Numerical Integration Algorithm Research Articles

Predicting the forming limits in the tube hydroforming process by coupling the cyclic plasticity model with ductile fracture criteria

This study presents a one-surface plasticity constitutive model combined with phenomenological and micromechanical damage models to be applied to the large plastic deformation during tube hydroforming. These factors lead to simulating the Bauschinger effect and transient behavior in large plastic strain ranges and predicting fracture onset. By using the integration approach, a numerical integration algorithm of implementation by explicit schemes was carried out. A comparison of several cyclic loading tests between experiments and numerical simulation was accomplished. The accuracy of these models is verified, which could be expanded to a large strain range forming process. Adopting the proposed models, finite element modeling is performed in the tube hydroforming process. The simulation and experimental results comparison showed that the proposed framework accurately predicted the tube formability in the hydroforming process.

The curse of dimensionality for the Lp-discrepancy with finite p

The Lp-discrepancy is a quantitative measure for the irregularity of distribution of an N-element point set in the d-dimensional unit-cube, which is closely related to the worst-case error of quasi-Monte Carlo algorithms for numerical integration. It's inverse for dimension d and error threshold ε∈(0,1) is the minimal number of points in [0,1)d such that the minimal normalized Lp-discrepancy is less or equal ε. It is well known, that the inverse of L2-discrepancy grows exponentially fast with the dimension d, i.e., we have the curse of dimensionality, whereas the inverse of L∞-discrepancy depends exactly linearly on d. The behavior of inverse of Lp-discrepancy for general p∉{2,∞} has been an open problem for many years. In this paper we show that the Lp-discrepancy suffers from the curse of dimensionality for all p in (1,2] which are of the form p=2ℓ/(2ℓ−1) with ℓ∈N.This result follows from a more general result that we show for the worst-case error of numerical integration in an anchored Sobolev space with anchor 0 of once differentiable functions in each variable whose first derivative has finite Lq-norm, where q is an even positive integer satisfying 1/p+1/q=1.

Comments on “Fuzzy Simulations for Expected Values of Functions of Fuzzy Numbers and Intervals”

It must be pointed out that the conclusion about the perfect performance of a recently proposed fuzzy simulation technique for expected values of fuzzy functions by Liu <i>et al.</i> (IEEE Trans. Fuzzy Syst., 29(6), 2021), the <i>improved stochastic discretization algorithm</i> (iSDA), is absolute and dangerous, which requires an urgent correction. Therefore, in this note, first and foremost, several theorems are initiated to address and specify the usage scope of the iSDA, which is related to the type of fuzzy variables involved in the functions. Further, examples and counter-examples with schematic diagrams based on the proposed theorems are provided to illustrate the performance of the iSDA. It is seen that the simulation results of the iSDA would be uncontrollable and lead to unbearable error for counter-examples, which immediately shows the necessity of this note. Second and incidentally, the respective convergence of the iSDA and another algorithm, the <i>special numerical integration algorithm</i> (NIA-S), are proved as supplementary contents for the original paper. Finally, the conclusion about the comparison between the iSDA and NIA-S is also restated.

1D modelling and preliminary analysis of the coupled DYNASTY–eDYNASTY natural circulation loop

In the continuous strive to improve the safety of current-generation and next-generation nuclear power plants, natural circulation can be used to design passive safety systems to remove the decay heat during the shutdown. The Molten Salt Fast Reactor (MSFR) is a peculiar type of Gen-IV nuclear facility, where the fluid fuel is homogeneously mixed with the coolant. This design leads to natural circulation in the presence of an internally distributed heat source during the shutdown. Furthermore, to shield the environment from the highly radioactive fuel, an intermediate loop between the primary and the secondary loops, able to operate in natural circulation, is required. To analyze the natural circulation with a distributed heat source and to study the natural circulation of coupled systems and the influence of the intermediate loop on the behaviour of the primary, Politecnico di Milano designed and built the DYNASTY-eDYNASTY facility. The two facilities are coupled with a double-pipe heat exchanger, which siphons heat from DYNASTY and delivers it to the eDYNASTY loop. This work focuses on modelling the coupled DYNASTY-eDYNASTY natural circulation loops using DYMOLA2023®, an integrated development environment based on the Modelica Object-Oriented a-causal simulation language. The 1D Modelica approach allows for building highly reusable and flexible models easing the design effort on a complex system such as the DYNASTY-eDYNASTY case without the need to rewrite the whole model from scratch. The coupled models were developed starting from the already-validated single DYNASTY model and the double-pipe heat exchanger coupling. The models were tested during the whole development process, studying the influence of the numerical integration algorithm on the simulation behaviour. A preliminary analysis of both the adiabatic and the heat loss models analyzed the effect of the secondary natural circulation loop on the behaviour of the DYNASTY loop. The simulation results showed that the eDYNASTY loop dampens the behaviour of the primary DYNASTY loop. Furthermore, a parametric analysis of the DYNASTY and the eDYNASTY coolers highlighted the influence of the cooling configuration on the facility’s behaviour. Finally, the simulation results identified the most critical aspects of the models in preparation for an experimental comparison.

