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ENME4332 Finite Element Analysis 1
Fall 2024
Final Project [100 points]
1 Project Description
In this project, we will solve a steady-state heat conduction problem within an arbitrary 2D domain subjected to a combination of prescribed temperature and flux boundary conditions. As is shown in Fig. 1, this 2D structure consists of two parts made of different materials (material 1: gray; material 2: pink) with different conductance. Please note that the heat source s is applied to the inner subdomains highlighted in pink. The natural boundaries are highlighted in red and blue for inward and outward heat fluxes, respectively. The essential boundaries are highlighted in green.
Please rigorously follow the dimensions shown in Fig. 1. With the origin of the coordinate system located at the left bottom corner, the center coordinates of the two circles are (0.4,0.7) and (1.6,0.7), and the radius is 0.1. The parameters a and b are unique values assigned to each student in a separate list.
The strong form of the governing equations for this problem is as follows:
2 Abaqus Modeling
Construct the 2D model in Abaqus and perform the following tasks:
• Mesh the domain with linear triangular elements.
• Consider two mesh sizes. The coarse mesh has around 250 - 300 elements, and the fine mesh has around 950-1000 elements.
• Solve the problem with these two meshes separately.
• Save the mesh details (nodal coordinates and element connectivity) for both mesh sizes into separate spreadsheets or text files for the subsequent usage in our own matlab code.
3 Matlab Implementation
Write a MATLAB-based FEM program to solve the described problem, and this set of code has three major steps, namely pre-processing, processing and post-processing. In the pre-processing step, we will:
Figure 1: 2D domain with boundary condition, source, and dimensions. Note that all the listed dimensions are in meters.
• Define the material properties.
• Read the mesh generated by Abaqus to construct the nodal coordinate matrix and element connectivity matrix.
• We also need to label the essential/natural boundaries and flag the two subdomains. Then, in the processing step, we will:
• Implement the element conductance matrix Ke (Eq. 8.10 in the textbook) and element flux matrix fe (Eq. 8.11)
• Assemble the global conductance matrix K and flux matrix f
• Apply essential boundary conditions and solve for the unknown temperature.
4 Post-processing and submissions
With the solutions obtained with our own FEM code, perform the following tasks.
Note: All the previously listed procedures are the hints that guide you through the project. The following questions are where your project will get accessed. Please organize your submission by answering each of the questions clearly with correct numbering.
1. Plot both coarse and fine mesh you generated and report the total number of elements.
2. Create 2D plots of temperature field T(x,y) with both coarse mesh and fine mesh results. Hint: Use the ’patch’command in MATLAB.
3. Plot the temperature T(x,y = 0.4) along the line y = 0.4m for both mesh sizes and compare with Abaqus results. Describe your observations. Hint: Use x coordinate as horizontal axis. On each graph, you will have four lines: MATLAB coarse, MATLAB fine, Abaqus coarse, and Abaqus fine.
4. Create 2D patch plots of flux magnitude |q|(x,y) and 2D arrow plots of flux at center of each element for both mesh sizes.
5. Plot flux magnitude |q|(x,y = 0.4) along the line y = 0.4m for both mesh sizes and compare with Abaqus results. Describe your observations.
6. Provide a better mesh strategy.
Due date: 10-Dec-2024
Please submit a zip folder containing your answers in PDF and MATLAB code to Canvas.