Fast solution of problems with multiple load cases by using wavelet‐compressed boundary element matrices

Bucher, Henrique F.; Wrobel, L.C.; Mansur, Webe João; Magluta, Carlos

doi:10.1002/cnm.598

Cited by 11 publications

(4 citation statements)

References 22 publications

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“…The elements smaller than a thresholding parameter are discarded using hard thresholding. Coordinate format is used to store the location and value of these elements (Bucher et al, 2003).…”

Section: Dwt To Compress Bem Matricesmentioning

confidence: 99%

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A novel way of using fast wavelet transforms to solve dense linear systems arising from boundary element methods

Ebrahimnejad

Attarnejad

2009

Engineering Computations

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Purpose -The purpose of this paper is to introduce a novel approach to solving linear systems arising from applying a Boundary Element Method (BEM) to elasticity problems. Design/methodology/approach -The key idea is based on using wavelet transforms as a tool to change dense and fully populated matrices of BEM systems into sparse matrices. Wavelets are then used again to produce an algorithm to solve the resultant sparse linear systems. The wavelet transformation part of the method can be added as a black box to existing BEM codes. Findings -Numerical results focusing on the sensitivity of the solution for various physical variables to the thresholding parameters, and savings in computer time and memory are presented. The results show that the proposed method is efficient for large problems. Research limitations/implications -Application of the proposed method is restricted to problems with number of DOF equal to an integer power of 2. Originality/value -The novel algorithm to solve transformed algebraic linear equations uses NSform of the modified matrix, taking the advantage of the hierarchical nature of Multi-Resolution Analysis (MRA) to decompose a parent system into descendant systems with reduced size. These smaller systems are then solved iteratively using generalized minimal residual method.

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“…The elements smaller than a thresholding parameter are discarded using hard thresholding. Coordinate format is used to store the location and value of these elements (Bucher et al, 2003).…”

Section: Dwt To Compress Bem Matricesmentioning

confidence: 99%

“…Applying wavelet compression to BEM for Laplace equation was developed by Bucher, et al (2004Bucher, et al ( , 2003 and Bucher and Wrobel (2002). In the earlier work of authors, development of method to elasticity problems was presented (Ebrahimnejad, 2007;Ebrahimnejad and Attarnejad, 2009).…”

Section: Introductionmentioning

confidence: 99%

A novel way of using fast wavelet transforms to solve dense linear systems arising from boundary element methods

Ebrahimnejad

Attarnejad

2009

Engineering Computations

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“…In order to solve the aforementioned drawbacks of BEM and extend its application to real industrial problems, many techniques have been proposed so far in the literature. One can mention the well-known Fast Multipole Boundary Element Method (FM/BEM), (Liu (2009)), the Fast Wavelet/BEM, (Bucher et al (2003), Ntalaperas et al (2010)), the Precorrected Fast Fourier Transform (FFT) Accelerated BEM (Xiao et al (2012)) and the combination of Hierarchical Matrices along with Adaptive Cross Approximation (ACA)/BEM (Rjasanow and Steinbach (2007)).…”

Section: Fast Bem Techniques and Their Application To Large-scale Problemsmentioning

confidence: 99%

“…Although its use can be considered as a black box in a BEM code, appears the disadvantage of requiring the knowledge of the final system of algebraic equations the construction of which is in general computational expensive. More details can be found in the representative works of Bucher et al (2003), Ravnik et al (2009), Ebrahimnejad and Attarnejad (2010) and Wang et al (2013).…”

