Finite Element Analysis (FEA) — The Matrix-vector Form

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In this course, we derive the matrix-vector weak form, which is used in the finite element method. We begin by deriving the matrix-vector form for a single general element, then proceed to derive it for the entire problem domain by assembling the matrix-vector forms for all the elements in the domain.
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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Finite Element Analysis (FEA) — Analysis of the Finite Element Method

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 discuss how the infinite-dimensional weak form of a 1D linear elliptic PDE can be transformed to a finite-dimensional weak form which forms the basis of the finite element method. We discuss basic Hilbert spaces, which are functional spaces in which acceptable solutions to the weak form exist. We also discuss basis functions and how the finite-dimensional weak form can be written as a sum over the finite subdomains (or elements) of the problem.

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

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.

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.