Smoothness Indicator Research Articles

Development of a Modified Seventh-Order WENO Scheme with New Nonlinear Weights

The aim of this paper is the presentation of a new modified weighted essentially non-oscillatory scheme with a seventh-order of accuracy, and, a novel nonlinear weighting method. Local smoothness indicators of this scheme constructed using Lagrange’s interpolation polynomial. We developed a new high-order global smoothness indicator, which guarantees the optimal order of accuracy of the scheme at critical points. We used the linear advection equation for verification of the order of accuracy at critical points. Later, we used a series of one- and two-dimensional benchmark problems to verify the shock-capturing capabilities and improved numerical resolution of the new scheme.

An order-adaptive compact approximation Taylor method for systems of conservation laws

We present a new family of high-order shock-capturing finite difference numerical methods for systems of conservation laws. These methods, called Adaptive Compact Approximation Taylor (ACAT) schemes, use centered (2p+1)-point stencils, where p may take values in {1,2,…,P} according to a new family of smoothness indicators in the stencils. The methods are based on a combination of a robust first order scheme and the Compact Approximate Taylor (CAT) methods of order 2p-order, p=1,2,…,P so that they are first order accurate near discontinuities and have order 2p in smooth regions, where (2p+1) is the size of the biggest stencil in which large gradients are not detected. CAT methods, introduced in [3], are an extension to nonlinear problems of the Lax-Wendroff methods in which the Cauchy-Kovalesky (CK) procedure is circumvented following the strategy introduced in [22] that allows one to compute time derivatives in a recursive way using high-order centered differentiation formulas combined with Taylor expansions in time. The expression of ACAT methods for 1D and 2D systems of conservative laws is given and the performance is checked in a number of test cases for several linear and nonlinear systems of conservation laws, including Euler equations for gas dynamics.

Towards shape optimisation of fluid flows using lattice Boltzmann methods and automatic differentiation

A flexible framework for shape optimisation is presented for incompressible Newtonian fluids using lattice Boltzmann methods. It is proposed to solve optimisation problems using line search methods, with design sensitivities obtained through forward propagation automatic differentiation. The underlying fluid flow problems are modelled by homogenised lattice Boltzmann methods, wherein permeability is varied to propagate derivative information at parametrised boundaries. The parametrisation is realised by describing the geometry using smooth indicator functions that have analytically differentiable boundaries. A number of simulation results are presented using the open-source software OpenLB (Krause et al., 2017), validating the approach and evaluating its application to domain identification and drag minimisation.

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.

High-order mapped WENO methods with improved efficiency

This paper is devised to improve the efficiency of mapped WENO (WENO-M) methods. The WENO-M methods apply a mapping function on the nonlinear weights to recover the optimal accuracy at extremal points and meanwhile improve the resolution of numerical solutions at non-smooth regions. But this mapping process increases the computational costs as well. To improve the efficiency of WENO-M methods, we develop a simple mapping function which is applied on the indicators of smoothness (not on the nonlinear weights). The mapped indicators of smoothness are sufficient to obtain the optimal accuracy at extremal points. The computational costs are also deceased since the nonlinear weights are only required to be computed once in each WENO reconstruction. Finally, several numerical tests confirm that the improved WENO-M (WENO-IM) methods have the similar numerical accuracy with WENO-M methods, but less computational costs.

A nonnegative and shape‐preserving global transport model on cubed sphere using high‐order conservative collocation scheme

AbstractA numerical model for global tracer transport is implemented by using Gauss–Legendre‐point conservative collocation (GLPCC) scheme on cubed sphere. Three‐point GLPCC scheme can achieve fifth‐order accuracy in the spherical geometry. To remove the spurious oscillations in the regions with strong gradients or discontinuities, the hierarchical reconstruction (HR) is applied to the “troubled cells” identified by a smoothness indicator based on the WENO concept. To assure the positivity‐preserving property, a flux correction operation is adopted to impose the restriction on the mass reduction due to the fluxes leaving the cell edges. The proposed numerical scheme aims to remove the nonphysical oscillations while preserving the accuracy in the smooth regions and maintain the compact spatial stencil as far as possible by making full use of degrees of freedom. The proposed schemes are evaluated by simulating the widely used benchmark tests for advection equation in one dimension and multidimensions of spherical geometry. Numerical results show that the fifth‐order accuracy is well preserved for smooth problems, while numerical oscillations can be effectively removed and the nonnegativity of the solutions is strictly guaranteed. It is very promising to develop the practical numerical model for tracer transport computation in atmospheric general circulation models using the proposed numerical framework.