Experimental Comparison of the Sample Sizes of the Two-Sample Tests of Wilcoxon and Student under the Arcsine Distribution

When statistical experiments are performed, a sample size should be chosen in some optimum way so that we should use a sample size no larger than necessary. This paper compares the minimum sample sizes of two-sample t-test and the Wilcoxon rank-sum test under the arcsine distribution, based on their power [1, 2]. To accomplish this task, some essential probabilities that are useful in the power and sample size determination of the Wilcoxon rank-sum test were derived by the author for computing the approximated formula given by Lehmann [2]. The composite numerical integration algorithm is used to compute these probabilities, which are related to the arcsine distribution. In this study, a computer program was built by the author to find the exact (simulated) minimum sample sizes n for any significant level and power by iterating on n with starting points for n provided by the approximated formulas of [2, 3]. The scientific novelty of this research paper is determining the minimum sample sizes by considering a new set of the arcsine distribution shift alternatives of the forms giving the left-hand endpoint of the displaced distribution as the quantile of order p, 0 &lt; p &lt; 1, of the second distribution rather than using alternatives that specify as the quantile of order p, p &lt; 0.5. As considered by [1], a choice that prevents losing some important alternative hypotheses is an extension to the set of alternative hypotheses considered by Guenther [1]. The exact (simulated) minimum sample sizes were computed and compared with each other and with the corresponding approximated formulas given by Lehmann [2] and Guenther [3]. Numerical results showed that the approximated formulas are very accurate and the Wilcoxon rank-sum test is more efficient when the sample size is more than 45. Otherwise, the Student two-sample t-test is better.

(Quasi-)Monte Carlo Improtance Sampling with QMCPY

(Quasi-)Monte Carlo, (Q)MC, methods are a class of powerful numerical integration algorithms that have been proven to scale well to high dimensions. Various techniques exist to decrease the computational cost of (Q)MC methods. This article focuses on importance sampling, a technique that performs variable transformations to make the integral easier for (Q)MC approximation. The build up to composed importance sampling is paralleled by code from our QMCPy package that implements these concepts.

Numerical algorithm for quaternion integration based on three independent parameters with no need for re-normalization

Numerical algorithm for quaternion integration based on three independent parameters with no need for re-normalization

Fast calculation of scattering patterns using hypergeometric function algorithms

The scattering of light, X-rays, electrons or neutrons by matter is used widespread for structural characterization from atomic to macroscopic length scales. With the advent of high-brilliance beam sources and the development fast, large area pixelated detectors, scattering patterns are now acquired at unprecedented frame rates and frame sizes. The slow analysis of these scattering patterns has evolved into a severe bottleneck retarding scientific insight. Here we introduce an algorithm based on the use of hypergeometric functions providing gains in computational speed of up to 105 compared to present numerical integration algorithms. Hypergeometric functions provide analytical descriptions of geometrical shapes, can be rapidly computed as series and asymptotic expansions, and can be efficiently implemented in GPUs. The algorithm provides the necessary computational speed to calculate scattering patterns on timescales required for real-time experiment feedback, the analysis of large volumes of scattering data, and for the generation of training data sets for machine learning.

Customization of the angular spectrum method for calculating the acoustic piston field transmitted through a solid plate using MATLAB

The angular spectrum (AS) model is customized to calculate the spatial acoustic pressure field generated by a piston source and transmitted through a steel plate immersed in water, for normal beam incidence. A MATLAB program is developed for this specific problem combining use of Gauss quadrature and a generalized Filon method. The program calculates the pressure wave number spectrum generated by the piston source and transforms the wave number spectrum into the spatial domain. Convergence analysis show that the MATLAB program is far more efficient than the more traditional approach of using the fast Fourier transform algorithm to transform the pressure wavenumber spectrum into the spatial domain. The MATLAB program is published here and free for others to use. The methods and MATLAB algorithms are obtained by•Converting the original 2D AS model to a 1D model using cylindrical coordinates.•Combining use of Gauss quadrature and a generalized Filon method for more accurate pressure calculations compared with use of the fast Fourier transform.•Introducing adaptive numerical integration algorithms in MATLAB and error control parameters which are easy to use.

