Discrete Vector Calculus and Helmholtz Hodge Decomposition for Classical Finite Difference Summation by Parts Operators

In this article,discrete variants of several results from vector calculus are studied for clas-sical finite difference summation by parts operators in two and three space dimensions.It is shown that existence theorems for scalar/vector potentials of irrotational/solenoidal vector fields cannot hold discretely because of grid oscillations,which are characterised explicitly.This results in a non-vanishing remainder associated with grid oscillations in the discrete Helmholtz Hodge decomposition.Nevertheless,iterative numerical methods based on an interpretation of the Helmholtz Hodge decomposition via orthogonal projections are pro-posed and applied successfully.In numerical experiments,the discrete remainder vanishes and the potentials converge with the same order of accuracy as usual in other first-order partial differential equations.Motivated by the successful application of the Helmholtz Hodge decomposition in theoretical plasma physics,applications to the discrete analysis of magnetohydrodynamic(MHD) wave modes are presented and discussed.

Conservative Vector Fields and the Intersect Rule

This paper covers the concept of a conservative vector field, and its application in vector physics and Newtonian mechanics. Conservative vector fields are defined as the gradient of a scalar-valued potential function. Gradient fields are irrotational, as in the curl in all conservative vector fields is zero, by Clairaut’s Theorem. Additionally, line integrals in conservative vector fields are path-independent, and line integrals over closed paths are always equal to zero, properties proved by the Gradient Theorem of multivariable calculus. Gradient fields represent conservative forces, and the associated potential function is analogous to potential energy associated with said conservative forces. The Intersect Rule provides a new, unique shortcut for determining if a vector field is conservative and deriving potential functions, by treating the indefinite integral as a set of infinitely many functions which satisfy the integral.

Analysis of the Results Obtained from the Application of the Two-Stage Method with Calculations of Some Statically Indeterminate Trusses

The paper presents results of calculations of forces in members of selected types of statically indeterminate trusses carriedout by application of the two-stage method of computations of such structural systems. The method makes possible to do the simple andapproximate calculations of the complex trusses in two stages, in each of which is calculated a statically determinate truss being anappropriate counterpart of the basic form of the statically indeterminate truss structure. Systems of the statically determinate trussesconsidered in the both stages are defined by cancelation of members, number of which is equal to the statically indeterminacy of thebasic truss. In the paper are presented outcomes obtained in the two-stage method applied for two different shapes of trusses and carriedout for various ways of removing of appropriate members from the basic trusses. The results are compared with outcomes gained due toapplication of suitable computer software for computation of the same types of trusses and for the same structural conditions.

Non-Destructive Crack Detection of Preserved Eggs Using a Machine Vision and Multivariate Analysis

Pidan or century egg, also known as preserved egg, is one of the most traditional and popular egg products in China. The crack detection of preserved eggshell is very important to guarantee its quality. In this study, we develop an image algorithm for preserved eggshell＇s crack detection by using natural light and polarized image. Four features including crack length, crack state coefficient, maximum projection and angular point are extracted from the natural light image by morphology calculus algorithms. The support vector machines（SVM） model with radial basis kernel function is established using the four features with an accuracy of about 92%. The detection accuracy is improved to 94% by using a new characteristic parameter of crack length on polarization image. The Multi-information fusion analysis indicates the potential for cracks detection by a real-time synthesis imaging system.