Finite Elements: Computational Engineering Sciences

A. J. Baker

Academic research confirms that practically all computational engineering sciences algorithms can be formulated as the extremum of the mathematician’s weak form theory. The resultant weak statement (WS) process enables theorization completion in the continuum, using calculus, vector field theory and modern approximation concepts. When finished, the WS theory discrete implementation can be formed using FE, FD and/or FV procedures. The FE construction is typically guaranteed optimal in its performance, i.e., accuracy, asymptotic convergence rate, etc, as it leads to precise constructions devoid of heurism associated with difference algebra schemes.

This text develops discrete implementations of WS theory for a diverse set of problem statements in the computational engineering sciences. Unique to the FE discrete implementation, the resulting algorithms can be immediately rendered computable using a transparent, object-oriented programming syntax. The engineering science problem classes covered in this text include

• heat conduction
• structural mechanics
• mechanical vibrations
• heat transfer, with convection and radiation
• fluid mechanics
• heat and mass convective transport

The text is organized into twelve chapters. Following an Introduction, and some pertinent classical overview material, an elementary heat conduction tutorial clearly illustrates all FE matrix constructs, the “famous” assembly algorithm, and the concept of quantitatively precise error estimation with. Subsequent chapter pairs develop expository one-dimensional, then general n-dimensional FE implementations in each engineering sciences category. The sequence of developments illustrates, examines and generalizes the available theoretical error estimates based on solution norms. In moving to the convective transport problem class, and non-linearity, a modified conservation principle theory, still expressed in the continuum, leads to a firm understanding of the phase lag and dispersive error mechanisms dominating applications in CFD.

This is Prof. Baker’s fourth text in the computational engineering sciences, unique in the field with its fully integrated hands-on computer experiments supported by Matlab® and the COMSOL® Problem Solving Environment (PSE). Reader interaction with text content is enhanced via the interactive website at www.wiley.com/go/baker/finite containing complete support materials for all computer labs as well as several of Prof. Baker’s topical lectures viewable on your PC via video streaming.