scholarly journals Cs-smooth isogeometric spline spaces over planar bilinear multi-patch parameterizations

2021 ◽  
Vol 47 (3) ◽  
Author(s):  
Mario Kapl ◽  
Vito Vitrih
Keyword(s):  
Isogeometric Analysis ◽  
Spline Space ◽  
Numerical Results ◽  
Optimal Approximation ◽  
Spline Functions ◽  
Approximation Power ◽  
Appl Math ◽  
A Current ◽  
Smooth Spline

AbstractThe design of globally Cs-smooth (s ≥ 1) isogeometric spline spaces over multi-patch geometries with possibly extraordinary vertices, i.e. vertices with valencies different from four, is a current and challenging topic of research in the framework of isogeometric analysis. In this work, we extend the recent methods Kapl et al. Comput. Aided Geom. Des. 52–53:75–89, 2017, Kapl et al. Comput. Aided Geom. Des. 69:55–75, 2019 and Kapl and Vitrih J. Comput. Appl. Math. 335:289–311, 2018, Kapl and Vitrih J. Comput. Appl. Math. 358:385–404, 2019 and Kapl and Vitrih Comput. Methods Appl. Mech. Engrg. 360:112684, 2020 for the construction of C1-smooth and C2-smooth isogeometric spline spaces over particular planar multi-patch geometries to the case of Cs-smooth isogeometric multi-patch spline spaces of degree p, inner regularity r and of a smoothness s ≥ 1, with p ≥ 2s + 1 and s ≤ r ≤ p − s − 1. More precisely, we study for s ≥ 1 the space of Cs-smooth isogeometric spline functions defined on planar, bilinearly parameterized multi-patch domains, and generate a particular Cs-smooth subspace of the entire Cs-smooth isogeometric multi-patch spline space. We further present the construction of a basis for this Cs-smooth subspace, which consists of simple and locally supported functions. Moreover, we use the Cs-smooth spline functions to perform L2 approximation on bilinearly parameterized multi-patch domains, where the obtained numerical results indicate an optimal approximation power of the constructed Cs-smooth subspace.

scholarly journals Construction of a normalized basis of a univariate quadratic $C^1$ spline space and application to the quasi-interpolation

2021 ◽  
Vol 40 ◽  
pp. 1-21
Author(s):  
A. Rahouti ◽  
Abdelhafid Serghini ◽  
A. Tijini
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Approximation Order ◽  
Spline Space ◽  
Numerical Results ◽  
Optimal Approximation ◽  
Polar Form ◽  
Piecewise Polynomial ◽  
The Finite Element Method ◽  
Element Method

In this paper, we use the finite element method to construct a new normalized basis of a univariate quadratic $C^1$ spline space. We give a new representation of Hermite interpolant of any piecewise polynomial of class at least $C^1$ in terms of its polar form. We use this representation for constructing several superconvergent and super-superconvergent discrete quasi-interpolants which have an optimal approximation order. This approach is simple and provides an interesting approximation. Numerical results are given to illustrate the theoretical ones.

scholarly journals Likelihood weighting of partial structure factors using spline coefficients

2005 ◽  
Vol 38 (1) ◽  
pp. 193-198 ◽  
Author(s):  
Kevin Cowtan
Keyword(s):  
Spline Functions ◽  
Partial Structure ◽  
Structure Factors ◽  
The Stability ◽  
Smooth Spline

A method for the weighting of structure factors from an incomplete and inaccurate model is described which relies on the fitting of smooth spline functions of resolution. The use of smooth spline functions avoids the problems of discontinuities introduced when performing calculations in resolution shells. The complexity of the functions to be fit may be varied by changing the number of spline parameters. This approach is used to investigate the stability of the problem when data are limited.

Numerical results for the classical free convection flow problem in a square porous cavity using spline functions

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Sanda Micula ◽  
Ioan Pop
Keyword(s):  
Saturated Porous Medium ◽  
Heat Transfer Process ◽  
Numerical Results ◽  
Spline Functions ◽  
Convection Flow ◽  
Content Type ◽  
Nusselt Numbers ◽  
Side Walls ◽  
Convective Cells ◽  
The Mathematical Model

Purpose This paper aims to present the problem of natural convection in a square cavity filled with a fluid-saturated porous medium having constant temperatures on the side walls, and the numerical results are obtained. Design/methodology/approach Dimensionless equations governing the mathematical model together with the boundary conditions are obtained, and the problem is solved by spline functions. Findings The numerical results of streamlines, isotherms and local and average Nusselt numbers are investigated and discussed for different values of the governing parameters. The Rayleigh number is proposed to be control parameter for heat and fluid flow inside the cavity. Originality/value Interesting results with this new numerical method have been obtained, such as the behaviour of the convective cells and the local and average Nusselt number. The obtained results are compared and successfully validated with previous reported results from the open literature. The present numerical results are new and original. The reported results can contribute to other researchers on electing the relevant parameters to optimize the heat transfer process in the modern industry.

