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Christos Xenophontos - One of the best experts on this subject based on the ideXlab platform.
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The Singular Function Boundary Integral Method for Singular Laplacian problems over circular sections
Applied Mathematics and Computation, 2010Co-Authors: Evgenia Christodoulou, Christos Xenophontos, Georgios C GeorgiouAbstract:The Singular Function Boundary Integral Method (SFBIM) for solving two-dimensional elliptic problems with boundary Singularities is revisited. In this method the solution is approximated by the leading terms of the asymptotic expansion of the local solution, which are also used to weight the governing partial differential equation. The Singular coefficients, i.e., the coefficients of the local asymptotic expansion, are thus primary unknowns. By means of the divergence theorem, the discretized equations are reduced to boundary integrals and integration is needed only far from the Singularity. The Dirichlet boundary conditions are then weakly enforced by means of Lagrange multipliers, the discrete values of which are additional unknowns. In the case of two-dimensional Laplacian problems, the SFBIM converges exponentially with respect to the numbers of Singular Functions and Lagrange multipliers. In the present work the method is applied to Laplacian test problems over circular sectors, the analytical solution of which is known. The convergence of the method is studied for various values of the order p of the polynomial approximation of the Lagrange multipliers (i.e., constant, linear, quadratic, and cubic), and the exact approximation errors are calculated. These are compared to the theoretical results provided in the literature and their agreement is demonstrated.
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The Singular Function Boundary Integral Method for Elliptic Problems with Boundary Singularities
Recent Advances in Boundary Element Methods, 2009Co-Authors: Evgenia Christodoulou, Christos Xenophontos, Georgios C GeorgiouAbstract:We review the Singular Function Boundary Integral Method (SFBIM) for solving two-dimensional elliptic problems with boundary Singularities. In this method the solution is approximated by the leading terms of the asymptotic expansion of the local solution. The unknowns to be calculated are the Singular coefficients, i.e. the coefficients of the local asymptotic expansion, also called generalized stress intensity factors. The discretized Galerkin equations are reduced to boundary integrals by means of the divergence theorem. The Dirichlet boundary conditions are then weakly enforced by means of Lagrange multipliers, the values of which are additional unknowns. In the case of two-dimensional Laplacian problems, we have shown that this method converges exponentially with respect to the number of Singular Functions. This is demonstrated via several benchmark applications, including ones involving the biharmonic operator which can be viewed as an extension of the theory.
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the Singular Function boundary integral method for biharmonic problems with crack Singularities
Engineering Analysis With Boundary Elements, 2007Co-Authors: Miltiades Elliotis, Georgios C Georgiou, Christos XenophontosAbstract:We use the Singular Function boundary integral method (SFBIM) to solve two model fracture problems on the plane. In the SFBIM, the solution is approximated by the leading terms of the local asymptotic solution expansion, which are also used to weight the governing biharmonic equation in the Galerkin sense. The discretized equations are reduced to boundary integrals by means of the divergence theorem and the Dirichlet boundary conditions are weakly enforced by means of Lagrange multipliers. The main advantage of the method is that the leading stress intensity factors (SIFs) are calculated directly together with the Lagrange multipliers, i.e. no post-processing of the numerical solution is necessary. The numerical results for the two model problems show the fast convergence of the method and compare well with those of the collocation Trefftz method.
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a Singular Function boundary integral method for laplacian problems with boundary Singularities
SIAM Journal on Scientific Computing, 2006Co-Authors: Christos Xenophontos, Miltiades Elliotis, Georgios C GeorgiouAbstract:A Singular Function boundary integral method for Laplacian problems with boundary Singularities is analyzed. In this method, the solution is approximated by the truncated asymptotic expansion for the solution near the Singular point and the Dirichlet boundary conditions are weakly enforced by means of Lagrange multiplier Functions. The resulting discrete problem is posed and solved on the boundary of the domain, away from the point of Singularity. The main result of this paper is the proof of convergence of the method; in particular, we show that the method approximates the generalized stress intensity factors, i.e., the coefficients in the asymptotic expansion, at an exponential rate. A numerical example illustrating the convergence of the method is also presented.
