Dr. Dimitris Vartziotis

@article{PausingerVartziotis_2021,

title = {On the symmetry of finite sums of exponentials},

author = {Florian Pausinger and Dimitris Vartziotis},

doi = {10.4171/EM/446},

issn = {0013-6018},

year  = {2021},

date = {2021-04-01},

journal = {Elemente der Mathematik},

volume = {76},

number = {2},

pages = {62–73},

abstract = {In this note we are interested in the rich geometry of the graph of a curve $gamma_a, b : [0, 1] rightarrow mathbbC$ defined as $gamma_a, b(t) = exp(2pi iat) + exp(2pi ibt)$, in which $a$, $b$ are two different positive integers. It turns out that the sum of only two exponentials gives already rise to intriguing graphs. We determine the symmetry group and the points of self intersection of any such graph using only elementary arguments and describe various interesting phenomena that arise in the study of graphs of sums of more than two exponentials.},

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@article{VartziotisEtAl_2018,

title = {GETOpt mesh smoothing: Putting GETMe in the framework of global optimization-based schemes},

author = {Dimitris Vartziotis and Doris Bohnet and Benjamin Himpel},

url = {https://www.sciencedirect.com/science/article/pii/S0168874X17308880},

doi = {https://doi.org/10.1016/j.finel.2018.04.010},

issn = {0168-874X},

year  = {2018},

date = {2018-08-01},

journal = {Finite Elements in Analysis and Design},

volume = {147},

pages = {10–20},

abstract = {We present a novel approach to mesh smoothing, called GETOpt, by using a variation of the Geometric Element Transformation Method (GETMe) which falls within the framework of global optimization-based schemes and mesh quality improvement methods. We introduce a global mesh quality measure, which is inspired by rigorous convergence results concerning the size of the geometric elements (non-degeneracy and comparable sizing) and whose gradient flow yields a combination of GETMe and Laplace smoothing. Under this scheme, it can be shown rigorously that the tetrahedral mesh never degenerates. Furthermore, numerical tests show that the sizing of the mesh elements become more comparable and that the FEM gives better approximations in case the PDE problem being solved demonstrates isotropic physical behavior. More specifically, we present in this article an example where the mesh is used to solve Poisson's equation using the standard Galerkin finite element method with piecewise linear and order 2 functions on the low-order mesh.},

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@article{VartziotisBohnet_2018,

title = {Fractal Curves from Prime Trigonometric Series},

author = {Dimitris Vartziotis and Doris Bohnet},

url = {https://www.mdpi.com/2504-3110/2/1/2},

doi = {10.3390/fractalfract2010002},

issn = {2504-3110},

year  = {2018},

date = {2018-01-01},

journal = {Fractal and Fractional},

volume = {2},

number = {1},

abstract = {We study the convergence of the parameter family of series: $V_alpha, beta ( t ) = sum_p p^-alpha exp (2pi i p^beta t )$, $alpha ,beta ın mathbbR_ >0$, $t ın [ 0 , 1 )$ defined over prime numbers $p$ and, subsequently, their differentiability properties. The visible fractal nature of the graphs as a function of α , β is analyzed in terms of Hölder continuity, self-similarity and fractal dimension, backed with numerical results. Although this series is not a lacunary series, it has properties in common, such that we also discuss the link of this series with random walks and, consequently, explore its random properties numerically.},

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@article{VartziotisBohnet_2017,

title = {A geometric mesh smoothing algorithm related to damped oscillations},

author = {Dimitris Vartziotis and Doris Bohnet},

url = {https://www.sciencedirect.com/science/article/pii/S0045782517301676},

doi = {10.1016/j.cma.2017.07.019},

issn = {0045-7825},

year  = {2017},

date = {2017-11-01},

journal = {Computer Methods in Applied Mechanics and Engineering},

volume = {326},

pages = {102–121},

publisher = {Elsevier BV},

abstract = {We introduce a smoothing algorithm for triangle, quadrilateral, tetrahedral and hexahedral meshes whose centerpiece is a simple geometric triangle transformation. The first part focuses on the mathematical properties of the element transformation. In particular, the transformation gives rise directly to a continuous model given by a system of coupled damped oscillations. Derived from this physical model, adaptive parameters are introduced and their benefits presented. The second part discusses the mesh smoothing algorithm based on the element transformation and its numerical performance on example meshes.},

