High-order Interpolation Research Articles

On Order Elevation for Unstructured Finite Volume Solvers using Defect Correction

Abstract Much of the effort in improving the accuracy of CFD simulations is focused on mesh refinement and adaptation although studies have shown that the use of high-order methods are more efficient in improving accuracy. Stability issues, complexity of implementation and demand of computational resources are some of the key factors hindering the use of high-order methods in commercial CFD solvers. This paper demonstrates an improvement in the order of accuracy of finite volume solutions on unstructured meshes without using a high-order solver. Defect correction and the error transport equation method are the techniques discussed, along with the method for obtaining an appropriate estimate of the truncation error, which is crucial in both these techniques. Methods to obtain high-order interpolation of the control volume averages and high-order integral functionals along curved boundaries are also discussed. Third-order accurate results are obtained for a variety of problems, including the 2-D Euler equations, without using a third-order discretization scheme.

From bias to balance: Leverage representation learning for bias-free MoCap solving

From bias to balance: Leverage representation learning for bias-free MoCap solving

MainModelingStr: A Structural Dynamic Nonlinear Analysis Framework with Adaptive Beam-Column Finite Elements

This study introduces a new method for nonlinear structural analysis of buildings using high-order Hermitian interpolation function matrices for Timoshenko elements. This method aims to improve the accuracy of traditional beam-column finite element analyses by using higher-order shape function polynomials. Thus, a p-adaptive procedure is employed to address the potential issues of increased complexity and round-off error associated with higher-order functions. The study uses a technique for removing outliers in the p-adaptive method, which is more consistent with the theoretical approach. Moreover, a novel strategy based on the generalized alpha method is applied to reduce even more convergence problems, allowing its parameters to adapt in each instant of the time-history analysis. These adaptive parameters permit setting a low tolerance error and avoiding changing these parameters manually in time-consuming analyses. The efficiency of using fourth-order beam-column elements is also investigated. Finally, the importance of the proposed method is demonstrated through examples of building analyses, which is the base of a proposed software called MainModelingStr.

Uncertainty quantification for the drag reduction of microbubble-laden fluid flow in a horizontal channel

Uncertainty quantification for the drag reduction of microbubble-laden fluid flow in a horizontal channel

A Modified Algorithm to Generate Flow Nets from the Nodal Potential and Stream Values of Eight-Node Quadrilateral Elements

Eight-node quadrilateral isoparametric elements of the serendipity type have frequently been used in finite-element analyses of two-dimensional seepage problems. The shape functions for these elements are quadratic. Hence, nonlinear variation in the potential and stream function values across each element could be approximated to a high degree of accuracy. This also necessitates a commensurate high-order interpolation function to locate, in a straightforward way, equipotential lines and streamlines. In this paper, a quadratic interpolation algorithm for locating deformation contours is modified to suit flow net generation. The modification lies in the procedure for identifying the pairs of the points of intersection to be joined when there are four, six, or eight points of intersection of the contour segments of the same level and the edges of an element. The original algorithm finds the pairs of intersection points in a local coordinate system by testing all possible cases that may be encountered. The modified algorithm considers that in most, if not all, scenarios, equipotential lines and streamlines extend monotonically from one impervious boundary of the flow domain to another and from an inflow boundary to an outflow boundary, respectively. The intersection points are rapidly paired by converting their local coordinates to global coordinates and sorting the order of the intersection points according to their global coordinates. The modified algorithm eliminates the need for an exhaustive search and complex matching process, enhancing computational efficiency. The modified algorithm is verified against an exact analytical solution to the flow net for a levee under-seepage flow. Excellent agreement is obtained. Two additional illustrative examples are analyzed. One is unconfined seepage through a rectangular dam, and the other is confined seepage beneath unsymmetrical cofferdams. The equipotential lines and streamlines obtained from the modified algorithm are shown to be smoother and more accurate than those obtained using popular commercial software (GeoStudio 24.2.0), especially when a coarse finite-element mesh is adopted.

