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A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics and Symmetry/Asymmetry".

Deadline for manuscript submissions: closed (30 April 2020) | Viewed by 7090

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Special Issue Editors

Prof. Dr. Viktor A. Rukavishnikov
E-Mail Website

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Guest Editor

Computing Center of Far-Eastern Branch, Russian Academy of Sciences, Kim-Yu-Chen Str. 65, Khabarovsk 680000, Russia
Interests: boundary value problems with singularity; numerical methods in electrodynamics; hydrodynamics and theory of elasticity
Special Issues, Collections and Topics in MDPI journals

Prof. Pedro M. Lima
E-Mail Website
Guest Editor

Instituto Superior Tecnico, University of Lisbon, Lisbon, Portugal
Interests: singular boundary value problems; integro-differential equations; high-order methods

Prof. Ildar B. Badriev
E-Mail Website
Guest Editor

Kazan Federal University, 18 Kremlyovskaya Street, Kazan 420008, Russian Federation
Interests: numerical solution of the inetial boundary value problems; difference method; mathematical simulation

Special Issue Information

Dear Colleagues,

Mathematical models of different natural processes are described by differential equations, systems of PDEs and integral equations. In most cases, the exact solution of such problems cannot be determined, so we have to use mesh methods to calculate an approximate solution using high performance computational complexes. These methods include the finite element method, the finite difference method, the finite volume method and combined methods.

In this Special Issue, we propose to publish qualitative works on theoretical studies of grid methods on approximation, stability, and convergence, as well as the results of numerical experiments confirming the effectiveness of the developed methods. New methods for boundary value problems with singularity, with a complex geometry of the domain boundary, and for non-linear equations are of particular interest. Articles concerning analysis of the numerical methods developed for the computation of mathematical models in different areas of applied science and engineering applications will be welcome.

As a rule, symmetry ideas are present in the computational schemes and make the process harmonious and effective.

Prof. Viktor A. Rukavishnikov
Prof. Pedro M. Lima
Prof. Ildar B. Badriev
Guest Editors

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Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

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Keywords

finite element method
difference method
finite volume method
numerical experiments
numerical analysis
singularity
symmetry

Published Papers (7 papers)

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Research

Open AccessArticle

Finite Element Study on the Wear Performance of Movable Jaw Plates of Jaw Crushers after a Symmetrical Laser Cladding Path

and

Symmetry 2020, 12(7), 1126; https://doi.org/10.3390/sym12071126 - 07 Jul 2020

Cited by 1 | Viewed by 892

Abstract

At present, research on the influence of friction heat on the wear resistance of laser cladding layers is still lacking, and there is even less research on the temperature of laser cladding layers under different loads by a finite element program generator (FEPG). After a symmetrical laser cladding path, the wear performance of the moving jaw will change. The study of the temperature change of the moving jaw material in friction provides a theoretical basis for the surface modification of the moving jaw. The model of the column ring is built in a finite element program generator (FEPG). When the inner part of the column is WDB620 (material inside the cylinder) and the outer part is ceramic powder (moving jaw surface material), the relationship between the temperature and time of the contact surface is analyzed under the load between 100 and 600 N. At the same time, the stable temperature, wear amount, effective hardening layer thickness, strain thickness, and iron oxide content corresponding to different loads in a finite element program generator (FEPG) were analyzed. The results showed that when the load is 300 N, the temperature error between the finite element program generator (FEPG) and the movable jaw material is the largest, and the relative error is 4.3%. When the load increases, the stable temperature of the moving jaw plate increases after the symmetrical laser cladding path, and the wear amount first decreases and then increases. The minimum wear amount appears at a load of 400 N and a temperature of 340 °C; the strain thickness of the sample material increases gradually, and the effective hardening layer thickness increases. However, when the load reaches 400 N, the thickness of the effective hardening layer changes little; the content of Fe decreases gradually, and the content of FeO and Fe₂O₃ increases. The increase of the moving jaw increases in turn the temperature of the laser cladding layer of the test jaw material, which intensifies the oxidation reaction of the ceramic powder of the laser cladding layer. Full article

(This article belongs to the Special Issue Mesh Methods - Numerical Analysis and Experiments)

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Open AccessArticle

Scattered Data Interpolation Using Quartic Triangular Patch for Shape-Preserving Interpolation and Comparison with Mesh-Free Methods

Samsul Ariffin Abdul Karim

Azizan Saaban

and

Van Thien Nguyen

Symmetry 2020, 12(7), 1071; https://doi.org/10.3390/sym12071071 - 30 Jun 2020

Cited by 3 | Viewed by 1021

Abstract

Scattered data interpolation is important in sciences, engineering, and medical-based problems. Quartic Bézier triangular patches with 15 control points (ordinates) can also be used for scattered data interpolation. However, this method has a weakness; that is, in order to achieve

C^{1}

continuity, [...] Read more.

