Degree Of Interpolation Research Articles

Data-driven stabilized finite element solution of advection-dominated flow problems

In this article, we address the solution of advection-dominated flow problems by stabilized methods, by means of data-driven least-squares computed stabilized coefficients. As main methodological tool, we introduce a data-driven off-line/on-line strategy to compute them with low computational cost.We compare the errors provided by the data-driven stabilized coefficients to those provided by several previously established stabilized coefficients within the solution of advection–diffusion and Navier–Stokes flows, on structured and un-structured grids, with Lagrange Finite Elements up to third degree of interpolation. We obtain substantial error improvements for high-order finite element interpolation.We conclude that the data-driven procedure is a rewarding procedure, worth to be applied to general stabilized solutions of general flow problems.

An objective FE-formulation for Cosserat rods based on the spherical Bézier interpolation

A generalization of the spherical linear interpolation (or slerp) for the finite rotations to the case of more than two control variables on SO(3) is introduced to design an objective FE-formulation for the non-linear space Cosserat rod model. The interpolation uses the De Casteljau’s algorithm.In this way, the same interpolation degree can be used for the placement of the centroid curve and for the finite rotation of the cross-section. A recursive formula is obtained for the interpolation of the rotations. Similar recursive formulas are derived for the spin and the curvature vector, leading to a generalization of the Bézier basis functions on the manifold SO(3).The rod formulation so obtained is invariant under a rigid rotation (objective), in the sense that the patch-test with respect to the rigid body motion is satisfied. Furthermore, an optimal path-independence is achieved as verified by several numerical investigations.

Investigations on adapted interpolation orders for a mixed isogeometric plate formulation

AbstractIn order to overcome locking effects that especially occur for lower order finite element formulations, different methods can be employed. This can be conducted using mixed formulations or adapted approximation orders, for instance. Hence, in order to tackle shear locking that is caused by non‐matching interpolation degrees in the shear strain equation, an irreducible and a mixed Reissner‐Mindlin plate formulation with accordingly adapted conforming discretizations are derived within the scope of this contribution. In addition, non‐uniform rational B‐splines (NURBS) are employed therefore, in order to benefit from the properties and refinement strategies offered by isogeometric analysis (IGA) and to achieve more accurate results. The effect of various combinations of interpolation orders on the convergence behavior and the ability to alleviate locking is investigated for both the irreducible and the mixed isogeometric plate formulation and examined for a benchmark example. This is also supplemented by investigations on the stability of the considered variants, tested by the existence of the correct number of zero‐energy modes.

Construction and analysis of a HDG solution for the total-flux formulation of the convected Helmholtz equation

We introduce a hybridizable discontinuous Galerkin (HDG) method for the convected Helmholtz equation based on the total flux formulation, in which the vector unknown represents both diffusive and convective phenomena. This HDG method is constricted with the same interpolation degree for all the unknowns and a physically informed value for the penalization parameter is computed. A detailed analysis including local and global well-posedness as well as a super-convergence result is carried out. We then provide numerical experiments to illustrate the theoretical results.

The Pedagogical Primacy of Language in Mental Imagery: Pictorialism vs. Descriptionalism

Abstract This paper argues for the primacy of language over vision as a means of communication. Words convey information more clearly, accurately, reliably, and profoundly than images do. Images by themselves give only impressions; they do not denote, unless accompanied by some sort or level of description. Also, any visual image, whether physical or mental, unless it is eidetic, must involve some degree of interpretation, interpolation, or description for it to be capable of conveying information, having meaning, or even being intelligible. Pictorialism is the theory that mental imagery is visual, while descriptionalism holds that mental imagery is nonvisual. As an epistemology, pictorialism supports representationalism as a philosophy of art. The poverty and limitations of representational theories of art militate against pictorialism. Descriptionalism as an epistemology supports several versions of symbolist theories of art. The richness and versatility of symbolic interpretations of art works support descriptionalism. Pictorialism fails the test of consistency unless eidetic imagery exists. Any pictorialism that claims support from noneidetic imagery or invokes the photographic fallacy is in fact a crypto- or quasi-descriptionalism. Since the photographic fallacy denies eidetic imagery, it is invalid and misbegotten as a defense of pictorialism. The interplay between the richness of visual images and the precision of verbal descriptions is a key element in developing a comprehensive pedagogy of aesthetics.

