Gauss Points Research Articles

A data-driven CUF-based beam model based on the tree-search algorithm

Data-driven Computational Mechanics (DDCM) has been proposed as a new computational paradigm in recent years. Most of the DDCM models are discretised in the FEM framework. In this paper, two strategies are employed in the framework of DDCM to improve computational efficiency. Firstly, an advanced structural theory, Carrera Unified Formula, is used to build the numerical model, which reduces the computational cost (amount of gauss points) and the number of iterations. Secondly, the tree-search algorithm is employed to build the hierarchical database, which speeds up the data searching efficiency. The corresponding results are compared with the FEM model, FE2 model and FEM-based data-driven model to verify the proposed scheme. Some numerical cases have been conducted to demonstrate the validation and robustness of the hierarchical database.

Multi-step wind speed forecasting model using a compound forecasting architecture and an improved QPSO-based synchronous optimization

The intermittency and randomness of wind speed time series influence the forecasting accuracy. To figure out this problem and enhance the forecasting performance, a novel compound structure is developed for short-term wind speed forecasting. The developed compound method firstly eliminates the inherent noise from the original empirical wind speed time series using wavelet packet decomposition (WPD) and subsequently constructs appropriate input matrix by phase space reconstruction (PSR) for multi-kernel least square support vector machine (MKLSSVM). To take advantage of different kernel function, MKLSSVM with optimal combination of radial basis kernel function, polynomial kernel function and linear kernel function is constructed to make wind speed forecasting. Then, the proposed improved quantum particle swarm optimization algorithm (QPSO) based on the chaos initialization, Gaussian distribution local attraction points, precocity judgment, and disturbance operator, namely ADQPSO, is employed to optimize the decomposition level of WPD, reconstruction parameters of PSR, kernel parameters and weighted coefficients in MKLSSVM synchronously. To evaluate the forecasting performance of the proposed hybrid model, four sets of historical wind speed data samples from Weihai wind farm in China are utilized to make multi-step short-term wind speed forecasting tests. The experimental results illustrate that the proposed hybrid model outperforms the compared single and new recently developed forecasting models, thus, the proposed WPD-PSR-ADQPSO-MKLSSVM is an effective method for short-term wind speed forecasting.

A Legendre–Galerkin Chebyshev collocation method for the Burgers equation with a random perturbation on boundary condition

In the paper, we apply the generalized polynomial chaos expansion and spectral methods to the Burgers equation with a random perturbation on its left boundary condition. Firstly, the stochastic Galerkin method combined with the Legendre–Galerkin Chebyshev collocation scheme is adopted, which means that the original equation is transformed to the deterministic nonlinear equations by the stochastic Galerkin method and the Legendre–Galerkin Chebyshev collocation scheme is used to deal with the resulting nonlinear equations. Secondly, the stochastic Legendre–Galerkin Chebyshev collocation scheme is developed for solving the stochastic Burgers equation; that is, the stochastic Legendre–Galerkin method is used to discrete the random variable meanwhile the nonlinear term is interpolated through the Chebyshev–Gauss points. Then a set of deterministic linear equations can be obtained, which is in contrast to the other existing methods for the stochastic Burgers equation. The mean square convergence of the former method is analyzed. Numerical experiments are performed to show the effectiveness of our two methods. Both methods provide alternative approaches to deal with the stochastic differential equations with nonlinear terms.

Bézier extraction based isogeometric topology optimization with a locally-adaptive smoothed density model

This paper presents a Bézier extraction based isogeometric topology optimization framework, in which the pseudo-densities and the weights at control points (CPs) are simultaneously treated as the design variables. A locally-adaptive smoothed density field, that is dynamically updated at each design iteration, is first proposed by utilizing the values of the non-uniform rational B-splines (NURBS) basis functions. The locally-adaptive density model can naturally handle the non-uniform CPs of the curvilinear edge Bézier elements, which enhances the smoothness of structural shape. In the Bézier extraction based topology optimization, the density surface of the optimized structure can be exactly described by NURBS. Then, the optimal shape can be achieved by slicing the density surface with a predefined level-set. The sensitivity analyses of the structural compliance and the compliant mechanism are derived with respect to the design variables and evaluated from the Gaussian quadrature points instead of each element centroid. Some advantages concerning the proposed isogeometric topology optimization are illustrated with several numerical examples that are widely used in recent literature of topology optimization. Meanwhile, the influences of the design parameters on the optimal solution are also discussed to assess the effectiveness of the method.

