Strong and Weak Forms for One-Dimensional Problems

In this chapter, the strong and weak forms for several one-dimensional physical problems are developed. Thestrong formconsists of thegoverningequationsand theboundaryconditions for aphysical system.The governingequations areusuallypartial differential equations, but in theone-dimensional case theybecome ordinary differential equations. The weak form is an integral form of these equations, which is needed to formulate the finite element method. In some numerical methods for solving partial differential equations, the partial differential equations can be discretized directly (i.e. written as linear algebraic equations suitable for computer solution). For example, in the finite differencemethod, one can directlywrite the discrete linear algebraic equations from the partial differential equations. However, this is not possible in the finite element method. A roadmap for the development of the finite elementmethod is shown inFigure 3.1.As can be seen from the roadmap, there are three distinct ingredients that are combined to arrive at the discrete equations (also called the systemequations; for stress analysis theyare called stiffness equations),whichare then solvedby a computer. These ingredients are