Introduction to Finite Element Methods

This learning track was developed by Professor Krishna Garikipati and Dr. Gregory Teichert, University of Michigan, in partnership with Ansys. This learning track will discuss the mathematics behind finite element methods. The main goal of these lectures is to turn the viewer into a competent developer of finite element code. It is hoped that these lectures on finite element methods will complement the series on continuum physics to provide a point of departure from which the seasoned researcher or advanced graduate student can embark on work in (continuum) computational physics.

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Innovation Course

FEA — Heat Conduction and Mass Diffusion at Steady-State

In this course, we will discuss the strong form of steady-state heat conduction and mass diffusion. We will introduce Fourier's law of heat conduction and temperature, and heat flux will also be discussed, as well as boundary conditions like concen...Read more

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Innovation Course

FEA - Linear Elliptic PDEs in 1D

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 el...Read more

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Innovation Course

FEA — The Finite-dimensional Weak Form

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, w...Read more

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Innovation Course

FEA — The Matrix-vector Form

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 t...Read more

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Innovation Course

FEA — Boundary Conditions, Basis Functions, and Numerics

We begin this course with a discussion about boundary conditions. We discuss a pure Dirichlet problem using an example of a linear elastic bar. We then move on to developing higher-order polynomial basis functions for Lagrange polynomials and discuss some properties of Lagrange ...Read more

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Innovation Course

FEA — Analysis of the Finite Element Method

This course discusses why and how finite element analysis (FEA) works and the special properties of 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...Read more

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Innovation Course

FEA — Variational Principles

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 respec...Read more

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Innovation Course

FEA — Lagrange Basis Functions and Numerical Quadrature in 1D Through 3D

In this course, we will discuss the use of Lagrange polynomials in the basis functions in 1D through 3D. The formula for the basis functions is first written in 2D, then in 3D. We will further talk about the Gaussian quadrature...Read more

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Innovation Course

FEA — Linear and Elliptic Partial Differential Equations for a Scalar Variable in Two Dimensions

This course discusses the process of building two-dimensional problems for the linear and elliptic PDEs using a scalar variable. The strong and the weak form are discussed using constitutive relations and bou...Read more

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Innovation Course

FEA - Linear Elliptic PDE in 3D

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 derive the finite element equations for linear elliptic PDEs with vector variables in three dimensions. One s...Read more

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Innovation Course

FEA - Unsteady heat conduction

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 parabolic problems. The problems that will be discussed are linear parabolic PDEs in three dimensions for a scalar variable. Physical problems such as unsteady heat conduction and unsteady mass diffusion are considered here.

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Innovation Course

FEA - Linear Elastodynamics

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 how to solve linear, hyperbolic partial differential equations for an unknown vector in three dimensions ...Read more

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