MECH 6511 Mechanical forming of metals - Project with coding: Finite Element Method (FEM)
MECH 6511 Project with coding
Introduction
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 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 mechanical shaping. You will have to derive the mathematics formulations, do the coding, and apply with applications. You must choose a product or component with a manufacturing history, and you need to write about the material selection and the manufacturing technique explaining why they have been used. FEM can be applied for the manufacturing process, the usage or the inspection.
The Finite Element Method
Generally, the FEM has seven steps:
- Idealization 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!
- Discretization The problem domain is discretized into a collection of simple shapes, or elements.
- 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).
- 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.
- Solve for primary unknowns The modified global equations are solved for the primary unknowns at the nodes.
- Calculate derived variables Calculated using the nodal values of the primary variables. Implementation Phases To help you track your progress, the project has four phases as follows. Phase 1 – Problem definition & Interface This covers from step 1 to part of step 3. Specific tasks include but are not limited to: • Detail the application and define the engineering problem to be studied. • Apply idealization and discretization; and justify the choice of element type and dimension. • Implement deformation (change of coordinates) and stress map (change of colors). • If you are using other languages, you also need to implement a proper GUI and visualization, as well as all the functionalities such as mesh import/generation. • Example of advanced features: high-dimensional or non-linear elements. Deadline: Week 4
Reference marks: 5
Phase 2 – Local stiffness matrix & Coding of single element This completes step 3 and steps 5-7 for one element only. Tasks include but are not limited to: • Derive the mathematics formulations of the local stiffness matrix for the element chosen. • Implement the local stiffness matrix in the program and apply loads and boundary condition. • Solve and show the deformation and stress map for a one-element problem. • Example of advanced features: isoparametric mapping. Deadline: Week 7
Reference marks: 10
Phase 3 – Global stiffness matrix & Solving for a full structure This fully implements steps 4-7 for a full structure but only solving for small deformations and a static stress analysis. Specific tasks include but are not limited to: • Implement the assembly of global stiffness matrix for the entire structure. • Apply loads and boundary conditions to the complete system of linear equations. • Solve and show the deformations and stress map for the problem. • Example of advanced features: Stress recovery using Gauss points. Deadline: Week 10
Reference marks: 16
Phase 4 – Large/plastic deformation & Analysis This extends the static FEA to plastic/non-linear analysis for your problem defined and analyzes the results and findings. Specific tasks include but are not limited to: • Quasi-static analysis can be used. Apply flow curve and multiple time steps. • Example of advanced features: dynamic analysis Deadline: Week 13
Reference marks: 14
Note: the marks in each phase are just for reference, they may be updated with the actual project evolvement. Completeness is more important than complexity, so you should only propose something that you can complete.
Programming Language C++ is suggested and a basic MFC platform is provided for Visual Studio Community 2022. You can download VS2022 for free as a student through https://www.visualstudio.com/downloads/ You need to choose “Desktop development with C++” and add “C++ MFC”.
You can use other programming languages, e.g., Python, for the implementation, but you will have to take care all the implementations including the graphical user interface (GUI) for visualizing the analysis results. The grading method will also be different for other programming languages based on the complexity of the languages. Report A report describing all the implementation details should be submitted with the program. It should contain all the steps listed above and refer to the corresponding portions/codes in your program if necessary. Remark: You may be able to find a lot of resources for implementing the FEM online. It is not prohibited to use them as reference, but you cannot directly use them in your code. If you do, it will be treated as plagiarism, and you will get a 0 mark for this project as a penalty (no exception). Grading The grading is based on the completeness and the complexity that you have achieved. Completeness refers to the development of the finite element method (FEM) into window-based programs solving the real-world problems in mechanical shaping manufacturing. You can think of you are implementing a commercial FEA software, including the graphical user interface (GUI) and visualizing the analysis results. Besides, you should also refer to the seven steps listed in the implementation details, starting from a proper problem identification to the validation. Remember that finishing the programming doesn't mean the project is complete, as there are a number of steps not really point to programming. The more complete of your story, program, and the report, the higher marks you can get. Don't forget to make a good presentation and organization of your report and give conclusions and recommendations at the end. Marks are distributed to all the mentioned components. This portion determines if you will pass the project.
Complexity refers to the implementation of your program and functionality. In the course outline and implementation detail, you can find the information, e.g., triangular v.s. quadrilateral elements, 2D v.s. 3D FEA (i.e., tetrahedral/hexahedral elements), first v.s. second order elements, isoparametric mapping. For instance, solving everything in 1D (1D elements, 1D case) could be enough for the completeness, but not quite complex. You can treat this is the bonus part. If you pay more efforts making the program more general that can handle more complex requirements, then you will be awarded more marks and standing out from other students. This portion determines your grade of the project. If you are using Python or other languages, your program must also need to have a proper GUI and visualization, as well as all the functionalities such as mesh import/generation. The completeness and complexity should be comparable with those results implemented with MeshWorks. Submission For the submission, you will at least submit two files: 1) Report, 2) Archived Program Code (.zip, .rar, etc.). It is also highly recommended to submit 3) Executable Program. 1) In the report, you should give your story, results, the details of your implementation, and the user guide for your program (your code and the executable program), as complete as possible. 2) The program code is just similar to the MeshWork.rar you downloaded from Moodle. You archive all the files into one and submit it to Moodle. For those using MeshWork, you may want to clean your solution first, which can be done by "Build->Clean Solution" for both Release and Debug modes if you have used both. When you compile your code by "Build->Build Solution", there are number of libraries and executable files generated which are normally large, so your submission may exceed the size limit. Cleaning is an inverse process that removes those generated files. You may also want to delete the MeshWorks.VC.db (or something similar) and ipch folder (or .vc folder, if any) located just in the root MeshWork folder, which is very large but not necessary. You final archived file should be having similar size of MeshWork.rar in Moodle: ~6MB. 3) The executable program is one of the generated files mentioned just now, which is an exe file located in MeshWork\release\MeshWorks.exe. You can copy this and all the .dll files in the folder out before cleaning (once you clean, the exe file is removed). Other files (e.g., .lib, *.pdb) in the same folder are not needed and they are the generated files, so they will be gone too after cleaning. This exe and dll files together can be run by themselves without using Visual Studio. Actually, they are the software you developed. You should also archive them into one and submit to Moodle. For those using Python or other languages, you may not have the executable program, so you just need to archive all your codes for the submission. References •
A First Course in the Finite Element Method
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Advanced Mechanics of Materials and Applied Elasticity
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Elasticity: theory, applications, and numerics
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Roark's Formulas for stress and strain