Evaluation the interface damage influences on mechanical properties of fiber-reinforced composites based on subdomain boundary element theory

In this paper, a novel microscopic method is presented to investigate the effective mechanical properties and local stress distribution of the composites with respect to some specific interfacial cracks. To this end, a subdomain boundary element method (SBEM) is combined with the asymptotic homogenization theory to calculate their effective mechanical properties. In order to accurately capture the stress distribution on the boundary and inside points (Gaussian integral points) of representative volume element (RVE), the parametric sub-cell is firstly constructed to discretize the RVE. On this basis, a node-pair decoupling method is introduced with consideration of unilateral and bilateral damage. The influence on mechanical properties of composites is revealed, which provides a solid foundation for the design and manufacture of the composite interface.

An Efficient Orthogonal Polynomial Method for Auxetic Structure Analysis with Epistemic Uncertainties

Traditional approaches used for analyzing the mechanical properties of auxetic structures are commonly based on deterministic techniques, where the effects of uncertainties are neglected. However, uncertainty is widely presented in auxetic structures, which may affect their mechanical properties greatly. The evidence theory has a strong ability to deal with uncertainties; thus, it is introduced for the modelling of epistemic uncertainties in auxetic structures. For the response analysis of a typical double-V negative Poisson’s ratio (NPR) structure with epistemic uncertainty, a new sequence-sampling-based arbitrary orthogonal polynomial (SS-AOP) expansion is proposed by introducing arbitrary orthogonal polynomial theory and the sequential sampling strategy. In SS-AOP, a sampling technique is developed to calculate the coefficient of AOP expansion. In particular, the candidate points for sampling are generated using the Gauss points associated with the optimal Gauss weight function for each evidence variable, and the sequential-sampling technique is introduced to select the sampling points from candidate points. By using the SS-AOP, the number of sampling points needed for establishing AOP expansion can be effectively reduced; thus, the efficiency of the AOP expansion method can be improved without sacrificing accuracy. The proposed SS-AOP is thoroughly investigated through comparison to the Gaussian quadrature-based AOP method, the Latin-hypercube-sampling-based AOP (LHS-AOP) method and the optimal Latin-hypercube-sampling-based AOP (OLHS-AOP) method.

FE2 methodology for discrete cohesive crack propagation in heterogenous materials

To study the discrete crack growth of heterogeneous materials, a dual-scale discrete fracture calculation method is proposed in this paper. An equivalent cohesive energy criterion is proposed for the FE2 model based on the characteristics of the fracture zone. The representative volume element (RVE) with a cohesive zone is embedded into the Gauss points of the macro-element to replace the full FE model of the discrete crack. Therefore, the dual-scale method of discrete fractures not only improves the dual-scale modeling of discrete cracks but also greatly reduces the computational cost. We further demonstrate that the built-in capabilities of commercial software are naturally compatible with this method. Finally, the fracture process of a two-phase heterogeneous material containing a zirconium alloy matrix and hydride was studied using the dual-scale method. The macro-fracture direction was related to the precipitation direction of the brittle hydrides. This conclusion is consistent with existing experimental observations.

An improved morgenstern-price method using gaussian quadrature

The Morgenstern-Price (MP) method is an effective solution to compute the factor of safety (FOS) of a slope against sliding that has been used extensively for decades. In the MP method, the soil mass above the slip surface needs to be tediously divided into vertical slices to compute the force and moment integrals. The proposed research improves the computational efficiency of the MP method by employing the Gaussian quadrature for the integrals, where the soil mass division into a significant number of vertical slices is improved by aligning a few Gauss points along the slip surface. Consequently, the computational efficiency can be dramatically improved, resulting in a reduction in calculation efforts. Moreover, this research further refines the iteration scheme to satisfy the equilibrium conditions for the FOS using a robustly stable numerical algorithm with two initial scaling factors. The critical slip surface was determined by the simulated annealing algorithm. Finally, the proposed method was thoroughly validated via a series of benchmark examples. This research could improve the traditional MP method by enhancing its robustness and efficiency and provide an informative technique for force integrals that may be motivated by other problems in geotechnical engineering.