Analytical Solutions for Power Law varied FG laminate Subjected to Thermo-mechanical Loadings

This paper consists of a thermal stress analysis of power law varied functionally graded laminate. In this paper, the exact temperature field obtained by the heat conduction equation, and the same has used for thermal stress analysis. In addition, a simple power law varied temperature field also considered for thermal stress analysis. Heat conductivity and coefficient of thermal expansion have graded as power-law along with the thickness of the domain. The Poisson ratio has kept constant. The numerical solution scheme involved the conversion of the boundary value problem to the initial value problem and the use of the 4th-order Runge Kutta-Grill algorithm for numerical integration. The parametric study includes a thick to thin laminate by considering various aspect ratios.

Preliminary validation of the 1D modeling of the DYNASTY natural circulation loop against results from water experimental campaign

In the continuous improvements of the safety of nuclear power plants, the study of natural circulation is of primary importance for developing passive safety systems and Generation-IV reactors, such as the Molten Salt Fast Reactor (MSFR). In particular, removing the decay heat during the shutdown of the MSFR involves dealing with a circulating fuel driven by natural circulation; as such, the development and validation of natural circulation models are of primary importance. To validate models and provide experimental data on natural circulation with a distributed heat source, the DYNASTY (DYnamics of NAtural circulation for molten SalT internallY heated) natural circulation loop has been built on the premises of Politecnico di Milano. This paper focuses on improving the 1D modeling of natural circulation phenomena in the DYNASTY facility and validating them against the results (also reported in this work) found in the first experimental campaign, carried out with water as the working fluid. The modeling of DYNASTY is based on a 1D approach using DYMOLA®, a modeling environment based on the MODELICA simulation language. The models used are improved versions of the ones available in the literature, which better model the experimental facility as they now include the mass flow rate meter, the heat loss between the facility and the environment, and a more realistic cooler. The improved model was then used to study the outcomes of the model simulations and verify the influence of the different numerical integration algorithms on the simulation results. The first DYNASTY experimental campaign provides in-depth testing of its behavior, with different cooling fan speeds and considering all possible heating configurations. The collected results for the heating and cooling transients are the mass flow rate and four fluid temperatures measured by thermocouples placed at the beginning and the end of the four DYNASTY legs. The experimental results were compared with the model simulations, showing the good predicting capabilities of the latter one following the tuning of the heat exchange model. In particular, the comparison showed an excellent match between the model and the experimental results for the heating transient while showing improving margins for the cooling transient.

Numerical Implementation of the Barcelona Basic Model Based on Return-Mapping Integration

This paper implemented the Barcelona basic model (BBM) into the OpenGeoSys (OGS) platform for numerical modeling of the coupled hydro-mechanical (HM) behavior of unsaturated soil. Within the implicit integration approach in the OGS, the integration rule of the BBM was developed first and the integration form of the BBM under a return mapping algorithm was built. The closest point projection method was used for calculating the return mapping directions with the associative flow rule. The numerical simulation results show that the BBM is feasible in fitting the experimental results. The numerical integration algorithm can reflect the elastic–plastic mechanical behavior of materials, and improve the calculation accuracy. The material exhibits obvious elastic–plastic characteristics during numerical simulation and experiment, and the water absorption process can lessen the mixture’s compression stiffness while enhancing its recovery stiffness.

