Finite Element Analysis (FEA) — Variational Principles

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This course focuses on obtaining the weak form using variational methods. Further, it discusses potential energy, or Gibbs free energy. We will talk about ‘pi’ as free energy functional and how the extrema can be used to check equilibrium states. Next, we discuss the variation in ‘pi’ with respect to the variation in the field. Then, we will see how the extremization of free energy functional is performed.
This course was developed by Prof. Krishna Garikipati and Dr. Gregory Teichert, at the University of Michigan in partnership with Ansys.

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This course was developed by Prof. Krishna Garikipati and Dr. Gregory Teichert, at the University of Michigan in partnership with Ansys. In this course, we will discuss the use of Lagrange Polynomials in the basis functions in 1 through 3 dimensions. The formula for the basis functions is first written in 2D and then in 3D. We will further talk about the Gaussian Quadrature for numerical integration. Lastly, triangular and tetrahedral elements are discussed using these basis functions.

This course was developed by Prof. Krishna Garikipati and Dr. Gregory Teichert, at the University of Michigan in partnership with Ansys.

In this course, we will discuss the strong form of steady-state heat conduction and mass diffusion. We will introduce the Fourier Law of heat conduction and temperature and heat flux will be discussed. The boundary conditions like concentration and mass influx with respect to mass diffusion. Then we will derive an equivalent infinite-dimensional weak form from the strong form of steady-state heat conduction and mass diffusion. After that, an equivalent finite-dimensional weak form is derived from the infinite-dimensional weak form. Further, we also discuss a general eight-node brick element or hexahedral element and the trilinear basis functions used in its formulation. In the following lessons, we will talk about the Jacobian of the map, followed by the integrals in terms of the degree of freedom of an element. Lastly, we will conclude with the matrix-vector weak form.

This course was developed by Prof. Krishna Garikipati and Dr. Gregory Teichert, at the University of Michigan in partnership with Ansys.

This course discusses why and how the FEA works and what are the special properties of the FEA. The norms that represent the finite-dimensional trial solution will be discussed. The important properties of the finite element method – consistency and the best approximation - will be explained in detail. Later, we will discuss the Pythagorean Theorem. The Sobolev estimates and convergence of the FEM will be explained further. And lastly, we will talk about finite element error estimation.