Fast costate estimation for indirect trajectory optimization using Bezier-curve-based shaping approach

In this study, a fast costate estimation method derived from the Bezier-curve-based shaping approach is proposed for indirect low-thrust transfer trajectory optimization. The trajectory is designed rapidly by means of the shape-based method, which can be discretized on the Gauss points and substituted into the discrete first-order optimal condition equation for costate estimation. The estimated costate at the starting point is substituted into the indirect method to solve the optimal control problem. The costate estimation performance for two interplanetary rendezvous missions from Earth to Mars and Earth to Dionysus is compared via random initialization and Newton's iteration method. The initial guesses for the transfer time are given by the Hohmann transfer and the Lambert two-pulse traversal search. The results show that the Lambert traversal search can provide a more accurate launch time estimation in the Bezier-curve-based shaping approach, which can estimate the costate for indirect trajectory optimization in an order of magnitude less time than that required by random initialization.

Rannacher time-marching with orthogonal spline collocation method for retrieving the discontinuous behavior of hedging parameters

In this paper, we extensively study the orthogonal spline collocation method known as the spline collocation at Gauss points with Rannacher’s time-marching scheme for free boundary value option pricing problems. Such financial problems commonly feature non-smooth payoff functions that cause inaccuracies in approximating the solution and its derivatives. As a result, unlike the problems with the smooth initial data, the quadratic convergence is not realized by the Crank-Nicolson time-stepping scheme for these problems. Furthermore, the non-smoothness in the initial condition leads to severe degradation in the convergence rates and spurious oscillations near the discontinuity. The rationale is that classical schemes strongly rely on the smoothness of the initial data. A rigorous time-marching scheme referred to as Rannacher time-stepping scheme is introduced for the American option’s price diagnosed by a linear complementarity problem to smoothen the data. Moreover, with careful analysis, second and fourth orders of convergence are established for the present scheme in temporal and spatial directions, respectively. The numerical results for three test problems are presented in tables and graphs to validate the theory. These results show that the present scheme achieves higher accuracy and sufficiently restores the expected behavior.

An improved continuous damage model to estimate multiaxial fatigue life under strain control problems

In this contribution, a modification and application of the Lemaitre damage evolution law is proposed for the problem of fatigue life estimate under low number of cycles. For this, the denominator of damage, considered constant for Lemaitre, was replaced by a function dependent on the stress triaxiality. The new function has a nonlinear and thermodynamically consistent behavior. In addition, the evolution law of the backstress tensor is adopted as proposed by Desmorat, where there is no saturation of the limit value. The formulation is implemented in an academic tool for analysis at one Gauss point, through an implicit integration strategy of the flow equations. The Newton-Raphson method is used to solve a system of nonlinear equations, with two tensor and two scalar equations. Thus, in order to demonstrate the robustness of the model, experimental data from the literature for SAE 1045 and S460N steels are adopted. Proportional and nonproportional axial, torsional and multiaxial loading paths are simulated. Finally, the results have shown that the modification formulation is a good strategy to control the damage evolution in predicting fatigue life. In this sense, the calculated lives by the new proposal are within a scatter band of factor three, both under normal or torsional uniaxial loading conditions, as well as proportional or nonproportional multiaxial conditions.