On stress and deformation accumulation of continuously welded rails under cyclic thermal loading in high-speed railways

This paper focuses on the stress and deformation accumulation of continuously welded rails (CWR) in ballastless tracks under cyclic thermal loading, which could pose a potential threat to the stability and service life of CWR but related theoretical studies are yet to be conducted. Firstly, the nonlinear longitudinal resistance of fasteners (LRF) resulting from the CWR-rail pad contact interface is mathematically described by extending the classical Dhal friction model and verified via an in-door experiment. Both the hysteretic and softening properties of the LRF can be captured compared with the traditional ideal elastoplastic model. Most importantly, we originally propose a quasi-dynamic method (QDM) to tackle nonlinear static problems by extending an explicit numerical integration algorithm from the dynamics field to the statics. Afterwards, the quasi-dynamic equilibrium equations of CWR in the tunnel transition zone and on the simply-supported beam bridges subjected to thermal loads and LRF are constructed, and the reliability of QDM to obtain static responses is well-validated by existing analytical methods. Finally, the limitation of the ideal elastoplastic LRF model, distribution characteristics and accumulation process of rail longitudinal deformation and axial additional stress are fully unveiled. The current work is conducive to better recognition of the longitudinal mechanical behavior of CWR under cyclic thermal loads.

Structure and Analysis of Properties of Basic Comb Filters for Automatic Systems

The paper considers the issues of using comb filters (CF), known in radio engineering, as an alternative to continuous corrective filters, when they are implemented in controllers in automatic control systems. The convenience of their implementation in controllers is associated with the organization of a fixed number of sample passes in the processed signal. In this paper, the properties of two types of comb filters of the first order are highlighted and studied in detail: a high-frequency filter and a proportional (forming) filter. The characteristics of these filters are taken as a basis for the implementation of more complex characteristics of corrective devices in combination with numerical integration algorithms. The analysis of the properties of CF was carried out in the frequency domain. It is shown that the initial sections of the frequency characteristics of the CF at certain filter parameters coincide well with the continuous analogues of such devices. It is these areas of frequency characteristics, limited by the cutoff frequency of the open system, that determine the coincidence of static and dynamic properties of the synthesized system. It is shown that CF reduces on 30 % the level of the standard deviation (RMS) of uniformly distributed white noise at the filter output. The decrease in the level of RMS occurs due to the presence of periodically recurring dips in the amplitude-frequency response (frequency response) of filters forming a so-called comb. Due to the fact that the filters in question "cut out" harmonics in the output signal spectrum that are multiples of the inverse of the total duration of the transmitted reference intervals, then a special choice of this duration can not only reduce the level of the random signal at the CF output, but at the same time significantly reduce the level of regular interference, if it is present. Such filters can be simultaneously used as amplitude modulated signal demodulators.

Error Analysis of Digital Electric Energy Metering Based on FPGA

With the emergence and development of smart grid, digital meters have been widely used in substations, so the research on digital energy metering error has become a research hotspot. The purpose of this paper is to analyze the error of digital power measurement based on FPGA, mainly take the analog input fusion module and digital power meter in the digital power measurement system as the research object, and analyze the source of measurement error. An FPGA-based energy metering error detection platform is built to analyze the energy metering error when there are harmonics in the full energy metering method currently used by most digital energy meter manufacturers. The comparative analysis of numerical integration algorithms shows that the asymmetric distortion rates of the simulated waveforms are approximate. The results of measurement error analysis under harmonic conditions show that when the number of sampling points per cycle is 60, the power measurement errors of the 2nd to 4th harmonics are equivalent. The magnitude of the measurement error is still consistent with other low-order harmonics, and the simulation results are consistent with the theoretical analysis results.

Computational Differential Algebraic Equation Framework and Multi Spatial and Time Discretizations Preserving Consistent Second-Order Accuracy: Nonlinear Dynamics

Abstract We propose the novel design, development, and implementation of the well-known generalized single step single solve (GS4) family of algorithms into the differential algebraic equation framework which not only allows altogether different numerical time integration algorithms within the GS4 family in each of the different subdomains but also additionally allows for the selection of different space discretized methods such as the finite element method and particle methods, and other spatial methods as well in a single analysis unlike existing state-of-the-art. For the first time, the user has the flexibility and robustness to embed different algorithms for time integration and different spatial methods for space discretization in a single analysis. In addition, the GS4 family enables a wide variety of choices of time integration methods in a single analysis and also ensures the second-order accuracy in time of all primary variables and Lagrange multipliers. This is not possible to date. However, the present framework provides the fusion of a wide variety of choices of time discretized methods and spatial methods and has the bandwidth and depth to engage in various types of research investigations as well and features for fine tuning of numerical simulations. It provides generality/versatility of the computational framework incorporating subdomains with different spatial and time integration algorithms with improved accuracy. The robustness and accuracy of the present work is not feasible in the current state of technology. Various numerical examples illustrate the significant capabilities and generality and effectiveness for general nonlinear dynamics.