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the Singular Function boundary integral method for a two dimensional fracture problem
Engineering Analysis With Boundary Elements, 2006Co-Authors: Miltiades Elliotis, Georgios C Georgiou, Christos XenophontosAbstract:The Singular Function boundary integral method (SFBIM) originally developed for Laplacian problems with boundary Singularities is extended for solving two-dimensional fracture problems formulated in terms of the Airy stress Function. Our goal is the accurate, direct computation of the associated stress intensity factors, which appear as coefficients in the asymptotic expansion of the solution near the crack tip. In the SFBIM, the leading terms of the asymptotic solution are used to approximate the solution and to weight the governing biharmonic equation in the Galerkin sense. The discretized equations are reduced to boundary integrals by means of Green's theorem and the Dirichlet boundary conditions are weakly enforced by means of Lagrange multipliers. The numerical results on a model problem show that the method converges extremely fast and yields accurate estimates of the leading stress intensity factors.
Miltiades Elliotis - One of the best experts on this subject based on the ideXlab platform.
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the Singular Function boundary integral method for an elastic plane stress wedge beam problem with a point boundary Singularity
Applied Mathematics and Computation, 2014Co-Authors: Miltiades Elliotis, Dimos C Charmpis, Georgios C GeorgiouAbstract:The Singular Function boundary integral method (SFBIM) is applied for the numerical solution of a 2-D Laplace model problem of a perfectly elastic wedge beam under plane stress conditions. The beam has a point boundary Singularity, it includes a curved boundary part and is subjected to non-trivial distributed external loading. The implemented solution method converges for this special model problem extremely fast. The numerical estimates attained for the leading Singular coefficients of the local asymptotic expansion and the stress and strain fields are highly accurate, as verified by comparison with the available analytical solution.
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the Singular Function boundary integral method for biharmonic problems with crack Singularities
Engineering Analysis With Boundary Elements, 2007Co-Authors: Miltiades Elliotis, Georgios C Georgiou, Christos XenophontosAbstract:We use the Singular Function boundary integral method (SFBIM) to solve two model fracture problems on the plane. In the SFBIM, the solution is approximated by the leading terms of the local asymptotic solution expansion, which are also used to weight the governing biharmonic equation in the Galerkin sense. The discretized equations are reduced to boundary integrals by means of the divergence theorem and the Dirichlet boundary conditions are weakly enforced by means of Lagrange multipliers. The main advantage of the method is that the leading stress intensity factors (SIFs) are calculated directly together with the Lagrange multipliers, i.e. no post-processing of the numerical solution is necessary. The numerical results for the two model problems show the fast convergence of the method and compare well with those of the collocation Trefftz method.
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a Singular Function boundary integral method for laplacian problems with boundary Singularities
SIAM Journal on Scientific Computing, 2006Co-Authors: Christos Xenophontos, Miltiades Elliotis, Georgios C GeorgiouAbstract:A Singular Function boundary integral method for Laplacian problems with boundary Singularities is analyzed. In this method, the solution is approximated by the truncated asymptotic expansion for the solution near the Singular point and the Dirichlet boundary conditions are weakly enforced by means of Lagrange multiplier Functions. The resulting discrete problem is posed and solved on the boundary of the domain, away from the point of Singularity. The main result of this paper is the proof of convergence of the method; in particular, we show that the method approximates the generalized stress intensity factors, i.e., the coefficients in the asymptotic expansion, at an exponential rate. A numerical example illustrating the convergence of the method is also presented.
