Enrichment of the Boundary Element Method through the Partition of Unity Method for Fracture Analysis using Local and Global Formulations

The present thesis proposes an innovative technique of applying enrichment to the Boundary Element Method to allow accurate analysis of 2D crack problems. An overview of fracture mechanics is given, with particular emphasis given to numerical methods and the techniques used to extract the highly important stress intensity factors - a measure of the singularity of a crack tip. The Boundary Element Method framework is described and later, the implementation of the new technique of enrichment is defined in detail. Finally, several crack problems are used to verify the accuracy of the method where the results are shown to compare very favourably with other well-established numerical methods

Nonlinear Structural Analysis

Nonlinear structural analysis techniques for engine structures and components are addressed. The finite element method and boundary element method are discussed in terms of stress and structural analyses of shells, plates, and laminates

Integration of Polynomials Times Double Step Function in Quadrilateral Domains for XFEM Analysis

The numerical integration of discontinuous functions is an abiding problem addressed by various authors. This subject gained even more attention in the context of the extended finite element method (XFEM), in which the exact integration of discontinuous functions is crucial to obtaining reliable results. In this scope, equivalent polynomials represent an effective method to circumvent the problem while exploiting the standard Gauss quadrature rule to exactly integrate polynomials times step function. Certain scenarios, however, might require the integration of polynomials times two step functions (i.e., problems in which branching cracks, kinking cracks or crack junctions within a single finite element occur). In this context, the use of equivalent polynomials has been investigated by the authors, and an algorithm to exactly integrate arbitrary polynomials times two Heaviside step functions in quadrilateral domains has been developed and is presented in this paper. Moreover, the algorithm has also been implemented into a software library (DD_EQP) to prove its precision and effectiveness and also the proposed method’s ease of implementation into any existing computational software or framework. The presented algorithm is the first step towards the numerical integration of an arbitrary number of discontinuities in quadrilateral domains. Both the algorithm and the library have a wide application range, in addition to fracture mechanics, from mathematical computing of complex geometric regions, to computer graphics and computational mechanics

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

Application of the b-spline collocation method to a geometrically non-linear beam problem

Engineers are researching solutions to resolve many of today\u27s technical challenges. Numerical techniques are used to solve the mathematical models that arise in engineering problems. A numerical technique that is increasingly being used to solve mathematical models in engineering research is called the B-spline Collocation Method. The B-spline Collocation Method has a few distinct advantages over the Finite Element and Finite Difference Methods. The main advantage is that the B-spline Collocation Method efficiently provides a piecewise-continuous, closed form solution. Another advantage is that the B-spline Collocation Method procedure is very simple and easy to apply to many problems involving partial differential equations. The current research involves developing, and extensively documenting, a comprehensive, step-by-step procedure for applying the B-spline Collocation Method to the solution of Boundary Value problems. In addition, the current research involves applying the B-spline Collocation Method to solve the mathematical model that arises in the deflection of a geometrically nonlinear, cantilevered beam. The solution is then compared to a known solution found in the literature

Loads and aeroelasticity division research and technology accomplishments for FY 1985 and plans for FY 1986

The Langley Research Center Loads and Aeroelasticity Division's research accomplishments for FY85 and research plans for FY86 are presented. The rk under each branch (technical area) will be described in terms of highlights of accomplishments during the past year and highlights of plans for the current year as they relate to five year plans for each technical area. This information will be useful in program coordination with other government organizations and industry in areas of mutual interest

A Study Of Wave Equations In Five Dimensional Spacetimes With Computational Methods

Tez (Doktora) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 2008Thesis (PhD) -- İstanbul Technical University, Institute of Science and Technology, 2008Genel görelilikte kullanılan instantonlar, Yang-Mills denklemlerinin sonlu eylem çözümleri olan Yang-Mills instantonlarına karşılık gelen çözümlerdir. Weierstarss’ın genel yerel en-küçük yüzeyler çözümü, genel bir instanton metriği verir. Nutku helikoit metriği de bu genel metriğin helikoit en-küçük yüzeyine karşılık gelen özel bir durumudur. Dirac ve Laplace denklemleri dört boyutlu durumda Mathieu fonksiyonları cinsinden çözülebilir. Bir zaman koordinatı metriğe doğrudan eklenirse çözümler, literatürde yüksek boyutlu çözümlerde karşılaşılan çift konfluent Heun fonksiyonları olurlar. Bir dönüşüm yardımıyla Mathieu denkleminin tekillik yapısı elde edilir. Beş boyutlu durum, bu dönüşüm sayesinde Mathieu fonksiyonları cinsinden ifade edilebilir. Zaman koordinatından gelen ek terimle birlikte, radyal ve açısal kısımlar değişik sabitler içerdiğinden, dört boyutlu durumdaki gibi bir ilerletici yazmak için bu fonksiyonların toplanması oldukça zorlaşır. Metriğin orijinde bir eğrilik tekilliğine sahip olması, bu bölgenin dışarılanmasını gerektirir. Bu da uygun sınır koşullarının kullanımını önemli kılar. Tek sayılı boyutlarda, Atiyah, Patodi ve Singer tarafından tanımlanmış olan yerel olmayan spektral sınır koşulları, topolojik engeller sebebiyle zorunludur. Çift sayılı boyutlarda yerel sınır koşulları kullanılabilse de, Dirac operatörünün gama-5 ve yük eşleniği simetrilerinin korunması isteniyorsa yerel olmayan spektral sınır koşulları kullanılmalıdır. Bu problemde sınır koşulları uygulanırken Atiyah-Patodi-Singer formalizmi kullanılmıştır. Manifoldun sınırında yazılan denklemler, sınır tanımlanmadan yazılan denklemlerden daha tekildir ve bu da çözümü zorlaştırır. Bilgisayar, denklemlerin çıkarılması ve analizlerinde yoğun olarak kullanılmıştır. Newman-Penrose formalizmini kullanan bir Maple paketi, çalışmadaki analitik hesapları yapmak için geliştirilmiştir. Paket ayrıca instanton metrikleri için tam bir Newman-Penrose hesaplayıcısı olarak kullanılabilir.Instanton solutions of general relativity are the counterparts of Yang-Mills instantons which are finite-action solutions of the Yang-Mills equations. Nutku s helicoid metric is a special case which corresponds to the helicoid minimal surface of Weierstrass general local solution of minimal surfaces. Dirac and Laplace equations can be solved in terms of Mathieu functions in the four dimensional case. If a time coordinate is added to the metric, the solutions become double confluent Heun functions which are known to arise in higher dimensional solutions. Mathieu equation is obtained by a transformation. The main difference between the two cases is that, the constants of the radial and angular parts are different, modified by the presence of the new term coming from the time-dependence, which makes the summation of these functions to form the propagator quite difficult. The metric has a curvature singularity at the origin. Therefore, the usage of the appropriate boundary conditions is important. We excise the singular point. In odd dimensions, using the non-local spectral boundary conditions which are described by Atiyah, Patodi and Singer is obligatory because of topological obstructions. These boundary conditions are also the only ones which conserve gamma-5 and charge conjugation symmetries of the Dirac operator in even dimensions. The Atiyah-Patodi-Singer formalism is used to impose boundary conditions in this problem. The equations written on the boundary of the manifold are more singular than the ones in the bulk and this makes the solution more difficult. A Maple package using the Newman-Penrose formalism is developed for computations.DoktoraPh

Sound radiation from vibrating elastic structures of arbitrary shape

Imperial Users onl

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