Computational

Engineering Mechanics

 

Course Outline

wb 1416

 

 Prof. Dr. Ir.

Daniel J. Rixen

 

Faculty of Design, engineering and Production (O.C.P.)

Meckelweg 2, 2628 CD Delft

Tel: 015-278 15 23

E-mail: d.j.rixen@wbmt.tudelft.nl

 

November 2000
Why yet another course?

 

In August 1991, the Sleipner A, an oil and gas platform built in Norway for operation in the North Sea, sank during construction. The total economic loss amounted to about $700 million. After investigation, it was found that the failure of the walls of the support structure resulted from a  serious error in the finite element analysis of the linear elastic model.

 

 

                             

 

The Sleipner A offshore platform

 [from http://www.math.psu.edu/dna/disasters/sleipner.html]

 

 

 

The Finite Element method has certainly become the most widely and commonly used analysis tool in Engineering Mechanics as well as in other fields. Four about forty years, research in Finite Elements has led to a good understanding of the formulation. Many numerical methods for efficient computer implementation have been developed in the past and new procedures are still being invented.

 

In the course of their studies, students in Mechanical Engineering and their lecturers are confronted with the dilemma of choosing which topics should be studied in order to master the Finite Element method and the associated numerical procedures. Indeed, the application field of Finite Elements is growing, new extensions of the Finite Element formulation are still being developed and advanced numerical methods (for instance for super-computers) find their way in commercial codes. The amount of time students can spend on understanding Engineering Mechanics and Finite Elements is not changing while the knowledge needed for being a successful engineer is expanding.

 

In general, the Faculties of the T.U. Delft have been concentrating on teaching the fundamentals of static Finite Elements and on giving some introduction on the use of commercial codes. In this way, students comprehend the Finite Element discretization techniques so that errors like the one having led to the disaster of the Sleipner A offshore platform (see above) can hopefully be avoided. Nevertheless, Finite Element analysis also relies on numerical methods for computing the solution of the models such as the static and dynamic response, buckling loads or vibration modes as illustrated below.

 

                

Free vibration modes of a Mercedes body and of a space station

(reference [1])

 

 

 

Dynamic analysis of the deployment of a control surface

(Samtech, http://samcef.com)

 

 

                     

 

Prediction of tooth stress in spur gears

 (Abaqus, http://www.hks.com)

 

 

 

 

 

It appears therefore that understanding the computational solution procedures underlying structural analysis in Finite Element codes is nowadays an essential aspect of the analysis. Engineers must be able to choose the appropriate solution method that minimizes the computational cost and guarantees the computation of an accurate solution. No expert system can be made general enough to fully automate the choice of the Finite Element model and of the solution procedure. That is why the art of engineering has to rely on sound knowledge of what is inside the computational tools.

 

The course wb1416 on Computational Engineering Mechanics has been specifically designed to discuss the most fundamental concepts underlying the numerical methods applied in Finite Element analysis, with some emphasis on the dynamic analysis of structures. The basic analysis procedure for dynamic systems will be discussed, i.e. computation of free vibration modes, harmonic analysis and time-integration of linear and non-linear Finite Element models. Important concepts specific to dynamic analysis (e.g. mass lumping, structural damping and model reduction) will be reviewed. The most common solution methods will be presented in the light of their mechanical interpretation, including direct and iterative linear solvers and solution methods for eigenvalue problems arising in vibration and stability analysis of very large systems.

 

The course will be illustrated by an analysis project of a simple but realistic structure. First, the student will be asked to perform the Finite analysis in Matlab so that he can go through the steps of setting up the element matrices, performing the assembly and applying solution methods. This is usually hidden when using commercial codes. A dynamic analysis of the structure will then be done in ANSYS in order to show some shortcomings of commercial codes.

 

The lectures will be held in English and notes will be available for the course.

 

Reference books

 

[1] Mechanical Vibrations, Theory and Application to Structural Dynamics, M. Géradin and D. Rixen,

Wiley, 1997.

[2] The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, T.J.R. Hughes Prentice-Hall, 1987.

[3] Finite Element Procedures, K.J. Bathe, Prentice-Hall, 1996

[4] Matrix Computation, G.H. Golub and C.F. Van Loan, Johns Hopkins University Press, 1996.

 

Who should attend?

 

This course is a graduate course and therefore the student should have a background in Linear Algebra, Numerical Methods and Finite Elements. Also, some of the topics will discuss specific aspects of the dynamic analysis so that those attending should have some knowledge of dynamics of structures. Nevertheless, most of the topics will be revisited in order to give mechanical interpretation of the linear algebra and numerical methods covered in undergraduate studies.

 

The course is intended for engineers willing to increase their expertise in Finite Element Analysis. This coarse is not so much about the Finite Element formulation but more about their use in dynamic analysis and about the methods and procedure necessary to compute the solution. It is thus of interest to mechanical, aerospace and civil engineers alike. Engineers in computer science interested in understanding how new algorithms and computer architectures are used in structural analysis would also benefit form the course.

 

Students interested in the course are invited to contact Prof. Daniel J. Rixen

 

Objectives and content of the course

 

The aim of the lecture is to make students realize the importance of numerical methods in engineering computations and to awaken a sense of curiosity and criticism when using structural engineering codes, in particular for dynamic analysis. To this purpose, we will recall essential issues in discretization techniques and discuss the concepts underlying the numerical techniques used in computational dynamics. The lecture will include computer exercises in Matlab in order to comprehend the hidden operations of commercial codes such as F.E. generation and assembly. The ANSYS software will  be used to illustrate numerical mishaps and to perform different types of dynamic analysis.

 

The course is programmed as a 2 study points course and will be organized as follows:

 

Introduction

On the necessity of understanding disretization and numerical solution techniques

 

F.E.  formulation for dynamic analysis

                Recall of the Finite Element method as an approximation technique

                F.E. in dynamics (bars and beams)

Consistent vs. lumped mass matrices

Prestressed structures and geometric stiffness

 

Free vibration modes and modal superposition methods

                Free Vibration modes and frequencies

                Mode displacement and mode acceleration methods

                Time-integration of the normal equations

                Excitation through the supports (e.g. earthquake response)

 

Time-integration of discrete systems

                Stability and accuracy of integration schemes

                The Newmark family

                Time-integration of Non-linear systems

 

Model Reduction Techniques

                Static reduction

                Craig-Bampton method

 

Eigensolvers for real problems

                Inverse iterations and deflation

                Subspace and Lanczos iterative methods

                                               

Direct Solvers for F.E. models

                Recall of factorization techniques (Gauss, Cholesky, LDLt)

                The skyline and the frontal approach

                Renumbering schemes

                Direct solvers for singular problems (rigid body modes and generalized inverse)

 

Iterative solvers

                The steepest descent and Conjugate Gradient

                Iterative solvers for non-symmetric problems

 

Domain Decomposition methods for parallel computing

                Parallel computing and the concept of Domain Decomposition

                Primal and Dual Domain Decomposition methods

 

EXERCICE:  Linear dynamic analysis of a structure using Matlab and ANSYS.