Gauss Points Research Articles

One-dimensional parallel forward modeling for geophysical electromagnetic fields excited by sagging overhead transmission lines

Geophysicists often regard the 50/60 Hz electromagnetic (EM) waves and their harmonics, excited by overhead transmission lines, as noise that interferes with geophysical prospecting. However, some researchers view them as effective signals for geophysical exploration. Forward modeling can be used to study the characteristics of EM fields generated by overhead transmission lines. It can help in either eliminating or utilizing power-frequency EM fields. Nevertheless, current research on the forward modeling of EM fields excited by transmission lines neglects the sagging shape and direction of the actual transmission lines, simplifying them into infinitely long straight lines. In this research, we develop a one-dimensional forward modeling algorithm specifically designed for overhead transmission lines, which can accurately calculate the EM fields excited by realistic overhead transmission lines considering the three-phase system, design parameters, sag, and orientation of the transmission lines. To construct an accurate transmission line model, we represent the sagging shape of a single transmission line within a span by a catenary equation. We then derive analytical expressions for the EM fields excited by sagging overhead transmission lines. During forward modeling, an appropriate number of Gauss points must be selected to balance accuracy and speed. The optimal number of Gauss points is determined based on the distance between the observation point and the transmission line source, as well as the type of fields being calculated. OpenMP parallelization is used to improve computational speed. The forward modeling results reveal significant differences between the forward responses of a single infinitely long straight line and realistic transmission lines. This highlights the importance of fully incorporating the three-phase system, design parameters, and geometric shapes of transmission lines in the forward modeling process. Moreover, the inversion test demonstrates the potential utility of EM fields radiated by transmission lines for geophysical exploration.

A multi-domain hybrid spectral collocation method for nonlinear Volterra integral equations with weakly singular kernel

In this paper, we propose a multi-domain hybrid spectral collocation method for the nonlinear Volterra integral equations (VIEs), whose solutions exhibit weak singularities at the endpoint t=0. In order to efficiently approximate the weakly singular solutions, we divide the interval [0,T] into N subintervals and use the Gauss points of the generalized log orthogonal functions as the collocation points to approximate the weakly singular integral term in the first subinterval. In the remaining subintervals, we use Legendre and Jacobi Gauss points as the collocation points to approximate the corresponding integral terms. We also provide a rigorous hp-version convergence analysis for the hybrid spectral collocation method under L2-norm. A series of examples demonstrate our method is particularly suitable for solutions that have weak singularities at one endpoint.

Slope Stability Analysis Based on the Explicit Smoothed Particle Finite Element Method

A landslide is a common natural disaster that causes environmental damage, casualties and economic losses, which seriously affects the sustainable development of society. In geomechanics, it is one of the largest deformation problems. Herein, the GPU-accelerated explicit smoothed particle finite element method (eSPFEM) for large deformation analysis in geomechanics was developed on the CUDA platform based on high-performance computing using a self-designed eSPFEM program code. The eSPFEM combines the strain smoothing nodal integration techniques found in the particle finite element method (PFEM) framework, which allows for the use of low-order triangular elements without volume locking and avoids frequent information transfer and mapping errors between Gaussian points and particles in PFEM. A numerical simulation of slope instability using the eSPFEM and based on a strength reduction technique was conducted using various examples, including a cohesive homogeneous slope, a non-cohesive homogeneous slope, a non-homogeneous slope and a slope with a thin soft band. The calculation results show that the eSPFEM can be applied to slope stability analysis under different working conditions, simulating the entire process of slope instability initiation, sliding and reaccumulation, and obtaining reliable FOS values. A numerical simulation was conducted to analyse a landslide that occurred in the Zhangjiazhuang tunnel on the Lanzhou–Xinjiang high-speed railway line on 18 January 2016. A natural unsaturated soil slope, a soil slope with a high moisture content and a soil slope with a high moisture content subjected to an earthquake were analysed. The findings of this study are in good agreement with the actual slope failure conditions. The primary triggers identified for the landslide were heavy rainfall and earthquakes. The verification results indicate that the eSPFEM can effectively simulate an actual landslide case, showcasing high accuracy and applicability in simulating the large deformation behaviour of landslides.

