Gauss-Lobatto Quadrature Points Research Articles

High-order polynomial approximations for solving non-inertial particle size density in flames

A novel numerical framework is discussed to simulate the time evolution of non-inertial particle size distributions (or number density functions) in flames. The generic form of the population balance equation is considered featuring nucleation, surface growth/loss and agglomeration/coagulation. This balance equation is first recast in a form that is prone to minimize spurious numerical errors in the simulation of surface growth/loss and collision integrals. Formally, this is achieved classifying the terms of the equation into: (i) Lagrangian transport in size-space (surface growth/loss), (ii) relaxation rates of the particle density at a given size (non-uniform growth/loss and negative contribution of collision integrals) and (iii) sources (nucleation and positive contribution of collision integrals). To secure accuracy, a high-order modal decomposition of the particle size distribution is introduced within every section of size considered. A Legendre polynomials basis is used with Gauss-Lobatto quadrature points. By construction, the method performs very well for dealing with particle surface growth/loss and it is also highly accurate for the estimation of the collision integrals thanks to the high-order quadrature. This is confirmed simulating canonical test cases of the literature to compare the numerical results against exact and analytical solutions. With a discretisation based on about 40 sections of size and with Legendre interpolation at the 5th-order, very good accuracy is obtained up to the third moment of the distributions for particle size ranging over up to 8 orders of magnitude. The method is cast to minimize computing cost. Strategies to couple this novel numerical framework with the simulation of carbon particles dynamics in flames are discussed.

Superconvergence of $$C^0$$-$$Q^k$$ Finite Element Method for Elliptic Equations with Approximated Coefficients

We prove that the superconvergence of $$C^0$$-$$Q^k$$ finite element method at the Gauss–Lobatto quadrature points still holds if variable coefficients in an elliptic problem are replaced by their piecewise $$Q^k$$ Lagrange interpolants at the Gauss–Lobatto points in each rectangular cell. In particular, a fourth order finite difference type scheme can be constructed using $$C^0$$-$$Q^2$$ finite element method with $$Q^2$$ approximated coefficients.

Strain gradient differential quadrature Kirchhoff plate finite element with the C2 partial compatibility

This paper constructs a four-node Kirchhoff plate element considering dilatation, deviatoric stretch and rotation gradient effects to address the general boundary value problems of size-dependent isotropic thin micro-plates. This element benefits from the merits of differential quadrature method (DQM) and finite element method (FEM) and possesses four nodal displacement parameters at each node, i.e., deflection, its two first partial derivatives and one second mixed partial derivative with respect to two in-plane coordinates. To guarantee the C2 partial compatibility among neighboring elements, we establish a novel DQ-based geometric mapping scheme relating the deflection values at Gauss-Lobatto quadrature points to the displacement parameters at four nodes. By applying the DQ rule, the Gauss-Lobatto quadrature rule and the developed mapping scheme, the total potential energy of a generic gradient-elastic Kirchhoff plate element is represented as a function of nodal displacement parameters. The element formulation is derived using the minimum total potential energy principle. Several numerical examples are provided to demonstrate the validity of the proposed method and explore the static bending, free vibration and critical buckling behavior of thin micro-plates. It is validated that the size-dependence of vibration and critical buckling mode shapes of thin micro-plates can be observed in some cases.

Strain gradient differential quadrature beam finite elements

In this paper, the superiorities of finite element method (FEM) and differential quadrature method (DQM) are blended to construct two types of beam elements corresponding to modified strain gradient -Bernoulli and Timoshenko beam models respectively. The two elements, being independent of shape functions and introducing three kinds of strain gradient effects, possess 3-DOFs (degrees of freedom) and 4-DOFs separately at each node. The Lagrange interpolation formula is employed to establish the trial functions of deflection and or rotation at Gauss-Lobatto quadrature points. To realize the inner-element compatibility condition, displacement parameters of quadrature points are converted into those of element nodes through a DQ-based mapping strategy. Total potential energy functional for each beam model is discretized in terms of nodal displacement parameters. The associated differential quadrature finite element formulations are derived by the minimum total potential energy principle. Specific expressions of element stiffness and mass matrices and nodal vector are provided. Numerical examples concerning with static bending, free vibration and buckling of macro/micro-beams are presented to demonstrate the availability of the proposed elements.

