hybrid nearly singular integration for three dimensional isogeometric boundary element analysis of coatings and other thin structures abstract the isogeometric boundary element method igabem has great potential for the simulation of elasticity problems because of its exact geometric representation and good approximation properties these advantages can be extended to thin structures including coatings but the development of an accurate and efficient method to deal with the large number of nearly singular integrals existing in the igabem presents a great challenge for very thin sections in this paper we propose a new sinh scheme for weakly strongly and hyper near singular integrals arising in 3d igabem for thermoelastic problems based on the sinh transformation method and adaptive integral method the presented scheme is efficient since it combines the advantages of both methods 1 when the thickness of coatings thin structures is moderately small an accurate and efficient integral result will be obtained by the adaptive integral method 2 when is very small the nearly singular integrals are computed by the sinh scheme efficiently with the introduction of nurbs in igabem truncation errors arising in the taylor expansion cannot be ignored based on the values of these errors the computed knot spans are further divided into several sub knot spans and different methods will be used to evaluate the integral over each sub knot span in the new scheme in addition based on the analytical extension of the nurbs surface an adaptation of the sinh transformation method is proposed which can evaluate the near singular integrals accurately for cases in which the projection point lies outside of the considered knot span several numerical examples are presented to validate the accuracy and efficiency of the 3d igabem based on the sinh scheme in the analysis of thermoelastic problems

isogeometric boundary element analysis using unstructured t splines we couple collocated isogeometric boundary element methods and unstructured analysis suitable t spline surfaces for linear elastostatic problems we extend the definition of analysis suitable t splines to encompass unstructured control grids unstructured meshes and develop basis functions which are smooth rational polynomials defined in terms of the bezier extraction framework and which pass standard patch tests we then develop a collocation procedure which correctly accounts for sharp edges and corners extraordinary points and t junctions this approach is applied to several three dimensional problems including a real world t spline model of a propeller we believe this work clearly illustrates the power of combining new analysis suitable computer aided design technologies with established analysis methodologies in this case the boundary element method

local refinement of analysis suitable t splines we develop a local refinement algorithm for analysis suitable t splines which does not produce excessive propagation of control points we then demonstrate its use as an adaptive framework for isogeometric analysis analysis suitable t splines are a class of t splines which are linearly independent and form a partition of unity these properties coupled with local refinement make this class of t splines appealing as a basis for isogeometric analysis

a hierarchical approach to adaptive local refinement in isogeometric analysis abstract adaptive local refinement is one of the key issues in isogeometric analysis in this article we present an adaptive local refinement technique for isogeometric analysis based on extensions of hierarchical b splines we investigate the theoretical properties of the spline space to ensure fundamental properties like linear independence and partition of unity furthermore we use concepts well established in finite element analysis to fully integrate hierarchical spline spaces into the isogeometric setting this also allows us to access a posteriori error estimation techniques numerical results for several different examples are given and they turn out to be very promising

isogeometric analysis using polynomial splines over hierarchical t meshes for two dimensional elastic solids isogeometric analysis has become a powerful alternative to standard finite elements due to its flexibility in handling complex geometries one of the major drawbacks of nurbs based isogeometric finite elements is the inefficiency of local refinement in this study we present an alternative to nurbs based isogeometric analysis that allows for local refinement the idea is based on polynomial splines and exploits the flexibility of t meshes for local refinement the shape functions satisfy important properties such as non negativity local support and partition of unity several numerical examples are used to demonstrate the reliability of the present method

smooth cubic spline spaces on unstructured quadrilateral meshes with particular emphasis on extraordinary points geometric design and isogeometric analysis considerations abstract we present a framework for geometric design and isogeometric analysis on unstructured quadrilateral meshes acknowledging the differing requirements posed by design e g the convenience of an intuitive control net and analysis e g good approximation behavior we propose the construction of a separate smooth spline space for each while ensuring isogeometric compatibility requiring the geometric models to be members of the analysis suitable spaces the methodology is simple and is presented for bi cubic splines extensions to higher degrees are possible and are briefly discussed the presentation has been structured to show compatibility with t splines a state of the art cad technology but the approach should extend to other locally refinable spline technologies based on local tensor product structures an instantiation of the framework is presented and several numerical tests focused on geometric design and isogeometric analysis demonstrate the versatility of the developed framework and show significantly higher convergence rates than attained previously in the considered setting

