High-order Interpolation Research Articles

An hp-version adaptive finite element algorithm for eigensolutions of moderately thick circular cylindrical shells via error homogenisation and higher-order interpolation

PurposeThis study presents a novel hp-version adaptive finite element method (FEM) to investigate the high-precision eigensolutions of the free vibration of moderately thick circular cylindrical shells, involving the issues of variable geometrical factors, such as the thickness, circumferential wave number, radius and length.Design/methodology/approachAn hp-version adaptive finite element (FE) algorithm is proposed for determining the eigensolutions of the free vibration of moderately thick circular cylindrical shells via error homogenisation and higher-order interpolation. This algorithm first develops the established h-version mesh refinement method for detecting the non-uniform distributed optimised meshes, where the error estimation and element subdivision approaches based on the superconvergent patch recovery displacement method are introduced to obtain high-precision solutions. The errors in the vibration mode solutions in the global space domain are homogenised and approximately the same. Subsequently, on the refined meshes, the algorithm uses higher-order shape functions for the interpolation of trial displacement functions to reduce the errors quickly, until the solution meets a pre-specified error tolerance condition. In this algorithm, the non-uniform mesh generation and higher-order interpolation of shape functions are suitable for addressing the problem of complex frequencies and modes caused by variable structural geometries.FindingsNumerical results are presented for moderately thick circular cylindrical shells with different geometrical factors (circumferential wave number, thickness-to-radius ratio, thickness-to-length ratio) to demonstrate the effectiveness, accuracy and reliability of the proposed method. The hp-version refinement uses fewer optimised meshes than h-version mesh refinement, and only one-step interpolation of the higher-order shape function yields the eigensolutions satisfying the accuracy requirement.Originality/valueThe proposed combination of methodologies provides a complete hp-version adaptive FEM for analysing the free vibration of moderately thick circular cylindrical shells. This algorithm can be extended to general eigenproblems and geometric forms of structures to solve for the frequency and mode quickly and efficiently.

Nonequilibrium molecular dynamics method based on coarse-graining formalism: Application to a nonuniform temperature field system.

A nonequilibrium molecular dynamics method is proposed to produce nonequilibrium states flexibly. In this method, virtual points are set in a simulation box, and coarse-grained physical quantities at these points are constrained using Gauss's principle of least constraint. The coarse-grained physical quantities are evaluated by averaging microscopic quantities with an appropriate weight. To obtain the weight to evaluate the coarse-grained physical quantities, a shape function matrix is initially constructed from the particle configuration. This matrix expresses an interpolation of the physical quantities at particle positions from the coarse-grained quantities at the virtual points. Then, a matrix form of the weight is calculated as the Moore-Penrose pseudoinverse matrix for the shape function matrix. This method is applied to constrain the coarse-grained kinetic energy and produce a nonuniform temperature field in the system. The temperature profile at a nonequilibrium steady state depends on the method for constructing the shape function matrix. In particular, a local temperature coincides with the coarse-grained temperature when the shape function matrix is constructed based on a higher-order interpolation.

A Block Matrix-Based Precise Integration Algorithm for Solving Nonhomogeneous Dynamic Response

Abstract This article puts forward a computationally efficient block matrix based precise integration algorithm for solving vibration response subjected to time-variable excitation and nonlinearity, especially for nonhomogeneous dynamic response solution with large-scale degrees-of-freedom. In detail, the nonlinear parts and time-varying inputs of a dynamic system are separated from the original dynamic equations and then simulated within a computing time-step by employing a piecewise interpolation function. A novel closed-form iteration formula is presented in conjunction with the block matrix strategy and modified increment-dimensional precise integration technique. Interestingly, the presented approach is essentially a high-accuracy and parallel algorithm, which exhibits a high prediction accuracy without the limitation of matrix inversion, higher-order derivative, periodicity requirement nor cycle oscillation and instability of high-order interpolation. The feasibility and advantage of the proposed method are verified with two numerical examples.