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the Singular Function boundary integral method for a two dimensional fracture problem
Engineering Analysis With Boundary Elements, 2006Co-Authors: Miltiades Elliotis, Georgios C Georgiou, Christos XenophontosAbstract:The Singular Function boundary integral method (SFBIM) originally developed for Laplacian problems with boundary Singularities is extended for solving two-dimensional fracture problems formulated in terms of the Airy stress Function. Our goal is the accurate, direct computation of the associated stress intensity factors, which appear as coefficients in the asymptotic expansion of the solution near the crack tip. In the SFBIM, the leading terms of the asymptotic solution are used to approximate the solution and to weight the governing biharmonic equation in the Galerkin sense. The discretized equations are reduced to boundary integrals by means of Green's theorem and the Dirichlet boundary conditions are weakly enforced by means of Lagrange multipliers. The numerical results on a model problem show that the method converges extremely fast and yields accurate estimates of the leading stress intensity factors.
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solving laplacian problems with boundary Singularities a comparison of a Singular Function boundary integral method with the p hp version of the finite element method
Applied Mathematics and Computation, 2005Co-Authors: Miltiades Elliotis, Georgios C Georgiou, Christos XenophontosAbstract:We solve a Laplacian problem over an L-shaped domain using a Singular Function boundary integral method as well as the p/hp finite element method. In the former method, the solution is approximated by the leading terms of the local asymptotic solution expansion, and the unknown Singular coefficients are calculated directly. In the latter method, these coefficients are computed by post-processing the finite element solution. The predictions of the two methods are discussed and compared with recent numerical results in the literature.
Georgios C Georgiou - One of the best experts on this subject based on the ideXlab platform.
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the Singular Function boundary integral method for an elastic plane stress wedge beam problem with a point boundary Singularity
Applied Mathematics and Computation, 2014Co-Authors: Miltiades Elliotis, Dimos C Charmpis, Georgios C GeorgiouAbstract:The Singular Function boundary integral method (SFBIM) is applied for the numerical solution of a 2-D Laplace model problem of a perfectly elastic wedge beam under plane stress conditions. The beam has a point boundary Singularity, it includes a curved boundary part and is subjected to non-trivial distributed external loading. The implemented solution method converges for this special model problem extremely fast. The numerical estimates attained for the leading Singular coefficients of the local asymptotic expansion and the stress and strain fields are highly accurate, as verified by comparison with the available analytical solution.
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The Singular Function Boundary Integral Method for Singular Laplacian problems over circular sections
Applied Mathematics and Computation, 2010Co-Authors: Evgenia Christodoulou, Christos Xenophontos, Georgios C GeorgiouAbstract:The Singular Function Boundary Integral Method (SFBIM) for solving two-dimensional elliptic problems with boundary Singularities is revisited. In this method the solution is approximated by the leading terms of the asymptotic expansion of the local solution, which are also used to weight the governing partial differential equation. The Singular coefficients, i.e., the coefficients of the local asymptotic expansion, are thus primary unknowns. By means of the divergence theorem, the discretized equations are reduced to boundary integrals and integration is needed only far from the Singularity. The Dirichlet boundary conditions are then weakly enforced by means of Lagrange multipliers, the discrete values of which are additional unknowns. In the case of two-dimensional Laplacian problems, the SFBIM converges exponentially with respect to the numbers of Singular Functions and Lagrange multipliers. In the present work the method is applied to Laplacian test problems over circular sectors, the analytical solution of which is known. The convergence of the method is studied for various values of the order p of the polynomial approximation of the Lagrange multipliers (i.e., constant, linear, quadratic, and cubic), and the exact approximation errors are calculated. These are compared to the theoretical results provided in the literature and their agreement is demonstrated.
