FEA - Linear Elliptic PDE in 3D

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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 such application case is that of linearized elasticity which is discussed in this course. We begin by discussing the constitutive relation between Cauchy stress and strain and the kinematic relation between strains and displacements. Following this, we discuss several properties of the elasticity tensor. Using this formation, we derive the finite-dimensional weak form of linearized elasticity. We then discuss how the finite-dimensional weak form can be written as a summation of element integrals. Then using the basis functions we transform the weak form to its equivalent matrix-vector form. We finally discuss how Neumann and Dirichlet boundary conditions are handled for such problems.

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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 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. 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.