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# mohamedlamine's blog

## An efficient convergence test for the fixed point method

Wed, 2018-11-07 13:10 - mohamedlamineThe fixed point method consists to find the solution of F(X)=X.

One can not get fixed with the convergence condition |F'(X)|<1 because if the function has an optimum then |F'(X)|=0 even if the solution is not yet reached.

We introduce an efficient convergence test with the condition:

|Xn+1 - Xn| ≤ epsilon1 And |F(Xn+1)-Xn+1| ≤ epsilon2

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## Rigid Body Modes

Mon, 2018-10-15 12:44 - mohamedlamine- Read more about Rigid Body Modes
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## Concrete stress-strain curve

Sat, 2018-01-20 18:14 - mohamedlamineThe concrete stress-strain curve corresponds to the compression case with negative values plotted on positive axis. If one needs to plot the tensile curve for this concrete material this will be almost similar with {0.1*compression} stresses with positive values. Furthermore if one needs to plot on the same axis the tensile steel and concrete stresses for a composite mixture he needs to consider this last case. Fortunately the tensile and compression steel stresses are quite similar with opposite values.

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## Failure Envelope

Sun, 2016-10-09 15:47 - mohamedlamineA finite element software computes the stresses and/or strains sets values at each dynamic iteration step. These values can be plotted on a stress-space giving several failure envelopes. For an isotropic or an homogeneous mixture the Von Mises criterion represents an ideal envelope which may not coincide with the available envelopes but the stresses values can be checked with the material strength.

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## Transformation Matrix

Tue, 2016-09-27 04:59 - mohamedlamineHello,

The attached file describes the development of the transformation matrix of a beam element:

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## Publication

Sat, 2016-09-24 14:07 - mohamedlamineHi all,

I am pleased to post the references of my recent paper entitled :

"CONTRIBUTION TO BRIDGE DAMAGE ANALYSIS" Mohammed Lamine MOUSSAOUI, Abderrahmane KIBBOUA, Mohamed CHABAAT

APPLIED MECHANICS AND MATERIALS Journal,

Volume 704 (2015) pp 435-441

DOI: 10.4028/www.scientific.net/AMM.704.435

N.B: The equation (7) has been reviewed with:

[P2](nxg)=[[I](nxn)| [0](nxf)]

and is available at:

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## Bridge Damage Dynamic Analysis

Fri, 2015-11-20 14:39 - mohamedlamine- Read more about Bridge Damage Dynamic Analysis
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## Reinforced Concrete Strength

Wed, 2015-09-09 15:52 - mohamedlamineThe composite strength of Reinforced Concrete which is a mixture of two main materials can be calculated by the formula given by Tsai : X=(vf+vm*Em/Ef)*Xf where vf is the fiber (steel) fraction and vm is the concrete (cement+sands+gravels) fraction. Xf is the yield strength of Steel and Xm is the compressive strength of the Concrete matrix. For the application case: vf=0.01911, vm=0.98089, Xf=270 MPa, Xm=40 MPa, Ef=207.000 GPa, Em=30.798 GPa we obtain X=44.563 MPa.

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## Neutralization of errors

Fri, 2014-01-31 18:00 - mohamedlamineHello,

The problem in numerics is the neutralization of the errors for example plus (+)error and minus (-)error. For that reason it is difficult to locate a convergent process. Some numerical algorithms avoid these omissions like leat squares methods which handle the square of the errors in the minimization function. In the other applications like solving processes truncating errors and roundoff errors can be analyzed successfully.

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## Shape Functions of a Constant Strain Tetrahedron

Fri, 2013-11-01 14:14 - mohamedlamineHello,

I have developped a New Method to Compute Numerically the Shape Functions of a Constant Strain Tetrahedron from the Geometrical Information of its Vertices using the Planes' Equations Defining the Barycentric Coordinates versus the Cartesian Coordinates. The Algorithm for Solving the Linear Systems of Equations has been Implemented Efficiently with the Partial Pivoting Gauss LU Method for any arbitrary Shaped and Non-Degenerated Element.

References: FEOFS2013, CanCNSM2013 and FDM2013.

Best Regards.

Mohammed Lamine Moussaoui

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## Technical Publication

Sat, 2013-08-03 00:29 - mohamedlamineHi all,

Here are the references of the following technical publication:

Title: " Dynamic Detection of Reinforced Concrete Bridge Damages by Finite Element Model Updating ".

Author(s) : Mohammed Lamine MOUSSAOUI, Mohamed CHABAAT, Abderrahmane KIBBOUA.

Area: Non-Smooth Vibrations in Structural Dynamics II.

