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The vibration analysis of an axially functionally graded material non-prismatic Timoshenko beam under axial thermal variation in humid environment is carried out using the harmonic differential quadrature method. In this modeling, the length and width of the beam remains constant whereas thickness of the beam is linearly varied along the axis of the beam. The material properties are temperature dependent and are assumed to be varied continuously along the axial direction according to power law distribution. Three types of temperature variations are considered in this study, that is, uniform temperature rise, linear temperature rise, and non-linear temperature rise. The temperature of the beam remains constant under uniform temperature rise condition and it is varied linearly and nonlinearly along the length of beam for rest of the conditions. The beam is subjected to uniform moisture concentration to impose humidity. Hamiltonian’s approach is used to derive the governing equations of motion. The resultant governing equations are then solved using the harmonic differential quadrature method to obtain the natural frequencies of the axially functionally graded material non-prismatic beam. The results obtained using the harmonic differential quadrature method are compared with results obtained for special cases. The effects of thermal variation, humidity, non-homogeneity parameter, and end conditions on natural frequencies of the non-prismatic beam are reported.

The reliability of a complex mathematical problem is presented in this work. The problem is a rupture concrete beam reinforced by steel fibers with various orientation angle. The structure is simulated by sinusoidal shear deformation theory and energy method. Harmonic differential quadrature method is applied for the solution of the problem under the buckling load. The reliability analysis of the mentioned system is studied utilizing a three-term conjugate finite-step length (TCFS) approach. The TCFS is formulated utilizing the conjugate gradient method with a limited scalar parameter to reach the numerical stabilization. The implicit buckling limit state function of the structure includes different parameters of length to thickness ratio of the beam, volume percent and orientation angle of steel fibers, rupture constant and boundary conditions. The results show that the failure probabilities of the studied structure may be decreased by increasing the rupture constant. In addition, the orientation angle of zero is the best choice for steel fiber in the concrete beam.

Complex manifolds with compatible metric have a naturally defined subspace of harmonic differential forms that satisfy Serre, Hodge, and conjugation duality, as well as hard Lefschetz duality. This last property follows from a representation of $sl(2,\mathbb{C})$, generalizing the well known structure on the harmonic forms of compact K\"ahler manifolds. Some topological implications are deduced., Comment: 7 pages, to appear in Proc. AMS

The optimization and buckling analysis of the rupture beams reinforced by steel fibers are presented in this article. The structure is simulated by hyperbolic shear deformation theory (HSDT). Utilizing Harmonic differential quadrature (HDQ) method, the buckling load is calculated. Utilizing adaptive improved harmony search (AIHS) algorithm, the optimization process of the rupture beam is examined. The mentioned algorithm is improved adaptively utilizing dynamic variables. The harmony memory is corrected at first and second adjustments, respectively on the basis of dynamic bandwidth and step size randomly. The optimum conditions of the rupture beam are evaluated with various harmony existing search methods. The influences of various constants including beam length to thickness ratio, rupture parameter, steel fibers volume fraction, steel fibers orientation angle and boundary supports are shown on the buckling load. The outcomes indicate that the proposed AIHS is performed better with respect to other harmony search methods. Meanwhile, the buckling load is reduced by enhancing the rupture parameter. Furthermore, the buckling load of the rupture beam with horizontal steel fibers is better with respect to other orientation angles.

This paper proposes a particular type of Stokes-Dirac structure for describing a Laplacian used in Poisson’s equations on topologically non-trivial manifolds, i.e., not Euclidian. The operator matrix representation of the structure includes not only exterior differential operators, but also codifferential operators in the sense of the dual of the pairing between differential forms. Since the successive operation of the matrix is equivalent to the Laplace-Beltrami operator, we call it a Stokes-Dirac operator. Furthermore, the Stokes-Dirac operator is augmented by harmonic differential forms that reflect the topological geometry of manifolds. The extension enable us to describe a power balance of particular boundary energy flows on manifolds with a non-trivial shape.

