Finite Element Analysis (FEA) — The Finite-dimensional Weak Form

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In this course, we discuss how the infinite-dimensional weak form of a 1D linear elliptic partial differential equation (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.

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

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Finite Element Analysis (FEA) — Boundary Conditions, Basis Functions, and Numerics

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 begin by discussing linear elliptic partial differential equations (PDEs) in one dimension. We discuss various elements required for solving PDEs, such as the boundary conditions and the constitutive relations. We then discuss the strong and weak forms of the PDEs and show how the two forms are equivalent.

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 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 and 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.

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.