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@article{VartziotisWipper_2017,

title = {The Fractal Nature of an Approximate Prime Counting Function},

author = {Dimitris Vartziotis and Joachim Wipper},

url = {https://www.mdpi.com/2504-3110/1/1/10},

doi = {10.3390/fractalfract1010010},

issn = {2504-3110},

year  = {2017},

date = {2017-11-01},

journal = {Fractal and Fractional},

volume = {1},

number = {1},

pages = {10},

publisher = {MDPI AG},

abstract = {Prime number related fractal polygons and curves are derived by combining two different 

aspects. One is an approximation of the prime counting function build on an additive function. 

The other is prime number indexed basis entities taken from the discrete or continuous Fourier basis.},

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@article{VartziotisBohnet_2016,

title = {Existence of an attractor for a geometric tetrahedron transformation},

author = {Dimitris Vartziotis and Doris Bohnet},

url = {https://www.sciencedirect.com/science/article/pii/S0926224516300808},

doi = {10.1016/j.difgeo.2016.08.002},

issn = {0926-2245},

year  = {2016},

date = {2016-12-01},

journal = {Differential Geometry and its Applications},

volume = {49},

pages = {197–207},

publisher = {Elsevier BV},

abstract = {We analyze the dynamical properties of a tetrahedron transformation on the space of non-degenerate tetrahedra which can be identified with the non-compact globally symmetric 8-dimensional space SL(3,R)/SO(3,R). We establish the existence of a local attractor which coincides with the set of regular tetrahedra and identify conditions which imply that the basin of attraction is the entire space. In numerical tests, these conditions are fulfilled for a large set of random tetrahedra.},

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@article{VartziotisHimpel_2014a,

title = {Efficient Mesh Optimization Using the Gradient Flow of the Mean Volume},

author = {Dimitris Vartziotis and Benjamin Himpel},

url = {https://doi.org/10.1137/130914000},

doi = {10.1137/130914000},

year  = {2014},

date = {2014-01-01},

journal = {SIAM Journal on Numerical Analysis},

volume = {52},

number = {2},

pages = {1050–1075},

publisher = {Society for Industrial & Applied Mathematics (SIAM)},

abstract = {The signed volume function for polyhedra can be generalized to a mean volume function for volume elements by averaging over the triangulations of the underlying polyhedron. If we consider these up to translation and scaling, the resulting quotient space is diffeomorphic to a sphere. The mean volume function restricted to this sphere is a quality measure for volume elements. We show that the gradient ascent of this map regularizes the building blocks of hybrid meshes consisting of tetrahedra, hexahedra, prisms, pyramids, and octahedra, that is, the optimization process converges to regular polyhedra. We show that the (normalized) gradient flow of the mean volume yields a fast and efficient optimization scheme for the finite element method known as the geometric element transformation method. Furthermore, we shed some light on the dynamics of this method and the resulting smoothing procedure both theoretically and experimentally.},

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@article{VartziotisEtAl_2013,

title = {Improving mesh quality and finite element solution accuracy by GETMe smoothing in solving the Poisson equation},

author = {Dimitris Vartziotis and Joachimand Wipper and Manolis Papadrakakis},

url = {https://www.sciencedirect.com/science/article/pii/S0168874X12002077},

doi = {10.1016/j.finel.2012.11.004},

issn = {0168-874X},

year  = {2013},

date = {2013-04-01},

journal = {Finite Elements in Analysis and Design},

volume = {66},

pages = {36–52},

publisher = {Elsevier BV},

abstract = {Mesh quality plays an essential role in finite element applications, since it affects the efficiency of the simulation with respect to solution accuracy and computational effort. Therefore, mesh smoothing techniques are often applied for improving mesh quality while preserving mesh topology. One of these methods is the recently proposed geometric element transformation method (GETMe), which is based on regularizing element transformations. It will be shown numerically that this smoothing method is particularly suitable, from an applicational point of view, since it leads to a significant reduction of discretization errors within the first few smoothing steps requiring only little computational effort. Furthermore, due to reduced condition numbers of the stiffness matrices the performance of iterative solvers of the resulting finite element equations is improved. This is demonstrated for the Poisson equation with a number of meshes of different complexity and type as well as for piecewise linear and quadratic finite element basis functions. Results are compared to those obtained by two variants of Laplacian smoothing and a state of the art global optimization-based approach.},