Improved dynamic analysis of shear deformable shells using the hybrid displacement boundary element method

In this paper, the dynamic analysis of shear deformable shells was conducted using the hybrid displacement boundary element method (HD-BEM). The approach involved utilizing the fundamental solution as a trial function to derive the 8-node element stiffness matrix, while approximating the boundary variables with quadratic shape functions. To ensure efficient computations and avoid singular integrals, the source points were strategically positioned outside the domain. For accurate calculations of the proposed 8-node element mass matrix, only four Gauss points were relied upon. It is important to note that both the element mass and stiffness matrices demonstrate symmetry. Throughout the study, free, forced, and harmonic analyses were considered. Encouragingly, the results obtained from the study exhibited exceptional accuracy levels, even when employing a coarse discretization approach.

Facets of High-Dimensional Gaussian Polytopes

We study the number of facets of the convex hull of n independent standard Gaussian points in Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^d$$\\end{document}. In particular, we are interested in the expected number of facets when the dimension is allowed to grow with the sample size. We establish an explicit asymptotic formula that is valid whenever d/n→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d/n\\rightarrow 0$$\\end{document}. We also obtain the asymptotic value when d is close to n.

A measurement system for the assessment of the effectiveness of sound-insulating multilayers against structure-borne excitation: design, prototyping and validation

This paper proposes a measurement system to assess the effectiveness of sound-insulating multilayers against structure-borne excitation, meant to complement the well-known and established systems used in the automotive field for airborne excitation. The system consists of a shaker, a frame connected to it and a plate clamped along its edges to the frame. When the frame is put into vibration by the shaker, the plate radiates noise into a small anechoic cabin where microphones are positioned. This layout intends to mimic the noise radiation from an automotive body panel. The development of such a system, in spite of its simplicity, presents non-trivial aspects, such as the design of a frame sufficiently rigid not to have modes in the frequency range of interest and the definition of a metric to evaluate the effectiveness of an acoustic multilayer in a robust way based only on SPL measurements. The paper describes how these challenges were overcome, using topological optimization to design a frame made with magnesium suitable for tests up to about 1kHz and FE simulations to define a robust assessment metric by placing the receiving microphones at a fixed distance from the plate and at positions corresponding to integration Gauss points. Besides the design, the paper describes also how the system was prototyped and eventually validated.

Solver‐free classical computational homogenization for nonlinear periodic heterogeneous media

AbstractModeling the behavior of composite materials is an important application of computational homogenization methods. Classical computational homogenization (CCH), based on asymptotic analysis, is such a method. In CCH, equilibrium equations are separated into two length scales and solved numerically. Solving the fine‐scale equilibrium equations at every coarse‐scale Gauss point, in every iteration of a Newton–Raphson loop, is often too computationally expensive for real engineering applications. In this study, we propose a modified CCH approach that avoids solving the fine‐scale equilibrium equations. The proposed method, which we call solver‐free CCH, works by pre‐computing a set of eigenstrain influence function tensors based on data from a small number of numerical experiments. Then, during the online stage of the computation, these eigenstrain influence tensors are used in the fine‐scale problem to evaluate and homogenize strains and stresses. This article begins by formulating the solver‐free CCH approach for small‐deformation problems involving composites with nonlinear constituent phase material models, including computation of the eigenstrain influence tensors and their application within the online stage. To verify the proposed approach, we consider loading cases outside the training set used to derive the eigenstrain influence tensors. The combinational efficiency and accuracy of the solver‐free CCH in comparison to the conventional CCH is studied on a multilayer composite plate in three point bending (3pt‐bend) and open hole tension.

Consistent C-bar radial-basis Galerkin method for finite strain analysis

For finite-strain problems of quasi-incompressible materials, with hyperelastic and hyperelasto-plastic behavior, we develop a mixed polynomial-enriched radial-basis function (RBF) interpolation. It is characterized by decoupling quadrature and polynomials for the deviatoric and volumetric terms of the right Cauchy–Green tensor. With an enriched interpolation, first and second derivatives of the deformation gradient are continuous. A finite-element mesh is employed for quadrature purposes with the corresponding distribution of Gauss points. In terms of discretization, linear, quadratic and cubic polynomials are combined and function support is established from the number of pre-assigned nodes. Due to the adoption of a Lagrangian kernel, finite strain elasto-plastic constitutive developments are based on the Mandel stress. Combined with the C¯ technique, these developments are found to be convenient from an implementation perspective, as RBF formulations for finite strain plasticity have been limited in terms of generality and amplitude of deformations in the quasi-incompressibility case. Three benchmark tests are successfully solved, and it was found that a combination of reduced integration with a deviatoric cubic basis and volumetric quadratic basis provides superior results in the quasi-incompressible case. Numerical testing shows very good convergence behavior of the results. Besides absence of locking with selective interpolation, outstanding Newton convergence was observed regardless of strain levels.

