alternate bem formulations for 2 and 3 d anisotropic thermoelasticity abstract direct boundary element formulations are presented for the elastostatic analysis of two and three dimensional general anisotropic media subject to known temperature distributions the thermal effects are incorporated in the analysis through a volume integral or alternatively using particular integrals a multi region boundary element approach is adopted in which particular integrals are derived assuming a quadratic temperature distribution in each sub region a multiple regression scheme is used for the best fit quadratic temperature distribution in each sub region for temperature data given at discrete points the multi region analysis capability of the numerical implementation enhances its capacity to model discontinuous temperature fields as well as piecewise homogeneous media such as laminated composites the formulations presented are incorporated in a general purpose system and are illustrated with examples

random multifractals negative dimensions and the resulting limitations of the thermodynamic formalism the story described in this paper has started with the death or survival criterion which the author published in 1972 1974 and had obtained in 1968 while investigating kolmogorov s hypothesis that the turbulent dissipation epsilon dx in a box is log normally distributed using this criterion the present paper discusses the concrete significance of negative fractal dimensions they arise in those random multifractal measures for which the cramer function f alpha the spectrum of singularities satisfies f alpha 0 and concerns three aspects of such measures a the fine grained multifractal properties which are non random and the same for almost all realizations b the properties obtained by using the partition function formalism and c the typical coarse grained multifractal properties however the f alpha s in the range where f alpha 0 say nothing about the variability of coarse grained properties between samples a description of these fluctuations hence a fuller multifractal description of the measure is shown to be provided by the values of f alpha in the range where f alpha 0 we prefer to reserve the term thermodynamic for the fine grained and partition functional properties and to say that the coarse grained properties go beyond the thermodynamics i e are not macroscopic but mesoscopic

on a class of skewed self similar and hyperbolic fractals probabilistic recurrent renewal theory is used herein to describe the geometry of a new class of self similar distributions in a skewed multiplicative mass splitting procedure it is shown how this model class can account for a deterministic and conservative treelike structure in connection with recent developments on anomalous multiplicative multifractals coexistence of both micro and macroscopic scales in such objects is a consequence of the underlying renewal phenomena being transient as is shown in a second part of the article various entropylike measures of the intrinsic disorder prevailing in these structures are proposed

boundary element method for thermoelastic analysis of three dimensional transversely isotropic solids abstract thermal effects are well known to manifest themselves as additional volume integral terms in the direct formulation of the boundary integral equation bie for linear elastic solids when using the boundary element method bem this domain integral has been successfully transformed in an exact manner to surface ones only in isotropy and in 2d anisotropy thereby restoring the bem as a truly boundary solution technique the difficulties with extending it to 3d general anisotropic solids lie in the mathematical complexity of the green s function and its derivatives for such materials these quantities are required items in the bem formulation in this paper the exact analytical transformation of the volume integral associated with thermal effects to surface ones is achieved for a transversely isotropic material using a similar approach which the authors have previously employed for the same task in bem for 2d general anisotropy a numerical scheme however needs to be employed to evaluate some of the new terms introduced in the surface integrals that arise from this process here the mathematical soundness of the formulation is demonstrated by a few examples the numerical results obtained are checked by alternative means including those obtained from the commercial fem code ansys

treatment of body force volume integrals in bem by exact transformation for 2 d anisotropic elasticity in the boundary element method bem based on the direct formulation body force effects manifest themselves as an additional volume integral term in the boundary integral equation bie the numerical solution of the integral equation with this term destroys the notion of the bem as atruly boundary solution method this paper discusses the treatment of this volume integral for two dimensional anisotropic elasticity with body forces present the analytical basis for transforming this integral exactly into boundary ones is presented for geometrically convex regions this restores the application of the bem to such problems as a truly boundary solution technique numerical examples are presented to demonstrate the veracity of the transformation and implementation 1997 by john wiley sons ltd

diffusion limited aggregation a paradigm of disorderly cluster growth the purpose of this talk is to present a brief overview of our group s recent research into dynamic mechanisms of disorderly growth an exciting new branch of condensed matter physics in which the methods and concepts of modern statistical mechanics are proving to be useful our strategy has been to focus on attempting to understand a single model system diffusion limited aggregation dla this philosophy was the guiding principle for years of research in phase transitions and critical phenomena for example by focusing on the ising model steady progress was made over a period of six decades and eventually led to understanding a wide range of critical point phenomena since even systems for which the ising model was not appropriate turned out to be described by variants of the ising model such as the xy and heisenberg models so also we are optimistic that whatever we may learn in trying to understand dla will lead to generic information helpful in understanding general aspects of dynamic mechanisms underlying disorderly growth

phase transition in the topological properties of the generalized percolation model abstract within the generalized percolation model we study scaling behavior of the cluster size chemical distance distribution function gsl specifying the number of clusters of mass s and chemical distance l using the exact series analysis we show that the kth moments of the gsl function with exp l taken as a fractal measure for k 0 gsl possesses a power law dependence on s with a constant gap exponent which agrees with previous results for the mass chemical distance probability distribution function and the mean chemical distance of the percolation clusters and lattice animals

