High Order Finite Research Articles

Higher Order Finite and Infinite Acoustical Elements Based on Ultraspherical Polynomials

Higher Order Finite and Infinite Acoustical Elements Based on Ultraspherical Polynomials

Verification of a Higher-Order Finite Difference Scheme for the One-Dimensional Two-Fluid Model

The one-dimensional two-fluid model is widely acknowledged as the most detailed and accurate macroscopic formulation model of the thermo-fluid dynamics in nuclear reactor safety analysis. Currently the prevailing one-dimensional thermal hydraulics codes are only first-order accurate. The benefit of first-order schemes is numerical viscosity, which serves as a regularization mechanism for many otherwise ill-posed two-fluid models. However, excessive diffusion in regions of large gradients leads to poor resolution of phenomena related to void wave propagation. In this work, a higher-order shock capturing method is applied to the basic equations for incompressible and isothermal flow of the one-dimensional two-fluid model. The higher-order accuracy is gained by a strong stability preserving multi-step scheme for the time discretization and a minmod flux limiter scheme for the convection terms. Additionally the use of a staggered grid allows for several second-order centered terms, when available. The continuity equations are first tested by manipulating the two-fluid model into a pair of linear wave equations and tested for smooth and discontinuous initial data. The two-fluid model is benchmarked with the water faucet problem. With the higher-order method, the ill-posed nature of the governing equations presents severe challenges due to a growing void fraction jump in the solution. Therefore the initial and boundary conditions of the problem are modified in order to eliminate a large counter-current flow pattern that develops. With the modified water faucet problem the numerical models behave well and allow a convergence study. Using the L1 norm of the liquid fraction, it is verified that the first and higher-order numerical schemes converge to the quasi-analytical solution at a rate of O(1/2) and O(2/3), respectively. It is also shown that the growing void jump is a contact discontinuity, i.e. it is a linearly degenerate wave. The sub-linear convergence rates are in exact agreement with the theory for a contact discontinuity.

Discussion on Direct Numerical Simulation of Large-Scale Structures in High-Reynolds Number Channel Flow by Means of High-Order Accurate Finite Difference Method for Wall-Parallel Directions

Discussion on Direct Numerical Simulation of Large-Scale Structures in High-Reynolds Number Channel Flow by Means of High-Order Accurate Finite Difference Method for Wall-Parallel Directions

High-Order Spectral Stochastic Finite Element Analysis of Stochastic Elliptical Partial Differential Equations

This study presents an experiment of improving the performance of spectral stochastic finite element method using high-order elements. This experiment is implemented through a two-dimensional spectral stochastic finite element formulation of an elliptic partial differential equation having stochastic coefficients. Deriving this spectral stochastic finite element formulation couples a two-dimensional deterministic finite element formulation of an elliptic partial differential equation with generalized polynomial chaos expansions of stochastic coefficients. Further inspection of the performance of resulting spectral stochastic finite element formulation with adopting linear and quadratic (9-node or 8-node) quadrilateral elements finds that more accurate standard deviations of unknowns are surprisingly predicted using quadratic quadrilateral elements, especially under high autocorrelation function values of stochastic coefficients. In addition, creating spectral stochastic finite element results using quadratic quadrilateral elements is not unacceptably time-consuming. Therefore, this study concludes that adopting high-order elements can be a lower-cost method to improve the performance of spectral stochastic finite element method.

High-Order Compact Finite Difference Schemes for Option Pricing in Stochastic Volatility Models on Non-Uniform Grids

We derive high-order compact finite difference schemes for option pricing in stochastic volatility models on non-uniform grids. The schemes are fourth-order accurate in space and second-order accurate in time for vanishing correlation. In our numerical study we obtain high-order numerical convergence also for non-zero correlation and non-smooth payoffs which are typical in option pricing. In all numerical experiments a comparative standard second-order discretisation is significantly outperformed. We conduct a numerical stability study which indicates unconditional stability of the scheme.

