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A- INVERSE PROBLEMS

Problem 1:
In the framework of Functional Analysis, Modelization and Numerization of:
-  The birth of a defect and its detection in a plate of a shell structure

Application: Damage mechanics, Fracture mechanics

Problem 2:
From a given an inverse crack problem, how can we determine the reconstruction and the fractal geometry of the crack?

Application:  Cracked plates/shells structure or defects inside the engineering structures

Formulation of Inverse 3D Crack Problem in Elasticity:

Let be Ω  an R3 elastic, isotrope, homogeneous bounded domain with a smooth regular border,
under the volume force f and applied force g on Γ1 and the input of the current flux Φ on Γ2  .
The domain Ω contains an embedded flaw (surface Γ, length 2a).

The defect can be an obstacle, an inclusion, a cavity or a crack.

a- Mathematical Formulation:
Find the geometry of the crack Γ if  u(x,t) is known on Γ2  and such as the following equation is verified:
                              x
                                                         on Γ1
                                                          on Γ2
                                                            on Γ

Remark:

• Measure of u(x,t) known on Γ2
• Input  current flux known

b- Main questions:

• Uniqueness of the solution
• Stability of the solution
• Reconstruction of the fractal geometry of the defect

c- Suggestions: Resolution with FEM or BIE

d- Applications:

• Damage and Fracture Mechanics
• Non Destructive Control Mechanics
• Granular and Powders Mechanics
• Medical Imaging
• Seismic Imaging
• Land Mines Imaging
• Geophysics (Oil Detection)

3- Impact Mechanics:

The foundation of the Impact theory was established in the same time with the science of Mechanics. The theory is based primarily on the impulse-momentum law for rigid bodies, but it is incapable of describing the transient stresses, forces, size of contact surface, or local deformations produced.
In order to estimate these quantities, the Impact problem must be handled with a new approach:
- Direct and Inverse Impact Problem of colliding bodies

Application:  Impact Mechanics

4- Granular and Powders Mechanics

Granular matter, powders are poorly understood within the continuum mechanics. There is no general theory, but a few Newton’s laws are used in the research.

For mechanical problems, a constitutive law describes how a material strains when it is stressed, or vice-versa.
In continuum mechanics, we know that the constitutive relations are :
 e = S·s    or  s = S·e    where
C is the stiffness matrix, S is the compliance matrix, and S = C-1
e    deformation tensor
s    strain tensor
To understand the granular, powders in silo or containers, and to build simulation models, a new approach must be investigated.  At this level, three main questions arise:
1- From the perspective of  symmetry and objectivity principles, how can we deduce an adequate law constitutive of the gramular media flow under the gravity?

2- What is the mathematical formulation of this physical problem ?
3- The inverse problem formulation for dead zones of granular flow in a silo or containers.

Application: Granular and Powders mechanics in Silo or Containers

5-Networks
Nowadays, the networks (wired /wireless) play a key-role in the enterprise-life. The administrators and the managers must face many challenges when they consider their Network Resources Planning in dynamic and competitive environment.
To deal with these issues, an inverse problem related to networks must be formulated in order to have an optimal capacity planning and then to conceive a simulation model (Event-discrete simulation) for their own network

Application: Intranet, Extranet, Extranet in the IT business enterprises.

B- PLATES AND SHELLS