the generalized morse oscillator in the so 4 2 dynamical group scheme a family of the morse oscillators with certain quantized coupling constants are described as composite objects in the framework of the so 4 2 dynamical group scheme although a single morse oscillator can be solved by the subgroup so 2 1 of so 4 2 this so 2 1 is not the spectrum generating group the set of all energy levels is given by the representation of another particular one parameter subgroup of so 4 2 which is the dynamical group of a single morse oscillator the continuous spectra of this oscillator and other variations of the morse potential are also discussed by making an analytic continuation from the morse potential well to the morse barrier

isogeometric residual minimization igrm for non stationary stokes and navier stokes problems abstract we show that it is possible to obtain a linear computational cost fem based solver for non stationary stokes and navier stokes equations our method employs a technique developed by guermond and minev 2011 which consists of singular perturbation plus a splitting scheme while the time integration schemes are implicit we use finite elements to discretize the spatial counterparts at each time step we solve a pde having weak derivatives in one direction only which allows for the linear computational cost at the expense of handling strong second order derivatives of the previous time step solution on the right hand side of these pdes this motivates the use of smooth functions such as b splines for high reynolds numbers some of these pdes become unstable to deal robustly with these instabilities we propose to use a residual minimization technique we test our method on problems having manufactured solutions as well as on the cavity flow problem

periodic motion planning and control for underactuated mechanical systems we consider the problem of periodic motion planning and of designing stabilising feedback control laws for such motions in underactuated mechanical systems a novel periodic motion planning method

mapping of shape invariant potentials under point canonical transformations the authors give explicit point canonical transformations which map twelve types of shape invariant potentials which are known to be exactly solvable into two potential classes the eigenfunctions in these two classes are given by hypergeometric and confluent hypergeometric functions respectively

fast isogeometric solvers for hyperbolic wave propagation problems abstract we use the alternating direction method to simulate implicit dynamics our spatial discretization uses isogeometric analysis namely we simulate a hyperbolic wave propagation problem in which we use tensor product b splines in space and an implicit time marching method to fully discretize the problem we approximate our discrete operator as a kronecker product of one dimensional mass and stiffness matrices as a result of this algebraic transformation we can factorize the resulting system of equations in linear i e o n time at each step of the implicit method we demonstrate the performance of our method in the model p wave propagation problem we then extend it to simulate the linear elasticity problem once we decouple the vector problem using alternating triangular methods we prove theoretically and experimentally the unconditional stability of both methods

algebraic treatment of the hyper coulomb problem a completely algebraic treatment of the six dimensional hyper coulomb problem is discussed in terms of an oscillator realization of the dynamical algebra of closed expressions are derived for the energy spectrum and form factors

isogeometric analysis of the isothermal navier stokes korteweg equations this paper is devoted to the numerical simulation of the navier stokes korteweg equations a phase eld model for water water vapor two phase ows we develop a numerical formulation based on isogeometric analysis that permits straightforward treatment of the higher order partial dierenti al operator that represents capillarity we introduce a new renement methodology that desensitizes the numerical solution to the computational mesh and achieves mesh invariant solutions finally we present several numerical examples in two and three dimensions that illustrate the eectiveness and robustness of our approach

an isogeometric analysis approach to gradient damage models continuum damage formulations are commonly used for the simulation of diffuse fracture processes implicit gradient damage models are employed to avoid the spurious mesh dependencies associated with local continuum damage models the c 0 continuity of traditional finite elements has hindered the study of higher order gradient damage approximations in this contribution we use isogeometric finite elements which allow for the construction of higher order continuous basis functions on complex domains we study the suitability of isogeometric finite elements for the discretization of higher order gradient damage approximations

a generalized finite element formulation for arbitrary basis functions from isogeometric analysis to xfem many of the formulations of cm rent research interest including iosogeometric methods and the extended finite element method use nontraditional basis functions some such as subdivision surfaces may not have convenient analytical representations the concept of an element if appropriate at all no longer coincides with the traditional definition developing a new software for each new class of basis functions is a large research burden especially if the problems involve large deformations non linear materials and contact the objective of this paper is to present a method that separates as much as possible the generation and evaluation of the basis functions from the analysis resulting in a formulation that can be implemented within the traditional structure of a finite clement program but that permits the use of arbitrary sets of basis functions that are defined only through the input file elements ranging from a traditional linear four node tetrahedron through a higher order element combining xfem and isogeometric analysis may be specified entirely through an input file without any additional programming examples of this framework to applications with lagrange elements isogeometric elements and xfem basis functions for fracture are presented

computational cost estimates for parallel shared memory isogeometric multi frontal solvers abstract in this paper we present computational cost estimates for parallel shared memory isogeometric multi frontal solvers the estimates show that the ideal isogeometric shared memory parallel direct solver scales as o p 2 log n p for one dimensional problems o n p 2 for two dimensional problems and o n 4 3 p 2 for three dimensional problems where n is the number of degrees of freedom and p is the polynomial order of approximation the computational costs of the shared memory parallel isogeometric direct solver are compared with those corresponding to the sequential isogeometric direct solver being the latest equal to o n p 2 for the one dimensional case o n 1 5 p 3 for the two dimensional case and o n 2 p 3 for the three dimensional case the shared memory version significantly reduces both the scalability in terms of n and p theoretical estimates are compared with numerical experiments performed with linear quadratic cubic quartic and quintic b splines in one and two spatial dimensions