Interpolation method for solving Volterra integral equations with weakly singular kernel using an advanced barycentric Lagrange formula

We presented an interpolation method for solving weakly singular Volterra integral equations of the second kind (SVK2). The method based on the barycentric Lagrange interpolation.. For the chosen nodes of the two singular kernel variables, we created two rules that confirm that the denominator of the kernel will never become zero or have an imaginary value. The product of four matrices, one of which is the square coefficients matrix of the barycentric functions, expresses both the unknown and data functions. The singular kernel is interpolated twice with to its two variables. Furthermore, the kernel is represented by five matrices: two of which are the monomial basis of the kernel's variables, and one matrix expresses the kernel's values at the two sets of nodes for the kernel's variables. Without using the collocation approach, we get the matrix of the unknown coefficients by substituting the interpolated solution into the integral equation to obtain an equivalent algebraic system. By solving the algebraic system, we get the interpolated solution. We solved five examples, two for non-singular equations and three for weakly singular equations. By adopting lower interpolation degrees, the interpolated solutions for non-singular equations are shown to be equal to the exact ones, whereas, for singular equations, they are found to strongly converge to the exact ones and are superior to those obtained by other cited methods. This ensures the proposed method's uniqueness and high accuracy.

Tensor decomposition for learning Gaussian mixtures from moments

In data processing and machine learning, an important challenge is to recover and exploit models that can represent accurately the data. We consider the problem of recovering Gaussian mixture models from datasets. We investigate symmetric tensor decomposition methods for tackling this problem, where the tensor is built from empirical moments of the data distribution. We consider identifiable tensors, which have a unique decomposition, showing that moment tensors built from spherical Gaussian mixtures have this property. We prove that symmetric tensors with interpolation degree strictly less than half their order are identifiable and we present an algorithm, based on simple linear algebra operations, to compute their decomposition. Illustrative experimentations show the impact of the tensor decomposition method for recovering Gaussian mixtures, in comparison with other state-of-the-art approaches.

Less memory and high accuracy logarithmic number system architecture for arithmetic operations

<p>Interpolation is another important procedure for logarithmic number system (LNS) addition and subtraction. As a medium of approximation, the interpolation procedure has an urgent need to be enhanced to increase the accuracy of the operation results. Previously, most of the interpolation procedures utilized the first degree interpolators with special error correction procedure which aim to eliminate additional embedded multiplications. However, the interpolation procedure for this research was elevated up to a second degree interpolation. Proper design process, investigation, and analysis were done for these interpolation configurations in positive region by standardizing the same co-transformation procedure, which is the extended range, second order co-transformation. Newton divided differences turned out to be the best interpolator for second degree implementation of LNS addition and subtraction, with the best-achieved BTFP rate of +0.4514 and reduction of memory consumption compared to the same arithmetic used in european logarithmic microprocessor (ELM) up to 51%.</p>

Stress Wave Propagation in 2D Functionally Graded Media: Optimization of Materials Distribution

In this paper, the analysis and optimization of the effect of the materials distribution on the behavior of 2D functionally graded media subjected to impacted loading has been investigated. First, it is assumed that there are two cases for distributing the components in the FG material. In the first case, the power law is considered for materials distribution, and in the second case, the volume fractional changes of the components are made by third degree interpolation. Considering the elastodynamic behavior of the FG materials under loading, the general governing equations of the wave propagation are extracted for the case of properties variation in two dimensions and then the equations are solved using the finite difference method. Finally, an optimization has been made using a single objective genetic algorithm. The results show that the materials distribution has a considerable effect of stress wave propagation in FGMs.

