these three FEM assignments are all related to each other. and for you to be able to do assignment 2 and 3, you need to follow the information in assignment 1 file. I have attached the report template files for both assignment 2 and 3 to be completed.
MATH122 – Mathematics for Engineers Assignment 1 Report Title Abdurahman Hijji, s3763612 Date 6/10/20 1. Model geometry parameters (3%) (a) Your student ID number is _____s3763612__________ The last digit of the ID number:d1 = __2__ The 2nd last digit of the ID: d2 = __1__ (b) Your unique model geometry parameters are calculated as follows: L = 10 + d1 + d2 = 10 + __2_ + _1__ = ___13____ w1 = 5 + d1 = 5 + _2__ = __7___ w2 = 1.5 + d1 / 2 = 1.5 + _1___ = ___2.5___ 2. Model geometry (5%) (a) Present an isotropic view of your 3D model below. In the figure, clearly label L to the dimension and the value of L. b Figure 1 A 3D isotropic view of the model geometry (b) Present a 2D graph of the cross-section of you model here (i.e. the x-y plane model view in the SpaceClaim). The figure is required to show the values w1 and w2, together with other constant dimensions (ideal but not compulsory) of the cross-section of your beam. Note: Your 2D beam image can be captured in SpaceClaim, and then edited using any image editing tools to include w1 and w2 values. Figure 2 A x-y plane 2D view of the model geometry MATH122 – Mathematics for Engineers Assignment 2 Report Title Student Name and ID Date ______________ 1. Introduction This assignment is to solve a three-dimensional (3D) heat transfer problem conducting through a beam. The beam has a length L = ? and an L-shaped cross-section with a circular hole cutting through from its back face to the front face along the beam length direction. The following Figure 1 shows an isotropic view of the 3D beam model geometry. (Replace this line by your beam 3D figure, in which the length L must be clearly shown, together with other geometry parameters, similar to the figure on Slide 3 of the Assignment 2 Lab Guide – Part 1 ) Figure 1 A 3D isotropic view of the model geometry The beam is made of engineering steel. The thermal conductivity of the material is k = 60.5 W/m∙oC. The heat transfer through the beam is described by the following partial differential equation (PDE): (Eq.1) subject to the boundary conditions imposed on the beam boundary surfaces. At the steady-state thermal state, the temperature variable T(x,y,z) depends on the material thermal conductivity k, the three independent spatial variables x, y and z, as well as the heat source G (i.e. heat generation or loss). The objective of this assignment is to solve the PDE (i.e. Eq.1) for the temperature T(x,y,z) distributions in the beam subject to three sets of boundary conditions, respectively. Due to the complexities of the beam geometry and boundary conditions, there are no analytical solutions for the partial differential equation (PDE). In another word, Eq.1 has no exact solutions. Thus, finding the approximate solutions for the PDE (i.e. Eq.1) is the only way to estimate the temperature distributions. In fact, seeking an approximate solution for a complex problem, such as this one, is a common engineering practice. The Finite Element Method (FEM) is one of the most powerful numerical methods widely used in the engineering field. In this assignment, we will use the FEM to solve the partial differential equation (PDE), i.e. Eq.1, for its solutions T(x,y,z) subject to the following three sets of different boundary conditions, respectively. (A) Boundary conditions for Model 1 (Please replace this sentence by the list of the boundary conditions for Model 1, i.e. Conditions (1) – (6) on Page 1 of Assignment 2 paper) (B) Boundary conditions for Model 2 Model 2 has the same boundary conditions as Model 1, except on the following boundary surfaces: (Please replace this sentence by the relevant boundary surface names), where the boundary conditions on these surfaces are: (Please replace this line by the list of boundary conditions different from Model 1) (C) Boundary conditions for Model 3 Model 3 has the same boundary conditions as Model 2, except on the boundaries: (Please replace this sentence by the relevant boundary names), where the boundary condition on these surfaces is: (Please replace this line by the list of the boundary condition different from Model 2) 2. The Finite Element Analysis (FEA) Procedure A typical FEA procedure consists of three major stages: · Pre-processing stage: this is to create the Finite Element model. · Solving stage: this is to solve the Finite Element model · Post Processing stage: this is to check and verify the solution and then extract the numerical results of the Finite Element model 2.1. Pre-processing procedure involves the following 4 steps in building up a finite element analysis (FEA) model: Step 1 is to specify the material property of the beam. In this assignment this is done by the software default setting in the Cell 2 Engineering Data of the Steady-State Thermal system, where the Engineering Steel material property data, including the thermal conductivity k = 60.5 W/(m oC) are specified for the beam. Step 2 is to define the model geometry, which we have done in Assignment 1 and the beam model is shown in Figure 1 above. Step 3 is to generate a FEM mesh, which is to divide the model geometry