Fractional Fourier transform for quasi-periodic Bloch functions

The fractional Fourier transform (FRFT) for quasi-periodic Bloch functions is studied. An isomorphism between square-integrable functions on the real line and quasi-periodic Bloch functions is used to extend existing work on the fractional Fourier transform for the former functions to the latter. The properties of the FRFT for quasi-periodic Bloch functions are discussed, and various numerical examples are presented.     

A C<Superscript><Emphasis Type="Italic">k</Emphasis></Superscript> continuous generalized finite element formulation applied to laminated Kirchhoff plate model

A generalized finite element method based on a partition of unity (POU) with smooth approximation functions is investigated in this paper for modeling laminated plates under Kirchhoff hypothesis. The shape functions are built from the product of a Shepard POU and enrichment functions. The Shepard functions have a smoothness degree directly related to the weight functions adopted for their evaluation. The weight functions at a point are built as products of C ^∞ edge functions of the distance of such a point to each of the cloud boundaries. Different edge functions are investigated to generate C ^k functions. The POU together with polynomial global enrichment functions build the approximation subspace. The formulation implemented in this paper is aimed at the general case of laminated plates composed of anisotropic layers. A detailed convergence analysis is presented and the integrability of these functions is also discussed.     

Comparison of the performance of SSPH and MLS basis functions for two-dimensional linear elastostatics problems including quasistatic crack propagation

We use symmetric smoothed particle hydrodynamics (SSPH) and moving least squares (MLS) basis functions to analyze six linear elastostatics problems by first deriving their Petrov-Galerkin approximations. With SSPH basis functions one can approximate the trial solution and its derivatives by using different basis functions whereas with MLS basis functions the derivatives of the trial solution involve derivatives of the basis functions used to approximate the trial solution. The class of allowable kernel functions for SSPH basis functions includes constant functions which are excluded in MLS basis functions if derivatives of the trial solution are also to be approximated. We compare results for different choices of weight functions, size of the compact support of the weight function, order of complete polynomials, and number of particles in the problem domain. The two basis functions are also used to analyze crack initiation and propagation in plane stress mode-I deformations of a plate made of a linear elastic isotropic and homogeneous material with particular emphasis on the computation of the T-stress. The crack trajectories predicted by using the two basis functions agree well with those found experimentally.     

Weakly equilibrated basis functions for elasticity problems

In this paper weakly equilibrated basis functions (EqBFs) are introduced for the development of a boundary point method. This study is the extension of the one in (Int. J. Numer. Methods Engng. 81 (2010) 971–1018) using exponential basis functions (EBFs) which are available just for partial differential equations (PDEs) with constant coefficients. Here the EqBFs are evaluated numerically to solve more general PDEs with non-constant coefficients. The EqBFs are found through weighted residual integrals defined over a fictitious domain embedding the main domain. A series of Chebyshev polynomials are used for the construction of the basis functions. By properly choosing the weight functions as the product of two unidirectional functions, here with Gaussian distribution, the main 2D integrals are written as the product of the simpler 1D ones. The results of the integrals can be stored for further use; however in some particular cases the EqBFs may be stored as a set of library functions. The results may also be found useful for those who are interested in residual-free functions in other numerical methods. For the verification, we discuss on the validity of the solution through an essential and comprehensive test procedure followed by several numerical examples.     

Green functions for diffuse light in a medium comprising two turbid half-spaces

A review of Green functions for diffuse light in two semi-infinite scattering and absorbing half-spaces separated by a plane interface is presented. The frequency-domain Green functions for an intensity-modulated point source are derived within the diffusion approximation by the Hankel transform with respect to the variable in the plane of the interface. Green functions for a line source and a plane source parallel to the interface are obtained from the three-dimensional Green functions by the method of descent. Green functions for a steady state are obtained as a limit of zero modulation frequency. Connection of the frequency-domain Green functions with the time-domain Green functions is shown by use of the Fourier transform in time. The influence of the relative optical parameters, namely, the ratios of diffusion coefficients, absorption coefficients, and refractive indices of the two media on the shape of the contour lines of the specific intensity, is shown for the continuous and intensity-modulated point sources.     

