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

Altering the thermal regime of soils below heated buildings in the continuous and discontinuous permafrost zones of Alaska

Thesis (Ph.D.) University of Alaska Fairbanks, 2016This research investigates the impacts of thermal insulation on the thermal regime of soils below heated buildings in seasonally and perennially frozen soils. The research provides practical answers (A) for designing frost‐protected shallow foundations in unfrozen soils of the discontinuous permafrost zone in Alaska and (B) shows that applying seasonal thermal insulation can reduce the risk of permafrost thawing under buildings with open crawl spaces, even in warming climatic conditions. At seasonal frost sites, this research extends frost‐protected shallow foundation applications by providing design suggestions that account for colder Interior Alaska’s air freezing indices down to 4 400 °C∙d (8,000 °F∙d). This research includes field studies at six Fairbanks sites, mathematical analyses, and finite element modeling. An appendix includes frost‐protected shallow foundation design recommendations. Pivotal findings include the discovery of more pronounced impacts from horizontal frost heaving forces than are likely in warmer climates. At permafrost sites, this research investigates the application of manufactured thermal insulation to buildings with open crawl spaces as a method to preserve soils in the frozen state. This research reports the findings from using insulation to reduce permafrost temperature, and increase the bearing capacity of permafrost soils. Findings include the differing thermal results of applying insulation on the ground surface in an open crawl space either permanently (i.e., left in place), or seasonally (i.e., applied in warm months and removed in cold months). Research includes fieldwork in Fairbanks, and finite element analyses for Fairbanks, Kotzebue, and Barrow. Pivotal findings show that seasonal thermal insulation effectively cools the permafrost. By contrast, Fairbanks, Kotzebue, and Barrow investigations show that permanently applied thermal insulation decreases the active layer, while also increasing (not decreasing) the permafrost temperature. Using seasonal thermal insulation, in a controlled manner, satisfactorily alters the thermal regime of soils below heated buildings and provides additional foundation alternatives for arctic buildings.Chapter 1: General Introduction -- 1.1 Investigating Practical Insulation Methods for Arctic Soils -- 1.2 One Hypothesis, Two Parts, Different Applications -- 1.2.1 Part A. Seasonal frost sites – directed heat confinement to keep footing soils thawed -- 1.2.2 Part B. Permafrost sites – restricting summer heat gain to keep soils cold -- 1.3 Investigation Methods for Both Parts -- Chapter 2: General Background for Both Parts -- 2.1 Forces in Frost Susceptible Soils -- 2.1.1 Basal freezing pressures -- 2.1.2 Tangential freezing stresses -- 2.2 Air Freezing Index -- 2.3 Geothermal Heat Flux -- Part A. Frost protected shallow foundations – for seasonal frost sites -- Chapter 3: Frost‐Protected Shallow Foundations -- 3.1 Introduction and Literature Review -- 3.2 Field Studies, Six Sites in the Seasonal Frost Zone -- 3.2.1 Insulation types and long‐term effective thermal resistivity -- 3.2.2 Six field locations -- 3.2.3 Monitoring methods -- 3.2.4 Field results and discussions -- 3.2.4.1 Timberland results and discussion -- 3.2.4.2 Merlin results and discussion -- 3.2.4.3 Goshawk results and discussion -- 3.2.4.4 Violin results and discussion -- 3.2.4.5 Bonita results and discussion -- 3.2.4.6 Army results and discussion -- 3.3 Conformal Mapping