# Efficient Constant Strain Tetrahedron Shape Functions

Dear researchers,

Hi all,

I am pleased to post my following publication:

Study on Dynamic Detection of Reinforced Concrete Bridge Damage by Finite Element Model Updating

• In book: Advanced Aspects of Engineering Research, Volume 15,
• Chapter: 11,
• Publisher: B P International

I have cited the originality of this research in  Blog: Hex element V/S Tetrahedron elments on February 25, 2014

with the Subject research mohammedlamine

The Constant Strain Tetrahedron has 12 degrees of freedom (dof) but the brick element has 24 dof. The other thing is that in general you need to use Gaussian integration points for the brick element to calculate the terms of your matrices and vectors which takes more cpu time. Approximately, it takes the same time for the curved brick elements. Before 2013, I have not found a valid method to calculate the shape functions of a tetrahedron. Some existing formula give negative volumes !

The new method that I have developed in 2013 computes the shape functions of the constant strain tetrahedron efficiently with direct matrix solving and after that the corresponding matrices (stiffness, .......) with direct matrix products : see my recent publication :

Dynamic Detection of Reinforced Concrete Bridge Damage by Finite Element Model Updating, Mohammed Lamine Moussaoui, Mohamed Chabaat and Abderrahmane Kibboua, Journal of Mechanics Engineering and Automation, Vol 04, Number 1, PP 40-45, January 2014, David Publishing Company, USA

It is also important to notice that the choice of the tetrahedron corresponds to that the tetrahedral elements fit very well any arbitrary shaped volume, eventually with curvatures, to be analyzed. This is the economic one.

Development of an original, new and efficient method for determining the shape functions of the four nodes constant strain tetrahedron to solve the lack of the well known existing formulations which can generate negative volumes as it is proven in some simple cases like in (1,0,0), (0,1,0), (0,0,1) and (0,0,0) tetrahedron vertices;

Consideration of the structural damping which is an important structural self-damping process under dynamic vibrations and an hysteresis material process due to the slip connections;

Generalization of Hooke's law for the volumic case using the tetrahedral element with three degrees of freedom per node with a volumic deformation;

MMUM updating techniques to analyze large structural complex systems of equations;

Finite element detection of damage areas at exact locations;

Structural material properties include concrete matrix and reinforcement steel in the composite mixture rules.

It is also Important to cite that I have claimed Pr Aboulfazl Shirazi-Adl of Mc-Gill university to recognize and to correct this error with the efficient new formulas that I have developed.

The error has been proved.

The solution has been found.

Regardless of the means to invest, the truth is not far to be reached.

Best regards

M.L. Moussaoui