Fast multipole boundary element method for solving large ......called octree structure, which...
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Lehrstuhl für Akustik mobiler Systeme Fakultät für Maschinenwesen Technische Universität München Bachelor/Term/Master’s thesis MW or Interdisciplinary Project (IDP) Fast multipole boundary element method for solving large-scale acoustic wave problems The boundary element method (BEM) is a powerful computational technique, pro- viding numerical solutions to a range of scientific and engineering problems. In the field of linear acoustics it is an important alternative to more traditional methods li- ke the finite element method (FEM). This is especially true for exterior problems, whe- re the acoustic domain such as the open air or the ocean is so large it is acceptable to model it to be infinite in extent. Applying domain methods to such a problem clearly requires some careful thought, since the Fig. 1: Total surface pressure of submarine for 200Hz plane wave incidence at broadside calculated by FMBEM. D. R. Wilkes, PhD thesis, Curtin University, 2014 infinite domain doesn’t allow simple discretisation. BEM on the contrary only requires the discretisation of the radiating boundary leading to an easy to prepare surface mesh with highly reduced degrees of freedom. However, the reduced dimensionality comes at the cost of mayor drawbacks: The coefficient matrices of the BEM equation are non symmetric, fully populated and complex valued resulting in high computational costs and storage requirements. The fast multipole method (FMM) can reduce both, the memory requirement and expedite the calculation of the conventional BEM. This can be achieved by calculating interactions between well-separated groups of elements instead of pair-wise interactions and by determining the result of the matrix-vector multiplications without needing to explicitly form the full coefficient matrix. The aim of this thesis will be incorporating the fast multipole method in- to the in-house BEM code. First the 3D region of space occupied by the boundary surface needs to be systematically discretised using a so- called octree structure, which recursively subdivides each domain region into eight smaller regions in form of unit cubes; the receiver and source boxes. To combine pre-existing groups or to reduce the spatial range of not well-seperated sets the implementation of translation algorithms will be required. Last but not least the interactions between the source and receiver locations must be calculated. This can be done by factorising the fundamental solution of the differential equation, introducing an interme- diate point between the source and receiver. This factorisation separates the direct interaction between two boundary elements, making it possi- ble to independently calculate both components, which take the form of multipole series expansions. Fig. 2: Level 3 octree: Source expansions for well-separated boxes (light grey) are translated (black lines) to receiver box (dark grey). D. R. Wilkes, PhD thesis, Curtin University, 2014 Tasks • Literature research on FMM for BEM • Getting familiar with BEM theory, the in-house BEM code (Python) and an available FMBEM code (MATLAB) • Implementation of FMM including octree discretisation, translation algorithms, mulitpole expansion and truncation • Implementation and visualisation of introduced error, speed and sto- rage improvement • Testing the algorithms for large scale acoustic applications Prerequisites • Interest in numerical methods, own initiative, reliable, inde- pendent way of working • Programming experience (ideally Python) • Optional: knowledge in acoustics, fast multipole methods • Optional: experience with git, paraView, LRZ computing Interested? Please send your CV and transcript of records to Simone Preuss, M. Sc. ([email protected]). The thesis can start immediately and can be done in either german or english. In case you are interested in an IDP, the topic can be extended for up to 3 students. A suitable accompanying lecture might be Computational Acoustics.
Transcript of Fast multipole boundary element method for solving large ......called octree structure, which...
Lehrstuhl für Akustik mobiler SystemeFakultät für MaschinenwesenTechnische Universität München
Bachelor/Term/Master’s thesis MW or Interdisciplinary Project (IDP)
Fast multipole boundary element method for solving large-scaleacoustic wave problems
The boundary element method (BEM) isa powerful computational technique, pro-viding numerical solutions to a range ofscientific and engineering problems. In thefield of linear acoustics it is an importantalternative to more traditional methods li-ke the finite element method (FEM). This isespecially true for exterior problems, whe-re the acoustic domain such as the openair or the ocean is so large it is acceptableto model it to be infinite in extent. Applyingdomain methods to such a problem clearlyrequires some careful thought, since the
Fig. 1: Total surface pressure of submarine for 200Hz plane wave incidence atbroadside calculated by FMBEM. D. R. Wilkes, PhD thesis, Curtin University, 2014
infinite domain doesn’t allow simple discretisation. BEM on the contrary only requires the discretisation ofthe radiating boundary leading to an easy to prepare surface mesh with highly reduced degrees of freedom.However, the reduced dimensionality comes at the cost of mayor drawbacks: The coefficient matrices of theBEM equation are non symmetric, fully populated and complex valued resulting in high computational costsand storage requirements. The fast multipole method (FMM) can reduce both, the memory requirement andexpedite the calculation of the conventional BEM. This can be achieved by calculating interactions betweenwell-separated groups of elements instead of pair-wise interactions and by determining the result of thematrix-vector multiplications without needing to explicitly form the full coefficient matrix.
The aim of this thesis will be incorporating the fast multipole method in-to the in-house BEM code. First the 3D region of space occupied bythe boundary surface needs to be systematically discretised using a so-called octree structure, which recursively subdivides each domain regioninto eight smaller regions in form of unit cubes; the receiver and sourceboxes. To combine pre-existing groups or to reduce the spatial range ofnot well-seperated sets the implementation of translation algorithms willbe required. Last but not least the interactions between the source andreceiver locations must be calculated. This can be done by factorising thefundamental solution of the differential equation, introducing an interme-diate point between the source and receiver. This factorisation separatesthe direct interaction between two boundary elements, making it possi-ble to independently calculate both components, which take the form ofmultipole series expansions.
Fig. 2: Level 3 octree: Source expansionsfor well-separated boxes (light grey) aretranslated (black lines) to receiver box(dark grey). D. R. Wilkes, PhD thesis, Curtin University, 2014
Tasks• Literature research on FMM for BEM• Getting familiar with BEM theory, the in-house BEM code (Python)
and an available FMBEM code (MATLAB)• Implementation of FMM including octree discretisation, translation
algorithms, mulitpole expansion and truncation• Implementation and visualisation of introduced error, speed and sto-
rage improvement• Testing the algorithms for large scale acoustic applications
Prerequisites• Interest in numerical methods, own initiative, reliable, inde-
pendent way of working• Programming experience (ideally Python)• Optional: knowledge in acoustics, fast multipole methods• Optional: experience with git, paraView, LRZ computing
Interested? Please send your CV and transcript of records to Simone Preuss, M. Sc. ([email protected]). The thesis can startimmediately and can be done in either german or english. In case you are interested in an IDP, the topic can be extended for up to 3 students. Asuitable accompanying lecture might be Computational Acoustics.