An advanced method of interpolation of short-focus electron beams boundary trajectories using higher-order root-polynomial functions and its comparative study

The comparison of three advanced novel methods for estimating the boundary trajectory of electron beams propagated in ionized gas, including lower-order interpolation, self-connected interpolation, and extrapolation, as well as higher-order interpolation, is considered and discussed in the article. All estimations of the corresponding errors have been provided relative to numerically solving the set of algebra-differential equations that describe the boundary trajectory of the electron beam. By providing analysis, it is shown and proven that lower-order interpolation usually gives the minimal value of average error, using the method of self-connected interpolation and extrapolation gives the minimal error for estimation of focal beam parameters, and higher-order interpolation is suitable to obtain a uniform error value over the entire interpolation interval. All results of error estimation were obtained using original computer software written in Python.

Block Interface Flux Reconstruction Method with Nonlinear Scheme for Aeroacoustics Problem

Nonconforming grids, which can simplify the generation of meshes for complex shapes and reduce the number of meshes, are commonly employed in addressing multiscale aeroacoustics numerical simulation challenges. This approach allows for the simulation of sound propagation processes at a reasonable cost. By utilizing the sixth-order weighted compact nonlinear hybrid scheme (WCNHS) with a piecewise exponential mapping function (WCNHS_Pe) and incorporating ghost grid blocks, a method has been developed to reconstruct interface flux for nonsingle-scale multiblock structured grids. The variable values of the ghost grid at the grid boundary are determined through high-order optimization interpolation methods. Furthermore, different time steps can be implemented for each subdomain to accommodate corresponding grid scales. To address coupling across multiple grid sizes, an interface flux reconstruction method is applied to numerically simulate various scenarios, including Sod shock tube problem, the Shu–Osher problem, Gaussian pulse propagation on two-dimensional nonconforming grids, sound propagation on discontinuous interfaces, sound radiation from three cylinders, sound generated by weak shock–vortex interaction, Gaussian pulse propagation on three-dimensional nonconforming grids, and slat noise prediction. The results demonstrate that the high-order finite difference method established in this study exhibits superior capability in capturing discontinuities as well as higher resolution and lower dispersion on nonconforming multisize meshes, making it suitable for simulating linear and nonlinear acoustic problems.

An adaptive data-driven algorithm with machine learning for modeling high-order nonlinear beam-column finite elements

An adaptive data-driven algorithm with machine learning for modeling high-order nonlinear beam-column finite elements

On the Lagrange and Hermite Quadrature Formula

Abstract This paper builds on the error analysis method for Newton-Cotes quadrature formulas developed by D. R. Hayes and L. Rubin in 1970, which utilizes Lagrange interpolation polynomials. By adopting and extending their approach, this work derives the error estimate for Hermite interpolation quadrature. Specifically, we construct a polynomial P(x) analogous to the scaling function A(x) used by Hayes and Rubin, and prove its non-negativity over the interval. This allows us to establish a precise error formula for Hermite interpolation quadrature. The results provide a novel application of Hayes and Rubin’s methodology, offering new insights into error analysis for high-order interpolation quadrature formulas.

A novel post-processed finite element method and its convergence for partial differential equations

A novel post-processed finite element method and its convergence for partial differential equations

Topology optimization based on the improved high-order RAMP interpolation model and bending properties research for curved square honeycomb sandwich structures

Topology optimization based on the improved high-order RAMP interpolation model and bending properties research for curved square honeycomb sandwich structures

BundleMoCap++: Efficient, robust and smooth motion capture from sparse multiview videos

BundleMoCap++: Efficient, robust and smooth motion capture from sparse multiview videos

Mechanical-electric-magnetic-thermal coupled enriched finite element method for magneto-electro-elastic structures