Numerical integrators for Lagrangian oceanography

Abstract. A common task in Lagrangian oceanography is to calculate a large number of drifter trajectories from a velocity field precalculated with an ocean model. Mathematically, this is simply numerical integration of an ordinary differential equation (ODE), for which a wide range of different methods exist. However, the discrete nature of the modelled ocean currents requires interpolation of the velocity field in both space and time, and the choice of interpolation scheme has implications for the accuracy and efficiency of the different numerical ODE methods. We investigate trajectory calculation in modelled ocean currents with 800 m, 4 km, and 20 km horizontal resolution, in combination with linear, cubic and quintic spline interpolation. We use fixed-step Runge–Kutta integrators of orders 1–4, as well as three variable-step Runge–Kutta methods (Bogacki–Shampine 3(2), Dormand–Prince 5(4) and 8(7)). Additionally, we design and test modified special-purpose variants of the three variable-step integrators, which are better able to handle discontinuous derivatives in an interpolated velocity field. Our results show that the optimal choice of ODE integrator depends on the resolution of the ocean model, the degree of interpolation, and the desired accuracy. For cubic interpolation, the commonly used Dormand–Prince 5(4) is rarely the most efficient choice. We find that in many cases, our special-purpose integrators can improve accuracy by many orders of magnitude over their standard counterparts, with no increase in computational effort. Equivalently, the special-purpose integrators can provide the same accuracy as standard methods at a reduced computational cost. The best results are seen for coarser resolutions (4 and 20 km), thus the special-purpose integrators are particularly advantageous for research using regional to global ocean models to compute large numbers of trajectories. Our results are also applicable to trajectory computations on data from atmospheric models.

Reduction of positional error in a path of a 2 DOF serial planar manipulator

Tracing a path is one of the major applications of a robotic arm. In this paper, a 2 DOF serial planar manipulator which traces the desired path is discussed. Quite often the path is defined by accuracy points lying on it. To trace these points, joints of links of the manipulator are required to be given as a coordinated inputs θ 1 and θ 2 (input angles of two links). This path is traced with a desired speed indicated by the time at which the accuracy points are reached.In open-loop control, this is achieved by giving time, voltage/current input to driving motors. This path is determined based on the torque, acceleration, velocity of the linkages and individual joints. Hence the relationship between time and joint angles is necessary. When joint angles θ 1 and θ 2 are interpolated between two accuracy points, precise coordinated motion is not possible. Some deviation from the desired path will occur. This is the positional error for tracing the path. Here, analysis has been done to find the effect of degree of interpolation on the positional error.

A new method to dispatch split particles in Particle-In-Cell codes

Particle-In-Cell codes are widely used for plasma physics simulations. It is often the case that particles within a computational cell need to be split to improve the statistics or, in the case of non-uniform meshes, to avoid the development of fictitious self-forces. Existing particle splitting methods are largely empirical and their accuracy in preserving the distribution function has not been evaluated in a quantitative way. Here we present a new method specifically designed for codes using adaptive mesh refinement. Although we point out that an exact, distribution function preserving method does exist, it requires a large number of split particles and its practical use is limited. We derive instead a method that minimizes the cost function representing the distance between the assignment function of the original particle and that of the sum of split particles. Depending on the interpolation degree and on the dimension of the problem, we provide tabulated results for the weight and position of the split particles. This strategy represents no overhead in computing time and for a large enough number of split-particles it asymptotically tends to the exact solution.

Applying PML technique with additional differential equation method in non-uniform mixed-element DGTD method

In discontinuous Galerkin time domain method (DGTD), the computational region is divided into multiple subdomains instead of using a single mesh for the multiscale structure in finite-difference time-domain (FDTD) method. Accordingly, electrically coarse structures are modeled by hexahedral elements with higher interpolation degrees, while electrically fine structures with complex shapes are spatial discretized by tetrahedral elements with lower interpolation degrees. A nonconforming mixed-element DGTD method is implemented to simulate transient electromagnetic response of general 3-D targets in this paper. Perfectly matched layer (PML) boundary condition is utilized to simulate the propagation of electromagnetic waves in infinite space. PML is formulated by auxiliary differential equation with transformed field variables E and H. The numerical flux is employed to communicate fields between adjacent subdomains including physical region and PML region. The accuracy and efficiency of the proposed method is demonstrated by several numerical experiments.

