Accurate calculation of high order pseudo-Zernike moments and their numerical stability

The accuracy of pseudo-Zernike moments (PZMs) suffers from various errors, such as the geometric error, numerical integration error, and discretization error. Moreover, the high order moments are vulnerable to numerical instability. In this paper, we present a method for the accurate calculation of PZMs which not only removes the geometric error and numerical integration error, but also provides numerical stability to PZMs of high orders. The geometric error is removed by taking the square-grids and arc-grids, the ensembles of which maps exactly the circular domain of PZMs calculation. The Gaussian numerical integration is used to eliminate the numerical integration error. The recursive methods for the calculation of pseudo-Zernike polynomials not only reduce the computation complexity, but also provide numerical stability to high order moments. A simple computational framework to implement the proposed approach is also discussed. Detailed experimental results are presented which prove the accuracy and numerical stability of PZMs.     

电阻应变称重传感器蠕变的模型和补偿

借助于电阻应变式称重传感器在恒定负荷作用下的蠕变模型，探讨了蠕变误差补偿机理及其可行的技术方案，提出了通过改变敏感栅端部的受力状况来改变应变片蠕变值的蠕变补偿思路，试验证明是可行的。     

Boundary element formulation for two-dimensional creep problems using isoparametric quadratic elements

A boundary element formulation for creep and time-dependent material behaviour problems based on an initial strain approach is presented. The details of numerical algorithm are shown where isoparametric quadratic elements are used both for the boundary elements and the quadrilateral domain cells. The Euler method with automatic time-step control scheme is implemented for time integration. Two creep power laws, time-hardening and strain-hardening, are employed to analyse a number of problems, including a square plate, a plate with a circular hole and a plate with a semi-circular notch subjected to a uniaxial load. The results are compared with analytical solutions where available and the corresponding finite element solutions.     

An extension of 3-D procedure to large strain analysis of shells

A numerical stress integration procedure for general 3-D large strain problems in inelasticity, based on the total formulation and the governing parameter method (GPM), is extended to shell analysis. The multiplicative decomposition of the deformation gradient is adopted with the evaluation of the deformation gradient practically in the same way as in a general 3-D material deformation. The calculated trial elastic logarithmic strains are transformed to the local shell Cartesian coordinate system and the stress integration is performed according to the GPM developed for small strain conditions. The consistent tangent matrix is calculated as in case of small strain deformation and then transformed to the global coordinate system.A specific step in the proposed procedure is the updating of the left elastic Green–Lagrangian deformation tensor. Namely, after the stresses are computed, the principal elastic strains and the principal vectors corresponding to the stresses at the end of time step are determined. In this way the shell conditions are taken into account appropriately for the next step.Some details are given for the stress integration in case of thermoplastic and creep material model.Numerical examples include bulging of plate (plastic, thermoplastic, and creep models for metal) and necking of a thin sheet. Comparison of solutions with those available in the literature, and with solutions using other type of finite elements, demonstrates applicability, efficiency and accuracy of the proposed procedure.     

Maximum-entropy meshfree method for compressible and near-incompressible elasticity

Numerical integration errors and volumetric locking in the near-incompressible limit are two outstanding issues in Galerkin-based meshfree computations. In this paper, we present a modified Gaussian integration scheme on background cells for meshfree methods that alleviates errors in numerical integration and ensures patch test satisfaction to machine precision. Secondly, a locking-free small-strain elasticity formulation for meshfree methods is proposed, which draws on developments in assumed strain methods and nodal integration techniques. In this study, maximum-entropy basis functions are used; however, the generality of our approach permits the use of any meshfree approximation. Various benchmark problems in two-dimensional compressible and near-incompressible small strain elasticity are presented to demonstrate the accuracy and optimal convergence in the energy norm of the maximum-entropy meshfree formulation.     

On the error analysis of the numerical solution of linear Volterra integral equations of the second kind

In the numerical solution of linear Volterra integral equations, two kinds of errors occur. If we use the collocation method, these errors are the collocation and numerical quadrature errors. Each error has its own effect in the accuracy of the obtained numerical solution. In this study we obtain an error bound that is sum of these two errors and using this error bound the relation between the smoothness of the kernel in the equation and also the length of the integration interval and each of these two errors are considered. Concluded results also are observed during the solution of some numerical examples.     

Accurate Computation of Orthogonal Fourier-Mellin Moments

Orthogonal Fourier-Mellin moments (OFMMs) suffer from geometric error and the numerical integration error. The geometric error arises when the square image is mapped into a unit disk and the mapping does not become perfect. The numerical integration error arises when the double integration is approximated by the zeroth order summation. In this paper, we propose methods which reduce these errors. The geometric error is reduced by considering the arc-grids lying on the boundary of the unit disk and the square grids lying completely inside the disk. The numerical integration error is reduced by Gaussian numerical integration, for which a simple computational framework is provided. The relative contributions of geometric error and numerical integration error to the total error are also analyzed. It is observed that the geometric error is significant only for the small images whereas the magnitude of numerical integration is significantly high for all image sizes, which increases with the order of moments. A simple computational framework which is similar to the conventional zeroth order approximation is also proposed which not only reduces numerical integration error but also reduces geometric error without considering arc-grids. The improved accuracy of OFMMs are shown to provide better image reconstruction, numerical stability and rotation and scale invariance. Exhaustive experimental results on a variety of real images have shown the efficacy of the proposed methods.     

