Domain decomposition methods for domain composition purpose: Chimera, overset, gluing and sliding mesh methods

Domain composition methods (DCM) consist in obtaining a solution to a problem, from the formulations of the same problem expressed on various subdomains. These methods have therefore the opposite objective of domain decomposition methods (DDM). Indeed, in contrast to DCM, these last techniques are usually applied to matching meshes as their purpose consists mainly in distributing the work in parallel environments. However, they are sometimes based on the same methodology as after decomposing, DDM have to recompose. As a consequence, in the literature, the term DDM has many times substituted DCM. DCM are powerful techniques that can be used for different purposes: to simplify the meshing of a complex geometry by decomposing it into different meshable pieces; to perform local refinement to adapt to local mesh requirements; to treat subdomains in relative motion (Chimera, sliding mesh); to solve multiphysics or multiscale problems, etc. The term DCM is generic and does not give any clue about how the fragmented solutions on the different subdomains are composed into a global one. In the literature, many methodologies have been proposed: they are mesh-based, equation-based, or algebraic-based. In mesh-based formulations, the coupling is achieved at the mesh level, before the governing equations are assembled into an algebraic system (mesh conforming, Shear-Slip Mesh Update, HERMESH). The equation-based counterpart recomposes the solution from the strong or weak formulation itself, and are implemented during the assembly of the algebraic system on the subdomain meshes. The different coupling techniques can be formulated for the strong formulation at the continuous level, for the weak formulation either at the continuous or at the discrete level (iteration-by-subdomains, mortar element, mesh free interpolation). Although the different methods usually lead to the same solutions at the continuous level, which usually coincide with the solution of the problem on the original domain, they have very different behaviors at the discrete level and can be implemented in many different ways. Eventually, algebraic- based formulations treat the composition of the solutions directly on the matrix and right-hand side of the individual subdomain algebraic systems. The present work introduces mesh-based, equation-based and algebraicbased DCM. It however focusses on algebraic-based domain composition methods, which have many advantages with respect to the others: they are relatively problem independent; their implicit implementation can be hidden in the iterative solver operations, which enables one to avoid intensive code rewriting; they can be implemented in a multi-code environment

Unstructured and semi-structured hexahedral mesh generation methods

Discretization techniques such as the finite element method, the finite volume method or the discontinuous Galerkin method are the most used simulation techniques in ap- plied sciences and technology. These methods rely on a spatial discretization adapted to the geometry and to the prescribed distribution of element size. Several fast and robust algorithms have been developed to generate triangular and tetrahedral meshes. In these methods local connectivity modifications are a crucial step. Nevertheless, in hexahedral meshes the connectivity modifications propagate through the mesh. In this sense, hexahedral meshes are more constrained and therefore, more difficult to gener- ate. However, in many applications such as boundary layers in computational fluid dy- namics or composite material in structural analysis hexahedral meshes are preferred. In this work we present a survey of developed methods for generating structured and unstructured hexahedral meshes.Peer ReviewedPostprint (published version

Doctor of Philosophy

dissertationOne of the fundamental building blocks of many computational sciences is the construction and use of a discretized, geometric representation of a problem domain, often referred to as a mesh. Such a discretization enables an otherwise complex domain to be represented simply, and computation to be performed over that domain with a finite number of basis elements. As mesh generation techniques have become more sophisticated over the years, focus has largely shifted to quality mesh generation techniques that guarantee or empirically generate numerically well-behaved elements. In this dissertation, the two complementary meshing subproblems of vertex placement and element creation are analyzed, both separately and together. First, a dynamic particle system achieves adaptivity over domains by inferring feature size through a new information passing algorithm. Second, a new tetrahedral algorithm is constructed that carefully combines lattice-based stenciling and mesh warping to produce guaranteed quality meshes on multimaterial volumetric domains. Finally, the ideas of lattice cleaving and dynamic particle systems are merged into a unified framework for producing guaranteed quality, unstructured and adaptive meshing of multimaterial volumetric domains

Doctor of Philosophy

dissertationComputational simulation has become an indispensable tool in the study of both basic mechanisms and pathophysiology of all forms of cardiac electrical activity. Because the heart is comprised of approximately 4 billion electrically active cells, it is not possible to geometrically model or computationally simulate each individual cell. As a result computational models of the heart are, of necessity, abstractions that approximate electrical behavior at the cell, tissue, and whole body level. The goal of this PhD dissertation was to evaluate several aspects of these abstractions by exploring a set of modeling approaches in the field of cardiac electrophysiology and to develop means to evaluate both the amplitude of these errors from a purely technical perspective as well as the impacts of those errors in terms of physiological parameters. The first project used subject specific models and experiments with acute myocardial ischemia to show that one common simplification used to model myocardial ischemia-the simplest form of the border zone between healthy and ischemic tissue-was not supported by the experimental results. We propose a alternative approximation of the border zone that better simulates the experimental results. The second study examined the impact of simplifications in geometric models on simulations of cardiac electrophysiology. Such models consist of a connected mesh of polygonal elements and must often capture complex external and internal boundaries. A conforming mesh contains elements that follow closely the shapes of boundaries; nonconforming meshes fit the boundaries only approximately and are easier to construct but their impact on simulation accuracy has, to our knowledge, remained unknown. We evaluated the impact of this simplification on a set of three different forms of bioelectric field simulations. The third project evaluated the impact of an additional geometric modeling error; positional uncertainty of the heart in simulations of the ECG. We applied a relatively novel and highly efficient statistical approach, the generalized Polynomial Chaos-Stochastic Collocation method (gPC-SC), to a boundary element formulation of the electrocardiographic forward problem to carry out the necessary comprehensive sensitivity analysis. We found variations large enough to mask or to mimic signs of ischemia in the ECG

