L3 - CUPGE Mechanics

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The aim of this course is to provide an introduction to numerical tools for solving partial differential equations from various fields of engineering. The spectral method applied to the equation of heat diffusion in a bar will be covered, as well as the development of codes based on this technique. In particular, students will be asked to implement this method in Python, integrating the basics of this language and versioning tools. Documents submitted by students will be produced using Latex word processing.  

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Emphasis is placed on the performance and limitations of engineering calculation methods, so that the student is able to use them correctly "in situation". This situational approach is certainly the most challenging aspect of this introduction to scientific computing, as it calls not only for a certain physical sense, but also for mathematical modelling and a minimum of computer skills.

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In the first part: deepen the basic notions of differential calculus seen in L2.

In the second part: introduce the qualitative study of differential equations.

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Resistance of materials (RoM) is a special discipline of continuum mechanics, used to calculate stresses and strains in slender structures made of different materials (machinery, mechanical engineering, building and civil engineering). It involves 1D static modeling of a deformable solid, assimilated to a beam linked to a frame and subjected to external mechanical loads.

RoM allows us to reduce the study of a structure's overall behavior (the relationship between stresses - forces or moments - and displacements) to that of the local behavior of the materials making it up(the relationship between stresses and strains). Mechanical stresses can be seen as " cohesive forces " in matter. The deformations of a physical object can be observed by a variation in its dimensions or overall shape.

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The aim of this first Fluid Mechanics module is to provide a basic understanding of the behavior of industrial fluids (air, water, hydraulic fluids), with a view to dimensioning simple systems involving static or dynamic fluids (flow rates, pressure, velocity, pressure drops, etc.). Emphasis is placed on the study and design of hydraulic installations.

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Rheology is the study of material deformation and flow under applied mechanical stress. In the field of materials, this science particularly concerns the following areas:

            - Viscoelasticity

            - Plasticity

            - Viscoplasticity

            - Non-Newtonian fluids

In practice, rheology is used to characterize the macroscopic mechanical properties of materials whose behavior escapes the classical theories of elastic solids and Newtonian fluids (with constant viscosity). Such materials can thus be considered as having an intermediate behavior between solid and fluid, between elastic and viscous.

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The aim of this course is to introduce the basics of physical hydrodynamics. The kinematic aspects are dealt with first: Euler and Lagrange formalism, analysis of the motion of an element of fluid volume, introduction of current and potential velocity functions, and applications to different types of flow. In the following section on fluid dynamics, we establish Euler's equation and Bernoulli's relation for the flow of perfect fluids, followed by the Navier-Stokes equation describing the flow of viscous Newtonian fluids. This will lead to the definition of the stress tensor and the Reynolds number, enabling us to deduce whether a flow is laminar or turbulent. The course concludes with an introduction to the mechanics of deformable solids : displacement field, expansion and deformation tensors.

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The course links scientific computing and variational methods for mechanics and is designed to model simple physical equations and implement numerical methods to solve these equations.

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Acquire basic notions of numerical methods for differential equations

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TD language courses aimed at training the 5 language skills;

Oral comprehension & expression

Written comprehension & expression

Oral interaction

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• read a technical drawing of medium difficulty
• identify standard mechanical components on a drawing
• identify the operation of a mechanical system of moderate difficulty from an overall drawing
• draw a part in several views, following the rules of projection
• identify and draw cylinder-plane, cylinder-cylinder, plane-cone, cylinder-cone intersections
• draw cross-sections and sectional views
• extract a part from an overall drawing of moderate difficulty
• use of basic Solidwork functions (part mode, assembly and drawing)

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Research & Development (R&D) is both the product of teamwork and individual talent, all at the service of innovation (applied research) and knowledge (fundamental research). The themes are diverse and varied, but a certain methodology is required to tackle any R&D problem.

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The aim of this course is to model continuous solid media, initially restricting ourselves to elastostatics under the assumption of small perturbations. The fundamental principle of statics is applied to deformable solids. The following concepts are introduced: tensors and tensor fields, algebra and tensor analysis, boundary problems, fundamental principle of statics, virtual power principle. Techniques for analytically solving classical problems and energy approaches are also covered. This course is fundamental to the training of students in mechanics, whether they are going into design or R&D.

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Project assigned to a group of two to three students, supervised by a tutor. Weekly meetings to monitor progress and help students write a report and give an oral presentation. Work is spread over a semester, culminating in a report and oral presentation.

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