Performance of fast fluid dynamics with a semi-Lagrangian scheme and an implicit upwind scheme in simulating indoor/outdoor airflow

Computational fluid dynamics can be time consuming for predicting indoor airflows and pollutant transport in large-scale problems or emergency management. Fast fluid dynamics (FFD) is able to accomplish efficient and accurate simulation of indoor/outdoor airflow. FFD solves the advection term of the Navier–Stokes equations either by a semi-Lagrangian (SL) scheme or an implicit upwind (IU) scheme. The SL scheme can be highly efficient, but its first-order version is not conservative and introduces significant numerical diffusion. To improve its accuracy, a high-order temporal and interpolation scheme that not only reduces dissipation and dispersion errors but also guarantees the convergence speed should be applied. Otherwise, an IU scheme instead could be used to solve the advection term. The IU scheme is conservative and introduces minor numerical diffusion, but it may increase the computation time. Therefore, this study investigated the performance of FFD with SL scheme using high-order temporal and interpolation schemes and that with IU scheme. The comparisons used experimental data of two indoor airflows and one outdoor airflow. The results showed that FFD with IU scheme was overall more accurate than FFD with SL scheme. In simulating indoor airflow, both methods were robust and the predictions were independent of time step sizes if the Courant number was less than or equal to one. In simulating the outdoor airflow, the FFD with SL scheme performed better than the FFD with IU scheme for large time step sizes. The FFD with IU scheme consumed 44%–61% computing time of the FFD with SL scheme.

Parametric interpolation framework for scalar conservation laws

In this paper we present a novel framework for obtaining high-order numerical methods for scalar conservation laws in one-space dimension for both the homogeneous and non-homogeneous cases (or balance laws). The numerical schemes for these two settings are somewhat different in the presence of shocks, however at their core they both rely heavily on the solution curve being represented parametrically. By utilizing high-order parametric interpolation techniques we succeed to obtain fifth order accuracy (in space) everywhere in the computation domain, including the shock location itself. In the presence of source terms a slight modification is required, yet the spatial order is maintained but with an additional temporal error appearing. We provide a detailed discussion of a sample scheme for non-homogeneous problems which obtains fifth order in space and fourth order in time even in the presence of shocks.

On the accuracy, robustness, and performance of high order interpolation schemes for the overset method on unstructured grids

AbstractA comprehensive study on interpolation schemes used in overset grid techniques is here presented. Based on a literature review, numerous schemes are implemented, and their robustness, accuracy, and performance are assessed. Two code verification exercises are performed for this purpose: a 2D analytical solution of a laminar Poiseuille steady flow; and an intricate manufactured solution of a turbulent flow case, characteristic of a boundary layer flow combined with an unsteady separation bubble. For both cases, the influence of grid layouts, grid refinement, and time‐step is investigated. Local and global errors, convergence orders, and mass imbalance are quantified. In terms of computational performance, strong scalability, cpu timings, load imbalancing, and domain connectivity information (DCI) overhead are reported. The effect of the overset‐grid interpolation schemes on the numerical performance of the solver, that is, number of nonlinear iterations, is also scrutinized. The results show that, for a second order finite volume code, once diffusion is dominant (low Reynolds number), interpolation schemes higher than second order, for example, least squares of degree 2, are needed not to increase the total discretization errors. For convection dominated flows (high Reynolds numbers), the results suggest that second order schemes, for example, nearest cell gradient, are sufficient to prevent overset grid schemes to taint the underlying discretization errors. In terms of performance, by single‐process and parallel communication optimization, the total overset‐grid overhead (with DCI done externally to the CFD code) may be less than 4% of the total run time for second‐order schemes and 8% for third‐order ones, therefore empowering higher‐order schemes and more accurate solutions.

