S. Aizikovich, A. Nikolaev, E. Sadyrin, L. Krenev, V. Irkha, A. Galybin
An effective mathematical model is proposed for describing the experiment on indentation of samples with layered or functionally graded coatings. It is based on the solution of the contact problem of the theory of elasticity of indentation of a punch into elastic half-space with a coating. The results of mathematical modelling and experiments on indentation of the ZnO coating manufactured by the method of pulsed laser deposition on a silicon substrate are compared. The microgeometrical characteristics, as well as the chemical composition of the coating, were studied.
{"title":"INDENTATION OF THIN COATINGS: THEORETICAL AND EXPERIMENTAL INVESTIGATION","authors":"S. Aizikovich, A. Nikolaev, E. Sadyrin, L. Krenev, V. Irkha, A. Galybin","doi":"10.2495/be450141","DOIUrl":"https://doi.org/10.2495/be450141","url":null,"abstract":"An effective mathematical model is proposed for describing the experiment on indentation of samples with layered or functionally graded coatings. It is based on the solution of the contact problem of the theory of elasticity of indentation of a punch into elastic half-space with a coating. The results of mathematical modelling and experiments on indentation of the ZnO coating manufactured by the method of pulsed laser deposition on a silicon substrate are compared. The microgeometrical characteristics, as well as the chemical composition of the coating, were studied.","PeriodicalId":23647,"journal":{"name":"WIT transactions on engineering sciences","volume":"201 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82813594","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, a creative collocation-type numerical method, the Finite Line Method (FLM), is proposed for solving general convection–diffusion equations. The method is based on the use of a finite number of lines crossing each collocation point, and the Lagrange polynomial interpolation formulation to construct the shape functions over each line. The directional derivative technique is proposed to derive the first-order partial derivatives of any physical variables with respect to the global coordinates for the high-dimensional problems from the lines’ ones and the high-order derivatives are evaluated from a recurrence formulation. The derived spatial partial derivatives are directly substituted into the governing partial differential equations and related boundary conditions of the convection–diffusion equations to set up the system of equations. The finite number of lines crossing each collocation point is called the line set. To evaluate the convection and diffusion terms accurately, two different line sets are used for these two terms, which are called the convection line set and central line set, respectively. The former is formed according to the velocity direction and is used for performing the upwind scheme in the computation of the convection term, and the latter is formed by the crossed lines including the collocation point at the center. A numerical example will be given to verify the correctness and stability of the proposed method.
{"title":"FINITE LINE METHOD FOR SOLVING CONVECTION–DIFFUSION EQUATIONS","authors":"Xiaowei Gao, Hua‐Yu Liu, Jingdong Ding","doi":"10.2495/be450041","DOIUrl":"https://doi.org/10.2495/be450041","url":null,"abstract":"In this paper, a creative collocation-type numerical method, the Finite Line Method (FLM), is proposed for solving general convection–diffusion equations. The method is based on the use of a finite number of lines crossing each collocation point, and the Lagrange polynomial interpolation formulation to construct the shape functions over each line. The directional derivative technique is proposed to derive the first-order partial derivatives of any physical variables with respect to the global coordinates for the high-dimensional problems from the lines’ ones and the high-order derivatives are evaluated from a recurrence formulation. The derived spatial partial derivatives are directly substituted into the governing partial differential equations and related boundary conditions of the convection–diffusion equations to set up the system of equations. The finite number of lines crossing each collocation point is called the line set. To evaluate the convection and diffusion terms accurately, two different line sets are used for these two terms, which are called the convection line set and central line set, respectively. The former is formed according to the velocity direction and is used for performing the upwind scheme in the computation of the convection term, and the latter is formed by the crossed lines including the collocation point at the center. A numerical example will be given to verify the correctness and stability of the proposed method.","PeriodicalId":23647,"journal":{"name":"WIT transactions on engineering sciences","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75068769","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper focuses on deriving the local variant of the singular boundary method (SBM) to solve the convection–diffusion equation. Adopting the combination of an SBM and finite collocation, one obtains the localized variant of SBM. Unlike the global variant, local SBM leads to a sparse matrix of the resulting system of equations, making it much more efficient to solve large-scale tasks. It also allows solving velocity vector variable tasks, which is a problem with global SBM. The article presents the steady numerical example for the convection–diffusion problem with variable velocity field and examines the dependence of the accuracy of the solution on the nodal grid’s density and the subdomain’s size.