Analysis, Results and Discussion -- 3.4 Finite Element Modeling -- 3.4.1 Software program -- 3.4.2 Boundary conditions -- 3.4.3 Material properties -- 3.4.4 Modeling results and discussion -- 3.5 General Discussion for Frost‐Protected Shallow Foundations in Interior Alaska -- 3.5.1 Freezing isotherm shape is vertical and deep -- 3.5.3 Insulation discussion -- 3.5.4 Thermistor and temperature readout discussion -- 3.5.5 Field monitoring considerations -- 3.5.6 Other frost protected shallow foundation considerations -- 3.6 Part A – Pivotal Findings for Frost‐Protected Shallow Foundations in Interior Alaska -- 3.6.1 Using frost‐protected shallow foundation systems requires some cautions -- 3.6.2 Recommendations -- 3.6.3 Summary, frost‐protected shallow foundations for non‐permafrost sites in cold climate -- Part B. Permafrost protection – by restricting summer heat gain -- Chapter 4: Insulation Methods for Permafrost Zone -- 4.1 Introduction and Literature Review -- 4.1.1 Building distress concerns with warmer climates -- 4.1.2 Temperature dependent adfreeze bond may be unrepairable if broken -- 4.1.3 History and insulation methods for Arctic foundations -- 4.1.4 Current arctic foundation design methods -- 4.2 Analyses Without Buildings, Permafrost Zone -- 4.2.1 Means and methods for testing permafrost sites without buildings -- 4.2.1.1 Finite‐element thermal analyses -- 4.2.1.2 Boundary conditions -- 4.2.1.3 Current climate temperature input -- 4.2.1.4 Surface temperature adjustments (“n‐factors”) -- 4.2.1.5 Analyses startups and run times -- 4.2.2 Results for permafrost sites without buildings -- 4.2.2.1 Fairbanks -- 4.2.2.2 Kotzebue -- 4.2.2.3 Barrow -- 4.3 Analyses with Buildings in Place, Permafrost Zone -- 4.3.1 Means and methods for testing permafrost sites with buildings in place -- 4.3.2 Field study, one site, Willow House -- 4.3.3 Field study, results and discussion -- 4.3.3.1 Site work, thermistor discussion -- 4.3.3.2 Site work, insulation discussion -- 4.3.4 Thermal analyses by finite‐element program -- 4.3.5 Print screen results, Fairbanks -- 4.3.6 Graphic results and discussion for Fairbanks -- 4.3.6.1 Comparative results, center of building with edge of building -- 4.3.6.2 Comparative results, permanent or seasonal insulation, with or without snow -- 4.3.6.3 Fairbanks discussion -- 4.3.7 Graphic results and discussion for Kotzebue -- 4.3.8 Graphic results and discussion for Barrow -- 4.4 Climate Change Impacts -- 4.4.1 Testing means and methods for climate change -- 4.4.2 Results and discussion for changed climate -- 4.4.3 Important caution -- 4.5 Discussion, Multiple Investigations, Permafrost Protection via Thermal Insulation -- 4.5.1 Variability in n‐factors not investigated -- 4.5.2 Snow drifting not investigated -- 4.5.2 Insulation reduces surface thermal amplitude and reduces active layer depth -- 4.5.3 Possible additional usage for shallow footings founded on permafrost -- 4.5.4 Research considerations and uncertainties -- 4.6 Part B – Pivotal Findings for Permafrost Sites -- 4.6.1 Results and recommendations, permafrost sites, permanent thermal insulation -- 4.6.2 Results and recommendations, permafrost sites, seasonal thermal insulation -- 4.6.3 Results and recommendations, permafrost concerns for warming climate -- 4.6.4 Results and recommendations, open crawl space and snow removal -- 4.6.5 Summary, thermal insulation methods for permafrost sites -- Chapter 5: Conclusions -- 5.1 Frost‐Protected Shallow Foundations – for Seasonal Frost Sites -- 5.2 Seasonal Frost Sites – Pivotal Findings -- 5.3 Permafrost Protection ─ Summary -- 5.4 Permafrost Sites ─ Pivotal Findings -- 5.5 Future Studies -- References

