Gaussian Quadrature Points Research Articles

Assumed natural strain and stabilized quadrilateral Lobatto spectral elements for C0 plate/shell analysis

SummaryLobatto elements are high order or spectral elements featured by non‐equispaced Lobatto nodes and the Lobatto nodal quadrature. The tensor product nature of the element interpolation allows the derivatives at the node be computed through the derivatives along the nodal line which involve only 1D interpolations. In this paper, generic schemes for formulating assumed natural strain and stabilized Lobatto Lagrange C0 plate/shell elements of order 2 are presented. Two assumed natural strain schemes are devised. Both schemes sample the normal membrane and transverse shear natural strain components at the 1D reduced‐order Gaussian quadrature points along the nodal lines. The difference is that the first and second schemes sample the membrane shear natural strain component at the Lobatto nodes and the 2D reduced‐order Gaussian quadrature points, respectively. Meanwhile, all the other natural strain components in both schemes can be obtained by 1D interpolation along the nodal lines. In the stabilization scheme for reduced‐integrated elements, only five stabilization vectors are required regardless of the element order. The new elements outperform the standard Lobatto element except when membrane‐dominated curved thin‐shell problems with strong boundary layer effect are considered by coarse meshes. Copyright © 2016 John Wiley & Sons, Ltd.

A Point-Adaptive Grouping Scheme of MLFMA for Electrically Large Scattering Simulation

A Gaussian point-adaptive grouping scheme for the multilevel fast multipole algorithm (MLFMA) has been proposed in this communication to significantly reduce the heavy memory cost resulted from using the basis functions defined on large patches without sacrificing the accuracy of the numerical solutions. The grouping process in the MLFMA has been considered as a single-objective optimization problem and is solved by using the clustering algorithm for Gaussian quadrature points on each patch. Meanwhile, the constraint of the addition theorem for MLFMA is still satisfied. As a result, the presented scheme is able to acquire an optimal number of multipoles for each basis function. Compared to the conventional octree grouping scheme or other grouping schemes used in the past, the method in this communication is advantageous in terms of the memory efficiency and the solution accuracy. Numerical examples have been given to demonstrate the validity and effectiveness of the proposed scheme.

Accuracy improvement of the most probable point‐based dimension reduction method using the hessian matrix

SummaryThis paper proposes a most probable point (MPP)‐based dimension reduction method (DRM) using the Hessian matrix called HeDRM to improve accuracy of reliability analysis in existing MPP‐based DRM methods. Conventional MPP‐based DRMs contain two types of errors: (1) error due to eliminating cross‐terms of a performance function by using the univariate DRM; (2) error because of dependency of an axis direction after a rotational transformation. The proposed method minimizes the aforementioned errors by utilizing the Hessian matrix of a performance function. By performing an orthogonal transformation using the eigenvectors of the Hessian matrix, the cross‐term effect of the performance function is minimized and the axis direction that results in the most accurate calculation is obtained because the Gaussian quadrature points for numerical integration are arranged along the eigenvector directions. In this way, the error incurred by exiting MPP‐based DRMs can be reduced that leads to more accurate probability of failure estimation. In addition, this paper proposes to allocate the Gaussian quadrature points using the magnitude of the eigenvalues of the Hessian matrix. This allocation makes it possible to predetermine the number of function evaluations required to estimate the probability of failure accurately and efficiently. Copyright © 2016 John Wiley & Sons, Ltd.

Analysis of Shallow Water Problems Using Element-Free Galerkin Method

A numerical meshless method is proposed to investigate the solution to shallow water equations. The numerical solution to pure convection equations, such as shallow water equations, is an important component of many problems. An element-free Galerkin (EFG) method has been developed and implemented to solve these equations. In this method, there is no need for nodal connectivity; it simply uses nodal data that are similar to those used in the finite element method. The essential boundary condition is enforced by the penalty method and the moving least squares approximation is used for the interpolation scheme. The numerical efficiency of the proposed method is demonstrated by solving several benchmark examples. Sensitivity analysis on the parameters of the EFG method was carried out and the results confirm its ability to solve shallow water equations and reveal that increasing the number of field nodes and Gauss quadrature points can improve the accuracy of the proposed method. It can be concluded that the average distance of the Gauss points should be less than one-third of the average distance of the nodal points. To produce the best results, the radius of influence domain should be three times the distance of the nodal points.

