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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 ...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 vari...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 spa...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...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 Lag...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 th...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 ...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 quad...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 a...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....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 dimen...Read more
