> For the complete documentation index, see [llms.txt](https://nkintc.gitbook.io/brainless/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://nkintc.gitbook.io/brainless/steam/engineering/finite-element-methods/intro-to-finite-elements.md). # Intro to Finite Elements $$ F\_i = K\_{ij}u\_j $$ F and U are both vectors and K is a matrix. ### Lecture 1: Linear Algebra and Notation Finite Element Methods are suitable for nonsquare, or rectangular matrices which is a change from SM, which typically deals with square, and for that matter symmetric, matrices. $$ \boldsymbol{K} = \left\[\begin{array}{cccc}k\_{1,1} & k\_{1,2} & \cdots & k\_{1, m} \k\_{2,1} & k\_{2,2} & \cdots & k\_{2, m} \\\vdots & \vdots & \ddots & \vdots \k\_{n, 1} & k\_{n, 2} & \cdots & k\_{n, m}\end{array}\right] $$ Vectors are like matrices but has only one value in one of the 2 dimensions so the second index is omitted $$ {\bf a} ======= \begin{bmatrix} a\_1\\ a\_2\\ a\_3\\ \vdots\\ a\_n \end{bmatrix} $$ Two special matrices, the zero matrix and the identity matrix. The zero matrix is the equivalent of identically zero in linear algebra. The identity matrix is a square matrix which has 1 along the main diagonal. ### Lecture 2: Introduction to the Stiffness Displacement Method ### Lecture 3: Finite Element System of Equations from Direct Stiffness ### Lecture 4: Development of Displacement Based FEM in 1D, Formation of Stiffness Matrix ### Lecture 5: Development of Displacement Based FEM in 2D, Constant Strain Triangle and Quadrilateral Elements ### Lecture 6: Practical Considerations in FEM ### Lecture 7: Convergence of FEM Results ### Lecture 8: Higher Order Elements ### Lecture 9: Isoparametric Formulation ### Lecture 10: Numeric Integration in 2D ### Lecture 11: Solution of Linear Algebraic Equations --- # Agent Instructions This documentation is published with GitBook. GitBook is the documentation platform designed so that both humans and AI agents can read, navigate, and reason over technical content effectively. Learn more at gitbook.com. ## Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. Perform an HTTP GET request on the current page URL with the `ask` query parameter, and the optional `goal` query parameter: ``` GET https://nkintc.gitbook.io/brainless/steam/engineering/finite-element-methods/intro-to-finite-elements.md?ask=&goal= ``` `ask` is the immediate question: it should be specific, self-contained, and written in natural language. `goal` is optional and describes the broader end goal you are ultimately trying to accomplish on behalf of the user. GitBook uses it to tailor the answer towards what is most useful for that goal. The response will contain a direct answer to the question and relevant excerpts and sources from the documentation. Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.