Implementation Of Numerical Algorithms Research Articles

A data-driven ductile fracture criterion for high-speed impact

Data-driven methods based on machine learning (ML) models offer new approaches for characterizing the fracture behavior of advanced elastoplastic materials. In this paper, a ML-based data-driven ductile fracture criterion is proposed to characterize the fracture behavior of elastoplastic materials under high-speed impact loading conditions. To reduce the required training dataset and enhance the predictability capability, several assumptions are used. Firstly, utilizing the decoupled assumption, two separate artificial neural network (ANN) models are employed to establish the fundamental fracture model and characterize the strain rate effect of ductile fracture behavior, respectively. In addition, the enhanced method with a logarithmic function is introduced to improve predictability capability of the proposed data-driven criterion under unknown high strain rates. To establish a complete numerical implementation framework, an enhanced rate-dependent data-driven constitutive model and a compatible numerical implementation algorithm are additionally introduced. Eventually, to assess the applicability of the proposed data-driven fracture criterion, numerical simulations of notched specimens and ballistic impact conditions of Ti-6Al-4V material are conducted, respectively. These investigation results demonstrate the effectiveness of the proposed data-driven ductile fracture criterion.

Large post-liquefaction deformation of sand: Mechanisms and modeling considering water absorption in shearing and seismic wave conditions

Large deformation of sand due to soil liquefaction is a major cause for seismic damage. In this study, the mechanisms and modeling of large post-liquefaction deformation of sand considering the significant influence of water absorption in shearing and seismic wave conditions. Assessment of case histories from past earthquakes and review of existing studies highlight the importance of the two factors. Based on the micro and macro scale mechanisms for post-liquefaction shear deformation, the mechanism for water absorption in shearing after initial liquefaction is revealed. This is aided by novel designed constant water-absorption-rate shear tests. Water absorption in shearing can be classified into three types, including partial water absorption, complete water absorption, and compulsory water absorption. Under the influence of water absorption in shearing, even a strongly dilative sand under naturally drained conditions could experience instability and large shear deformation. The mechanism for amplification of post-liquefaction deformation under surface wave load is also explained via element tests and theoretical analysis. This shows that surface wave–shear wave coupling can induce asymmetrical force and resistance in sand, resulting in asymmetrical accumulation of deformation, which is amplified by liquefaction. A constitutive model, referred to as CycLiq, is formulated to capture the large deformation of sand considering water absorption in shearing and seismic wave conditions, along with its numerical implementation algorithm. The model is comprehensively calibrated based on various types of element tests and validated against centrifuge shaking table tests in the liquefaction experiments and analysis projects (LEAP). The model, along with various numerical analysis methods, is adopted in the successful simulation of water absorption in shearing and Rayleigh wave-shear wave coupling induced large liquefaction deformation. Furthermore, the model is applied to high-performance simulation for large-scale soil-structure interaction in liquefiable ground, including underground structures, dams, quay walls, and offshore wind turbines.

Post-fire mechanical properties and constitutive model of Q690 high-strength structural steel

Fire hazard is one of the main threats to the integrity of steel structural buildings. Developing an accurate post-fire constitutive model for structural steel is crucial for enhancing the failure assessment of steel structures exposed to fire. In this study, the post-fire mechanical behavior of Q690 high-strength structural steel is investigated. Eighteen round bar specimens made of Q690 steel were heated with temperatures ranging from 20 °C to 900 °C. Tensile tests were conducted on these specimens to obtain the post-fire stress-strain curves and associated mechanical parameters of the material. To accurately characterize the post-fire behavior of Q690 steel, a temperature-dependent elastoplastic constitutive model combined with its numerical implementation algorithm is proposed. This model consists of four components: a temperature-dependent elasticity criterion, the von Mises yield criterion, an associated flow rule, and a temperature-dependent isotropic hardening law. The proposed constitutive model has a notable feature, i.e. the material parameters in the model can be defined as a function of temperature according to the test results. As a result, when utilizing this model for post-fire assessments of steel members, only three routine parameters, i.e. elastic modulus at room temperature, Poisson's ratio, and the temperature experienced by the material, are required in the numerical simulation to generate the material’s stress-strain relationships. To verify the proposed constitutive model, the new model was introduced into the numerical simulations of the tested round bar specimens after exposure to various temperature conditions. Both the load-displacement curves and local deformation of each round bar specimen obtained from the numerical simulations are compared with the test results to validate the accuracy of the new model. The constitutive model presented in this study provides a useful tool for assessing the post-fire behaviors of steel components.

