1. Lesson 3 Basic Concepts

2. Fundamentals Any continuous quantity (temperature, displacement, etc.) can be approximated by a discrete model composed of a set of piecewise continuous functions Functions defined using values of continuous quantities at a finite number of points (nodes) The FEM is virtually independent of geometry and loading Different classes of problems can be analyzed using very similar programs Concepts are simple – details in software complex

3. Solving for 2-D Heat Transfer

4. 2-D heat transfer cont.

5. Analytical approach – separation of variables Assume we wish to solve Laplace’s Eq. with the boundary conditions To obtain a solution by the Separations of Variables approach, assume Thus the original Laplace Eq. is transformed to the form Rearranging

6. continued Hence, The solution to Eq. (1) is with boundary conditions

7. continued The solutions to Eq. (2) is with boundary conditions

8. continued Consequently, Now one must find E n. Employ orthogonality of trigonometric functions Note that

9. continued Thus, the solution for T(x,y) is While analytical solutions exist for some problems, there are many that have no clear cut analytical forms.