Closed-form stiffness matrices for the linear strain and quadratic strain tetrahedron finite elements

Shiakolas, Panos S.; Nambiar, R.V.; Lawrence, K. L.; Rogers, William A.

doi:10.1016/0045-7949(92)90407-q

Cited by 23 publications

(5 citation statements)

References 11 publications

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“…Rathod [8] presented analytical integration formulas for a 4-node isoparametric FE and showed that all the integration formulas could be obtained on the basis of four simple integrals. Generation of Fortran code by CAS was discussed by Wang et al [9] and Lawrence et al [10] for triangles, and Shiakolas et al [11] for tetrahedra.…”

Section: Introductionmentioning

confidence: 99%

Elastic stiffness of straight‐sided triangular finite elements by analytical and numerical integration

Griffiths¹,

Huang²,

Schiermeyer³

2008

Commun. Numer. Meth. Engng.

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SUMMARYMost finite element (FE) software uses numerical integration to compute FE stiffness matrices. In the case of straight-sided isoparametric triangular elements, numerical integration is exact, provided a sufficient number of integration points are used. In this paper, the same integrations are performed analytically in closed form with the help of computer algebra software for 3-, 6-, 10-and 15-noded triangles. The resulting analytical expressions have been converted into Fortran 95 subroutines and compared with the conventional numerical integration methods. The analytical routines produce exactly the same results as their numerical counterparts, but run significantly faster. All Fortran 95 code developed in this study is available on the first author's Web site.

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Section: Introductionmentioning

confidence: 99%

Elastic stiffness of straight‐sided triangular finite elements by analytical and numerical integration

Griffiths¹,

Huang²,

Schiermeyer³

2008

Commun. Numer. Meth. Engng.

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“…It is believed that the formulations are advantageous in situations where the closed-form rectangular element cannot be used. [5] It examines the problem of smoothing the numerically discontinuous model of a physically continuous system. Several solutions using least squares smoothing are suggested and some practical recommendations are made.…”

Section: Introductionmentioning

confidence: 99%

An Efficient Method to Generate Element Stiffness Matrix of Quadrilateral Elements in Closed Form on Application of Vehicle Analysis

Jeyakarthikeyan

Yogeshwaran

Sridharan

2016

AMM

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This paper presents about generating elemental stiffness matrix for quadrilateral elements in closed form solution method for application on vehicle analysis which is convenient and simple as long as Jacobian is matrix of constant. The interpolation function of the field variable to be found can integrate explicitly once for all, which gives the constant universal matrices A, B and C. Therefore, stiffness matrix is no longer integration of the given functional, it is simple calculation of universal matrices and local co-ordinates of the element. So time taken for generation of element stiffness can be reduced considerably compared to Gauss numerical integration method. For effective use of quadrilateral elements hybrid grid generation is recommended that contains all interior element edges are parallel to each other (rectangle or square elements) and outer boundary elements are quadrilaterals with distortion. So in the Proposed method, the closed form and Gauss numerical method is used explicitly for interior elements and outer elements respectively. The time efficiency of proposed method is compared with conventional Gauss quadrature that is used for entire domain. It is found that the proposed method is much efficient than Gauss Quadrature.

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“…Ou seja, as matrizes de massa de rigidez, compressibilidade, volumétrica e de acoplamento não são mais integradas numericamente para cada elemento. Não é um procedimento novo, as referências [52,53,54,55,56,57,58,59,60,61] apresentam técnicas de integração, de redução no armazenamento dos termos das matrizes, as expressões destas matrizes para elementos triangulares e tetraédricos [53,58,61,59,60,54],…”

Section: Estimadores De Erro Para Problemas De Acústica Internaunclassified

“…O emprego de elementos triangulares neste trabalho, em parte deve-se a decisão de usar matrizes fechadas. A simplicidade das expressões para integrais de área e de linha quando expressas em coordenadas de área ou naturais recomendavam este elemento [53,58,61,59,60,54]. Expressões mais simples também facilitam a elaboração de programas em linguagem de matemática simbólica.…”

Section: Estimadores De Erro Para Problemas De Acústica Internaunclassified

“…Além disso as taxas de convergência para formulação-p são superiores a formulação-h [46,38,48]. Por outro lado, a integração numérica de funções de ordem elevada consome muito tempo de processamento [52,53,54,55,56,57,58,59,60,61]. Portanto, é mais adequado o uso de matrizes fechadas, com integração analítica neste tipo de problemas.…”

Section: Funções De Formaunclassified

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Estimador de erro para a formulação p do metodo dos elementos finitos aplicado ao problema fluido-estrutura

Duarte¹,

Pavanello²

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Gostaria de agradecer às pessoas e instituições que colaboraram para a conclusão deste trabalho:-Ao meu orientador, Professor Renato Pavanello, pelo apoio, pelas discussões, pelas sugestões e pela paciência durante esta longa jornada em que trabalhamos juntos.-À minha família pela paciência e compreensão pelas longas ausências.-Aos professores da Faculdade de Engenharia Mecânica que colaboraram dando sugestões durante a realização deste trabalho. Pelo companheirismo e pela participação especial que tiveram neste período agredeço ao professor Iguti e à professora Silvia.-Aos colegas que trabalharam comigo, aos que participaram apoiando de alguma forma, por tudo isto ou apenas pela convivência e amizade.-Ao Sergio, Allan, Juan, Daniel e ao Ivan.-Aos colegas de departamento da UFMG que, de forma solidária, assumiram meus encargos tornando possível meu afastamento.-Ao Departamento de Mecânica Computacional pela infra-estrutura fornecida durante a realização deste trabalho.-À CAPES por intermédio do programa PICD da UFMG pelo apoio financeiro. vi ResumoDuarte, Horacio V., Estimador de Erro para a Formulação p do Método dos Elementos

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Closed-form stiffness matrices for the linear strain and quadratic strain tetrahedron finite elements

Cited by 23 publications

References 11 publications

Elastic stiffness of straight‐sided triangular finite elements by analytical and numerical integration

Elastic stiffness of straight‐sided triangular finite elements by analytical and numerical integration

An Efficient Method to Generate Element Stiffness Matrix of Quadrilateral Elements in Closed Form on Application of Vehicle Analysis

Estimador de erro para a formulação p do metodo dos elementos finitos aplicado ao problema fluido-estrutura

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