A high-order weighted compact nonlinear scheme for compressible flows

This paper presents a weighted compact nonlinear scheme (WCNS) with high-order accuracy. The WCNS has the flexibility to select a numerical flux and is easily applied to general coordinates. In this study, the specific substencils of the original targeted essentially non-oscillatory (TENO) scheme, which is effective against discontinuities with a high-order interpolation, are combined with WCNS. The accuracy of the interpolation can be easily extended, and the expressions of the fifth, seventh, ninth, and eleventh-order accuracies are provided in this paper. For two types of weight coefficients (WCNS-JS and WCNS-T), the present scheme is investigated using a various benchmark-test problems including strong shock and high-frequency waves. Within substencils, the WCNS-JS can capture strong discontinuities; however, it cannot resolve small-scale fluctuations because of the dissipation effect. On the contrary, the WCNS-T with a TENO weighing coefficient can capture strong discontinuities and high-frequency waves because high-order accuracy works effectively on the premise that a suitable grid resolution is satisfied.

Finite variation sensitivity analysis for discrete topology optimization of continuum structures

This paper proposes two novel approaches to perform more suitable sensitivity analyses for discrete topology optimization methods. To properly support them, we introduce a more formal description of the Bi-directional Evolutionary Structural Optimization (BESO) method, in which the sensitivity analysis is based on finite variations of the objective function. The proposed approaches are compared to a naive strategy; to the conventional strategy, referred to as First-Order Continuous Interpolation (FOCI) approach; and to a strategy previously developed by other researchers, referred to as High-Order Continuous Interpolation (HOCI) approach. The novel Woodbury approach provides exact sensitivity values and is a better alternative to HOCI. Although HOCI and Woodbury approaches may be computationally prohibitive, they provide useful expressions for a better understanding of the problem. The novel Conjugate Gradient Method (CGM) approach provides sensitivity values with arbitrary precision and is computationally viable for a small number of steps. The CGM approach is a better alternative to FOCI since, for appropriate initial conditions, it is always more accurate than the conventional strategy. The standard compliance minimization problem with volume constraint is considered to illustrate the methodology. Numerical examples are presented together with a broad discussion about BESO-type methods.

Development of adaptive multi-resolution MPS method for multiphase flow simulation

In this study, an adaptive multi-resolution moving particle semi-implicit (MPS) method is developed aiming for efficient and accurate multiphase flow simulation. The least-square MPS(LSMPS) scheme is adopted for high-order accurate operator discretization. An adaptive multi-resolution framework is developed to realize adaptive particle splitting and merging near the interface of multiphase flow. LSMPS also provides high-order interpolation of physical quantities such as velocity and pressure during particle splitting and merging. Accuracy and efficiency of the proposed method is demonstrated through convergence study and numerical simulation of static two-fluid pool, Rayleigh–Taylor instability and single rising bubble.

A three-dimensional multi-grid staggered finite-difference time-domain method for underwater target acoustic scattering

Solving the linearized equations of continuum mechanics to model the acoustic scattering of targets close to the seafloor is a challenging task. Among the traditional algorithms, staggered second-order finite-difference time-domain (FDTD) schemes combine simplicity and ease of implementation and parallelization, especially on modern GPUs. They have been successfully applied in two-dimensional configurations. Their major drawback is their inherent low resolution, due to their formal low order and the stair-step representation of the interfaces between media (e.g., target/water). To avoid excessive computational costs in three-dimensional configurations, while maintaining high numerical accuracy, we propose a FDTD multi-grid method. The mesh is refined locally near the interfaces, while a coarse grid is employed in homogeneous regions. In the fine grids, the aforesaid FDTD is employed, whereas, in the coarse grid, higher-order staggered schemes are used to compensate for the decrease in resolution due to the larger spacing. High-order interpolation allows matching the solutions at the grid interfaces. In this presentation, the performances of the new method on classical benchmark tests are analyzed and discussed. The numerical and computational capabilities are then illustrated by applying the proposed technique to the scattering of buried objects.

A Hierarchical Grid Solver for Simulation of Flows of Complex Fluids.

