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

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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 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, we derive an equivalent finite-dimensional weak form from the infinite-dimensional weak form. 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.

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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.

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 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.