New way to construct high order Hamiltonian variational integrators

This paper develops a new approach to construct variational integrators. A simplified unconventional Hamilton’s variational principle corresponding to initial value problems is proposed, which is convenient for applications. The displacement and momentum are approximated with the same Lagrange interpolation. After the numerical integration and variational operation, the original problems are expressed as algebraic equations with the displacement and momentum at the interpolation points as unknown variables. Some particular variational integrators are derived. An optimal scheme of choosing initial values for the Newton-Raphson method is presented for the nonlinear dynamic system. In addition, specific examples show that the proposed integrators are symplectic when the interpolation point coincides with the numerical integration point, and both are Gaussian quadrature points. Meanwhile, compared with the same order symplectic Runge-Kutta methods, although the accuracy of the two methods is almost the same, the proposed integrators are much simpler and less computationally expensive.

A serendipity triangular patch for evaluating weakly singular boundary integrals

Element subdivision is the most widely used method for the numerical evaluation of weakly singular integrals in three-dimensional boundary element analyses. In the traditional subdivision method, the sub-elements, which are called patches in this paper, are obtained by simply connecting the singular point with each vertex of the element. Patches with large angles at the source point may be produced and thus, a large number of Gaussian quadrature points are needed to achieve acceptable accuracy. In this paper, a serendipity triangular patch with four-node is presented to solve the problem. Case studies have been made to investigate the effect of the location of the middle node of the serendipity patch on accuracy, and an optimal location is determined. Moreover, theoretical analysis validating the optimal location is also given with a new form of polar coordinate transformation. Numerical examples are presented to compare the new patch with the conventional linear patch with respect to both accuracy and efficiency. In all cases, the results are encouraging.

SAMBA: Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos

A new arbitrary Polynomial Chaos (aPC) method is presented for moderately high-dimensional problems characterised by limited input data availability. The proposed methodology improves the algorithm of aPC and extends the method, that was previously only introduced as tensor product expansion, to moderately high-dimensional stochastic problems. The fundamental idea of aPC is to use the statistical moments of the input random variables to develop the polynomial chaos expansion. This approach provides the possibility to propagate continuous or discrete probability density functions and also histograms (data sets) as long as their moments exist, are finite and the determinant of the moment matrix is strictly positive. For cases with limited data availability, this approach avoids bias and fitting errors caused by wrong assumptions. In this work, an alternative way to calculate the aPC is suggested, which provides the optimal polynomials, Gaussian quadrature collocation points and weights from the moments using only a handful of matrix operations on the Hankel matrix of moments. It can therefore be implemented without requiring prior knowledge about statistical data analysis or a detailed understanding of the mathematics of polynomial chaos expansions. The extension to more input variables suggested in this work, is an anisotropic and adaptive version of Smolyak's algorithm that is solely based on the moments of the input probability distributions. It is referred to as SAMBA (PC), which is short for Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos. It is illustrated that for moderately high-dimensional problems (up to 20 different input variables or histograms) SAMBA can significantly simplify the calculation of sparse Gaussian quadrature rules. SAMBA's efficiency for multivariate functions with regard to data availability is further demonstrated by analysing higher order convergence and accuracy for a set of nonlinear test functions with 2, 5 and 10 different input distributions or histograms.

Multiscale analysis of shear failure of thick-walled hollow cylinder in dry sand

A novel hierarchical multiscale model has been applied to simulate the thick-walled hollow cylinder tests in dry sand and to investigate the corresponding shear failures. The combined finite-element method and discrete-element method (FEM/DEM) model employs the FEM as a vehicle to advance the solution for a macroscopic non-linear boundary value problem incrementally. It is, meanwhile, free of conventional macroscopic phenomenological constitutive law, which is replaced by discrete-element simulations conducted with representative volume elements (RVEs) associated with the Gauss quadrature points of the FEM mesh. Numerical simulations proposed by the authors indicate that this multiscale approach is capable of replicating the evolution of cavity pressure during cavity expansion – before and after the onset of strain localisation – in qualitative agreement with laboratory tests. In particular, the curvilinear shear bands observed from experiments have been reproduced numerically. The information provided by the mesoscale DEM and the macroscale FEM reveals a close linkage between significant particle rotations taking place inside the dilative shear bands and the highly anisotropic microstructural attributes of the associated RVEs.