Modeling fracture propagation in a rock‐water‐air system with the assumed enhanced strain method

AbstractThis article presents a fully coupled finite element method for the single cohesive fracture propagation in a deformable porous medium (rock) interacting with two immiscible pore fluids, that is, water and air. The cohesive fracture is represented with the assumed enhanced strain method which can be cast into a classical plasticity framework. The governing equations for the solid skeleton deformation, pore water flow, and pore air flow are derived based on the mixture theory. The coupled nonlinear global system is then solved by the standard Galerkin finite element method. The numerical implementation is verified against the numerical and analytical solutions previously reported in the literature. The proposed model is capable of capturing fluid transfer between the fracture and the host rock due to the pressure difference. The numerical examples demonstrate that the behavior of fracture propagation can be significantly influenced by the capillary pressure inside fracture and vice versa. Since the fracture is represented at the level of Gauss points, the proposed numerical model can be easily implemented on any standard finite element code base.

Superconvergent Nyström and Degenerate Kernel Methods for Integro-Differential Equations

The aim of this paper is to carry out an improved analysis of the convergence of the Nyström and degenerate kernel methods and their superconvergent versions for the numerical solution of a class of linear Fredholm integro-differential equations of the second kind. By using an interpolatory projection at Gauss points onto the space of (discontinuous) piecewise polynomial functions of degree ⩽r−1, we obtain convergence order 2r for degenerate kernel and Nyström methods, while, for the superconvergent and the iterated versions of theses methods, the obtained convergence orders are 3r+1 and 4r, respectively. Moreover, we show that the optimal convergence order 4r is restored at the partition knots for the approximate solutions. The obtained theoretical results are illustrated by some numerical examples.

An improved quadrature scheme in B-spline material point method for large-deformation problem analysis

The B-spline material point method (BSMPM) has proved to be a promising numerical method for modeling problems with large deformations. While the cell-crossing noise is alleviated by using high-order continuous B-spline basis functions, the BSMPM still suffers from reduced accuracy arising from quadrature errors when simulating large-deformation problems. In this work, a quadrature scheme for the information mapping and the internal force calculations in the BSMPM is developed to substitute for the widely adopted numerical integration at the material particles or the Gauss points within the particle domain. Representative numerical examples of elastic and elasto-plastic large-deformation problems demonstrate the highly enhanced accuracy and convergence of the BSMPM simulations for solid mechanics problems involving large deformations by the proposed quadrature scheme. Moreover, it is shown that, compared to the algorithm of applying numerical Gauss quadrature in the particle domain, the proposed integration scheme is a more favorable option for improving the capability and efficiency of the BSMPM in the analysis of large-deformation problems.

Computational material stability analysis in finite thermo-electro-mechanics

In the paper at hand, computational analysis of material stability in electro-active polymers (EAP) is considered with taking into account coupling between the thermal, electrical and mechanical fields. To this end, a mixed Hu–Washizu-type thermo-electro-mechanical finite element formulation is proposed. Both, fully monolithic and partitioned finite element formulations are briefly described and employed to simulate the coupled response of homogeneous and inhomogeneous EAP, under varying temperatures. The partitioned finite element approach, which is based on a split of the problem into electro-mechanical and thermal sub-problems, is utilized to carry out material stability analysis on the Gauss point level. The structural response beyond the material-instability point is traced.

A shape function based interpolation approach for nodal design variable based structural optimization

AbstractIn this article, a novel density interpolation scheme for topological optimization based on nodal density variables is presented. The method relies on the definition of an independent interpolation mesh for the description of the density field. This mesh is used for the interpolation of the density field at the Gauss points of the design domain mesh. Provided that the interpolation mesh is realized by linear elements, the proposed scheme is range restricted and assures physically meaningful values of the interpolated density. Explicit analytical sensitivity expressions can be derived. The utilization of an independent interpolation mesh allows for the utilization of any element type and geometry on the body for the computation of the displacements. It is also shown that the chosen interpolation scheme has interesting properties of volume and material volume conservation with respect to the mesh utilized on the body and the domain of the interpolation mesh can differ from the domain of the mesh for the displacements computation. This allows for the realization of very regular interpolation meshes even in case of very complex shapes of the domain of the body to be optimized. Examples are provided showing the performance of the proposed interpolation scheme.