A third-order WENO scheme based on exponential polynomials for Hamilton-Jacobi equations

In this study, we provide a novel third-order weighted essentially non-oscillatory (WENO) method to solve Hamilton-Jacobi equations. The key idea is to incorporate exponential polynomials to construct numerical fluxes and smoothness indicators. First, the new smoothness indicators are designed by using the finite difference operator annihilating exponential polynomials such that singular regions can be distinguished from smooth regions more efficiently. Moreover, to construct numerical flux, we employ an interpolation method based on exponential polynomials which yields improved results around steep gradients. The proposed scheme retains the optimal order of accuracy (i.e., three) in smooth areas, even near the critical points. To illustrate the ability of the new scheme, some numerical results are provided along with comparisons with other WENO schemes.

Technical and economic aspects of the use of soft starters for asynchronous electric drives

The paper presents data on the results of computer-aided mathematical modeling and experimental studies of transient processes accompanying the start of asynchronous electric drives in order to search for the methods that enable the adjustment of a soft starter to achieve minimum oscillation amplitude of the electric motor dynamic momentum in terms of the current characteristic. The indicators of smoothness, applicable to some types of electric drives of turbine mechanisms, are revealed. It is found that the correlation between the maximum value of the starting current of the electric motor and the maximum amplitude of the momentum fluctuation, as well as between the starting time and the maximum amplitude of the momentum fluctuation, is insignificant. The most informative indicator, providing the function of adaptive self-tuning of the soft starter and maximum technical and economic efficiency of its use, is the oscillation of the dynamic momentum during the start-up. It is established that the amplitude of the dynamic momentum fluctuation can be judged by the displacement of the motor housing during start-up, which will require the modernization of the electric drive. The necessity of additionally calculating the implicit components of costs in terms of the costs of entrepreneurial risk when assessing the economic effect from the modernization of asynchronous electric drives using a soft starter is substantiated.

Improved weighted NND scheme for shock-capturing

Aimed at decreasing the numerical dissipation of weighted non-oscillatory and non-free-parameter dissipation (WNND) scheme, we present an improved counterpart for shock-capturing. The new algorithm is based on the framework of Z-type weighting procedure with new local and global smoothness indicators. The performances of the proposed scheme are evaluated on several numerical tests governed by one-dimensional Euler equations. Numerical results indicate that the improved WNND scheme has advantages over the original WNND and third-order WENO-JS and WENO-Z schemes.

Very high order WENO schemes using efficient smoothness indicators

Very high order weighted essentially non-oscillatory (WENO) schemes can obtain more accurate results than those relatively low order WENO schemes for many problems. However, as the order increases, the candidate sub-stencils of WENO schemes will contain more node points. Then, the classical smoothness indicators become very complex and need much more floating point operations. Although they could be written in a more succinct form which is a sum of perfect squares, only a little improvement of the computational efficiency is obtained. Furthermore, for the power parameter p in the nonlinear weights of the previous (2r-1)th order WENO schemes, only the range of choices is given and the optimal value could be problem dependent. In this article, another class of very high order WENO schemes is constructed by using more efficient smoothness indicators. Compared with very high order WENO schemes using classical smoothness indicators, the new ones are more time efficient, and less error prone for programming. Furthermore, they have the same approximate dispersion relation (ADR) as their underlying linear schemes. And a more important advantage is that they have both good behavior near critical points and good stability near discontinuities. For Euler equations, a predetermined order reduction technique including two inequalities is proposed to deal with the problems caused by interactions between discontinuities of different characteristic fields. One inequality is constructed using this kind of smoothness indicators of pressure to suppress the non-physical oscillations. And the other is proposed according to the governing equation of the pressure to prevent the emergence of negative pressure. Several test problems are presented to demonstrate the good properties of the proposed WENO schemes.

Human–Robot Cooperation Control Based on Trajectory Deformation Algorithm for a Lower Limb Rehabilitation Robot

Although many studies have certified the advantages of human–robot cooperation control with admittance model (AM), robot compliance, and movement smoothness need to be further improved. In this article, a trajectory deformation algorithm (TDA) is developed as a high-level trajectory planner, which can plan subject's desired trajectory based on interaction force during physical human–robot interaction (pHRI). A low-level proportional-derivative (PD) position controller is selected to ensure the lower limb rehabilitation robot can track the desired trajectory. Then, the validity of TDA is verified through simulation and experiment studies. The energy per unit distance (EPUD) and dimensionless squared jerk (DSJ) are chosen as indicators of robot compliance and movement smoothness, respectively. The experimental results demonstrated that both the EPUD and the DSJ values using TDA are smaller than that using the AM, indicating the TDA can improve robot compliance and movement smoothness. Therefore, it may have great potential in fields involving pHRI, such as robot-aided rehabilitation.

