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Derivation of Shape Functions for Nine-Node Rectangular Element Based on Steady-State Heat Conduction
This paper presents a novel method for deriving shape functions of a nine-node rectangular element based on solving the steady-state heat conduction equation. Unlike traditional methods using Lagrange polynomials, this approach utilizes a linear combination of temperature distribution (TD) coefficients to extract second-order shape functions for a one-dimensional (1D) quadratic element. These functions are then extended to the nine-node rectangular element through multiplication in two orthogonal directions. This method allows for the derivation of shape functions within the local coordinate system of a quadrilateral element, without solving Laplace’s equation. To validate the performance of the nine-node element, a thermal analysis of a square plate was conducted using simple square and triangular elements, as well as an analytical method. Sensitivity analysis for boundary conditions indicated a negligible error margin. Numerical studies further demonstrate the high accuracy of the nine-node rectangular element in modeling various scenarios, including the numerical analysis of torsion and TD on rectangular shapes. The single nine-node quadrilateral element (Q9) effectively models shear deformation effects, providing higher accuracy in capturing complex stress distributions compared to four-node elements (Q4). Its ability to consider shear deformation makes it more suitable for detailed stress analysis, with the error reduced to 18%.
Derivation of Shape Functions for Nine-Node Rectangular Element Based on Steady-State Heat Conduction
This paper presents a novel method for deriving shape functions of a nine-node rectangular element based on solving the steady-state heat conduction equation. Unlike traditional methods using Lagrange polynomials, this approach utilizes a linear combination of temperature distribution (TD) coefficients to extract second-order shape functions for a one-dimensional (1D) quadratic element. These functions are then extended to the nine-node rectangular element through multiplication in two orthogonal directions. This method allows for the derivation of shape functions within the local coordinate system of a quadrilateral element, without solving Laplace’s equation. To validate the performance of the nine-node element, a thermal analysis of a square plate was conducted using simple square and triangular elements, as well as an analytical method. Sensitivity analysis for boundary conditions indicated a negligible error margin. Numerical studies further demonstrate the high accuracy of the nine-node rectangular element in modeling various scenarios, including the numerical analysis of torsion and TD on rectangular shapes. The single nine-node quadrilateral element (Q9) effectively models shear deformation effects, providing higher accuracy in capturing complex stress distributions compared to four-node elements (Q4). Its ability to consider shear deformation makes it more suitable for detailed stress analysis, with the error reduced to 18%.
Derivation of Shape Functions for Nine-Node Rectangular Element Based on Steady-State Heat Conduction
Ramin Tabatabaei Mirhosseini (author) / Ramin Shamsi (author)
2025
Article (Journal)
Electronic Resource
Unknown
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