This course is an introduction to the finite element method as applicable to a range of problems in physics and engineering sciences. The treatment is mathematical, but only for the purpose of clarifying the formulation. The emphasis is on coding up the formulations in a modern, open-source environment that can be expanded to other applications, subsequently.
The course includes about 45 hours of lectures covering the material I normally teach in an introductory graduate class at University of Michigan. 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. We do spend time in rudimentary functional analysis, and variational calculus, but this is only to highlight the mathematical basis for the methods, which in turn explains why they work so well. Much of the success of the Finite Element Method as a computational framework lies in the rigor of its mathematical foundation, and this needs to be appreciated, even if only in the elementary manner presented here. A background in PDEs and, more importantly, linear algebra, is assumed, although the viewer will find that we develop all the relevant ideas that are needed.
The development itself focuses on the classical forms of partial differential equations (PDEs): elliptic, parabolic and hyperbolic. At each stage, however, we make numerous connections to the physical phenomena represented by the PDEs. For clarity we begin with elliptic PDEs in one dimension (linearized elasticity, steady state heat conduction and mass diffusion). We then move on to three dimensional elliptic PDEs in scalar unknowns (heat conduction and mass diffusion), before ending the treatment of elliptic PDEs with three dimensional problems in vector unknowns (linearized elasticity). Parabolic PDEs in three dimensions come next (unsteady heat conduction and mass diffusion), and the lectures end with hyperbolic PDEs in three dimensions (linear elastodynamics). Interspersed among the lectures are responses to questions that arose from a small group of graduate students and post-doctoral scholars who followed the lectures live. At suitable points in the lectures, we interrupt the mathematical development to lay out the code framework, which is entirely open source, and C++ based.
A working knowledge of linear algebra is needed. Some exposure to partial differential equations would be very helpful. Experience with programming is a must, even if it is only Matlab or a language such as Fortran, C or Python.
Learn about statics through real life engineering examples. Engage with the theory to grasp the full understanding of simple machines and complex mathematical models. Are you interested in improving your mechanics or introducing yourself to the subject all together? Join our unique course, devised by the Ural Federal University. Through our innovative approach, you will receive the basic traditional material by engaging in practically-oriented tasks and learn the strictly theoretical mathematical analysis of basic concepts.
This course will help the students who have taken intro-level high-school physics to get ready for more advanced courses including AP Physics C: Mechanics. This short course is intended for the high-school students who have taken an introductory-level physics course, acquired some background in Mechanics and intend to take a more advanced course – for instance, AP Physics C.
The science of stars and stones. Archaeoastronomy is the “science of stars and stones”: it studies the relationships between the ancient monuments and the sky, in order to gain a better understanding of the ideas of the architects of the past and of their religious and symbolic world. The course provides the first complete, easy introduction to this fascinating discipline.
Il corso affronta le tematiche dell’elettromagnetismo e dell’ottica partendo dall’applicazione del metodo sperimentale, per prepararsi al meglio all'ingresso all'università.
The course addresses the thematic of electromagnetism and optics, starting with the applications of the experimental method, to better start the academic learning path.
Learn about the science behind the current exploration of the solar system in this free class. Use principles from physics, chemistry, biology, and geology to understand the latest from Mars, comprehend the outer solar system, ponder planets outside our solar system, and search for habitability in our neighborhood and beyond.
In this course, you will learn how to characterize the energy state of a system and the mechanisms for transferring energy from one system to another. These are the tools necessary to understand stationary and transportation power systems from small scale, like batteries, to large scale, like nuclear power plants.
In this course you will learn a whole lot of modern physics (classical and quantum) from basic computer programs that you will download, generalize, or write from scratch, discuss, and then hand in. Join in if you are curious (but not necessarily knowledgeable) about algorithms, and about the deep insights into science that you can obtain by the algorithmic approach.
Il corso affronta le tematiche della meccanica e della termodinamica partendo dall’applicazione del metodo sperimentale, per prepararsi al meglio all'ingresso all'università.
The course addresses the thematic of Mechanics and thermodynamics starting from the applications of the experimental method, to better prepare to start the college experience with the fundamental knowledge.