Tree-based grids bring the advantage of using fast Cartesian discretizations, such as finite differences, and the flexibility and accuracy of local mesh refinement. The main challenge is how to adapt the discretization stencil near the interfaces between grid elements of different sizes, which is usually solved by local high-order geometrical interpolations. Most methods usually avoid this by limiting the mesh configuration (usually to graded quadtree/octree grids), reducing the number of cases to be treated locally. In this work, we employ a moving least squares meshless interpolation technique, allowing for more complex mesh configurations, still keeping the overall order of accuracy. This technique was implemented in the HiG-Flow code to simulate Newtonian, generalized Newtonian and viscoelastic fluids flows. Numerical tests and application to viscoelastic fluid flow simulations were performed to illustrate the flexibility and robustness of this new approach.

A roadmap for Generalized Plane Waves and their interpolation properties

This work focuses on the study of partial differential equation (PDE) based basis function for Discontinuous Galerkin methods to solve numerically wave-related boundary value problems with variable coefficients. To tackle problems with constant coefficients, wave-based methods have been widely studied in the literature: they rely on the concept of Trefftz functions, i.e. local solutions to the governing PDE, using oscillating basis functions rather than polynomial functions to represent the numerical solution. Generalized Plane Waves (GPWs) are an alternative developed to tackle problems with variable coefficients, in which case Trefftz functions are not available. In a similar way, they incorporate information on the PDE, however they are only approximate Trefftz functions since they don’t solve the governing PDE exactly, but only an approximated PDE. Considering a new set of PDEs beyond the Helmholtz equation, we propose to set a roadmap for the construction and study of local interpolation properties of GPWs. Identifying carefully the various steps of the process, we provide an algorithm to summarize the construction of these functions, and establish necessary conditions to obtain high order interpolation properties of the corresponding basis.

Discrete Kirchhoff–Love shell quadrilateral finite element designed from cubic Hermite edge curves and Coons surface patch

We present a nonlinear discrete Kirchhoff–Love four-node shell finite element that is based on the cubic Hermite edge curves and the bilinear Coons surface patch spanning the surface between them. The cubic Hermite edge curves are constructed by minimizing the bending curvature of a spatial curve connecting two adjacent nodes of the element. The G1-continuity is obtained at each node of the finite element mesh. Namely, the tangent vectors of the set of the edge curves attached to a given node of the mesh share the same tangent plane to the shell mid-surface for any configuration. To avoid the membrane locking, common in shell elements with higher-order interpolations, the assumed natural strains are adopted, solving the plate compatibility equation. The derived element has 5 degrees of freedom per node, 3 mid-surface displacements and 2 rotations of the mid-surface normal vector, which also rotate the corresponding mid-surface tangent plane. Several numerical examples illustrate its performance in linear and nonlinear tests, for both regular and distorted meshes.

A novel framework for high-speed cam curve synthesis: piecewise high-order interpolation, pointwise scaling and piecewise modulation

This study proposes a novel mathematical framework for optimal synthesis of high-speed cam curves, comprising three key aspects: piecewise high-order interpolation (PHOI), pointwise scaling and piecewise modulation. In this framework, PHOI is responsible for constructing a smooth and faithful cam curve under multiorder motion constraints (including displacement, velocity, acceleration and jerk), and pointwise scaling and piecewise modulation are further introduced for optimal tuning of nodal and internodal motion parameters. Mathematically, the combination of PHOI, pointwise scaling and/or piecewise modulation will enable a full and direct control of local/global motion parameters for enhanced high-speed performance of the synthesized cam curve. Concerning the validation of this framework, three detailed numerical examples (curve reconstruction, kinematic and dynamic optimizations) are performed. The excellent interpolation accuracy of PHOI, improved motion continuity by pointwise scaling, lowered motion peaks by pointwise scaling and piecewise modulation, and reduced dynamic error of the optimized cam curve collectively demonstrate the effectiveness and versatility of the proposed framework in data-driven reconstruction and application-oriented optimization of high-speed precision cam profiles.

