scholarly journals LOCALIZATION OF SOLUTION OF THE PROBLEM OF ISOTROPIC PLATE ANALYSIS WITH THE USE OF B-SPLINE DISCRETE-CONTINUAL FINITE ELEMENT METHOD

2021 ◽  
Vol 17 (1) ◽  
pp. 55-74
Author(s):  
Marina Mozgaleva ◽  
Pavel Akimov ◽  
Taymuraz Kaytukov
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Isotropic Plate ◽  
Basis Functions ◽  
Numerical Implementation ◽  
B Spline ◽  
Plate Analysis ◽  
Element Method

Localization of solution of the problem of isotropic plate analysis with the use of B-spline discrete-continual finiteelement method (specificversion of wavelet-based discrete-continual finiteelement method) is under consideration in the distinctive paper. The original operational continual and discrete-continual formulations of the problem are given, some actual aspects of construction of normalized basis functions of a B-spline are considered, the corresponding local constructions for an arbitrary discrete-continual finiteelement are described, some information about the numerical implementation and an example of analysis are presented.

scholarly journals LOCALIZATION OF SOLUTION OF THE PROBLEM FOR POISSON’S EQUATION WITH THE USE OF B-SPLINE DISCRETE-CONTINUAL FINITE ELEMENT METHOD

2021 ◽  
Vol 17 (3) ◽  
pp. 157-172
Author(s):  
Marina Mozgaleva ◽  
Pavel Akimov
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Basis Functions ◽  
Poisson’S Equation ◽  
Numerical Implementation ◽  
Poisson's Equation ◽  
B Spline ◽  
Element Method

Localization of solution of the problem for Poisson’s equation with the use of B-spline discrete-continual finiteelement method (specificversion of wavelet-based discrete-continual finiteelement method) is under consideration in the distinctive paper. The original operational continual and discrete-continual formulations of the problem are given, some actual aspects of construction of normalized basis functions of a B-spline are considered, the corresponding local constructions for an arbitrary discrete-continual finiteelement are described, some information about the numerical implementation and an example of analysis are presented.

scholarly journals LOCALIZATION OF SOLUTION OF THE PROBLEM OF TWO-DIMENSIONAL THEORY OF ELASTICITY WITH THE USE OF B-SPLINE DISCRETE-CONTINUAL FINITE ELEMENT METHOD

2021 ◽  
Vol 17 (2) ◽  
pp. 83-104
Author(s):  
Marina Mozgaleva ◽  
Pavel Akimov ◽  
Taymuraz Kaytukov
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Theory Of Elasticity ◽  
Basis Functions ◽  
Numerical Implementation ◽  
Two Dimensional ◽  
Dimensional Theory ◽  
B Spline ◽  
Element Method

Localization of solution of the problem of two-dimensional theory of elasticity with the use of B-spline discrete-continual finite element method (specific version of wavelet-based discrete-continual finite element method) is under consideration in the distinctive paper. The original operational continual and discrete-continual formulations of the problem are given, some actual aspects of construction of normalized basis functions of a B-spline are considered, the corresponding local constructions for an arbitrary discrete-continual finite element are described, some information about the numerical implementation and an example of analysis are presented.

scholarly journals About the B-spline wavelet discrete-continual finite element method of the local plate analysis

Vestnik MGSU ◽  
2021 ◽  
pp. 666-675
Author(s):  
Pavel A. Akimov ◽  
Marina L. Mozgaleva ◽  
Taymuraz B. Kaytukov
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Analytical Solution ◽  
Element Approximation ◽  
Plate Material ◽  
Basic Direction ◽  
Spline Wavelet ◽  
B Spline ◽  
Plate Analysis ◽  
Element Method

Introduction. This distinctive paper addresses the local semi-analytical solution to the problem of plate analysis. Isotropic plates featuring the regularity (constancy) of physical and geometric parameters (modulus of elasticity of the plate material, Poisson’s ratio of the plate material, dimensions of the cross section of the plate) along one direction (dimension) are under consideration. This direction is conventionally called the basic direction. Materials and methods. The B-spline wavelet discrete-continual finite element method (DCFEM) is used. The initial operational formulation of the problem was constructed using the theory of distribution and the so-called method of extended domain, proposed by Prof. Alexander B. Zolotov. Results. Some relevant issues of construction of normalized basis functions of the B-spline are considered; the technique of approximation of corresponding vector functions and operators within DCFEM is described. The problem remains continual if analyzed along the basic direction, and its exact analytical solution can be obtained, whereas the finite element approximation is used in combination with a wavelet analysis apparatus in respect of the non-basic direction. As a result, we can obtain a discrete-continual formulation of the problem. Thus, we have a multi-point (in particular, two-point) boundary problem for the first-order system of ordinary differential equations with constant coefficients. A special correct analytical method of solving such problems was developed, described and verified in the numerous papers of the co-authors. In particular, we consider the simplest sample analysis of a plate (rectangular in plan) fixed along the side faces exposed to the influence of the load concentrated in the center of the plate. Conclusions. The solution to the verification problem obtained using the proposed version of wavelet-based DCFEM was in good agreement with the solution obtained using the conventional finite element method (the corresponding solutions were constructed with and without localization; these solutions almost completely coincided, while the advantages of the numerical-analytical approach were quite obvious). It is shown that the use of B-splines of various degrees within wavelet-based DCFEM leads to a significant reduction in the number of unknowns.