x fem in isogeometric analysis for linear fracture mechanics the extended finite element method x fem has proven to be an accurate robust method for solving problems in fracture mechanics x fem has typically been used with elements using linear basis functions although some work has been performed using quadratics in the current work the x fem formulation is incorporated into isogeometric analysis to obtain solutions with higher order convergence rates for problems in linear fracture mechanics in comparison with x fem with conventional finite elements of equal degree the nurbs based isogeometric analysis gives equal asymptotic convergence rates and equal accuracy with fewer degrees of freedom dof results for linear through quartic nurbs basis functions are presented for a multiplicity of one or a multiplicity equal the degree

bifurcation phenomena near homoclinic systems a two parameter analysis the bifurcations of periodic orbits in a class of autonomous three variable nonlinear differential equation systems possessing a homoclinic orbit associated with a saddle focus with eigenvalues i where 1 sil nikov s condition are studied in a two parameter space the perturbed homoclinic systems undergo a countable set of tangent bifurcations followed by period doubling bifurcations leading to periodic orbits which may be attractors if 1 2 the accumulation rate of the critical parameter values at the homoclinic system is exp 2 a global mechanism for the onset of homoclinicity in strongly contractive flows is analyzed cusp bifurcations with bistability and hysteresis phenomena exist locally near the onset of homoclinicity a countable set of these cusp bifurcations with scaling properties related to the eigenvalues i of the stationary state are shown to occur in infinitely contractive flows in the two parameter space the periodic orbit attractor domain exhibits a spiral structure globally around the set of homoclinic systems in which all the different periodic orbits are continuously connected

application of pht splines in bending and vibration analysis of cracked kirchhoff love plates abstract in this work we present an extended geometry independent field approximation x gift formulation for cracked kirchhoff love plates the plate geometry is modeled by non uniform rational b splines nurbs while the solution is approximated by polynomial splines over hierarchical t meshes pht splines and enriched by the heaviside function and crack tip asymptotic expansions the adaptive refinement is driven by a recovery based error estimator the formulation is employed for bending and vibration analysis we compare different strategies for refinement enrichment and evaluation of fracture parameters the obtained results are shown to be in a good agreement with the reference solutions

isogeometric boundary element analysis of steady incompressible viscous flow part 1 plane problems abstract a novel approach is presented to the boundary element analysis of steady incompressible flow nurbs basis functions are used for describing the geometry of the problem and for approximating the unknowns in addition the arising volume integrals are treated differently to published work that is volumes are described by bounding nurbs curves instead of cells and a mapping is used the advantage of our approach is that non trivial boundary shapes can be described with very few parameters and that no generation of cells is required for the solution of the non linear equations both classical and modified newton raphson methods are used a comparison of the two methods is made on the classical example of a forced cavity flow where accurate solutions are available in the literature the results obtained agree well with published ones for moderate reynolds numbers using both methods but it is found that the latter requires a relaxation scheme and considerably more iterations to converge finally it is shown on a practical example of an airfoil how more complex boundary shapes can be approximated with few parameters and a solution obtained with a small number of unknowns

properties of a bursting model with two slow inhibitory variables models for certain excitable cells such as the pancreatic beta cell must reproduce bursting oscillations of the membrane potential this has previously been done using one slow variable to drive bursts the dynamics of such models have been analyzed however new models for the beta cell often include additional slow variables and therefore the previous analysis is extended to two slow variables using a simplified version of a beta cell model some unusual time courses of this model motivated a geometric singular perturbation analysis and the application of averaging to reduce the dynamics to the slow variable phase plane a geometric understanding of the solutionstructure and of transitions between various modes of behavior was then developed a novel use of the bifurcation code auto finds nullclines for the slow variables when the fast variables are periodic by averaging over the fast oscillations in contrast with the parabolic neuronal burster this model requires bistability in th