Algebraic level sets for CAD/CAE integration and moving boundary problems

Boundary representation (B-rep) of CAD models obtained from solid modeling kernels are commonly used in design, and analysis applications outside the CAD systems. Boolean operations between interacting B-rep CAD models as well as analysis of such multi-body systems are fundamental operations on B-rep geometries in CAD/CAE applications. However, the boundary representation of B-rep solids is, in general, not a suitable representation for analysis operations which lead to CAD/CAE integration challenges due to the need for conversion from B-rep to volumetric approximations. The major challenges include intermediate mesh generation step, capturing CAD features and associated behavior exactly and recurring point containment queries for point classification as inside/outside the solid. Thus, an ideal analysis technique for CAD/CAE integration that can enable direct analysis operations on B-rep CAD models while overcoming the associated challenges is desirable. ^ Further, numerical surface intersection operations are typically necessary for boolean operations on B-rep geometries during the CAD and CAE phases. However, for non-linear geometries, surface intersection operations are non-trivial and face the challenge of simultaneously satisfying the three goals of accuracy, efficiency and robustness. In the class of problems involving multi-body interactions, often an implicit knowledge of the boolean operation is sufficient and explicit intersection computation may not be needed. Such implicit boolean operations can be performed by point containment queries on B-rep CAD models. However, for complex non-linear B-rep geometries, the point containment queries may involve numerical iterative point projection operations which are expensive. Thus, there is a need for inexpensive, non-iterative techniques to enable such implicit boolean operations on B-rep geometries. ^ Moreover, in analysis problems with evolving boundaries (ormoving boundary problems), interfaces or cracks, blending functions are used to enrich the underlying domain with the known behavior on the enriching entity. The blending functions are typically dependent on the distance from the evolving boundaries. For boundaries defined by free form curves or surfaces, the distance fields have to be constructed numerically. This may require either a polytope approximation to the boundary and/or an iterative solution to determine the exact distance to the boundary. ^ In this work a purely algebraic, and computationally efficient technique is described for constructing signed distance measures from Non-Uniform Rational B-Splines (NURBS) boundaries that retain the geometric exactness of the boundaries while eliminating the need for iterative and non-robust distance calculation. The proposed technique exploits the NURBS geometry and algebraic tools of implicitization. Such a signed distance measure, also referred to as the Algebraic Level Sets, gives a volumetric representation of the B-rep geometry constructed by purely non-iterative algebraic operations on the geometry. This in turn enables both the implicit boolean operations and analysis operations on B-rep geometries in CAD/CAE applications. Algebraic level sets ensure exactness of geometry while eliminating iterative numerical computations. Further, a geometry-based analysis technique that relies on hierarchical partition of unity field compositions (HPFC) theory and its extension to enriched field modeling is presented. The proposed technique enables direct analysis of complex physical problems without meshing, thus, integrating CAD and CAE. The developed techniques are demonstrated by constructing algebraic level sets for complex geometries, geometry-based analysis of B-rep CAD models and a variety of fracture examples culminating in the analysis of steady state heat conduction in a solid with arbitrary shaped three-dimensional cracks. ^ The proposed techniques are lastly applied to investigate the risk of fracture in the ultra low-k (ULK) dies due to copper (Cu) wirebonding process. Maximum damage induced in the interlayer dielectric (ILD) stack during the process steps is proposed as an indicator of the reliability risk. Numerical techniques based on enriched isogeometric approximations are adopted to model damage in the ULK stacks using a cohesive damage description. A damage analysis procedure is proposed to conduct damage accumulation studies during Cu wirebonding process. Analysis is carried out to identify weak interfaces and potential sites for crack nucleation as well as damage nucleation patterns. Further, the critical process condition is identified by analyzing the damage induced during the impact and ultrasonic excitation stages. Also, representative ILD stack designs with varying Cu percentage are compared for risk of fracture

Aeronautical engineering, a continuing bibliography with indexes

This bibliography lists 567 reports, articles and other documents introduced into the NASA scientific and technical information system in January 1984