On representations of anisotropic invariants

Simple relations are established between isotropic functions and anisotropic functions through some vectors or tensors which characterized the anisotropy group. The results enable us to obtain representations of anisotropic functions using the much well-known tables for representations of isotropic functions. Transverse isotropy, orthotropy and crystal classes of triclinic, monoclinic and rhombic systems are considered.     

Gaussian Surrogates for Computer Models With Time-Varying Inputs and Outputs

Computer models of dynamic systems produce outputs that are functions of time; models that solve systems of differential equations often have this character. In many cases, time series output can be usefully reduced via principal components to simplify analysis. Time-indexed inputs, such as the functions that describe time-varying boundary conditions, are also common with such models. However, inputs that are functions of time often do not have one or a few “characteristic shapes” that are more common with output functions, and so, principal component representation has less potential for reducing the dimension of input functions. In this article, Gaussian process surrogates are described for models with inputs and outputs that are both functions of time. The focus is on construction of an appropriate covariance structure for such surrogates, some experimental design issues, and an application to a model of marrow cell dynamics.     

The intrinsic XFEM: a method for arbitrary discontinuities without additional unknowns

A new method for treating arbitrary discontinuities in a finite element (FE) context is presented. Unlike the standard extended FE method (XFEM), no additional unknowns are introduced at the nodes whose supports are crossed by discontinuities. The method constructs an approximation space consisting of mesh‐based, enriched moving least‐squares (MLS) functions near discontinuities and standard FE shape functions elsewhere. There is only one shape function per node, and these functions are able to represent known characteristics of the solution such as discontinuities, singularities, etc. The MLS method constructs shape functions based on an intrinsic basis by minimizing a weighted error functional. Thereby, weight functions are involved, and special mesh‐based weight functions are proposed in this work. The enrichment is achieved through the intrinsic basis. The method is illustrated for linear elastic examples involving strong and weak discontinuities, and matches optimal rates of convergence even for crack‐tip applications. Copyright © 2006 John Wiley & Sons, Ltd.     

A new shear-flexible FRP-reinforced concrete slab element

In this paper, a simple shear-flexible rectangular layered FRP-reinforced concrete slab element is developed based on Mindlin–Reissner plate theory and Timoshenko’s composite beam functions for nonlinear finite element analysis of FRP-reinforced concrete slabs. The Timoshenko’s composite beam functions are developed for FRP-reinforced concrete slabs based on those for composite laminates. The plane displacement interpolation functions of a quadrilateral isoparametric element with drilling degrees of freedom are employed to describe the membrane effects and the bending effects of the slab element are represented by the rotation functions of the slab element derived from Timoshenko’s composite beam functions. Both geometric nonlinearity and material nonlinearity of the materials are included in the new element. The efficiency of the element for nonlinear finite element analysis of FRP-reinforced concrete slabs is validated by comparing the computed results of two numerical examples with those obtained from lab tests.     

On enrichment functions in the extended finite element method

This paper presents mathematical derivation of enrichment functions in the extended finite element method for numerical modeling of strong and weak discontinuities. The proposed approach consists in combining the level set method with characteristic functions as well as domain decomposition and reproduction technique. We start with the simple case of a triangular linear element cut by one interface across which displacement field suffers a jump. The main steps towards the derivation of enrichment functions are as follows: (1) extension of the subfields separated by the interface to the whole element domain and definition of complementary nodal variables; (2) construction of characteristic functions for describing the geometry and physical field; (3) determination of the sets of basic nodal variables; (4) domain decompositions according to Step 3 and then reproduction of the physical field in terms of characteristic functions and nodal variables; and (5) comparison of the piecewise interpolations formulated at Steps 3 and 4 with the standard extended finite element method form, which yields enrichment functions. In this process, the physical meanings of both the basic and complementary nodal variables are clarified, which helps to impose Dirichlet boundary conditions. Enrichment functions for weak discontinuities are constructed from deeper insights into the structure of the functions for strong discontinuities. Relationships between the two classes of functions are naturally established. Improvements upon basic enrichment functions for weak discontinuities are performed so as to achieve satisfactory convergence and accuracy. From numerical viewpoints, a simple and efficient treatment on the issue of blending elements is also proposed with implementation details. For validation purposes, applications of the derived functions to heterogeneous problems with imperfect interfaces are presented. Copyright © 2012 John Wiley & Sons, Ltd.     