Modeling of Surface Waves in a Fluid Saturated Poro-Elastic Medium under Initial Stress Using Time-Space Domain Higher Order Finite Difference Method

In this present context, mathematical modeling of the propagation of surface waves in a fluid saturated poro-elastic medium under the influence of initial stress has been considered using time dependent higher order finite difference method (FDM). We have proved that the accuracy of this finite-difference scheme is 2M when we use 2nd order time domain finite-difference and 2M-th order space domain finite-difference. It also has been shown that the dispersion curves of Love waves are less dispersed for higher order FDM than of lower order FDM. The effect of initial stress, porosity and anisotropy of the layer in the propagation of Love waves has been studied here. The numerical results have been shown graphically. As a particular case, the phase velocity in a non porous elastic solid layer derived in this paper is in perfect agreement with that of Liu et al. (2009).

A Parallel Discontinuous Galerkin Method with Physical Orthogonal Basis on Curved Elements

This paper describes a parallel high-order Discontinuous Galerkin method based on orthogonal basis functions in physical space for arbitrary three-dimensional curved boundary geometry. The physical orthogonal basis are obtained by solving multi-stage linear systems which satisfy the orthogonal properties on arbitrary grids. On curved-boundary surfaces, the high-order Finite Element type of grids are used to represent real local curvature. An efficient parallel strategy is devised for the resulting solver, which based on OPENMP (Open Multi-Processing) automatic parallelization of independent elemental and face-based operations. The convergence accuracy of the method is tested for subsonic flow over a sphere. To further demonstrate the robustness of the method for complex curved geometry, external flow over a whole airplane configure is tested. The validation cases demonstrate the accuracy and robustness of the method.

Simulation of Dynamic Earthquake Ruptures in Complex Geometries Using High-Order Finite Difference Methods

We develop a stable and high-order accurate finite difference method for problems in earthquake rupture dynamics in complex geometries with multiple faults. The bulk material is an isotropic elastic solid cut by pre-existing fault interfaces that accommodate relative motion of the material on the two sides. The fields across the interfaces are related through friction laws which depend on the sliding velocity, tractions acting on the interface, and state variables which evolve according to ordinary differential equations involving local fields. The method is based on summation-by-parts finite difference operators with irregular geometries handled through coordinate transforms and multi-block meshes. Boundary conditions as well as block interface conditions (whether frictional or otherwise) are enforced weakly through the simultaneous approximation term method, resulting in a provably stable discretization. The theoretical accuracy and stability results are confirmed with the method of manufactured solutions. The practical benefits of the new methodology are illustrated in a simulation of a subduction zone megathrust earthquake, a challenging application problem involving complex free-surface topography, nonplanar faults, and varying material properties.

Erratum To: "High-Order Finite Difference Schemes for Solving the Advection-Diffusion Equation, Mathematical and Computational Applications, Vol. 15, No. 3, 449-460, 2010."

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Guidelines and Recommendations on the Use of Higher Order Finite Elements for Bending Analysis of Plates

This paper compares and evaluates various plate finite elements to analyse the static response of thick and thin plates subjected to different loading and boundary conditions. Plate elements are based on different assumptions for the displacement distribution along the thickness direction. Classical (Kirchhoff and Reissner-Mindlin), refined (Reddy and Kant), and other higher-order displacement fields are implemented up to fourth-order expansion. The Unified Formulation UF by the first author is used to derive finite element matrices in terms of fundamental nuclei which consist of 3×3 arrays. The MITC4 shear-locking free type formulation is used for the FE approximation. Accuracy of a given plate element is established in terms of the error vs. thickness-to-length parameter. A significant number of finite elements for plates are implemented and compared using displacement and stress variables for various plate problems. Reduced models that are able to detect the 3D solution are built and a Best Plate Diagram (BPD) is introduced to give guidelines for the construction of plate theories based on a given accuracy and number of terms. It is concluded that the UF is a valuable tool to establish, for a given plate problem, the most accurate FE able to furnish results within a certain accuracy range. This allows us to obtain guidelines and recommendations in building refined elements in the bending analysis of plates for various geometries, loadings, and boundary conditions.