Symmetries & tensor networks in two-dimensional quantum physics

The most general description of a quantum many-body system is given by a wave- function that lives in a Hilbert space with dimension exponential in the number of particles. This makes it extremely hard to study strongly correlated phenomena like the fractional quantum Hall effect and high-temperature superconductivity. Whenever interactions are sufficiently local and temperature is low, the system does not explore the full Hilbert space, but its ground state resides in the small corner of Hilbert space described by the area law. Containing little entanglement, the states can then be expressed as tensor networks, a family of wavefunctions with a polynomial number of parameters. On the one hand, tensor networks can be used as a variational manifold in nu- merical computations. On the other hand, they allow building model wavefunctions much like locality allows writing down physically realistic Hamiltonians. Besides allowing for an analytical treatment, these models grant access both to the physical and the entanglement degrees of freedom. This is particularly useful in classifying phases of matter. A large number of phases can be explained in terms of Landau’s symmetry-breaking paradigm. This framework, however, is not complete, as exemplified by the existence of phases with intrinsic topological order in two dimensions. It was a major conceptual advance when tensor networks could explain (non-chiral) topological phases as those where the symmetry resides in the entanglement degrees of freedom. The symmetries corresponding to those topological phases act as discrete, finite groups on the virtual degrees of freedom. The purpose of this Thesis is to generalize this program to include other symmetries. We investigate a class of tensor networks with continuous symmetries and find that they cannot describe gapped physics with a unique ground state. The abelian case is found to describe a non-Lorentz invariant phase transition point into a topologically ordered phase. The physics of the non- abelian case is that of a plaquette state that spontaneously breaks the translation symmetry of the lattice. The non-abelian PEPS arises as the ground state of a local parent Hamiltonian whose ground state manifold is completely characterized by the tensor network. In both cases, we find two types of corrections to the entanglement entropy: first there is a correction that is logarithmic in the size of the boundary and independent of the shape. A further correction depends only on the shape of the partition, imposing further restrictions on regions that are suffciently thin. Finally, we investigate symmetries that mix the virtual with the physical degrees of freedom and are furthermore anisotropic. Their physics is described by subsystem symmetry protected topological order. In particular, we focus on the entanglement entropy in the cluster phase and show that there is a universal constant correction to the entropy throughout the phase. This is important in the program of establishing the entanglement entropy as a detection mechanism for topologically ordered phases. We put forward a numerical algorithm to compute the correction and use it to discover a novel phase of matter in which the cluster phase is embedded.Die allgemeinste Beschreibung eines Quanten-Vielteilchensystems ergibt sich aus einer Wellenfunktion, die in einem Hilbert-Raum lebt, dessen Dimension exponentiell in der Anzahl der Teilchen ist. Dies macht es äußerst schwierig, stark korrelierte Phänomene wie den fraktionalen Quanten-Hall-Effekt und die Hochtemperatursupraleitung zu untersuchen. Wenn die Wechselwirkungen ausreichend lokal sind und die Temperatur niedrig ist, steht dem System nicht der gesamte Hilbert-Raum zur Verfügung. Sein Grundzustand befindet sich in der kleinen "Ecke" des Hilbert-Raums, die durch das area law beschrieben wird. Mit wenig Verschränkung können die Zustände dann als Tensornetzwerke ausgedrückt werden, eine Familie von Wellenfunktionen mit einer polynomiellen Anzahl von Parametern. Einerseits können Tensornetzwerke als variationelle Ansätze bei numerischen Berechnungen verwendet werden. Auf der anderen Seite ermöglichen sie das Erstellen von Modellwellenfunktionen. Diese Modelle ermöglichen nicht nur eine analytische Behandlung, sondern gewähren auch Zugang zu den physikalischen und den Verschränkungsfreiheitsgraden. Dies ist besonders nützlich bei der Klassifizierung von Phasen der Materie. Eine große Anzahl von Phasen kann mit Landaus Theorie der Symmetriebrechung erklärt werden. Diese Beschreibung ist jedoch nicht vollständig, was durch die Existenz von Phasen mit intrinsischer topologischer Ordnung in zwei Dimensionen veranschaulicht wird. Es war ein großer konzeptioneller Fortschritt, als Tensornetzwerke (nicht-chirale) topologische Phasen als solche identifizieren konnten, bei denen die Symmetrie in den Verschränkungsfreiheitsgraden liegt. Die diesen topologischen Phasen entsprechenden Symmetrien wirken als diskrete, endliche Gruppen auf den virtuellen Freiheitsgraden. Der Zweck dieser Arbeit ist es, dieses Programm auf andere Symmetrien zu verallgemeinern. Wir untersuchen eine Klasse von Tensornetzwerken mit kontinuierlichen Symmetrien und stellen fest, dass sie keine mit eindeutigen Grundzustand unter einer Energielücke beschreiben können. Der abelsche Fall beschreibt einen nicht-Lorentz-invarianten Phasenübergangspunkt in eine topologisch geordnete Phase. Die Physik des nicht-abelschen Falls ist die eines Plaquette-Zustands, der spontan die Translationssymmetrie des Gitters bricht. Der nicht-abelsche PEPS entsteht als Grundzustand eines lokalen \textit{parent}-Hamiltonians, dessen Grundzustandsunterraum vollständig durch das Tensornetzwerk beschrieben wird. In beiden Fällen finden wir zwei Arten von Korrekturen an der Verschränkungsentropie: Erstens gibt es eine Korrektur, die in der Größe der Grenze logarithmisch und unabhängig von der Form ist. Eine weitere Korrektur hängt nur von der Form des Schnitts ab ab, wodurch ausreichend dünne Bereiche weiter eingeschränkt werden. Schließlich untersuchen wir Symmetrien, die virtuelle und physikalische Freiheitsgraden mischen und darüber hinaus anisotrop sind. Ihre Physik wird durch topologische Ordnung beschrieben, die stabil ist solange bestimmte Subsystem-Symmetrien nicht gebrochen werden. Insbesondere konzentrieren wir uns auf die Verschränkungsentropie in der Clusterphase und zeigen, dass die Entropie in der gesamten Phase universell eine konstante Korrektur erhält. Dies ist wichtig im Programm zur Etablierung der Verschränkungsentropie als Detektionsmechanismus für topologisch geordnete Phasen. Wir schlagen einen numerischen Algorithmus vor, um die Korrektur zu berechnen und entdecken eine neue Phase der Materie, in die die Clusterphase eingebettet ist

