The role of the multiquadric shape parameters in solving elliptic partial differential equations

This study examines the generalized multiquadrics (MQ), φ_j(x) = [(x−x_j)²+c_j²]^β in the numerical solutions of elliptic two-dimensional partial differential equations (PDEs) with Dirichlet boundary conditions. The exponent β as well as c_j² can be classified as shape parameters since these affect the shape of the MQ basis function. We examined variations of β as well as c_j² where c_j² can be different over the interior and on the boundary. The results show that increasing ,β has the most important effect on convergence, followed next by distinct sets of (c_j²)Ω∂Ω ≪ (c_j²)∂Ω. Additional convergence accelerations were obtained by permitting both (c_j²)Ω∂Ω and (c_j²)∂Ω to oscillate about its mean value with amplitude of approximately 1/2 for odd and even values of the indices. Our results show high orders of accuracy as the number of data centers increases with some simple heuristics.     

Interpretation of regional geochemistry using optimal interpolation parameters

A collection of data analysis procedures is presented which are derived from estimation of geographic interpolation parameters. Several interpolation models are discussed along with a procedure to obtain the best model. The power parameter, p, and the search radius, c, are the standard parameters in inverse distance weighting interpolation which is appropriate for sampling patterns that are not highly irregular. The power parameter is shown to characterize the regional behavior of geochemical measurements. This characterization process can be used to associate similar types of geochemical measurements, produce optimal contour maps, derive meaningful residual maps, and highlight unusual geochemical areas by a weighted sum variable. The computer program. BESTP, (used to estimate the optimum inverse distance weighting interpolation parameters) is presented, along with an example using reconnaissance groundwater data from the Plainview Quadrangle, Texas.     

High accuracy geometric Hermite interpolation

We describe a parametric cubic spline interpolation scheme for planar curves which is based on an idea of Sabin for the construction of C¹ bicubic parametric spline surfaces. The method is a natural generalization of [standard] Hermite interpolation. In addition to position and tangent, the curvature is prescribed at each knot. This ensures that the resulting interpolating piecewise cubic curve is twice continuously differentiable with respect to arclength and can be constructed locally. Moreover, under appropriate assumptions, the interpolant preserves convexity and is 6-th order accurate.     

Identification of non-linear processes using reciprocal multiquadric functions

In this paper radial basis function (RBF) networks are used to model general non-linear discrete-time systems. In particular, reciprocal multiquadric functions are used as activation functions for the RBF networks. A stepwise regression algorithm based on orthogonalization and a series of statistical tests is employed for designing and training of the network. The identification method yields non-linear models, which are stable and linear in the model parameters. The advantages of the proposed method compared to other radial basis function methods and backpropagation neural networks are described. Finally, the effectiveness of the identification method is demonstrated by the identification of two non-linear chemical processes, a simulated continuous stirred tank reactor and an experimental pH neutralization process.     

Adjusting shape parameters using model-based optical flow residuals

We present a method for estimating the shape of a deformable model using the least-squares residuals from a model-based optical flow computation. This method is built on top of an estimation framework using optical flow and image features, where optical flow affects only the motion parameters of the model. Using the results of this computation, our new method adjusts all of the parameters so that the residuals from the flow computation are minimized. We present face tracking experiments that demonstrate that this method obtains a better estimate of shape compared to related frameworks     

Assessing interpolation accuracy in elevation models

Methods that identify the spatial variation in elevation model accuracy and highlight relative variation are proposed. Visualization within the geographic resources analysis support system (GRASS) is used to identify the accuracy preserved in interpolation digital contour data to produce elevation models. The interpolation routines are inverse distance weighting, contour flood filling, simultaneous over-relaxation, and one-dimensional spline fitting. The results of the interpolation process are presented as colored contours, shaded relief maps, aspect maps, and products of Laplacian filtering, profile and plan convexity, and visualizing root mean square error methods     

Improved sensitivity analysis using a complex variable semi-analytical method

The semi-analytical method (SAM) is a computationally efficient and easy to implement approach often used for the sensitivity analysis of finite element models. However, it is known to exhibit serious inaccuracy for shape sensitivity analysis for structures modeled by beam, frame, plate, or shell elements. In the present paper, we use a semi-analytical approach based on complex variables (SACVM) to compute the sensitivity of finite element models composed of beam and plate elements. The SACVM combines the complex variable method (CVM) with the semi-analytical method (SAM) to obtain the response sensitivity accurately and efficiently. The current approach maintains the computational efficiency of the semi-analytical method but with higher accuracy. In addition, the current approach is insensitive to the choice of step size, a feature that simplifies its use in practical problems. The method is applicable to any structural elements including beam, frame, plate, or shell elements and only requires minor modifications to existing finite element codes.     

