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. The treatment is mathematical, which is natural for a topic whose roots lie deep in functional analysis and variational calculus. It is not formal, however, because 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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Finite Element Analysis (FEA) — The Finite-dimensional Weak Form

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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

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Finite Element Analysis (FEA) — Analysis of the Finite Element Method

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Finite Element Analysis (FEA) — Variational Principles

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Finite Element Analysis (FEA) — Heat Conduction and Mass Diffusion at Steady-State

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Finite Element Analysis (FEA) — Lagrange Basis Functions and Numerical Quadrature in 1D Through 3D

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Finite Element Analysis (FEA) — Linear and Elliptic Partial Differential Equations for a Scalar Variable in Two Dimensions

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 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.

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.

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.

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.

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. 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.

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 boundary conditions. Then, it talks about the gradient of the trial solution and the weighting function. Lastly, it discussed 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. In this course, we will discuss how to solve linear, hyperbolic partial differential equations for an unknown vector in three dimensions and apply the findings to study linear elastodynamics. The problem is time-dependent and so are the boundary conditions. Apart from boundary conditions, we also need initial conditions to solve this class of problems. We discuss various time discretization methods and the Newmark family of algorithms used to solve time-dependent problems. We then write the time-discretized problem in its modal form and learn how to solve it. We end the course with a discussion about the stability analysis and amplification matrix for such problems.

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.

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 and apply the findings to study linear elastodynamics. The problem is time-dependent and so are the boundary conditions. Apart from boundary conditions, we also need initial conditions to solve this class of problems. We discuss various time discretization methods and the Newmark family of algorithms used to solve time-dependent problems. We then write the time-discretized problem in its modal form and learn how to solve it. We end the course with a discussion about the stability analysis and amplification matrix for such problems.