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Dependence of contact area on the resolution of fractal interfaces in elastic and inelastic problems

ENGINEERING COMPUTATIONS 2011年第5-6期28卷 717-746页

作者： Panagouli, Olympia Mistakidis, Euripidis Univ Thessaly Dept Civil Engn Lab Struct Anal & Design Volos Greece

Purpose - The purpose of this paper is to investigate the influence of the resolution with which interfaces of fractal geometry are represented, on the contact area and consequently on the contact interfacial stresses. The study is based on a numerical approach. The paper focuses on the differences between the cases of elastic and inelastic materials having as primary parameter the resolution of the interface. Design/methodology/approach - A multi-resolution parametric analysis is performed for fractal interfaces dividing a plane structure into two parts. On these interfaces, unilateral contact conditions are assumed to hold. The computer-generated surfaces adopted here are self-affine curves, characterized by a precise value of the resolution delta of the fractal set. Different contact simulations are studied by applying a horizontal displacements on the upper part of the structure. For every value of s, a solution is taken in terms of normal forces and displacements at the interface. The procedure is repeated for different values of the resolution delta. At each scale, a classical Euclidean problem is solved by using finite element models. In the limit of the finest resolution, fractal behaviour is achieved. Findings - The paper leads to a number of interesting conclusions. In the case of linear elastic analysis, the contact area and, consequently, the contact interfacial stresses depend strongly on the resolution of the fractal interface. Contrary, in the case of inelastic analysis, this dependence is verified only for the lower resolution values. As the resolution becomes higher, the contact area tends to become independent from the resolution. Originality/value - The originality of the paper lies on the results and the corresponding conclusions obtained for the case of inelastic material behaviour, while the results for the case of elastic analysis verify the findings of other researchers.

关键词： Rough interfaces fractal geometry fractal interpolation functions Unilateral contact fractal resolution Topography Stress (materials)

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fractal Haar system

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NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 2011年第12期74卷 4152-4165页

作者： Navascues, M. A. Univ Zaragoza Dept Matemat Aplicada E-50009 Zaragoza Spain

The Haar system is an alternative to the classical Fourier bases, being particularly useful for the approximation of discontinuities. The article tackles the construction of a set of fractal functions close to the Haar set. The new system holds the property of constitution of bases of the Lebesgue spaces of p-integrable functions on compact intervals. Likewise, the associated fractal series of a continuous function is uniformly convergent. The case p = 2 owns some peculiarities and is studied separately. (C) 2011 Elsevier Ltd. All rights reserved.

关键词： fractal interpolation functions Haar wavelets Bases of functional spaces

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fractal INTERPOLANTS ON THE s-SETS

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fractalS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY 2010年第3期18卷 343-348页

作者： Paramanathan, P. Uthayakumar, R. Deemed Univ Dept Math Gandhigram Rural Inst Gandhigram 624302 Tamil Nadu India

In this paper, we mainly study the s-sets (regular 1-sets), which is the most important fractal in the study of fractal geometry. The regular 1-sets are subsets of countable collection of rectifiable curves. Also we define new real maps on the s-sets by using the methodology based on fractal interpolation functions. In addition, some results are applied to the waveform signals for interpolation.

关键词： fractal interpolation functions s-Set Regular 1-Sets Iterated Function Systems Open Set Condition

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Reconstruction of Sampled Signals with fractal functions

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ACTA APPLICANDAE MATHEMATICAE 2010年第3期110卷 1199-1210页

作者： Navascues, M. A. Univ Zaragoza Dept Matemat Aplicada Ctr Politecn Super Ingenieros Zaragoza 50018 Spain

This paper proposes a procedure to build a fractal model for real sampled signals like financial series, climatic data, bioelectric recordings, etc. The mapping constructed owns in general a rich geometric structure. In a first step, the method provides a truncate Chebyshev approximant which performs a low-pass filtering of the signal, displaying in this way the leading cycles of the phenomenon observed. In the second, the polynomial is transformed in a fractal object. The Lipschitz properties of the original signal guarantee a good approximation of the represented variable, whenever the sampling frequency is high enough.

关键词： fractal interpolation functions Chebyshev series Signal processing

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fractal Approximation

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COMPLEX ANALYSIS AND OPERATOR THEORY 2010年第4期4卷 953-974页

作者： Navascues, M. A. Univ Zaragoza Ctr Politecn Super Ingn Zaragoza 50018 Spain

In the present article every complex square integrable function defined in a real bounded interval is approached by means of a complex fractal function. The approximation depends on a partition of the interval and a vectorial parameter of the iterated function system providing the fractal attractor. The original may be discontinuous or undefined in a set of zero measure. The fractal elements can modify the features of the originals, for instance their character of smooth or non-smooth. The properties of the operator mapping every function into its fractal analogue are studied in the context of the uniform and least square norms. In particular, the transformation provides a decomposition of the set of square integrable maps. An orthogonal system of fractal functions is constructed explicitly for this space. Sufficient conditions for the uniform convergence of the fractal series expansion corresponding to this basis are also deduced. The fractal approximation of real functions is obtained as a particular case.