Research on a 3-DOF Planar Cable-Driven Parallel Robot Based on Fuzzy Adaptive PD Computed Torque Control

Cable-driven parallel robots (CDPR) with spatial configuration are widely used while ones with plannar configuration are rarely applied. Therefore, this paper studies the modeling and control characteristics of a three degree-of-freedom cable-driven planar parallel robot with great application prospects. Aiming at the shortcomings of the existing control methods towards CDPR, a new method based on fuzzy adaptive PD computing torque control is proposed which contains a reasonable optimal distribution of cable forces. To validate the method, the tracking control simulation research and analysis are carried out for the trajectory of high-order interpolation planning, the results of which show that the control method has favorable dynamic tracking accuracy and mechanical properties. Moreover, compared with the fixed-gain computed torque control, the proposed control method can overcome the influence of model inaccuracy more effectively.

Cavitation modelling using real gas state equation: A conceptual study

The prediction of the onset of cavitation to allow investigations of the flow in off-design conditions during design is unavoidable for a variety of applications. Numerical modeling of the cavitation phenomena is however computationally challenging due to the nature of the phenomena. High density ratios, steep pressure gradients, the presence of multiple phase including phase change and hence to occurrence of real gas thermophysical behavior would cause a computational effort unmanageable during design. Today’s models are therefore greatly simplified representations of the reality. Conservation equations are usually solved using a mixture approach with flow properties being a volume fraction weighted average of two incompressible fluids thus neglecting real gas effects. The source term from the phase exchange however usually poses significant stability problems. In this article we will therefore present an approach for cavitation modeling using high-order polynomial real-gas state equations to accurately predict the properties of water at any state. Computational performance is improved by the use of structured tabulation using high order interpolation methods. The numerical results are then compared against measurement data and conventional cavitation model results.

A Two-Way Nesting Unstructured Quadrilateral Grid, Finite-Differencing, Estuarine and Coastal Ocean Model with High-Order Interpolation Schemes

The balance between the need of improving horizontal resolution in simulating local small-scale ocean processes and computational costs makes it desirable to refine model mesh locally. A three-dimensional, two-way nesting unstructured quadrilateral grid, primitive equations, finite-differencing, estuarine and coastal ocean model is developed for multi-scale modeling. Because the model grid is capable of multi-area nesting and multi-level refinement at each subdomain, the model is highly compatible with simulations involved in complex topography and studies of local small-scale ocean processes. The two-way information exchange is achieved by a virtual grid method, and its basic idea is to implement numerical integrations of variables at nesting interfaces with the support of virtual grid variables, which are interpolated or updated from actual grid variables. The model is novel for two interpolation schemes: the high-order spatial interpolation at the middle temporal level (HSIMT) parabolic interpolation scheme and HSIMT advection-equivalent interpolation scheme, and they have high-order accuracy and good consistency with the advection scheme applied to solving the tracer equations. The conservation of both volume and tracer contents is ensured via a flux correction algorithm. The two original interpolation schemes are examined in an ideal salinity advection experiment in the peak preservation skill, stability, and conservation properties. A realistic application to the Deep Waterway Project area in the Changjiang Estuary showed that the nested grid model can reproduce the hydrodynamic processes at the observed sites successfully while it failed to maintain the performance with the structured grid model in simulating the variance of salinity, for which the enforced conservation had primary responsibility. The HSIMT parabolic interpolation scheme was distinguished from other schemes for its outstanding performances in simulating salinity.

