Non-Parametric Approximations for Anisotropy Estimation in Two-dimensional Differentiable Gaussian Random Fields

Spatially referenced data often have autocovariance functions with elliptical isolevel contours, a property known as geometric anisotropy. The anisotropy parameters include the tilt of the ellipse (orientation angle) with respect to a reference axis and the aspect ratio of the principal correlation lengths. Since these parameters are unknown a priori, sample estimates are needed to define suitable spatial models for the interpolation of incomplete data. The distribution of the anisotropy statistics is determined by a non-Gaussian sampling joint probability density. By means of analytical calculations, we derive an explicit expression for the joint probability density function of the anisotropy statistics for Gaussian, stationary and differentiable random fields. Based on this expression, we obtain an approximate joint density which we use to formulate a statistical test for isotropy. The approximate joint density is independent of the autocovariance function and provides conservative probability and confidence regions for the anisotropy parameters. We validate the theoretical analysis by means of simulations using synthetic data, and we illustrate the detection of anisotropy changes with a case study involving background radiation exposure data. The approximate joint density provides (i) a stand-alone approximate estimate of the anisotropy statistics distribution (ii) informed initial values for maximum likelihood estimation, and (iii) a useful prior for Bayesian anisotropy inference.Comment: 39 pages; 8 figure

Post-processing Polygonal Voxel Data from Numerical Simulation

Many applications dealing with geometry acquisition and processing produce polygonal meshes that carry artifacts like discretization noise. While there are many approaches to remove the artifacts by smoothing or filtering the mesh, they are not tailored to any specific application subject to·certain restrictive objectives. We show how to incorporate smoothing schemes based on the general Laplacian approximation to satsify all those objectives at the same time for the results of flow simulation in the application field of car manufacturing. In the presented application setting the major restrictions come from the bounding volume of the flow simulation, the so-called installation space. In particular, clean mesh regions (without noise) should not be smoothed while at the same time the installation space must not be violated by the smoothing of the noisy mesh regions. Additionally, aliasing effects at the boundary between clean and noisy mesh regions must be prevented. To address the fact that the meshes come from flow simulation, the presented method is versatile enough to preserve their exact volume and to apply anisotropic filters using the flow information. Although the paper focuses on the results of a specific application, most of its findings can be transferred to different settings as well

Post-processing Polygonal Voxel Data from Numerical Simulation

Many applications dealing with geometry acquisition and processing produce polygonal meshes that carry artifacts like discretization noise. While there are many approaches to remove the artifacts by smoothing or filtering the mesh, they are not tailored to any specific application subject to·certain restrictive objectives. We show how to incorporate smoothing schemes based on the general Laplacian approximation to satsify all those objectives at the same time for the results of flow simulation in the application field of car manufacturing. In the presented application setting the major restrictions come from the bounding volume of the flow simulation, the so-called installation space. In particular, clean mesh regions (without noise) should not be smoothed while at the same time the installation space must not be violated by the smoothing of the noisy mesh regions. Additionally, aliasing effects at the boundary between clean and noisy mesh regions must be prevented. To address the fact that the meshes come from flow simulation, the presented method is versatile enough to preserve their exact volume and to apply anisotropic filters using the flow information. Although the paper focuses on the results of a specific application, most of its findings can be transferred to different settings as well

Natural neighbor concepts in scattered data interpolation and discrete function approx- imation

The concept of natural neighbors employs the notion of distance to define local neighborhoods in discrete data. Especially when querying and accessing large scale data, it is important to limit the amount of data that has to be processed for an answer. Because of its implicit definition on distances, the natural neighbor concept is extremely well suited to provide meaningful neighborhoods in spatial data with a scattered, inhomogeneous distribution. This paper revisits some unique properties of natural neighbor based methods and summarizes important findings for their successful application to scattered data interpolation, and the computation of discrete harmonic functions

