Introduction to Finite Elements in Engineering

Thoroughly updated with improved pedagogy, the fifth edition of this classic textbook continues to provide students with a clear and comprehensive introduction the fundamentals of the finite element method. New features include coverage of core topics – including mechanics and heat conduction, energy and Galerkin approaches, convergence and adaptivity, time-dependent problems, and computer implementation – in the context of simple 1D problems, before advancing to 2D and 3D problems; expanded coverage of reduction of bandwidth, profile and fill-in for sparse solutions, time-dependent problems, plate bending, and nonlinearity; over thirty additional solved problems; and downloadable Matlab, Python, C, Javascript, Fortran and Excel VBA code providing students with hands-on experience. Accompanied by online solutions for instructors, this is the definitive text for senior undergraduate and graduate students studying a first course in the finite element method, and for professional engineers keen to shore up their understanding of finite element fundamentals.

Algorithm 1016

In this article, we introduce the Python framework PyMGRIT, which implements the multigrid-reduction-in-time (MGRIT) algorithm for solving (non-)linear systems arising from the discretization of time-dependent problems. The MGRIT algorithm is a reduction-based iterative method that allows parallel-in-time simulations, i.e., calculating multiple time steps simultaneously in a simulation, using a time-grid hierarchy. The PyMGRIT framework includes many different variants of the MGRIT algorithm, ranging from different multigrid cycle types and relaxation schemes, various coarsening strategies, including time-only and space-time coarsening, and the ability to utilize different time integrators on different levels in the multigrid hierachy. The comprehensive documentation with tutorials and many examples and the fully documented code allow an easy start into the work with the package. The functionality of the code is ensured by automated serial and parallel tests using continuous integration. PyMGRIT supports serial runs suitable for prototyping and testing of new approaches, as well as parallel runs using the Message Passing Interface (MPI). In this manuscript, we describe the implementation of the MGRIT algorithm in PyMGRIT and present the usage from both a user and a developer point of view. Three examples illustrate different aspects of the package itself, especially running tests with pure time parallelism, as well as space-time parallelism through the coupling of PyMGRIT with PETSc or Firedrake.