Cross-Points in Domain Decomposition Methods with a Finite Element Discretization

Non-overlapping domain decomposition methods necessarily have to exchange Dirichlet and Neumann traces at interfaces in order to be able to converge to the underlying mono-domain solution. Well known such non-overlapping methods are the Dirichlet-Neumann method, the FETI and Neumann-Neumann methods, and optimized Schwarz methods. For all these methods, cross-points in the domain decomposition configuration where more than two subdomains meet do not pose any problem at the continuous level, but care must be taken when the methods are discretized. We show in this paper two possible approaches for the consistent discretization of Neumann conditions at cross-points in a Finite Element setting

Overlapping schwarz methods for isogeometric analysis

We construct and analyze an overlapping Schwarz preconditioner for elliptic problems discretized with isogeometric analysis. The preconditioner is based on partitioning the domain of the problem into overlapping subdomains, solving local isogeometric problems on these subdomains, and solving an additional coarse isogeometric problem associated with the subdomain mesh. We develop an $h$-analysis of the preconditioner, showing in particular that the resulting algorithm is scalable and its convergence rate depends linearly on the ratio between subdomain and \u201eoverlap sizes\u201d for fixed polynomial degree $p$ and regularity $k$ of the basis functions. Numerical results in two- and three-dimensional tests show the good convergence properties of the preconditioner with respect to the isogeometric discretization parameters $h, p, k$, number of subdomains $N$, overlap size, and also jumps in the coefficients of the elliptic operator

Grid generation for the solution of partial differential equations

A general survey of grid generators is presented with a concern for understanding why grids are necessary, how they are applied, and how they are generated. After an examination of the need for meshes, the overall applications setting is established with a categorization of the various connectivity patterns. This is split between structured grids and unstructured meshes. Altogether, the categorization establishes the foundation upon which grid generation techniques are developed. The two primary categories are algebraic techniques and partial differential equation techniques. These are each split into basic parts, and accordingly are individually examined in some detail. In the process, the interrelations between the various parts are accented. From the established background in the primary techniques, consideration is shifted to the topic of interactive grid generation and then to adaptive meshes. The setting for adaptivity is established with a suitable means to monitor severe solution behavior. Adaptive grids are considered first and are followed by adaptive triangular meshes. Then the consideration shifts to the temporal coupling between grid generators and PDE-solvers. To conclude, a reflection upon the discussion, herein, is given

A Velocity Decomposition Approach for Three-Dimensional Unsteady Viscous Flow at High Reynolds Number

A velocity decomposition method is developed for the solution of three-dimensional, unsteady flows. The velocity vector is decomposed into an irrotational component (viscous-potential velocity) and a vortical component (vortical velocity). The vortical velocity is selected so that it is zero outside of the rotational region of the flow field and the flow in the irrotational region can thus be solely described by the viscous- potential velocity. The formulation is devised to employ both the velocity potential and the Navier-Stokes-based numerical methods such that the field discretization required by the Navier-Stokes solver can be reduced to only encompass the rotational region of the flow field and the number of unknowns that are to be solved by the Navier-Stokes solver is greatly reduced. A higher-order boundary-element method is used to solve for the viscous potential by applying a viscous boundary condition to the body surface. The finite-volume method is used to solve for the total velocity on a reduced domain, using the viscous-potential velocity as the boundary condition on the extent of the domain. The viscous-potential velocity and the total velocity are time dependent due to the unsteadiness in the boundary layer and the wake. A two-way coupling algorithm is developed to tightly couple the two solution procedures in time. The velocity-decomposition-coupled solver developed in this work is used to solve three-dimensional, laminar and turbulent unsteady flows. For turbulent flows, the solver is applied for both Unsteady-Reynolds-Averaging-Navier-Stokes and Large-Eddy-Simulation computations. The solver is demonstrated to be capable of solving problems with realistic geometries. Preliminary results for 3D lifting flow are also presented. By using the velocity-decomposition-coupled solver, the solution can be determined on a greatly reduced domain and have the same accuracy as that calculated by a conventional Navier-Stokes solver on a large domain. For some test cases, the number of unknowns in the computational mesh is reduced up to 50%.PHDNaval Architecture & Marine EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/137005/1/chanyang_1.pd

Computational fracture modelling by an adaptive cracking particle method

Conventional element-based methods for crack modelling suffer from remeshing and mesh distortion, while the cracking particle method is meshless and requires only nodal data to discretise the problem domain so these issues are addressed. This method uses a set of crack segments to model crack paths, and crack discontinuities are obtained using the visibility criterion. It has simple implementation and is suitable for complex crack problems, but suffers from spurious cracking results and requires a large number of particles to maintain good accuracy. In this thesis, a modified cracking particle method has been developed for modelling fracture problems in 2D and 3D. To improve crack description quality, the orientations of crack segments are modified to record angular changes of crack paths, e.g. in 2D, bilinear segments replacing straight segments in the original method and in 3D, nonplanar triangular facets instead of planar circular segments, so continuous crack paths are obtained. An adaptivity approach is introduced to optimise the particle distribution, which is refined to capture high stress gradients around the crack tip and is coarsened when the crack propagates away to improve the efficiency. Based on the modified method, a multi-cracked particle method is proposed for problems with branched cracks or multiple cracks, where crack discontinuities at crack intersections are modelled by multi-split particles rather than complex enrichment functions. Different crack propagation criteria are discussed and a configurational-force-driven cracking particle method has been developed, where the crack propagating angle is directly given by the configuration force, and no decomposition of displacement and stress fields for mixed-mode fracture is required. The modified method has been applied to thermo-elastic crack problems, where the adaptivity approach is employed to capture the temperature gradients around the crack tip without using enrichment functions. Several numerical examples are used to validate the proposed methodology

ISOGEOMETRIC OVERLAPPING ADDITIVE SCHWARZ PRECONDITIONERS IN COMPUTATIONAL ELECTROCARDIOLOGY

In this thesis we present and study overlapping additive Schwarz preconditioner for the isogeometric discretization of reaction-diffusion systems modeling the heart bioelectrical activity, known as the Bidomain and Monodomain models. The cardiac Bidomain model consists of a degenerate system of parabolic and elliptic PDE, whereas the simplified Monodomain model consists of a single parabolic equation. These models include intramural fiber rotation, anisotropic conductivity coefficients and are coupled through the reaction term with a system of ODEs, which models the ionic currents of the cellular membrane. The overlapping Schwarz preconditioner is applied with a PCG accelerator to solve the linear system arising at each time step from the isogeometric discretization in space and a semi-implicit adaptive method in time. A theoretical convergence rate analysis shows that the resulting solver is scalable, optimal in the ratio of subdomain/element size and the convergence rate improves with increasing overlap size. Numerical tests in three-dimensional ellipsoidal domains confirm the theoretical estimates and additionally show the robustness with respect to jump discontinuities of the orthotropic conductivity coefficients