APPLICATIONS OF MATHEMATICS IN ENGINEERING AND ECONOMICS (AMEE '11): Proceedings of the 37th International Conference

Oscillations of a Hybrid Systems with Resistance
View Description Hide DescriptionThe hybrid system in the paper means a mechanical system which consists of two parts with different structure—a part with distributed parameters and a part with discrete parameters. The most simple type of these systems is discussed in [1] and [2], where the free oscillations and the forced oscillations of a rod with a linear oscillator are studied. In [3] a more complicated example—the bending oscillations of a beam connected with a simple oscillator is considered and it is shown how the approach proposed in [1] could be extended in this case. In all these papers there was no resistance (proportional to the velocity) attached on the system with distributed parameters. In the present paper just this case will be considered. It turned out that in this case the approach developed in the previous papers does not work. To obtain an analytical solution of the problem another approach is developed. More concrete the investigated hybrid system is a rod (whose longitudinal oscillations are considered) connected with a single linear oscillator.

Model and Predictive Control for a Wind Turbine
View Description Hide DescriptionA mathematical model of the system consisting of wind turbine, gear box and asynchronous generator is presented in this work. The model is linearized. Then a controller, which provides a desire mode of frequency stabilization, is developed using the predictive control theory.

An analytical study of the dual mass mechanical system stability
View Description Hide DescriptionIn this paper an autonomous, nonlinear model of five ordinary differential equations modeling the motion of a dual mass mechanical system with universal joint is studied. The model is investigated qualitatively. On the base of the stability analysis performed, we obtain that the system is: i) in an equilibrium state, or ii) in a structurally unstable behavior when equilibrium states disappear. In case (i) the system is in a normal technical condition and in case (ii) hard break‐downs take place.

Finite Element Analysis of a Wind Turbine Response when the Tower and the Blades are Modeled as Distributed Parameter Systems
View Description Hide DescriptionFinite element (FE) model of a wind turbine (WT) system is constructed where the tower and the blades are assumed as distributed parameter systems, the nacelle—as a concentrated mass and the wind loading is generated from fluid simulations. The system response is simulated by a program implementation of the model and the results obtained are compared to the results from ANSYS 3‐D simulations. The constructed WT model could be utilized for identification and optimization of the system parameters as well as for defining modal characteristics and generating reduced models based on the first several natural frequencies and modal shapes.

Mode III crack problems in functionally graded piezoelectric/piezomagnetic composites by BIEM
View Description Hide DescriptionCracked functionally graded magnetoelectroelastic (MEE) medium is considered. The crack is electromagnetically permeable and subjected to an incident time‐harmonic SH‐wave. The boundary value problem (BVP) for the coupled system of governing equations is reduced to a non‐hypersingular traction boundary integral equation.
Software based on the Boundary Integral Equation Method (BIEM) is developed using FORTRAN. Comparison with the results obtained by the dual integral equation method is given. The numerical examples show the dependence of the Stress Intensity Factor (SIF) on the normalized frequency of the external loading, type of the material and magnitude of the inhomogeneity gradient of the magnetolelectroelastic solid.

First Instances of Generalized Expo‐Rational Finite Elements on Triangulations
View Description Hide DescriptionIn this communication we consider a construction of simplicial finite elements on triangulated two‐dimensional polygonal domains. This construction is, in some sense, dual to the construction of generalized expo‐rational B‐splines (GERBS). The main result is in the obtaining of new polynomial simplicial patches of the first several lowest possible total polynomial degrees which exhibit Hermite interpolatory properties. The derivation of these results is based on the theory of piecewise polynomial GERBS called Euler Beta‐function B‐splines. We also provide 3‐dimensional visualization of the graphs of the new polynomial simplicial patches and their control polygons.