Vectorized Numerical Algorithms to Solve Internal Problems of Computational Fluid Dynamics

The opportunities provided by new information technologies, object-oriented programming tools, and modern operating systems for solving boundary value problems in CFD described by partial differential equations are discussed. An approach to organizing vectorized calculations and implementing finite-difference methods for solving boundary value problems in CFD is considered. Vectorization in CFD problems, eliminating nested loops, is ensured through the appropriate data organization and the use of vectorized operations with arrays. The implementation of numerical algorithms with vectorized mesh structures, including access to internal and boundary mesh cells, is discussed. Specific examples are reported and the implementation of the developed computational algorithms is discussed. Despite the fact that the capabilities of the developed algorithms are illustrated by solving benchmark CFD problems, they enable a relatively simple generalization to more complex problems described by three-dimensional equations.

A data-driven approach for predicting the ballistic resistance of elastoplastic materials

Data-driven methods and machine learning methods provide efficient and accurate approaches for solving impact problems. In this paper, a data-driven approach is proposed for numerical simulations of ballistic impact behavior for elastoplastic materials. An enhanced rate-dependent scheme is employed for improving the predictability of the data-driven constitutive model. A new method that introduces a stress triaxiality indicator to three separate constitutive models is proposed to consider the discrepancy between the mechanical responses of materials under tension, compression, and shear. Additionally, a modified Bai-Wierzbicki fracture criterion considering the strain rate effect and the stress-state effect is used to evaluate the fracture behavior of the materials during impact simulations. Subsequently, a compatible numerical implementation algorithm that considers loading, unloading, and reverse loading is established to enable the application of the data-driven approach in finite element simulations. Numerical validation of the proposed data-driven approach is conducted through several simple loading examples, such as cyclic loading, torsional loading, and tension–torsion combined loading. The data-driven approach is then employed to simulate the ballistic impact behavior of Ti-6Al-4V targets of different thicknesses that are struck by blunt projectiles. Impact properties—including the relationship between residual velocity and impact velocity, ballistic limit velocities, and fracture paths—are comprehensively studied. The results demonstrate the reasonable predictability and accuracy of the data-driven approach when applied to ballistic impact simulations.

Numerical analysis on failure of sheet metals with non-ordinary state-based peridynamics

In this paper, a general anisotropic plastic model for sheet metals is proposed in the non-ordinary state-based peridynamics. The anisotropy plasticity of metallic material is described by the Hill48 yield criterion, and the associated flow rule and non-linear mixed-isotropic-kinematic hardening are included as well. Accordingly, the Newton-Raphson method is used here, which is more efficient than the dichotomy method. Meanwhile, a widely used continuum damage model is coupled with a new bond-breaking method based on the classic Hash principles. A detailed algorithm for numerical implementation is provided for a better understanding. The proposed model is verified by several typical examples and shows its great potential in describing anisotropic plasticity-ductile deformation and failure of sheet metal. In the last, the efficiency of the proposed failure criterion is analyzed and discussed through the numerical analysis of a uniaxial tensile failure test.

