On stiffness matrices for C0 continuous tetrahedra

doi:10.1016/0045-7949(83)90109-8

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On stiffness matrices for C0 continuous tetrahedra

Journal Article•DOI•

On stiffness matrices for C0 continuous tetrahedra

G. Subramanian¹, C.Jeyachandra Bose¹•Institutions (1)

Indian Institute of Technology Madras¹

01 Jan 1983-Computers & Structures (Pergamon)-Vol. 16, Iss: 5, pp 603-611

TL;DR: In this article, a simple method for the determination of stiffness matrices for the family of C 0 continuous tetrahedral elements is presented, where two basic matrices of numbers are obtained once and for all globally and stored as data to programs.

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About: This article is published in Computers & Structures.The article was published on 1983-01-01. It has received 4 citations till now.

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Journal Article•DOI•

An alternate stable midpoint quadrature to improve the element stiffness matrix of quadrilaterals for application of functionally graded materials (FGM)

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P.V. Jeyakarthikeyan¹, G. Subramanian¹, R. Yogeshwaran¹•Institutions (1)

SRM University¹

01 Jan 2017-Computers & Structures

TL;DR: In this paper, a weighted integration route with robust stable one-point integration is recommended, which is an alternative to Gauss quadrature to obtain stiffness matrix of quadrilaterals.

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18 citations

Journal Article•DOI•

Fast numerical algorithms with universal matrices for finding element matrices of quadrilateral and hexahedral elements

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P.V. Jeyakarthikeyan¹, M. Radha Jeyakarthikeyan¹, Hamza Sulayman Abdullahi²•Institutions (2)

SRM University¹, Bayero University Kano²

01 Mar 2021-Ain Shams Engineering Journal

TL;DR: In this article, 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.

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4 citations

Journal Article•DOI•

Weighted Integration Route to Stiffness Matrix of Quadrilaterals for Speed, Accuracy and Functionally Graded Material Application

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G. Subramanian, P.V. Jeyakarthikeyan

01 Apr 2018-American Journal of Applied Sciences

TL;DR: In this article, a weighted integration route with robust one-point integration (hourglass-controlled) is proposed as efficient, time saving alternative to Gauss quadrature for stiffness matrix of bilinear quadrilaterals.

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Abstract: A weighted integration route with robust one-point integration (hourglass-controlled) is proposed as efficient, time saving alternative to Gauss quadrature for stiffness matrix of bilinear quadrilaterals. One-point rule relies on sampling at the center of the element to linearize the geometric transformation and average the material property over it. This enables, for a given element, explicit integration of stiffness matrix yielding a first approximation. For a second and better approximation, this procedure is applied independently to each of the four sub-squares of the mapped 2-square of the element and the matrices are assembled. A weighted addition of the two approximations produces a stiffness matrix as accurate as from 3-point Gauss-quadrature (G9P). Whereas, due to explicit integrations, obtaining stiffness matrix in this way demands less than a third of the time needed for 2-point Gauss-quadrature (G4P). On both counts (speed and accuracy) this approach outperforms Gauss-quadrature. Sampling (material and geometry) at 5-points makes this element superior to G4P for Functionally Graded Material (FGM) applications. Bench mark examples by this approach are validated with Gauss quadrature and analytical solutions.

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2 citations

Cites background from "On stiffness matrices for C0 contin..."

...Thus, these linear substitute shape functions return the corner nodes of the 2-square in the transform plane as a set of corner nodes for a parallelogram instead of the original quadrilateral (Subramanian and Bose, 1982; 1983) for a similar observation)....
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...Simple exactly integrated numerical universal matrices like those for triangles and tetrahedrons (Subramanian and Bose, 1982; 1983) also are possible only for parallelograms....
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Dissertation•

Closed-form Development Of A Family Of Higher Order Tetrahedral Elements Through The Fourth Order

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Sara E. McCaslin

08 Aug 2008

TL;DR: McCaslin et al. as mentioned in this paper investigated the efficiency of closed-form implementation of stiffness matrices and error estimators compared to numerical implementation, and concluded that closed form implementation is more efficient by a factor of at least 4 when compared with numerically integrated elements.

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Abstract: CLOSED-FORM DEVELOPMENT OF A FAMILY OF HIGHER ORDER TETRAHEDRAL ELEMENTS THROUGH THE FOURTH ORDER Sara Elizabeth McCaslin, PhD. The University of Texas at Arlington, 2008 Supervising Professor: Kent Lawrence This research is concerned with the development and implementation of a family of tetrahedral elements through the fourth order. The straight-sided tetrahedral elements are developed in closed-form. This work investigates the efficiency of closedform implementation of stiffness matrices and error estimators compared to numerical implementation. An additional objective is the compaction of closed-form source-code files which require as little storage space as possible, a more pronounced requirement at high p-levels. For the straight-sided elements through p-level 4, the stiffness matrix, equivalent nodal load vectors, and error estimators (based on nodal averaging) are developed using closed-form equations obtained through the use of a computer algebra iv system. The stiffness matrix and error estimators are also implemented using numerical integration so that a timing comparison between the numerical and the closed-form approaches could be performed. The curved-sided elements, including the stiffness matrix, equivalent nodal load vectors, and error estimators are also implemented using Gaussian quadrature only. A test conducted on a model of all curved-sided elements is used to verify that the elements are working correctly. Results indicate that the closed-form implementation solutions are comparable to the numerical solutions. For all p-levels the closed-form stiffness matrix is more efficient by a factor of at least 4 when compared with numerically integrated elements.

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Cites methods from "On stiffness matrices for C0 contin..."