multiple reciprocity boundary element analysis of two dimensional anisotropic thermoelasticity involving an internal arbitrary non uniform volume heat source abstract in the direct boundary element method bem formulation of anisotropic thermoelasticity thermal loads manifest themselves as additional volume integral terms in the boundary integral equation bie conventionally this requires internal cell discretisation throughout the whole domain in this paper the multiple reciprocity method in bem analysis is employed to treat the general 2d thermoelasticity problem when the thermal loading is due to an internal non uniform volume heat source by successively performing the volume to surface integral transformation the general formulation of the associated bie for the problem is derived the successful implementation of such a scheme is illustrated by three numerical examples

boundary element analysis of uncoupled transient thermo elastic problems with time and space dependent heat sources abstract a boundary element method bem for the analysis of two and three dimensional uncoupled transient thermo elastic problems involving time and space dependent heat sources is presented the domain integrals are efficiently treated using the cartesian transformation and the radial integration methods without considering any internal cells similar to the dual reciprocity method drm some internal points without any connectivity are considered however in contrast to the drm any arbitrary mesh free interpolation method can be used in the present formulation there is no need to find any particular solutions and the shape functions in the mesh free interpolation method can be arbitrary and sufficiently complicated unlike the drm the generated system of equations contains the unknowns only on the boundary after finding the primary unknowns on the boundary the temperature displacement and stress components at all internal points can directly be found without solving any system of equations three examples with different forms of heat sources are presented to demonstrate the efficiency and accuracy of the proposed method although the proposed bem is mathematically more complicated than domain methods such as the finite element method fem it is more efficient from a modelling viewpoint since only the surface mesh has to be generated in the presented method

determination of interior point stresses in two dimensional bem thermoelastic analysis of anisotropic bodies abstract in conventional boundary element method bem analysis the displacements and stresses at an interior point of an elastic body are obtained using the respective somigliana s identities after the boundary solutions for the displacements and tractions have been solved for this paper presents the derivation of somigliana s identity for the strains at any interior point of an anisotropic domain subjected to thermal effects from these strains the corresponding stresses at the interior point may be calculated directly using the duhamel neumann relation this identity is obtained in terms of integrals over the boundary of the solution domain only difficulties arising from the exact transformation of the volume integral term associated with thermal loading in bem analysis into surface integrals in the anisotropic case are discussed and the procedures to overcome them are described three numerical examples are presented to demonstrate the veracity of the formulations developed

negative fractal dimensions and multifractals abstract a new notion of fractal dimension is defined when it is positive it effectively falls back on known definitions but its motivating virtue is that it can take negative values which measure usefully the degree of emptiness of empty sets the main use concerns random multifractals for which f

self similarity of harmonic measure on dla abstract the right hand side of the curve of the harmonic measure on dla is undefined this does not necessarily imply that the harmonic measure and the dla geometry are not self similar we show for off lattice dla that the right hand tail satisfies a different rescaling rule this cauchy rescaling is compatible with self similarity the analysis is done on off off lattice dla in which both the brownian motion and the laplace equation are off lattice the cluster sizes range between 32 and 50 000 atoms the square lattice used to numerically estimate the laplacian potential introduces a lower cutoff on the spatial resolution of this potential we find a dependence of the right tail of the distribution of holders on this ultraviolet cutoff whereas the shape of the tail does depend on this ultraviolet lattice cutoff the applicability of the collapse rules do not

on the properties of random multiplicative measures with the multipliers exponentially distributed under the formalism of annealed averaging of the partition function a type of random multifractal measures with the multipliers exponentially distributed is investigated in detail branching emerges in the curve of generalized dimensions and negative values of generalized dimensions arise three equivalent methods of classification of the random multifractal measures are proposed which is based on i the properties of the curves of generalized dimensions ii the solution properties of equation q 0 and iii the relative position of the curve f and the diagonal f in the first quadrant these classes of measures correspond to 0 1 0 1 1 and 0 1 0 respectively phase diagram is introduced to illustrate the diverse performance of the random measures that is multiplicatively generated

new anomalous multiplicative multifractals left sided ƒ α and the modelling of dla abstract a sharp distinction is drawn between general multiplicative multifractals as originally introduced by the author and the more familiar but more restricted class defined by frisch and parisi and by halsey et al all the general multiplicative multifractals are exactly renormalizable by design however this is important and perhaps suprising it is shown that the scaling relations that serve to define the restricted class fail to extend to certain multiplicative multifractals as a result in addition to the familiar restricted class the general multifractals are shown to allow for a wide variety of anomalous behaviors several of which are described we believe that suitable examples of these anomalies are of the most important concrete applications to fully developed turbulence and to diffusion limited aggregates dla this paper concerns several new classes of multifractals we have recently proposed as tools to model the anomalies of dla they are left sided that is characterized by max with max d 0 related multifractals characterized by max 0 and even by max and min 0 are encountered along the way

