High-order Polynomial Approximation Research Articles

Advancing instantaneous detonation prediction of condensed energetic materials based on high-order compressible multiphase fluid dynamics

The instantaneous detonation model (IDM) is widely used in simulating underwater explosions due to its efficiency and ability to ignore the detonation reaction process. In this study, we propose a new IDM to predict the fluid structure in the detonation zone of an RS211 explosive charge. This model is based on high-order solutions provided by the detonation shock dynamics model, where the spatial term is discretized using fifth-order WENO reconstruction in characteristic space and Lax–Friedrichs’s splitting and the temporal terms are discretized using a third-order TVD Runge–Kutta scheme. The interface motion is captured using the level-set method combined with MGFM, and a programmed burn model is provided to describe the generation and propagation of the detonation wave. The self-similarity of detonation wave propagation is validated, and the quantitative calculation formula of the instantaneous detonation model is obtained by averaging or curve fitting the dimensionless results. Consequently, the IDM of the RS211 charge is established using high-order polynomial approximations of the Taylor rarefaction zone and a constant static zone for 1D planar, cylindrical, and spherical RS211 charges. The application of the IDM involves direct mapping from the radial direction to the spatial structured grid for 1D planar, 2D cylindrical, and 3D spherical charges. Numerical results demonstrate that the IDM proposed in this paper shows good accuracy and high computational efficiency.

Graph Neural Networks With High-Order Polynomial Spectral Filters.

General graph neural networks (GNNs) implement convolution operations on graphs based on polynomial spectral filters. Existing filters with high-order polynomial approximations can detect more structural information when reaching high-order neighborhoods but produce indistinguishable representations of nodes, which indicates their inefficiency of processing information in high-order neighborhoods, resulting in performance degradation. In this article, we theoretically identify the feasibility of avoiding this problem and attribute it to overfitting polynomial coefficients. To cope with it, the coefficients are restricted in two steps, dimensionality reduction of the coefficients' domain and sequential assignment of the forgetting factor. We transform the optimization of coefficients to the tuning of a hyperparameter and propose a flexible spectral-domain graph filter, which significantly reduces the memory demand and the adverse impacts on message transmission under large receptive fields. Utilizing our filter, the performance of GNNs is improved significantly in large receptive fields and the receptive fields of GNNs are multiplied as well. Meanwhile, the superiority of applying a high-order approximation is verified across various datasets, notably in strongly hyperbolic datasets. Codes are publicly available at: https://github.com/cengzeyuan/TNNLS-FFKSF.

The Extended Diamond Difference - Constant Nodal Method with Decoupled Cell Iteration Scheme in Two-Dimensional Discrete Ordinate Transport Problems

Analytical solutions for the Boltzmann transport equation are typically limited to specific scenarios, prompting the research transport community to extensively explore numerical approaches with special attention to the discrete ordinates (SN) formulation. The spectral nodal methods are widely recognized for their capability to preserve the analytical solution of the SN mathematical model. However, these formulations exhibit notable complexity with high-order polynomial approximations for the transverse leakage terms, motivating researchers to pursue simpler hybrid approaches by partially preserving the analytical solution. Both spectral nodal and hybrid spectral nodal methods introduce nonstandard auxiliary equations not amenable to standard iterative schemes due to a highly coupling in angular directions. Following these ideas, we present a finite element – spectral nodal hybrid method referred to as Extended Diamond Difference-Constant Nodal (EDD-CN) method for two-dimensional fixed-source discrete ordinate transport problems with constant nodal approximation for the leakage terms. The proposed method seeks to preserve eigenfunctions associated with the dominant pair of eigenvalues in the transverse integrated SN equations. In addition, we propose a novel iterative scheme called Decoupled Cell Iteration (DCI) scheme which decouples the angular directions in the nonstandard auxiliary equations and follows conventional iteration transport sweeps for solving SN problems.