shape optimization and its extension to topological design based on isogeometric analysis in most of structural optimization approaches finite element method fem has been employed for structural response analysis and sensitivity calculation however the approaches generally suffer certain drawbacks in shape optimization cumbersome parameterization of design domain is required and time consuming remeshing task is also necessary in topology optimization design results are generally restricted on the initial design space and additional post processing is required for communication with cad systems these drawbacks are due to the use of different mathematical languages in design or geometric modeling and numerical analysis spline basis functions are used in design and geometric modeling whereas lagrangian and hermitian polynomials in analysis isogeometric analysis is a very attractive and promising alternative to overcome the limitations resulting from the use of the conventional fem in structural optimization in isogeometric analysis the same spline information such as control points and spline basis functions which represent geometries in cad systems are also used in numerical analysis such unification of the mathematical languages in cad analysis and design optimization can resolve the issues mentioned above in this work structural shape optimization using isogeometric analysis is studied on 2d and shell problems the proposed framework is extended to topology optimization using trimming techniques new inner fronts are introduced by trimming spline curves in topology optimization trimmed surface analysis which was recently proposed to analyze arbitrary complex topology problems is employed for topology optimization some benchmarking problems in shape and topology optimization are treated using the proposed approach

realization of quaternionic discrete series on the unit ball in h d abstract we give a realization for the quaternionic discrete series for the group sp 1 d as the kernel of certain elementary cauchy riemann fueter operator and we compute the corresponding reproducing kernels

an isogeometric design through analysis methodology based on adaptive hierarchical refinement of nurbs immersed boundary methods and t spline cad surfaces we explore hierarchical refinement of nurbs as a basis for adaptive isogeometric and immersed boundary analysis we use the principle of b spline subdivision to derive a local refinement procedure which combines full analysis suitability of the basis with straightforward implementation in tree data structures and simple generalization to higher dimensions we test hierarchical refinement of nurbs for some elementary fluid and structural analysis problems in two and three dimensions and attain good results in all cases using the b spline version of the finite cell method we illustrate the potential of immersed boundary methods as a seamless isogeometric design through analysis procedure for complex engineering parts defined by t spline cad surfaces specifically a ship propeller and an automobile wheel we show that hierarchical refinement considerably increases the flexibility of this approach by adaptively resolving local features

homoclinic orbits and mixed mode oscillations in far from equilibrium systems nonlinear autonomous dynamical systems with ahomoclinic tangency to a periodic orbit are investigated we study the bifurcation sequences of the mixed mode oscillations generated by the homoclinicity which are shown to belong to two different types depending on the nature of the liapunov numbers of the basic periodic orbit a detailed numerical analysis is carried out to show how the existence of a tangent homoclinic orbit allows us to understand in a quantitative way a particular and regular sequence of cool flame ignition oscillations observed in a thermokinetic model of hydrocarbon oxidation chaotic cool flame oscillations are also observed in the same model when the control parameter crosses a critical value this chaotic set of trajectories becomes globally unstable and forms a cantor like hyperbolic repellor and the ignition mechanism generates ahomoclinic tangency to the cantor set of trajectories the complex bifurcation diagram may be globally reconstructed from a one dimensional dynamical system thanks to the strong contractivity of thermokinetics it is found that a symbolic dynamics with three symbols is necessary to classify the periodic windows of the complex bifurcation sequence observed numerically in this system

fourier jacobi expansion of automorphic forms on sp 1 q generating quaternionic discrete series abstract the aim of this paper is to develop the notion of the fourier expansion of automorphic forms on sp 1 q generating quaternionic discrete series which are non holomorphic forms there is such an expansion given by tsuneo arakawa assuming the boundedness of the forms and the integrability of the discrete series we study these automorphic forms without such assumptions when q 1 we prove the koecher principle for such automorphic forms whose validity is known for holomorphic automorphic forms except elliptic modular forms

skew symmetric nitsche s formulation in isogeometric analysis dirichlet and symmetry conditions patch coupling and frictionless contact abstract a simple skew symmetric nitsche s formulation is introduced into the framework of isogeometric analysis iga to deal with various problems in small strain elasticity essential boundary conditions symmetry conditions for kirchhoff plates patch coupling in statics and in modal analysis as well as signorini contact conditions for linear boundary or interface conditions the skew symmetric formulation is parameter free for contact conditions it remains stable and accurate for a wide range of the stabilization parameter several numerical tests are performed to illustrate its accuracy stability and convergence performance we investigate particularly the effects introduced by nitsche s coupling including the convergence performance and condition numbers in statics as well as the extra outlier frequencies and corresponding eigenmodes in structural dynamics we present the hertz test the block test and a 3d self contact example showing that the skew symmetric nitsche s formulation is a suitable approach to simulate contact problems in iga