multifractals in physics successes dangers and challenges abstract the multifractal formalism for describing distributions of measures on sets which may be fractal is reviewed with particular attention to random resistor networks diffusion limited aggregates and mass fractals the formalism has been very successful in describing crossover phenomena but its use is somewhat dangerous due to problems with negative moments with averaging procedures and with finite size effects

multifractal formalism for self similar bridges we derive the thermodynamics of self similar paths or bridges joining the two points and of the plane these paths may be constituted with both macroscopic and microscopic fragments each deserving its specific statistics while remaining continuous such discontinuous paths are also studied with some information related to the statistics of their jumps if the bridges under study are bound to be non decreasing paths this study coincides with the one of multifractal measures on the unit interval relaxing this condition leads to an extension of the multifractal formalism whose main lines are derived here

behaviour of the harmonic measure at the bottom of fjords diffusion limited aggregates are among many important fractal shapes that involve deep indentations usually called fjords to estimate the harmonic measure at the bottom of a fjord seems a prohibitive task but the authors find that a new mathematical equality due to beurling carleson and jones makes it easy they find that the harmonic measure at the bottom of a fjord as a function of its euclidean depth can exhibit a wide range of behaviours they introduce an infinite family of model fjords for which the equality takes a very simple form in this family the decay of the harmonic measure at their bottoms can be for example power law semi exponential stretched exponential and exponentially stretched exponential they show that self affinity or randomness can lead to faster than power law decays of the minimal growth probability on boundaries

degenerate scales for boundary value problems in anisotropic elasticity abstract degenerate scales usually refer to a size effect which causes non unique solutions of boundary integral equations for certain type of boundary value problems with a unique solution they are closely connected to the presence of a logarithmic function in the integral kernel of the single layer potential operator the equations of the elasticity theory provide one of the known application fields where degenerate scales appear the paper discusses conditions and formula for controlling and detection of the degenerate scales in the case of fully anisotropic analysis no restrictions are considered for the material only the loading should cause two dimensional deformation of the anisotropic body a technique for the evaluation of the degenerate scales is discussed and tested the examples provide results of special simple cases and demonstrate suitability of the proposed technique in relation to calculation of degenerate scales by numerical solution of pertinent boundary integral equation by the boundary element method

numerical investigations of discrete scale invariance in fractals and multifractal measures fractals and multifractals and their associated scaling laws provide a quantification of the complexity of a variety of scale invariant complex systems here we focus on lattice multifractals which exhibit complex exponents associated with observable log periodicity we perform detailed numerical analyses of lattice multifractals and explain the origin of three different scaling regions found in the moments a novel numerical approach is proposed to extract the log frequencies in the non lattice case there is no visible log periodicity i e no preferred scaling ratio since the set of complex exponents spreads irregularly within the complex plane a non lattice multifractal can be approximated by a sequence of lattice multifractals so that the sets of complex exponents of the lattice sequence converge to the set of complex exponents of the non lattice one an algorithm for the construction of the lattice sequence is proposed explicitly

multifractality of the harmonic measure on fractal aggregates and extended self similarity abstract we show that dla follows a surprising new scaling rule it expresses that the screened region in which the harmonic measure is tiny increases more than proportionately as the cluster grows this scaling rule also gives indirect evidence that the harmonic measure of lattice dla follows a hyperbolic probability distribution of exponent equal to 1 this distribution predicts that sample moments behave erratically hence explains why the common restricted multifractal formalism fails to apply to dla

multifractal analysis of complex random cascades we achieve the multifractal analysis of a class of complex valued statistically self similar continuous functions for we use multifractal formalisms associated with pointwise oscillation exponents of all orders our study exhibits new phenomena in multifractal analysis of continuous functions in particular we find examples of statistically self similar such functions obeying the multifractal formalism and for which the support of the singularity spectrum is the whole interval 0

multifractal analysis of first order phase transitions in an ising system with four spin interactions abstract the multifractal formalism is applied for investigating probability measures of energy levels of an ising model with four spin interactions on finite size hexagonal lattices the holder exponent characterizing singularities of these measures is determined as a function of the temperature variable and the ratio of the strength of four spin interactions to the strength of two spin couplings it is shown that for the variable values at which the system reveals the existence of a precursor of the first order phase transition the maximal holder exponent associated with the energy region of the most concentrated probability measure possesses a minimum which is nondifferentiable with respect to temperature

complex multifractal measures and a generalized multifractal formalism abstract strongly inhomogeneous multifractal measures involving extremely diverse local probability scales and exhibiting left sided multifractal spectra are considered the standard multifractal formalism is extended to such drastically inhomogeneous multifractal measures by introducing additional filtering variables and scaling exponents conjugated to these variables it is shown that the generalized multifractal formalism provides a more adequate and complete description of complex structures of strongly inhomogeneous multifractal measures in comparison with conventional formalism