Conceptual Model Summary Report Simulation Framework for Regional Geologic CO{sub 2} Storage Along Arches Province of Midwestern United States

A conceptual model was developed for the Arches Province that integrates geologic and hydrologic information on the Eau Claire and Mt. Simon formations into a geocellular model. The conceptual model describes the geologic setting, stratigraphy, geologic structures, hydrologic features, and distribution of key hydraulic parameters. The conceptual model is focused on the Mt. Simon sandstone and Eau Claire formations. The geocellular model depicts the parameters and conditions in a numerical array that may be imported into the numerical simulations of carbon dioxide (CO{sub 2}) storage. Geophysical well logs, rock samples, drilling logs, geotechnical test results, and reservoir tests were evaluated for a 500,000 km{sup 2} study area centered on the Arches Province. The geologic and hydraulic data were integrated into a three-dimensional (3D) grid of porosity and permeability, which are key parameters regarding fluid flow and pressure buildup due to CO{sub 2} injection. Permeability data were corrected in locations where reservoir tests have been performed in Mt. Simon injection wells. The final geocellular model covers an area of 600 km by 600 km centered on the Arches Province. The geocellular model includes a total of 24,500,000 cells representing estimated porosity and permeability distribution. CO{sub 2} injection scenarios were developed for on-site and regional injection fields at rates of 70 to 140 million metric tons per year

Aeronautical Engineering: A special bibliography with indexes, supplement 48

This special bibliography lists 291 reports, articles, and other documents introduced into the NASA scientific and technical information system in August 1974