Topology optimization of continuum structures with displacement constraints based on meshless method

In this paper, the element free Galerkin method (EFG) is applied to carry out the topology optimization of continuum structures with displacement constraints. In the EFG method, the matrices in the discretized system equations are assembled based on the quadrature points. In the sense, the relative density at Gauss quadrature point is employed as design variable. Considering the minimization of weight as an objective function, the mathematical formulation of the topology optimization subjected to displacement constraints is developed using the solid isotropic microstructures with penalization interpolation scheme. Moreover, the approximate explicit function expression between topological variables and displacement constraints are derived. Sensitivity of the objective function is derived based on the adjoint method. Three numerical examples are used to demonstrate the feasibility and effectiveness of the proposed method.

Weighted discrete least-squares polynomial approximation using randomized quadratures

We discuss the problem of polynomial approximation of multivariate functions using discrete least squares collocation. The problem stems from uncertainty quantification (UQ), where the independent variables of the functions are random variables with specified probability measure. We propose to construct the least squares approximation on points randomly and uniformly sampled from tensor product Gaussian quadrature points. We analyze the stability properties of this method and prove that the method is asymptotically stable, provided that the number of points scales linearly (up to a logarithmic factor) with the cardinality of the polynomial space. Specific results in both bounded and unbounded domains are obtained, along with a convergence result for Chebyshev measure. Numerical examples are provided to verify the theoretical results.

Enabling High-Dimensional Hierarchical Uncertainty Quantification by ANOVA and Tensor-Train Decomposition

Hierarchical uncertainty quantification can reduce the computational cost of stochastic circuit simulation by employing spectral methods at different levels. This paper presents an efficient framework to simulate hierarchically some challenging stochastic circuits/systems that include high-dimensional subsystems. Due to the high parameter dimensionality, it is challenging to both extract surrogate models at the low level of the design hierarchy and to handle them in the high-level simulation. In this paper, we develop an efficient ANOVA-based stochastic circuit/MEMS simulator to extract efficiently the surrogate models at the low level. In order to avoid the curse of dimensionality, we employ tensor-train decomposition at the high level to construct the basis functions and Gauss quadrature points. As a demonstration, we verify our algorithm on a stochastic oscillator with four MEMS capacitors and 184 random parameters. This challenging example is simulated efficiently by our simulator at the cost of only 10 minutes in MATLAB on a regular personal computer.

Numerical integration techniques for discontinuous manufactured solutions

When applying the method of manufactured solutions (MMS) on computational fluid dynamic software, determining the exact solutions and source terms for finite volume codes where the stored value is an integrated average over the control volume is non-trivial and not frequently discussed. MMS with discontinuities further complicates the problem of determining these values. In an effort to adapt the standard MMS procedure to solutions that contain discontinuities we show that Newton–Cotes and Gauss quadrature numerical integration methods exhibit high error, first order limitations. We propose a new method for determining the exact solutions and source terms on a uniform structured grid containing shock discontinuities by performing linearly and quadratically exact transformations on split cells. Transformations are performed on triangular and quadrilateral elements of a systematically divided discontinuous cell. Using a quadratic transformation in conjunction with a nine point Gauss quadrature method, a minimum of 4th order accuracy is achieved for fully general solutions and shock shapes. A linear approximation of curved shocks is also experimentally shown to be 2nd order accurate. The numerical integration method is then applied to a CFD code using simple discontinuous manufactured solutions which return consistent 1st order convergence values. The result is an important step towards being able to use MMS to verify solutions with discontinuities. This work also highlights the use of higher order numerical integration techniques for continuous and discontinuous solutions that are required for MMS on higher order finite volume codes.