Diffusion Synthetic Acceleration for High-Order Discontinuous Finite Element SN Transport Schemes and Application to Locally Refined Unstructured Meshes

Diffusion synthetic acceleration (DSA) schemes compatible with adaptive mesh refinement (AMR) grids are derived for the SN transport equations discretized using high-order discontinuous finite elements. These schemes are directly obtained from the discretized transport equations by assuming a linear dependence in angle of the angular flux along with an exact Fick’s law and, therefore, are categorized as partially consistent. These schemes are akin to the symmetric interior penalty technique applied to elliptic problems and are all based on a second-order discontinuous finite element discretization of a diffusion equation (as opposed to a mixed or P1 formulation). Therefore, they only have the scalar flux as unknowns. A Fourier analysis has been carried out to determine the convergence properties of the three proposed DSA schemes for various cell optical thicknesses and aspect ratios. Out of the three DSA schemes derived, the modified interior penalty (MIP) scheme is stable and effective for realistic problems, even with distorted elements, but loses effectiveness for some highly heterogeneous configurations. The MIP scheme is also symmetric positive definite and can be solved efficiently with a preconditioned conjugate gradient method. Its implementation in an AMR SN transport code has been performed for both source iteration and GMRes-based transport solves, with polynomial orders up to 4. Numerical results are provided and show good agreement with the Fourier analysis results. Results on AMR grids demonstrate that the cost of DSA can be kept low on locally refined meshes.

Spectral Method and High-Order Finite Differences for the Nonlinear Cable Equation

We use high-order approximation schemes for the space derivatives in the nonlinear cable equation and investigate the behavior of numerical solution errors by using exact solutions, where available, and grid convergence. The space derivatives are numerically approximated by means of differentiation matrices. Nonlinearity in the equation arises from the Hodgkin-Huxley dynamics of the gating variables for ion channels. We have investigated in particular the effects of synaptic current distribution and compared the accuracy of the spectral solutions with that of finite differencing. A flexible form for the injected current is used that can be adjusted smoothly from a very broad to a narrow peak, which furthermore leads, for the passive cable, to a simple, exact solution. We have used three distinct approaches to assess the numerical solutions: comparison with exact solutions in an unbranched passive cable, the convergence of solutions with progressive refinement of the grid in an active cable, and the simulation of spike initiation in a biophysically realistic single-neuron model. The spectral method provides good numerical solutions for passive cables comparable in accuracy to those from the second-order finite difference method and far greater accuracy in the case of a simulated system driven by inputs that are smoothly distributed in space. It provides faster convergence in active cables and in a realistic neuron model due to better approximation of propagating spikes.

Stable and Accurate Interpolation Operators for High-Order Multiblock Finite Difference Methods

Block-to-block interface interpolation operators are constructed for several common high-order finite difference discretizations. In contrast to conventional interpolation operators, these new interpolation operators maintain the strict stability, accuracy, and conservation properties of the base scheme even when nonconforming grids or dissimilar operators are used in adjoining blocks. The stability properties of the new operators are verified using eigenvalue analysis, and the accuracy properties are verified using numerical simulations of the Euler equations in two spatial dimensions.

High-Order Compact Finite Difference Scheme for Option Pricing in Stochastic Volatility Models

High-Order Compact Finite Difference Scheme for Option Pricing in Stochastic Volatility Models

Obtaining and Verifying High-Order Unstructured Finite Volume Solutions to the Euler Equations

Interest in high-order unstructured-mesh finite volume methods continues to grow, as researchers are attracted both to the possibilities of high-accuracy solutions on coarse meshes and to applications for which high-order accuracy is all but mandatory, including large-eddy simulations for complex geometries. A major obstacle for those interested in developing a high-order solver, however, is that there are many more details that must be exactly right for a nominally-high-order scheme to be genuinely-high-order than for a second-order scheme to be second-order. In many cases, these details are either omitted or glossed over in the literature. This paper attempts to close this gap by illuminating a number of obscure, or often overlooked, aspects of correctly implementing a high-order-accurate unstructured-mesh finite volume solver. We discuss reconstruction, flux integration, curved-boundary treatment, and demonstrating order of accuracy. In each case, we describe test cases that are appropriate for confirming the correct behavior of a high-order code, as well as demonstrating some of the common pitfalls in implementation and showing their effects. We hope that this paper will serve other researchers as a practical guide to creating their own high-order-accurate solvers.