关键词： fractal interpolation functions Least squares approximation Orthogonal systems

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fractal interpolation on a Torus

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ACTA APPLICANDAE MATHEMATICAE 2009年第1期106卷 93-104页

作者： Navascues, M. A. Univ Zaragoza Dept Matemat Aplicada Ctr Politecn Super Ingn Zaragoza 50018 Spain

A very general method of fractal interpolation on T (1) is proposed in the first place. The approach includes the classical cases using trigonometric functions, periodic splines, etc. but, at the same time, adds a diversity of fractal elements which may be more appropriate to model the complexity of some variables. Upper bounds of the committed error are provided. The arguments avoid the use of derivatives in order to handle a wider framework. The Lebesgue constant of the associated partition plays a key role. The procedure is proved convergent for the interpolation of specific functions with respect to some nodal bases. In a second part, the approximation is then extended to bidimensional tori via tensor product of interpolation spaces. Some sufficient conditions for the convergence of the process in the Fourier case are deduced.

关键词： fractal interpolation functions interpolation on surfaces Torus fractals

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Fitting fractal surfaces on non-rectangular grids

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fractalS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY 2008年第2期16卷 151-158页

作者： Navascues, M. A. Sebastian, M. V. Univ Zaragoza Dept Matemat Aplicada Ctr Politecn Super Ingn Zaragoza 50018 Spain Univ Zaragoza Dept Matemat Aplicada Escuela Univ Ingn Tecn Ind Zaragoza 50018 Spain

In a complex society, the visualization and interpretation of large amounts of data acquires an increasing importance. This can be done at once by means of two- or three-dimensional maps. To approach this problem, we undertake the construction of several variable fractal functions. In the first place, we introduce real fractal functions defined as perturbations of the classical ones. These basic mappings allow us to compute multidimensional approximations of experimental variables by means of linear combinations of products of fractal functions of Legendre type. The paper proposes a method of non-smooth representation for a large number of three-dimensional data on non-uniform grids. The procedures described are applied in the last part of the paper to the implementation of fitting maps for brain electrical activity.

关键词： fractal interpolation functions Legendre polynomials multivariant polynomial approximation

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fractal interpolants on the unit circle

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APPLIED MATHEMATICS LETTERS 2008年第4期21卷 366-371页

作者： Navascues, M. A. Univ Zaragoza Dpto Matemat Aplicada Ctr Politech Super Ingenieros Zaragoza 50018 Spain

A methodology based on fractal interpolation functions is used in this work to define new real maps on the circle generalizing the classical ones. A partition on the circle and a scale vector enable the modification of the definition and properties of the standard periodic functions. The fractal analogues can be constructed even if the originals are not continuous. The new functions provide Hilbert bases for the square integrable maps on the circle. (C) 2007 Elsevier Ltd. All rights reserved.

关键词： fractal interpolation functions harmonic analysis trigonometric polynomials circular functions iterated function systems

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Natural bicubic spline fractal interpolation

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NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 2008年第11期69卷 3679-3691页

作者： Chand, A. K. B. Navascues, M. A. Univ Zaragoza Dept Matemat Aplicada Ctr Politecn Super Ingn Zaragoza 50018 Spain

fractal interpolation functions provide natural deterministic approximation of complex phenomena. Cardinal cubic splines are developed through moments (i.e. second derivative of the original function at mesh points). Using tensor product, bicubic spline fractal interpolants are constructed that successfully generalize classical natural bicubic splines. An upper bound of the difference between the natural cubic spline blended fractal interpolant and the original function is deduced. In addition, the convergence of natural bicubic fractal interpolation functions towards the original function providing the data is studied. (C) 2007 Elsevier Ltd. All rights reserved.

关键词： fractals Iterated function systems fractal interpolation functions Cardinal splines Blending function Surface approximation

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Fundamental sets of fractal functions

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ACTA APPLICANDAE MATHEMATICAE 2008年第3期100卷 247-261页

作者： Navascues, M. A. Chand, A. K. B. Univ Zaragoza Ctr Politecn Super Ingn Dept Matemat Aplicada Zaragoza 50018 Spain

fractal interpolants constructed through iterated function systems prove more general than classical interpolants. In this paper, we assign a family of fractal functions to several classes of real mappings like, for instance, maps defined on sets that are not intervals, maps integrable but not continuous and may be defined on unbounded domains. In particular, based on fractal interpolation functions, we construct fractal Muntz polynomials that successfully generalize classical Muntz polynomials. The parameters of the fractal Muntz system enable the control and modification of the properties of original functions. Furthermore, we deduce fractal versions of classical Muntz theorems. In this way, the fractal methodology generalizes the fundamental sets of the classical approximation theory and we construct complete systems of fractal functions in spaces of continuous and p-integrable mappings on bounded domains.

关键词： fractal interpolation functions Muntz space Muntz theorems density fundamental sets

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