The Finite Element Method (FEM) has developed into a key and indispensable technology in the modeling and simulation of advanced engineering systems in various fields, e.g., housing, transportation, and communications. In building such advanced engineering systems, engineers and designers go through a sophisticated process of modeling, simulation, visualization, analysis, designing, prototyping, testing, and lastly fabrication. There are a many works involved before the fabrication of the final product or system. This is to ensure the workability of the finished product, as well as for cost effectiveness.
There are numerous physical engineering problems in a particular system, many of them have been formulated for the many physical phenomena in engineering systems, including mechanics for solids and structures, heat transfer, acoustic, fluid mechanics, and others. The FEM is a numerical method seeking an approximated solution of field variables in the problem domain that is difficult to be obtained analytically, e.g., the problems of stress analysis, thermal analysis, fluid flow analysis, piezoelectric analysis. The analyst can determine the distribution of some field variables like the displacement in stress analysis, the temperature or heat flux, and the electrical change.
This project aims to help you understanding how and why FEM is applied to material and manufacturing studies by implementing it yourself. We will focus on the FEM for the stress analysis within the metal forming. You will have to derive the mathematics formulations, do the coding, and apply with applications.
The Finite Element Method
Generally, the FEM has eight steps:
The "real" problem is idealized by making assumptions to simplify the problem:
by reducing the dimensions (all real problems are 3D, but may be idealized with 1D,
2D or 3D models),
by idealizing the support conditions,
by suppressing details, such as small holes and fillets, that are insignificant from the
analysis point of view, but which complicate matters during mesh generation. This step can be dramatically important if the assumptions are not correct!
The problem domain is discretized into a collection of simple shapes, or elements.
3. Choice of the type of element and compute the local stiffness matrices
1D: Truss, beam, frame, and in their higher orders;
2D: Triangular or quadrilateral (or other) elements and in their higher orders
3D: Tetrahedral or hexahedral (or other) elements and in their higher orders
The results can be very different from one type to another. This is due to the theory hidden behind those elements. After that, compute all stiffness matrices for the discrete elements (with isoparametric mapping).
4. Assembly of the discrete elements
The element equations for each element in the FEM mesh are assembled into a set of global equations that model the properties of the entire system.
5. Application of boundary conditions
Solution cannot be obtained unless boundary conditions are applied. They reflect the known values for certain primary unknowns. Imposing the boundary conditions modifies the global equations.
6. Solve for primary unknowns
The modified global equations are solved for the primary unknowns at the nodes.
7. Calculate derived variables
Calculated using the nodal values of the primary variables.
C++ is suggested and a basic MFC platform will be provided for Visual Studio 2015