In this article, the authors used two different numerical approaches for frequency response of annular sector and sector plates with functionally graded materials and laminated composite cases. First-order shear deformation (FSDT) and Love's conical shell theories are used for obtaining the annular sector plate equations via some suitable angles and geometric parameters. The method of harmonic differential quadrature (HDQ) and discrete singular convolution (DSC) have been used for numerical solution of the resulting governing equations for modal analysis. Simple power-law and four-parameter power law distributions in terms of the volume fractions of constituents have been used for FGM composites. Comparison and convergence study for the present numeral methods have been made via existing results available in the literature for sector and annular sector plates. Frequencies values for annular sector/sector plates have been obtained for different material and geometric parameters, boundary conditions and sector angles.

We give a type-independent construction of an explicit basis for the semi-invariant harmonic differential forms of an arbitrary pseudo-reflection group in characteristic zero. Our "top-down" approach uses the methods of Cartan's exterior calculus and is in some sense dual to related work of Solomon, Orlik--Solomon, and Shepler describing (semi-)invariant differential forms. We apply our results to a recent conjecture of Zabrocki which provides a representation theoretic-model for the Delta conjecture of Haglund--Remmel--Wilson in terms of a certain non-commutative coinvariant algebra for the symmetric group. In particular, we verify the alternating component of a specialization of Zabrocki's conjecture., Comment: 28 pages. Supercedes arXiv:1908.00196

The aim of this study is to give a numerical solution for free vibration problem of functionally graded and carbon nanotube reinforced (CNTR) circular cylindrical panel. Based on the Love’s shell theory and first-order shear deformation theory (FSDT), related equations of motion have been obtained. Then, the methods of harmonic differential quadrature (HDQ) and discrete singular convolution (DSC) are used for numerical computations . Two different material properties are assumed to change continuously in the thickness direction according to the volume fraction power law and the general four-parameter power law distributions. Accuracy and convergence of the methods have been validated by comparing the obtained results with the existing results available in the open literature. Frequency values have been given for different material and geometric parameters , modes and boundary cases for circular cylindrical panels with FGM and CNTR cases. The effects of types of distributions of CNTR material are also discussed.

The priority of this paper is to explore the computational characteristics of Harmonic Differential Quadrature (HDQ) method for free flexural vibration analysis of functionally graded piezoelectric material (FGPM) beam. The modified Timoshenko beam theory is used in this study where electric potential is assumed to have sinusoidal variation across the depth. The equations of motion and boundary conditions are derived by employing Hamilton‘s principle. The material properties are assumed to have a power law or sigmoid law variation across the depth. The available equations of motion are then solved using the Harmonic Differential Quadrature (HDQ) method to obtain the natural frequencies of the FGPM beam. The efficacy of the present method is validated by comparing the results with the previous published work. It is observed that the results obtained by HDQ method are at the same level of accuracy than those of previous analysis using GDQ method.

A theorem of Matsushima–Shimura shows that the space of harmonic differential forms on the quotient of products of upper half planes under the action of certain groups, when the quotient is compact, is the direct sum of two subspaces called the universal and cuspidal subspaces. We generalize this result to compact quotients of products of upper half planes and upper half spaces (hyperbolic three spaces) under the action of certain groups to obtain a similar decomposition.

Consider the bi-harmonic differential expression of the form $ A=\triangle _{M}^{2}+q\ $ on a manifold of bounded geometry (M,g) with metric g, where △M is the scalar Laplacian on M and q≥0 is a locally integrable function on M. In the terminology of Everitt and Giertz, the differential expression A is said to be separated in Lp(M), if for all u∈Lp(M) such that Au∈Lp(M), we have qu∈Lp(M). In this paper, we give sufficient conditions for A to be separated in Lp(M),where 1

This paper proposes an extended Stokes-Dirac structure for describing a global structure of port-Hamiltonian systems defined on manifolds with non-trivial topology under consistent boundary conditions. For the aim, the relationship between the Stokes-Dirac structure and the topological geometry of the manifolds is clarified in terms of harmonic differential forms.