On a Retarded Nonlocal Ordinary Differential System with Discrete Diffusion Modeling Life Tables

In this paper, we consider a system of ordinary differential equations with non-local discrete diffusion and finite delay and with either a finite or an infinite number of equations. We prove several properties of solutions such as comparison, stability and symmetry. We create a numerical simulation showing that this model can be appropriate to model dynamical life tables in actuarial or demographic sciences. In this way, some indicators of goodness and smoothness are improved when comparing with classical techniques.

Lax-Wendroff Approximate Taylor Methods with Fast and Optimized Weighted Essentially Non-oscillatory Reconstructions

The goal of this work is to introduce new families of shock-capturing high-order numerical methods for systems of conservation laws that combine Fast WENO (FWENO) and Optimal WENO (OWENO) reconstructions with Approximate Taylor methods for the time discretization. FWENO reconstructions are based on smoothness indicators that require a lower number of calculations than the standard ones. OWENO reconstructions are based on a definition of the nonlinear weights that allows one to unconditionally attain the optimal order of accuracy regardless of the order of critical points. Approximate Taylor methods update the numerical solutions by using a Taylor expansion in time in which, instead of using the Cauchy–Kovalevskaya procedure, the time derivatives are computed by combining spatial and temporal numerical differentiation with Taylor expansions in a recursive way. These new methods are compared between them and against methods based on standard WENO implementations and/or SSP-RK time discretization. A number of test cases are considered ranging from scalar linear 1d problems to nonlinear systems of conservation laws in 2d.

A New Smoothness Indicator of Adaptive Order Weighted Essentially Non-Oscillatory Scheme for Hyperbolic Conservation Laws

Adaptive order weighted essentially non-oscillatory scheme (WENO-AO(5,3)) has increased the computational cost and complexity of the classic fifth-order WENO scheme by introducing a complicated smoothness indicator for fifth-order linear reconstruction. This smoothness indicator is based on convex combination of three third-order linear reconstructions and fifth-order linear reconstruction. Therefore, this paper proposes a new simple smoothness indicator for fifth-order linear reconstruction. The devised smoothness indicator linearly combines the existing smoothness indicators of third-order linear reconstructions, which reduces the complexity of that of WENO-AO(5,3) scheme. Then WENO-AO(5,3) scheme is modified to WENO-O scheme with new and simple formulation. Numerical experiments in 1-D and 2-D were run to demonstrate the accuracy and efficacy of the proposed scheme in which WENO-O scheme was compared with original WENO-AO(5,3) scheme along with WENO-AO-N, WENO-Z, and WENO-JS schemes. The results reveal that the proposed WENO-O scheme is not only comparable to the original scheme in terms of accuracy and efficacy but also decreases its computational cost and complexity.

A note on WENO-Z scheme

In this paper we recover a latent advantage of WENO-Z schemes. Taking the fifth-order WENO-Z scheme for instance, we realize that the scheme can be regarded as a nonlinear combination of a five-cell stencil and three three-cell stencils. The five-cell stencil is allotted a global higher-order indicator of smoothness than three-cell stencils. Then the five-cell stencil dominates the nonlinear combination and ensures the optimal accuracy in the smooth regions even at extremal points. In non-smooth regions, the three-cell stencils dominate the combination and compress the nonphysical oscillations. As the adaptive order WENO schemes which release the requirement of linear optimal weights, we will show that there is no requirement of linear optimal weights for the WENO-Z schemes as well, and even it is unnecessary to require the sum of linear optimal weights to be one.

A new ninth‐order central Hermite weighted essentially nonoscillatory scheme for hyperbolic conservation laws

AbstractIn this article, we propose a new ninth‐order central Hermite weighted essentially nonoscillatory (HWENO) scheme, for solving hyperbolic conservation laws. The new scheme consists of the following: ninth‐order reconstruction using only five points stencil; to calculate the linear weights we used the central WENO (CWENO) technique and for nonlinear weights we used a new weighting technique. The numerical solution is advanced in time by using the ninth‐order linear strong‐stability‐preserving Runge–Kutta (ℓSSPRK) scheme and for computing the numerical flux, we used the central‐upwind flux which is efficient, simple and can be used for nonconex fluxes problems. The resulting scheme is ninth order in both smooth regions and at critical points with very small numerical dissipation near discontinuities, this is due to using new smoothness indicators. Several numerical examples are presented for one‐ and two‐dimensional problems to confirm that the new scheme is superior to the other high‐order WENO schemes.

Discovery of topological metamaterials by symmetry relaxation and smooth topological indicators

Robustness against small perturbations is a crucial feature of topological properties. This robustness is both a source of theoretical interest and a drive for technological applications, but presents a challenge when looking for new topological systems: Small perturbations cannot be used to identify the global direction of change in the topological indices. Here, we overcome this limitation by breaking the symmetries protecting the topology. The introduction of symmetry-breaking terms causes the topological indices to become non-quantized variables, which are amenable to efficient design algorithms based on gradient methods. We demonstrate this capability by designing discrete and continuous phononic systems realizing conventional and higher-order topological insulators.