Unsymmetric multi-level hanging nodes and anisotropic polynomial degrees in [formula omitted]-conforming higher-order finite element methods

The implementation of higher-order finite element schemes that can handle multi-level hanging nodes is known to be a difficult task. In fact, most of the available literature on hanging nodes in finite element schemes restricts to one-level hanging nodes resulting from symmetric bisections. The intent of this paper is to provide all data structures and algorithms that are necessary for an implementation of a H1-conforming higher-order finite element method. The corresponding finite element spaces are defined via tensor products of hierarchic as well as nodal polynomials for quadrilateral and hexahedron based meshes with unsymmetric, multi-level hanging nodes and arbitrary anisotropic polynomial degree distributions, where special care is given to possible orientation problems. The meshes may even result from non-matching refinements. Given these data structures and algorithms, an extension for Serendipity spaces is described in detail along with some other techniques to improve computational efficiency. Numerical results from an implementation based on these data structures and algorithms serve as a validation and show the broad possibilities that highly flexible higher-order finite element schemes have to offer. To the best of our knowledge, this paper offers the most comprehensive numerical results so far for various three-dimensional benchmark problems using in particular finite element spaces defined for hierarchical as well as nodal polynomials on meshes with unsymmetric refinement ratios as well as (multi-level) hanging nodes. Most notably, the numerical results indicate that unsymmetric refinements are indeed favorable over symmetric refinements with respect to convergence rates. However, the actual optimal refinement ratio for a given problem seems to depend on the type and magnitude of singularities to be resolved as well as on the chosen (full) tensor product or Serendipity finite element spaces. In addition to these numerical results, we find that systems of equations defined via finite element spaces using the nodal Lagrange polynomials with Gauss–Lobatto quadrature points as support points yield drastically improved condition numbers.

A coupled finite and boundary spectral element method for linear water-wave propagation problems

• A coupled finite and boundary spectral element method for linear water-wave propagation problems is proposed. • Boundary spectral element method (BSEM) is a new technique that combines the advantages of the spectral approach and the BEM. • BSEM has been applied to the mild-slope equation with variable bathymetry in one direction. • A convergence study has been made for the BSEM alone and coupled with finite spectral elements. • The proposed formulation has been validated by solving classical water-wave propagation problems. A coupled boundary spectral element method (BSEM) and spectral element method (SEM) formulation for the propagation of small-amplitude water waves over variable bathymetries is presented in this work. The wave model is based on the mild-slope equation (MSE), which provides a good approximation of the propagation of water waves over irregular bottom surfaces with slopes up to 1:3. In unbounded domains or infinite regions, space can be divided into two different areas: a central region of interest, where an irregular bathymetry is included, and an exterior infinite region with straight and parallel bathymetric lines. The SEM allows us to model the central region, where any variation of the bathymetry can be considered, while the exterior infinite region is modelled by the BSEM which, combined with the fundamental solution presented by Cerrato et al. [A. Cerrato, J. A. González, L. Rodríguez-Tembleque, Boundary element formulation of the mild-slope equation for harmonic water waves propagating over unidirectional variable bathymetries, Eng. Anal. Boundary Elem. 62 (2016) 22–34.] can include bathymetries with straight and parallel contour lines. This coupled model combines important advantages of both methods; it benefits from the flexibility of the SEM for the interior region and, at the same time, includes the fulfilment of the Sommerfeld’s radiation condition for the exterior problem, that is provided by the BSEM. The solution approximation inside the elements is constructed by high order Legendre polynomials associated with Legendre–Gauss–Lobatto quadrature points, providing a spectral convergence for both methods. The proposed formulation has been validated in three different benchmark cases with different shapes of the bottom surface. The solutions exhibit the typical p -convergence of spectral methods.