We present a new numerical method for pricing credit default swaps under fully correlated multifactor reduced-form models. In particular, the proposed approach combines an implicit/explicit operator splitting procedure with the harmonic differential quadrature scheme, and is so efficient that it can be applied to models with up to six stochastic factors. This is a remarkable advantage, as we can use two factors to describe the interest rate, other two factors to describe the default probability, and other two factors to take into account, for example, the so-called counterparty risk. The performances of the novel method are demonstrated by extensive simulation, in which various kinds of models with four and six fully correlated factors are considered.

Generalising a conjecture of Singerman, it is shown that there are orientably regular chiral hypermaps (equivalently regular chiral dessins d'enfants) of every non-spherical type. The proof uses the representation theory of automorphism groups of Riemann surfaces acting on homology and on various spaces of differentials. Some examples are given.

International audience; For sufficiently smooth bounded plane domains, the equivalence between the inequalities of Babuška –Aziz for right inverses of the divergence and of Friedrichs on conjugate harmonic functions was shown by Horgan and Payne in 1983 [7]. In a previous paper [4] we proved that this equivalence, and the equality between the associated constants, is true without any regularity condition on the domain. In three dimensions, Velte [9] studied a generalization of the notion of conjugate harmonic functions and corresponding generalizations of Friedrich's inequality, and he showed for sufficiently smooth simply-connected domains the equivalence with inf-sup conditions for the divergence and for the curl. For this equivalence, Zsuppán [10] observed that our proof can be adapted, proving the equality between the corresponding constants without regularity assumptions on the domain. Here we formulate a generalization of the Friedrichs inequality for conjugate harmonic differential forms on bounded open sets in any dimension that contains the situations studied by Horgan–Payne and Velte as special cases. We also formulate the corresponding inf-sup conditions or Babuška –Aziz inequalities and prove their equivalence with the Friedrichs inequalities, including equality between the corresponding constants. No a-priori conditions on the regularity of the open set nor on its topology are assumed.

Sandwich structures are widely used in practice and thus various engineering theories adopting simplifying assumptions are available. However, most engineering theories of beams, plates and shells cannot recover all stresses accurately through their constitutive equations. Therefore, the soft-core is directly modeled by two-dimensional (2D) elasticity theory without any pre-assumption on the displacement field. The top and bottom faces act like the elastic supports on the top and bottom edges of the core. The differential equations of the 2D core are then solved by the harmonic differential quadrature method (HDQM). To circumvent the difficulties in dealing with the locally distributed load by point discrete methods such as the HDQM, a general and rigorous way is proposed to treat the locally distributed load. Detailed formulations are provided. The static behavior of sandwich panels under different locally distributed loads is investigated. For verification, results are compared with data obtained by ABAQUS with very fine meshes. A high degree of accuracy on both displacement and stress has been observed.

We construct differential forms of all orders and a covariant derivative together with its adjoint on the probability space of a standard Poisson process, using derivation operators. In this framewok we derive a de Rham–Hodge–Kodaira decomposition as well as Weitzenböck and Clark–Ocone formulas for random differential forms. As in the Wiener space setting, this construction provides two distinct approaches to the vanishing of harmonic differential forms.

This study reports the results of an investigation into the static analysis of microbeams based on nonlocal thermal elasticity theory by differential quadrature (DQ) and harmonic differential quadrature (HDQ) methods. To show the accuracy of the method, the results of present work are compared with those of other works. Different parameters such as temperature, length-to-thickness ratio, length-to-width ratio and boundary conditions are studied, too. From the knowledge of author, it is the first time that results for static analysis of microbeams by nonlocal thermal elasticity theory in thermal environment using two types of differential quadrature method are investigated and the results may be used as benchmarks for the future works.