{"title":"LOCALIZED SINGULAR BOUNDARY METHOD FOR SOLVING THE CONVECTION–DIFFUSION EQUATION WITH VARIABLE VELOCITY FIELD","authors":"J. Mužík, R. Bulko","doi":"10.2495/be450101","DOIUrl":"https://doi.org/10.2495/be450101","url":null,"abstract":"This paper focuses on deriving the local variant of the singular boundary method (SBM) to solve the convection–diffusion equation. Adopting the combination of an SBM and finite collocation, one obtains the localized variant of SBM. Unlike the global variant, local SBM leads to a sparse matrix of the resulting system of equations, making it much more efficient to solve large-scale tasks. It also allows solving velocity vector variable tasks, which is a problem with global SBM. The article presents the steady numerical example for the convection–diffusion problem with variable velocity field and examines the dependence of the accuracy of the solution on the nodal grid’s density and the subdomain’s size.","PeriodicalId":23647,"journal":{"name":"WIT transactions on engineering sciences","volume":"106 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86979846","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
V. Gnitko, Artem Karaiev, Kyryl Degtyariov, I. Vierushkin, E. Strelnikova
The paper presents new computational techniques based on coupled boundary and finite element methods to study fluid–structure interaction problems. Thin shells and plates are considered as structure elements interacting with an ideal and incompressible liquid. To describe the motion of both structural elements and the fluid, the basic relations of the continuous mechanics are incorporated. The liquid pressure is determined by applying the Laplace equation. Two kinds of boundary value problems are considered corresponding to one-sided and two-sided contact of structural elements with the liquid. Integral equations for numerical simulation of pressure are obtained. For a two-sided contact of the structural element with the liquid, hypersingular integral equations are received, whereas singular integral equations with logarithmic singularities describe the problems of one-sided contact. Considering the structure axial symmetry, the integral equations are reduced to one-dimensional ones. The finite element method for determining modes and frequencies of the elastic structure coupled with boundary element method for the hypersingular integral equation is implemented to find the fluid pressure on the structure element with two-sided contact with the liquid. The liquid pressure evaluation in axisymmetric problems is reduced to one-dimensional integral equations with kernels in the form of elliptic integrals. The effective technique is developed for numerical simulation of obtained singular integrals. The same technique is extended to hypersingular integral equations. The frequencies and modes of structure vibrations taking into account the added masses of the liquid are obtained. Thin circular plates and shells of revolution are considered as structure elements in numerical simulations. The accuracy and reliability of the proposed method are ascertained.
{"title":"SINGULAR AND HYPERSINGULAR INTEGRAL EQUATIONS IN FLUID–STRUCTURE INTERACTION ANALYSIS","authors":"V. Gnitko, Artem Karaiev, Kyryl Degtyariov, I. Vierushkin, E. Strelnikova","doi":"10.2495/be450061","DOIUrl":"https://doi.org/10.2495/be450061","url":null,"abstract":"The paper presents new computational techniques based on coupled boundary and finite element methods to study fluid–structure interaction problems. Thin shells and plates are considered as structure elements interacting with an ideal and incompressible liquid. To describe the motion of both structural elements and the fluid, the basic relations of the continuous mechanics are incorporated. The liquid pressure is determined by applying the Laplace equation. Two kinds of boundary value problems are considered corresponding to one-sided and two-sided contact of structural elements with the liquid. Integral equations for numerical simulation of pressure are obtained. For a two-sided contact of the structural element with the liquid, hypersingular integral equations are received, whereas singular integral equations with logarithmic singularities describe the problems of one-sided contact. Considering the structure axial symmetry, the integral equations are reduced to one-dimensional ones. The finite element method for determining modes and frequencies of the elastic structure coupled with boundary element method for the hypersingular integral equation is implemented to find the fluid pressure on the structure element with two-sided contact with the liquid. The liquid pressure evaluation in axisymmetric problems is reduced to one-dimensional integral equations with kernels in the form of elliptic integrals. The effective technique is developed for numerical simulation of obtained singular integrals. The same technique is extended to hypersingular integral equations. The frequencies and modes of structure vibrations taking into account the added masses of the liquid are obtained. Thin circular plates and shells of revolution are considered as structure elements in numerical simulations. The accuracy and reliability of the proposed method are ascertained.","PeriodicalId":23647,"journal":{"name":"WIT transactions on engineering sciences","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82010403","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this study, we consider a special incorrectly posed boundary value problem of the theory of cracks, which arises when modelling the initiation and development of fracture on the interface between poroelastic materials. The main feature of the problem is the formulation of boundary conditions, which is different from the standard formulations. The problem is considered for a strip, where three conditions are set on one side of the strip and one on the other side, which makes it possible to classify this formulation as a semi-inverse one.
