On Hermitian interpolation of first-order data with locally generated $\mathcal{C}^1$-splines over triangular meshes

In the numerical methods of couple stress theory for elasticity, C 1 continuity on the shape functions is required, because the second-order derivatives of displacement appear in the variational equation. N. Sukumar constructed a C 1 natural neighbor interpolant by embedding Sibson’s interpolant in the Bernstein-Bezier surface representation of a cubic simplex and used it to solve the two-dimensional biharmonic problems of a circular plate under a biaxial state of stress. In this paper, all of the natural neighbors of the introduced point are determined based on the extended Delaunay triangulation which can be used to guarantee the uniqueness of the triangulation compared to traditional Delaunay triangulation, and the convex hull of the set of natural neighbors is defined to be the domain of the introduced point, so nodes outside of the domain have no influence on the introduced point. On account of the computational efficiency, the non-Sibson interpolant is calculated and introduced into a cubic Bernstein polynomials to obtain the Bernstein-Bezier basis function .According to the fact that the vertex Bezier ordinates are identical to the nodal function values, and the tangent Bezier ordinates and centre Bezier ordinates are related to the nodal gradient values, the relations between Bezier ordinates and the nodal function and gradient values are first presented, and the structure and computational algorithm are then outlined to construct the transformation matrix. When the construction of the transformation matrix is finished, a transformation matrix- Bernstein-Bezier basis function product is carried to compute the new C 1 natural neighbor interpolant and its derivatives. The new C 1 natural neighbor interpolant has quadratic completeness, interpolates to nodal function and nodal gradient values, and reduces to a cubic polynomial on the boundary of domain. The new C 1 natural element method based on Galerkin scheme is introduced to couple stress theory for elasticity to obtain the discrete form of governing equation. Numerical analyses of the method are presented, covering the influences of the couple stress on stress concentration of an infinite plate with central hole subjected to the uniaxial pull and biaxial pull respectively. Numerical results of typical problems show that when solving fourth-order elliptic problems such as those arising in couple stress theories, the C 1 natural neighbor interpolant based on the extended Delaunay triangulation is valid and feasible.

In computer graphics, shape recognition is widely solved problem. The shape functions, shape distribution and Minkowski norm are the standard methods for the determination of similarity measure. In this paper, the shape functions D2, D3 and new C1 are applied to five triangular meshes of a half-sphere, a cylinder and a plane obtained from ball-bar, ring, and gauge block after trimming. In the using of optical scanners the calibration is necessary and it is done using calibration artefacts with known dimensions (according to which the calibration is executed). So, it is useful to find the algorithm, where the known dimensions are not necessary for calibration. Therefore, the aim of this paper (and the first step for finding the algorithm) is to define whether each shape function is competent to measure the similarity and whether the new shape function C1 is as good as the standard functions. To determine it, Measurement System Analysis (MSA) was used.KeywordsShape functionShape distributionShape matching methodMeasurement system analysis (MSA)Shape recognitionTriangular mesh

SUMMARYWe present a new shell model and an accompanying discretisation scheme that is suitable for thin and thick shells. The deformed configuration of the shell is parameterised using the mid‐surface position vector and an additional shear vector for describing the out‐of‐plane shear deformations. In the limit of vanishing thickness, the shear vector is identically zero and the Kirchhoff–Love model is recovered. Importantly, there are no compatibility constraints to be satisfied by the shape functions used for discretising the mid‐surface and the shear vector. The mid‐surface has to be interpolated with smooth C1‐continuous shape functions, whereas the shear vector can be interpolated with C0‐continuous shape functions. In the present paper, the mid‐surface as well as the shear vector are interpolated with smooth subdivision shape functions. The resulting finite elements are suitable for thin and thick shells and do not exhibit shear locking. The good performance of the proposed formulation is demonstrated with a number of linear and geometrically non‐linear plate and shell examples. Copyright © 2012 John Wiley & Sons, Ltd.

In the previous papers (Kim et al. Submitted for publication, Oh et al. in press), for uniformly or locally non-uniformly distributed particles, we constructed highly regular piecewise polynomial particle shape functions that have the polynomial reproducing property of order k for any given integer k ≥ 0 and satisfy the Kronecker Delta Property. In this paper, in order to make these particle shape functions more useful in dealing with problems on complex geometries, we introduce smooth-piecewise-polynomial Reproducing Polynomial Particle shape functions, corresponding to the particles that are patch-wise non-uniformly distributed in a polygonal domain. In order to make these shape functions with compact supports, smooth flat-top partition of unity shape functions are constructed and multiplied to the shape functions. An error estimate of the interpolation associated with such flexible piecewise polynomial particle shape functions is proven. The one-dimensional and the two-dimensional numerical results that support the theory are resented.