Volumetric colon wall unfolding is a novel method for virtual colon analysis and visualization with valuable applications in virtual colonoscopy (VC) and computer-aided detection (CAD) systems. A volumetrically unfolded colon enables doctors to visualize the entire colon structure without occlusions due to haustral folds, and is critical for performing efficient and accurate texture analysis on the volumetric colon wall. Though conventional colon surface flattening has been employed for these uses, volumetric colon unfolding offers the advantages of providing the needed quantities of information with needed accuracy. This work presents an efficient and effective volumetric colon unfolding method based on harmonic differentials. The colon volumes are reconstructed from CT images and are represented as tetrahedral meshes. Three harmonic 1-forms, which are linearly independent everywhere, are computed on the tetrahedral mesh. Through integration of the harmonic 1-forms, the colon volume is mapped periodically to a canonical cuboid. The method presented is automatic, simple, and practical. Experimental results are reported to show the performance of the algorithm on real medical datasets. Though applied here specifically to the colon, the method is general and can be generalized for other volumes.

In this paper, the differential quadrature (DQ) method is presented for easy and effective analysis of isotropic functionally graded (FG) and functionally graded coated (FGC) thin plates with constant Poisson’s ratio and varying Young’s modulus in the thickness direction. The bending of FG and FGC plates under transverse loading has been studied using the polynomial differential quadrature (PDQ) and the harmonic differential quadrature (HDQ) methods. A three-dimensional elasticity solution for a moderately thick FG plate with exponential Young’s modulus is used as the benchmark. Two examples, including a thin FG rectangular plate and a thin FGC rectangular plate with sigmoidal Young’s modulus, are investigated. The numerical results of PDQ and HDQ methods reveal good agreement with other solutions. Also, it is shown that the formulations for thin FG plates and homogeneous plates are similar, except that the plane strain components of the middle surface in FG plates are not zero.

An examination of the accuracy and convergence behaviors of polynomial basis function differential quadrature (PDQ) and harmonic basis function differential quadrature (HDQ) for free vibration analysis of variable thickness thick skew plates will be carried out. The plate governing equations are based on the first-order shear deformation theory including the effects of rotary inertia. Arbitrary thickness variations will be assumed yielding a system of equations with nonlinear spatial dependent coefficients. Differential quadrature (DQ) analogs of the equations are obtained by transforming the governing equations and boundary conditions into the computational domains. Studies are carried out to examine the effects of different types of boundary conditions, skew angles, and thickness-to-length ratios for thin as well as moderately thick plates. The thickness is simulated by bilinear or nonlinear functions. The results are compared with those of other numerical schemes. It is concluded that both PDQ and HDQ yield accurate solutions for natural frequencies both at low and high modes of vibration. Some new results are presented for skew plates with bilinearly varying thickness and for different sets of boundary conditions, which can be used as benchmarks for future works.

The main purpose of this study is to compare the methods of differential quadrature (DQ) and harmonic differential quadrature (HDQ). For this purpose, DQ and HDQ methods are presented for buckling, bending, and free vibration analysis of thin isotropic plates and columns. Plates of different shapes such as rectangular, circular, square, skew, trapezoidal, annular, and sectorial plate subjected to different boundary conditions are selected to demonstrate the accuracy of the method. Four different support conditions are taken into consideration for columns. Numerical results are presented to illustrate the method and demonstrate its efficiency. It is emphasized that the HDQ method gives more accurate results and needs less grid points than the DQ method. (C) 2003 Elsevier Ltd. All rights reserved.

Let X=G/P be a homogeneous space of a complex semisimple Lie group G equipped with a hermitian metric. We study the action of the Hodge star operator on the space of harmonic differential forms on X. We obtain explicit combinatorial formulas for this action when X is an irreducible hermitian symmetric space of compact type., Comment: 15 pages, LaTeX, 4 figures

The concept of behavior spaces introduced by Shiba plays an important role of systematic investigation of abelian differentials on an open Riemann surface. A Shiba behavior space consists of harmonic differentials which satisfy a certain period condition and boundary behavior. In this paper, for any open Riemann surface of infinite genus we construct Shiba behavior spaces with arbitrarily prescribed period condition and with specific boundary behavior.