A comparison between physics-based Li-ion battery models

Physics-based electrochemical battery models, such as the Doyle-Fuller-Newman (DFN) model, are valuable tools for simulating Li-ion battery behavior and understanding internal battery processes. However, the complexity and computational demands of such models limit their applicability for battery management systems and long-term aging simulations. Reduced-order models (ROMs), such as the Extended Single Particle Model (ESPM), Single Particle Model (SPM) and Polynomial and Padé approximations, here all referred to as simplifications, lead to faster computational speeds. Choosing the appropriate simplification method for a specific cell type and operating condition is a challenge. This study investigates the simulation accuracy and calculation speed of various simplifications for high-energy (HE) and high-power (HP) batteries at different current loading conditions and compares those to the full-order DFN model. The results indicate that among the ROMs, the ESPM consistently offers the best combination of high computational speed and relatively good accuracy in most conditions in comparison to the full-order DFN model. Among the approximations, higher-order polynomial approximation, third and fourth-order Padé approximation perform the best in terms of accuracy. The higher-order polynomial approximation shows an advantage in terms of computing speed, while the fourth-order Padé approximation achieves the highest overall accuracy among the different approximations.

Data assimilation finite element method for the linearized Navier-Stokes equations with higher order polynomial approximation

In this article, we design and analyze an arbitrary-order stabilized finite element method to approximate the unique continuation problem for laminar steady flow described by the linearized incompressible Navier–Stokes equation. We derive quantitative local error estimates for the velocity, which account for noise level and polynomial degree, using the stability of the continuous problem in the form of a conditional stability estimate. Numerical examples illustrate the performances of the method with respect to the polynomial order and perturbations in the data. We observe that the higher order polynomials may be efficient for ill-posed problems, but are also more sensitive for problems with poor stability due to the ill-conditioning of the system.

A novel numerical method based on a high order polynomial approximation of the fourth order Steklov equation and its eigenvalue problems

<p style='text-indent:20px;'>Based on high order polynomial approximation and dimension reduction technique, we propose a novel numerical method for the fourth order Steklov problems in the circular domain. We first decompose the primal problem into a set of 1D problems via polar coordinate transformation and Fourier basis functions expansion. Then, by introducing a non-uniformly weighed Sobolev space, the variational form and corresponding discrete scheme are derived. Employing the Lax-Milgram lemma and approximation properties of the projection operators, we further prove existence and uniqueness of weak solutions and approximation solutions for each one-dimensional problems, and the error estimation between them, respectively. We also carry out ample numerical experiments which illustrate that the numerical algorithm is efficient and highly accurate.</p>

Totally volume integral of fluxes for discontinuous Galerkin method-two dimensional Euler equations

In this paper, the scheme of constructing high-order accurate totally volume discontinuous finite element method for the numerical solution of the 1D Euler equations is extended to 2D Euler equations on Cartesian meshes. In the present work, the boundary integral fluxes are transformed into volume integral by applying divergence theorem to the boundary integral of the Riemann fluxes. Therefore, the totally volume discontinuous finite element is independent on the boundary integral fluxes at the element boundaries as opposed to the classical discontinuous Galerkin method. The accuracy is obtained by applying high-order polynomial approximations within elements using the tensor product of Lagrange polynomial. For temporal integration, strong stability preserving Runge-Kutta method SSPRK (3, 3) is applied. The scheme is stabilized by using Streamline Upwind Petrov Galerkin (SUPG) stabilization technique. For the spatial discretization, the polynomial of order 1and 2 are used, the shape function is constructed for the master (computational) element after applying the coordinate transformation for the physical domain, the transformation for the governing equations is performed to get it in the function of computational Cartesian, then the governing equations are put in conservative form. The numerical results of applying totally volume integral discontinuous Galerkin method for two-dimensional Euler equations presented in this paper show that the scheme is very accurate, fast, and effective even with shock appearance.