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The Singular Function Boundary Integral Method for Elliptic Problems with Boundary Singularities
Recent Advances in Boundary Element Methods, 2009Co-Authors: Evgenia Christodoulou, Christos Xenophontos, Georgios C GeorgiouAbstract:We review the Singular Function Boundary Integral Method (SFBIM) for solving two-dimensional elliptic problems with boundary Singularities. In this method the solution is approximated by the leading terms of the asymptotic expansion of the local solution. The unknowns to be calculated are the Singular coefficients, i.e. the coefficients of the local asymptotic expansion, also called generalized stress intensity factors. The discretized Galerkin equations are reduced to boundary integrals by means of the divergence theorem. The Dirichlet boundary conditions are then weakly enforced by means of Lagrange multipliers, the values of which are additional unknowns. In the case of two-dimensional Laplacian problems, we have shown that this method converges exponentially with respect to the number of Singular Functions. This is demonstrated via several benchmark applications, including ones involving the biharmonic operator which can be viewed as an extension of the theory.
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the Singular Function boundary integral method for biharmonic problems with crack Singularities
Engineering Analysis With Boundary Elements, 2007Co-Authors: Miltiades Elliotis, Georgios C Georgiou, Christos XenophontosAbstract:We use the Singular Function boundary integral method (SFBIM) to solve two model fracture problems on the plane. In the SFBIM, the solution is approximated by the leading terms of the local asymptotic solution expansion, which are also used to weight the governing biharmonic equation in the Galerkin sense. The discretized equations are reduced to boundary integrals by means of the divergence theorem and the Dirichlet boundary conditions are weakly enforced by means of Lagrange multipliers. The main advantage of the method is that the leading stress intensity factors (SIFs) are calculated directly together with the Lagrange multipliers, i.e. no post-processing of the numerical solution is necessary. The numerical results for the two model problems show the fast convergence of the method and compare well with those of the collocation Trefftz method.
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a Singular Function boundary integral method for laplacian problems with boundary Singularities
SIAM Journal on Scientific Computing, 2006Co-Authors: Christos Xenophontos, Miltiades Elliotis, Georgios C GeorgiouAbstract:A Singular Function boundary integral method for Laplacian problems with boundary Singularities is analyzed. In this method, the solution is approximated by the truncated asymptotic expansion for the solution near the Singular point and the Dirichlet boundary conditions are weakly enforced by means of Lagrange multiplier Functions. The resulting discrete problem is posed and solved on the boundary of the domain, away from the point of Singularity. The main result of this paper is the proof of convergence of the method; in particular, we show that the method approximates the generalized stress intensity factors, i.e., the coefficients in the asymptotic expansion, at an exponential rate. A numerical example illustrating the convergence of the method is also presented.
Lluís Bibiloni - One of the best experts on this subject based on the ideXlab platform.
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A New Singular Function
The American Mathematical Monthly, 2011Co-Authors: Jaume Paradís, Pelegrí Viader, Lluís BibiloniAbstract:A new continuous strictly increasing Singular Function is described with the help of the ternary and binary systems for real number representation; in this, our Function is similar to Cantor's func...
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Riesz-Nágy Singular Functions revisited
SSRN Electronic Journal, 2006Co-Authors: Jaume Paradís, Pelegrí Viader, Lluís BibiloniAbstract:In 1952 F. Riesz and Sz.Nagy published an example of a monotonic continuous Function whose derivative is zero almost everywhere, that is to say, a Singular Function. Besides, the Function was strictly increasing. Their example was built as the limit of a sequence of deformations of the identity Function. As an easy consequence of the definition, the derivative, when it existed and was finite, was found to be zero. In this paper we revisit the Riesz-N´agy family of Functions and we relate it to a system for real number representation which we call (t, t-1)–expansions. With the help of these real number expansions we generalize the family. The Singularity of the Functions is proved through some metrical properties of the expansions used in their definition which also allows us to give a more precise way of determining when the derivative is 0 or infinity.
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A Total Order in (0, 1] Defined Through a ‘Next’ Operator
Order, 1999Co-Authors: Jaume Paradís, Pelegrí Viader, Lluís BibiloniAbstract:A ‘next’ operator, σ, is built on the set R _1 =(0, 1] − {1 − 1 / e } defining a partial order that, with the help of the axiom of choice, can be extended to a total order in R _1. In addition, the orbits {σ^ n (α)}_ n ∈ Z are all dense in R _1 and are constituted by elements of the same arithmetical character: if α is an algebraic irrational of degree k , all the elements in α's orbit are algebraic of degree k ; if α is transcendental, all are transcendental. Moreover, the asymptotic distribution Function of the sequence formed by the elements in any of the half-orbits is a continuous, strictly increasing, Singular Function very similar to the well-known Minkowski's ?(⋅) Function.