An endeavor to exploit three-dimensional elasticity solutions for bending and buckling of rectangular plates via the differential quadrature (DQ) and harmonic differential quadrature (HDQ) methods is performed. Unlike other works, the priority of this paper is to examine the computational characteristics of the two methods; therefore, we focus our studies only on the simply supported and clamped rectangular plates. To start with, we first outline the basic equations and boundary conditions describing the bending and buckling of rectangular plates followed by normalizing and discretizing them according to the DQ and HDQ algorithms. The resulting algebraic equation systems are then solved to obtain the solutions. Based on these solutions, the computational characteristics of the DQ and HDQ methods are investigated in terms of their numerical performances. It is found that the DQ method displays obvious superior convergence characteristics over the HDQ method for the three-dimensional static analysis of rectangular plates.

The structure of weighting coefficient matrices of Harmonic Differential Quadrature (HDQ) is found to be either centrosymmetric or skew centrosymmetric depending on the order of the corresponding derivatives. The properties of both matrices are briefly discussed in this paper. It is noted that the computational effort of the harmonic quadrature for some problems can be further reduced up to 75 per cent by using the properties of the above-mentioned matrices., 12 pages, 1 table, Original MS. Word format, published in Common. Numer. Methods. Engrg

Stress analyses for orthotropic composite materials containing a through crack under remote shear loads (Mode II) are conducted. By employing the complex theory, a harmonic differential equation was established for the orthotropic plates with axes normal to the three orthogonal planes of material symmetry. An analytical complex function was introduced by following the Westergaard approach. Stress around a mode II crack in the orthotropic composite plate is deduced to have an analytical form. In addition, the analytical solution for a mode II crack was examined in the case of isotropic materials. It demonstrated that the analytical solution obtained is correct for the mode II cracked orthotropic composite plates.

A layer potential based approach for boundary value problems for harmonic differential forms in nonsmooth domains is developed. This allows a complete and unified treatment of several fundamental problems in potential theory.

Harmonic differential quadrature method is developed for the free vibration analysis of linear elastic beams. In the method of differential quadrature, partial space derivatives of a function appearing in a differential equation are approximated by means of a polynomial expressed as the weighted linear sum of the function values at a preselected grid of discrete points. The weighting coefficients are treated as the unknowns. Applying this concept to the governing differential equation of beam gives a set of linear simultaneous equations, which are solved for the unknown weighting coefftcients by accounting for the boundary conditions. Beams of different support combinations such as clamped, simply supported, guided, and free are selected to demonstrate the accuracy of the method. Flexural vibration case is taken into consideration. First two frequencies are obtained in the applications. Numerical results are presented to illustrate the method and demonstrate its efficiency.

Recently, Bert, Wang and Striz [1, 2] applied the differential quadrature (DQ) and harmonic differential quadrature (HDQ) methods to analyze static and dynamic behaviors of anisotropic plates. Their studies showed that the methods were conceptually simple and computationally efficient in comparison to other numerical techniques. Based on some recent work by the present author [3, 4], the purpose of this note is to further simplify the formulation effort and improve computing efficiency in applying the DQ and HDQ methods for these cases.

This paper contains the proof of the assertion in its title. My motivation for considering this problem was the following. Let M be a compact differentiable manifold and let D be an elliptic differential operator between the spaces of C ~ sections of two bundles on M. In [4] M. F. Atiyah proves that if index D>0, then, for every Galois covering AT/---~M, the operator / ) induced by D on ~ /has nontrivial L 2 solutions/)u=0. Atiyah goes on to ask whether the same is true for coverings which are not Galois. His guess is that the answer is negative in general but that the counter-example will be difficult to construct. The simplest situation to consider in this connection is that of a surface and the operator whose index is the Euler characteristic. For infinite coverings, since every L 2 harmonic function would be constant and nonzero constants are not in L 2, Atiyah's question reduces to the question of existence of L 2 harmonic forms of degree one. It is shown here that a counter-example will not be found in this simple setting. From now on S will denote an oriented, compact surface of genus g=g(S)>l equipped with a smooth Riemannian metric. S--%S will be an arbitrary (usually infinite) covering of S, and A = A will be the Laplace operator on S with respect to the pull back metric. A differential (exterior form of degree one) on S is harmonic, i.e. Ao)=0, and in L 2 if and only if (cf. [15], Theorem 26)