A new technique to derive the Green’s type integral formula in thermoelasticity

New closed-form influence functions of a unit point heat source on elastic displacements and new Green’s type integral formula for a boundary-value problem (BVP) for a thermoelastic half-space are presented. The main difficulties in obtaining such results are observed in deriving the influence functions of a concentrated unit force onto elastic volume expansion and, also, in Green’s functions in heat conduction. For canonical Cartesian domains, these functions have been derived successfully for hundreds of BVPs and were published in a handbook. So, this paper shows the way to derive not only thermoelastic influence functions and Green’s type integral formulas for the half-space, but also for many new BVPs in thermoelasticity in other Cartesian canonical domains. Moreover, the technique proposed here may be applied in any orthogonal canonical domain provided by the lists of Green’s functions in heat conduction and influence functions for elastic volume expansion that are known.     

The use of bubble functions for the post-local buckling of plate assemblies by the finite strip method

The finite strip displacement functions for post-local buckling are augmented with so-called bubble functions, which are extra modes associated with internal or nodeless degrees of freedom. A non-linear finite strip method of analysis including bubble functions is described for the post-local buckling of geometrically imperfect plate assemblies. It is shown that the use of bubble functions improves significantly the convergence of the method. The non-linear finite strip method is then used to study channel and I-section members in compression and bending.     

A MIXED-HYBRID FINITE ELEMENT MODEL BASED ON ORTHOGONAL FUNCTIONS

A mixed-hybrid formulation for stress finite elements is presented. The stresses and the displacements in the domain of the element and the displacements on the boundary are simultaneously and independently approximated using orthogonal functions. The stress approximation functions are used as weighting functions in the weighted residual enforcement of the local compatibility and constitutive equations. Similarly, the displacement approximation functions in the domain and on the boundary are used as weighting functions in the weighting residual enforcement of the local equilibrium equation and of the static boundary conditions, respectively. Legendre polynomials and Fourier series are used to illustrate the performance of the finite element formulation when applied to elastostatic problems.     

Scattering of visible and near infrared radiation by NaCl ice and glacier ice

Measurements are presented for the phase functions of glacier ice, columnar NaCl ice, and grease ice. Results for glacier ice show a shoulder at 80° characteristic of scattering from vapor bubbles. The NaCl ice types show similar forward scattering but significantly stronger backscattering. The scattering is predominantly due to the brine inclusions, and in the visible region the phase functions are independent of wavelength. For columnar NaCl ice samples, the phase functions depend on sample orientation as well as deflection angle; however, this effect can be ignored for many applications. Orientation-averaged values are presented for general use, and the empirical results are fitted to bimodal Henyey-Greenstein functions.     

Solutions of the FPK equation for time-delayed dynamical systems with the continuous time approximation method

This paper presents a method of finite-dimensional Markov process (FDMP) approximation for stochastic dynamical systems with time delay and numerical solutions of probability density functions of the systems. Solutions of probability density functions of time-delayed systems are rare in the literature. The FDMP method preserves the standard state space format of the system, and allows us to apply all the existing methods and theories for analysis and control of stochastic dynamical systems and to compute the probability density functions efficiently. The solutions of the FPK equation for a linear time-delayed stochastic system are presented. The effects of different spectral differentiation schemes for the FDMP method on the probability density functions are compared.     

Causal FFT treatment applicable to singularity functions

The objective of this paper is to present a method of causal FFT treatment which is available to not only approximately time-limited and band-limited functions but also singularity functions appearing frequently in earthquake engineering problems such as time domain response analyses and seismic record processing. By means of the causal FFT treatment for a singularity function, it is possible to determine several causal functions. The properties of the obtained causal functions are discussed in detail. The 0 second and 0 Hz value problems of engineering importance concerning the determination of the discrete singularity values of non-time-limited or non-band-limited functions are presented. Finally, by presenting numerical examples, several problems existing in the FFT analyses of singularity functions and a method to solve the problems are explained in detail.     