into a finite number of smaller pieces that are called elements (this is where the name: Finite Element Method came from). This process is called Discretization, during which the model geometry is divided into a set of finite number of elements called a FE mesh. In this assignment, we use ANSYS built-in meshing algorithm to generate a 3D mesh of a medium density. Figure 2 presents the mesh generated in ANSYS Meshing. (please replace this line by your mesh figure here) Figure 2 The finite element mesh of the model The mesh statistics (i.e. numbers of notes and elements in the mesh) are provided in Table 1 below. Table 1 The FEM mesh statistics (please replace this line by your mesh statistics table) All the elements have six faces, and the shapes of majority elements are like regular bricks except in the region near the hole where the element shape becomes irregular due to the curvature surfaces of the pipe channel. This type of elements is called hexahedral elements. In general, the more regular the shape of the elements is, the more accurate solution the FEM modelling provides. For an irregular geometry, it is common to have irregular 3D elements with only 4 faces, and this type of elements is called tetrahedral elements. These 3D elements are also referred as prisms, wedges and pyramids. Step 4 is to specify the loading, supporting and environmental conditions prescribed on the boundary surfaces that enclose the body of the model geometry. All the external influences on the model boundary surfaces are called boundary conditions. Please note that after a FEM model is solved, the first thing we need to do is to check if the solution agrees with the model physics. If the solution does not agree with the prescribed boundary conditions, i.e. the model physics is not satisfied. We will need to go back to the 3rd stage of building FEM model to check the boundary conditions of the problem and correct the wrong settings, and then re-solve the FEM model. We shall repeat this self-checking procedure until the solution agrees with all the prescribed boundary conditions. This 4-step process of building a FEM model is called Pre-processing in a typical Finite Element Analysis (FEA) procedure. By end of the Pre-processing stage, ANSYS has gathered all the information about the geometry and physics of the problem required by the next stage – Solving. 2.2. Solving is the 2nd stage of a typical Finite Element Analysis (FEA) procedure. During this stage, the software - ANSYS converts all the model information into its code and establishes several sets of large algebraic equations based on the Finite Element Method algorithms. The number of sets depends on the number of physical variables to be solved, and the number of equations in each set is proportional to the number of nodes in the FEM mesh. The FEM model is solved by computer that executes the FEM algorithms developed by the software provider ANSYS Inc. The development of all engineering simulation software companies has been devoted on the Solving stage by continuously implementing new mathematical algorithms into the code to improve the software solving accuracy and capacity. However, as an end-user, all we need to do is to click on the Solve button and wait for the solution. For a small problem like this assignment, it takes only a few seconds to get the solution. For some large problems, it generally takes a few hours, or days, or weeks to obtain the solutions. 2.3. Post Processing is the last stage of a typical Finite Element Analysis (FEA) procedure After the solution is obtained, we will need to check and verify the solution results and then extract the result for reporting. At this stage, firstly we will need to view the results and verify the solution by going through the self-checking procedure described previously. Only after the results are confirmed by the model physics, we can then move forward to extract the model results in the forms of figures, tables, charts and animations, etc. for reporting the solution of the FEM model. 3. FEM solution verification As described above, this is the first step of the Post processing of the FEM results. (A) Model 1 result verification · Check if the temperature on the Front face equals the prescribed temperature value, and do the same for the Back face (please replace this line by the relevant figure (or figures) here) Figure 3 Model 1 Temperature distributions on the Front face and Back face of the beam with the labels showing the temperature values, respectively) (Please replace this line by your comment here) · Check if the temperature on the Pipe inner face varies linearly (approximately) as the prescribed linear expression of your Model 1. (please replace this line by the relevant figure of your here) Figure 4 Model 1 Temperature variations along the Pipe inner face (Please replace this line by your comment here) · Check the energy balance of the model using concept of energy conservation (Please replace this line by the energy balance calculations, including the absolute error and relative error in the energy balance. Hint: Refer to Slides 24 and 28 of Assignment 2 Lab Guide Part 1 for the calculations; you also need to present your table of Steady-State Thermal > Solution > Probes, here to support your calculations as they are based on the information given in the table).