur gear pair with force-dependent mesh stiffness

Time-varying mesh stiffness is one of the main excitation sources of a gear system, and it is also considered as an important factor for the vibration and noise of gears. Thus, this excitation is usually taken as an input into the gear dynamic model to obtain the system dynamic responses. However, the mesh stiffness of a gear pair is actually nonlinear with respect to the dynamic mesh force (DMF) that fluctuates during the operation of gears. Therefore, the dynamic model of gears with the quasi-static mesh stiffness calculated under a constant load is not accurate sufficiently. In this paper, a dynamic model of spur gear is established with considering the effect of the force-dependent time-varying mesh stiffness, backlash and profile deviation. Due to the nonlinear relationship between the mesh stiffness and the load for each tooth pair, it needs first to determine the load sharing among tooth pairs and then calculate the overall mesh stiffness of the gear pair. As the mesh stiffness and DMF are related, the mesh stiffness is no longer directly taken into the gear dynamic model as an input, but is jointly solved with the numerical integration process using the gear dynamic model. Finally, the dynamic responses predicted from the established gear dynamic model are compared with the experimental results for validation and compared with the traditional models to reveal their differences. The results indicate that the established dynamic model of spur gear transmission has a wider application range than the traditional models.

Numerical analysis of the dynamics of rigid blocks subjected to support excitation

The dynamic behaviour of rigid blocks subjected to support excitation is represented by discontinuous differential equations with state jumps. In the numerical simulation of these systems, the jump times corresponding to the numerical trajectory do not coincide with the ones of the given problem. When multiple state jumps occur, this approximation may affect the accuracy of the solution and even cause an order reduction in the method. Focus here is on the error behaviour in the numerical dynamic. The basic idea is to investigate how the error propagates in successive impacts by decomposing the numerical integration process of the overall system into a sequence of discretized perturbed problems.

Calculation Formulas and Simulation Algorithms for Entropy of Function of LR Fuzzy Intervals.

Entropy has continuously arisen as one of the pivotal issues in optimization, mainly in portfolios, as an indicator of risk measurement. Aiming to simplify operations and extending applications of entropy in the field of LR fuzzy interval theory, this paper first proposes calculation formulas for the entropy of function via the inverse credibility distribution to directly calculate the entropy of linear function or simple nonlinear function of LR fuzzy intervals. Subsequently, to deal with the entropy of complicated nonlinear function, two novel simulation algorithms are separately designed by combining the uniform discretization process and the numerical integration process with the proposed calculation formulas. Compared to the existing simulation algorithms, the numerical results show that the advantage of the algorithms is well displayed in terms of stability, accuracy, and speed. On the whole, the simplified calculation formulas and the effective simulation algorithms proposed in this paper provide a powerful tool for the LR fuzzy interval theory, especially in entropy optimization.