Spectral element method for 3-D controlled-source electromagnetic forward modelling using unstructured hexahedral meshes

SUMMARYFor forward modelling of realistic 3-D land-based controlled-source electromagnetic (EM) problems, we develop a parallel spectral element approach, blending the flexibility and versatility of the finite element method in using unstructured grids with the accuracy of the spectral method. Complex-shaped structures and topography are accommodated by using unstructured hexahedral meshes, in which the elements can have curved edges and non-planar faces. Our code is the first spectral element algorithm in EM geophysics that uses the total field formulation (here that of the electric field). Combining unstructured grids and a total field formulation provides advantages in dealing with topography, in particular, when the transmitter is located on rough surface topography. As a further improvement over existing spectral element methods, our approach does not only allow for arbitrary distributions of conductivity, but also of magnetic permeability and dielectric permittivity. The total electric field on the elements is expanded in terms of high-order Lagrangian interpolants, and element-wise integration in the weak form of the boundary value problem is accomplished by Gauss–Legendre–Lobatto quadrature. The resulting complex-valued linear system of equations is solved using the direct solver MUMPS, and, subsequently, the magnetic field is computed at the points of interest by Faraday’s law. Five numerical examples comprehensively study the benefits of this algorithm. Comparisons to semi-analytical and finite element results confirm accurate representation of the EM responses and indicate low dependency on mesh discretization for the spectral element method. A convergence study illuminates the relation between high order polynomial approximation and mesh size and their effects on accuracy and computational cost revealing that high-order approximation yields accurate modelling results for very coarse meshes but is accompanied by high computational cost. The presented numerical experiments give evidence that 2nd and 3rd degree polynomials in combination with moderately discretized meshes provide better trade-offs in terms of computational resources and accuracy than lowest and higher order spectral element methods. To our knowledge, our final example that includes pronounced surface topography and two geometrically complicated conductive anomalies represents the first successful attempt at using 2nd order hexahedral elements supporting curved edges and non-planar faces in controlled-source EM geophysics.

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.

A new algorithm based on compressed Legendre polynomials for solving boundary value problems

<abstract><p>In this paper, we discuss a novel numerical algorithm for solving boundary value problems. We introduce an orthonormal basis generated from compressed Legendre polynomials. This basis can avoid Runge phenomenon caused by high-order polynomial approximation. Based on the new basis, a numerical algorithm of two-point boundary value problems is established. The convergence and stability of the method are proved. The whole analysis is also applicable to higher order equations or equations with more complex boundary conditions. Four numerical examples are tested to illustrate the accuracy and efficiency of the algorithm. The results show that our algorithm have higher accuracy for solving linear and nonlinear problems.</p></abstract>

Effectiveness of Floating-Point Precision on the Numerical Approximation by Spectral Methods

With the fast advances in computational sciences, there is a need for more accurate computations, especially in large-scale solutions of differential problems and long-term simulations. Amid the many numerical approaches to solving differential problems, including both local and global methods, spectral methods can offer greater accuracy. The downside is that spectral methods often require high-order polynomial approximations, which brings numerical instability issues to the problem resolution. In particular, large condition numbers associated with the large operational matrices, prevent stable algorithms from working within machine precision. Software-based solutions that implement arbitrary precision arithmetic are available and should be explored to obtain higher accuracy when needed, even with the higher computing time cost associated. In this work, experimental results on the computation of approximate solutions of differential problems via spectral methods are detailed with recourse to quadruple precision arithmetic. Variable precision arithmetic was used in Tau Toolbox, a mathematical software package to solve integro-differential problems via the spectral Tau method.

Efficient projection kernels for discontinuous Galerkin simulations of disperse multiphase flows on arbitrary curved elements

In this work, we develop projection kernels for Euler-Lagrange point-particle simulations of disperse multiphase flows on arbitrary curved elements. These kernels are employed in a high-order discontinuous Galerkin framework for projecting the action of the particles to the Eulerian mesh. Instead of commonly used isotropic kernels, such as a Gaussian-type kernel, we construct an anisotropic polynomial-based smoothing function that preserves the compactness of the discontinuous Galerkin method on high-aspect-ratio elements and maintains an acceptable computational cost. At the same time, it mitigates inaccuracies associated with larger particles and numerical instabilities arising from the Dirac delta function and low-order kernels. Specifically, the geometric mapping of a physical element to the reference element is exploited to construct a kernel that is elliptical in 2D and ellipsoidal in 3D. We also develop a strategy to conserve interphase transfer near boundaries, particularly on curved elements. This strategy employs a high-order polynomial approximation to appropriately rescale the interphase source terms in an efficient manner. The compatibility of the proposed methodology with different types of meshes is investigated. We then apply it to a number of particle-laden flow configurations, including supersonic dusty flow over a flat plate, moving shocks interacting with clouds of particles, and hypersonic dusty flows over blunt bodies.