Constrained curve drawing using trigonometric splines having shape parameters

The problem of drawing a smooth obstacle avoiding curve has attracted the attention of many people working in the area of CAD/CAM and its applications. In the present paper we propose a method of constrained curve drawing using certain C¹-quadratic trigonometric splines having shape parameters, which have been recently introduced in [Han X. Quadratic trigonometric polynomial curves with a shape parameter. Computer Aided Geometric Design 2002;19:503–12]. Besides this, we have also presented a simpler approach for studying the approximation properties of the trigonometric spline curves.     

On the accuracy of shape sensitivity

The calculation of sensitivity of the response of a structure modeled by finite elements to shape variation is known to be subject to numerical difficulties. The accuracy of a given method is typically measured against the yard stick of finite-difference sensitivity calculation. The present paper demonstrates with a simple example that this approach may be flawed because of discretization errors associated with the finite element mesh. Seven methods for calculating sensitivity derivatives are compared for a two-material beam problem with a moving interface. It is found that as the mesh is refined, displacement sensitivity derivatives converge more slowly than the displacements. Six of the methods agree fairly well, but the adjoint variational surface method provides substantially different results. However, the difference is found to reflect convergence from another direction to the same answer rather than reduced accuracy. Additionally, it is observed that small derivatives are particularly prone to accuracy problems.     

Personalized face-pose estimation network using incrementally updated face shape parameters

In this paper, a deep learning method is proposed for human image processing that incorporates a mechanism to update target-specific parameters. The aim is to improve system performance in situations where the target can be continuously observed. Network-based algorithms typically rely on offline training processes that use large datasets, while trained networks typically operate in a one-shot fashion. That is, each input image is processed one by one in the static network. On the other hand, many practical applications can be expected to use continuous observation rather than observation of a single image. The proposed method employs dynamic use of multiple observations to improve system performance. In this paper, the effectiveness of the proposed method adopting an iterative update process is clarified through its implementation in the task of face-pose estimation. The method consists of two separate processes: 1) sequential estimation and updating of face-shape parameters (target-specific parameters) and 2) face-pose estimation for every single image using the updated parameters. Experimental results indicate the effectiveness of the proposed method.

     

Multizone decomposition for simulation of time-dependent problems using the multiquadric scheme

This paper discusses the application of the multizone decomposition technique with multiquadric scheme for the numerical solutions of a time-dependent problem. The construction of the multizone algorithm is based on a domain decomposition technique to subdivide the global region into a number of finite subdomains. The reduction of ill-conditioning and the improvement of the computational efficiency can be achieved with a smaller resulting matrix on each subdomain. The proposed scheme is applied to a hypothetical linear two-dimensional hydrodynamic model as well as a real-life nonlinear two-dimensional hydrodynamic model in the Tolo Harbour of Hong Kong to simulate the water flow circulation patterns. To illustrate the computational efficiency and accuracy of the technique, the numerical results are compared with those solutions obtained from the same problem using a single domain multiquadric scheme. The computational efficiency of the multizone technique is improved substantially with faster convergence without significant degradation in accuracy.     

Statistics of eigensolutions for systems with random shape parameters using BEM

This paper presents a probabilistic boundary element method for analysis of the statistics of structural eigenvalues and eigenvectors, when the shape parameters of the structures are considered as random variables. Using this method, engineers are able to estimate the errors of the structural eigenvalues and eigenvectors resulting from manufacturing errors, and evaluate the differences between the experimental results and numerical results, which are given by the finite element method or boundary element method, etc. This method can be used to design and analyse the components of engineering structures because of its simplicity and effectiveness.     

High accuracy spline interpolation for 5-axis machining

This article presents a new algorithm for simultaneous 5-axis spline interpolation. The algorithm basically merges two concepts: (1) the interpolation of the toolpath with Pythagorean Hodograph (PH) curves and (2) the analytic solution of the inverse kinematic problem using the template equation method. The first method allows one to obtain a analytic relation between the arclength and the parameter of the toolpath curve. This one enables one to control the velocity of the tool on the workpiece. The second method allows one to determine the analytic solution of the parameterized inverse kinematic problem that permits us to introduce arbitrary number of geometric parameters. A natural selection of the possible parameters can be the parameters of tool geometry and the workpiece placement. This way, the off-line generated inverse solutions—that transform the cutter contact curve into axis values—can be online compensated, as soon as the exact parameter values are become known. Based on these two approaches, a robust and fast method for the simultaneous 5-axis spline interpolation is developed. The result of this new algorithm is time-dependent axis splines which represent the given toolpath with high accuracy.     