C^{1}

continuity, the three inner points can only be determined using an optimization method. Thus, we cannot obtain the exact Bézier ordinates, and the quartic scheme is global and not local. Therefore, the quartic Bézier triangular has received less attention. In this work, we use Zhu and Han’s quartic spline with ten control points (ordinates). Since there are only ten control points (as for cubic Bézier triangular cases), all control points can be determined exactly, and the optimization problem can be avoided. This will improve the presentation of the surface, and the process to construct the scattered surface is local. We also apply the proposed scheme for the purpose of positivity-preserving scattered data interpolation. The sufficient conditions for the positivity of the quartic triangular patches are derived on seven ordinates. We obtain nonlinear equations that can be solved using the regula-falsi method. To produce the interpolated surface for scattered data, we employ four stages of an algorithm: (a) triangulate the scattered data using Delaunay triangulation; (b) assign the first derivative at the respective data; (c) form a triangular surface via convex combination from three local schemes with

C^{1}

continuity along all adjacent triangles; and (d) construct the scattered data surface using the proposed quartic spline. Numerical results, including some comparisons with some existing mesh-free schemes, are presented in detail. Overall, the proposed quartic triangular spline scheme gives good results in terms of a higher coefficient of determination (R²) and smaller maximum error (Max Error), requires about 12.5% of the CPU time of the quartic Bézier triangular, and is on par with Shepard triangular-based schemes. Therefore, the proposed scheme is significant for use in visualizing large and irregular scattered data sets. Finally, we tested the proposed positivity-preserving interpolation scheme to visualize coronavirus disease 2019 (COVID-19) cases in Malaysia. Full article

(This article belongs to the Special Issue Mesh Methods - Numerical Analysis and Experiments)

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Open AccessArticle

Numerical Solution of Direct and Inverse Problems for Time-Dependent Volterra Integro-Differential Equation Using Finite Integration Method with Shifted Chebyshev Polynomials

Ratinan Boonklurb

Ampol Duangpan

and

Phansphitcha Gugaew

Symmetry 2020, 12(4), 497; https://doi.org/10.3390/sym12040497 - 30 Mar 2020

Cited by 5 | Viewed by 863

Abstract

In this article, the direct and inverse problems for the one-dimensional time-dependent Volterra integro-differential equation involving two integration terms of the unknown function (i.e., with respect to time and space) are considered. In order to acquire accurate numerical results, we apply the finite integration method based on shifted Chebyshev polynomials (FIM-SCP) to handle the spatial variable. These shifted Chebyshev polynomials are symmetric (either with respect to the point

x = \frac{L}{2}

or the vertical line

x = \frac{L}{2}

depending on their degree) over

[0, L]

, and their zeros in the interval are distributed symmetrically. We use these zeros to construct the main tool of FIM-SCP: the Chebyshev integration matrix. The forward difference quotient is used to deal with the temporal variable. Then, we obtain efficient numerical algorithms for solving both the direct and inverse problems. However, the ill-posedness of the inverse problem causes instability in the solution and, so, the Tikhonov regularization method is utilized to stabilize the solution. Furthermore, several direct and inverse numerical experiments are illustrated. Evidently, our proposed algorithms for both the direct and inverse problems give a highly accurate result with low computational cost, due to the small number of iterations and discretization. Full article

(This article belongs to the Special Issue Mesh Methods - Numerical Analysis and Experiments)