Research on the Inverse Problem of Reliability Evaluation–Model and Algorithm

Due to the high dependence of economic and social development on power systems, the demand for reliable operation of power systems is increasing. Considering the popularity and widespread installation of smart meters, accurate system/node reliability indexes can be obtained. The inverse problem of reliability evaluation (IPRE) refers to the use of known system/node reliability indexes to obtain component reliability parameters. In this paper, a novel method of solving the IPRE is proposed. First, based on a nonsequential Monte Carlo (NSMC) method, analytical expressions for system reliability indexes in terms of component reliability parameters are derived, and then, the nonlinear equations of the IPRE are constructed. Second, a high-order polynomial approximation based on the conjugate gradient algorithm is used to calculate the unknown component reliability parameters, and the results are compared with those obtained using traditional neural networks method. Finally, a continuation method is used to correct the errors of the obtained component reliability parameters. Three cases, namely, the IEEE 1979 Reliability Test System (IEEE RTS-79), the Roy Billinton Test System (RBTS) and the Chuanyu power system in Southwest China, are used to test the method proposed in this paper to verify its feasibility and accuracy.

Condensed generalized finite element method

AbstractGeneralized or extended finite element methods (GFEM/XFEM) are in general badly conditioned and have numerous additional degrees of freedom (DOF) compared with the FEM due to introduction of enriched functions. In this paper, we develop an approach to establish a subspace of a conventional GFEM/XFEM approximation space using partition of unity (PU) techniques and local least‐square procedures. The proposed GFEM is referred to as condensed GFEM (CGFEM), which (i) possesses as many DOFs as the preliminary FEM, (ii) enjoys similar approximation properties with the GFEM/XFEM, and (iii) is well‐conditioned in the sense that its conditioning is of the same order as that of the FEM. The fundamental approximation properties of CGFEM are proven mathematically. The CGFEM is applied to a problem of high order polynomial approximations and a Poisson crack problem. The optimal convergence orders of the former are proven rigorously. Numerical experiments and comparisons with the conventional GFEM/XFEM and FEM are presented to verify the theory and effectiveness of CGFEM.

ILES of an array of three subsonic counter-flow jets issuing from a wing leading edge exposed to hypersonic aerodynamic heating

The flow of an active thermal protection system exploiting subsonic counter-flow jets for wing leading edges of hypersonic vehicles is numerically studied on the basis of the three dimensional Navier-Stokes equations. The coolant air issuing from around the stagnation point as an array of three jets spreads over both the upper and the lower sides of the cylinder surface and about 40 ~ 60% cooling effectiveness is achieved in the range up to 5 degrees angle of attack despite the occurrence of various three-dimensional fluid-dynamic instabilities. The numerical scheme is second order accurate but simple inclusion of high order polynomial approximation in the reconstruction enables the capturing of finer structure of the flow field.

A Posteriori Error Estimates for the Laplace--Beltrami Operator on Parametric $C^2$ Surfaces

We prove new a posteriori error estimates for surface finite element methods (SFEMs). SFEMs approximate solutions to PDEs posed on surfaces. Prototypical examples are elliptic PDEs involving the Laplace--Beltrami operator. Typically the surface is approximated by a polyhedral or higher-order polynomial approximation. The resulting finite element method exhibits both a geometric consistency error due to the surface approximation and a standard Galerkin error. A posteriori estimates for SFEMs require practical access to geometric information about the surface in order to computably bound the geometric error. It is thus advantageous to allow for maximum flexibility in representing surfaces in practical codes when proving a posteriori error estimates for SFEMs. However, previous a posteriori estimates using general parametric surface representations are suboptimal by one order on $C^2$ surfaces. Proofs of error estimates optimally reflecting the geometric error instead employ the closest point projection, which is defined using the signed distance function. Because the closest point projection is often unavailable or inconvenient to use computationally, a posteriori estimates using the signed distance function have notable practical limitations. We merge these two perspectives by assuming practical access only to a general parametric representation of the surface, but using the distance function as a theoretical tool. This allows us to derive sharper geometric estimators which exhibit improved experimentally observed decay rates when implemented in adaptive surface finite element algorithms.

