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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.

In this work, the harmonic differential quadrature (HDQ) method is employed to investigate the vibration characteristics of axially functionally graded (AFG) tapered beam. The material parameters of beam are assumed to be varied continuously along beam length according to power law function. The Hamilton’s principle is used to obtain the equation of motion considering Timoshenko beam theory. Then the governing equations are solved using harmonic differential quadrature method. The non-dimensional frequencies are obtained for clamped–clamped type of boundary condition. The competency of the present method is then validated by the other available numerical techniques in the literature. The influence of non-homogeneity parameter, taper ratio and aspect ratio on the frequencies of the AFG tapered beam is discussed in detail.

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 aim of this investigation is to study the effect of selection of mesh distribution on efficacy of natural frequency of Timoshenko beam. The harmonic differential quadrature (HDQ) method is used to compute the natural frequency of Timoshenko beam. In this regards, three different types of meshing are used namely uniform meshing, normalized meshing and non-uniform. The results obtained using these meshing are compared with results obtained in previous published work under different types of boundary conditions. It is found that non-uniform meshing gives better convergence as well as good results as compare to results using uniform and normalized meshing under all boundary conditions.

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.

A highly accurate method for obtaining static deflections of a thin annular plate is helpful to effectively design the complicated structures with these plates. There have been numerous methods to achieve such a target. However, there is no direct technical literature for comparing these methods comprehensively. Therefore, the current study aims at performing comparison of three methods, optimization method (OM), finite element method (FEM), and harmonic differential quadrature (HDQ) method. Combining an instance, the comparisons give us insight into high accuracy and consistency of each other, showing high accuracy of the methods in this field. Compared with the results of FEM, the maximum error, less than 1%, demonstrates that the accuracy of the OM is high enough. Combining the small errors, the excellent stability of those brings a reliable method in this field. The maximum error and fluctuation drawn from the HDQ are evidently larger than those of the OM, and it is difficult to obtain results with higher accuracy based on the HDQ. Finally, the work described here suggests that the OM can be utilized to deal with such a complex problem in view of engineering and theory, and the HDQ method is more suitable to study the method for solving very complex partial differential systems of high order.

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.

In this paper we obtain sufficient conditions for the bi-harmonic differential operator A = ΔE2 + q to be separated in the space L2 (M) on a complete Riemannian manifold (M,g) with metric g, where ΔE is the magnetic Laplacian onM and q ≥ 0 is a locally square integrable function on M. Recall that, in the terminology of Everitt and Giertz, the differential operator A is said to be separated in L2 (M) if for all u ∈ L2 (M) such that Au ∈ L2 (M) we have ΔE2u ∈ L2 (M) and qu ∈ L2 (M).

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.

Using harmonic differential quadrature method, an approach to analyze sandwich cylindrical shell panels with any sort of boundary conditions under a generally distributed static loading, undergoing elasto-plastic deformation is proposed. The faces of the sandwich shell panel are made of some isotropic materials with linear work hardening behavior while the core is assumed to be an isotropic material experiencing only elastic behavior. The faces are modeled as thin cylindrical shells obeying the Kirchhoff–Love assumptions. For the core material, it is assumed to be thick and the in-plane stresses are negligible. Upon application of an inner and outer general lateral loading, the governing equations are derived using the principle of virtual displacements. Using an iterative approach, named elasto-plastic harmonic differential quadrature method (EP-HDQM), the equations are solved. The obtained results are compared with the results from finite element software Ansys for different sandwich shell panel configurations. Then, the effects of changing different parameters on the stress and displacement components of sandwich cylindrical shell panels in different elasto-plastic conditions are investigated.

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.

Surface stress and small-scale effects on nonlinear vibration analysis of a single-layer boron nitride sheet are investigated based on theories of nonlocal and surface piezoelasticity. The single-layer boron nitride sheet is embedded in an elastic medium which is simulated by Pasternak model. Considering electromechanical coupling, the discretize governing equations of motion are obtained by Hamilton’s principal. Two numerical methods, harmonic differential quadrature and differential cubature, are employed to determine the nondimensional nonlinear frequency of single-layer boron nitride sheet. The detailed study parameters are conducted to investigate the influences of the small-scale, temperature change, surface effects, elastic medium, vibrational mode, aspect ratio, and smartness of sheet on the nonlinear nonlocal vibration characteristics of single-layer boron nitride sheet. It is concluded that ignoring surface and small-scale effects lead to inaccurate results in vibrational response of the single-layer boron nitride sheet. Furthermore, differential cubature method yields accurate results with less computational effort with respect to harmonic differential quadrature method. The obtained results of this study may be useful for designing of nano-electro mechanical system or micro-electro mechanical system and other nano-/micro-smart structures.