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@article{VartziotisPapadrakakis_2013,

title = {Improved GETMe by adaptive mesh smoothing},

author = {Dimitris Vartziotis and Manolis Papadrakakis},

url = {https://cames.ippt.pan.pl/index.php/cames/article/view/80},

issn = {2956-5839},

year  = {2013},

date = {2013-01-01},

journal = {Computer Assisted Methods in Engineering and Science},

volume = {20},

number = {1},

pages = {55–71},

abstract = {Mesh smoothing improves mesh quality by node relocation without altering mesh topology. Such methods play a vital role in finite element mesh improvement with a direct consequence on the quality of the discretized solution. In this work, an improved version of the recently proposed geometric element transformation method (GETMe) for mesh smoothing is presented. Key feature is the introduction of adaptive concepts, which improve the resulting mesh quality, reduce the number of parameters, and enhance the parallelization capabilities. Implementational aspects are discussed and results of a more efficient version are presented, which demonstrate that GETMe adaptive smoothing yields high quality meshes, is particularly fast, and has a comparably low memory profile. Furthermore, results are compared to those of other state-of-the-art smoothing methods.},

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@article{VartziotisWipper_2012,

title = {Fast smoothing of mixed volume meshes based on the effective geometric element transformation method},

author = {Dimitris Vartziotis and Joachim Wipper},

doi = {10.1016/j.cma.2011.09.008},

year  = {2012},

date = {2012-01-01},

journal = {Computer Methods in Applied Mechanics and Engineering},

volume = {201-204},

pages = {65–81},

publisher = {Elsevier BV},

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pubstate = {published},

tppubtype = {article}

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@article{VartziotisHuggenberger_2012,

title = {Iterative geometric triangle transformations},

author = {Dimitris Vartziotis and Simon Huggenberger},

url = {https://ems.press/journals/em/articles/11208},

doi = {10.4171/em/196},

year  = {2012},

date = {2012-01-01},

journal = {Elemente der Mathematik},

volume = {67},

number = {2},

pages = {68–83},

publisher = {European Mathematical Society - EMS - Publishing House GmbH},

abstract = {Seit über einem Jahrhundert sind geometrische Transformationen von Dreiecken Gegenstand mathematischer Untersuchungen. In dem vorliegenden Artikel wird ein Ausgangsdreieck in ein neues Dreieck transformiert, indem auf jeder der Dreiecksseiten ein gleichschenkliges, nach aussen gerichtetes Aufsatzdreieck errichtet wird. Die Scheitelpunkte der Aufsatzdreiecke bilden die Eckpunkte des neuen Dreiecks. Durch wiederholtes Anwenden dieser Transformation erhält man eine Folge von Dreiecken. Die Form der Dreiecke dieser Folge konvergiert gegen eine charakteristische Dreiecksform, von der nachgewiesen wird, dass sie mit jedem Transformationsschritt echt angenähert wird.},

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@article{VartziotisEtAl_2012,

title = {Roadscanner: Feasibility Study and Development of a GNSS-Probe for Creating Digital Maps of High Accuracy and Integrity},

author = {Dimitris Vartziotis and Iraklis Goudas and Spyros Savvas and Athanasios Batagiannis and Florian Dittmann and Francesca Zanier},