isogeometric analysis of the cahn hilliard phase field model abstract the cahn hilliard equation involves fourth order spatial derivatives finite element solutions are not common because primal variational formulations of fourth order operators are only well defined and integrable if the finite element basis functions are piecewise smooth and globally c 1 continuous there are a very limited number of two dimensional finite elements possessing c 1 continuity applicable to complex geometries but none in three dimensions we propose isogeometric analysis as a technology that possesses a unique combination of attributes for complex problems involving higher order differential operators namely higher order accuracy robustness two and three dimensional geometric flexibility compact support and most importantly the possibility of c 1 and higher order continuity a nurbs based variational formulation for the cahn hilliard equation was tested on two and three dimensional problems we present steady state solutions in two dimensions and for the first time in three dimensions to achieve these results an adaptive time stepping method is introduced we also present a technique for desensitizing calculations to dependence on mesh refinement this enables the calculation of topologically correct solutions on coarse meshes opening the way to practical engineering applications of phase field methodology

a finite strain eulerian formulation for compressible and nearly incompressible hyperelasticity using high order b spline finite elements summary we present a numerical formulation aimed at modeling the nonlinear response of elastic materials using large deformation continuum mechanics in three dimensions this finite element formulation is based on the eulerian description of motion and the transport of the deformation gradient when modeling a nearly incompressible solid the transport of the deformation gradient is decomposed into its isochoric part and the jacobian determinant as independent fields a homogeneous isotropic hyperelastic solid is assumed and b splines based finite elements are used for the spatial discretization a variational multiscale residual based approach is employed to stabilize the transport equations the performance of the scheme is explored for both compressible and nearly incompressible applications the numerical results are in good agreement with theory illustrating the viability of the computational scheme copyright 2011 john wiley sons ltd

dispersion optimized quadrature rules for isogeometric analysis modified inner products their dispersion properties and optimally blended schemes abstract this paper introduces optimally blended quadrature rules for isogeometric analysis and analyzes the numerical dispersion of the resulting discretizations to quantify the approximation errors when we modify the inner products we generalize the pythagorean eigenvalue theorem of strang and fix the proposed blended quadrature rules have advantages over alternative integration rules for isogeometric analysis on uniform and non uniform meshes as well as for different polynomial orders and continuity of the basis the optimally blended schemes improve the convergence rate of the method by two orders with respect to the fully integrated galerkin method the proposed technique increases the accuracy and robustness of isogeometric analysis for wave propagation problems

nonunitary representations of the su 2 algebra in the dirac equation with a coulomb potential a novel realization of the classical su 2 algebra is introduced for the dirac relativistic hydrogen atom defining a set of operators that allow the factorization of the problem an extra phase is needed as a new variable in order to define the algebra we take advantage of the operators to solve the dirac equation using algebraic methods a similar path to the one used in the angular momentum case is used hence the radial eigenfunctions so calculated comprise nonunitary representations of the algebra one of the interesting properties of such nonunitary representations is that they are not labeled by integer nor by half integer numbers as occurs in the usual angular momentum representation

periodic motion planning for virtually constrained euler lagrange systems the paper suggests an explicit form of a general integral of motion for some classes of dynamical systems including n degrees of freedom euler lagrange systems subject to n 1 virtual holonomic

a parallel multifrontal algorithm and its implementation abstract in this paper we describe a multifrontal method for solving sparse systems of linear equations arising in finite element and finite difference methods the method proposed in this study is a combination of the nested dissection ordering and the frontal method it can significantly reduce the storage and computational time required by the conventional direct methods and is also a natural parallel algorithm in addition the method inherits major advantages of the frontal method which include a simple interface with finite element codes and an effective data structure so that the entire computation is performed element by element on a series of small linear systems with dense stiffness matrices the numerical implementation targets both distributed memory machines as well as conventional sequential machines its performance is tested through a series of examples

a new realisation of dynamical groups and factorisation method a new method of algebraisation of quantum mechanical eigenvalue equations is presented in this method the dynamical algebra is represented on the space of group matrix elements the ladder operators of the dynamical algebra are obtained from infeld hull miller factorisations the method is used to study the first poschl teller equation even in the non symmetric case the energy spectrum and the exact normalised solutions are obtained in agreement with the results of non algebraic methods

an su 1 1 algebraic method for the hydrogen atom an algebraic solution for the hydrogen atom analogous to the one recently proposed to solve the relativistic version of the system is presented we add to the usual radial description of the problem an additional angular variable and an associated operator which can be considered as part of an su 1 1 lie algebra the operators of the algebra define radial ladder operator relating the eigenfunctions of the system in unit steps of the principal quantum number we conclude that the radial bound states of the hydrogen atom in our extended configuration space can be regarded as spanning the minimal m representation of the su 1 1 lie algebra the method can also be extended to solve the s wave morse problem and the three dimensional harmonic oscillator