Section: Fast Bem Techniques and Their Application To Large-scale Problemsmentioning

confidence: 99%

Solution of large scale elastic problems with the boundary element method

Γκορτσάς¹

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Η Μέθοδος των Συνοριακών Στοιχείων (ΜΣΣ) είναι ιδανική για την επίλυση προβλημάτων που σχετίζονται με άπειρα εκτεινόμενα χωρία. Ένα τέτοιο πρόβλημα είναι η εκτίμηση χαμηλής συχνότητας θορύβου και του εύρους μικρο-σεισμικών κυμάτων που εκπέμπονται λόγω της λειτουργίας μιας μεγάλης ανεμογεννήτριας. Όμως λόγω του μεγέθους αυτού του προβλήματος, η συμβατή ΜΣΣ αδυνατεί να το επιλύσει διότι απαιτούνται χρονοβόροι υπολογισμοί και σχεδόν απαγορευτικές απαιτήσεις αποθήκευσης δεδομένων. Ως εκ τούτου ο πρώτος στόχος της παρούσας διατριβής είναι η ανάπτυξη μιας προηγμένης ΜΣΣ η οποία θα μπορεί να λύνει με ακρίβεια και ταχύτητα το εν λόγω πρόβλημα. Για το σκοπό αυτό προτείνεται μια ΜΣΣ η οποία βασίζει την διαχείριση του συστήματος αλγεβρικών εξισώσεων που παράγει σε μεθόδους Ιεραρχικών Μητρών (Hierarchical Matrices Method) σε συνδυασμό με αλγεβρικές χαμηλής τάξης προσέγγισης τεχνικές, γνωστές ως Adaptive Cross Approximation (ACA). Για την αποτελεσματική αλγεβρική συμπίεση των χρησιμοποιούμενων ιεραρχικών μητρώων, διάφορες ACA τεχνικές χρησιμοποιούνται και αξιολογούνται, ενώ για την ταχεία επίλυση του τελικού συστήματος αλγεβρικών εξισώσεων ένας προ επιλεγμένος GMRES αλγόριθμος διαδοχικών λύσεων (preconditioned GMRES iterative solver) χρησιμοποιείται. Σε επόμενο βήμα, η αναπτυχθείσα ACA /ΜΣΣ χρησιμοποιείται για την επίλυση ελαστικών, ακουστικών και αλληλεπίδρασης ελαστικών-ακουστικών προβλημάτων που σχετίζονται με την γέννεση θορύβου χαμηλής συχνότητας και επιφανειακών εδαφικών ταλαντώσεων από την λειτουργία μεγάλων ανεμογεννητριών. Η ταυτόχρονη μελέτη των ακουστικών και ελαστικών κυμάτων που παράγει μια ανεμογεννήτρια και η μεταξύ τους αλληλεπίδραση εμφανίζεται για πρώτη φορά στη διεθνή βιβλιογραφία.

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Application of a generic domain‐decomposition strategy to solve shell‐like problems through 3D BE models

Araújo

Silva

Telles

2006

Commun. Numer. Meth. Engng.

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SUMMARYEfficient integration algorithms and solvers specially devised for boundary-element procedures have been established over the last two decades. A good deal of quadrature techniques for singular and quasisingular boundary-element integrals have been developed and reliable Krylov solvers have proven to be advantageous when compared to direct ones, also in case of non-Hermitian matrices. The former has implied in CPU-time reduction during the assembling of the system of equations and the latter in its faster solution. Here, a triangular polar co-ordinate transformation and the Telles co-ordinate transformation are employed separately and combined to develop the matrix-assembly routines (integration routines). In addition, the Jacobi-preconditioned Biconjugate Gradient solver (J-BiCG) is used along with a generic substructuring boundary element algorithm. Thus, solution CPU time and computer memory can be considerably reduced. Discontinuous boundary elements are also included to simplify the coupling of the BE models (substructures). Numerical experiments involving 3D thin-walled domains (shell-like structural elements) are carried out to show the performance of the computer code with respect to accuracy and efficiency of the system solution. Precision, CPU-time and potential applications of the BE code developed are commented upon.

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Fast solution of problems with multiple load cases by using wavelet‐compressed boundary element matrices

Cited by 11 publications

References 22 publications

A novel way of using fast wavelet transforms to solve dense linear systems arising from boundary element methods

A novel way of using fast wavelet transforms to solve dense linear systems arising from boundary element methods

Solution of large scale elastic problems with the boundary element method

Application of a generic domain‐decomposition strategy to solve shell‐like problems through 3D BE models

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