scholarly journals Solution of large scale elastic problems with the boundary element method

2017 ◽  
Author(s):  
Θεόδωρος Γκορτσάς
Keyword(s):  
Boundary Element Method ◽  
Boundary Element ◽  
Large Scale ◽  
Hierarchical Matrices ◽  
Iterative Solver ◽  
Adaptive Cross Approximation ◽  
Preconditioned Gmres ◽  
Element Method

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.

A Boundary Element Method Based on the Hierarchical Matrices and Multipole Expansion Theory for Acoustic Problems

2018 ◽  
Vol 15 (03) ◽  
pp. 1850009 ◽  
Author(s):  
Xiujuan Liu ◽  
Haijun Wu ◽  
Weikang Jiang
Keyword(s):  
Boundary Element Method ◽  
Boundary Element ◽  
Large Scale ◽  
Multipole Expansion ◽  
Hierarchical Matrices ◽  
Low Rank ◽  
Scale Model ◽  
Degenerate Kernel ◽  
Set Up ◽  
Element Method

The coefficient matrices of conventional boundary element method (CBEM) are dense and fully populated. Special techniques such as hierarchical matrices (H-matrices) format are required to extent its ability of handling large-scale problems. Adaptive cross approximation (ACA) algorithm is a widely adopted algorithm to obtain the H-matrices. However, the accuracy of the ACA boundary element method (ACABEM) cannot be adjusted by changing the tolerance [Formula: see text] when it exceeds a certain value. In this paper, the degenerate kernel approximation idea for the low-rank matrices is developed to build a fast BEM for acoustic problems by exploring the multipole expansion of the kernel, which is referred as the multipole expansion H-matrices boundary element method (ME-H-BEM). The newly developed algorithm compresses the far-field submatrices into low rank submatrices with the expansion terms of Green’s function. The obtained H-matrices are applied in conjunction with the generalized minimal residual method (GMRES) to solve acoustic problems. Numerical examples are carefully set up to compare the accuracy, efficiency as well as memory consumption of the CBEM, ACABEM, fast multipole boundary element method (FMBEM) and ME-H-BEM. The results of a pulsating sphere indicate that the ME-H-BEM keeps both storage and operation logarithmic-linear complexity of the H-matrices format as the ACABEM does. Moreover, the ME-H-BEM can achieve better convergence and higher accuracy than the ACABEM. For the analyzed complicated large-scale model, the ME-H-BEM with appropriate number of expansion terms has an advantage in terms of efficiency as compared with the ACABEM. Compared with the FMBEM, the ME-H-BEM is easier to be implemented.

Recent Development of the Fast Multipole Boundary Element Method for Modeling Acoustic Problems

2009 ◽  
Author(s):  
Yijun Liu ◽  
Milind Bapat
Keyword(s):  
Boundary Element Method ◽  
Acoustic Wave ◽  
Boundary Element ◽  
Large Scale ◽  
Boundary Integral ◽  
Practical Significance ◽  
Fast Multipole ◽  
Tree Structures ◽  
Wave Problems ◽  
Element Method

Some recent development of the fast multipole boundary element method (BEM) for modeling acoustic wave problems in both 2-D and 3-D domains are presented in this paper. First, the fast multipole BEM formulation for 2-D acoustic wave problems based on a dual boundary integral equation (BIE) formulation is presented. Second, some improvements on the adaptive fast multipole BEM for 3-D acoustic wave problems based on the earlier work are introduced. The improvements include adaptive tree structures, error estimates for determining the numbers of expansion terms, refined interaction lists, and others in the fast multipole BEM. Examples involving 2-D and 3-D radiation and scattering problems solved by the developed 2-D and 3-D fast multipole BEM codes, respectively, will be presented. The accuracy and efficiency of the fast multipole BEM results clearly demonstrate the potentials of the fast multipole BEM for solving large-scale acoustic wave problems that are of practical significance.

