# Browsing by Subject "Isogeometric analysis"

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Item Advances in saccular aneurysm biomechanics : enlargement via rate-sensitive inelastic growth, bio-mathematical stages of aneurysm disease, and initiation profiles.(2016-12) Nugen, Frederick Theodore; Hughes, Thomas J. R.; Moser, Robert deLancey; Sacks, Michael S; Barr, Ronald; Gonzalez, Oscar; Kemper, Craig; Beasley, Haley KShow more I have created the first simulation of saccular aneurysm initiation and development from a healthy artery geometry. It is capable of growing saccular aneurysm geometries from patient-specific data. My model describes aneurysm behavior in a way that bridges fields. I assume arteries are made of a rate-sensitive inelastic material which produces irreversible deformation when it is overstressed. The material is assumed to consist of a 3D hyperelastic background material embedded with 1D transversely-isotropic fibers. I optionally use a Winkler foundation term to model support of external organs and distinguish healthy tissue from diseased tissue. Lesions are defined as a local degradation of artery wall structure. My work suggests passive mechanisms of growth are insufficient for predicting saccular aneurysms. Furthermore, I identify a new concept of stages of aneurysm disease. The stages connect mathematical descriptions of the simulation with clinically-relevant changes in the modeled aneurysm. They provide an evocative framework through which clinical descriptions of arteries can be neatly matched with mathematical features of the model. The framework gives a common language of concepts---e.g., collagen fiber, pseudoelastic limit, inelastic strain, and subclinical lesion---through which researchers in different fields, with different terminologies, can engage in an ongoing dialog: under the model, questions in medicine can be translated into equivalent questions in mathematics. A new stage of “subclinical lesion” has been identified, with a suggested direction for future biomechanics research into early detection and treatment of aneurysms. This stage defines a preclinical aneurysm-producing lesion which occurs before any artery dilatation. It is a stage of aneurysm development involving microstructural changes in artery wall makeup. Under the model, this stage can be identified by its reduced strength: its structural support is still within normal limits, but presumably would perform more poorly in ex vivo failure testing than healthy tissue from the same individual. I encourage clinicians and biomechanicians to measure elastin degradation, and to build detailed multiscale models of elastin degradation profiles as functions of aging and tortuosity; and similarly for basal tone. I hope such measurements will to lead to early detection and treatment of aneurysms. I give specific suggestions of biological tissue experiments to be performed for improving and reinforming constitutive modeling techniques.Show more Item Isogeometric analysis of phase-field models for dynamic brittle and ductile fracture(2012-08) Borden, Michael Johns; Hughes, Thomas J. R.; Ghattas, Omar; Landis, Chad M.; Ravi-Chandar, Krshnaswamy; Wheeler, Mary F.Show more To date, efforts to model fracture and crack propagation have focused on two broad approaches: discrete and continuum damage descriptions. The discrete approach incorporates a discontinuity into the displacement field that must be tracked and updated. Examples of this approach include XFEM, element deletion, and cohesive zone models. The continuum damage, or smeared crack, approach incorporates a damage parameter into the model that controls the strength of the material. An advantage of this approach is that it does not require interface tracking since the damage parameter varies continuously over the domain. An alternative approach is to use a phase-field to describe crack propagation. In the phase-field approach to modeling fracture the problem is reformulated in terms of a coupled system of partial differential equations. A continuous scalar-valued phase-field is introduced into the model to indicate whether the material is in the unfractured or fractured ''phase''. The evolution of the phase-field is governed by a partial differential equation that includes a driving force that is a function of the strain energy of the body in question. This leads to a coupling between the momentum equation and the phase-field equation. The phase-field model also includes a length scale parameter that controls the width of the smooth approximation to the discrete crack. This allows discrete cracks to be modeled down to any desired length scale. Thus, this approach incorporates the strengths of both the discrete and continuum damage models, i.e., accurate modeling of individual cracks with no interface tracking. The research presented in this dissertation focuses on developing phase-field models for dynamic fracture. A general formulation in terms of the usual balance laws supplemented by a microforce balance law governing the evolution of the phase-field is derived. From this formulation, small-strain brittle and large-deformation ductile models are then derived. Additionally, a fourth-order theory for the phase-field approximation of the crack path is postulated. Convergence and approximation results are obtained for the proposed theories. In this work, isogeometric analysis, and particularly T-splines, plays an important role by providing a smooth basis that allows local refinement. Several numerical simulations have been performed to evaluate the proposed theories. These results show that phase-field models are a powerful tool for predicting fracture.Show more Item T-splines as a design-through-analysis technology(2011-08) Scott, Michael Andrew; Hughes, Thomas J. R.; Sederberg, Thomas W.; Taylor, Robert L.; Ghattas, Omar; Landis, Chad M.; Ying, LexingShow more To simulate