In this paper, based on the high-order theory (HOT) of sandwich structures, the response of sandwich cylindrical shells with flexible core and any sort of boundary conditions under a general distributed static loading is investigated. The faces and the core are made of isotropic materials. The faces are modeled as thin cylindrical shells obeying the Kirchhoff–Love assumptions. For the core material, it is assumed to be thick and the in-plane stresses are negligible. The governing equations are derived using the principle of the stationary potential energy. Using harmonic differential quadrature method (HDQM), the equations are solved for deformation components. The obtained results are compared with finite element results for different sandwich shell configurations. Then, the effects of changing different parameters on the stress and displacement components of sandwich cylindrical shells are investigated. A comparison between HOT-HDQM and finite element results is presented for different sandwich shell configurations.

We present a framework for the construction of Weitzenbock and Clark-Ocone formulae for differential forms on the probability space of a normal martingale. This approach covers existing constructions based on Brownian motion, and extends them to other normal martingales such as compensated Poisson processes. It also applies to the path space of Brownian motion on a Lie group and to other geometries based on the Poisson process. Classical results such as the de Rham-Hodge-Kodaira decomposition and the vanishing of harmonic differential forms are extended in this way to finite difference operators by two distinct approaches based on the Weitzenbock and Clark-Ocone formulae.

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 we express the theory of harmonic differential forms on conformal Euclidean manifolds in terms of the so called Clifford multivector fields. The aim is to give good definitions for d and d* operators in Clifford multivector case. Using these definitions we derive a formula for the Laplace operator. Three fundamental examples are included in the end of the paper and connections to existing theory is discussed.

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.

Given a complete ortho-normal system ϕ = (ϕ0,ϕ1,ϕ2,…) of L2H(\( \mathcal{D} \)), the space of holomorphic and absolutely square integrable functions in the bounded domain \( \mathcal{D} \) of ℂn, we construct a holomorphic imbedding \( \iota _\phi :\mathcal{D} \to \mathfrak{F}(n,\infty ) \), the complex infinite dimensional Grassmann manifold of rank n. It is known that in \( \mathfrak{F}(n,\infty ) \) there are n closed (μ, μ)-forms τμ (μ = 1,…,n) which are invariant under the holomorphic isometric automorphism of \( \mathfrak{F}(n,\infty ) \) and generate algebraically all the harmonic differential forms of \( \mathfrak{F}(n,\infty ) \). So we obtain in \( \mathcal{D} \) a set of (μ, μ)-forms ιϕ*τμ (μ = 1,…, n), which are independent of the system ϕ chosen and are invariant under the bi-holomorphic transformations of \( \mathcal{D} \). Especially the differential metric ds21 associated to the Kahler form ιϕ*τ1 is a Kahler metric which differs from the Bergman metric ds2 of \( \mathcal{D} \) in general, but in case that the Bergman metric is an Einstein metric, ds12 differs from ds2 only by a positive constant factor.

Let Fr (0 < r < m + 1) be a smooth r-vector valued function in a suitable open domain of \({\mathbb{R}}^{m+1}\) satisfying \(\partial F_r = 0\) in Ω, where ∂ is the Dirac operator in \({\mathbb{R}}^{m+1}\) . Then it is proved that there exists Hr, an r-vector valued harmonic function in Ω, such that Fr = \(\partial H_r\partial\). Two proofs of this structure theorem are given, one based on properties of harmonic differential forms and one relying upon primitivation of monogenic functions.

The space HFk(Ω) of harmonic multi-vector fields in a domain \(\Omega \subset \mathbb{R}^{n}\) as introduced in [1] is closely connected to the space of harmonic forms. The main aim of this paper is to characterize the dual space of HFk(E) being \(\mathbf{E} \subset \mathbb{R}^{n}\) a compact set. It is proved that HFk(E)* is isomorphic to a certain quotient space of so-called harmonic pairs outside E vanishing at infinity.

This paper presents a semi-analytical numerical method for surface location error (SLE) prediction in milling processes, governed by a time-periodic delay-differential equation (DDE) in state-space form. The time period is discretized as a set of sampling grid points. By using the harmonic differential quadrature method (DQM), the first-order derivative in the DDE is approximated by the linear sums of the state values at all the sampling grid points. On this basis, the DDE is discretized as a set of algebraic equations. A dynamic map can then be constructed to simultaneously determine the stability and the steady-state SLE of the milling process. To obtain optimal machining parameters, an optimization model based on the milling dynamics is formulated and an interior point penalty function method is employed to solve the problem. Experimentally validated examples are utilized to verify the accuracy and efficiency of the proposed approach.

In the present paper, axisymmetric vibrations of a circular sandwich plate with relatively stiff core of exponentially varying thickness have been investigated. The face sheets are treated as membranes of constant thickness and the core is assumed to be solid as well as moderately thick. The equations of motion have been derived using Hamilton’s energy principle. The frequency equation for clamped boundary condition is obtained by employing harmonic differential quadrature method. The lowest three roots of this equation have been reported as the frequencies for the first three modes of vibration. The effect of various plate parameters on the natural frequencies has been studied. Three-dimensional mode shapes for a specified sandwich plate have been illustrated. A comparison of the results with published work has been made.