Analysis and Optimization of an Adaptive Interpolation Algorithm for the Numerical Solution of a System of Ordinary Differential Equations with Interval Parameters

We consider an adaptive interpolation algorithm for numerical integration of systems of ordinary differential equations (ODEs) with interval parameters and initial conditions. At each time moment, a piecewise polynomial function of a prescribed degree is constructed in the course of algorithm operation that interpolates the dependence of the solution on particular values of interval uncertainties with a controlled accuracy. We study the question of computational costs of the algorithm. An analytic estimate is derived for the number of operations; it depends on the algorithm parameters, in particular, the degree of interpolation and the specific features of the ODE system being integrated. Using a number of representative examples of various dimension and containing a varying number of interval uncertainties, we show that there exists an optimum value of interpolation degree from the viewpoint of computational costs.

A high-order non field-aligned approach for the discretization of strongly anisotropic diffusion operators in magnetic fusion

In this work we present a hybrid discontinuous Galerkin scheme for the solution of extremely anisotropic diffusion problems arising in magnetized plasmas for fusion applications. Unstructured meshes, non-aligned with respect to the dominant diffusion direction, allow an unequalled flexibility in discretizing geometries of any shape, but may lead to spurious numerical diffusion. Curved triangles or quadrangles are used to discretize the poloidal plane of the machine, while a structured discretization is used in the toroidal direction. The proper design of the numerical fluxes guarantees the correct convergence order at any anisotropy level. Computations performed on well-designed 2D and 3D numerical tests show that non-aligned discretizations are able to provide spurious diffusion free solutions as long as high-order interpolations are used. Introducing an explicit measure of the numerical diffusion, a careful investigation is carried out showing an exponential increase of this latest with respect to the non-alignment of the mesh with the diffusion direction, as well as an exponential decrease with the polynomial degree of interpolation. A brief assessment of the method with respect to two finite-difference schemes using non-aligned discretization, but classically used in fusion modeling, is also presented. Program summaryProgram Title: Laplace-HDG (Laplace Hybrid Discontinuous Galerkin)CPC Library link to program files:http://dx.doi.org/10.17632/c3dhycyvj8.1Licensing provisions: GPLv3Programming language: Fortran 95Nature of problem: Anisotropic Laplace problem in 2D with Dirichlet boundary conditionsSolution method: Hybrid discontinuous Galerkin scheme

Convergence rates of a Hermite generalization of Floater–Hormann interpolants

Cirillo and Hormann (2018) introduce an iterative approach to the Hermite interpolation problem, which, starting from the Lagrange interpolant, successively adds m corrections terms to interpolate the data up to the mth derivative. The method is general enough to be applied to any interpolant in linear form with a sufficiently continuous set of basis functions, but Cirillo and Hormann focus their attention on Floater–Hormann interpolants, a family of barycentric rational interpolants that are based on a particular blend of local polynomial interpolants of degree d. They show that the resulting iterative rational Hermite interpolants converge at the rate of O(h(m+1)(d+1)) as the mesh size h converges to zero for m=1,2, and their numerical results suggest that the same rate holds for m>2. In this paper we prove this convergence rate for any m≥1.

Comparison of remote sensing based approaches for mapping bathymetry of shallow, clear water rivers

Shallow rivers provide important habitat for various aquatic and terrestrial species. The bathymetry of such environments is, however, difficult to measure as devices and approaches have been traditionally developed mainly for deeper waters. This study addresses the mapping of shallow water bathymetry with high spatial resolution and accuracy by comparing three remote sensing (RS) approaches: one based on echo sounding (active RS) and two on photogrammetry (passive RS): bathymetric Structure from Motion (SfM) and optical modelling. The tests were conducted on a 500 m long and ~30 m wide reach of sand-bedded meandering river: (1) during a rising spring flood (Q = 10–15 m3/s) with medium turbidity and high water color and; (2) during autumn low discharge (Q = 4 m3/s) with low turbidity and color. Each method was used to create bathymetric models. The models were compared with high precision field measurements with a mean point spacing of 0.86 m. Echo sounding provided the most accurate (ME~−0.02 m) and precise (SDE = ± 0.08 m) bathymetric models despite the high degree of interpolation needed. However, the echo sounding-based models were spatially restricted to areas deeper than 0.2 m and no small scale bathymetric variability was captured. The quality of the bathymetric SfM was highly sensitive to flow turbidity and color and therefore depth. However, bathymetric SfM suffers less from substrate variability, turbulent flow or large stones and cobbles on the river bed than optical modelling. Color and depth did affect optical model performance, but clearly less than the bathymetric SfM. The optical model accuracy improved in autumn with lower water color and turbidity (ME = −0.05) compared to spring (ME = −0.12). Correlations between the measured and modelled depth values (r = 0.96) and the models precision (SDE = 0.09–0.11) were close to those achieved with echo sounding. Shadows caused by riparian vegetation restricted the spatial extent of the optical models.

