Numerical Simulation in Junior Design Course in
Engineering Physics
Mehmet Sözen[1]
Abstract
This paper discusses the use of numerical modeling and simulation of engineering problems using finite element and finite difference methods in an introductory junior design course in an engineering physics curriculum. The methodology used in teaching this course and the modern engineering tools used in preparing the students for their professional careers are discussed. The nature of the course contents and typical mini-projects assigned to the students are presented. The objectives of the course and assignments and the overall student reaction to this course are discussed.
Introduction
The Engineering Physics program at Embry-Riddle Aeronautical University is a four-year program designed to produce graduates who can operate at the interface between scientists and engineers. The students in this program complete the fundamental engineering science courses common to most traditional engineering programs as well as most of the courses of traditional physics programs. In addition, they complete a two-course sequence of space systems engineering, a two-course sequence of one credit hour junior design courses and a two-course sequence of senior design courses (two and three credit hours). The latter is the capstone design course sequence that focuses on space mission/system design.
This paper deals with the second course of the junior design sequence, which is titled Introduction to Design II (EP 397). This course is designed to introduce the students to numerical modeling and simulation of engineering systems. An introduction to finite element method (FEM) and finite difference method (FDM) and their applications are covered in the course. As this is a course for second semester of junior year, the students are expected to have completed the courses such as mechanics of materials before taking this course, so that the course can be effectively used for introducing the students to the structural and thermal analysis of simple parts or systems using the FEM and/or FDM. The students are required to use commercial finite element analysis (FEA) package (COSMOS/M) and develop finite difference equations, which can be solved by using mathematical software tools such as MATLAB or MAPLE, or choose to write their own simple codes in their choice of computer language.
The course is conducted as one-hour lecture and one hour laboratory session per week. While the theory is covered in the lecture session, the lab session is utilized for training the students not only for using the software tools but also to provide the students an opportunity to work on their mini-projects in the presence of an experienced teaching assistant who is responsible for teaching the lab sessions. The primary responsibility of the teaching assistant is to teach the students how to use COSMOS so that they can set up simple problems with simple mesh or element types, and implement the boundary conditions. Assistance with difficulties in problem set up and analysis is also provided occasionally. Sixty percent of the weight of the course grade is allocated to mini-projects and the remaining forty percent is based on in-class exams. Approximately half of the course is devoted to numerical simulation by finite element method while the other half is devoted to finite difference method. A total of approximately four mini-projects are roughly divided in the same ratio among these topics even though in some projects both methods are used for comparison. Class notes are the main source of reference in addition to references for supplemental materials such as Moaveni[1], Hoffman[2], and Incropera and DeWitt[3].
Finite Element Method
The first part of the course is devoted to an introduction to finite element method. In this section, direct formulation method is discussed and applied to one-dimensional problems. These include both structural problems, as will be discussed in the next section, as well as heat transfer problems such as steady state conduction through a composite wall or in a fin. More emphasis is given to structural problems though. The course also includes discussion of finite element methods other than direct formulation method. These include minimum total potential formulation and weighted residual methods (e.g., collocation method, subdomain method, Galerkin method, and least squares method). These provide the students some insight about the mathematics behind the analysis tools offered by the commercial FEA codes. Simple one-dimensional problems for which closed form analytical solutions are obtainable are discussed and assigned first. This is followed by two-dimensional problems for which closed form analytical solutions are either very complicated or are nonexistent. The objective is to make the students aware of the power of numerical methods of solving otherwise unsolvable engineering problems. In this section of the course, types of one-dimensional elements (e.g., linear, quadratic, and cubic) are discussed and an introduction to two-dimensional elements is also covered.
Mini-projects on Finite Element Method
Typical mini-projects assigned to the students in a semester are discussed here. The first assignment involves a problem that can be analyzed with one-dimensional approximation. It requires solving for the nodal displacements for a bar of fixed thickness and variable width (see Fig. 1) supporting a load P. The students are asked to solve for the nodal displacements, from which the normal stresses can be found, by
Figure 1[1]
using direct formulation method, and by using COSMOS FEA code and comparing the solutions obtained by the two methods for cases of 2 elements and 8 elements used in modeling. They are also asked to compare these solutions with the exact solution from the closed form analytical solution. The objective of this assignment is to demonstrate the fact that FEM can be used to obtain very accurate solutions even with relatively small number of elements when the geometry is simple.
In the first part of the second assignment the power of FEM is explored further by using the same problem used in the first assignment (see Fig. 1). This time the students perform a two-dimensional analysis for obtaining the nodal displacements by using COSMOS package with quadrilateral elements as well as with quadrilateral elements. The results obtained by using FEA are compared at representative nodes with the results of one-dimensional analytical solution. The results obtained from COSMOS by using coarse and finer mesh are also compared against each other. Here one of the objectives is to demonstrate that two-dimensional effects can be pretty significant and although such effects could be quite complex to account for in closed form analytical solutions, they can be taken into account with much less effort by the use of a FEA package. The second objective is to demonstrate the requirement of systematic mesh refinement in obtaining an acceptable accuracy in numerical simulation by finite element method. The second part of the assignment deals with a more complex geometry. In this problem (see Fig. 2), there is a rectangular section cut out from a plate of fixed thickness and variable width. Again, a simplified one-dimensional analysis with direct formulation method is required together with two-dimensional analysis using COSMOS. In this assignment, a two-dimensional closed form analytical solution would simply be impractical while numerical simulation by FEM would be a simple matter for a relatively adept user of the code. The objective is to demonstrate the dominance of the two dimensionality of the problem and how significantly the one-dimensional analysis deviates from a two-dimensional one.