{"title":"INTEGRAL EQUATIONS FOR MODELLING OF FRACTURE INITIATION AND DEVELOPMENT IN LAYERED POROELASTIC MEDIA","authors":"A. Galybin, S. Aizikovich","doi":"10.2495/be450081","DOIUrl":"https://doi.org/10.2495/be450081","url":null,"abstract":"In this study, we consider a special incorrectly posed boundary value problem of the theory of cracks, which arises when modelling the initiation and development of fracture on the interface between poroelastic materials. The main feature of the problem is the formulation of boundary conditions, which is different from the standard formulations. The problem is considered for a strip, where three conditions are set on one side of the strip and one on the other side, which makes it possible to classify this formulation as a semi-inverse one.","PeriodicalId":23647,"journal":{"name":"WIT transactions on engineering sciences","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81642037","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper presents a brief overview of recently developed localized collocation solvers and their various engineering applications. The research progress of localized collocation solvers is discussed, and their basic mathematical formulations are summarized. Finally, applications for thermal analysis in functionally graded materials and steady-state convection-diffusion equation with nonhomogeneous term are given.
{"title":"RECENT ADVANCES IN LOCALIZED COLLOCATION SOLVERS BASED ON SEMI-ANALYTICAL BASIS FUNCTIONS","authors":"Wenzhi Xu, Zhuojia Fu","doi":"10.2495/be450121","DOIUrl":"https://doi.org/10.2495/be450121","url":null,"abstract":"This paper presents a brief overview of recently developed localized collocation solvers and their various engineering applications. The research progress of localized collocation solvers is discussed, and their basic mathematical formulations are summarized. Finally, applications for thermal analysis in functionally graded materials and steady-state convection-diffusion equation with nonhomogeneous term are given.","PeriodicalId":23647,"journal":{"name":"WIT transactions on engineering sciences","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78499618","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Direct boundary integral equation (BIE) formalisms for wave radiation and scattering have been stable and universally accepted for decades. Yet, the classic separation of variables (SOV) solutions for acoustic radiation and scattering from spheres do not always agree with BEM results. For certain conditions, the boundary acoustic field predicted by low-frequency SOV and BEM methods match exactly and for other situations predicted fields by the two methods are complex-conjugates of each other. While this difference is subtle, modern BEM literature has not cited the transfer of known mathematics to this engineering application. Tracing signs within BEM code is daunting. To create a lucid and reproducible record of the issue and its resolution, this paper presents an analytical BIE solution for spherical geometry based on a Legendre polynomial simplex element and a power series of the spatial phase term of the Helmholtz operator Fundamental Solution. Optical theorem reasoning suggests the traditional BIE approach is the method in error. The core of this issue is the application of the divergence theorem (strictly true only for real-valued functions) to time-harmonic (complex-valued) formulations. The conjugation of spatial derivatives of a complex-valued field can be understood from Wirtinger derivatives and Dolbeault operators. This issue manifests itself when the Sommerfeld radiation condition is applied for unbounded domains. Exterior calculus ideas properly unite, generalize and extend a variety of related classical theorems including divergence, Cauchy’s integral theorem from complex analysis, and Green’s identities used in constructing a BIE. The resulting Stokes–Cartan theorem is properly applied to acoustic scattering in 3D within this paper and invokes corrections which match BIE and SOV solutions for the low frequency problems investigated.