A note on the convergence of finite element approximations for problems formulated in curvilinear coordinate systems

The asymptotic accuracies of structural shell theories are reviewed. Several finite element models to solve arch problems are formulated by utilizing the shell theories. The asymptotic rate of energy convergence is determined by the ability of the approximate strains to assume arbitray polynomial states. The optimal choice of interpolation functions for tangential and normal displacement is treated. Stress resultants and stress couples are evaluated by using nodal forces (strain integration method) or internal strain patterns (strain method). The accuracy of these methods are investigated. In particular the distribution of errors in the strains calculated by the strain method is determined and utilized for accuracte stress evaluation. The theoretical considerations are supported by a vaste number of numerical experiments, which confirm the theoretical results.     

Three algorithms to compute covariance matrices: Comparison of their computational complexity

Three numerical algorithms for computing the solution of the covariance matrix differential equations of states of a linear time-invariant dynamical system forced by white Gaussian noise are analyzed. Estimates of errors due to truncation and roundoff are derived for each algorithm. The error analyses are based on the assumption that computation is performed in floating point mode and that it is not numerically ill-conditioned. Computational complexity of each algorithm is also discussed. Two numerical examples are presented to evaluate the performance of each algorithm.     

Efficient numerical integration of an elastic–plastic damage law within a mixed velocity–pressure formulation

This study focuses on numerical integration of constitutive laws in numerical modeling of cold materials processing that involves large plastic strain together with ductile damage. A mixed velocity–pressure formulation is used to handle the incompressibility of plastic deformation. A Lemaitre damage model where dissipative phenomena are coupled is considered. Numerical aspects of the constitutive equations are addressed in detail. Three integration algorithms with different levels of coupling of damage with elastic–plastic behavior are presented and discussed in terms of accuracy and computational cost. The implicit gradient formulation with a non-local damage variable is used to regularize the localization phenomenon and thus to ensure the objectivity of numerical results for damage prediction problems. A tensile test on a plane plate specimen, where damage and plastic strain tend to localize in well-known shear bands, successfully shows both the objectivity and effectiveness of the developed approach.     

Stochastic P-type/D-type iterative learning control algorithms

This paper presents stochastic algorithms that compute optimal and sub-optimal learning gains for a P-type iterative learning control algorithm (ILC) for a class of discrete-time-varying linear systems. The optimal algorithm is based on minimizing the trace of the input error covariance matrix. The state disturbance, reinitialization errors and measurement errors are considered to be zero-mean white processes. It is shown that if the product of the input-output coupling matrices C ( t + 1 ) B ( t ) is full column rank, then the input error covariance matrix converges to zero in presence of uncorrelated disturbances. Another sub-optimal P-type algorithm, which does not require the knowledge of the state matrix, is also presented. It is shown that the convergence of the input error covariance matrices corresponding to the optimal and sub-optimal P-type and D-type algorithms are equivalent, and all converge to zero at a rate inversely proportional to the number of learning iterations. A transient-response performance comparison, in the domain of learning iterations, for the optimal and sub-optimal P- and D-type algorithms is investigated. A numerical example is added to illustrate the results.     

Calculation of turbine rotors in secondary creep range

Turbine rotors in power plants are exposed to triaxial stresses by centrifugal forces at high temperatures which induce long time creep effects. In the first step, we implemented an algorithm for centrifugal volume forces in ADINA. In the next step, we tested the numerical behaviour of the modified Newton-algorithm in ADINA within the long time secondary creep range for simplified examples. Isotropie strain hardening was assumed in most cases of creep calculations.

The estimate of creep properties is based on the minimum least square method. The numerical stability in creep calculations is dependent on the magnitude of time step size and on a good fit between creep law properties and the real experimental material data.

In the examples of turbine rotor models with rotational symmetry we were able to estimate the state of stress and strain under creep conditions for life time periods up to 2 × 10⁵ hr.     

Preplanning of disk merges

A method is presented for developing Runge-Kutta integration algorithms with built-in estimates of the accumulated truncation error. Several new 2-nd, 3-rd, and 4-th order algorithms are given. The computation per step of the new algorithms is identical to that of algorithms which provide only an estimate of the local truncation error. Numerical experimentation with the new algorithms shows that the estimated error compares very well with the true accumulated error. Further, the error is of the same order as that incurred using traditional Runge-Kutta algorithms.     

Moving horizon numerical observers of nonlinear control systems

In this note, we develop moving horizon numerical observers and analyze the error. In the error estimation, we take into consideration both the integration error and the optimization error. The design facilitates the use of a variety of numerical algorithms to form different observers. As a special case, an Euler-Newton observer is introduced. The numerical observer is independent of any optimization software or toolbox. Furthermore, the observer is formulated in a way that is especially efficient for systems with sampled measurement.     

