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Generalization to 1D Finite Element Method: Strong and Weak Formulations Explained

This lecture provides an in-depth exploration of the generalization of the 1D finite element method. Key topics include the strong formulation and the weak formulation, detailing how to derive each. The session also covers the discretization process of finite element methods, showcasing exemplary shape functions for two finite elements over the interval [0, l]. Additionally, attendees will learn about the composition of global shape functions from local ones and how the transition from finite differences to finite elements affects local systems of equations.

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Generalization to 1D Finite Element Method: Strong and Weak Formulations Explained

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Presentation Transcript

  1. Graph grammarand algorithmic transformations Lecture topics: Generalization to 1D finite element method

  2. GENERALIZATION TO 1D FINITE ELEMENT METHOD Strong formulation Find such that Weak formulation Find such that

  3. GENERALIZATION TO 1D FINITE ELEMENT METHOD Finite element method disretization Exemplary shape functions for [0,l] = [0,1], for two finite elements

  4. GENERALIZATION TO 1D FINITE ELEMENT METHOD Exemplary shape functions for [0,l] = [0,1], for two finite elements

  5. GENERALIZATION TO 1D FINITE ELEMENT METHOD Exemplary shape functions for [0,l] = [0,1], for two finite elements

  6. GENERALIZATION TO 1D FINITE ELEMENT METHOD Global shape functionsare composed with local shape functions, e.g. Local system of equations generated over the element K

  7. GENERALIZATION TO 1D FINITE ELEMENT METHOD Notice that when we switch from finite difference to finite elements, it only changes the local systems of equations at tree nodes

  8. GENERALIZATION TO 1D FINITE ELEMENT METHOD Notice that when we switch from finite difference to finite elements, it only changes the local systems of equations at tree nodes

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