Directly solving the Hamilton–Jacobi equations by Hermite WENO Schemes

In this paper, we present a class of new Hermite weighted essentially non-oscillatory (HWENO) schemes based on finite volume framework to directly solve the Hamilton-Jacobi (HJ) equations. For HWENO reconstruction, both the cell average and the first moment of the solution are evolved, and for two dimensional case, HWENO reconstruction is based on a dimension-by-dimension strategy which is the first used in HWENO reconstruction. For spatial discretization, one of key points for directly solving HJ equation is the reconstruction of numerical fluxes. We follow the idea put forward by Cheng and Wang (2014) 3 to reconstruct the values of solution at Gauss-Lobatto quadrature points and numerical fluxes at the interfaces of cells, and for neither the convex nor concave Hamiltonian case, the monotone modification of numerical fluxes is added, which can guarantee the precision in the smooth region and converge to the entropy solution when derivative discontinuities come up. The third order TVD Runge-Kutta method is used for the time discretization. Extensive numerical experiments in one dimensional and two dimensional cases are performed to verify the efficiency of the methods.

A collocated C0 finite element method: Reduced quadrature perspective, cost comparison with standard finite elements, and explicit structural dynamics

SummaryWe demonstrate the potential of collocation methods for efficient higher‐order analysis on standard nodal finite element meshes. We focus on a collocation method that is variationally consistent and geometrically flexible, converges optimally, embraces concepts of reduced quadrature, and leads to symmetric stiffness and diagonal consistent mass matrices. At the same time, it minimizes the evaluation cost per quadrature point, thus reducing formation and assembly effort significantly with respect to standard Galerkin finite element methods. We provide a detailed review of all components of the technology in the context of elastodynamics, that is, weighted residual formulation, nodal basis functions on Gauss–Lobatto quadrature points, and symmetrization by averaging with the ultra‐weak formulation. We quantify potential gains by comparing the computational efficiency of collocated and standard finite elements in terms of basic operation counts and timings. Our results show that collocation is significantly less expensive for problems dominated by the formation and assembly effort, such as higher‐order elastostatic analysis. Furthermore, we illustrate the potential of collocation for efficient higher‐order explicit dynamics. Throughout this work, we advocate a straightforward implementation based on simple modifications of standard finite element codes. We also point out the close connection to spectral element methods, where many of the key ideas are already established. Copyright © 2014 John Wiley & Sons, Ltd.

Numerical Simulation of Infrared Signature Emitted by Liquid Rocket Plume Using Wide Band K-Distribution Model

A new wide band k-distribution model has been developed and used to investigate infrared radiation signatures of liquid rocket plume in some detectors’ working spectrum regions, in which the temperature and the gas molar concentration fields are solved using engineering empirical formulations. Based on the line parameters in HITEMP database, absorption coefficient variables of water vapor and carbon dioxide corresponding to 12-point Gauss-Lobatto quadrature points are derived and expressed as a simple formulation. The absorption coefficient variables of mixture are obtained by summing the absorption coefficient variables of individual gas species under the hypothesis that the k-distributions are statistically uncorrelated. Its validity for the numerical prediction of liquid rocket plume radiation is verified by comparison with line-by-line approach (LBL). Compared with line-by-line calculations, the maximum relative errors of the new band model are less than 10 % in the detectors’ working spectrum regions, while the computational time of the new band model is less than 1/1000 of LBL. The effects of fly parameter on infrared radiation signatures of liquid rocket plume are studied using the new model. The results showed that the integral radiation intensities of liquid rocket plume increase with the nozzle exit temperature, Mach number and the ratio of nozzle exit pressure to atmospheric pressure. The radiation intensity variation trend for liquid rocket plume with altitude is different for different detectors’ working spectrum regions.

