# Browsing by Subject "Boundary element methods"

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Item Boundary/finite element meshing from volumetric data with applications(2005) Zhang, Yongjie; Bajaj, ChandrajitShow more The main research work during my Ph.D. study is to extract adaptive and quality 2D (triangular or quadrilateral) meshes over isosurfaces and 3D (tetrahedral or hexahedral) meshes with isosurfaces as boundaries directly from volumetric imaging data. The software named LBIE-Mesher (Level Set Boundary Interior and Exterior Mesher) is developed. LBIE-Mesher generates 3D meshes for the volume interior to an isosurface, the volume exterior to an isosurface, or the interval volume between two isosurfaces. An algorithm has been developed to extract adaptive and quality 3D meshes directly from volumetric imaging data. The extracted tetrahedral meshes are extensively used in the Finite Element Method (FEM). A top-down octree subdivision coupled with the dual contouring method is used to rapidly extract adaptive 3D finite element meshes with correct topology from volumetric imaging data. The edge contraction and smoothing methods are used to improve the mesh quality. The main contribution is extending the dual contouring method to crack-free interval volume 3D meshing with feature sensitive adaptation. Compared to other tetrahedral extraction methods from imaging data, our method generates adaptive and quality 3D meshes without introducing any hanging nodes. Furthermore, another algorithm has been developed to extract adaptive and quality quadrilateral or hexahedral meshes directly from volumetric data. First, a bottom-up surface topology preserving octree-based algorithm is applied to select a starting octree level. Then the dual contouring method is used to extract a preliminary uniform quad/hex mesh, which is decomposed into finer quads/hexes adaptively without introducing any hanging nodes. The positions of all boundary vertices are recalculated to approximate the boundary surface more accurately. Mesh adaptivity can be controlled by a feature sensitive error function, the regions that users are interested in, or finite element calculation results. Finally, a relaxation based technique is deployed to improve mesh quality. Several demonstration examples are provided from a wide variety of application domains. An approach has been described to smooth the surface and improve the quality of surface/volume meshes with feature preserved using geometric flow. For triangular and quadrilateral surface meshes, the surface diffusion flow is selected to remove noise by relocating vertices in the normal direction, and the aspect ratio is improved with feature preserved by adjusting vertex positions in the tangent direction. For tetrahedral and hexahedral volume meshes, besides the surface vertex movement in the normal and tangent directions, interior vertices are relocated to improve the aspect ratio. Our method has the properties of noise removal, feature preservation and quality improvement of surface/volume meshes, and it is especially suitable for biomolecular meshes because the surface diffusion flow preserves sphere accurately if the initial surface is close to a sphere. A comprehensive approach has been proposed to construct quality meshes for imviii plicit solvation models of biomolecular structures starting from atomic resolution data in the Protein Data Bank (PDB). First, a smooth volumetric synthetic electron density map is constructed from parsed atomic location data of biomolecules in the PDB, using Gaussian isotropic kernels. An appropriate parameter selection is made for constructing an error bounded implicit solvation surface approximation to the Lee-Richards molecular surface. Next, a modified dual contouring method is used to extract triangular meshes for the molecular surface, and tetrahedral meshes for the volume inside or outside the molecule within a bounding sphere/box of influence. Finally, geometric flows are used to improve the mesh quality. Some of our generated meshes have been successfully used in finite element simulations. Techniques have been developed to generate an adaptive and quality tetrahedral finite element mesh of a human heart. An educational model and a patient-specific model are constructed. There are three main steps in our mesh generation: model acquisition, mesh extraction and boundary/material layer detection. (1) Model acquisition. Beginning from an educational polygonal model, we edit and convert it to volumetric gridded data. A component index for each cell edge and grid point is computed to assist the boundary and material layer detection. For the patient-specific model, some boundary points are selected from MRI images, and connected using cubic splines and lofting to segment the MRI data. Different components are identified. (2) Mesh extraction. We extract adaptive and quality tetrahedral meshes from the volumetric gridded data using our LBIE-Mesher. The mesh adaptivity is controlled by regions or using a feature sensitive error function. (3) Boundary/material layer detection. The boundary of each component and multiple material layers are identified and meshed. The extracted tetrahedral mesh of the educational model is being