Delayed Genz–Keister Sequences-Based Sparse-Grid Quadrature Nonlinear Filter With Application to Target Tracking

An improved quadrature nonlinear filter named delayed Genz-Keister sequences-based sparse-grid quadrature filter (DGKSGQF) is developed for the target tracking problems. The filter changes the non-nested Gaussian quadrature points of the quadrature filters to the nested Genz-Keister points for selecting the unvariate points, which are the basis point sets extended to form a multidimensional grid using the sparse-grid theory. As a result, the points used for lower accuracy levels DGKSGQF can be reused for any higher accuracy level. Thus, it can further reduce the number of total points used for the conventional Gauss-Hermite SGQF without sacrificing performance. The proposed filter is applied to the reentry ballistic target tracking problem. The simulation results show that the DGKSGQF achieves higher accuracy than the EKF and the UKF. In addition, it can more flexibly control the performance in terms of the number of points and accuracy level.

An Efficient Technique Based on Polynomial Chaos to Model the Uncertainty in the Resonance Frequency of Textile Antennas Due to Bending

The generalized polynomial chaos theory is combined with a dedicated cavity model for curved textile antennas to statistically quantify variations in the antenna's resonance frequency under randomly varying bending conditions. The nonintrusive stochastic method solves the dispersion relation for the resonance frequencies of a set of radius of curvature realizations corresponding to the Gauss quadrature points belonging to the orthogonal polynomials having the probability density function of the random variable as a weighting function. The formalism is applied to different distributions for the radius of curvature, either using a priori known or on-the-fly constructed sets of orthogonal polynomials. Numerical and experimental validation shows that the new approach is at least as accurate as Monte Carlo simulations while being at least 100 times faster. This makes the method especially suited as a design tool to account for performance variability when textile antennas are deployed on persons with varying body morphology.

A generalized framework for nodal first derivative summation-by-parts operators

A generalized framework is presented that extends the classical theory of finite-difference summation-by-parts (SBP) operators to include a wide range of operators, where the main extensions are (i) non-repeating interior point operators, (ii) nonuniform nodal distribution in the computational domain, (iii) operators that do not include one or both boundary nodes. Necessary and sufficient conditions are proven for the existence of nodal approximations to the first derivative with the SBP property. It is proven that the positive-definite norm matrix of each SBP operator must be associated with a quadrature rule; moreover, given a quadrature rule there exists a corresponding SBP operator, where for diagonal-norm SBP operators the weights of the quadrature rule must be positive. The generalized framework gives a straightforward means of posing many known approximations to the first derivative as SBP operators; several are surveyed, such as discontinuous Galerkin discretizations based on the Legendre–Gauss quadrature points, and shown to be SBP operators. Moreover, the new framework provides a method for constructing SBP operators by starting from quadrature rules; this is illustrated by constructing novel SBP operators from known quadrature rules. To demonstrate the utility of the generalization, the Legendre–Gauss and Legendre–Gauss–Radau quadrature points are used to construct SBP operators that do not include one or both boundary nodes.

Fast integral equation solver for Maxwell’s equations in layered media with FMM for Bessel functions

The paper presents a new fast integral equation solver for Maxwell’s equations in 3-D layered media. First, the spectral domain dyadic Green’s function is derived, and the 0-th and the 1-st order Hankel transforms or Sommerfeld-type integrals are used to recover all components of the dyadic Green’s function in real space. The Hankel transforms are performed with the adaptive generalized Gaussian quadrature points and window functions to minimize the computational cost. Subsequently, a fast integral equation solver with O(Nz2NxNy log(NxNy)) in layered media is developed by rewriting the layered media integral operator in terms of Hankel transforms and using the new fast multipole method for the n-th order Bessel function in 2-D. Computational cost and parallel efficiency of the new algorithm are presented.

