Noise levels have become an issue for urban communities for many years due to the rapid growth of air and ground traffic densities. Additionally to these noise sources, many other machines producing significant noise levels surround our daily activities and contribute to deterioration of quality of life. A quite large part of this noise is generated by vibrating structures and manufactures have considered the noise level of their products as a relevant design parameter. Therefore, the demand towards reliable and computational efficient numerical simulation programs is strongly growing, so that these tools can be used within a virtual prototyping development cycle.

In this course we present the state-of-the-art overview of numerical schemes to efficiently solve the acoustic conservation equations (unknowns are acoustic pressure and particle velocity) and the acoustic wave equation (pressure or acoustic potential formulation). Thereby, the different equations model both vibrational and or induced sound generation and its propagation. The course will contain both the physical / mathematical modelling, advanced numerical schemes, such as higher order and spectral finite elements elements as well as discontinuous Galerkin methods to solve the underlying partial differential equations together with their implementation focusing on relevant practical applications.

The lectures and the computer labs will allow the student to develop:

- a wide knowledge of numerical methods for hyperbolic differential equations

- the theoretical properties of numerical methods and the algorithms for solving various mathematical problems arising in acoustics;

and to

- critically apply advanced numerical models to mathematical problems stemming from real world engineering applications independently of they own field of specialization;

- motivate the choice of the adopted methods and tools;

- critical reasoning and interpretation of the results obtained in the light of the theory through the implementation and/or the use of the algorithms;

It is expected a critical knowledge and understanding and a fully comprehensive ability to distinguish the different situations and make reasoned choices and justify the procedures followed. It also expected correctness a rigorous exposition of the theory.

1. Mathematical models for acoustics. Review of mathematical modeling with partial (hyperbolic) differential equations. General Properties of Waves. One Dimensional Waves on a String. Waves in Elastic Solids. Waves in Ideal Fluids. Second-Order Differential models. Waves in thin Rods and Plates. Helmholtz’s equation. Elasto-acoustic coupling.

2. Finite elements for hyperbolic equations. Review of finite element formulation for the stationary case. Low order finite elements (FE) versus high order spectral elements (SE) discretization. Continuous Galerkin (CG) and Discontinuous Galerkin (DG) formulations. Analysis of the proposed method: stability, accuracy, dispersion and dissipation properties.

3. Multidimensional hyperbolic problems. Semidiscretization with finite elements of some relevant models in Section 1. Strong and weak treatment of the boundary conditions. Radiation conditions for problems in unbounded domains. Numerical treatment of boundary conditions for hyperbolic systems. Temporal discretization: the forward and backward Euler schemes, the upwind, Lax-Friedrichs and Lax-Wendroff schemes. Newmark and Runge-Kutta methods.

4. Nonlinear hyperbolic problems. Scalar equations and hyperbolic systems (Burgers and Euler equations). Approximation by DG finite elements. Choice of the numerical fluxes. Temporal discretization.

Labs sessions. Most of the methods presented during the lectures will be numerically investigated during the laboratory sessions based on employing computer codes.

Who is going to attend the course is expected to have rooted basis in mathematical analysis and some basic/medium knowledge of numerical analysis.

Evaluation

The course will offer lectures and computer labs. Computer labs will be focused on specific methodologies, case studies or practical applications discussed during lectures. Course attendance is warmly suggested. The exam consists of an oral discussion of homework exercises given during the lectures and on the main topics of the course.

Intended learning outcomes.

Students are expected to

- know and understand numerical methods for differential problems, as well as evaluate their properties;
- write and interpret algorithms;
- choose the most appropriate numerical scheme for solving a differential problem;
- solve mathematical problems with computer codes ;
- implement algorithms and use them appropriately;
- critically reason and interpret the results obtained in the light of the theory;
- model and numerically simulate Engineering problems.