Numerical Simulation of Acoustic-Acoustics- and Structural Mechanics-Acoustics-Couplings on Nonmatching Grids

• Applicant: Professorin Dr. Barbara Wohlmuth
• Co-Applicant: Professor Manfred Kaltenbacher, Ph.D.
• Subject Area: Mechanics ; Acoustics
• Term: from 2006 to 2013
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 32173990

• Final Report Year: 2013

Final Report Abstract

In many technical applications a sensor or/and an actuator is immersed in an acoustic fluid, e.g., ultrasound transducers for non-destructive testing as well as therapy and electrodynamic loudspeakers. Quite often physical effects on different scales have to be captured in the mathematical and numerical model to obtain reliable simulation results. In order to keep as much flexibility as possible, the abstract framework of the Mortar Finite Element method (FEM) allows independently generated grids which are well suited to approximate the solution of the decoupled subproblems. To obtain a consistent formulation, suitable tearing and interconnecting concepts at the physically interfaces between the sub-models have to be designed. Originally, non-conforming mortar techniques have been introduced as a new discretization scheme and analyzed within the setting of the Laplace operator in 2D based on simple artificial subdomain decompositions. This research project has significantly contributed to push these techniques beyond the state of being a class of stable and optimal discretization schemes. The full potential of the method can be seen in complex coupled problems where the subdomain decomposition naturally stems from the multi-physics character of the problem. The first project period was more focussed on fundamental questions including new stability and optimal a priori results whereas the second period was mainly devoted to the successful co-design of stable algorithms handling non-matching meshes and highly complex multi-physics problems. The main achievements compared to the state of art at the date of the proposal can be summarized as follows. First of all, we have successfully applied the Mortar FEM to real complex engineering applications in 3D and illustrated the high potential of this method. The now possible individual meshing of the involved subdomains results in a considerably reduction of the unknowns and thus in the computational time. As has been shown, the range of possible multi-physics applications for which mortar techniques on non-matching meshes can be efficiently used is broad. As typical for acoustic applications, the problems are quite often formulated on unbounded domains, and thus suitable tools are required to handle it efficiently within the numerical simulation. Although a lot of research has been done before on the construction of absorbing boundary conditions and perfectly matched layer techniques, not too many results are available for non-linear wave equations. Within this project newly designed absorbing boundary conditions for the wave equation with temperature dependent speed of sound have been investigated. The multi-physics structure of the considered problems brings in effects on different time and space scales. To increase the efficiency of our possibly highly non-conforming meshes in space, we have been introducing a multi-time stepping approach for an ultrasound heating model problem. The combination of the three ingredients, non-matching meshes in space to handle different physical sub-models and space scales, absorbing boundary conditions to replace unbounded by bounded space domains and multi-time stepping algorithms to face different time scales, allows us to design efficient simulation strategies for a broad range of multi-physics applications in acoustics.

Publications

• Applications of the mortar finite element method in vibroacoustics and flow induced noise computations. Acta Acustica united with Acustica, 96(3):536–553(18), 2010
  S. Triebenbacher, M. Kaltenbacher, B. Wohlmuth, and B. Flemisch
  
• The equivalence of standard and mixed finite element methods in applications to elasto-acoustic interaction. SIAM J. Sci. Comput., 32:1980–2006, 2010
  B. Flemisch, M. Kaltenbacher, S. Triebenbacher, and B. Wohlmuth
  
• Non-matching grids for a flexible discretization in computational acoustics. Commun. Comput. Phys., 11(2):472– 488, 2011
  B. Flemisch, M. Kaltenbacher, S. Triebenbacher, and B. Wohlmuth
  
• A multi-time stepping integration method for the ultrasonic heating problem. ZAMM, 92:869–881, 2012
  I. Shevchenko, M. Kaltenbacher, and B. Wohlmuth
  
• Non-matching Grids for the Numerical Simulation of Problems a from Aeroacoustics and Vibroacoustics. PhD thesis, Alpen Adria Universität Klagenfurt, September 7, 2012
  S. Triebenbacher
  
• Self-adapting absorbing boundary conditions for the wave equation. Wave motion, 49(4):461–473, 2012
  I. Shevchenko and B. Wohlmuth
  
• Absorbing boundary conditions for a wave equation with a temperature dependent speed of sound. J. Comput. Acoust., 2013
  I. Shevchenko, M. Kaltenbacher, and B. Wohlmuth
  
• Aeroacoustics of the edge tone: 2D-3D coupling between CFD and CAA. Acta acustica united with Acustica, 99:245–259, 2013
  I. Vaik, G. Paál, M. Kaltenbacher, S. Triebenbacher, S. Becker, and I. Shevchenko