Functional data are defined as realizations of random functions (mostly smooth functions) varying over a continuum, which are usually collected on discretized grids with measurement errors. In order to accurately smooth noisy functional observations and deal with the issue of high-dimensional observation grids, we propose a novel Bayesian method based on the Bayesian hierarchical model with a Gaussian-Wishart process prior and basis function representations. We first derive an induced model for the basis-function coefficients of the functional data, and then use this model to conduct posterior inference through Markov chain Monte Carlo methods. Compared to the standard Bayesian inference that suffers serious computational burden and instability in analyzing high-dimensional functional data, our method greatly improves the computational scalability and stability, while inheriting the advantage of simultaneously smoothing raw observations and estimating the mean-covariance functions in a nonparametric way. In addition, our method can naturally handle functional data observed on random or uncommon grids. Simulation and real studies demonstrate that our method produces similar results to those obtainable by the standard Bayesian inference with low-dimensional common grids, while efficiently smoothing and estimating functional data with random and high-dimensional observation grids when the standard Bayesian inference fails. In conclusion, our method can efficiently smooth and estimate high-dimensional functional data, providing one way to resolve the curse of dimensionality for Bayesian functional data analysis with Gaussian-Wishart processes.

A New Unconstrained Optimization Method for Imprecise Function and Gradient Values

Shape analyses and similarity measuring is a very often solved problem in computer graphics. The shape distribution approach based on shape functions is frequently used for this determination. The experience from a comparison of ball-bar standard triangular meshes was used to match hip bones triangular meshes. The aim is to find relation between similarity measures obtained by shape distributions approach.

We classify all possible local linear procedures on triangular meshes resulting in polynomial C1-spline functions with affinely uniform shape conditions for the basic functions at the edges, and fitting the 9 value and gradient data at the vertices of the mesh triangles. There is a unique procedure among them with shape functions and basic polynomials of degree 5 and all other admissible procedures are its perturbations with higher degree.

Two-point approximations are developed by utilizing both the function and gradient information of two data points. The objective of this work is to build a high-quality approximation to realize computational savings in solving complex optimization and reliability analysis problems. Two developments are proposed in the new twopoint approximations: 1) calculation of a correction term by matching with the previous known function value and supplementing it to the first-order approximation for including the effects of higher order terms and 2) development of a second-order approximation without the actual calculation of second-order derivatives by using an approximate Hessian matrix. Several highly nonlinear functions and structural examples are used for demonstrating the new two-point approximations that improved the accuracy of existing first-order methods. EVELOPMENTof new and improved constraint approximations has been an active area of research in the mathematical optimization community over the last two decades.13 More accurate function approximations reduce the finite element analysis cost and increase the search domain in each iteration of an optimization run. Recent developments in constraint approximations addressed methods for eigenvalues, dynamic response, static forces, structural reliability, etc. Some of the typical methods are summarized in the following. Kirsch4 developed a scaling factor for a stiffness matrix and approximated the displacements, stresses, and forces with respect to sizing and topology variables. Vanderplaats and Salajegheh5 used two levels of approximations for discrete sizing and shape design of structures, first for solving the continuous problem and next the discrete one using a dual theory. Canfield6 developed an approach for eigenvalues by approximating the modal strain and kinetic energies independently. Thomas et al.7 presented an approximation for the frequency response of damped structures by using the concept of Ref. 6. A multivariate spline approximation was developed by Wang and Grandhi8 making use of the previously generated exact finite element analyses. Similarly, a Hermite approximation for n-dimensional problems used several data points and had the property of reproducing the function and gradient values at the known data points.9 Canfield10 used multipoint data in building an approximate Hessian matrix and constructed a second-order approximation for improving the accuracy. Toropov et al.11 approximated functions as a linear combination or products of variables, and the coefficients of the polynomial were computed by applying a least squared approach on multiple data points. Fadel et al. 12 considered intermediate variables in terms of exponentials and they were computed by matching the approximate function gradients with the previous point's exact values. This approach did not compare the function values of the previous point. Wang and Grandhi13 used a single exponential for all of the variables and calculated it by matching the approximate function with the previous point exact value, and the gradient information of the previous point was not utilized. The proposed paper develops an improved two-point approximation utilizing both function and gradient information of two data points. In the new two-point approximation, two developments are proposed: 1) calculation of a correction term based on two function values and

