## Periodic Report Summary 2 - WAVE PROPAGATION (Realistic computational modelling of large-scale wave propagation problems in unbounded domains)

Realistic computational modelling of large-scale wave propagation problems in unbounded domains

Peter Ruge, Carolin Birk, TU Dresden, Germany; Chongmin Song, UNSW, Australia

The correct prediction of the propagation of noise and vibration in soil or air is of great relevance with respect to environmental protection. Here, the numerical modelling of time-dependent processes in unbounded domains is required. This task belongs to the major research fields in computational mechanics. The overall aim of this project is to further develop a numerical method to represent unbounded media in transient analyses. It is hoped for numerical models, which represent radiation damping accurately, are local in time, and are suitable for the analysis of large-scale problems. To achieve the overall aim, the scaled boundary finite element method (SBFEM) is used to model unbounded domains. The SBFEM is a semi-analytical method which is formulated in the frequency-domain originally. The project is devoted to further developing the method. The work carried out is summarised as follows:

(1) The mixed-variables technique and the SBFEM have been combined in one programme. Elastic wave propagation in two and three dimensions, acoustic-wave propagation in a three-dimensional (3D) prismatic reservoir and diffusion in a prism embedded in half-space have been addressed.

(2) The concept of expanding the SBFE dynamic stiffness into a series of continued fractions has been further developed. This is based on the idea of using a doubly-asymptotic expansion for scalar wave propagation (S. Prempramote et al., Int. J. Numer. Meth. Eng. 2009, 79). The concept of doubly-asymptotic expansion was implemented for diffusion in a semi-infinite layer. The modelling of diffusion in unbounded domains leads to fractional differential equations in the time-domain. Considerable progress was made in developing a novel non-classical method for the solution of fractional differential equations. Further, the doubly-asymptotic expansion of the multi-degree-of freedom SBFE dynamic stiffness was developed and implemented for scalar waves in semi-infinite layered systems.

(3) The two approaches have been evaluated with respect to numerical performance. Very accurate approximations in the frequency-domain are obtained, but instability problems can occur for large systems. The reason of these numerical problems has been investigated by studying the scalar wave equation in spherical coordinates. The analytical solution of this problem reveals why the approaches mentioned earlier fail in certain situations. A modified doubly-asymptotic continued fraction expansion has been developed to overcome this problem and implemented for the scalar wave equation in spherical coordinates.

(4) An improved continued-fraction solution of the SBFE equations for multi-dimensional problems and vector waves has been developed. A matrix-valued scaling factor is introduced, which is chosen such that singularities are removed. The proposed procedure can be used for large-scale systems, no ill-conditioning is observed.

(5) Based on a novel coordinate transformation, a modified SBFEM has been derived, which is suitable for 3D layered systems with parallel top and bottom. The proposed method has been validated analysing various 3D foundations resting on or embedded in soil layers over rigid bedrock.

Conclusions

The novel concepts greatly extend the area of applicability of the SBFEM. Direct time-domain models of unbounded domains are obtained. The tools to analyse transient, large-scale multifield problems are thus available. The developed SBFEM has the potential be combined with commercial software in the future. A sophisticated computational tool for the prediction of traffic-induced noise and vibration in air and soil will then be available. These applications are of relevance with respect to the European noise policy. Thus, the research will have a notable long-term socio-economic impact on the civil society.

Peter Ruge, Carolin Birk, TU Dresden, Germany; Chongmin Song, UNSW, Australia

The correct prediction of the propagation of noise and vibration in soil or air is of great relevance with respect to environmental protection. Here, the numerical modelling of time-dependent processes in unbounded domains is required. This task belongs to the major research fields in computational mechanics. The overall aim of this project is to further develop a numerical method to represent unbounded media in transient analyses. It is hoped for numerical models, which represent radiation damping accurately, are local in time, and are suitable for the analysis of large-scale problems. To achieve the overall aim, the scaled boundary finite element method (SBFEM) is used to model unbounded domains. The SBFEM is a semi-analytical method which is formulated in the frequency-domain originally. The project is devoted to further developing the method. The work carried out is summarised as follows:

(1) The mixed-variables technique and the SBFEM have been combined in one programme. Elastic wave propagation in two and three dimensions, acoustic-wave propagation in a three-dimensional (3D) prismatic reservoir and diffusion in a prism embedded in half-space have been addressed.

(2) The concept of expanding the SBFE dynamic stiffness into a series of continued fractions has been further developed. This is based on the idea of using a doubly-asymptotic expansion for scalar wave propagation (S. Prempramote et al., Int. J. Numer. Meth. Eng. 2009, 79). The concept of doubly-asymptotic expansion was implemented for diffusion in a semi-infinite layer. The modelling of diffusion in unbounded domains leads to fractional differential equations in the time-domain. Considerable progress was made in developing a novel non-classical method for the solution of fractional differential equations. Further, the doubly-asymptotic expansion of the multi-degree-of freedom SBFE dynamic stiffness was developed and implemented for scalar waves in semi-infinite layered systems.

(3) The two approaches have been evaluated with respect to numerical performance. Very accurate approximations in the frequency-domain are obtained, but instability problems can occur for large systems. The reason of these numerical problems has been investigated by studying the scalar wave equation in spherical coordinates. The analytical solution of this problem reveals why the approaches mentioned earlier fail in certain situations. A modified doubly-asymptotic continued fraction expansion has been developed to overcome this problem and implemented for the scalar wave equation in spherical coordinates.

(4) An improved continued-fraction solution of the SBFE equations for multi-dimensional problems and vector waves has been developed. A matrix-valued scaling factor is introduced, which is chosen such that singularities are removed. The proposed procedure can be used for large-scale systems, no ill-conditioning is observed.

(5) Based on a novel coordinate transformation, a modified SBFEM has been derived, which is suitable for 3D layered systems with parallel top and bottom. The proposed method has been validated analysing various 3D foundations resting on or embedded in soil layers over rigid bedrock.

Conclusions

The novel concepts greatly extend the area of applicability of the SBFEM. Direct time-domain models of unbounded domains are obtained. The tools to analyse transient, large-scale multifield problems are thus available. The developed SBFEM has the potential be combined with commercial software in the future. A sophisticated computational tool for the prediction of traffic-induced noise and vibration in air and soil will then be available. These applications are of relevance with respect to the European noise policy. Thus, the research will have a notable long-term socio-economic impact on the civil society.