Bezsíťové metody reprezentují alternativu ke standardním diskretizačním technikám, které pro svůj chod vyžadují síť. V posledních desetiletích bylo vynaloženo mnoho úsilí k ověření konkurenceschopnosti bezsíťových metod v různých inženýrských odvětvích. Diplomová práce je zaměřena na aplikaci vhodné bezsíťové metody ve výpočetní aeroakustice. Stěžejní část této práce se zabývá úlohami šíření zvuku, které lze modelovat pomocí linearizovaných Eulerových rovnic. Obecně lze tyto rovnice zařadit mezi lineární hyperbolické systémy. Pro úlohy aeroakustiky se jako vhodná bezsíťová metoda jeví Finite point method (FPM), která byla úspěšně použita pro řešení úloh dynamiky tekutin. Odvozením této metody a návrhy k dosažení vysoké přesnosti se věnuje další část práce. Úlohy šíření zvuku se známým řešením jsou testovány vlastním softwarem vyvinutým speciálně pro tyto účely.

Realistic and real-time computational simulation of biological organs (e.g., human kidneys, human liver) is a necessity when one tries to build a quality surgical simulator that can simulate surgical procedures involving these organs. Currently deformable models, spring-mass models, or finite element models are widely used to achieve the realistic simulations and/or the real-time performance. It is widely agreed that continuum mechanics based numerical techniques are preferred over deformable models or spring-mass models, but those techniques are computationally expensive and hence the higher accuracy offered by those numerical techniques come at the expense of speed. Hence there is a need to study the speed of different numerical techniques, while keeping an eye on the accuracy offered by those numerical techniques. Such studies are available for the Finite Element Method (FEM) but rarely available for the Boundary Element Method (BEM). Hence the present work aims to conduct a study on the viability of BEM for the real-time simulation of biological organs, and the present study is justified by the fact that BEM is considered to be inherently efficient when compared to mesh based techniques like FEM. A significant portion of literature on the real-time simulation of biological organs suggests the use of BEM to achieve better simulations. When one talks about the simulation of biological organs, one needs to have the geometry of a biological organ in hand. Geometry of biological organs of interest is not readily available many a times, and hence there is a need to extract the three dimensional (3D) geometry of biological organs from a stack of two dimensional (2D) scanned images. Software packages that can readily reconstruct 3D geometry of biological organs from 2D images are expensive. Hence, a novel procedure that requires only a few free software packages to obtain the geometry of biological organs from 2D image sequences is presented. The geometry of a pig liver is extracted from CT scan images for illustration purpose. Next, the three dimensional geometry of human kidney (left and right kidneys of male, and left and right kidneys of female) is obtained from the Visible Human Dataset (VHD). The novel procedure presented in this work can be used to obtain patient specific organ geometry from patient specific images, without requiring any of the many commercial software packages that can readily do the job. To carry out studies on the speed and accuracy of BEM, a source code for BEM is needed. Since the BEM code for 3D elasticity is not readily available, a BEM code that can solve 3D linear elastostatic problems without accounting for body forces is developed from scratch. The code comes in three varieties: a MATLAB version, a Fortran version (sequential version), and a Fortran version (parallelized version). This is the first free and open source BEM code for 3D elasticity. The developed code is used to carry out studies on the viability of BEM for the real-time simulation of biological organs, and a few representative problems involving kidneys and liver are found to give accurate solutions. The present work demonstrates that it is possible to simulate linear elastostatic behaviour in real-time using BEM without resorting to any type of precomputations, on a computer cluster by fully parallelizing the simulations and by performing simulations on different number of processors and for different block sizes. Since it is possible to get a complete solution in real-time, there is no need to separately prove that every type of cutting, suturing etc. can be simulated in real-time. Future work could involve incorporating nonlinearities into the simulations. Finally, a BEM based simulator may be built, after taking into account details like rendering.