scholarly journals A priori error for unilateral contact problems with Lagrange multipliers and isogeometric analysis

2018 ◽  
Vol 39 (4) ◽  
pp. 1627-1651 ◽  
Author(s):  
Pablo Antolin ◽  
Annalisa Buffa ◽  
Mathieu Fabre
Keyword(s):  
Isogeometric Analysis ◽  
A Priori ◽  
Contact Problems ◽  
Unilateral Contact ◽  
Spline Space ◽  
Three Dimensions ◽  
B Spline ◽  
A Priori Error Estimate ◽  
The Stability ◽  
Unilateral Contact Problems

Abstract In this paper we consider a unilateral contact problem without friction between a rigid body and a deformable one in the framework of isogeometric analysis. We present the theoretical analysis of the mixed problem. For the displacement, we use the pushforward of a nonuniform rational B-spline space of degree $p$ and for the Lagrange multiplier, the pushforward of a B-spline space of degree $p-2$. These choices of space ensure the proof of an inf–sup condition and so on, the stability of the method. We distinguish between contact and noncontact sets to avoid the classical geometrical hypothesis of the contact set. An optimal a priori error estimate is demonstrated without assumption on the unknown contact set. Several numerical examples in two and three dimensions and in small and large deformation frameworks demonstrate the accuracy of the proposed method.

scholarly journals Active Cloaking and Illusion of Electric Potentials in Electrostatics

2021 ◽  
Author(s):  
Andreas Helfrich-Schkabarenko ◽  
Alik Ismail-Zadeh ◽  
Aron Sommer
Keyword(s):  
Electric Potential ◽  
Superposition Principle ◽  
Current Source ◽  
Three Dimensional ◽  
Numerical Results ◽  
Research Fields ◽  
Electric Potentials ◽  
Three Dimensional Models ◽  
First Time ◽  
A Current

Abstract Cloaking and illusion has been demonstrated theoretically and experimentally in several research fields. Here we present for the first time an active exterior cloaking device in electrostatics operating in a two-horizontally-layered electroconductive domain, and use the superposition principle to cloak electric potentials. The device uses an additional current source pattern introduced on the interface between two layers to cancel the total electric potential to be measured. Also, we present an active exterior illusion device allowing for detection of a signal pattern corresponding to any arbitrarily chosen current source instead of the existing current source. The performance of the cloaking/illusion devices is demonstrated by three-dimensional models and numerical experiments using synthetic measurements of the electric potential. Sensitivities of numerical results to a noise in measured data and to a size of cloaking devices are analysed. The numerical results show quite reasonable cloaking/illusion performance, which means that a current source can be hidden electrostatically. The developed active cloaking/illusion methodology can be used in subsurface geo-exploration studies, electrical engineering, live sciences, and elsewhere.

scholarly journals Numerical Modeling Based on Spline Basis Functions: Application to Groundwater Flow Modeling in Karst Aquifers and Advection Dominated Problems

2019 ◽  
Author(s):  
◽  
Luka Malenica
Keyword(s):  
Numerical Model ◽  
Groundwater Flow ◽  
Local Time ◽  
Adaptive Algorithm ◽  
Isogeometric Analysis ◽  
Basis Functions ◽  
Spline Functions ◽  
The Novel ◽  
Karst Aquifers ◽  
Spline Basis

The main objective of this thesis is to utilize the powerful approximation properties of spline basis functions for numerical solutions of engineering problems that arise in the field of fluid mechanics. Special types of spline functions, the so-called Fup basis functions, are used as representative members of the spline family. However, the techniques developed in this work are quite general with respect to the choice of different spline functions. The application of this work is twofold. The first practical goal is the development of a novel numerical model for groundwater flow in karst aquifers. The concept of isogeometric analysis (IGA) is presented as a unified framework for multiscale representation of the geometry, material heterogeneity and solution. Moreover, this fundamentally higher-order approach enables the description of all fields as continuous and smooth functions by using a linear combination of spline basis functions. Since classical IGA uses the Galerkin and collocation approach, in this thesis, a third concept, in the form of control volume isogeometric analysis (CV-IGA), is developed and set as the foundation for the development of a karst flow numerical model. A discrete-continuum (hybrid) approach is used, in which a three-dimensional laminar matrix flow is coupled with a one-dimensional turbulent conduit flow. The model is capable of describing variably saturated conditions in both flow domains. Since realistic verification of karst flow models is an extremely difficult task, the particular contribution of this work is the construction of a specially designed 3D physical model (dimensions: 5.66 x 2.95 x 2.00 m) to verify the developed numerical model under controlled laboratory conditions. As a second application, this thesis presents the development of a full space-time adaptive collocation algorithm with particular application to advection-dominated problems. Since these problems are usually characterized by numerical instabilities, the novel adaptive algorithm accurately resolves small-scale features while controlling the numerical error and spurious numerical oscillations without need for any special stabilization technique. The previously developed spatial adaptive strategy dynamically changes the computational grid at each global time step, while the novel adaptive temporal strategy uses different local time steps for different collocation points based on the estimation of the temporal discretization error. Thus, in parts of the domain where temporal changes are demanding, the algorithm uses smaller local time steps, while in other parts, larger local time steps can be used without affecting the overall solution accuracy and stability. In contrast to existing local time stepping methods, the developed method is applicable to implicit discretization and resolves all temporal scales independently of the spatial scales. The efficiency and accuracy of the full space-time adaptive algorithm is verified with some classic 1D and 2D advection-diffusion benchmark test cases.

