Fast numerical algorithms with universal matrices for finding element matrices of quadrilateral and hexahedral elements
| Autor: | M. Radha Jeyakarthikeyan, Hamza Sulayman Abdullahi, P.V. Jeyakarthikeyan |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
020209 energy
Richardson extrapolation 02 engineering and technology Gauss quadrature symbols.namesake Stiffness matrix 0202 electrical engineering electronic engineering information engineering medicine Weighted Richardson extrapolation Mathematics Quadrilateral 020208 electrical & electronic engineering Mathematical analysis Geometric transformation FGM General Engineering Stiffness Engineering (General). Civil engineering (General) Closed form formulation Quadrature (mathematics) symbols Gaussian quadrature Hexahedron Hourglass TA1-2040 medicine.symptom |
| Zdroj: | Ain Shams Engineering Journal, Vol 12, Iss 1, Pp 847-863 (2021) |
| ISSN: | 2090-4479 |
| Popis: | In this paper, a new closed-form formulation with universal matrices strongly recommends that Weighted Richardson Extrapolation (WRE) with robust and hourglass controlled one-point quadrature can absolutely replace the conventional Gauss Quadrature in terms of efficiency, accuracy, and speed to find element stiffness matrices of quadrilateral and hexahedral elements. Linearizing the geometric transformation and averaging the material property over an element, using sampling point at the origin (0,0) of a standard mapped 2-square ( ξ , η ) plane for two-dimensional analysis and origin (0,0,0) of a standard mapped 2-cube ( ξ , η , ζ ) system in the hexahedral element (mid-point rule), helps to obtain element stiffness matrix quickly and explicitly through constant universal matrices. This technique is used independently to each of the eight sub-cubes of the mapped 2-cube ( ξ , η , ζ ) of the element for three-dimensional problems and four sub-squares of the mapped 2-square of the element for two-dimensional problems. These matrices are assembled appropriately for a second and better approximation. A weighted addition of the two approximations produces a stiffness matrix as accurate as from conventional Gauss Quadrature. However, finding the stiffness matrix in this way, due to explicit integrations, demands only a third of the time needed for Gauss Quadrature. |
| Databáze: | OpenAIRE |
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