An unsupervised machine learning approach to reduce nonlinear FE[formula omitted] multiscale calculations using macro clustering

Solving nonlinear multiscale methods with history-dependent behaviors and fine macroscopic meshes is a well-known challenge. In this work, an unsupervised machine learning-based clustering approach is developed to reduce nonlinear Multilevel Finite Element-FE2 calculations. In contrast with most available techniques which aim at developing Reduced Order Models (ROM) or AI-based surrogate models for the microscale nonlinear problems, the present technique reduces the problem from the macro scale by creating clusters of macro Gauss points which are assumed to be in close mechanical states. Then, a single micro nonlinear Representative Volume Element (RVE) calculation is performed for each cluster. A linear approximation of the macro stress is used in each cluster. Handling internal variables is carried out by using anelastic macro strains in the clustering vectors in addition to the macro strains components. Finally, some convergence issues related to the use of clusters at the macro scale are addressed through a cluster freezing algorithm. The technique is applied to nonlinear hyperelastic, viscoelastic and elastoplastic composites. In contrast to available ROM or machine-learning -based acceleration techniques, the present method does not require neither preliminary off-line calculations, nor training, nor data base, nor reduced basis at the macro scale, while maintaining typical speed-up factors about 20 as compared to classical FE2.

CECM: A continuous empirical cubature method with application to the dimensional hyperreduction of parameterized finite element models

We propose a method for finding optimal quadrature/cubature rules with positive weights for parameterized functions in 1D, 2D or 3D spatial domains. The method takes as starting point the values of the functions at the Gauss points of a finite element (FE) mesh of the spatial domain for a representative sample of input parameters, and then construct an elementwise continuous orthogonal basis for such functions using the truncated Singular Value Decomposition (SVD) along with element polynomial fitting. To avoid possible memory bottlenecks in computing the SVD, we propose a Sequential Randomized SVD (SRSVD) in which the matrix is provided in a column-partitioned format, and which uses randomization to accelerate the processing of each individual block. After computing the basis functions, the method determines an exact integration rule for such functions, featuring as many points as functions, and in which the points are selected among the Gauss points of the FE mesh. Finally, the desired optimal rule is obtained by an sparsification process in which the algorithm zeroes one weight at a time while readjusting the positions and weights of the remaining points so that the constraints of the problem are satisfied. We apply this methodology to multivariate polynomials in cartesian domains to demonstrate that the method is indeed able to produce optimal rules – i.e., Gauss product rules –, and to a 2-parameters, 3D sinusoidal–exponential function to illustrate the use of the SRSVD in scenarios in which the standard SVD cannot handle the operation because of memory limitations. Lastly, the fact that the method does not require the analytical expression of the integrand functions – just their values at the FE Gauss points – makes it suitable for dealing with the so-called hyperreduction of parameterized finite element models. We exemplify this by showing its performance in the derivation of low-dimensional surrogate models in the context of the multiscale FE method. The Matlab source codes of both the CECM and the SRSVD, along with the scripts for launching the numerical tests, are openly accessible in the public repository https://github.com/Rbravo555/CECM-continuous-empirical-cubature-method.

Free vibration analysis of three-dimensional solids with arbitrary geometries using discrete Ritz method

A numerical method, discrete Ritz method (DRM), is proposed for the free vibration analysis of three-dimensional (3D) solids with arbitrary geometries. The problem is formulated within a cuboidal domain, and arbitrarily shaped 3D solid can be simulated by assigning cutouts within the cuboidal domain. Geometries of 3D solids are characterized by using level set functions. DRM transforms the problem into a variable stiffness system in which Chebyshev polynomials are used to approximate vibrational behavior and Gauss points are used to discretize the cuboidal domain. The vibration behaviors of arbitrarily shaped solids can be numerically simulated by setting the stiffness and thickness of Gauss points within the cutouts to zero. New formulations in DRM have resulted in completely standard energy functionals and computation procedures for arbitrarily shaped 3D solids. Free vibration behaviors of 3D solids, including polygonal cross-section cylinders, circular cylinders, rectangular plates with cutouts, perforated super-elliptical plates, ellipsoids, and pyramids, are investigated, and results are compared with those reported in the literature. DRM is shown to be capable of solving free vibration problems of complex-shaped 3D solids with high precision and stability.