MIMC-VINS: A Versatile and Resilient Multi-IMU Multi-Camera Visual-Inertial Navigation System

As cameras and inertial sensors are becoming ubiquitous in mobile devices and robots, it holds great potential to design visual-inertial navigation systems (VINS) for efficient versatile 3-D motion tracking, which utilize any (multiple) available cameras and inertial measurement units (IMUs) and are resilient to sensor failures or measurement depletion. To this end, rather than the standard VINS paradigm using a minimal sensing suite of a single camera and IMU, in this article, we design a real-time consistent multi-IMU multi-camera (MIMC) VINS estimator that is able to seamlessly fuse multimodal information from an arbitrary number of uncalibrated cameras and IMUs. Within an efficient multi-state constraint Kalman filter framework, the proposed MIMC-VINS algorithm optimally fuses asynchronous measurements from all sensors while providing smooth, uninterrupted, and accurate 3-D motion tracking even if some sensors fail. The key idea of the proposed MIMC-VINS is to perform high-order on-manifold state interpolation to efficiently process all available visual measurements without increasing the computational burden due to estimating additional sensors’ poses at asynchronous imaging times. In order to fuse the information from multiple IMUs, we propagate a joint system consisting of all IMU states while enforcing rigid-body constraints between the IMUs during the filter update stage. Finally, we estimate online both spatiotemporal extrinsic and visual intrinsic parameters to make our system robust to errors in prior sensor calibration. The proposed system is extensively validated in both Monte Carlo simulations and real-world experiments.

Nonlinear weighting process in ghost-cell immersed boundary methods for compressible flow

Computational challenges arise for the immersed boundary method (IBM) when dealing with compressible flow, where discontinuous and smoothly varying flow regions appear near the immersed boundary. The conventional ghost-cell IBM provides inaccurate results for smoothly varying regions when a low-order interpolation is used, or it suffers from a numerical instability for the discontinuous flow when a high-order interpolation is used. In this study, a new ghost-cell approach, nonlinear-weighted IBM (NWIBM), is developed to address the abovementioned issues. Inspired by a variety of weighted nonoscillatory interpolation methods, this work combines the high- and low-order polynomials during ghost-cell value estimation to enforce proper boundary conditions in the immersed boundary. A multidimensional smoothness indicator is designed to evaluate flow discontinuities. The nonlinear weighting obtained from the smoothness indicator makes the high-order polynomial dominant and the low-order polynomial negligible in the smoothly varying region, and vice versa in the discontinuous region. The enhanced performance and applicability of the proposed method were validated through various numerical tests in compressible flow. It was demonstrated that the NWIBM provides more stable and more accurate numerical solutions compared with conventional ghost-cell approaches.

3D Airborne EM Forward Modeling Based on Time-Domain Spectral Element Method

Airborne electromagnetic (AEM) method uses aircraft as a carrier to tow EM instruments for geophysical survey. Because of its huge amount of data, the traditional AEM data inversions take one-dimensional (1D) models. However, the underground earth is very complicated, the inversions based on 1D models can frequently deliver wrong results, so that the modeling and inversion for three-dimensional (3D) models are more practical. In order to obtain precise underground electrical structures, it is important to have a highly effective and efficient 3D forward modeling algorithm as it is the basis for EM inversions. In this paper, we use time-domain spectral element (SETD) method based on Gauss-Lobatto-Chebyshev (GLC) polynomials to develop a 3D forward algorithm for modeling the time-domain AEM responses. The spectral element method combines the flexibility of finite-element method in model discretization and the high accuracy of spectral method. Starting from the Maxwell's equations in time-domain, we derive the vector Helmholtz equation for the secondary electric field. We use the high-order GLC vector interpolation functions to perform spectral expansion of the EM field and use the Galerkin weighted residual method and the backward Euler scheme to do the space- and time-discretization to the controlling equations. After integrating the equations for all elements into a large linear equations system, we solve it by the multifrontal massively parallel solver (MUMPS) direct solver and calculate the magnetic field responses by the Faraday's law. By comparing with 1D semi-analytical solutions for a layered earth model, we validate our SETD method and analyze the influence of the mesh size and the order of interpolation functions on the accuracy of our 3D forward modeling. The numerical experiments for typical models show that applying SETD method to 3D time-domain AEM forward modeling we can achieve high accuracy by either refining the mesh or increasing the order of interpolation functions.