Two-point approximations are developed by utilizing both the function and gradient information of two data points. The objective of this work is to build a high-quality approximation to realize computational savings in solving complex optimization and reliability analysis problems. Two developments are proposed in the new twopoint approximations: 1) calculation of a correction term by matching with the previous known function value and supplementing it to the first-order approximation for including the effects of higher order terms and 2) development of a second-order approximation without the actual calculation of second-order derivatives by using an approximate Hessian matrix. Several highly nonlinear functions and structural examples are used for demonstrating the new two-point approximations that improved the accuracy of existing first-order methods. EVELOPMENTof new and improved constraint approximations has been an active area of research in the mathematical optimization community over the last two decades.13 More accurate function approximations reduce the finite element analysis cost and increase the search domain in each iteration of an optimization run. Recent developments in constraint approximations addressed methods for eigenvalues, dynamic response, static forces, structural reliability, etc. Some of the typical methods are summarized in the following. Kirsch4 developed a scaling factor for a stiffness matrix and approximated the displacements, stresses, and forces with respect to sizing and topology variables. Vanderplaats and Salajegheh5 used two levels of approximations for discrete sizing and shape design of structures, first for solving the continuous problem and next the discrete one using a dual theory. Canfield6 developed an approach for eigenvalues by approximating the modal strain and kinetic energies independently. Thomas et al.7 presented an approximation for the frequency response of damped structures by using the concept of Ref. 6. A multivariate spline approximation was developed by Wang and Grandhi8 making use of the previously generated exact finite element analyses. Similarly, a Hermite approximation for n-dimensional problems used several data points and had the property of reproducing the function and gradient values at the known data points.9 Canfield10 used multipoint data in building an approximate Hessian matrix and constructed a second-order approximation for improving the accuracy. Toropov et al.11 approximated functions as a linear combination or products of variables, and the coefficients of the polynomial were computed by applying a least squared approach on multiple data points. Fadel et al. 12 considered intermediate variables in terms of exponentials and they were computed by matching the approximate function gradients with the previous point's exact values. This approach did not compare the function values of the previous point. Wang and Grandhi13 used a single exponential for all of the variables and calculated it by matching the approximate function with the previous point exact value, and the gradient information of the previous point was not utilized. The proposed paper develops an improved two-point approximation utilizing both function and gradient information of two data points. In the new two-point approximation, two developments are proposed: 1) calculation of a correction term based on two function values and

In this paper, a meshfree shell adaptive procedure is developed for the applications in the sheet metal forming simulation. The meshfree shell formulation is based on the first-order shear deformable shell theory and utilizes the degenerated continuum and updated Lagrangian approach for the nonlinear analysis. For the sheet metal forming simulation, an h-type adaptivity based on the meshfree background cells is considered and a geometric error indicator is adopted. The enriched nodes in adaptivity are added to the centroids of the adaptive cells and their shape functions are computed using a first-order generalized meshfree (GMF) convex approximation. The GMF convex approximation provides a smooth and non-negative shape function that vanishes at the boundary, thus the enriched nodes have no influence outside the adapted cells and only the shape functions within the adaptive cells need to be re-computed. Based on this concept, a multi-level refinement procedure is developed which does not require the constraint equations to enforce the compatibility. With this approach the adaptive solution maintains the order of meshfree approximation with least computational cost. Two numerical examples are presented to demonstrate the performance of the proposed method in the adaptive shell analysis.

In the optical scanning, the processed data are in the form of point clouds that can be organised into the triangular meshes and can represent the surface with the given inaccuracy of the scanner. Meshes of special surfaces (such as sphere or ball bar) are used for the scanner calibration, qualification and verification. These artefacts restrict the calibration process, because their sizes must be periodically calibrated by the relevant institutions.The aim of this paper is to present new possibilities for optical laser scanner accuracy comparison. Three types of laser scanners with different accuracies were used. The calibration sphere was used to compare the results of the deviations (the commonly used method for scanner accuracy verification) from calibrated value with the results of new methods that use shape functions measuring distances, cone heights or curvatures. The measurement system analysis was used to verify that these functions can be used for scanners comparison. The comparison of shape distributions obtained from results of shape functions in the form of polylines was used to compare scanners according to their accuracies. The comparison of shape distribution results was done using ranges and Minkowski L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${L}_{1}$$\\end{document} norm.

A local tangential lifting differential method for triangular meshes

In this paper, we develop an effective and robust adaptive time-frequency analysis method for signals with intra-wave frequency modulation. To handle this kind of signals effectively, we generalize our data-driven time-frequency analysis by using a shape function to describe the intra-wave frequency modulation. The idea of using a shape function in time-frequency analysis was first proposed by Wu (Wu 2013 Appl. Comput. Harmon. Anal. 35, 181-199. (doi:10.1016/j.acha.2012.08.008)). A shape function could be any smooth 2π-periodic function. Based on this model, we propose to solve an optimization problem to extract the shape function. By exploring the fact that the shape function is a periodic function with respect to its phase function, we can identify certain low-rank structure of the signal. This low-rank structure enables us to extract the shape function from the signal. Once the shape function is obtained, the instantaneous frequency with intra-wave modulation can be recovered from the shape function. We demonstrate the robustness and efficiency of our method by applying it to several synthetic and real signals. One important observation is that this approach is very stable to noise perturbation. By using the shape function approach, we can capture the intra-wave frequency modulation very well even for noise-polluted signals. In comparison, existing methods such as empirical mode decomposition/ensemble empirical mode decomposition seem to have difficulty in capturing the intra-wave modulation when the signal is polluted by noise.

A three-dimensional ES-FEM for fracture mechanics problems in elastic solids