Improved multiquadric method for elliptic partial differential equations via PDE collocation on the boundary

The multiquadric radial basis function (MQ) method is a recent meshless collocation method with global basis functions. It was introduced for discretizing partial differential equations (PDEs) by Kansa in the early 1990s. The MQ method was originally used for interpolation of scattered data, and it was shown to have exponential convergence for interpolation problems.In [1], we have extended the Kansa-MQ method to numerical solution and detection of bifurcations in 1D and 2D parameterized nonlinear elliptic PDEs. We have found there that the modest size nonlinear systems resulting from the MQ discretization can be efficiently continued by a standard continuation software, such as auto. We have observed high accuracy with a small number of unknowns, as compared with most known results from the literature.In this paper, we formulate an improved Kansa-MQ method with PDE collocation on the boundary (MQ PDECB): we add an additional set of nodes (which can lie inside or outside of the domain) adjacent to the boundary and, correspondingly, add an additional set of collocation equations obtained via collocation of the PDE on the boundary. Numerical results are given that show a considerable improvement in accuracy of the MQ PDECB method over the Kansa-MQ method, with both methods having exponential convergence with essentially the same rates.     

Improved accuracy of time-resolved micro-Particle Image Velocimetry using phase-correlation and confocal microscopy

Micro-Particle Image Velocimetry (μPIV) measurements often suffer from poor image quality because of volume illumination effects, out of focus particles, and low seeding densities. As a result, measurements are typically ensemble averaged in time to improve the signal-to-noise ratio (SNR) of the resulting cross correlations. To achieve reliable, time-accurate μPIV measurements we need to improve the SNR of the recorded images and/or the SNR treatment of the resulting cross correlations. In this paper, we improve image quality and cross correlation SNR by comparing the use of confocal microscopy with spectral filtering. Steady-state spatiotemporally resolved data from widefield and confocal μPIV experiments were used and cross correlations were performed using standard techniques and the Robust Phase Correlation (RPC) method that employs a PIV-optimized spectral filter on the cross-correlation planes. The accuracy improvements were assessed by comparison against the time-averaged ensemble cross correlation, which currently represents the most accurate and accepted approach for steady-state μPIV measurements. Results show 24.77 % erroneous vectors for two-pass standard cross correlation with widefield imaging, which was reduced to 9.08 % erroneous vectors when using the RPC and confocal imaging. Furthermore, a 59.2 % reduction of error referenced to the ensemble correlation was observed when using RPC with confocal imaging over baseline cases. Improvements seen for RPC and confocal cases result from synergistically improving the correlation signal-to-noise ratio, resulting in correlation planes with sharper primary peaks and lower background levels.     

Detecting objects of variable shape structure with hidden state shape models

This paper proposes a method for detecting object classes that exhibit variable shape structure in heavily cluttered images. The term "variable shape structure" is used to characterize object classes in which some shape parts can be repeated an arbitrary number of times, some parts can be optional, and some parts can have several alternative appearances. Hidden State Shape Models (HSSMs), a generalization of Hidden Markov Models (HMMs), are introduced to model object classes of variable shape structure using a probabilistic framework. A polynomial inference algorithm automatically determines object location, orientation, scale and structure by finding the globally optimal registration of model states with the image features, even in the presence of clutter. Experiments with real images demonstrate that the proposed method can localize objects of variable shape structure with high accuracy. For the task of hand shape localization and structure identification, the proposed method is significantly more accurate than previously proposed methods based on chamfer-distance matching. Furthermore, by integrating simple temporal constraints, the proposed method gains speed-ups of more than an order of magnitude, and produces highly accurate results in experiments on non-rigid hand motion tracking.     

An interactive procedure for shape preserving cubic spline interpolation

An interactive system is presented which allows flexible shape preserving cubic spline interpolation. The corresponding parameter-depending minimization problem is considered, but similar results are then obtained by means of an interactive adjustment procedure for which the total amount in arithmetic operation is only 0(n).     

IFS分形造型的参数插值研究

根据迭代函数系统理论,结合计算机图形学方法,研究自然景物的分形造型问题,建立一类具有可预见性和可控性的参数控制模型并讨论了参数插值的收敛条件.基于静态插值、动态插值和插值等3种插值方式,将一迭代函数系统的分形图形作为种子图形,应用插值参数控制种子图形的演化变形,得到一系列新的变形图形,实现对IFS吸引子分形的可预见性和可控性的控制.实验结果表明,利用该方法能直观有效地获得大量变化多端的分形图.     

Generate mesh with shape parameters

In CAD/CAM,mesh rather than smooth surface is only needed sometimes.A mesh-generating method from permanence principle of Coons patch is developed.A new mesh point is defined through local small subpatch and all mesh points are computed by a linear system with special symmetric block tridiagonal coefficient matrix.By simplification,the determinant of coefficient matrix is determined by determinants of submatrices.Condition of existence of solution is given.Whether coefficient matrix is singular can be judged by a simple polynomial function with the eigenvalue of submatrix as variable.Numerical examples demonstrate the effects of shape parameters.