Conference: 4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM 2013)

Location: McGill University, Montreal, Canada

Date: July 23-26, 2013

Status: Accepted / Done

Best regards.

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## Weakness

Sun, 2013-07-28 05:02 - mohamedlamineHi all,

There are several formulae allowing the computation of tetrahedral volume elements. The weakness of some of them is that they give negative volumes. This is due to the vector dot product and its direction. You must take care when using them.

You have to check your formula with some simple examples like the vertices of the following one: (1,0,0) , (0,1,0) , (0,0,1) , (0,0,0).

Mohammed lamine Moussaoui

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## academia

Sun, 2013-07-07 15:51 - mohamedlamineHi all,

If you have published works before the era of internet or want to share research, find below a useful link:

Best regards.

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## Convergence of a Solution

Wed, 2013-07-03 10:47 - mohamedlamine

The Solution of a Differential Equation or a Set of Differential Equations Converges vers the Exact if it is Consistant and it is Stable : Lax's Theorem. Since the Exact is not always known it is convenient to apply this Theorem. Numerical instabilities are a result of Roundoff errors and Truncation errors. The Domain of Stability can be obtained from a Von Newman analysis in the Complex domain. This implies a condition (relation) between the variation steps.

Mohammed Lamine

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## Publication

Sat, 2013-06-08 21:12 - mohamedlamine" Damage Detection in Bridges using a Mathematical Model by an Updating Method "

Mohammed Lamine MOUSSAOUI , Mohamed CHABAAT , Abderrahmane KIBBOUA

The 9th International Conference on Fracture and Strength of Solids, FEOFS 2013, June 2013, Jeju, South Korea

Status: Accepted

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## free-free vibrations v.s F.E. softwares

Sat, 2013-03-02 07:35 - mohamedlamineHi All,

Do you know why some finite element softwares can not solve free-free vibrations of many applications unless you fix at least one degree of freedom ?

In this case I do not see how they are able to analyze free-free structures as planes.

Mohammed Lamine

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## Structural Damping

Thu, 2013-02-21 04:32 - mohamedlamine

Which best Describes Structural Damping : Dissipative Energy, Linearization from Stiffness and Mass Matrices, or Strain Energy

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## Are Finite Element Softwares Reliable ?

Sat, 2013-01-26 06:50 - mohamedlamineHello,

Some Finite Element Softwares use the Formulation of an Element and Jump to another one

by Merging Nodes : Quadrilateral to Triangular and Hexahedral to Tetrahedral without modifying

the Shape Functions.

This Method is Mathematically and Numericaly Forbidden because they use the Same** **Shape Functions

when Merging the Nodes for the Defined First Element (see Element Definition and after Mesh Process).

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## Tetrahedron

Mon, 2012-12-31 07:22 - mohamedlamineThe Simplex K with 4 Vertices in Dimension 3 is a Tetrahedron and :

K = determinant((x2-x1) , (x3-x2) , (x4-x3) ; (y2-y1) , (y3-y2) , (y4-y3) ; (z2-z1) , (z3-z2) , (z4-z3))

The Volume of this element is :

V(K) = abs(K) / 3!

If V(K) ~= 0 then K is Nearly Degenerated

Moussaoui

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## Inverse

Mon, 2012-12-17 07:19 - mohamedlamineIt is not recommended to Invert a Matrix in Numerical Analysis. Solving with Factorization methods (L.U, L.D.Lt, ..) is better than Solution with Direct Inverse. This is due to Numerical Errors (roundoff) and/or ill-Conditioned problems. In case of you need to use an Inverse, you are advised to take care about using existing algorithms which can't be exact in all applications.

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## Conditionning

Mon, 2012-04-30 21:24 - mohamedlamineConvergence of Numerical solution may depend on the conditionning of the problem's matrices.

An ill-conditionned matrix produce singulariy cases. This is due to bad shaped elements in

structural mechanics: nearly degenerated elements with zero area (in 2D) or zero volume (in 3D)

The conditionning can be analyzed by numerical algorithms or existing commands in order to

detect these problems.

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## Well-Posed Problem in Partial Differentrial Equations

Sun, 2010-01-24 06:08 - mohamedlamineTo avoid the Ambiguity appearing in the Formulation Process using the Numerical Methods of Partial Differentrial Equations it is Important to Satisfy the following Conditions given by Fletcher C.A.J. 1989, p18 :

"The governing Equations and Auxiliary (Initial and Boundary) Conditions are Well-Posed Mathematically if the Following three Condtions are met:

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## Mechanical Vibrations of the Gear Teeth

Sun, 2009-12-13 04:38 - mohamedlamineRecent research work available on request.

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