Station-keeping of libration point orbits by means of projecting to the manifolds

There are many different types of (quasi)periodic orbits in the vicinity of collinear libration points. These libration point orbits (LPOs) are inherently unstable and must be controlled. This paper studies general station-keeping strategies that are suitable for various types of LPOs. Two complementary methods are proposed by exploiting the hyperbolic dynamics of the collinear libration points. One main advantage of these methods is that no information of the nominal orbit is needed in order to do the station-keeping. The first is a semi-analytical projection method which is based on the high order polynomial approximation of the center manifold. After the transformation from synodic coordinates to center manifold coordinates is derived, the error state can be projected to center manifold or its stable manifold by introducing a station-keeping Delta-V, effectively eliminating the unstable component. The second is a fully numerical method. An escape time algorithm is designed to find the intersection of stable and unstable manifolds, effectively projecting the error state to the center manifold. By applying these techniques to the collinear libration points of Earth–Moon system, it has been demonstrated that these proposed methods can achieve better maneuver cost performance and they can handle station-keeping problems of various types of LPOs in a unified manner.

Asymptotic safety of quantum gravity beyond Ricci scalars

We investigate the asymptotic safety conjecture for quantum gravity including curvature invariants beyond Ricci scalars. Our strategy is put to work for families of gravitational actions which depend on functions of the Ricci scalar, the Ricci tensor, and products thereof. Combining functional renormalisation with high order polynomial approximations and full numerical integration we derive the renormalisation group flow for all couplings and analyse their fixed points, scaling exponents, and the fixed point effective action as a function of the background Ricci curvature. The theory is characterised by three relevant couplings. Higher-dimensional couplings show near-Gaussian scaling with increasing canonical mass dimension. We find that Ricci tensor invariants stabilise the UV fixed point and lead to a rapid convergence of polynomial approximations. We apply our results to models for cosmology and establish that the gravitational fixed point admits inflationary solutions. We also compare findings with those from $f(R)$-type theories in the same approximation and pin-point the key new effects due to Ricci tensor interactions. Implications for the asymptotic safety conjecture of gravity are indicated.

A multiple-scale higher order polynomial collocation method for 2D and 3D elliptic partial differential equations with variable coefficients

In this paper, we present a multiple-scale higher order polynomial collocation method for the numerical solution of 2D and 3D elliptic partial differential equations (PDEs) with variable coefficients. The collocation method with higher order polynomial approximation is very simple for solving PDEs, but it has not become the mainstream method. The main reason is that its resultant algebraic equations have highly ill-conditioned behavior. In our scheme, the multiple-scale coefficients are introduced in the polynomial approximation to overcome the ill-conditioned problem. Based on the concept of the equilibrate matrix, the multiple scales are automatically determined by the collocation points. We find these scales can largely reduce the condition number of the coefficient matrix. Numerical results confirm the accuracy, effectiveness and stability of the present method for smoothed and near-singular 2D and 3D elliptic problems on various irregular domains.

A p-Adaptive Local Discontinuous Galerkin Level Set Method for Willmore Flow

The level set method is often used to capture interface behavior in two or three dimensions. In this paper, we present a combination of local discontinuous Galerkin (LDG) method and level set method for simulating Willmore flow. The LDG scheme is energy stable and mass conservative, which are good properties comparing with other numerical methods. In addition, to enhance the efficiency of the proposed LDG scheme and level set method, we employ a p-adaptive local discontinuous Galerkin technique, which applies high order polynomial approximations around the zero level set and low order ones away from the zero level set. A major advantage of the level set method is that the topological changes are well defined and easily performed. In particular, given the stiffness of Willmore flow, a high order semi-implicit Runge-Kutta method is employed for time discretization, which allows larger time step. The equations at the implicit time level are linear, we demonstrate an efficient and practical multi-grid solver to solve the equations. Numerical examples are given to illustrate the combination of the LDG scheme and level set method provides an efficient and practical approach when simulating the Willmore flow.