A k‐means clustering machine learning‐based multiscale method for anelastic heterogeneous structures with internal variables

AbstractA new machine‐learning based multiscale method, called k‐means FE, is introduced to solve general nonlinear multiscale problems with internal variables and loading history‐dependent behaviors, without use of surrogate models. The macro scale problem is reduced by constructing clusters of Gauss points in a structure which are estimated to be in the same mechanical state. A k‐means clustering—machine learning technique is employed to select the Gauss points based on their strain state and sets of internal variables. Then, for all Gauss points in a cluster, only one micro nonlinear problem is solved, and its response is transferred to all integration points of the cluster in terms of mechanical properties. The solution converges with respect to the number of clusters, which is weakly depends on the number of macro mesh elements. Accelerations of FE calculations up to a factor 50 are observed in typical applications. Arbitrary nonlinear behaviors including internal variables can be considered at the micro level. The method is applied to heterogeneous structures with local quasi‐brittle and elastoplastic behaviors and, in particular, to a nuclear waste package structure subject to internal expansions.

An FE-DMN method for the multiscale analysis of thermomechanical composites

We extend the FE-DMN method to fully coupled thermomechanical two-scale simulations of composite materials. In particular, every Gauss point of the macroscopic finite element model is equipped with a deep material network (DMN). Such a DMN serves as a high-fidelity surrogate model for full-field solutions on the microscopic scale of inelastic, non-isothermal constituents. Building on the homogenization framework of Chatzigeorgiou et al. (Int J Plast 81:18–39, 2016), we extend the framework of DMNs to thermomechanical composites by incorporating the two-way thermomechanical coupling, i.e., the coupling from the macroscopic onto the microscopic scale and vice versa, into the framework. We provide details on the efficient implementation of our approach as a user-material subroutine (UMAT). We validate our approach on the microscopic scale and show that DMNs predict the effective stress, the effective dissipation and the change of the macroscopic absolute temperature with high accuracy. After validation, we demonstrate the capabilities of our approach on a concurrent thermomechanical two-scale simulation on the macroscopic component scale.

Free vibrations analysis of composite and hybrid axisymmetric shells

The free vibration of laminated composite (C) and hybrid axisymmetric shell structures, consisting of a composite laminated material sandwiched between two functionally graded material laminas (F1/C/F2), is analysed in the present work. The numerical solutions are obtained by expanding the variables in Fourier series in the circumferential direction and using conical frustum finite elements in the meridional direction. The implemented finite element is a simple conical frustum with two nodal circles, with ten degrees of freedom per nodal circle. This model requires only a reduced number of finite elements to model the geometry of axisymmetric structures, the integration procedures use one Gauss point, and the through the thickness properties variation in FGM laminas is modelled by a small number of virtual layers, resulting a very high computational efficiency. The in-house developed code presents very good solutions when compared with results obtained by alternative available models.

FEM×DEM multi‐scale model for cemented granular materials: Inter‐ and intra‐granular cracking induced strain localisation

AbstractIn this paper, we present a multi‐scale model that combines the finite element method (FEM) and the discrete element method (DEM) to study the behaviour of cemented granular materials (CGM) at both the sample (macroscopic) and particle (microscopic) scales, taking into account inter‐ and intra‐granular cracking. At the microscopic scale, the material is made up of Volume Elements (VE) composed of particles. Their mechanical behaviour is modelled by the DEM. In the VEs, we find circular grains (single particles) and meso‐grains; the meso‐grains being made up of clusters of particles linked by strong cohesive bonds. All particles interact via normal/tangential contact and rolling resistance laws with cohesive bonds. At the macroscopic scale, the sample is modelled using the FEM. A VE is assigned to each Gauss point of the mesh; the mechanical response of the VE is used to numerically derive the constitutive response of the material to the strain increment exerted at each point. In the models reported in this paper, localised failures in shear band mode are observed at the macroscopic scale. By performing a microscopic analysis, the results show that the occurrence and development of shear bands give rise locally to a strong evolution of microscopic characteristics such as void ratio, number of contacts (total and debonding contacts), and remarkably by inter‐ and intra‐granular cracking in the case of meso‐grains. Furthermore, the stress and strain tensors non‐coaxiality is clearly demonstrated inside the shear band, but almost negligible outside.