Meshfree point collocation method with intrinsic enrichment for interface problems

A meshfree collocation method with an intrinsic wedge enrichment is presented for solving interface problems. To approximate the class of functions with discontinuous derivatives on the interface, the wedge is asymptotically added to the basis functions. A general class of wedge basis functions with specified orders of asymptotic behavior at the interface is developed for moving least square approximations. These are implemented in diffuse derivative methods where the shape functions are approximately differentiated. The reproducing properties of these approximations for the polynomial part and for the wedge function along straight boundaries of the basis are demonstrated. For curved boundaries, the reproducing properties of the wedge functions are more restricted. Numerical results show the ease of constructing the intrinsic enrichment and the robustness of the numerical scheme in solving interface problems.     

Achieving finite element mesh quality via optimization of the Jacobian matrix norm and associated quantities. Part II—A framework for volume mesh optimization and the condition number of the Jacobian matrix

Three‐dimensional unstructured tetrahedral and hexahedral finite element mesh optimization is studied from a theoretical perspective and by computer experiments to determine what objective functions are most effective in attaining valid, high‐quality meshes. The approach uses matrices and matrix norms to extend the work in Part I to build suitable 3D objective functions. Because certain matrix norm identities which hold for 2×2 matrices do not hold for 3×3 matrices, significant differences arise between surface and volume mesh optimization objective functions. It is shown, for example, that the equality in two dimensions of the smoothness and condition number of the Jacobian matrix objective functions does not extend to three dimensions and further, that the equality of the Oddy and condition number of the metric tensor objective functions in two dimensions also fails to extend to three dimensions. Matrix norm identities are used to systematically construct dimensionally homogeneous groups of objective functions. The concept of an ideal minimizing matrix is introduced for both hexahedral and tetrahedral elements. Non‐dimensional objective functions having barriers are emphasized as the most logical choice for mesh optimization. The performance of a number of objective functions in improving mesh quality was assessed on a suite of realistic test problems, focusing particularly on all‐hexahedral ‘whisker‐weaved’ meshes. Performance is investigated on both structured and unstructured meshes and on both hexahedral and tetrahedral meshes. Although several objective functions are competitive, the condition number objective function is particularly attractive. The objective functions are closely related to mesh quality measures. To illustrate, it is shown that the condition number metric can be viewed as a new tetrahedral element quality measure. Published in 2000 by John Wiley & Sons, Ltd.     

Wide‐range and accurate mixed‐mode weight functions and stress intensity factors for cracked discs

The Wu‐Carlsson displacement‐based weight function method is extended to obtain the mode I and mode II weight functions for the edge‐ and centre‐cracked discs. Compared with Fett's direct adjustment weight functions for the edge‐cracked discs, the present weight functions are more accurate and are applicable for a wider range of crack lengths. Using the present weight functions, extensive and highly accurate mixed‐mode stress intensity factors are obtained for the cracked discs subjected to diametrically compressive forces. Assuming perfect contact and using Coulomb friction law and the present weight functions, the mode II stress intensity factors for the cracked discs with consideration of friction are obtained and widely compared with the corresponding results from finite element analyses.     

Wave propagations in exponentially graded transversely isotropic half-space with potential function method

Time-harmonic response of a vertically graded transversely isotropic, linearly elastic half-space is analytically determined by introducing a new set of potential functions. The potential functions are set in such a way that the governing equations be simple and with physical meaning as well. In addition, the potential functions introduced in this paper are degenerated to a complete set of potential functions used frequently for wave propagations in homogeneous transversely isotropic media. Utilizing Fourier series and Hankel integral transforms, the governing equations for the potential functions are solved, after which the displacements and stresses are presented in the form of line integrals. Both the displacements and stresses determined here are collapsed on the solution previously reported for the constant profile transversely isotropic material. Because of complicated integrand functions, the integrals are evaluated numerically and presented graphically, where the effect of degree of change of material properties plays a major role, which may be recognized easily.