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Open AccessArticle

Discrete Symmetry Group Approach for Numerical Solution of the Heat Equation

Khudija Bibi

and

Tooba Feroze

Symmetry 2020, 12(3), 359; https://doi.org/10.3390/sym12030359 - 02 Mar 2020

Cited by 3 | Viewed by 822

Abstract

In this article, an invariantized finite difference scheme to find the solution of the heat equation, is developed. The scheme is based on a discrete symmetry transformation. A comparison of the results obtained by the proposed scheme and the Crank Nicolson method is carried out with reference to the exact solutions. It is found that the proposed invariantized scheme for the heat equation improves the efficiency and accuracy of the existing Crank Nicolson method. Full article

(This article belongs to the Special Issue Mesh Methods - Numerical Analysis and Experiments)

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Open AccessArticle

Numerical Solution of the Navier–Stokes Equations Using Multigrid Methods with HSS-Based and STS-Based Smoothers

and

Symmetry 2020, 12(2), 233; https://doi.org/10.3390/sym12020233 - 04 Feb 2020

Cited by 3 | Viewed by 814

Abstract

Multigrid methods (MGMs) are used for discretized systems of partial differential equations (PDEs) which arise from finite difference approximation of the incompressible Navier–Stokes equations. After discretization and linearization of the equations, systems of linear algebraic equations (SLAEs) with a strongly non-Hermitian matrix appear. Hermitian/skew-Hermitian splitting (HSS) and skew-Hermitian triangular splitting (STS) methods are considered as smoothers in the MGM for solving the SLAE. Numerical results for an algebraic multigrid (AMG) method with HSS-based smoothers are presented. Full article

(This article belongs to the Special Issue Mesh Methods - Numerical Analysis and Experiments)

Open AccessArticle

Numerical Method for Dirichlet Problem with Degeneration of the Solution on the Entire Boundary

Viktor A. Rukavishnikov

and

Elena I. Rukavishnikova

Symmetry 2019, 11(12), 1455; https://doi.org/10.3390/sym11121455 - 26 Nov 2019

Cited by 8 | Viewed by 1039

Abstract

The finite element method (FEM) with a special graded mesh is constructed for the Dirichlet boundary value problem with degeneration of the solution on the entire boundary of the two-dimensional domain. A comparative numerical analysis is performed for the proposed method and the classical finite element method for a set of model problems in symmetric domain. Experimental confirmation of theoretical estimates of accuracy is obtained and conclusions are made. Full article

(This article belongs to the Special Issue Mesh Methods - Numerical Analysis and Experiments)

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Open AccessArticle

Utilization of the Brinkman Penalization to Represent Geometries in a High-Order Discontinuous Galerkin Scheme on Octree Meshes

and

Symmetry 2019, 11(9), 1126; https://doi.org/10.3390/sym11091126 - 05 Sep 2019

Cited by 4 | Viewed by 1069

Abstract

We investigate the suitability of the Brinkman penalization method in the context of a high-order discontinuous Galerkin scheme to represent wall boundaries in compressible flow simulations. To evaluate the accuracy of the wall model in the numerical scheme, we use setups with symmetric reflections at the wall. High-order approximations are attractive as they require few degrees of freedom to represent smooth solutions. Low memory requirements are an essential property on modern computing systems with limited memory bandwidth and capability. The high-order discretization is especially useful to represent long traveling waves, due to their small dissipation and dispersion errors. An application where this is important is the direct simulation of aeroacoustic phenomena arising from the fluid motion around obstacles. A significant problem for high-order methods is the proper definition of wall boundary conditions. The description of surfaces needs to match the discretization scheme. One option to achieve a high-order boundary description is to deform elements at the boundary into curved elements. However, creating such curved elements is delicate and prone to numerical instabilities. Immersed boundaries offer an alternative that does not require a modification of the mesh. The Brinkman penalization is such a scheme that allows us to maintain cubical elements and thereby the utilization of efficient numerical algorithms exploiting symmetry properties of the multi-dimensional basis functions. We explain the Brinkman penalization method and its application in our open-source implementation of the discontinuous Galerkin scheme, Ateles. The core of this presentation is the investigation of various penalization parameters. While we investigate the fundamental properties with one-dimensional setups, a two-dimensional reflection of an acoustic pulse at a cylinder shows how the presented method can accurately represent curved walls and maintains the symmetry of the resulting wave patterns. Full article

(This article belongs to the Special Issue Mesh Methods - Numerical Analysis and Experiments)

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