Successfully completed a course at the University of Michigan

Today I received a certificate confirming the successful completion of the course “The Finite Element Method for Problems in Physics” of the University of Michigan.

The course was aimed at studying the mathematical techniques underlying the Finite Element Method. Prof. Krishna Garikipati described in great detail the difficulties of formalization of mathematical physics problems in linear and nonlinear formulation. The nuances of constructing finite-dimensional weak form for the multi-dimensional problem, numerical methods of approximation and solving partial differential equations were considered especially carefully.

The emphasis was on the development of algorithms for solving differential equations and their coding in C++. The basis was an open source FEM library - deal.II, which can later be expanded for other applications.

The course was quite difficult for me, but I was able to successfully complete all practical classes and get a score of 64.5%.

https://coursera.org/share/7126e9f564641c09a022a042409fcc11 — Course certificate

Here is a short list of topics that aroused my greatest interest:

Derivation of the weak form using a variational principle;
The strong and weak forms of steady state heat conduction and mass diffusion;
Lagrange basis functions in 1 through 3 dimensions;
Analysis of the integration algorithms for first order, parabolic equations;
Modal decomposition.

Also I hope that there are further materials on shell elements, nonlinear analysis (geometric nonlinearity, plasticity and hyperelasticity).

Finite Element Analysis (FEA) is a good choice for problem analysis in complex domains when the area changes (such as during a solid-state reaction with a moving boundary), when the desired accuracy changes over the entire area, or when the solution is not smooth.

There are many good books on FEM/FEA:

(Hughes, 2000)
(Zienkiewicz et al., 2013)
(Fish & Belytschko, 2007)

This text is geared toward assisting engineering and physical science students in cultivating comprehensive skills in linear static and dynamic finite element methodology. Based on courses taught at Stanford University and the California Institute of Technology, it ranges from fundamental concepts to practical computer implementations. Additional sections touch upon the frontiers of research, making the book of potential interest to more experienced analysts and researchers working in the finite element field. In addition to its examination of numerous standard aspects of the finite element method, the volume includes many unique components, including a comprehensive presentation and analysis of algorithms of time-dependent phenomena, plus beam, plate, and shell theories derived directly from three-dimensional elasticity theory. It also contains a systematic treatment of "weak," or variational, formulations for diverse classes of initial/boundary-value problems. Directed toward students without in-depth mathematical training, the text incorporates introductory material on the mathematical theory of finite elements and many important mathematical results, making it an ideal primer for more advanced works on this subject.

The Finite Element Method: Its Basis and Fundamentals offers a complete introduction to the basis of the finite element method, covering fundamental theory and worked examples in the detail required for readers to apply the knowledge to their own engineering problems and understand more advanced applications.

The text material evolved from over 50 years of combined teaching experience it deals with a formulation and application of the finite element method. A meaningful course can be constructed from a subset of the chapters in this book for a quarter course; instructions for such use are given in the preface. The course material is organized in three chronological units of one month each: 1) the finite element formulation for one-dimensional problems, 2) the finite element formulation for scalar field problems in two dimensions and 3) finite element programming and application to scalar field problems; and finite element formulation for vector field problems in two dimensions and beams. In conjunction with the book there will be the access and use of ABAQUS software and MATLAB exercises.