NURBS-Enhanced Finite Element Method (NEFEM)

The development of NURBS-Enhanced Finite Element Method (NEFEM) is revisited. This technique allows a seamless integration of the CAD boundary representation of the domain and the finite element method (FEM). The importance of the geometrical model in finite element simulations is addressed and the benefits and potential of NEFEM are discussed and compared with respect to other curved finite element techniques

NURBS-Enhanced Finite Element Method (NEFEM)

The development of NURBS-Enhanced Finite Element Method (NEFEM) is revisited. This technique allows a seamless integration of the CAD boundary representation of the domain and the finite element method (FEM). The importance of the geometrical model in finite element simulations is addressed and the benefits and potential of NEFEM are discussed and compared with respect to other curved finite element techniques.Peer ReviewedPostprint (published version

The treatment of corner and pole-type singularities in numerical conformal mapping techniques

This paper is a report of recent developments concerning the nature and the treatment of singularities that affect certain numerical conformal mapping techniques. The paper also includes some new results on the nature of singularities that the mapping function may have in the complement of the closure of the domain under consideration

On the spine of a PDE surface

yesThe spine of an object is an entity that can characterise the object¿s topology and describes the object by a lower dimension. It has an intuitive appeal for supporting geometric modelling operations. The aim of this paper is to show how a spine for a PDE surface can be generated. For the purpose of the work presented here an analytic solution form for the chosen PDE is utilised. It is shown that the spine of the PDE surface is then computed as a by-product of this analytic solution. This paper also discusses how the of a PDE surface can be used to manipulate the shape. The solution technique adopted here caters for periodic surfaces with general boundary conditions allowing the possibility of the spine based shape manipulation for a wide variety of free-form PDE surface shapes

Structural Response Analyses of Piezoelectric Composites using NURBS

Variational method deduced on the basis of the minimum potential energy is an efficient method to find solutions for complex engineering problems. In structural mechanics, the potential energy comprises strain energy, kinetic energy and the work done by external actions. To obtain these, the displacement are required as a priori. This research is concerned with the development of a numerical method based on variational principles to analyze piezoelectric composite plates and solids. A Non-Uniform Rational B-Spline (NURBS) function is used for describing both the geometry and electromechanical displacement fields. Two dimensional plate models are formulated according to the first order shear deformable plate theory for mechanical displacement. The electric potential varies non-linearly through the thickness, this variation is modelled by a discrete layer-wise linear variation. The matrix equations of motion are reported for piezoelectric sensors, actuator, and power harvester. Normal mode summation technique is applied to study the frequency response of displacement, voltage and the power output. A full three dimensional model is also developed to study the dynamics of piezoelectric sandwich structures. Simulations are provided for thick plates using plate theory and three dimensional models to verify the applicability of those theories in their regime. Newmark’s direct integration technique and a fourth order Runge-Kutta method were used to study the transient vibration. The variational method developed in this thesis can be applied to other structural mechanics problem

Vector Geometry and Applications to Three-Dimensional Computer Graphics

The mathematics behind algorithms involved in generating three-dimensional images on a computer has stemmed from the analysis of the processes of sight and vision. These processes have been modeled to provide methods of visualising three-dimensional data sets. The applications of such visualisations are varied. This project will study some of the mathematics that IS used in three-dimensional graphics application

Modeling and Simulation of Random Processes and Fields in Civil Engineering and Engineering Mechanics

This thesis covers several topics within computational modeling and simulation of problems arising in Civil Engineering and Applied Mechanics. There are two distinct parts. Part 1 covers work in modeling and analyzing heterogeneous materials using the eXtended Finite Element Method (XFEM) with arbitrarily shaped inclusions. A novel enrichment function, which can model arbitrarily shaped inclusions within the framework of XFEM, is proposed. The internal boundary of an arbitrarily shaped inclusion is first discretized, and a numerical enrichment function is constructed "on the fly" using spline interpolation. This thesis considers a piecewise cubic spline which is constructed from seven localized discrete boundary points. The enrichment function is then determined by solving numerically a nonlinear equation which determines the distance from any point to the spline curve. Parametric convergence studies are carried out to show the accuracy of this approach, compared to pointwise and linear segmentation of points, for the construction of the enrichment function in the case of simple inclusions and arbitrarily shaped inclusions in linear elasticity. Moreover, the viability of this approach is illustrated on a Neo-Hookean hyperelastic material with a hole undergoing large deformation. In this case, the enrichment is able to adapt to the deformation and effectively capture the correct response without remeshing. Part 2 then moves on to research work in simulation of random processes and fields. Novel algorithms for simulating random processes and fields such as earthquakes, wind fields, and properties of functionally graded materials are discussed. Specifically, a methodology is presented to determine the Evolutionary Spectrum (ES) for non-stationary processes from a prescribed or measured non-stationary Auto-Correlation Function (ACF). Previously, the existence of such an inversion was unknown, let alone possible to compute or estimate. The classic integral expression suggested by Priestley, providing the ACF from the ES, is not invertible in a unique way so that the ES could be determined from a given ACF. However, the benefits of an efficient inversion from ACF to ES are vast. Consider for example various problems involving simulation of non-stationary processes or non-homogeneous fields, including non-stationary seismic ground motions as well as non-homogeneous material properties such as those of functionally graded materials. In such cases, it is sometimes more convenient to estimate the ACF from measured data, rather than the ES. However, efficient simulation depends on knowing the ES. Even more important, simulation of non-Gaussian and non-stationary processes depends on this inversion, when following a spectral representation based approach. This work first examines the existence and uniqueness of such an inversion from the ACF to the ES under a set of special conditions and assumptions (since such an inversion is clearly not unique in the most general form). It then moves on to efficient methodologies of computing the inverse, including some established optimization techniques, as well as proposing a novel methodology. Its application within the framework of translation models for simulation of non-Gaussian, non-stationary processes is developed and discussed. Numerical examples are provided demonstrating the capabilities of the methodology. Additionally in Part 2, a methodology is presented for efficient and accurate simulation of wind velocities along long span structures at a virtually infinite number of points. Currently, the standard approach is to model wind velocities as a multivariate stochastic process, characterized by a Cross-Spectral Density Matrix (CSDM). In other words, the wind velocities are modeled as discrete components of a vector process. To simulate sample functions of the vector process, the Spectral Representation Method (SRM) is used. The SRM involves a Cholesky decomposition of the CSDM. However, it is a well known issue that as the length of the structure, and consequently the size of the vector process, increases, this Cholesky decomposition breaks down (from the numerical point of view). To avoid this issue, current research efforts in the literature center around approximate techniques to simplify the decomposition. Alternatively, this thesis proposes the use of the frequency-wavenumber (F-K) spectrum to model the wind velocities as a stochastic "wave," continuous in both space and time. This allows the wind velocities to be modeled at a virtually infinite number of points along the length of the structure. In this work, the relationship between the CSDM and the F-K spectrum is first examined, as well as simulation techniques for both. The F-K spectrum for wind velocities is then derived. Numerical examples are then carried out demonstrating that the simulated wave samples exhibit the desired spectral and coherence characteristics. The efficiency of this method, specifically through the use of the Fast Fourier Transform, is demonstrated