url = {https://www.sciencedirect.com/science/article/pii/S1877042812029618},

doi = {10.1016/j.sbspro.2012.06.1218},

issn = {1877-0428},

year  = {2012},

date = {2012-01-01},

journal = {Procedia - Social and Behavioral Sciences},

volume = {48},

pages = {2473–2481},

publisher = {Elsevier BV},

abstract = {This paper describes the main outcome of the Roadscanner project, in developing a Vehicle Probe prototype equipped with a Global Navigation Satellite System (GNSS) receiver, an Inertial Navigation System (INS) and a Laser Scanner (LS) unit to create digital maps of high accuracy and integrity. GNSS have become an important factor in transport with limitless possibilities. The GNSS concept is based on a constellation of satellites that provides autonomous geo-spatial positioning with global coverage. This enables GNSS receivers to calculate a Position Velocity Time (PVT) solution using the signals transmitted from satellites along the line-of-sight to the receiver. Following the obvious application of navigation and guidance for various types of vehicles, the advances in accuracy and integrity of PVT have enabled other uses, such as Advanced Driver Assistance Systems (ADAS) and digital automotive simulations. However, GNSS also has intrinsic limitations (subject for example to radio-frequency interference, signal availability and vehicle dynamics), which can be a serious hindrance for demanding applications, where accuracy and integrity are critical. In order to remedy GNSS drawbacks, the satellite receiver data can be integrated with additional sensors, such as an INS. The sensor fusion provides a solution of enhanced accuracy, robustness and integrity and also yields attitude information. Furthermore, adding a laser scanner (LS) provides road surface measurements, which are important for simulation, or road surface quality inspection.},

note = {Transport Research Arena 2012},

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@article{VartziotisWipper_2011,

title = {A dual element based geometric element transformation method for all-hexahedral mesh smoothing},

author = {Dimitris Vartziotis and Joachim Wipper},

url = {https://www.sciencedirect.com/science/article/pii/S0045782510002719},

doi = {https://doi.org/10.1016/j.cma.2010.09.012},

year  = {2011},

date = {2011-02-01},

journal = {Computer Methods in Applied Mechanics and Engineering},

volume = {200},

number = {9-12},

pages = {1186–1203},

abstract = {Due to their increased complexity hexahedral elements are more challenging with respect to mesh generation and mesh improvement techniques than tetrahedral elements. In particular, there is a lack of geometry-based all-hexahedral smoothing methods for mesh quality improvement being easy to implement, practicable, and efficient. The recently introduced geometric element transformation method represents a new promising element oriented smoothing concept to resolve this deficiency. By giving a dual octahedron based regularizing transformation this new approach is adapted in order to smooth all-hexahedral meshes. First numerical tests indicate that the resulting smoothing method yields high quality results at least comparable to those of a state of the art global optimization-based approach while being significantly faster.},

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@article{VartziotisWipper_2010,

title = {Characteristic parameter sets and limits of circulant Hermitian polygon transformations},

author = {Dimitris Vartziotis and Joachim Wipper},

url = {https://www.sciencedirect.com/science/article/pii/S0024379510001990},

doi = {https://doi.org/10.1016/j.laa.2010.04.022},

issn = {0024-3795},

year  = {2010},

date = {2010-01-01},

journal = {Linear Algebra and its Applications},

volume = {433},

number = {5},

pages = {945–955},

abstract = {Polygon transformations based on taking the apices of similar triangles constructed on the sides of an initial polygon are analyzed as well as the limit polygons obtained by iteratively applying such transformations. In contrast to other approaches, this is done with respect to two construction parameters representing a base angle and an apex perpendicular subdivision ratio. Furthermore, a combined transformation leading to circulant Hermitian matrices is proposed, which eliminates the rotational effect of the basic transformation. A finite set of characteristic parameter subdomains is derived for which the sequence converges to specific eigenpolygons. Otherwise, limit polygons turn out to be linear combinations of up to three eigenpolygons. This leads to a full classification of circulant Hermitian similar triangles based polygon transformations and their limit polygons. As a byproduct classical results as Napoleon's theorem and the Petr–Douglas–Neumann theorem can be easily deduced.},