Figure 2[1]
Finite Difference Method
The second part of the course that deals with the finite difference method is discussed and applied mainly to heat conduction (diffusion) problems. This starts with a one-dimensional boundary value problem (steady state one-dimensional fin problem with convective boundaries). It is extended to steady two-dimensional conduction problem, which results in an elliptic partial differential equation (Laplace equation), and finally into one-dimensional unsteady conduction problem resulting in a parabolic partial differential equation. Energy balance method is also covered to give the students a powerful tool for obtaining finite difference equations for boundary nodes with complexity in shape or exposure to different types of boundary conditions as well as for problems with energy generation in the conducting medium. Simple methods of obtaining finite difference approximations (forward, central, backward differencing) and their order of accuracy are discussed. For unsteady conduction problems, the solution by both explicit finite difference schemes as well as implicit finite difference schemes is discussed.
Mini-projects on Finite Difference Method
In this section, the typical assignments dealing with the finite difference method are discussed. These assignments are designed to encourage the students to use mathematical tools such as MATLAB and MAPLE for matrix operations after establishing the system of algebraic equations by finite difference approximations. The first assignment involves both FEM and FDM. In this assignment, the first part deals with a one-dimensional fin for which the steady state temperature distribution is already obtained by FEA using one-dimensional quadratic elements (see Fig. 3). This is an exercise of interpolating the temperatures of several off-nodal points by the use of local, global, and natural coordinates. The second part of the assignment deals with the solution of steady state temperature distribution for this fin geometry for a given fin material with a given base temperature and surrounding convective conditions using COSMOS. The last part of the assignment calls for modeling the fin as one-dimensional using finite difference method. It
Figure 3[1]
requires that the numerical solution be compared with the closed form analytical solution as well as the solution obtained by using COSMOS. The objective is again to demonstrate the accuracy of both FDM and FEM by comparison with the exact closed form analytical solution.
The first part of the second assignment deals with a two-dimensional steady conduction problem in a rectangular slab two sides of which are exposed to constant temperature boundary conditions and the other two sides are exposed to convective boundary conditions (see Fig. 4). Solutions with both FDM and FEM (using COSMOS) are required together with a comparison between the two solutions. The objective of this assignment is to demonstrate implementation of different types of boundary conditions in both FDM and FEM and to compare the solutions obtained from the two numerical solution methods.
Figure 4
The second part of the assignment requires the determination of the steady state temperature distribution in a simplified air-cooled gas turbine blade (see Fig. 5) by using finite difference method. Given a maximum allowable temperature in the blade, the students are to determine whether the design will be satisfactory.
Figure 5[3]
Here the symmetry conditions that exist in the given problem extensively are utilized to significantly reduce the computational domain required for finite difference formulation of the problem. The problem is to be solved by both FDM as well as by using COSMOS and the results are to be compared.
Conclusions
The author taught this course in the format discussed for the first time in spring 2001. Formal written reports were required from each student separately for each project. The student response to the course was mostly positive even though this was a course of completely new nature to the students. The power and value of the numerical simulation in engineering problem solving were well appreciated. The students had not only a good insight but also fun in observing the animation of the results of behavior of the parts in the first two assignments by the post-processing tools of COSMOS. At the end of the semester, there were student comments in the course evaluation by students suggesting that this course was very useful and should have been a three credit hour course. On the other hand, a few students found the workload of this course excessive for a one credit hour course. The nature of the mini-projects involving occasional tedious formulations and calculations may have been the reason for this.
One of the valuable experiences for the students was the requirement of the use of mathematical tools such as MATLAB and MAPLE, which are widely used in academia and industry today. Recent student surveys in the program show that there is somewhat low satisfaction in the computer science course, which deals with C language, in the Engineering Physics curriculum. In order to better utilize this knowledge, simple assignments requiring the students to write simple codes are being considered and planned for this course.
References
[1] S. Moaveni, (1999) Finite Element Analysis, Theory and Application with ANSYS, Prentice Hall, Upper Saddle River, New Jersey.
[2] J. D. Hoffman, (1992) Numerical Methods for Engineers and Scientists, McGraw-Hill, Inc., New York
[3] F. P. Incropera, and D. P. DeWitt, (1996) Introduction to Heat Transfer, John Wiley and Sons, New York.
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ASEE Southeast Section Conference
6
Mehmet Sözen
Dr. Mehmet Sözen is an associate professor of engineering physics in the Department of Physical Science at Embry-Riddle Aeronautical University. He teaches junior and senior design courses, and space systems engineering courses as well as physics for engineering majors. He has a doctoral degree in mechanical engineering from The Ohio State University. His expertise and interest are in transport phenomena in porous media, high heat flux applications, space systems engineering and design. His current research focuses on investigation of water mist fire suppression effectiveness in non-premixed counter-flow diffusion flames, and spacecraft thermal control. He has received research grants from NASA and NSF. He has been a summer faculty fellow at Argonne National Laboratory (twice) and the Naval Research Laboratory.
[1] Associate Professor of Engineering Physics, Department of Physical Science, Embry-Riddle Aeronautical University, 600 S. Clyde Morris Blvd., Daytona Beach, FL 32114.