{"title":"IMPLICATIONS OF STOKES–CARTAN THEOREM TO TIME-HARMONIC ACOUSTIC BOUNDARY INTEGRAL EQUATION FORMULATIONS","authors":"P. Schafbuch","doi":"10.2495/be450071","DOIUrl":"https://doi.org/10.2495/be450071","url":null,"abstract":"Direct boundary integral equation (BIE) formalisms for wave radiation and scattering have been stable and universally accepted for decades. Yet, the classic separation of variables (SOV) solutions for acoustic radiation and scattering from spheres do not always agree with BEM results. For certain conditions, the boundary acoustic field predicted by low-frequency SOV and BEM methods match exactly and for other situations predicted fields by the two methods are complex-conjugates of each other. While this difference is subtle, modern BEM literature has not cited the transfer of known mathematics to this engineering application. Tracing signs within BEM code is daunting. To create a lucid and reproducible record of the issue and its resolution, this paper presents an analytical BIE solution for spherical geometry based on a Legendre polynomial simplex element and a power series of the spatial phase term of the Helmholtz operator Fundamental Solution. Optical theorem reasoning suggests the traditional BIE approach is the method in error. The core of this issue is the application of the divergence theorem (strictly true only for real-valued functions) to time-harmonic (complex-valued) formulations. The conjugation of spatial derivatives of a complex-valued field can be understood from Wirtinger derivatives and Dolbeault operators. This issue manifests itself when the Sommerfeld radiation condition is applied for unbounded domains. Exterior calculus ideas properly unite, generalize and extend a variety of related classical theorems including divergence, Cauchy’s integral theorem from complex analysis, and Green’s identities used in constructing a BIE. The resulting Stokes–Cartan theorem is properly applied to acoustic scattering in 3D within this paper and invokes corrections which match BIE and SOV solutions for the low frequency problems investigated.","PeriodicalId":23647,"journal":{"name":"WIT transactions on engineering sciences","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78006558","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is part of a research work to implement, test, and apply a novel numerical tool that can simulate on a personal computer and in just a few minutes a problem of potential or elasticity with up to tens of millions of degrees of freedom. The first author’s group has already developed their own version of the fast multipole method (FMM) for two-dimensional problems, which relies on a consistent construction of the single-layer potential matrix of the collocation boundary element method so that ultimately only polynomial terms (as for the double-layer potential matrix) are required to be integrated along generally curved segments related to a given field expansion pole. The core of the present paper is the mathematical assessment of the double expansions needed in the 3D FMM. The 3D implementation is combined with a particular formulation for linear triangle elements in which all integrations for adjacent source point and boundary element are carried out analytically. As a result, numerical approximations are due exclusively to the FMM series truncations. This allows isolating and testing truncation errors incurred in the series expansions and thus for the first time properly assessing the mathematical features of the FMM, as illustrated by means of two examples. Adaptive numerical quadratures as well as the complete solution of a mixed boundary problem using a GMRES solver, for instance, are just additional tasks and, although already implemented, are not reported herein.
{"title":"ACCURATE FAST MULTIPOLE SCHEME FOR THE BOUNDARY ELEMENT ANALYSIS OF THREE-DIMENSIONAL LINEAR POTENTIAL PROBLEMS","authors":"N. Dumont, Hilton Marques SOUZA SANTANA","doi":"10.2495/be450011","DOIUrl":"https://doi.org/10.2495/be450011","url":null,"abstract":"This paper is part of a research work to implement, test, and apply a novel numerical tool that can simulate on a personal computer and in just a few minutes a problem of potential or elasticity with up to tens of millions of degrees of freedom. The first author’s group has already developed their own version of the fast multipole method (FMM) for two-dimensional problems, which relies on a consistent construction of the single-layer potential matrix of the collocation boundary element method so that ultimately only polynomial terms (as for the double-layer potential matrix) are required to be integrated along generally curved segments related to a given field expansion pole. The core of the present paper is the mathematical assessment of the double expansions needed in the 3D FMM. The 3D implementation is combined with a particular formulation for linear triangle elements in which all integrations for adjacent source point and boundary element are carried out analytically. As a result, numerical approximations are due exclusively to the FMM series truncations. This allows isolating and testing truncation errors incurred in the series expansions and thus for the first time properly assessing the mathematical features of the FMM, as illustrated by means of two examples. Adaptive numerical quadratures as well as the complete solution of a mixed boundary problem using a GMRES solver, for instance, are just additional tasks and, although already implemented, are not reported herein.","PeriodicalId":23647,"journal":{"name":"WIT transactions on engineering sciences","volume":"3 1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72965121","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Kansa method is one of the most popular meshless methods today. Its ease of implementation, high order of interpolation and ease of application to problems with complex geometry constitute its advantage over many other methods for solving partial differential equation-based problems. However, the Kansa method has a significant disadvantage – the need to find the shape parameter value despite these undeniable advantages. There are dozens of algorithms for finding a good shape parameter value, but none of them is proven to be optimal. Therefore, there is still a great scientific need to research new algorithms and improve those already known. In this work, an algorithm based on the study of the oscillation of certain shape parameter functions concerning the problems of two-dimensional heat flow in a material with spatially variable thermophysical parameters was investigated. It has been shown that algorithms of this type allow this class of problems to achieve solutions with high accuracy. At the same time, it was indicated that this direction of development of algorithms for searching for a good value of the shape parameter is auspicious. It is because this algorithm can be extended to a wide range of functions whose oscillation is studied and, consequently, its application to a broader range of problems.