Geometric Properties of the Adaptive Delaunay Tessellation

Recently, the Adaptive Delaunay Tessellation (Adt) was introduced in the context of computational mechanics as a tool to support Voronoi-based nodal integration schemes in the finite element method. While focusing on applications in mechanical engineering, the former presentation lacked rigorous proofs for the claimed geometric properties of the Adt necessary for the computation of the nodal integration scheme. This paper gives pending proofs for the three main claims which are uniqueness of the Adt, connectedness of the Adt, and coverage of the Voronoi tiles by adjacent Adt tiles. Furthermore, this paper provides a critical assessment of the Adt for arbitrary point sets

Issues and Implementation of C 1 and C 2 Natural Neighbor Interpolation

Abstract. Smooth local coordinates have been proposed by Hiyoshi and Sugihara 2000 to improve the classical Sibson’s and Laplace coordinates. These smooth local coordinates are computed by integrating geometric quantities over weights in the power diagram. In this paper we describe how to efficiently implement the Voronoi based C 2 local coordinates. The globally C 2 interpolant that Hiyoshi and Sugihara presented in 2004 is then compared to Sibson’s and Farin’s C 1 interpolants when applied to scattered data interpolation.

Generation of Accurate Integral Surfaces in Time-Dependent Vector Fields

We present a novel approach for the direct computation of integral surfaces in general vector fields. As opposed to previous work, which we analyze in detail, our approach is based on a separation of integral surface computation into two stages: surface approximation and generation of a graphical representation. This allows us to overcome several limitations of previous techniques. We first describe an algorithm for surface integration that approximates a series of timelines using iterative refinement and computes a skeleton of the integral surface. In a second step, we generate a well-conditioned triangulation. The presented approach allows a highly accurate treatment of very large time-varying vector fields in an efficient, streaming fashion. We examine the properties of the presented methods on several example datasets and perform a numerical study of its correctness and accuracy. Finally, we examine some visualization aspects of integral surfaces

Comparative Tensor Visualisation within the Framework of Consistent Time-Stepping Schemes

Nowadays, the design of so-called consistent time-stepping schemes that basically feature a physically correct time integration, is still a state-of-the-art topic in the area of numerical mechanics. Within the proposed framework for finite elastoplasto- dynamics, the spatial as well as the time discretisation rely both on a Finite Element approach and the resulting algorithmic conservation properties have been shown to be closely related to quadrature formulas that are required for the calculation of time-integrals. Thereby, consistent integration schemes, which allow a superior numerical performance, have been developed based on the introduction of an enhanced algorithmic stress tensor, compare [MMS06]-[MMS07c]. In this contribution, the influence of this consistent stress enhancement, representing a modified time quadrature rule, is analysed for the first time based on the spatial distribution of the tensor-valued difference between the standard quadrature rule, relying on a specific evaluation of the well-known continuum stresses, and the favoured nonstandard quadrature rule, involving the mentioned enhanced algorithmic stresses. This comparative analysis is carried out using several visualisation tools tailored to set apart spatial and temporal patterns that allow to deduce the influence of both step size and material constants on the stress enhancement. The resulting visualisations indeed confirm the physical intuition by pointing out locations where interesting changes happen in the data

Comparative Tensor Visualisation within the Framework of Consistent Time-Stepping Schemes

Nowadays, the design of so-called consistent time-stepping schemes that basically feature a physically correct time integration, is still a state-of-the-art topic in the area of numerical mechanics. Within the proposed framework for finite elastoplasto- dynamics, the spatial as well as the time discretisation rely both on a Finite Element approach and the resulting algorithmic conservation properties have been shown to be closely related to quadrature formulas that are required for the calculation of time-integrals. Thereby, consistent integration schemes, which allow a superior numerical performance, have been developed based on the introduction of an enhanced algorithmic stress tensor, compare [MMS06]-[MMS07c]. In this contribution, the influence of this consistent stress enhancement, representing a modified time quadrature rule, is analysed for the first time based on the spatial distribution of the tensor-valued difference between the standard quadrature rule, relying on a specific evaluation of the well-known continuum stresses, and the favoured nonstandard quadrature rule, involving the mentioned enhanced algorithmic stresses. This comparative analysis is carried out using several visualisation tools tailored to set apart spatial and temporal patterns that allow to deduce the influence of both step size and material constants on the stress enhancement. The resulting visualisations indeed confirm the physical intuition by pointing out locations where interesting changes happen in the data