scholarly journals Multiscale Boundary Element Method for Acoustic Wave Model

MATEMATIKA ◽  
2019 ◽  
Vol 35 (3) ◽  
Author(s):  
Nor Afifah Hanim Zulkefli ◽  
Yeak Su Hoe ◽  
Munira Ismail
Keyword(s):  
Boundary Element Method ◽  
Numerical Methods ◽  
Acoustic Wave ◽  
Boundary Element ◽  
Computational Efficiency ◽  
Large Scale ◽  
Wave Model ◽  
Large Scale Problems ◽  
Speed Up ◽  
Element Method

In numerical methods, boundary element method has been widely used to solve acoustic problems. However, it suffers from certain drawbacks in terms of computational efficiency. This prevents the boundary element method from being applied to large-scale problems. This paper presents proposal of a new multiscale technique, coupled with boundary element method to speed up numerical calculations. Numerical example is given to illustrate the efficiency of the proposed method. The solution of the proposed method has been validated with conventional boundary element method and the proposed method is indeed faster in computation.

The Mixed Fast Multipole Boundary Element Method for Solving Strip Cold Rolling Process

2010 ◽  
Vol 20-23 ◽  
pp. 76-81 ◽  
Author(s):  
Hai Lian Gui ◽  
Qing Xue Huang
Keyword(s):  
Variational Inequality ◽  
Boundary Element Method ◽  
Boundary Element ◽  
Large Scale ◽  
Mixed Boundary ◽  
Contact Problems ◽  
Boundary Integral ◽  
Fast Multipole ◽  
Mixed Variational Inequality ◽  
Element Method

Based on fast multipole boundary element method (FM-BEM) and mixed variational inequality, a new numerical method named mixed fast multipole boundary element method (MFM-BEM) was presented in this paper for solving three-dimensional elastic-plastic contact problems. Mixed boundary integral equation (MBIE) was the foundation of MFM-BEM and obtained by mixed variational inequality. In order to adapt the requirement of fast multipole method (FMM), Taylor series expansion was used in discrete MBIE. In MFM-BEM the calculation time was significant decreased, the calculation accuracy and continuity was also improved. These merits of MFM-BEM were demonstrated in numerical examples. MFM-BEM has broad application prospects and will take an important role in solving large-scale engineering problems.

Application Incompatible Element in Mixed Fast Multipole Boundary Element Method

2010 ◽  
Vol 439-440 ◽  
pp. 80-85
Author(s):  
Hai Lian Gui ◽  
Qing Xue Huang ◽  
Ya Qin Tian ◽  
Zhi Bing Chu
Keyword(s):  
Boundary Element Method ◽  
Boundary Element ◽  
Large Scale ◽  
Fast Multipole ◽  
Mixed Variational Inequality ◽  
Calculation Time ◽  
Incompatible Elements ◽  
Improve Calculation ◽  
Compatible Elements ◽  
Element Method

Based on fast multipole boundary element method (FM-BEM) and mixed variational inequality, a new method named mixed fast multipole boundary element method (MFM-BEM) was presented in this paper. In order to improve calculation time and accuracy, incompatible elements as interpolation functions were used in the algorithm. Elements were optimized by mixed incompatible elements and compatible elements. On the one hand, the difficult to satisfy precise coordinate was avoided which caused by compatible elements; on the other hand, the merits of MFM-BEM were retained. Through analysis of example, it was conclusion that calculation time and accuracy were improved by MFM-BEM, calculation continuity was also better than traditional FM-BEM. With increasing of degree of freedom, calculation time of MFM-BEM grew slower than the time of traditional FM-BEM. So MFM-BEM provided a theoretical basis for solving large-scale engineering problems.

scholarly journals Efficient Boundary Element Method for Large Scale Corrosion Problem.

1999 ◽  
Vol 65 (635) ◽  
pp. 1493-1497 ◽  
Author(s):  
Kenji AMAYA ◽  
Naoki NARUSE ◽  
Shigeru AOKI ◽  
Matsuho MIYASAKA
Keyword(s):  
Boundary Element Method ◽  
Boundary Element ◽  
Large Scale ◽  
Corrosion Problem ◽  
Element Method
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