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1

Application of isogeometric approach to dynamics of curved beam

Łasecka-Plura Magdalena, Białasik Jan, Kuczma Mieczysław, Tabrizikahou Alireza

Vibrations in Physical Systems

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2023

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Vol. 34, nr 1

art. no. 2023117

EN

In the paper dynamics of a free-form Timoshenko curved beam is investigated. The considered problem is solved using isogeometric analysis. Non-uniform rational B-spline (NURBS) basis functions are applied to describe both geometry and displacement field of the considered beam. The Timoshenko beam theory is used to derive the element stiffness and mass matrices. The application of the presented method is shown in numerical examples. The correctness of the presented approach is proved by comparing the obtained results to those available in the literature and calculated by the finite element method. Analysis of convergence is presented for different orders of NURBS basis functions.

2

Prediction of random vibration fatigue damage using isogeometric modelling

Wang Shubiao, Khalij Leila, Troian Renata, Shi Lujie

Computer Assisted Methods in Engineering and Science

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2021

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Vol. 28, no. 3

193--223

EN

The finite element analysis (FEA) method is indispensable in simulation technology, as itcan help engineers predict results to avoid the cost of experimental testing. However, thefinite element mesh generation process can be time-consuming, and the approximate meshmodel can lead to inaccurate stress results. Improving the accuracy of stress estimationleads to a better assessment of damage or life of mechanical components. In this study, weapplied the isogeometric analysis (IGA) implemented in LS-DYNA software to study twospecimens subjreted to the stationary Gaussian random loads. These geometric modelswere represented by non-uniform rational B-spline (NURBS) to assess the damage andfatigue life in the frequency domain by using Dirlik’s distribution and cumulative damage.A comparison with FEA was conducted to highlight the main differences. Experimentalfatigue tests with an electrodynamic shaker were also carried out to check if the fatiguelives predicted by numerical models are consistent. The study showed that IGA predictssimilar results to FEA with an acceptable relative error and reduced the time for meshgeneration, requiring fewer integration points and mesh elements.

3

Three-dimensional simulations of the airborne COVID-19 pathogens using the advection-diffusion model and alternating-directions implicit solver

Łoś Marcin, Woźniak Maciej, Muga Ignacio, Paszynski Maciej

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2021

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Vol. 69, nr 4

art. no. e137125

EN

In times of the COVID-19, reliable tools to simulate the airborne pathogens causing the infection are extremely important to enable the testing of various preventive methods. Advection-diffusion simulations can model the propagation of pathogens in the air. We can represent the concentration of pathogens in the air by “contamination” propagating from the source, by the mechanisms of advection (representing air movement) and diffusion (representing the spontaneous propagation of pathogen particles in the air). The three-dimensional time-dependent advection-diffusion equation is difficult to simulate due to the high computational cost and instabilities of the numerical methods. In this paper, we present alternating directions implicit isogeometric analysis simulations of the three-dimensional advection-diffusion equations. We introduce three intermediate time steps, where in the differential operator, we separate the derivatives concerning particular spatial directions. We provide a mathematical analysis of the numerical stability of the method. We show well-posedness of each time step formulation, under the assumption of a particular time step size. We utilize the tensor products of one-dimensional B-spline basis functions over the three-dimensional cube shape domain for the spatial discretization. The alternating direction solver is implemented in C++ and parallelized using the GALOIS framework for multi-core processors. We run the simulations within 120 minutes on a laptop equipped with i7 6700 Q processor 2.6 GHz (8 cores with HT) and 16 GB of RAM.

4

Linear computational cost implicit solver for parabolic problems

Gurgul Grzegorz, Łoś Marcin, Paszynski Maciej, Calo Victor

Computer Science

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2020

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T. 21 (3)

335–352

EN

In this paper, we use the alternating direction method for isogeometric finite elements to simulate transient problems. Namely, we focus on a parabolic problem and use B-spline basis functions in space and an implicit time-marching method to fully discretize the problem. We introduce intermediate time-steps and separate our differential operator into a summation of the blocks that act along a particular coordinate axis in the intermediate time-steps. We show that the resulting stiffness matrix can be represented as a multiplication of two (in 2D) or three (in 3D) multi-diagonal matrices, each one with B-spline basis functions along the particular axis of the spatial system of coordinates. As a result of these algebraic transformations, we get a system of linear equations that can be factorized in a linear O(N) computational cost at every time-step of the implicit method. We use our method to simulate the heat transfer problem. We demonstrate theoretically and verify numerically that our implicit method is unconditionally stable for heat transfer problems (i.e., parabolic). We conclude our presentation with a discussion on the limitations of the method.

