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

Current Status
Not Enrolled
Price
Free
Get Started

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 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.

Recommended Courses

STRUCTURES
Learn Physics

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

STRUCTURES
Learn Physics

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 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.

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.