increasingly complex physical phenomena and systems, tightly integrated design-through-analysis (DTA) tools are essential. In this dissertation, the complementary strengths of isogeometric analysis and T-splines are coupled and enhanced to create a seamless DTA framework. In all cases, the technology de- veloped meets the demands of both design and analysis. In isogeometric analysis, the smooth geometric basis is used as the basis for analysis. It has been demonstrated that smoothness offers important computational advantages over standard finite elements. T-splines are a superior alternative to NURBS, the current geometry standard in computer-aided design systems. T-splines can be locally refined and can represent complicated designs as a single watertight geometry. These properties make T-splines an ideal discretization technology for isogeometric analysis and, on a higher level, a foundation upon which unified DTA technologies can be built. We characterize analysis-suitable T-splines and develop corresponding finite element technology, including the appropriate treatment of extraordinary points (i.e., unstructured meshing). Analysis-suitable T-splines form a practically useful subset of T-splines. They maintain the design flexibility of T-splines, including an efficient and highly localized refinement capability, while preserving the important analysis-suitable mathematical properties of the NURBS basis. We identify Bézier extraction as a unifying paradigm underlying all isogeometric element technology. Bézier extraction provides a finite element representation of NURBS or T-splines, and facilitates the incorporation of T-splines into existing finite element programs. Only the shape function subroutine needs to be modified. Additionally, Bézier extraction is automatic and can be applied to any T-spline regardless of topological complexity or polynomial degree. In particular, it represents an elegant treatment of T-junctions, referred to as "hanging nodes" in finite element analysis We then detail a highly localized analysis-suitable h-refinement algorithm. This algorithm introduces a minimal number of superfluous control points and preserves the properties of an analysis-suitable space. Importantly, our local refinement algorithm does not introduce a complex hierarchy of meshes. In other words, all local refinement is done on one control mesh on a single hierarchical “level” and all control points have similar influence on the shape of the surface. This feature is critical for its adoption and usefulness as a design tool. Finally, we explore the behavior of T-splines in finite element analysis. It is demonstrated that T-splines possess similar convergence properties to NURBS with far fewer degrees of freedom. We develop an adaptive isogeometric analysis framework which couples analysis-suitable T-splines, local refinement, and Bézier extraction and apply it to the modeling of damage and fracture processes. These examples demonstrate the feasibility of applying T-spline element technology to very large problems in two and three dimensions and parallel implementations.Show more Item Thermodynamically consistent modeling and simulation of multiphase flows(2014-12) Liu, Ju; Hughes, Thomas J. R.Show more Multiphase flow is a familiar phenomenon from daily life and occupies an important role in physics, engineering, and medicine. The understanding of multiphase flows relies largely on the theory of interfaces, which is not well understood in many cases. To date, the Navier-Stokes-Korteweg equations and the Cahn-Hilliard equation have represented two major branches of phase-field modeling. The Navier-Stokes-Korteweg equations describe a single component fluid material with multiple states of matter, e.g., water and water vapor; the Cahn-Hilliard type models describe multi-component materials with immiscible interfaces, e.g., air and water. In this dissertation, a unified multiphase fluid modeling framework is developed based on rigorous mathematical and thermodynamic principles. This framework does not assume any ad hoc modeling procedures and is capable of formulating meaningful new models with an arbitrary number of different types of interfaces. In addition to the modeling, novel numerical technologies are developed in this dissertation focusing on the Navier-Stokes-Korteweg equations. First, the notion of entropy variables is properly generalized to the functional setting, which results in an entropy-dissipative semi-discrete formulation. Second, a family of quadrature rules is developed and applied to generate fully discrete schemes. The resulting schemes are featured with two main properties: they are provably dissipative in entropy and second-order accurate in time. In the presence of complex geometries and high-order differential terms, isogeometric analysis is invoked to provide accurate representations of computational geometries and robust numerical tools. A novel periodic transformation operator technology is also developed within the isogeometric context. It significantly simplifies the procedure of the strong imposition of periodic boundary conditions. These attributes make the proposed technologies an ideal candidate for credible numerical simulation of multiphase flows. A general-purpose parallel computing software, named PERIGEE, is developed in this work to provide an implementation framework for the above numerical methods. A comprehensive set of numerical examples has been studied to corroborate the aforementioned theories. Additionally, a variety of application examples have been investigated, culminating with the boiling simulation. Importantly, the boiling model overcomes several challenges for traditional boiling models, owing to its thermodynamically consistent nature. The numerical results indicate the promising potential of the proposed methodology for a wide range of multiphase flow problems.Show more