72762 - Numerical Methods For Civil Engineering M

Learning outcomes

After the course the student has the knowledge of the most important computational tools in civil engineering, with particular reference to numerical methods for the solution of algebraic and differential systems.

Course contents

Modules 1-3

1-Definition of numerical algorithms and main sources of errors.

Sources of error in scientific computation. Representation of real numbers in a calculator. Floating point system. IEEE arithmetic, rounding errors. Floating-point operations. Calculation in finite arithmetic. phenomenon of numerical cancellation. Numerical algorithms and their properties.

Good position and conditioning a problem. Notes on the analysis of an approximate problem: stability, conditioning, convergence.

2-Numerical solution of nonlinear equations (and systems).

Nonlinear equations. Bisection method, Newton's method and secant method. Speed of convergence. Systems of nonlinear equations.

3- Numerical solution of linear systems

Direct methods: LU factorization algorithm. Solution of triangular systems. Pivoting (exchange of rows or columns). Symmetric and positive definite matrices. Cholesky decomposition. Conditioning. Stability and computational cost of algorithms. Iterative methods for linear systems: Jacobi and Gauss-Siedel. Convergence. Notes on Conjugate Gradient and Krylov Methods.

4- Elements of function approximations, numerical integration and numerical differentiation

Lagrange polynomial interpolation. Runge example. Composite polynomial interpolation: linear spline. Newton-Cotes formulas for numerical integration: midpoint formula, trapezium formula, Cavalieri-Simpson formula. Simple integration formulas and composite formulas, degree of precision. Numerical differentiation and Divided differences. Notes on the finite difference method. Consistency, Stability and Convergence. Lax theorem.

5-Approximation of functions and data

Approximation in the sense of least squares: existence and uniqueness. System of normal equations. Use and application of QR Factorization and SVD.

Module 4

4 1 - Virtual Work and Energy Principles in Mechanics

• The principle of virtual displacements

• The Principle of Minimum of Total Potential Energy

2 - Dynamical systems: Hamilton's principle

• Hamilton's principle for discrete systems

• Hamilton's principle for the continuum

3 - Direct Variational Methods

• Strong, Weak and Variational formulation

• The Ritz Method

• The Methods of Weighted Residuals (Petrov-Galerkin, Galerkin, Collocation, Least-Squares, Subdomains)

4 - The Finite Element Method

• 1D Isoparametric Elements

• 2D Isoparametric Elements

 

Readings/Bibliography

Slides and notes available on https://virtuale.unibo.it/

• A. QUARTERON, R. SACCO, F.SALERI MATEMATICA NUMERICA. SPRINGER-VERLAG ITALIA MILANO 2008
• C. D'Angelo, A. Quarteroni: Matematica Numerica Esercizi, Laboratori e Progetti, 1st Edition., 2010, Springer Italia. ISBN: 978-88-470-1639-2
• J.N. Reddy, Energy Principles and Variational Methods in Applied Mechanics, 3rd edition, Wiley, 2017. ISBN: 978-1-119-08737-3
• A.J.M. Ferreira, N. Fantuzzi, MATLAB Codes for Finite Element Analysis, 2nd Edition, Springer, 2020. ISBN: 978-3-030-47951-0

Teaching methods

The course involves practical work based on computational tools using Matlab.

Assessment methods

The exam consists of a written test divided into two parts.

The first part contains questions on the theoretical part and exercises that require the analysis and preparation of Matlab scripts (Modules 1-3). The test is aimed at verifying the acquisition of the knowledge required by the course program, with particular regard to the theoretical aspects and at assessing the skills acquired in solving mathematical problems using Matlab.

The second part consists of a written test relating to Module 4. The student can take this second test only if he has carried out all the exercises assigned during the course according to the methods communicated by the teacher.

Notes, codes and personal texts are not allowed during the exam.

The final grade is the weighted average of the two tests:

• evaluation of the first part (2/3)
• evaluation of the second part (1/3)

Teaching tools

Lectures, exercises, laboratory activities using Matlab.

Office hours

See the website of Margherita Porcelli

See the website of Fiorella Sgallari

See the website of Germana Landi

See the website of Nicholas Fantuzzi