Finite-element analysis of errors on stress and strain measurements in dynamic tensile testing of low-ductile materials

Tensile dynamic tests are essential experiments to develop and validate constitutive equations. In this paper, we studied the errors on stress and strain measurement in dynamic tensile tests by using finite-element analysis. Two strain and one stress measures were discussed. Mainly, we considered the influence of the material and multiple geometrical and testing parameters. We were limited to the case of elastic behaviour of the material. We observed that the errors on the strain measures are essentially influenced by the geometrical parameters. On the other hand, the error on the stress measure are highly correlated to stress field homogeneity.     

Algorithms for fast computation of Zernike moments and their numerical stability

Accuracy, speed and numerical stability are among the major factors restricting the use of Zernike moments (ZMs) in numerous commercial applications where they are a tool of significant utility. Often these factors are conflicting in nature. The direct formulation of ZMs is prone to numerical integration error while in the recent past many fast algorithms are developed for its computation. On the other hand, the relationship between geometric moments (GMs) and ZMs reduces numerical integration error but it is observed to be computation intensive. We propose fast algorithms for both the formulations. In the proposed method, the order of time complexity for GMs-to-ZMs formulation is reduced and further enhancement in speed is achieved by using quasi-symmetry property of GMs. The existing q-recursive method for direct formulation is further modified by incorporating the recursive steps for the computation of trigonometric functions. We also observe that q-recursive method provides numerical stability caused by finite precision arithmetic at high orders of moment which is hitherto not reported in the literature. Experimental results on images of different sizes support our claim.     

Time integration errors and some new functionals for the dynamics of a free mass

We study the numerical integration of the Poisson second-order ordinary differential equation which describes, for instance, the dynamics of a free mass. Classical integration algorithms, when applied to such an equation, furnish solutions affected by a significant “drift” error, apparently not studied so far. In the first part of this work we define measures of such a drift. We then proceed to illustrate how to construct both classical and extended functionals for the equation of motion of a free mass with given initial conditions. These tools allow both the derivation of new variationally-based time integration algorithms for this problem, and, in some cases, the theoretical isolation of the source of the drift. While we prove that this particular error is unavoidable in any algorithmic solution of this problem, we also provide some new time integration algorithms, extensions at little added cost of classical methods, which permit to substantially improve numerical predictions.     

Algorithms of increasing the calculation accuracy for an aircraft’s orientation angle

Algorithms for calculating an aircraft’s orientation angles based on the results of the numerical integration of Poisson’s and quaternion equations are proposed. The algorithms in the presence of random errors of the matrix elements of direction cosines and quaternions are characterized by a significantly higher accuracy in comparison to the formulas for solving the formulated problem. The results of testing the mathematical modeling data under random errors, which confirm the increased accuracy of the calculation of the orientation angles, are given.     

Application of a posteriori error estimation to finite element simulation of compressible Navier-Stokes flow

In this paper, we discuss how to improve the adaptive finite element simulation of compressible Navier-Stokes flow via a posteriori error estimate analysis. We use the moving space-time finite element method to globally discretize the time-dependent Navier-Stokes equations on a series of adapted meshes. The generalized compressible Stokes problem, which is the Stokes problem in its most generalized form, is presented and discussed. On the basis of the a posteriori error estimator for the generalized compressible Stokes problem, a numerical framework of a posteriori error estimation is established corresponding to the case of compressible Navier-Stokes equations. Guided by the a posteriori errors estimation, a combination of different mesh adaptive schemes involving simultaneous refinement/unrefinement and point-moving are applied to control the finite element mesh quality. Finally, a series of numerical experiments will be performed involving the compressible Stokes and Navier-Stokes flows around different aerodynamic shapes to prove the validity of our mesh adaptive algorithms.     

Motion and structure from two perspective views: algorithms, erroranalysis, and error estimation

Deals with estimating motion parameters and the structure of the scene from point (or feature) correspondences between two perspective views. An algorithm is presented that gives a closed-form solution for motion parameters and the structure of the scene. The algorithm utilizes redundancy in the data to obtain more reliable estimates in the presence of noise. An approach is introduced to estimating the errors in the motion parameters computed by the algorithm. Specifically, standard deviation of the error is estimated in terms of the variance of the errors in the image coordinates of the corresponding points. The estimated errors indicate the reliability of the solution as well as any degeneracy or near degeneracy that causes the failure of the motion estimation algorithm. The presented approach to error estimation applies to a wide variety of problems that involve least-squares optimization or pseudoinverse. Finally the relationships between errors and the parameters of motion and imaging system are analyzed. The results of the analysis show, among other things, that the errors are very sensitive to the translation direction and the range of field view. Simulations are conducted to demonstrate the performance of the algorithms and error estimation as well as the relationships between the errors and the parameters of motion and imaging systems. The algorithms are tested on images of real-world scenes with point of correspondences computed automatically