...1.5.1 Closed-form Stiffness Matrices Tinawi [23] used closed-form integration to obtain the stiffness matrix of non- hierarchic triangular elements in 1972; Subramanian and Bose [24, 25] developed stiffness matrices without the use of numerical approximation for the family of plane triangular elements in 1982 and for C0 continuous tetrahedra in 1983....
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...113 nn[[11]] = Simplify[L2 L3 Ei[2,L2,L3]]; nn[[12]] = Simplify[L1 L3 Ei[2,L1,L3]]; nn[[13]] = Simplify[L1 L2 Ei[2,L1,L2]]; nn[[14]] = Simplify[L1 L4 Ei[2,L1,L4]]; nn[[15]] = Simplify[L2 L4 Ei[2,L2,L4]]; nn[[16]] = Simplify[L3 L4 Ei[2,L3,L4]]; (* Face modes *) nn[[17]] = Simplify[ L2 L3 L4 Fi[0, 0, L2, L3, L4 ]]; nn[[18]] = Simplify[ L3 L4 L1 Fi[0, 0, L1, L3, L4 ]]; nn[[19]] = Simplify[ L4 L1 L2 Fi[0, 0, L1, L2, L4 ]]; nn[[20]] = Simplify[ L1 L2 L3 Fi[0, 0, L1, L2, L3 ]]; nn[[21]] = Simplify[L2 L3 Ei[3,L2,L3]]; nn[[22]] = Simplify[L1 L3 Ei[3,L1,L3]]; nn[[23]] = Simplify[L1 L2 Ei[3,L1,L2]]; nn[[24]] = Simplify[L1 L4 Ei[3,L1,L4]]; nn[[25]] = Simplify[L2 L4 Ei[3,L2,L4]]; nn[[26]] = Simplify[L3 L4 Ei[3,L3,L4]]; (* Face modes *) nn[[27]] = Simplify[ L2 L3 L4 Fi[1, 0, L2, L3, L4 ]]; nn[[28]] = Simplify[ L3 L4 L1 Fi[1, 0, L1, L3, L4 ]]; nn[[29]] = Simplify[ L4 L1 L2 Fi[1, 0, L1, L2, L4 ]]; nn[[30]] = Simplify[ L1 L2 L3 Fi[1, 0, L1, L2, L3 ]]; nn[[31]] = Simplify[ L2 L3 L4 Fi[0, 1, L2, L3, L4 ]]; nn[[32]] = Simplify[ L3 L4 L1 Fi[0, 1, L1, L3, L4 ]]; nn[[33]] = Simplify[ L4 L1 L2 Fi[0, 1, L1, L2, L4 ]]; nn[[34]] = Simplify[ L1 L2 L3 Fi[0, 1, L1, L2, L3 ]]; (* Bubble mode *) nn[[35]] = L1 L2 L3 L4; (* From Shiakolas *) (* Put into appropriate format for use with developed equations *) NT = Flatten[Table[i3 * nn[[i]], {i,1,nTot}],1]; NN = Transpose[NT]; NN = Simplify[NN]; (* Form the R matrix *) RL1 = D[NN,L1];RL2 = D[NN,L2];RL3 = D[NN,L3]; R = Flatten[{RL1, RL2, RL3}, 1];...
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...1 Closed-form Stiffness Matrices Tinawi [23] used closed-form integration to obtain the stiffness matrix of nonhierarchic triangular elements in 1972; Subramanian and Bose [24, 25] developed stiffness matrices without the use of numerical approximation for the family of plane triangular elements in 1982 and for C0 continuous tetrahedra in 1983....
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...Subramanian, G., and Bose, C.J....
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...Subramanian, G., and Bose, C.J. Convenient generation of stiffness matrices for the family of plane triangular elements, Computers and Structures 1982; 10:119 – 124....
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References

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Open Access

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Book•

The finite element method

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O. C. Zienkiewicz

01 Jan 1989

TL;DR: In this article, the methodes are numeriques and the fonction de forme reference record created on 2005-11-18, modified on 2016-08-08.

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Abstract: Keywords: methodes : numeriques ; fonction de forme Reference Record created on 2005-11-18, modified on 2016-08-08

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Theory of matrix structural analysis

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Journal Article•DOI•

The TET 20 and TEA 8 Elements for the Matrix Displacement Method

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J. H. Argyris¹, I. Fried¹, D. W. Scharpf¹•Institutions (1)

Imperial College London¹

01 Jul 1968-Aeronautical Journal

TL;DR: In this paper, it was shown that third order complete polynomials yielding a second order or parabolic strain distribution fit into a tetrahedron element with 20 nodal points and 60 degrees of freedom, denoted within ASKA as TET 20.

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Abstract: Following on from the LUMINA and HERMES elements discussed in TN’s 11 and 12, this note analyses two further three-dimensional elements outlined in ref. 1. These new elements possess the desirable features of completeness and consequent invariance of the polynomials for the displacement fields. It has been shown in ref. 2 and stated in ref. 1 that third order complete polynomials yielding a second order or parabolic strain distribution fit into a tetrahedron element with 20 nodal points and 60 degrees of freedom, denoted within ASKA as TET 20 (Fig. 1).

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38 citations

Journal Article•DOI•

Convenient generation of stiffness matrices for the family of plane triangular elements

[...]

G. Subramanian¹, C.Jeyachandra Bose¹•Institutions (1)

Indian Institute of Technology Madras¹

01 Jan 1982-Computers & Structures

TL;DR: A simple and convenient way to generate stiffness matrices in closed-form for all members of the family of plane triangular elements is presented in this article, where numerical integration is completely avoided.

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23 citations