Seismic Analysis of Post-tensioned Gravity Dams using Scaled Boundary Finite Element Method

Dams are hydraulic structures built across rivers to create reservoirs, which provide essential services to society such as flood control, human water supply, and electricity generation. A dam shall be designed to ensure stability against overturning and sliding caused by the hydro-pressure of the reservoir. A common type of dam is the concrete gravity dam that mainly relies on its self-weight and resistance to sliding on the foundation to maintain its stability. Installing post-tensioned anchors (PTAs) is a practical and cost-effective technique in dam engineering. It provides an additional stabilizing force and improves the shear resistance at the dam-foundation interface. Seismic safety evaluation of post-tensioned concrete gravity dams is necessary for new dam designs or strengthened existing dams to guarantee that the structures will survive at specified seismic hazard levels. This thesis presents the development of an efficient numerical framework for the seismic analysis of post-tensioned concrete gravity dam-reservoir-foundation systems. This framework is realized by implementing the scaled boundary finite element method (SBFEM) in the well-known commercial FEM software ABAQUS as user elements (UEL). Polytope elements (polygonal elements in 2D and polyhedral elements in 3D) are as versatile as standard FEM solid elements, while they provide greater flexibility in mesh generation for bounded domains. Unbounded user elements (UEL) are derived to model wave propagation in far-fields. An unbounded UEL only requires discretization with a small number of faces at the near-field/far-field interface and can rigorously satisfy the radiation condition at infinity. The ABAQUS software enhanced with the UELs is employed for two-dimensional seismic analysis of gravity dams, overcoming the difficulties encountered in standard FEM, for example, local mesh refinement for geometrical features, generating matching interfacial meshes for weak joints, and simulation of anchor-structure interactions. The overall system consists of a near-field containing the dam body and its neighboring reservoir and foundation, and a far-field of the reservoir and foundation continua. The near-field dam and foundation are discretized as quadtree meshes assigned with polygonal UELs. Quadtree meshes allow rapid and smooth transitions in element size, which facilitates the local mesh refinement for dam lift joints, anchor boreholes, drainage systems, etc. An unbounded UEL represents the far-field foundation in terms of displacement unit-impulse response matrices. It captures free-field motions and transfers them as equivalent seismic inputs acting at the near-field/far-field interface. The reservoir is modeled by ABAQUS built-in acoustic elements. At the far end of the reservoir, a non-reflecting acoustic boundary embedded in ABAQUS is employed to satisfy the radiation condition of the unbounded reservoir. Comprehensive considerations have been taken in the numeral simulation of post-tensioned gravity dams, such as weak joint behaviors, anchor-structure interaction, and concrete damage. Weak joints in a concrete gravity dam, such as the dam-foundation interface and the dam lift joints, are the most likely places where the sliding and cracking occur. A cohesive-frictional contact scheme is utilized to simulate the non-linear behaviors of these weak joints. A PTA is usually grouted with the structure along a portion of the length, called bond length. At the grouting interface, the bond stress develops with the slippage between the anchor and structure, and then transfers the prestressing in the anchor to the structure. Cohesive elements connected with the anchor and structure are generated along the bond length to simulate the bond-slip interaction. A Mazars' damage evolution law for dynamic loading is applied to simulate the quasi-brittle behaviors of the concrete. To avoid mesh sensitivity, a partially regularized local damage model is introduced into this application. Automatic re-meshing algorithms to generate conforming interfacial meshes are developed for the sake of the simulation of interfacial problems. For the weak joints, the domains in contact are allowed to be discretized individually, and then the existing meshes at the interfaces are re-meshed to be node-to-node matching. The anchor is embedded automatically in the structure by inserting additional nodes into the existing structural meshes along the anchor layout. By duplicating the inserted nodes and connecting the duplicated nodes, beam elements conforming with structural meshes are formed naturally. These re-meshing procedures are easily operated on the polygonal meshes allowing arbitrary numbers of nodes and edges. Cohesive elements can be generated with the matching nodes at interfaces, and no constraints are required to connect them with the surrounding elements. The proposed approach is verified by performing seismic analysis of a post-tensioned gravity dam with simple geometry, and comparing the results obtained from the model using ABAQUS built-in elements. The advantages of the proposed approach in handling complex problems are demonstrated through dams with multiple inclined anchors. Applications of this method can be extended to three-dimensional cases, and composite materials with randomly spread fiber inclusions

Lattice cleaving: a multimaterial tetrahedral meshing algorithm with guarantees

pre-printWe introduce a new algorithm for generating tetrahedral meshes that conform to physical boundaries in volumetric domains consisting of multiple materials. The proposed method allows for an arbitrary number of materials, produces high-quality tetrahedral meshes with upper and lower bounds on dihedral angles, and guarantees geometric fidelity. Moreover, the method is combinatoric so its implementation enables rapid mesh construction. These meshes are structured in a way that also allows grading, to reduce element counts in regions of homogeneity. Additionally, we provide proofs showing that both element quality and geometric fidelity are bounded using this approach