scholarly journals UMERICAL SOLUTION OF THE PROBLEM OF BEAM ANALYSIS WITH THE USE OF B-SPLINE FINITE ELEMENT METHOD

2020 ◽  
Vol 16 (3) ◽  
pp. 12-22
Author(s):  
Pavel Akimov ◽  
Marina Mozgaleva ◽  
Taymuraz Kaytukov
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Solution ◽  
Basis Functions ◽  
Numerical Implementation ◽  
Bernoulli Beam ◽  
B Spline ◽  
Bending Analysis ◽  
Beam Analysis ◽  
Spline Finite Element

Numerical solution of the problem of beam analysis (bending analysis of the Bernoulli beam) with the use of B-spline finiteelement method is under consideration in the distinctive paper. The original continual and finiteelement formulations of the problem are given, some actual aspects of construction of normalized basis functions of a B-spline are considered, the corresponding local constructions for an arbitrary finiteelement are described, some information about the numerical implementation and an example of analysis are presented.

scholarly journals An Online Generalized Multiscale Finite Element Method for Unsaturated Filtration Problem in Fractured Media

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1382
Author(s):  
Denis Spiridonov ◽  
Maria Vasilyeva ◽  
Aleksei Tyrylgin ◽  
Eric T. Chung
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Model Reduction ◽  
Basis Functions ◽  
Computational Domain ◽  
Filtration Problem ◽  
Spectral Problems ◽  
Multiscale Finite Element Method ◽  
Multiscale Finite Element ◽  
Element Method

In this paper, we present a multiscale model reduction technique for unsaturated filtration problem in fractured porous media using an Online Generalized Multiscale finite element method. The flow problem in unsaturated soils is described by the Richards equation. To approximate fractures we use the Discrete Fracture Model (DFM). Complex geometric features of the computational domain requires the construction of a fine grid that explicitly resolves the heterogeneities such as fractures. This approach leads to systems with a large number of unknowns, which require large computational costs. In order to develop a more efficient numerical scheme, we propose a model reduction procedure based on the Generalized Multiscale Finite element method (GMsFEM). The GMsFEM allows solving such problems on a very coarse grid using basis functions that can capture heterogeneities. In the GMsFEM, there are offline and online stages. In the offline stage, we construct snapshot spaces and solve local spectral problems to obtain multiscale basis functions. These spectral problems are defined in the snapshot space in each local domain. To improve the accuracy of the method, we add online basis functions in the online stage. The construction of the online basis functions is based on the local residuals. The use of online bases will allow us to get a significant improvement in the accuracy of the method. We present results with different number of offline and online multisacle basis functions. We compare all results with reference solution. Our results show that the proposed method is able to achieve high accuracy with a small computational cost.

Generation and Modification of Non-Uniform B-Spline Surface Approximations to PDE Surfaces Using the Finite Element Method

1990 ◽  
Author(s):  
Joanna M. Brown ◽  
Malcolm I. G. Bloor ◽  
M. Susan Bloor ◽  
Michael J. Wilson
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Spline Approximation ◽  
The Finite Element Method ◽  
B Spline ◽  
B Splines ◽  
Spline Surface ◽  
Spline Surfaces ◽  
Spline Basis ◽  
Element Method

Abstract A PDE surface is generated by solving partial differential equations subject to boundary conditions. To obtain an approximation of the PDE surface in the form of a B-spline surface the finite element method, with the basis formed from B-spline basis functions, can be used to solve the equations. The procedure is simplest when uniform B-splines are used, but it is also feasible, and in some cases desirable, to use non-uniform B-splines. It will also be shown that it is possible, if required, to modify the non-uniform B-spline approximation in a variety of ways, using the properties of B-spline surfaces.

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