A lode-dependent plasticity model for high-strength structural steel

In this study, a new plasticity model was developed to characterize the mechanical behavior of high-strength structural steel under multiaxial stress states. This model utilizes a Lode-dependent yield criterion in conjunction with a non-associated flow rule and isotropic hardening law to determine the evolution of the plastic flow stress. Further, a detailed numerical implementation algorithm for this plasticity model is presented, including the integration algorithm of the constitutive equations using an implicit elastic predictor-return mapping method and formulation of the consistent tangent modulus. To validate the proposed plasticity model, six types of Q690 high-strength steel specimens were tested under monotonic loading. These specimens can be used to obtain the elastoplastic response of the material under uniaxial tension, compression, pure shear, tensile shear, and plane strain state, respectively. Corresponding numerical simulations were performed for each specimen using ABAQUS/Standard with the proposed model being introduced into the analysis via the UMAT user subroutine. Then, the prediction performance of the proposed plasticity model was investigated by comparing the load-displacement curves of each test specimen obtained from both experiments and numerical simulations. It is shown that the proposed plasticity model can accurately predict the mechanical response of the high-strength steel under different stress states and enables to visualize the key stress state parameters and material strength variation at the failure region of each specimen, which confirms the promising prospect of this new model in practical applications.

Unified periodic boundary condition for homogenizing the thermo-mechanical properties of composites

This work emphasizes on the Periodic Boundary Condition (PBC) utilized in the Finite Element Homogenization (FEH) method to numerically determine the Thermo-mechanical (T-M) properties of composites. The unified numerical implementation algorithms of the PBC used for accurately predicting the elastic properties, coefficients of thermal expansion and elasto-plastic behaviors of composites and available for representative volume elements with conformal and non-conformal surface meshes, respectively, are proposed and detailed. Regarding not only the thermo-elastic properties of composites, but also the elasto-plastic behaviors of composites under both simple uniaxial tensile, shear, uniaxial cycle loading paths and complex non-radial loading paths, the unified numerical implementation algorithms are verified to accurately predict the T-M properties of composites, through comparison with the results of the analytical models, the DIGIMAT-FE method, the experimental test and the results from the literature. The unified numerical implementation algorithms of the PBC can be extended to the predictions of other properties of various composites. This work provides a comprehensive insight into the PBC used for predicting the T-M properties of composites and benefits development of future FE codes implemented in commercial software packages.

An enhanced data-driven constitutive model for predicting strain-rate and temperature dependent mechanical response of elastoplastic materials

Data-driven and machine-learning based approaches provide a highly compatible and efficient fundamentals for the mechanical constitutive modeling of engineering materials. In this work, an enhanced data-driven constitutive model is developed to predict the stress–strain relationship of an elastoplastic material through the integration of a data-driven concept with fundamental plasticity theory. A novel strain reconfiguration strategy is proposed to improve the learning capability and predictability of the data-driven model, along with a two-step training method. A compatible numerical implementation algorithm is developed to incorporate the data-driven approach into a finite element calculation. This developed data-driven constitutive model is applied to learn and predict the mechanical response of Ti-6Al-4V titanium alloy under multiple loading conditions, including five different loading rates, four different temperatures, and thirteen different stress states. The excellent correlation with the experimental results demonstrates the high accuracy and generality of the presented approach, especially its capability for predicting unknown nonlinear stress–strain response. The presented theory reveals the great potential of employing such a data-driven approach in computational mechanics.