A new perspective on flux and slope limiting in discontinuous Galerkin methods for hyperbolic conservation laws

In this work, we discuss and develop multidimensional limiting techniques for discontinuous Galerkin (DG) discretizations of scalar hyperbolic problems. To ensure that each cell average satisfies a local discrete maximum principle (DMP), we impose inequality constraints on the local Lax–Friedrichs fluxes of a piecewise-linear (P1) approximation. Since the piecewise-constant (P0) version corresponds to a property-preserving low-order finite volume method, the validity of DMP conditions can always be enforced using slope and/or flux limiters. We show that the (currently rather uncommon) use of direct flux limiting makes it possible to construct more accurate DMP-satisfying approximations in which a weak form of slope limiting is used to prevent unbounded growth of solution gradients. After presenting two flux limiters that ensure the validity of local DMPs for cell averages, we discuss the design of slope limiters based on different kinds of inequality constraints. In particular, we derive new limiting procedures based on flux constraints and constraints for directional derivatives. The latter approach makes it possible to preserve directional monotonicity in applications to problems that require different treatment of different space directions. At the flux limiting stage, the anisotropy of the problem at hand can be taken into account by using a customized definition of local bounds for the DMP constraints. At the slope limiting stage, we adjust the magnitude of individual directional derivatives using low-order reconstructions from cell averages to define the bounds. In this way, we avoid unnecessary limiting of well-resolved derivatives at smooth peaks and in internal/boundary layers. The properties of selected algorithms are explored in numerical studies for DG-P1 discretizations of two-dimensional test problems. In the context of hp-adaptive DG methods, the new limiting procedures can be used in P1 subcells of macroelements marked as ‘troubled’ by a smoothness indicator.

Collision-free and smooth joint motion planning for six-axis industrial robots by redundancy optimization

Abstract Planning collision-free and smooth joint motion is crucial in robotic applications, such as welding, milling, and laser cutting. Kinematic redundancy exists when a six-axis industrial robot performs five-dimensional tasks, and there are infinite joint configurations for a six-axis industrial robot to realize a cutter location data of the tool path. The robot joint motion can be optimized by taking advantage of the kinematic redundancy, and the collision-free joint motion with minimum joint movement is determined as the optimal. However, most existing redundancy optimization methods do not fully exploit the redundancy of the six-axis industrial robots when they conduct five-dimensional tasks. In this paper, we present an optimization method to solve the problem of inverse kinematics for a six-axis industrial robot to synthesize the joint motion that follows a given tool path, while achieving smoothness and collision-free manipulation. B-spline is applied for the joint configuration interpolation, and the sum of the squares of the first, second, and third derivatives of the B-spline curves are adopted as the smoothness indicators. Besides, the oriented bounding boxes are adopted to simplify the shape of the robot joints, robot links, spindle unit, and fixtures to facilitate collision detections. Dijkstra's shortest path technique and Differential Evolution algorithm are combined to find the optimal joint motion efficiently and avoid getting into a local optimal solution. The proposed algorithm is validated by simulations on two six-axis industrial robots conducting five-axis flank milling tasks respectively.

A high order positivity-preserving conservative WENO remapping method on 2D quadrilateral meshes

In this paper, we present high order accurate positivity-preserving conservative remapping algorithm which is based on the multi-resolution weighted essentially non-oscillatory (WENO) reconstruction. We use a third-order method on 2D quadrilateral meshes as an example to present the algorithm. The method can effectively remap the physical variables after mesh rezoning in the ALE algorithm. By calculating the intersection exactly, this method does not require the same connectivity between the old and new meshes. By reconstructing a quadratic polynomial and a zero-order polynomial for each cell in a two-dimensional domain, this method assigns nonlinear weights for these polynomials accordingly after calculating the smoothness indicators over the integration area, yielding third order accuracy without numerical oscillations. After calculating the overlaps between the old and new meshes and integrating the polynomials over the intersections, the remapping is completed. Furthermore, to ensure the positivity-preserving property of relevant physical variables in hydrodynamics numerical simulation such as density and internal energy, a simple and efficient positive-preserving limiter is adopted to slightly modify the reconstructed polynomials, which can maintain the original order of accuracy and conservation. The algorithm can be extended to higher order accuracy using higher order reconstruction and higher order integration formula over the intersection areas. A series of numerical experiments are performed to test the properties of the multi-resolution WENO conservative remapping algorithm. Numerical results show that the algorithm is conservative, positivity-preserving, highly efficient, third-order accurate for smooth problems, and essentially non-oscillatory for discontinuous problems.