The geometrically nonlinear static and dynamic analysis of thin rectangular plates resting on elastic foundation has been studied. Winkler–Pasternak foundation model is considered. Dynamic analogues Von Karman equations are used. The governing nonlinear partial differential equations of the plate are discretized in space and time domains using the harmonic differential quadrature (HDQ) and finite differences (FD) methods, respectively. The analysis provides for both clamped and simply supported plates with immovable inplane boundary conditions at the edges. Various types of dynamic loading, namely a step function, a sinusoidal pulse and an N-wave, are investigated and the results are presented graphically. The accuracy of the proposed HDQ–FD coupled methodology is demonstrated by the numerical examples.

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 non-linear static and dynamic response of doubly curved thin isotropic shells has been studied for the step and sinusoidal loadings. Dynamic analogues Von Karman-Donnel type shell equations are used. Clamped immovable and simply supported immovable boundary conditions are considered. The governing nonlinear partial differential equations of the shell are discretized in space and time domains using the harmonic differential quadrature (HDQ) and finite differences (FD) methods, respectively. The accuracy of the proposed HDQ-FD coupled methodology is demonstrated by the numerical examples. Numerical examples demonstrate the satisfactory accuracy, efficiency and versatility of the presented approach.

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.

A superfluid film flowing along the walls of a porous material can be modeled by a harmonic differential on a punctured Riemann surface satisfying integrality conditions on its periods and residues. In this paper, we use classical Riemann surface theory to mathematically investigate the observed physical phenomenon of the appearance of pairs of vortices in the fluid and the resulting patterns of circulation. We show that the existence of strings of circulation in the superfluid depends on a quantized divisibility condition between the number of vortex–antivortex pairs and the circulation of the fluid.

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.

An accuracy study between the Differential Quadrature (DQ) and Harmonic Differential Quadrature (HDQ) methods for three-dimensional elasticity solutions of free vibration of rectangular plates is carried out. The solution capability of the DQ and HDQ methods is first studied. Then the numerical performance of both the methods is compared. It is found that the DQ method displays more superior convergence characteristics over the HDQ method for the lower modes of vibration. However, the HDQ method is advantageous over the DQ method for computing higher modes of vibration. It is also discovered that the DQ and HDQ methods produce better convergent solutions than the Finite Element Method (FEM) when a similar number of discrete points/nodes are used. Copyright © 1999 John Wiley & Sons, Ltd.

A variant of the classical theorem of Runge is established for harmonic differenrial forms on an open subset of ℝn. It generalizes the case of analytic functions for n=2. Harmonic forms with point singularities are introduced, and a theorem of displacement of poles is proved. An integral representation analogous to the Cauchy formula is constructed. Bibliography: 5 titles.

In this paper we consider harmonic differential forms and Clifford multi-vector functions on the hyperbolic upper half-space. We see how the operators and their solutions are related and present a Moisil-Theodorescu-type system related to the harmonic multi-vectors.

Analytical stress analyses are presented for orthotropic composite materials containing a through crack under uniaxial normal loads (mode I). A harmonic differential equation has been established for the orthotropic plates with axes normal to the three orthogonal planes of material symmetry by introducing new complex variables. The complex theory was employed to find stress functions to satisfy the equilibrium equation, compatibility equation and boundary condition at infinite and crack surfaces. An analytical solution was examined in the case of isotropic materials. It is demonstrated that the analytical solution obtained is correct for the orthotropic composite 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.

A global method of generalised differential quadrature is applied to solve the two-dimensional incompressible Navier-Stokes equations in the vorticity-stream-function formulation. Numerical results for the flow past a circular cylinder were obtained using just a few grid points. A good agreement is found with the experimental data.

A theoretical solution is given for the linearized buckling problem of an orthotropic toroidal shell with an elliptical cross-section under external pressure loading. The solution is based on the Sanders-Budiansky shell theory, and makes use of the harmonic differential quadrature method. Theory developed earlier for the buckling of orthotropic shells of revolution, and the vibration of orthotropic elliptical toroidal shells, is incorporated in the present work. Numerical results obtained from the solution are compared with results given in the literature, and good correspondence is generally observed. A parametric study is then conducted, covering a wide range of material and geometric parameters. Regression formulas are derived, indicating the variation of the buckling pressure with the degree of orthotropy of the material. Overall, the study introduces a new tool for the buckling analysis of elliptical toroidal shells, and extends the information available for orthotropic toroidal shells.Copyright © 2009 by ASME

A classical subject in the theory of Riemann surfaces is the study of harmonic and analytic differentials. In particular it is well-known that the dimensions of the spaces of harmonic and analytic differentials with respect to the corresponding fields are 2g, where g is the topological genus of the surface. A further study of this classical setting about the connection between the topological and conformal structure with different spaces of differential forms is due to A. Pfluger, who obtained some important decomposition theorems in this connection derived from the Dirichlet principle. Here we shall consider some of these questions for Klein surfaces, that is, we allow the surface to be non-orientable or to have a boundary.

We study a generalization of the Hadamard theorem on three circles to harmonic differential forms. An inequality for the L 2-norms of a harmonic form over concentric spheres is proved. Also, we obtain an estimate for the L ∞-norms.