5

Isogeometric Fast Multipole Boundary Element Method Based on Burton-Miller Formulation for 3D Acoustic Problems

Chen Leilei, Zhao Wenchang, Liu Cheng, Chen Haibo, Marburg Steffen

Archives of Acoustics

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2019

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Vol. 44, No. 3

475--492

EN

An isogeometric boundary element method is applied to simulate wave scattering problems governed by the Helmholtz equation. The NURBS (non-uniform rational B-splines) widely used in the CAD (computer aided design) field is applied to represent the geometric model and approximate physical field variables. The Burton-Miller formulation is used to overcome the fictitious frequency problem when using a single Helmholtz boundary integral equation for exterior boundary-value problems. The singular integrals existing in Burton-Miller formulation are evaluated directly and accurately using Hadamard’s finite part integration. Fast multipole method is applied to accelerate the solution of the system of equations. It is demonstrated that the isogeometric boundary element method based on NURBS performs better than the conventional approach based on Lagrange basis functions in terms of accuracy, and the use of the fast multipole method both retains the accuracy for isogeometric boundary element method and reduces the computational cost.

6

Applications of alternating direction solver for simulations of time-dependent problems

Łoś M., Paszyński M.

Computer Science

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2017

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Vol. 18 (2)

117--128

EN

This paper deals with the application of an Alternating Direction Solver (ADS) to a non-stationary linear elasticity problem solved with the isogeometric finite element method (IGA-FEM). Employing a tensor product B-spline basis in isogeometric analysis under some restrictions leads to a system of linear equations with a matrix possessing a tensor product structure. The ADI algorithm is a direct method that exploits this Kronecker product structure to solve the system in O (N), where N is the number of degrees of freedom (basis functions). This is asymptotically faster than state-of-the-art, general-purpose, multi-frontal direct solvers when applied to explicit dynamics. In this paper, we also present a complexity analysis of the ADS incorporating dependence on the B-spline basis of order p.

7

Elastic-plastic analysis of pressure vessels and rotating disks made of functionally graded materials using the isogeometric approach

Kalali A. T., Hassani B., Hadidi-Moud S.

Journal of Theoretical and Applied Mechanics

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2016

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Vol. 54 nr 1

113--125

EN

An NURBS-based isogeometric analysis for elastic-plastic stress in a cylindrical pressure vessel is presented. The vessel is made of a ceramic/metal functionally graded material, i.e. a particle-reinforced composite. It is assumed that the material plastic deformation follows an isotropic strain-hardening rule based on the von Mises yield criterion. The mechanical properties of the graded material are modelled by the modified rule of mixtures. Selected finite element results are also presented to establish the supporting evidence for validation of the isogeometric analysis. Similar analyses are performed and solutions for spherical pressure vessel and rotating disk made of FGMs are also provided.

8

One-dimensional fully automatic h-adaptive isogeometric finite element method package

Lipski P., Paszyński M

Computer Science

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2016

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Vol. 17 (4)

439--459

EN

his paper deals with an adaptive finite element method originally developed by Prof. Leszek Demkowicz for hierarchical basis functions. In this paper, we investigate the extension of the adaptive algorithm for isogeometric analysis performed with B-spline basis functions. We restrict ourselves to h-adaptivity, since the polynomial order of approximation must be fixed in the isogeometric case. The classical variant of the adaptive FEM algorithm, as delivered by the group of Prof. Demkowicz, is based on a two-grid paradigm, with coarse and fine grids (the latter utilized as a reference solution). The problem is solved independently over a coarse mesh and a fine mesh. The fine-mesh solution is then utilized as a reference to estimate the relative error of the coarse-mesh solution and to decide which elements to refine. Prof. Demkowicz uses hierarchical basis functions, which (though locally providing Cp−1 continuity) ensure only C0 on the interfaces between elements. The CUDA C library described in this paper switches the basis to B-spline functions and proposes a one-dimensional isogeometric version of the h-adaptive FEM algorithm to achieve global Cp−1 continuity of the solution.

9

Multi-frontal parallel direct solver for one dimensional isogeometric collocation method

Lipski P., Paszyński M

Computer Methods in Materials Science

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2015

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Vol. 15, No.1

192--197

EN

In this paper wc present a new multi-frontal solver for the isogeometric collocation method (ISO-C) on GPU. The ISO-C method constitutes an alternative for the isogeometric finite element method (ISO-FEM). The key advantage of ISO-C over ISO-FEM is that it does not include the computationally intensive operation of integrating the variational formulation. The ISO-C method requires using only a single collocation point per one basis function, whereas in ISO-FEM, Gaussian quadrature is applied on many points at each finite element. The presented multi-frontal solver for collocation method results in logarithmic execution time assuming that large enough number of GPU processors is available. In this article, the method is employed for an exemplary ID nanolithography problem of Step-and-Flash Imprint Lithography (SFIL). The algorithm, however, may be applied to a wide class of 2D and 3D problems.