Framework for a chemo‐mechanical model of the breathing effect in lithium–silicon batteries

AbstractWe present a chemo‐mechanical finite element method model for the breathing effect of the silicon anode during lithiation and delithiation. The model is based on the framework for gradient extended standard dissipative solids. It couples diffusion of lithium ions, phase transformation with a moving phase boundary, and the related volume changes, which lead to mechanical stresses in the material. The driving force for diffusion is given by the gradient of the electrochemical potential. Phase transformation and phase boundary movement are described by the Allen–Cahn equation in combination with a constrained double obstacle potential. To shift the constraint from the nodes to the Gauss points, the model makes use of the micromorphic approach.

A smoothed assumed enhanced strain method for frictional contact with constant strain elements

This paper presents a finite element framework for imposing frictional contact conditions on embedded fracture faces, implemented by the constant-strain assumed enhanced strain (AES) method, where penalty method is used to impose both non-penetration constraint and Coulomb's law of friction. The proposed constant-strain AES method for modeling embedded frictional contact can be cast into an integration algorithm similar to those used in the classical plasticity theory, where displacement jump is calculated from the local traction equilibrium at Gauss point, so the method does not introduce any additional global degrees of freedom. Moreover, constant-strain elements are often desirable in practice because they can be easily created automatically for large-scale engineering applications with complicated geometries. As encountered in other enriched finite element methods for frictional contact, the problem of normal contact pressure oscillations is also observed in the constant-strain AES method. Therefore, we developed a strain-smoothing procedure to effectively mitigate the oscillations. We investigated and verified the proposed AES framework through several numerical examples, and illustrated the capability of this method in solving challenging nonlinear frictional contact problems.

Local repulsion of planar Gaussian critical points

We study the local repulsion between critical points of a stationary isotropic smooth planar Gaussian field. We show that the critical points can experience a soft repulsion which is maximal in the case of the random planar wave model, or a soft attraction of arbitrary high order. If the type of critical points is specified (extremum, saddle point), the points experience a hard local repulsion, that we quantify with the precise magnitude of the second factorial moment of the number of points in a small ball.

A meshfree large-deformation analysis method for geotechnical engineering based on the RBF field variable mapping technology

In this paper, a practical meshfree large deformation method (MFLDM) is proposed for numerical analysis in geotechnical engineerings, including: soil foundation, slop, dam, etc. The MFLDM leverages both the flexible nodal distribution in the meshfree method and the high stability in the arbitrary Lagrangian–Eulerian (ALE) framework. In each calculation step, two sets of Gauss points, fixed and moving Gauss points, are generated in the background mesh. In addition, the radial basis function (RBF) is used to map field variables, including stress, stain, and constitutive variables, between the fixed and moving Gauss points to achieve the field variable redistribution during the large-deformation analysis. The proposed MFLDM, which is written in C++ using the object-oriented programming approach, can be completely integrated into the self-development calculating system named GEODYNA and coupled with the finite element method (FEM) at the matrix level, which significantly broadens its practical application. The proposed model is verified by several numerical examples and compared with different constitutive models, including the linear elasticity model, ideal elastic–plastic model, and generalized elastic–plastic model. The comparison results verify the high accuracy, fast convergence, and good robustness of the proposed MFLDM. Finally, the proposed MFLDM is applied to a local large deformation analysis between the cut-off wall and the core wall on a deep overburden.

Band structures analysis of fluid–solid phononic crystals using wavelet‐based boundary element method and frequency‐independent fundamental solutions

AbstractA novel method of combination of wavelet‐based boundary element method (WBEM) with frequency‐independent fundamental solutions is proposed to determine the band structures of fluid–solid phononic crystals (PCs) with square and triangular lattices. Integral equations established are based on the frequency‐independent fundamental solutions, which can avoid nonlinear eigenvalue problems and reduce computing time. Domain integral terms arising from the use of frequency‐independent fundamental solutions are handled with the radial integration method (RIM) and dual reciprocity method (DRM), respectively. The results show the lower precision in high frequency domain of using RIM to handle domain integral terms than that of using DRM, which can be solved by increasing Gauss point. The B‐spline wavelet on the interval and wavelet coefficients are applied to approximate the physical boundary conditions. It is proved that coupling conditions between matrix and scatterers and Bloch theorem are also applicable to wavelet coefficients. Some small matrix entries generated by wavelet vanishing moment characteristics are truncated by the provided matrix compression technique, and the influence of compressed matrices on the results is studied. Furthermore, the final Eigen equations constructed are modified to avoid numerical instability. Some examples are provided to demonstrate the accuracy and efficiency of the proposed method.