utilized in the analysis of cardiac fluid dynamics via immersed continuum method, and the generated patient-specific model will be used in simulating the electrical activity of the heart.Show more Item Fast algorithms for frequency domain wave propagation(2012-12) Tsuji, Paul Hikaru; Ying, Lexing; Ghattas, Omar N.; Engquist, Bjorn; Fomel, Sergey; Ren, KuiShow more High-frequency wave phenomena is observed in many physical settings, most notably in acoustics, electromagnetics, and elasticity. In all of these fields, numerical simulation and modeling of the forward propagation problem is important to the design and analysis of many systems; a few examples which rely on these computations are the development of metamaterial technologies and geophysical prospecting for natural resources. There are two modes of modeling the forward problem: the frequency domain and the time domain. As the title states, this work is concerned with the former regime. The difficulties of solving the high-frequency wave propagation problem accurately lies in the large number of degrees of freedom required. Conventional wisdom in the computational electromagnetics commmunity suggests that about 10 degrees of freedom per wavelength be used in each coordinate direction to resolve each oscillation. If K is the width of the domain in wavelengths, the number of unknowns N grows at least by O(K^2) for surface discretizations and O(K^3) for volume discretizations in 3D. The memory requirements and asymptotic complexity estimates of direct algorithms such as the multifrontal method are too costly for such problems. Thus, iterative solvers must be used. In this dissertation, I will present fast algorithms which, in conjunction with GMRES, allow the solution of the forward problem in O(N) or O(N log N) time.Show more Item Fast methods to model the response of fluid-filled fractures and estimate the fracture properties(2018-11-21) Alulaiw, Badr Abdullah; Sen, Mrinal K.; Spikes, Kyle T; Fomel, Sergey; Grand, Stephen P; Foster, DouglasShow more Estimation of fracture orientation and properties has become an important part of seismic reservoir characterization especially in unconventional reservoirs because of the crucial role of fractures in enhancing the permeability in tight reservoirs. The presence of fluid inside the fractures affects their seismic response. Using equivalent medium theories, seismic wave signatures such as Amplitude Variation with Offset and azimuth (AVOz), Normal Moveout (NMO) correction and shear waves splitting have been used to detect the presence of gas-filled and fluid-filled fractures. These methods, however, are unable to specify the type of fluid inside the fractures and cannot be used for thin beds and complex geology where the subsurface properties change laterally. Hence, modeling the seismic waveform using numerical methods is inevitable. The main limitation of those methods is their high computation costs. In this dissertation, I focus on developing two fast numerical methods to model the response of fluid-filled fractures as well as one fast global optimization method to estimate the fracture properties. Although local optimization methods are computationally cheap, the probability of being trapped in a local minimum becomes high when the initial model is not close to the global minimum especially when applied to highly nonlinear problems. Quantum Annealing (QA) is a recent global optimization method that was shown to be faster than Simulated Annealing (SA) in many situations. QA has been recently applied to geophysical problems. In this research, I modify QA by proposing using a new kinetic term that helps QA converge faster to the global minimum. With a synthetic dataset, I illustrate that QA is faster than Very Fast Simulated Annealing (VFSA) using a highly non-linear forward model that computes the response of seismic Amplitude Variation with Angle (AVA) for spherical waves. Most AVA inversion algorithms are based on plane wave solutions whereas seismic surveys use point sources to generate spherical waves. Although the plane wave solution is an excellent approximation for spherical waves, this approximation breaks down in the vicinity of the critical angle. Here, I implement an AVA inversion method for three parameters (P-wave velocity, S-wave velocity and density) based on analytical approximation for spherical waves. In addition, I apply this algorithm to a 2D seismic dataset from Cana field, Oklahoma with the primary objective of resolving the Woodford formation. I compare the results with those obtained by a local optimization method. The results clearly demonstrate superior performance of the proposed inversion method over that of local optimization. Specifically, the inverted images show clear delineation of the Woodford formation. For a reservoir containing vertical and rotationally invariant fractures, the linear slip model characterizes the reservoir using four properties: two elastic properties describing the isotropic host rock and two fracture properties – normal ΔN and tangential ΔT fracture weaknesses. This model, however, ignores the pore porosity effect on the anisotropy and hence the fracture properties might be inaccurate. In this work, I estimate the fracture properties