Transcription of Strong and Weak Forms for One-Dimensional Problems
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Strong and Weak Forms for One-Dimensional Problems
3. Strong and Weak < Strong >FormsStrong > for < Strong >One-DimensionalStrong > < Strong >ProblemsStrong > In this chapter, the Strong and weak < Strong >FormsStrong > for several < Strong >One-DimensionalStrong > physical < Strong >ProblemsStrong > are developed. The Strong form consists of the governing equations and the boundary conditions for a physical system. The governing equations are usually partial differential equations, but in the < Strong >One-DimensionalStrong > case they become ordinary differential equations. The weak form is an integral form of these equations, which is needed to formulate the nite element method. In some numerical methods for solving partial differential equations, the partial differential equations can be discretized directly ( written as linear algebraic equations suitable for computer solution). For example, in the nite difference method, one can directly write the discrete linear algebraic equations from the partial differential equations. However, this is not possible in the nite element method. A roadmap for the development of the nite element method is shown in Figure As can be seen from the roadmap, there are three distinct ingredients that are combined to arrive at the discrete equations (also called the system equations; for stress analysis they are called stiffness equations), which are then solved by a computer.
The weak form is the most intellectually challenging part in the development of finite elements, so a student may encounter some difficulties in understanding this concept; it …
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