Multivariate Hermite interpolation on scattered point sets using tensor‐product expo‐rational B‐splines
View Description Hide DescriptionAt the Seventh International Conference on Mathematical Methods for Curves and Surfaces, To/nsberg, Norway, in 2008, several new constructions for Hermite interpolation on scattered point sets in domains in combined with smooth convex partition of unity for several general types of partitions of these domains were proposed in [1]. All of these constructions were based on a new type of B‐splines, proposed by some of the authors several years earlier: expo‐rational B‐splines (ERBS) [3].
In the present communication we shall provide more details about one of these constructions: the one for the most general class of domain partitions considered.
This construction is based on the use of two separate families of basis functions: one which has all the necessary Hermite interpolation properties, and another which has the necessary properties of a smooth convex partition of unity. The constructions of both of these two bases are well‐known; the new part of the construction is the combined use of these bases for the derivation of a new basis which enjoys having all above‐said interpolation and unity partition properties simultaneously.
In [1] the emphasis was put on the use of radial basis functions in the definitions of the two initial bases in the construction; now we shall put the main emphasis on the case when these bases consist of tensor‐product B‐splines. This selection provides two useful advantages: (A) it is easier to compute higher‐order derivatives while working in Cartesian coordinates; (B) it becomes clear that this construction becomes a far‐going extension of tensor‐product constructions.
We shall provide 3‐dimensional visualization of the resulting bivariate bases, using tensor‐product ERBS. In the main tensor‐product variant, we shall consider also replacement of ERBS with simpler generalized ERBS (GERBS) [2], namely, their simplified polynomial modifications: the Euler Beta‐function B‐splines (BFBS). One advantage of using BFBS instead of ERBS is the simplified computation, since BFBS are piecewise polynomial, which ERBS are not.
One disadvantage of using BFBS in the place of ERBS in this construction is that the necessary selection of the degree of BFBS imposes constraints on the maximal possible multiplicity of the Hermite interpolation.

An approach to accelerate iterative methods for solving nonlinear operator equations
View Description Hide DescriptionWe propose and analyze a generalization of a Steffensen type acceleration method in case of extracting a locally unique solution of a nonlinear operator equation on a Banach space. In order, we use a special choice of a divided difference for operators. Convergence analysis and some applications of the obtained results are provided.

WAVELET‐BASED LOSSLESS ONE‐AND TWO‐DIMENSIONAL REPRESENTATION OF MULTIDIMENSIONAL GEOMETRIC DATA
View Description Hide DescriptionIn the present communication we develop a complete representation of whole multidimensional manifolds, with the Cantor diagonal type of algorithm [1, 2] replaced by a new and simpler type of Cartesian‐indexing basis‐matching algorithm [3]. We provide graphical comparison between the results obtained via the Cantor diagonal algorithm and the Cartesian‐indexing algorithm. For this purpose, we test the algorithms on several different types of ‘benchmark’ multidimensional manifolds: Green's functions for linear PDEs, Cartesian products of 3‐dimensional manifolds, intersections of multidimensional manifolds. One new type of intersection problems which can be solved invoking the new representation is computing the intersections of multidimensional manifolds in parametric form (rather than only in implicit form, as earlier [3]).
This work is based on research conducted within two consecutive Strategic Projects of the Norwegian Research Council: ‘GPGPU—Graphics Hardware as a High‐end Computational Resource’ (2004‐2007) and ‘Heterogeneous Computing’ (2008‐2010).

COMPARISON BETWEEN POLYNOMIAL, EULER BETA‐FUNCTION AND EXPO‐RATIONAL B‐SPLINE BASES
View Description Hide DescriptionEuler Beta‐function B‐splines (BFBS) are the practically most important instance of generalized expo‐rational B‐splines (GERBS) which are not true expo‐rational B‐splines (ERBS). BFBS do not enjoy the full range of the superproperties of ERBS but, while ERBS are special functions computable by a very rapidly converging yet approximate numerical quadrature algorithms, BFBS are explicitly computable piecewise polynomial (for integer multiplicities), similar to classical Schoenberg B‐splines.
In the present communication we define, compute and visualize for the first time all possible BFBS of degree up to 3 which provide Hermite interpolation in three consecutive knots of multiplicity up to 3, i.e., the function is being interpolated together with its derivatives of order up to 2.
We compare the BFBS obtained for different degrees and multiplicities among themselves and versus the classical Schoenberg polynomial B‐splines and the true ERBS for the considered knots. The results of the graphical comparison are discussed from analytical point of view.
For the numerical computation and visualization of the new B‐splines we have used Maple 12.