Free Vibration Analysis Of Curved Box Girder Bridges by Higher Order Finite Strip

Free Vibration Analysis Of Curved Box Girder Bridges by Higher Order Finite Strip

Higher order finite and infinite elements for the solution of Helmholtz problems

In recent years the focus in the field of acoustic finite element computations has shifted to the mid-frequency regime. At higher frequencies, however, the conventional finite elements with linear shape functions fail to provide reliable results due to so-called pollution effects. This motivates the use of higher order shape functions and related p-FEM concepts. The solution of large systems of equations arising from engineering problems often involves the use of iterative solution procedures, mostly by means of Krylov-subspace methods. The performance of these methods strongly depends on the spectrum of the resulting system matrices, which is affected by the polynomial approximation within the finite element formulation. The current work shows that finite elements based on Bernstein polynomials yield a favorable spectrum of the system matrix and particularly good performance in combination with commonly employed Krylov solvers. This is shown for interior as well as exterior Helmholtz problems, where an infinite element formulation is employed to account for the sound radiation.

3D Finite Volume simulation of acoustic waves in the earth atmosphere

We present a very high-order Finite Volume discretization of the 3D linearized convective Euler equations for the simulation of acoustic waves in the Earth atmosphere. We also derive a weakly nonlinear model for the approximation of N-waves. We discuss the use and the implementation of these methods on massively parallel computers based on our experience on the Bull-TERA 10 parallel HPC machine at the CEA. Verification–validation is done in dimension two and three with physical data obtained from a large-scale physical experiment.

High-order Finite Volume Methods and Multiresolution Reproducing Kernels

This paper presents a review of some of the most successful higher-order numerical schemes for the compressible Navier-Stokes equations on unstructured grids. A suitable candidate scheme would need to be able to handle potentially discontinuous flows, arising from the predominantly hyperbolic character of the equations, and at the same time be well suited for elliptic problems, in order to deal with the viscous terms. Within this context, we explore the performance of Moving Least-Squares (MLS) approximations in the construction of higher order finite volume schemes on unstructured grids. The scope of the application of MLS is threefold: 1) computation of high order derivatives of the field variables for a Godunov-type approach to hyperbolic problems or terms of hyperbolic character, 2) direct reconstruction of the fluxes at cell edges, for elliptic problems or terms of elliptic character, and 3) multiresolution shock detection and selective limiting. The proposed finite volume method is formulated within a continuous spatial representation framework, provided by the MLS approximants, which is “broken” locally (inside each cell) into piecewise polynomial expansions, in order to make use of the specialized finite volume technology for hyperbolic problems. This approach is in contrast with the usual practice in the finite volume literature, which proceeds bottom-up, starting from a piecewise constant spatial representation. Accuracy tests show that the proposed method achieves the expected convergence rates. Representative simulations show that the methodology is applicable to problems of engineering interest, and very competitive when compared to other existing procedures.

New Multigrid Method Including Elimination Algolithm Based on High-Order Vector Finite Elements in Three Dimensional Magnetostatic Field Analysis

A new multigrid method based on high-order vector finite elements is proposed in this paper. Low level discretizations in this method are obtained by using low-order vector finite elements for the same mesh. Gauss-Seidel method is used as a smoother, and a linear equation of lowest level is solved by ICCG method. But it is often found that multigrid solutions do not converge into ICCG solutions. An elimination algolithm of constant term using a null space of the coefficient matrix is also described. In three dimensional magnetostatic field analysis, convergence time and number of iteration of this multigrid method are discussed with the convectional ICCG method.