This lesson covers the fundamentals of weighted residual methods and their application in structural mechanics. It delves into the Galerkin method, least square method, and collocation method, explaining how these methods provide approximate solutions to differential equations. The lesson also discusses the concept of weighted functions and their role in different methods of analysis. It further illustrates the application of these methods through examples, including solving a governing differential equation using the Galerkin method and the least square method. The lesson concludes with an explanation of how weighted residual statements can reduce the governing differential equation to a weak form, offering a wider choice of trial functions.

- Weighted residual methods, including the Galerkin method, least square method, and collocation method, are powerful tools for finding approximate solutions to differential equations.
- Weighted functions play a crucial role in these methods, with different choices of weighted functions corresponding to different methods of analysis.
- The Galerkin method involves using the same weighted function and trial function.
- The least square method minimizes the square of the residue in the domain with respect to fitting coefficients.
- The collocation method assumes a weight function such that the residue vanishes identically at selected points.
- Weighted residual statements can reduce the governing differential equation to a weak form, offering a wider choice of trial functions.

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