Time-harmonic elasticity with controllability and higher-order discretization methods

The time-harmonic solution of the linear elastic wave equation is needed for a variety of applications. The typical procedure for solving the time-harmonic elastic wave equation leads to difficulties solving large-scale indefinite linear systems. To avoid these difficulties, we consider the original time dependent equation with a method based on an exact controllability formulation. The main idea of this approach is to find initial conditions such that after one time-period, the solution and its time derivative coincide with the initial conditions. The wave equation is discretized in the space domain with spectral elements. The degrees of freedom associated with the basis functions are situated at the Gauss–Lobatto quadrature points of the elements, and the Gauss–Lobatto quadrature rule is used so that the mass matrix becomes diagonal. This method is combined with the second-order central finite difference or the fourth-order Runge–Kutta time discretization. As a consequence of these choices, only matrix–vector products are needed in time dependent simulation. This makes the controllability method computationally efficient.

High-order triangle-based discontinuous Galerkin methods for hyperbolic equations on a rotating sphere

High-order triangle-based discontinuous Galerkin (DG) methods for hyperbolic equations on a rotating sphere are presented. The DG method can be characterized as the fusion of finite elements with finite volumes. This DG formulation uses high-order Lagrange polynomials on the triangle using nodal sets up to 15th order. The finite element-type area integrals are evaluated using order 2 N Gauss cubature rules. This leads to a full mass matrix which, unlike for continuous Galerkin (CG) methods such as the spectral element (SE) method presented in Giraldo and Warburton [A nodal triangle-based spectral element method for the shallow water equations on the sphere, J. Comput. Phys. 207 (2005) 129–150], is small, local and efficient to invert. Two types of finite volume-type flux integrals are studied: a set based on Gauss–Lobatto quadrature points (order 2 N − 1) and a set based on Gauss quadrature points (order 2 N). Furthermore, we explore conservation and advection forms as well as strong and weak forms. Seven test cases are used to compare the different methods including some with scale contractions and shock waves. All three strong forms performed extremely well with the strong conservation form with 2 N integration being the most accurate of the four DG methods studied. The strong advection form with 2 N integration performed extremely well even for flows with shock waves. The strong conservation form with 2 N − 1 integration yielded results almost as good as those with 2 N while being less expensive. All the DG methods performed better than the SE method for almost all the test cases, especially for those with strong discontinuities. Finally, the DG methods required less computing time than the SE method due to the local nature of the mass matrix.

A pseudospectral method of solution of Fisher's equation

In this paper, we develop an accurate and efficient pseudospectral solution of Fisher's equation, a prototypical reaction–diffusion equation. The solutions of Fisher's equation are characterized by propagating fronts that can be very steep for large values of the reaction rate coefficient. There is an ongoing effort to better adapt pseudospectral methods to the solution of differential equations with solutions that resemble shock waves or fronts typical of hyperbolic partial differential equations. The collocation method employed is based on Chebyshev–Gauss–Lobatto quadrature points. We compare results for a single domain as well as for a subdivision of the main domain into subintervals. Instabilities that occur in the numerical solution for a single domain, analogous to those found by others, are attributed to round-off errors arising from numerical features of the discrete second derivative matrix operator. However, accurate stable solutions of Fisher's equation are obtained with a multidomain pseudospectral method. A detailed comparison of the present approach with the use of the sinc interpolation is also carried out.

Tensor product Gauss-Lobatto points are Fekete points for the cube

Tensor products of Gauss-Lobatto quadrature points are frequently used as collocation points in spectral element methods. Unfortunately, it is not known if Gauss-Lobatto points exist in non-tensor-product domains like the simplex. In this work, we show that the n n -dimensional tensor-product of Gauss-Lobatto quadrature points are also Fekete points. This suggests a way to generalize spectral methods based on Gauss-Lobatto points to non-tensor-product domains, since Fekete points are known to exist and have been computed in the triangle and tetrahedron. In one dimension this result was proved by Fejér in 1932, but the extension to higher dimensions in non-trivial.

Spectral collocation methods for a partial integro-differential equation with a weakly singular kernel

AbstractWe propose and analyze the spectral collocation approximation for the partial integro-differential equations with a weakly singular kernel. The space discretization is based on the pseudo-spectral method, which is a collocation method at the Gauss-Lobatto quadrature points. We prove unconditional stability and obtain the optimal error bounds which depend on the time step, the degree of polynomial and the Sobolev regularity of the solution.