An extended and revised Lake Suigetsu varve chronology from ∼50 to ∼10 ka BP based on detailed sediment micro-facies analyses

Lake Suigetsu (Japan) is a key site for radiocarbon (14C) calibration and palaeo-environmental reconstruction in East Asia. Here we present a description of the sediment (micro)facies, which in combination with a new approach to varve interpolation allows construction of a revised varve based chronology that extends the previous 2012 varve based chronology by ∼10 ka, back to ∼50 ka BP. Challenges in varve counting and interpolation, which were previously discussed in detail only for the Last Glacial-Interglacial Transition, are described here back to ∼50 ka BP. Furthermore, the relative merits of varve counting by μXRF scanning and by thin-section microscopy are discussed. Facies analysis reveals four facies zones, their transitions driven by both local and climatic controls. The lamination quality of the sediment is highly variable and varve interpolation reveals that in the analysed time interval, on average, only 50% of the annual cycles are represented by seasonal layers. In the remaining years seasonal layers are indistinguishable, i.e. either did not form or were not preserved. For varve interpolation an advanced version of the Varve Interpolation Program was used, which enabled the construction of the longest, purely varve dated chronology published, despite long intervals of poor lamination quality. The calculated interpolation uncertainty is +8.9% and −4.6%, which is well within expectations considering the high degree of interpolation and the length of the record.

Default mode network is mechanism for semantic color-emotional interdomain integration

Progressing toward more accurate and more efficient numerical codes for the simulation of transport and turbulence in the edge plasma of tokamaks, we propose in this work a new hybrid discontinuous Galerkin solver. Based on 2D advection–diffusion conservation equations for the ion density and the particle flux in the direction parallel to the magnetic field, the code simulates plasma transport in the poloidal section of tokamaks, including the open field lines of the Scrape-off Layer (SOL) and the closed field lines of the core region. The spatial discretization is based on a high-order hybrid DG scheme on unstructured meshes, which provides an arbitrary high-order accuracy while reducing considerably the number of coupled degrees of freedom with a local condensation process. A discontinuity sensor is employed to identify critical elements and regularize the solution with the introduction of artificial diffusion. Based on a finite-element discretization, not constrained by a flux-aligned mesh, the code is able to describe plasma facing components of any complex shape using Bohm boundary conditions and to simulate the plasma in versatile magnetic equilibria, possibly extended up to the center. Numerical tests using a manufactured solution show appropriate convergence orders when varying independently the number of elements or the degree of interpolation. Validation is performed by benchmarking the code with the well-referenced edge transport code SOLEDGE2D (Bufferand et al., 2013, 2015 [1], [2]) in the WEST geometry. Final numerical experiments show the capacity of the code to deal with low-diffusion solutions.

Extensions of planar GC sets and syzygy matrices

The geometric characterization, introduced by Chung and Yao, identifies node sets for total degree interpolation such that the Lagrange fundamental polynomials are products of linear factors. Sets satisfying the geometric characterization are usually called GC sets. Gasca and Maeztu conjectured that planar GC sets of degree n contain n + 1 collinear points. It has been shown that the conjecture holds for degrees not greater than 5 but it is still unsolved for general degree. One promising approach consists of studying the syzygies of the ideal of polynomials vanishing at the nodes. In order to describe syzygy matrices of GC sets, we analyze the extension of a GC set of degree n to a GC set of degree n + 1, by adding a n + 2 nodes on a line.