as well as pore porosity using a new expression for the stiffness tensor for a porous fractured medium. I use the ray-Born approximation to calculate the seismic response of a laterally varying porous reservoir and QA to estimate the fracture properties. Using numerical experiments, I compare the inversion results from both unconstrained and constrained simultaneous (PP and PSV components) seismic inversion as well as constrained inversion using only the PP component. I explain the importance of including a constraint to mitigate the effect of the equivalence problem between ΔN and porosity. Unlike the unconstrained inversion, the estimated properties from the constrained inversion are acceptable. Also, I illustrate that the simultaneous constrained inversion is more robust than using the PP component alone. I apply this algorithm to a 3D multicomponent seismic dataset acquired in Saudi Arabia. The estimated fracture orientation agrees with those obtained in previous studies using borehole image logs, oriented cores, drilling observation and seismic in the same area. Also, the computed porosity using available well logs matches the inverted porosity very well. Computationally cheap analytical methods and equivalent medium theories available to model seismic wavefields diffracted by multiple fluid-filled fractures are not capable of handling complex fracture models or wave multi-scattering. Hence, using expensive numerical methods is inevitable. The advantages of boundary element method (BEM) over domain methods, such as finite difference and finite element methods, include the ease of handling irregular fracture geometry and reduction of the problem dimensions making the computation fast. Moreover, BEM models the complete wavefield including multiples, reverberations and refracted waves inside the fractures. The downside of BEM is that the computation cost increases rapidly whenever we increase the number of boundary elements making these methods computationally inefficient to model a large number of 2D cracks or 3D fractures. By combining the Indirect Boundary Element Method (IBEM) and a Generalized Born Series (GBS), I propose a new algorithm that can compute the response of 3D fluid-filled fracture sets effectively. In addition, when I consider equally spaced fractures that have the same geometry within a fracture set, computation can be performed even more rapidly. I compare the wavefield obtained using this approximation in five numerical experiments with those obtained from IBEM and show that the results are accurate in many situations.Show more Item Isogeometric Analysis for boundary integral equations(2015-12) Taus, Matthias Franz; Rodin, G. J. (Gregory J.); Hughes, Thomas J. R.; Demkowicz, Leszek F.; Biros, George; Sayas, Francisco JavierShow more Since its emergence, Isogeometric Analysis (IgA) has initiated a revolution within the field of Finite Element Methods (FEMs) for two reasons: (i) geometry descriptions originating from Computer Aided Design (CAD) can be used directly for analysis purposes, and (ii) the availability of smooth exact geometry descriptions and smooth basis functions can be used to develop new, highly accurate and highly efficient numerical methods. Whereas in FEMs the first issue is still open, it has already been shown that Isogeometric BEMs (IBEMs) provide a complete design-through-analysis framework. However, in contrast to FEMs, the effect of smoothness provided by IgA has not yet been explored in IBEMs. In this dissertation, we address this aspect of IgA. We show that the smoothness and exactness properties provided by the IgA framework can be used to design highly accurate and highly efficient BEMs which are not accessible with conventional BEMs. We develop Collocation IBEMs on piecewise smooth geometries. This allows us to show that IBEMs converge in the expected rates and result in system matrices with mesh-independent condition numbers. The latter property is particularly beneficial for large-scale problems that require iterative linear solvers. However, using conventional Collocation BEMs, this approach is not accessible because hyper-singular integrals have to be evaluated. In contrast, using Collocation IBEMs, the smoothness properties of the IgA framework can be used to regularize the hyper-singular integrals and reduce them to weakly singular integrals which can be evaluated using well-known techniques. We perform several numerical examples on canonical shapes to show these results. In addition, we use well-known mathematical results to develop a sound theoretical foundation to some of our methods, a result that is very rare for Collocation discretizations. Finally, using the exactness of IgA geometry descriptions, we design Patch Tests that allow one to rigorously test IBEM implementations. We subject our implementation to these Patch Tests which not only shows the reliability of our method but also shows that IBEMs can be as accurate as machine precision. We apply our IBEMs to Laplace's equation and the equations of linear elasticity. In addition, input files for our implementation can be automatically obtained from commercial CAD packages. These practical aspects allow us to apply IBEMs to analyze a propeller under a wind load.Show more