Conservative Finite Difference Schemes to a Nonlinear Parabolic System of Cancer Invasion
View Description Hide DescriptionWe consider a nonlinear parabolic system of equations that describes interactions between tumor cells and microenvironmental factors such as extracellular matrix and matrix degrada‐tive enzymes.
Given the application, conservation of nonnegativity and conservation or evolution laws of total cell density, concentration of total MDEs and ECM densities are of paramount importance.
We propose finite difference approximations that maintains these properties of the continuous solution. Numerical experiments are provided to demonstrate the behaviour of the model and to illustrate some important invasive mechanisms of cancer cells.

Efficient numerical method for solving Cauchy problem for the Gamma equation
View Description Hide DescriptionIn this work we consider Cauchy problem for the so called Gamma equation, derived by transforming the fully nonlinear Black‐Scholes equation for option price into a quasilinear parabolic equation for the second derivative (Greek) of the option price V. We develop an efficient numerical method for solving the model problem concerning different volatility terms. Using suitable change of variables the problem is transformed on finite interval, keeping original behavior of the solution at the infinity. Then we construct Picard‐Newton algorithm with adaptive mesh step in time, which can be applied also in the case of non‐differentiable functions. Results of numerical simulations are given.

First Instances of Univariate and Tensor‐product Multivariate Generalized Expo‐Rational Finite Elements
View Description Hide DescriptionWe consider a construction of finite elements (FE) which is, in some sense, dual to the construction of generalized expo‐rational B‐splines (GERBS)[1, 2]. The main result is the introduction of univariate and multivariate tensor‐product GERBS‐based FE which exhibit Hermite interpolatory properties. The derivation of these results is based on the theory of ‐smooth expo‐rational B‐splines (ERBS) and ‐smooth, piecewise polynomial GERBS called Euler Beta‐function B‐splines. We provide visualization of the approximations of some model curves and surfaces using the new FE, as well as of the size and distribution of the error of these approximations.

A TWO‐LEVEL SECOND‐ORDER FINITE DIFFERENCE SCHEME FOR THE SINGLE TERM STRUCTURE EQUATION
View Description Hide DescriptionIn the paper [6] the classical single factor term structure equation for models that predict non‐negative interest rates is numerically studied. For these models the authors proposed a second order accurate three‐level finite difference scheme (FDs) using the appropriate boundary conditions at zero. For the same problem we propose a two‐level second‐order accurate FDs. We also propose an effective algorithm for solving the difference schemes, for which also follows the positivity of the numerical solution. The flexibility of our FDs makes it easy to change the drift and diffusion terms in the model. The numerical experiments confirm the second‐order of accuracy of the scheme and the positivity‐convexity property.

Matrix‐Vector Multiplication in Adaptive Wavelet Methods
View Description Hide DescriptionWavelet based methods are an established tool in signal and image processing and a promising tool for the numerical solution of operator equations. They have namely some interesting properties which may provide an advantage over classical methods. It is well‐known fact that representations of smooth functions and also representations of a wide class of operators are sparse in wavelet coordinates. Further advantage of wavelet methods consists in the existence of a diagonal preconditioner. The condition number of the preconditioned stiffness matrices does not depend on the size of matrices. Although the stiffness matrices in wavelet coordinates are only quasi sparse, an approximate multiplication of these matrices with given sparse vectors can be performed in the linear complexity. These are crucial parts to design efficient adaptive wavelet schemes. In this contribution, we focus on biorthogonal spline wavelets and we compare different approximate matrix‐vector multiplication techniques.

Uniqueness and symmetry of minimizers to repulsive nonlocal functionals
View Description Hide DescriptionIn the present article we study the minimizers of the action functional, corresponding to the repulsive Hartree equation in the presence of external Coulomb potential. Using a modified reflection method and Pohozaev integral identities we prove that the action minimizer is radially symmetric and unique.

Operational calculi for multidimensional nonlocal evolution boundary value problems
View Description Hide DescriptionHere we propose a direct approach to the construction of operational calculi connected with linear nonlocal boundary value problems for a large class of linear evolution equations with several space variables and one time variable.

Boundary Value Problems With Integral Conditions
View Description Hide DescriptionThe weakly perturbed nonlinear boundary value problems (BVP) for almost linear systems of ordinary differential equations (ODE) are considered. We assume that the nonlinear part contain an additional function, which defines the perturbation as singular. Then the Poincare method is not applicable. The problem of existence, uniqueness and construction of a solution of the posed BVP with integral condition is studied.