What is Multiscale Modelling? Problem we try to solve! Atomistic/Continuum Coupling Quasicontinuum Method What is FEM? Steps in performing Finite Element Analysis : HINT: Don't even think of this!! 1.Multiscale modeling refers to a style of modeling in which multiple models at different scales are used simultaneously to describe a system. Why Multiscale Modelling? ` USING ---FINITE ELEMENT METHODS 1. Many engineering phenomena can be expressed by "governing equations" and "boundary conditions". 2. The governing equations are often in the form of partial differential equations(PDE) or ordinary differential equations (ODE). 3. From mathematical standpoint, Finite Element Method (FEM) is a numerical method used for solving a set of related differential equations such as 2. The different models usually focus on different scales of resolution. Why use FEM ? 1. Because FEM can be adapted to problems of great complexity and unusual geometry using grid or mesh.

2. It is an extremely powerful tool for solving critical problems in heat transfer, fluid mechanics, electrostatics, and structural and mechanical systems.

3. Availability of fast and inexpensive computers for daily engineering applications. Step 1 – Select Element Type: Element type affects the ease of modeling and the accuracy of solution. Step 2 – Discretize: The problem domain (structure) is divided into a collection of simple shapes, or elements. Step 3 – Derive Governing Equations for Each Element: Element matrix equation can be developed based on the physics of the problem using methods such as energy method or virtual work principle. Step 4 – Assemble Global Governing Equations: The element matrices are assembled into a global matrix equation that models the properties of the entire domain(structure). Step 5 – Apply Boundary Conditions: Boundary conditions reflect the known values for certain primary unknowns. Imposing the boundary conditions modifies the global matrix equations. Step 6 – Solve for Primary Unknowns: The modified global matrix equations are solved for the primary unknowns at the nodes. Step 7 – Calculate Derived Variables: Other field variables can be calculated using the nodal values of the primary variables. Step 8 – Check and Interpret the Results: This step is probably the most critical. While checking and interpreting the results, one may find results unreasonable, thus prompting revisions and re-analysis. Structure of a FEM program: ` Schedule for coding: November to December Prefered Programming language: C language Variables to be found: Deformation Applications: The QC method of has been used to investigate a wide range of problems including fracture, nano-indentation, three-dimensional dislocation junctions and grain boundary structure. The method is well-suited to automatic mesh adaption, allowing it to expand the atomistic region as required to track the progress of defects. USING ---MOLECULAR

DYNAMICS 3. In our project, one from Continuum Mechanics(FEM) and one from Molecular Dynamics. Atomistic simulations are able to describe defects on small scale, however the required no.of atoms and the computational costs becomes very large. Finite Element based continuum simulations are computationally less burdening and less accurate. The QC method defines two types of atoms, ‘local representative atoms’ and ‘non-local representative atoms’ Non-local representative atoms are atoms in fully atomistic region. Local representative atoms are atoms in the local FE region. The interface atoms are not only included in the atomistic energy but also in FE nodes of the continuum. The pad atom positions rp are dictated by interpolation from the FE nodal positions. To avoid overcounting of the energy of the interface atoms/nodes, the energies of the continuum elements adjacent to the interface are weighted differently, with the use of weight function "wμ". The energy of each element, Eμ, is obtained via the Cauchy–Born procedure. ` Schedule for coding: January to March

Prefered Programming language: C language Variables to be found: Deformation ` Schedule for coding: November to December Prefered Programming language: C language Variables to be passed: Displacement and Force f2 = 0, because the motion of pad atom P causes a force on atom A but the motion of atom A does not cause a force on P.

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# Multiscale Modelling

FEM - MD Coupling

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