adaptive isogeometric analysis by local h refinement with t splines isogeometric analysis based on non uniform rational b splines nurbs as basis functions preserves the exact geometry but suffers from the drawback of a rectangular grid of control points in the parameter space which renders a purely local refinement impossible this paper demonstrates how this difficulty can be overcome by using t splines instead t splines allow the introduction of so called t junctions which are related to hanging nodes in the standard fem obeying a few straightforward rules rectangular patches in the parameter space of the t splines can be subdivided and thus a local refinement becomes feasible while still preserving the exact geometry furthermore it is shown how state of the art a posteriori error estimation techniques can be combined with refinement by t splines numerical examples underline the potential of isogeometric analysis with t splines and give hints for further developments

efficient quadrature for nurbs based isogeometric analysis abstract we initiate the study of efficient quadrature rules for nurbs based isogeometric analysis a rule of thumb emerges the half point rule indicating that optimal rules involve a number of points roughly equal to half the number of degrees of freedom or equivalently half the number of basis functions of the space under consideration the half point rule is independent of the polynomial order of the basis efficient rules require taking into account the precise smoothness of basis functions across element boundaries several rules of practical interest are obtained and a numerical procedure for determining efficient rules is presented we compare the cost of quadrature for typical situations arising in structural mechanics and fluid dynamics the new rules represent improvements over those used previously in isogeometric analysis

fourier series expansion in a non orthogonal system of coordinates for the simulation of 3d dc borehole resistivity measurements abstract we describe a new method to simulate 3d borehole resistivity measurements at zero frequency dc the method combines the use of a fourier series expansion in a non orthogonal system of coordinates with an existing 2d goal oriented higher order self adaptive hp finite element algorithm the new method is suitable for simulating measurements acquired with borehole logging instruments in deviated wells it delivers high accuracy simulations and it enables a considerable reduction of the computational complexity with respect to available 3d simulators since the number of fourier modes basis functions needed to solve practical applications is limited typically below 10 furthermore numerical results indicate that the optimal 2d grid based on the 0th fourier mode also called central fourier mode can be employed to efficiently solve the final 3d problem thereby avoiding the expensive construction of optimal 3d grids specifically for a challenging through casing resistivity application we reduce the computational time from several days using a 3d simulator to just 2 h with the new method while gaining accuracy the new simulation method can be easily extended to different physical phenomena with similar geometries as those arising in the simulation of 3d borehole electrodynamics and sonic acoustics coupled with elasticity measurements in addition the method is especially suited for inversion since we demonstrate that the number of fourier modes needed for the exact representation of the materials is limited to only one the central mode for the case of borehole measurements acquired in deviated wells

variational multiscale residual based turbulence modeling for large eddy simulation of incompressible flows we present an les type variational multiscale theory of turbulence our approach derives completely from the incompressible navier stokes equations and does not employ any ad hoc devices such as eddy viscosities we tested the formulation on forced homogeneous isotropic turbulence and turbulent channel flows in the calculations we employed linear quadratic and cubic nurbs a dispersion analysis of simple model problems revealed nurbs elements to be superior to classical finite elements in approximating advective and diffusive processes which play a significant role in turbulence computations the numerical results are very good and confirm the viability of the theoretical framework

the p version of the finite element method in the p version of the finite element method the triangulation is fixed and the degree p of the piecewise polynomial approximation is progressively increased until some desired level of precision is reached in this paper we first establish the basic approximation properties of some spaces of piecewise polynomials defined on a finite element triangulation these properties lead to an a priori estimate of the asymptotic rate of convergence of the p version the estimate shows that the p version gives results which are not worse than those obtained by the conventional finite element method called the h version in which h represents the maximum diameter of the elements when quasi uniform triangulations are employed and the basis for comparison is the number of degrees of freedom furthermore in the case of a singularity problem we show under conditions which are usually satisfied in practice that the rate of convergence of the p version is twice that of the h version with quasi uniform mesh inve

the cost of continuity a study of the performance of isogeometric finite elements using direct solvers we study the performance of direct solvers on linear systems of equations resulting from isogeometric analysis the problem of choice is the canonical laplace equation in three dimensions from this study we conclude that for a fixed number of unknowns and polynomial degree of approximation a higher degree of continuity k drastically increases the cpu time and ram needed to solve the problem when using a direct solver this paper presents numerical results detailing the phenomenon as well as a theoretical analysis that explains the underlying cause

generating self excited oscillations via two relay controller a tool for the design of a self excited oscillation of a desired amplitude and frequency in linear plants by means of the variable structure control is proposed an approximate approach based on the describing function method is given which requires that the mechanical plant should be a low pass filter the hypothesis that usually holds when the oscillations are relatively fast the proposed approach is demonstrated via the controller design and experiments on the furuta pendulum

comment on new non unitary representations in a dirac hydrogen atom nonunitary representations of a novel realization of the su 2 algebra recently introduced for the dirac relativistic hydrogen atom are found to be actually unitary representations of a related su 1 1 algebra