Jaume Paradís - One of the best experts on this subject based on the ideXlab platform.
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A Singular Function with a non-zero finite derivative on a dense set with Hausdorff dimension one
Journal of Mathematical Analysis and Applications, 2016Co-Authors: Juan Fernández Sánchez, Pelegrí Viader, Jaume Paradís, Manuel Díaz CarrilloAbstract:Abstract This article closes a trilogy on the existence of Singular Functions with non-zero finite derivatives. In two previous papers, the authors had exhibited a continuous strictly increasing Singular Function from [ 0 , 1 ] into [ 0 , 1 ] with a derivative that takes non-zero finite values at two different zero-measure sets: first, at the points of an uncountable set; then at the points of a dense set in [ 0 , 1 ] . In the present paper, the possibilities are further stretched as the construction is improved to extend it to an uncountable dense set whose intersection with any interval ( a , b ) has Hausdorff dimension one. Another feature of this third article is the construction of the required Function using the most paradigmatic of the Singular Functions: the Cantor–Lebesgue one.
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A Singular Function with a non-zero finite derivative on a dense set
Nonlinear Analysis: Theory Methods & Applications, 2014Co-Authors: Juan Fernández Sánchez, Pelegrí Viader, Jaume Paradís, Manuel Díaz CarrilloAbstract:Abstract The authors had exhibited in a previous paper a continuous strictly increasing Singular Function from [ 0 , 1 ] into [ 0 , 1 ] with a derivative that takes non-zero finite values at the points of an uncountable set. In this article, the construction is improved to encompass a dense set.
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A Singular Function with a non-zero finite derivative
Nonlinear Analysis: Theory Methods & Applications, 2012Co-Authors: Juan Fernández Sánchez, Pelegrí Viader, Jaume Paradís, Manuel Díaz CarrilloAbstract:Abstract This paper exhibits, for the first time in the literature, a continuous strictly increasing Singular Function with a derivative that takes non-zero finite values at some points. For all the known “classic” Singular Functions—Cantor’s, Hellinger’s, Minkowski’s, and the Riesz–Nagy one, including its generalizations and variants—the derivative, when it existed and was finite, had to be zero. As a result, there arose a strong suspicion (almost a conjecture) that this had to be the case for any Singular Function. We present here a Singular Function, constructed as a patchwork of known classic Singular Functions, with derivative 1 on a subset of the Cantor set.
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A New Singular Function
The American Mathematical Monthly, 2011Co-Authors: Jaume Paradís, Pelegrí Viader, Lluís BibiloniAbstract:A new continuous strictly increasing Singular Function is described with the help of the ternary and binary systems for real number representation; in this, our Function is similar to Cantor's func...
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Riesz-Nágy Singular Functions revisited
SSRN Electronic Journal, 2006Co-Authors: Jaume Paradís, Pelegrí Viader, Lluís BibiloniAbstract:In 1952 F. Riesz and Sz.Nagy published an example of a monotonic continuous Function whose derivative is zero almost everywhere, that is to say, a Singular Function. Besides, the Function was strictly increasing. Their example was built as the limit of a sequence of deformations of the identity Function. As an easy consequence of the definition, the derivative, when it existed and was finite, was found to be zero. In this paper we revisit the Riesz-N´agy family of Functions and we relate it to a system for real number representation which we call (t, t-1)–expansions. With the help of these real number expansions we generalize the family. The Singularity of the Functions is proved through some metrical properties of the expansions used in their definition which also allows us to give a more precise way of determining when the derivative is 0 or infinity.