Element Free Galerkin Method for Static Analysis of Thin Micro/Nanoscale Plates based on the Nonlocal Plate Theory

A B S T R A C T In this article, element free Galerkin method is used for static analysis of thin orthotropic micro/nanoscale plates based on the nonlocal plate theory. Equilibrium equation is obtained based on the nonlocal Kirchoff plate theory. Weak form of the equilibrium equation is discretized based on the moving least square (MLS) approximation functions. Since MLS approximation functions do not satisfy the Kronecker’s delta property, the penalty method is used to impose the essential boundary conditions. Discrete form of the weak form is then solved and the plate deflection is obtained. Numerical results show that the number of nodes scattered in the plate domain, support domain radius and the number of Gauss quadrature points affect the results. Therefore, before presentation of the final results, the method is calibrated using some exact results. Finally, the plate deflection is obtained for various boundary conditions and the small scale effect is studied. In addition, as an example bending problem of nano graphene sheets is solved for different boundary conditions.

High-order central ENO finite-volume scheme for ideal MHD

A high-order accurate finite-volume scheme for the compressible ideal magnetohydrodynamics (MHD) equations is proposed. The high-order MHD scheme is based on a central essentially non-oscillatory (CENO) method combined with the generalized Lagrange multiplier divergence cleaning method for MHD. The CENO method uses k-exact multidimensional reconstruction together with a monotonicity procedure that switches from a high-order reconstruction to a limited low-order reconstruction in regions of discontinuous or under-resolved solution content. Both reconstructions are performed on central stencils, and the switching procedure is based on a smoothness indicator. The proposed high-order accurate MHD scheme can be used on general polygonal grids. A highly sophisticated parallel implementation of the scheme is described that is fourth-order accurate on two-dimensional dynamically-adaptive body-fitted structured grids. The hierarchical multi-block body-fitted grid permits grid lines to conform to curved boundaries. High-order accuracy is maintained at curved domain boundaries by employing high-order spline representations and constraints at the Gauss quadrature points for flux integration. Detailed numerical results demonstrate high-order convergence for smooth flows and robustness against oscillations for problems with shocks. A new MHD extension of the well-known Shu–Osher test problem is proposed to test the ability of the high-order MHD scheme to resolve small-scale flow features in the presence of shocks. The dynamic mesh adaptation capabilities of the approach are demonstrated using adaptive time-dependent simulations of the Orszag–Tang vortex problem with high-order accuracy and unprecedented effective resolution.

An improved meshless method with almost interpolation property for isotropic heat conduction problems

In the paper an improved element free Galerkin method is presented for heat conduction problems with heat generation and spatially varying conductivity. In order to improve computational efficiency of meshless method based on Galerkin weak form, the nodal influence domain of meshless method is extended to have arbitrary polygon shape. When the dimensionless size of the nodal influence domain approaches 1, the Gauss quadrature point only contributes to those nodes in whose background cell the Gauss quadrature point is located. Thus, the bandwidth of global stiff matrix decreases obviously and the node search procedure is also avoided. Moreover, the shape functions almost possess the Kronecker delta function property, and essential boundary conditions can be implemented without any difficulties. Numerical results show that arbitrary polygon shape nodal influence domain not only has high computational accuracy, but also enhances computational efficiency of meshless method greatly. (C) 2013 Elsevier Ltd. All rights reserved.

A simple technique to improve computational efficiency of meshless methods

In order to improve computational efficiency of meshless methods based on Galerkin weak form, in the paper a simple technique is proposed, that is, the nodal influence domain of meshless methods is extended to arbitrary shape. Specifically, circle and rectangle nodal influence domains which are primarily used in meshless methods are generalized to arbitrary convex polygon. When the dimensionless size of the nodal influence domain approaches to 1, the Gauss quadrature point only contributes to these nodes in whose background cell the Gauss quadrature point is located. Thus, the band width of stiff matrix decreases obviously. Meanwhile, the node search process is not needed. The results obtained using the current technique have been compared with those obtained using the finite element method and meshless method with rectangle nodal influence domain, and they present that the provided technique not only has high calculation accuracy, but also enhances computational efficiency of meshless methods greatly. In addition, the technique simplifies imposition of essential boundary conditions as that of the finite element method.