PL

W artykule przedstawiamy nowy solwer wielofrontalny dla izogeometrycznej metody kolokacji (ISO-C) na GPU. Metoda ISO-C stanowi alternatywę dla izogeometrycznej metody elementów skończonych (ISO-FEM). Główną zaletą metody ISO-C jest redukcja znacznego kosztu obliczeniowego całkowania sformułowania wariacyjnego występującego w metodzie ISO-FEM. Metoda ISO-C wymaga bowiem użycia tylko jednego punktu kolokacji dla jednej funkcji bazowej, podczas gdy metoda ISO-FEM wiąże się z zastosowaniem kwadratury Gaussa w wielu punktach na każdym elemencie skończonym. Prezentowany solwer wielofrontalny dla metody kolokacji uzyskuje logarytmiczną złożoność obliczeniową przy założeniu odpowiednio dużej liczby procesorów graficznych GPU. Niniejsza publikacja przedstawia proste wykorzystanie metody dla jednowymiarowego przykładowego problemu nanolitografii Step-and-Flash Imprint Lithography (SFIL). Zaprezentowany algorytm znajduje jednak ogólnie zastosowanie dla szerokiej klasy problemów w dwóch i trzech wymiarach.

10

Application of isogeometric analysis to modeling of elastic deformation in dual phase materials by using Statistically Similar Representative Volume Element on heterogeneous hardware devices

Bachniak D., Rauch Ł.

Computer Methods in Materials Science

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2015

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Vol. 15, No.4

459--468

EN

Statistically Similar Representative Volume Element (SSRVE) is a methodology applied for reduction of complexity of material microstructure representation for dual phase materials like DP steels or composites. It is based on assumption that typical RVE can be further reduced into simplified form, which joined together periodically behaves the same as its larger equivalent. SSRVE is based on Non-Uniform Relational B-Splines representation and determined by using optimization procedure, where objective function includes comparison of mechanical properties, shape coefficients and statistical characteristics. The first of these elements requires application of Finite Element Method (FEM) allowing to simulate deformation of pattern RVE and current SSRVE within elastic or elastic-plastic range. This paper presents approach allowing to replace time consuming FEM with more efficient Isogeometric Analysis (IGA). The performance of new approach is analysed and compared to conventional FEM-based methodology. Special attention is put on possibilities of IGA implementation on heterogeneous hardware devices allowing to improve computational efficiency and decrease overall power consumption.

PL

Statystycznie Podobny Reprezentatywny Element Objętościowy (ang. Statistically Similar Representative Volume Element, SSRVE) jest metodyką stosowaną w celu redukcji złożoności obliczeniowej reprezentacji mikrostruktury materiałów wielofazowych jak np.: stale DP oraz kompozyty. Podejście to bazuje na założeniu, że typowa reprezentacja RVE może być jeszcze bardziej uproszczona do elementu, który połączony periodycznie z samym sobą będzie zachowywał się podobnie jak jego bardziej złożony odpowiednik. SSRVE konstruowany jest w oparciu o krzywe NURBS (ang. Non-Uniform Rational B-Splines), a wyznaczany za pomocą procedury optymalizacji, gdzie funkcja celu zawiera własności mechaniczne, współczynniki kształtu oraz charakterystykę statystyczną analizowanej mikrostruktury. Pierwszy z wymienionych elementów funkcji celu wymaga zastosowania Metody Elementów Skończonych (MES), umożliwiającej symulację odkształcenia zarówno wzorca RVE jak i kolejnych rozwiązań SSRVE w zakresie sprężystym oraz sprężysto-plastycznym. Niniejszy artykuł przedstawia podejście pozwalające na podmianę kosztownej obliczeniowo procedury MES bardziej wydajną metodą analizy Izogeometrycznej (ang. Isogeometric Analysis, IGA). Wydajność nowego podejścia została przeanalizowana i porównana z konwencjonalnym rozwiązaniem opartym o MES. Ponadto, w artykule omówiona została możliwość zastosowania rozwiązania IGA z wykorzystaniem heterogenicznych architektur sprzętowych, co umożliwi poprawę wydajności obliczeniowej całego podejścia.