Study on electromagnetic transient numerical simulation of single-phase inverter circuit

With the continuous progress and development of new energy power generation technology, new energy grid and power electronic devices are widely used in power system, and the modeling requirements for electromagnetic transient simulation analysis of power system are higher and higher. Combined with the research status at home and abroad, this paper first analyzes the single-phase inverter circuit model, and establishes the mathematical model of single-phase inverter circuit in different switching states. Based on the stability, accuracy and efficiency of numerical integration algorithm, different numerical integration substitution methods are applied to the mathematical model established in each state of the circuit, and the electromagnetic transient analysis of single-phase inverter circuit is carried out the correctness and feasibility of the experimental results are verified by comparing with the results in Simulink.

Efficient Numerical Integration Algorithm of Probabilistic Risk Assessment for Aero-Engine Rotors Considering In-Service Inspection Uncertainties

Numerical integration methods have the characteristics of high efficiency and precision, making them attractive for aero-engine probabilistic risk assessment and design optimization of an inspection plan. One factor that makes the numerical integration method a suitable approach to in-service inspection uncertainties is the explicit derivation of the integration formula and integration domains. This explicit derivation ensures accurate characterization of a multivariable system’s failure risk evolution mechanism. This study develops an efficient numerical integration algorithm for probabilistic risk assessment considering in-service inspection uncertainties. The principle of probability conservation is applied to the transformation of the integration domain from the current flight cycle to the initial (N = 0) computational space. Consequently, the integration formula of failure probability is deduced, and a detailed mathematical demonstration of the proposed method is provided. An actual compressor disk is evaluated using the efficient numerical integration algorithm and the Monte Carlo simulation to validate the accuracy and efficiency of the proposed method. Results show that the time cost of the proposed algorithm is dozens of times lower than that of the Monte Carlo simulation, with a maximum relative error of 5%. Thus, the efficient numerical integration algorithm can be applied to failure analysis in the airworthiness design of commercial aero-engine components.

Vibro-Impact Response Analysis of Collision with Clearance: A Tutorial

A collision with clearance causes obvious nonlinearity in structures, and dynamic response analysis plays an important role in predicting the mechanical performance of the structure. The general form of the nonlinear dynamic equation of a structure and the clearance modeling method are introduced, and the clearance-caused nonlinear term is expressed by nonlinear impact forces. Different clearance collision models of local nonlinear structures are presented. The relationships between different impact forces and clearances are analyzed by two rigid sphere models. The solution methods of the nonlinear dynamic equation are compared by a vibro-impact response, such as the Newmark-β method combined with the Newton–Raphson method, generalized α method and precise integration method. The single degree of freedom model is adopted to compare the efficiency of the different numerical integration algorithms. Taking the beam structure model as a case study, the accurate nonlinear collision model with clearance is established by using the impact force model with high accuracy, and the accuracy of the model is verified by comparing the reference model with the numerical model.

Generation of gravitational waves from freely decaying turbulence

We study the stochastic gravitational wave background (SGWB) produced by freely decaying vortical turbulence in the early Universe. We thoroughly investigate the time correlation of the velocity field, and hence of the anisotropic stresses producing the gravitational waves. With hydrodynamical simulations, we show that the unequal time correlation function (UETC) of the Fourier components of the velocity field is Gaussian in the time difference, as predicted by the “sweeping” decorrelation model. We introduce a decorrelation model that can be extended to wavelengths around the integral scale of the flow. Supplemented with the evolution laws of the kinetic energy and of the integral scale, this provides a new model UETC of the turbulent velocity field consistent with the simulations. We discuss the UETC as a positive definite kernel, and propose to use the Gibbs kernel for the velocity UETC as a natural way to ensure positive definiteness of the SGWB. The SGWB is given by a 4-dimensional integration of the resulting anisotropic stress UETC with the gravitational wave Green's function. We perform this integration using a Monte Carlo algorithm based on importance sampling, and find that the result matches that of the simulations. Furthermore, the SGWB obtained from the numerical integration and from the simulations show close agreement with a model in which the source is constant in time and abruptly turns off after a few eddy turnover times. Based on this assumption, we provide an approximate analytical form for the SGWB spectrum and its scaling with the initial kinetic energy and integral scale. Finally, we use our model and numerical integration algorithm to show that including an initial growth phase for the turbulent flow heavily influences the spectral shape of the SGWB. This highlights the importance of a complete understanding of the turbulence generation mechanism.