scholarly journals The Magnetic Field of a Current Loop Encircling an Axisymmetric Iron Core

1976 ◽  
Vol 29 (2) ◽  
pp. 1 ◽  
Author(s):  
RL Dewar
Keyword(s):  
Convergent Series ◽  
Numerical Results ◽  
Current Loop ◽  
Iron Core ◽  
Air Gaps ◽  
The Core ◽  
Asymptotic Expressions ◽  
Transformer Core ◽  
A Current ◽  
Straight Rod

The effect of an iron transformer core on the field of a current loop is examined for two models of the core: (1) An infinite straight rod of high permeability aligned along the axis of symmetry, for which asymptotic expressions for the effect of the core are obtained and compared with numerical results. (2) A rectangular toroidal iron casing surrounding the loop. The latter model is more realistic because a return path is provided for the flux. For this model, the effect of air gaps is considered, and rapidly convergent series are obtained and numerical results are given. The significance of these results for tokamak equilibrium is indicated.

Hydrodynamic Performance of a Current Energy Generator Based on WIG

2019 ◽  
Author(s):  
Jian Wang ◽  
Guanghua He ◽  
Weijie Mo ◽  
Shijun Zhang ◽  
Jiangtao Man
Keyword(s):  
Boundary Conditions ◽  
Drag Force ◽  
Lift Force ◽  
Numerical Results ◽  
Hydrodynamic Performance ◽  
Published Data ◽  
Energy Extraction ◽  
Time Step ◽  
Current Energy ◽  
A Current

Abstract The hydrodynamic performance of a novel current energy generator is studied with consideration of the effect of Wing in Ground (WIG) by Star CCM+. The pitch and heave motions of a turbine with a 2D single oscillating wing and two parallelized oscillating wings in uniform flow are simulated, and the numerical results including the lift force, drag force and moment coefficients of the hydrofoil are calculated to analyze the hydrodynamic performance of the generator. First, the convergence studies with respect to the mesh and time step are firstly carried out by compared with the published data. Secondly, the hydrodynamic performance of the WIG-based current energy extraction is investigated, and a good performance of the current energy extraction is confirmed. Finally, the effect of boundary conditions of wing and wall on the performance of the current energy generator is investigated.

scholarly journals Determination of paramagnetic and ferromagnetic phases of an Ising model on a third-order Cayley tree

2021 ◽  
Vol 24 (1) ◽  
pp. 13001
Author(s):  
H. Akιn
Keyword(s):  
Ising Model ◽  
Exact Solutions ◽  
Cayley Tree ◽  
Numerical Results ◽  
Coupling Constants ◽  
Partition Functions ◽  
Third Order ◽  
Recurrence Equations ◽  
Appl Math

In this present paper, the recurrence equations of an Ising model with three coupling constants on a third-order Cayley tree are obtained. Paramagnetic and ferromagnetic phases associated with the Ising model are characterized. Types of phases and partition functions corresponding to the model are rigorously studied. Exact solutions of the mentioned model are compared with the numerical results given in Ganikhodjaev et al. [J. Concr. Appl. Math., 2011, 9, No. 1, 26-34].

ANALYSIS-SUITABLE T-SPLINES OF ARBITRARY DEGREE: DEFINITION, LINEAR INDEPENDENCE AND APPROXIMATION PROPERTIES

2013 ◽  
Vol 23 (11) ◽  
pp. 1979-2003 ◽  
Author(s):  
L. BEIRÃO DA VEIGA ◽  
A. BUFFA ◽  
G. SANGALLI ◽  
R. VÁZQUEZ
Keyword(s):  
Linear Independence ◽  
Spline Space ◽  
Optimal Approximation ◽  
Dual Basis ◽  
Approximation Properties ◽  
Local Refinement ◽  
Arbitrary Degree ◽  
Fundamental Properties ◽  
Blending Functions

T-splines are an important tool in IGA since they allow local refinement. In this paper we define analysis-suitable T-splines of arbitrary degree and prove fundamental properties: Linear independence of the blending functions and optimal approximation properties of the associated T-spline space. These are corollaries of our main result: A T-mesh is analysis-suitable if and only if it is dual-compatible. Indeed, dual compatibility is a concept already defined and used in L. Beirão da Veiga et al.5 Analysis-suitable T-splines are dual-compatible which allows for a straightforward construction of a dual basis.

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