{"title":"CERTAIN RELATIONS BETWEEN THE MAIN MATRIX CONDITION NUMBER AND MULTIQUADRIC SHAPE PARAMETER IN THE NON-SYMMETRIC KANSA METHOD","authors":"O. Popczyk, G. Dziatkiewicz","doi":"10.2495/be450111","DOIUrl":"https://doi.org/10.2495/be450111","url":null,"abstract":"The Kansa method is one of the most popular meshless methods today. Its ease of implementation, high order of interpolation and ease of application to problems with complex geometry constitute its advantage over many other methods for solving partial differential equation-based problems. However, the Kansa method has a significant disadvantage – the need to find the shape parameter value despite these undeniable advantages. There are dozens of algorithms for finding a good shape parameter value, but none of them is proven to be optimal. Therefore, there is still a great scientific need to research new algorithms and improve those already known. In this work, an algorithm based on the study of the oscillation of certain shape parameter functions concerning the problems of two-dimensional heat flow in a material with spatially variable thermophysical parameters was investigated. It has been shown that algorithms of this type allow this class of problems to achieve solutions with high accuracy. At the same time, it was indicated that this direction of development of algorithms for searching for a good value of the shape parameter is auspicious. It is because this algorithm can be extended to a wide range of functions whose oscillation is studied and, consequently, its application to a broader range of problems.","PeriodicalId":23647,"journal":{"name":"WIT transactions on engineering sciences","volume":"71 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86888525","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Rocío Velázquez Mata, Antonio Romero ORÕNEZ, Pedro GALIN Barrera
This paper describes a general method to compute the boundary integral equation for common engineering problems. The proposed procedure consists of a new quadrature rule to evaluate singular and weakly singular integrals. The methodology is based on the computation of the quadrature weights by solving an undetermined system of equations in the minimum norm sense. The B´ezier–Bernstein form of a polynomial is also implemented as an approximation basis to represent both geometry and field variables. Therefore, exact boundary geometry is considered, and arbitrary high-order elements are allowed. This procedure can be used for any element node distribution and shape function. The validity of the method is demonstrated by solving a two-and-a-half-dimensional elastodynamic benchmark problem.
{"title":"QUADRATURE RULE FOR SINGULAR INTEGRALS IN COMMON ENGINEERING PROBLEMS","authors":"Rocío Velázquez Mata, Antonio Romero ORÕNEZ, Pedro GALIN Barrera","doi":"10.2495/be450051","DOIUrl":"https://doi.org/10.2495/be450051","url":null,"abstract":"This paper describes a general method to compute the boundary integral equation for common engineering problems. The proposed procedure consists of a new quadrature rule to evaluate singular and weakly singular integrals. The methodology is based on the computation of the quadrature weights by solving an undetermined system of equations in the minimum norm sense. The B´ezier–Bernstein form of a polynomial is also implemented as an approximation basis to represent both geometry and field variables. Therefore, exact boundary geometry is considered, and arbitrary high-order elements are allowed. This procedure can be used for any element node distribution and shape function. The validity of the method is demonstrated by solving a two-and-a-half-dimensional elastodynamic benchmark problem.","PeriodicalId":23647,"journal":{"name":"WIT transactions on engineering sciences","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83589008","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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