МОДЕЛИРОВАНИЕ АЭРОДИНАМИЧЕСКОГО ВЗАИМОДЕЙСТВИЯ ПРИ ТРЕКОВЫХ ИСПЫТАНИЯХ ИЗДЕЛИЙ АВИАЦИОННОЙ ТЕХНИКИ

Ground track tests of modern ballistic-type aircraft products make it possible to simulate aerodynamic loads in conditions with a maximum medium density and having a significantly lower cost compared to flight tests. During track testing of products at supersonic speeds, intense vibrations of the structural elements of the movable track equipment and the product under test occur. The article deals with the processes of air flow around a rail installation moving with a high acceleration. Basic models are applied that describe the motion of a homogeneous medium at various velocities, accounting the effects of compressibility, turbulence, and heat transfer. Depending on the conditions, various flow turbulence models are used. An algorithm for numerical implementation on a rectangular grid with local adaptation and subgrid resolution of complex geometry is developed. The methodology for the numerical solution of the equations of dynamics describing the flow of a compressible gas around a curved surface is based on the integration of fluid motion and the transfer of scalar quantities in partial derivatives with respect to the volumes of computational cells-polyhedrons. Using the FlowVision software package, the simulation of the supersonic air flow around products of various shapes was performed in relation to track tests. This made it possible to determine previously unknown dependences of the coefficients of aerodynamic drag, lift force and lateral force on the speed of motion, as well as the moments of aerodynamic forces for estimating the contribution of airflow processes to the total vibration field acting on the track carriage with the product.

Constitutive modeling of the magnetic-dependent nonlinear dynamic behavior of isotropic magnetorheological elastomers

Isotropic magnetorheological elastomers (MREs) are smart materials fabricated by embedding magnetizable particles randomly into a polymer matrix. Under a magnetic field, its modulus changes rapidly, reversibly, and continuously, offering wide application potential in the vibration control area. Experimental observations show that there is a strong frequency, strain amplitude, and magnetic dependence of the dynamic behavior of isotropic MRE. Although important for potential applications, the magnetic-dependent nonlinear dynamic behavior of isotropic MRE has received little theoretical attention. To accurately evaluate the dynamic performance of isotropic MRE and to guide the design of isotropic MRE-based products, a new constitutive model based on continuum mechanics theory is developed to depict the magnetic-dependent nonlinear dynamic behavior of isotropic MRE. Subsequently, the numerical implementation algorithm is developed, and the prediction ability of the model is examined. The model provides a deeper understanding of the underlying mechanics of the magnetic-dependent nonlinear viscoelastic behavior of isotropic MRE. Furthermore, the model can be utilized to predict the magnetomechanical coupling behavior of isotropic MRE and therefore serves as a useful platform to promote the design and application of isotropic MRE-based devices.

Implementation of Numerical Methods for Solving Differential Equations using Python

Numerical analysis is a branch of mathematics that deals with the development, analysis, and implementation of numerical algorithms for solving mathematical problems. In particular, it focuses on finding approximate solutions to problems that cannot be solved analytically, often using computers and other numerical methods. The numerical analysis involves the use of various mathematical techniques, including linear algebra, calculus, optimization, and statistics, to develop numerical algorithms that can solve a wide range of problems in fields such as engineering, physics, finance, and computer science. These problems may involve the calculation of derivatives and integrals, the solution of differential equations, the optimization of functions, and the approximation of functions. The goal of numerical analysis is to provide accurate and efficient solutions to mathematical problems that are difficult or impossible to solve using traditional analytical methods. This field has become increasingly important in modern science and engineering, as it enables researchers and practitioners to simulate complex systems, design new products, and solve challenging mathematical problems.

On the Control of Numerical Results in the Problems of Identification of Dynamic Energy Objects

The article proposes an approach to solving actual problem of the obtained results’ control in the construction and implementation of al-gorithms for identifying energy objects based on integral dynamic models. The considered method based on the use of quadrature algo-rithms and splines for approximation of the kernel, with the transition to solving equations with a degenerate kernel using recurrent formulas, showed sufficiently high efficiency in solving the problems of obtain-ing high performance at provided control over the accuracy of the inte-gral dynamic model parameters’ calculation, as well as providing re-sistance to the experimental data errors. The proposed method makes it possible to solve the problem of accumulating calculations, which, in turn, leads the numerical implementation algorithm to a form in which it is feasible to obtain solutions in real-time. The obtained integrated models have a sufficient level of adequacy and can be usedin integrat-ed computing control systems for energy objects

Verification of Bison fission product species conservation under TRISO reactor conditions