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@article{VartziotisEtAl_2009,

title = {The geometric element transformation method for tetrahedral mesh smoothing},

author = {Dimitris Vartziotis and Joachim Wipper and Bernd Schwald},

url = {https://www.sciencedirect.com/science/article/pii/S004578250900334X},

doi = {https://doi.org/10.1016/j.cma.2009.09.027},

issn = {0045-7825},

year  = {2009},

date = {2009-12-01},

journal = {Computer Methods in Applied Mechanics and Engineering},

volume = {199},

number = {1},

pages = {169–182},

abstract = {The geometric element transformation method (GETMe) has been introduced as a new element driven approach to mesh smoothing. It is based on simple geometric transformations, which, if applied iteratively, lead to the regularization of mesh elements. Global mesh smoothing can be accomplished by successively improving the worst elements or by averaging node positions obtained by the simultaneous transformation of all elements. GETMe smoothing has been successfully applied in the case of surface meshes. As shown in this paper, this approach also naturally extends to tetrahedral mesh smoothing without major conceptual modifications. A regularizing transformation for tetrahedra is presented and a combined approach of simultaneous and sequential GETMe smoothing is described. First numerical examples yield high quality meshes superior to those obtained by other geometry-based methods. In fact, the presented results are in a majority of cases at least comparable to those obtained by a state of the art global optimization-based method.},

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@article{VartziotisWipper_2009b,

title = {The geometric element transformation method for mixed mesh smoothing},

author = {Dimitris Vartziotis and Joachim Wipper},

doi = {10.1007/s00366-009-0125-6},

year  = {2009},

date = {2009-06-01},

journal = {Engineering with Computers},

volume = {25},

number = {3},

pages = {287–301},

publisher = {Springer Science and Business Media LLC},

abstract = {The geometric element transformation method (GETMe) is a geometry-based smoothing method for mixed and non-mixed meshes. It is based on a simple geometric transformation applicable to elements bounded by polygons with an arbitrary number of nodes. The transformation, if applied iteratively, leads to a regularization of the polygons. Global mesh smoothing is accomplished by averaging the new node positions obtained by local element transformations. Thereby, the choice of transformation parameters as well as averaging weights can be based on the element quality which leads to high quality results. In this paper, a concept of an enhanced transformation approach is presented and a proof for the regularizing effect of the transformation based on eigenpolygons is given. Numerical examples confirm that the GETMe approach leads to superior mesh quality if compared to other geometry-based methods. In terms of quality it can even compete with optimization-based techniques, despite being conceptually significantly simpler.},

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@article{VartziotisWipper_2009a,

title = {On the construction of regular polygons and generalized napoleon vertices},

author = {Dimitris Vartziotis and Joachim Wipper},

url = {https://forumgeom.fau.edu/FG2009volume9/FG200920.pdf},

issn = {1534-1178},

year  = {2009},

date = {2009-01-01},

journal = {Forum Geometricorum},

volume = {9},

pages = {213–223},

abstract = {An algebraic foundation for the derivation of geometric construction schemes transforming arbitrary polygons with n vertices into $k$-regular $n$-gons is given. It is based on circulant polygon transformations and the associated eigenpolygon decompositions leading to the definition of generalized Napoleon vertices. Geometric construction schemes are derived exemplarily for different choices of $n$ and $k$.},

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@article{VartziotisWipper_2009,

title = {Classification of symmetry generating polygon-transformations and geometric prime algorithms},

author = {Dimitris Vartziotis and Joachim Wipper},

url = {http://mathematica-pannonica.ttk.pte.hu/articles/mp20-2/mp20-2-167-187.pdf},

issn = {2786-0752},

year  = {2009},

date = {2009-01-01},

journal = {Mathematica Pannonica},

volume = {20},

number = {2},

pages = {167–187},

abstract = {Infinite polygon transformations by iteratively taking the apices of similar isosceles triangles erected on the sides of a polygon are analyzed. In contrast to other approaches, this is done with respect to the base angle of the isosceles triangles. A finite set of characteristic angles is derived, which only depend on the number of vertices of the polygon. This results in a systematic and complete classification of consecutive base angle intervals and the associated limit polygons. The latter in turn are linear combinations of eigen-polygons obtained by diagonalizing the circulant matrix representation of the transformation. Analyzing the principles of the symmetry of eigenpolygons and thereby of the limit polygons reveals a connection to prime numbers. Based on these results, a geometric sieve as well as a factorization algorithm for prime numbers is presented.},

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