Text Box: Finite element method, error and uncertainty One day short course   Introduction  The finite element method is a numerical technique for solving partial differential equations. The objective of this course is to explain the fundamental mathematical procedures of the finite element method for the purpose of exposing the underlying sources of error. Corresponding, practical methods of error control are discussed and illustrated. The course is aimed at both designers and engineers who wish to understand the fundamental formulation, characteristics and sources of error in the finite element method to increase the reliability of their models. The mathematical content is clearly explained and illustrated. The course begins with an illustration of the fundamental principles of the finite element method by applying it to a simple engineering problem. Methods for setting up and solving the finite element equations are discussed and contrasted with the finite difference method. The next stage overviews the fundamental procedures of the finite element method, from its fundamental piecewise formulation and use of shape functions, through element formulation and assembly, to numerical solution and interpolation. The material is then extended to cover first and second order time-varying problems, where basic time-integration methods are discussed and their sources of error illustrated. Eigensolutions and their application in mode superposition solutions are also explained. The various sources of error revealed in earlier sections are then summarised and analysed. Theoretical and practical techniques of numerical error estimation and model adaptivity are discussed and illustrated using a simple finite element program (provided) which is used to study the impact of the errors on the solution. Some of the key numerical techniques used, specifically, natural coordinate systems, numerical integration and Jacobian transformations, are also explained. A comprehensive set of course notes and a simple finite element program for illustrating numerical errors are supplied. Course content  1 Boundary value problems 1.1 Example problem 1.2 Exact solution 1.3 Approximate solutions 1.4 Weighted residual and least squares methods 1.5 Finite difference method 2 Finite element method 2.1 Galerkin weighted residual method 2.2 Integration by parts 2.3 Natural boundary conditions 2.4 Element equations 2.5 Polynomial assumed functions 2.6 Global finite element assembly 2.7 Solution interpolation 3 First order propagation problems 3.1 Galerkin method for first-order problems 3.2 Finite difference solution in time 4 Second order propagation problems 4.1 Galerkin method for second-order problems 4.2 Finite difference solution in time 5 Eigensolution 5.1 Computation of eigenvalues and eigenvectors 5.2 Conversion to generalised coordinates 6 Mode superposition solution 6.1 Static mode superposition 6.2 Transient mode superposition 6.3 Time integration 6.4 Dynamic magnification 7 Discretisation error 7.1 Spatial discretisation error 7.2 Error analysis and estimation 7.3 Temporal discretisation error 8 Truncation error 8.1 Eigen series truncation 8.2 Iteration, ill-conditioning and round-off error 9 Uncertainty 9.1 Dealing with uncertainty 9.2 Minimising uncertainty 9.3 Sensitivity studies Appendix A : Natural coordinates and numerical integration Appendix B : Finite element solution example Appendix C : Eigensolution computation example

Presenter:      Jonathan Smith

Duration:          One day

 

Synopsis

The finite element method is a numerical technique for solving partial differential equations.

This course explains the fundamental mathematical procedures of the method and exposes the potential sources of error.

The course is aimed at both designers and engineers who wish to understand the fundamental formulation, characteristics and sources of error in the method. The mathematical content is clearly explained and illustrated.

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Finite element method, error and uncertainty

Verification, validation and quality management