Adaptive sinh transformation Gaussian quadrature for 2D potential problems using deep learning

In the boundary element method (BEM), the sinh transformation method is an effective method for evaluating nearly singular integrals, but a relationship between the integration accuracy and the number of Gaussian points is needed to achieve adaptive computation. Based on deep learning, we propose a novel integration scheme, adaptive sinh transformation Gaussian quadrature (ASTGQ), which can determine the number of Gaussian points according to the required accuracy. First, a large number of integration data samples of the sinh transformation method are generated in different cases, and the neural network is trained to establish the relationship between the number of Gaussian points and the integration accuracy. Then, based on the improved loss function and evaluation index, a better network model is obtained to ensure that the actual integration accuracy is slightly higher than the requirement of using the minimum Gaussian points. In this way, when the trained neural network is used in the sinh transformation method, the higher accuracy requirement can be met at a lower cost. Numerical examples demonstrate that, compared to the adaptive Gaussian quadrature (AGQ) method, the proposed scheme can significantly improve the computational efficiency when evaluating the nearly singular integrals for very thin coatings and other structures.

3D target tracking and shape reconstruction in clutter using Gaussian process and point completion network

AbstractExisting problems for 3D extended target tracking and contour reconstruction under clutter environment are thoroughly investigated in this article. Due to the sparsity of available point cloud data and interference from clutter, the shape completion method, such as Point Completion Network (PCN), cannot reconstruct 3D contour directly. Moreover, the traditional Gaussian process (GP) model suffered from computing overhead and cannot handle irregular non‐convex shapes. Here, a two‐stage tracking algorithm is proposed, which firstly combines the 3D GP measurement model with a probability data association filter to jointly estimate the kinematics and 3D Gaussian basis points, which contain partial shape information. Afterwards, this method input produced Gaussian basis points to a deep learning‐based PCN to reconstruct complete geometry of the contour. The effectiveness of the proposed algorithm is verified by extended target tracking with both simple and complex geometric shapes. The simulation results show that the proposed algorithm obtains accurate kinematic state estimation and produces complete point cloud estimation for 3D contour.

Entropy Stable DGSEM Schemes of Gauss Points Based on Subcell Limiting.

The discontinuous Galerkin spectral element method (DGSEM) is a compact and high-order method applicable to complex meshes. However, the aliasing errors in simulating under-resolved vortex flows and non-physical oscillations in simulating shock waves may lead to instability of the DGSEM. In this paper, an entropy-stable DGSEM (ESDGSEM) based on subcell limiting is proposed to improve the non-linear stability of the method. First, we discuss the stability and resolution of the entropy-stable DGSEM based on different solution points. Second, a provably entropy-stable DGSEM based on subcell limiting is established on Legendre-Gauss (LG) solution points. Numerical experiments demonstrate that the ESDGSEM-LG scheme is superior in non-linear stability and resolution, and ESDGSEM-LG with subcell limiting is robust in shock-capturing.

Two Simple Numerical Implementation Methods for Damage-Coupled Viscoplastic Constitutive Model

Abstract This paper is concerned with the two simple numerical implementation methods for a damage-coupled Chaboche-type viscoplastic constitutive model. By considering the damage variable as a constant in each incremental step, the return-mapping procedure is reduced to the solution of only one nonlinear scalar equation. Depending on the use of damage value in the current or prior incremental state, the two methods are named the backward difference implicit integration scheme and the two-step explicit integration scheme respectively. These two numerical algorithms are implemented into the ansys software by developing the usermat subroutine and verified by comparing them with available experimental data. Several numerical examples on the Gauss point level are studied in terms of stability, accuracy, computational efficiency, and applicability for further numerical observation. In addition to higher computational efficiency and lower memory requirements, the two methods can be easily extended to other damage models due to their simplicity.