When assessing the reliability and predictive capabilities of a simulation tool, code verification is used to ensure that the implemented numerical algorithm is a faithful representation of its underlying mathematical model, including partial differential or integral equations, initial and boundary conditions, and auxiliary relationships. During this process, numerical results in a discrete solution are compared to the analytical solution of the mathematical model. In this paper, the code verification process is applied to one-dimensional spatiotemporal problems that exercise partial differential equation governing the conservation of fission product species (or mass diffusion). Numerical experiments were performed in the Bison fuel performance code to evaluate its predictive capability under various TRISO reactor conditions such as base irradiation and safety heating test conditions for either short- or long-lived fission product species, as well as a case concerning evaporation from the outer surface of a particle. The code predictions were compared with the expected exact results obtained from the analytical expressions, and the fact that they demonstrate the correct analytical behavior provides strong evidence of proper numerical algorithm implementation.

Численное моделирование взаимодействия газовзвеси с ударной волной континуальными математическими моделями с идеальной и диссипативными несущими средами

This paper compares computer implementations of numerical algorithms for solving the equations of mathematical models of the dynamics of gas suspensions with viscous heat-conducting, inviscid heatconducting and ideal carrier media. Mathematical models are developed within the framework of the continuum technique for modeling the dynamics of multiphase media. In the study, the process of interaction of a shock wave moving from a homogeneous gas into a gas suspension, which is often encountered in the mining industry, was modeled. The relevance of the study of this flow of inhomogeneous media is associated with the shielding of industrial explosions by aerosol curtains. When modeling for a viscous medium, homogeneous Dirichlet boundary conditions were set, for an inviscid medium, homogeneous Neumann boundary conditions. The equations of the mathematical model were integrated by the McCormack finite difference method. To overcome numerical oscillations, a nonlinear scheme for correcting grid functions was used. The program that implements the continuum method for the dynamics of multiphase media consisted of a block for specifying boundary conditions, a block that implements a numerical solution, and a block for accounting for interfacial interaction. As a result of comparing numerical calculations of mathematical models of the dynamics of a gas suspension with an ideal, inviscid heat-conducting and viscous heat-conducting carrier media, it was found that during the movement of a gas suspension, the viscosity of the carrier medium of the gas suspension has the greatest influence on the intensity of interfacial momentum exchange.

Algorithm Implementation of Equivalent Expansive Force in Strength Reduction FEM and Its Application in the Stability of Expansive Soil Slope

Loss of matric suction during rain has an important effect on the instability of expansive soil slope. A new device was designed for testing the expansive force in order to propose and determine the equivalent expansive force corresponding to the matric suction. First, the internal relation between the matric suction and the corresponding equivalent expansive force of unsaturated expansive soil was analyzed. Then, numerical algorithm implementation of the equivalent expansive force was discussed, and the equivalent expansive force was introduced into the strength reduction finite element method (FEM). Finally, the equivalent effects of the matric suction and the equivalent expansive force were compared and analyzed by evaluating the stability of an unsaturated expansive soil slope. The results show that the new testing device significantly improves the accuracy of the expansive force test and shortens the testing time. The relation between the matric suction and the equivalent expansive force with the change in initial water content is obvious. The equivalent expansive force can reflect the macro contribution of matric suction to unsaturated expansive soil, and the developed strength reduction FEM based on the equivalent expansive force can be used to evaluate the rainfall-induced instability of an expansive soil slope caused by the decrease in matric suction resulting from the rainfall infiltration.

