Computational Modeling of Structural Problems Using Atomic Basis Functions

Kozulić, Vedrana; Gotovac, Blaž

doi:10.1007/978-3-319-19443-1_17

Cited by 5 publications

(5 citation statements)

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“…As well-known saying goes "new wine needs new skin bags". To develop lacunary interpolation new tools were neededthe splines [12,[16][17][18]. Numerous examples of applications of polynomial splines to lacunary interpolation are in [25][26][27][28][29][30][31][32][33][34][35][36].…”

Section:  mentioning

confidence: 99%

Tomic Functions and Lacunary Interpolation Series in Boundary Value Problems for Partial Derivatives Equations and Image Processing

Rvachov¹,

Rvachova²,

Tomilova³

2020

reks

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In the paper we consider and solve the problem of construction of the so called tomic functions – the systems of infinitely differentiable functions which while retaining many important properties of the shifts of atomic function up(x) such as locality and representation of algebraic polynomials and being based on the atomic functions nevertheless have nonuniform character and therefore allow to take into account the inhomogeneous and changing character of the data encountered in real world problems in particular in boundary value problems for partial differential equations with variable coefficients and complex geometry of domains in which these boundary value problems must be solved. The same class of tomic functions can be applied to processing,denoising and sparse storage of signals and images by lacunary interpolation. The lacunary or Birkhoff interpolation of functions in which the function is being restored by the values of derivatives of orderin points in which values of function and derivatives of order k<r are unknown is of great importance in many real world problems such as remote sensing. The lacunary interpolation methods using the tomic functions possesss important advantages over currently widely applied lacunary spline interpolation in view of infinite smoothness of tomic functions.The tomic functions can also be applied to connect (to stitch) atomic expansions with different steps on different intervals preserving smoothness and optimal approximation properties. The equations for of construction oftomic functions tofuj(x) –analogues of the basic functions of the generalized atomic Taylor expansions are obtained – which are needed for lacunary (Birkhoff) interpolation. For the applications in variational and collocation methods for solving bondary value problems for partial derivative and integral equations the tomic functions ftupr,j(x) are obtained that are analogues of B-splines and atomic functions fupn(x). Using similar methods, the tomic functions based on other atomic functions such as Ξn(x) can be obtained.

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Section:  mentioning

confidence: 99%

Tomic Functions and Lacunary Interpolation Series in Boundary Value Problems for Partial Derivatives Equations and Image Processing

Rvachov¹,

Rvachova²,

Tomilova³

2020

reks

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“…First, the problem (18) was analyzed by proposed method using solution structure (2) with a different density of uniform collocation points which make up a grid that covers the given two-dimensional domain but does not conform to it. Fig.…”

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confidence: 99%

“…Rvachev and Rvachev [13], in their pioneering work, called these basis functions ˝atomic˝ because they span the vector spaces of all three fundamental functions in mathematics: algebraic, exponential and trigonometric polynomials. In numerical modelling, we applied Fup basis functions that belong to the atomic functions of algebraic type [16], [17], [18]. All derivatives of atomic Fup basis functions required by differential operators in the solution structure can be used directly in the numerical procedure.…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution of Poisson’s Equation in an Arbitrary Domain by Using Meshless R-Function Method

Kozulić¹,

Gotovac²

2016

Proceedings of the 27th International DAAAM Symposium 2016

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This paper describes a numerical procedure that uses solution structure method, atomic basis functions and a collocation technique. Solution structure method is based on the theory of R-functions. The solution of a boundary value problem is expressed in the form of formulae called solution structure which depends on three components: the first component describes the geometry of the domain exactly in analytical form, the second describes all boundary conditions exactly, while the third component is called differential component because it contains information about governing equation. Unknown differential component of the solution structure is represented by a linear combination of basis functions. Here, we propose to use atomic basis functions because of their good approximation properties. To determine the coefficients of linear combination in the solution structure, a collocation technique is used. Combination of atomic basis functions and solution structure method gives the meshfree method that can be applied for solving boundary value problems in domains of arbitrarily complex geometry with complex boundary conditions. This paper summarizes the main principles of the proposed method and presents its application to solution of the torsion problem.

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“…The work in [16] gives a generalization of atomic functions to the multivariable case. The use of Fup basis functions, which are atomic functions of the algebraic type, has been shown to solve the problem of signal processing [17], the initial value problem [18], the boundary value problems using the Fup Collocation Method [19], the boundary-initial value problems [20], elasto-plastic analysis of prismatic bars subjected to torsion [21], and modeling of groundwater flow and transport problems [22]. Gotovac et al [23] presented a true multiresolution approach based on the Adaptive Fup Collocation Method (AFCM).…”

Section: Introductionmentioning

confidence: 99%

The Class of Atomic Exponential Basis Functions EFupn(x,ω)-Development and Application

Kurbaša¹,

Gotovac²,

Kozulić³

2023

Computer Modeling in Engineering &Amp; Sciences

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The purpose of this paper is to present the class of atomic basis functions (ABFs) which are of exponential type and are denoted by EFup n (x, ω). While ABFs of the algebraic type are already represented in the numerical modeling of various problems in mathematical physics and computational mechanics, ABFs of the exponential type have not yet been sufficiently researched. These functions, unlike the ABFs of the algebraic type Fup n (x), contain the tension parameter ω, which gives them additional approximation properties. Exponential monomials up to the nth degree can be described exactly by the linear combination of the functions EFup n (x, ω). The function EFup n for n = 0 is called the "mother" ABF of the exponential type, i.e., EFup 0 (x, ω) ≡ Eup(x, ω). In other words, the functions EFup n (x, ω) are elements of the linear vector space EUP n and retain all the properties of their "mother" function Eup(x, ω). Thus, this paper, in terms of its content and purpose, can be understood as a sequel of the article by Brajčić Kurbaša et al., which shows the basic properties and application of the basis function Eup (x, ω). This paper presents, in an analogous way, the development and application of the exponential basis functions EFup n (x, ω). Here, for the first time, expressions for calculating the values of the functions EFup n (x, ω) and their derivatives are given in a form suitable for application in numerical analyses, which is shown in the verification examples of the approximations of known functions.

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Computational Modeling of Structural Problems Using Atomic Basis Functions

Cited by 5 publications

References 1 publication

Tomic Functions and Lacunary Interpolation Series in Boundary Value Problems for Partial Derivatives Equations and Image Processing

Tomic Functions and Lacunary Interpolation Series in Boundary Value Problems for Partial Derivatives Equations and Image Processing

Numerical Solution of Poisson’s Equation in an Arbitrary Domain by Using Meshless R-Function Method

The Class of Atomic Exponential Basis Functions EFupn(x,ω)-Development and Application

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