Abstract Magneto-electro-elastic (MEE) materials possess the ability to convert mechanical, electrical, and magnetic energies, playing a critical role in smart devices. To improve the accuracy and efficiency of solving the mechanical properties of MEE structures in mechanical-electrical-magnetic-thermal (MEMT) environments, an MEMT coupled multiphysics enriched finite element method (MP-EFEM) is proposed. Based on the fundamental equations and boundary conditions of MEE materials, the interpolation coverage function is introduced into the MEMT coupled finite element method (FEM) to construct higher-order approximate interpolation displacement shape functions, electric potential shape functions, and magnetic potential shape functions. Combined with the variational principle, MP-EFEM is proposed, and the governing equations of MP-EFEM are derived. Numerical examples validate the accuracy and high efficiency of MP-EFEM in solving the mechanical properties of MEE structures in MEMT environments. When compared to the MEMT coupled FEM (MEMT-FEM), the results show that this method offers higher accuracy and efficiency. Therefore, MP-EFEM can effectively analyze the mechanical properties of MEE structures under multiphysics coupling, providing a new method for the design and development of smart devices.

Gaussian smoothed particle hydrodynamics: A high-order meshfree particle method

Gaussian smoothed particle hydrodynamics: A high-order meshfree particle method

Stepping into high-order interpolation in space

Stepping into high-order interpolation in space

Numerical investigation of the interaction between an interface and a decaying Lamb–Oseen vortex

The present study investigates the dynamics of the interface in the presence of a decaying Lamb–Oseen vortex, and four distinct wave patterns are observed: non-breaking waves with small periodic oscillations, plunging breakers, depression breakers, and gravity–capillary waves. The deformation of the interface is induced by a two-dimensional Lamb–Oseen vortex, and the study examines the influence of vortex strength and surface tension on the resulting flow. The wave dynamics are characterized as a function of the Reynolds and Weber numbers, and a phase diagram is presented in terms of (Re, We) to distinguish the different wave patterns. To ensure accurate reconstruction of the interface, the numerical methods used in this study feature a mass and momentum consistent advection method, high-order interpolation schemes, and a block-structured adaptive mesh refinement strategy. The study presents the characteristics of the air cavity entrained at the moment of wave impact for each wave pattern. Furthermore, the results provide insight into the nature of bubble entrainment within a vortex and reveals the bubble entrainment process via a breakup cascade. Meanwhile, it is also shown that the entrainment of bubble results in significant vortex distortion. Overall, this research contributes to enhance our understanding of wave dynamics and the intricate interaction between vortices and interfaces.

Micromechanical analysis of fiber-reinforced ceramic matrix composites by a geometrically nonlinear hierarchical quadrature element model

Micromechanical analysis of fiber-reinforced ceramic matrix composites by a geometrically nonlinear hierarchical quadrature element model

Splines finite element solver for one-dimensional time-dependent Maxwell's equations via Fourier Transform Discretization

In this article, we solve the time-dependent Maxwell coupled equations in their one-dimensional version relatively to space-variable. We effectuate a variable reduction via Fourier transform to make the time variable as a frequency parameter easy and quickly to manage. A Galerkin variational method based on higher-order spline interpolations is used to approximate the solution relatively to the spacial variable. So, the state of existence of the solution, its uniqueness, and its regularity are studied and proved, and the study is also provided by an error estimate and the order of convergence of the proposed method. Also, we use the critical Nyquist frequency to calculate numerically the solution of the Inverse Fourier Transform(IFT); and for all numerical computations, we consider several quadrature methods. Finally, we give some experiments to illustrate the success of such an approach.

Analysis of nonlinear problems using the Dual Mesh Control Domain Method with arbitrary meshes

Analysis of nonlinear problems using the Dual Mesh Control Domain Method with arbitrary meshes

Motion state factor driven for doubly-curved shallow shell deformation reconstruction

Motion state factor driven for doubly-curved shallow shell deformation reconstruction