Uniaxial tensile constitutive model of fiber reinforced concrete considering bridging effect and its numerical algorithm

This paper proposes a constitutive model of fiber reinforced concrete (FRC) composites under uniaxial tensile load considering fiber bridging effect, and the corresponding numerical algorithm is given. Due to the linear simplification on plain concrete’s constitutive model, and adoption of the explicit modified Euler integral algorithm, the solving process of this numerical model is greatly simplified. In addition, the numerical implementation algorithm of the model in ABAQUS through user-defined subroutine (UMAT) is given. The finite element calculation results and the experimental results under uniaxial tensile load are in good agreement, which proves that the model has good accuracy in predicting the peak stress of FRC under different fiber content. The simplification method proposed and the solution algorithm induced in this paper can ensure the stability and accuracy of the results, which provide a feasible development direction for FRC calculation in large-scale structures.

Slip-enhanced plastic-damage constitutive model for masonry structures

Masonry structures are widespread in earthquake-prone areas. In this study, a slip-enhanced plastic-damage constitutive model is proposed for the nonlinear analysis of masonry structures, aiming to accurately and comprehensively predict the responses of masonry structures under strong earthquakes. Based on a framework of a bi-scalar plastic-damage model, a shear damage variable is introduced to describe the slipping failure of the bed joints. The tensile and compressive elastoplastic damage energy release rates are adopted as the driving forces of the tensile and compressive damage, respectively. The slip activation stress is proposed and adopted to drive the evolution of the shear damage. Then, to explore the damage and plastic evolutions, parallel systems of masonry under different uniaxial loading conditions are introduced, comprising a series of micro-elements with stochastic fracture strains. The macro stochastic damage evolution laws and plastic evolution law of the proposed model are determined based on the parallel systems. In addition, the procedures for parameter identification of the proposed model are discussed. An explicit numerical algorithm implementation is presented and integrated in finite element software ABAQUS. Finally, three numerical examples of unreinforced masonry walls with different height–width ratios under cyclic loading demonstrate that the proposed model can obtain reliable results that are in good agreement with test results, and several numerical examples of masonry panels with orthotropy illustrate that the proposed model has good extensibility.

Применение концепции Q-детерминанта для эффективной реализации численных алгоритмов на примере метода сопряженных градиентов для решения систем линейных уравнений

The problem of improving the efficiency of parallel computing is very topical. The article demonstrates the application of the concept of Q -determinant for the effective implementation of numerical algorithms by the example of the conjugate gradient method for solving systems of linear equations. The concept of the Q -determinant is based on a unified representation of numerical algorithms in the form of the Q -determinant. Any numerical algorithm has a Q -determinant. The Q -determinant consists of Q -terms. Their number is equal to the number of output data items. Each Q -term describes all possible ways to compute one of the output data items based on the input data. The Q -determinant allows you to express and evaluate the internal parallelism of the algorithm, as well as to show the method of its parallel execution. The article gives the main notions of the Q -determinant concept necessary for better understanding of our research. Also, we describe a method of designing effective programs for numerical algorithms on the base of the concept of the Q -determinant. As a result, we obtain the program which uses the parallelism resource of the algorithm completely, and this program is called Q -effective. As application of the method for design of Q -effective programs, we describe the designing programs for conjugate gradient method for implementation on parallel computing systems with shared and distributed memory. Finally, for developed programs we present the results of experiments on a supercomputer “Tornado SUSU”.

Linear Chain Method for Numerical Modelling of Burnup Systems

The theoretical aspects of the linear chain method for the numerical modelling of nuclear transmutation systems, and particularly regarding the transmutation trajectory analysis (TTA), are presented. The theoretical background of the TTA method, as an advanced version of the linear chain method, with the detailed description of the applied mathematical set-up and graphical visualisation of transformation chains, is shown. As the TTA method was initially developed at the AGH University of Science and Technology almost 25 years ago, several numerical implementations were introduced worldwide, yet the mathematical improvements or alternative forms of solutions and numerical algorithms were reported since then. The method was also implemented and tested by different research groups, also in confrontation with alternative approaches to the nuclear transformation problem known as the matrix method. The aim of the paper is to present the background of the developed method and its advantages, clarify misunderstandings in the method perception and suggest unexplored options in numerical algorithm implementation.