Generalizing harmonic differential forms (rot ω=0, div ω=0 in M, M being a smooth riemannian manifold with boundary) of first and second kind (ω=0 and *ω=0 on ∂M resp.) within the framework of Hilbert space notation, it is possible to extend the meaning of the boundary conditions to non-smooth boundaries. It turns out that in this case the classical result is still valid for certain open subregions G of M: The dimension of the space of harmonic differential forms of second kind is given by the q-th Betti number of G; *-duality leads to the respective result for harmonic differential forms of first kind.

In this paper we introduce some polyhedra in Grassman manifolds which we call Grassmannian simplices. We study two aspects of these polyhedra: their combinatorial structure (Section 2) and their relation to harmonic differential forms on the Grassmannian (Section 3). Using this we obtain results about some new differential forms, one of which is the classical dilogarithm (Section 1). The results here unite two threads of mathematics that were much studied in the 19th century. The analytic one, concerning the dilogarithm, goes back to Leibnitz (1696) and Euler (1779) and the geometric one, concerning Grassmannian simplices, can be traced to Binet (1811) . In Section 4, we give some of this history along with some recent related results and open problems. In Section 0, we give as an introduction an account in geometric terms of the simplest cases.

On finite surfaces, the class of harmonic functions which are constant on each contour is a finite-dimensional vector space of functions with finite Dirichlet norm. This paper considers the corresponding class of functions on bordered surfaces of class SOg and generalizes some of the properties of harmonic measures on finite surfaces. In particular, for generalized harmonic measures, we investigate the level curves and their orthogonal trajectories. The principal results, Theorems 4.1 and 4.4, state, in a sense made precise, that almost all of the level curves of a generalized harmonic measure are analytic Jordan curves and almost all of their orthogonal trajectories begin and end on the border given in the definition of the surface. These results have application to the level curves of a Green's function via a theorem of Kuramochi. We also consider the question on a parabolic surface as to when a harmonic differential with finite norm and integral periods is a weak limit of period reproducing differentials.

1. Throughout this paper we denote by R an open Riemann surface and by Ro a relatively compact subdomain of R with the relative boundary aRo consisting of a finite number of mutually disjoint closed analytic Jordan curves. The open set R, = R Po can be considered to be a neighborhood of the ideal boundary 3 of R. For the sake of simplicity, we denote by a the common relative boundary aRo = aR and we fix the orientation of a positively with respect to the domain Ro. A harmonic differential a defined on R1 = R1 u a is called a harmonic singularity

There has been new interest developing in the study of the Hodge theory and p -harmonic differential forms in recent years, largely pertaining to applications in quasi-conformal analysis, electromagnetic theory, relativity theory, mathematical physics, nonlinear potential theory, etc. In this paper, we develop an effective method to find conjugate p -harmonic differential forms in R n by applying Hodge theory in calculations with the p -harmonic equation. As applications of the method, we obtain some interesting examples of conjugate p -harmonic differential forms.

Bending analysis of functionally graded single walled carbon nano tubes is presented in this paper. Carbon nano tubes are modeled as Euler-Bernoulli beam theory in this study. Harmonic differential quadrature (HDQ) method is used to discretize the governing equations. In order to show the accuracy of present work, the results are compared with those of other existing results. Then the effects of different parameters such as power law index, inner and outer radius of nano tubes and length nano tubes of are studied, too.