h and p adaptivity driven by recovery and residual based error estimators for pht splines applied to time harmonic acoustics abstract in this work we demonstrate the application of pht splines for time harmonic acoustic problems modeled by the helmholtz equation solutions of the helmholtz equation have two features global oscillations associated with the wave number and local gradients caused by geometrical irregularities we show that after a sufficient number of degrees of freedom is used to approximate global oscillations adaptive refinement can capture local features of the solution we compare residual based and recovery based error estimators and investigate the performance of p refinement the simulations are done in the context of recently introduced geometry independent field approximation gift where pht splines are only used to approximate the solution while the computational domain is parameterized with nurbs this approach builds on the natural adaptation ability of pht splines and avoids the re parameterization of the nurbs geometry during the solution refinement process

thb splines an effective mathematical technology for adaptive refinement in geometric design and isogeometric analysis local refinement with hierarchical b spline structures is an active topic of research in the context of geometric modeling and isogeometric analysis by exploiting a multilevel control structure we show that truncated hierarchical b spline thb spline representations support interactive modeling tools while simultaneously providing effective approximation schemes for the manipulation of complex data sets and the solution of partial differential equations via isogeometric analysis a selection of illustrative 2d and 3d numerical examples demonstrates the potential of the hierarchical framework

parametric study and shape optimization of piezoelectric energy harvesters by isogeometric analysis and kriging metamodeling abstract piezo electric energy harvesters pehs are widely used to recycle waste vibrations for powering small electronic devices in this work the performance of pehs of non conventional shapes is studied with the aim to optimize their design a peh is modeled as a bimorph cantilever kirchhoff love plate subjected to base acceleration the shape is described by non uniform rational b splines nurbs and the problem is solved by the isogeometric analysis iga accuracy of the iga solutions is demonstrated on two benchmark problems then we conduct a parametric study to investigate the effect of shape perturbations on the fundamental frequency the peak amplitude of the frequency response function frf and the peak amplitude of the frf per unit of area next we build the kriging meta model to obtain the boundary shapes that maximize the peak amplitude of the frf and the peak amplitude of the frf per unit of area for a target frequency the optimization is performed for three cases with a fixed tip mass without a tip mass and with a tip mass being one of the design parameters

on the condition number of high order finite element methods influence of p refinement and mesh distortion abstract in this article the condition number of the stiffness matrix k is compared for three high order finite element methods fems i e the p version of the fem the spectral element method sem and the nurbs based isogeometric analysis iga note that only problems in linear elasticity are considered in the analysis it is well known that the condition number is one factor strongly influencing the number of significant digits for direct solvers or the required iteration count for iterative solution schemes therefore it is important to investigate the effect of the choice of the shape functions and the element distortion on k based on numerous one and two and three dimensional examples these influences are comprehensively studied and the numerical results are compared with condition number estimates extracted from the literature overall a good agreement is observed for p fem sem and two dimensional iga while discrepancies are noted for three dimensional isogeometric elements these are important findings as theoretical results may be only available for very restricted scenarios where one element geometries constant jacobi matrices of the element maps etc are considered

isogeometric boundary element for analyzing steady state heat conduction problems under spatially varying conductivity and internal heat source abstract this paper proposes an isogeometric boundary element igabem with the aim to solve the 2d steady state heat conduction problems under spatially varying conductivity and internal heat source the igabem boundary domain integral equation is derived underlying the divergence theorem of gauss and the laplace equation the non uniform rational b spline nurbs basis used in the cad cae industries is applied to approximate both the boundary of problem domain and the unknown physical quantities and the radial integration method rim is employed to deal with the domain integrals induced by the non homogeneous thermal conductivity and the internal heat source the present boundary domain integral equation is firstly divided into several nurbs patches upon which the igabem can be further defined the main advantage of the nurbs basis in comparison with the standard piecewise polynomial basis is that there is almost no error in the boundary approximation with igabem which plays a significant role in controlling the accuracy of numerical calculation four numerical examples consisting of square regions with cubic and exponential thermal conductivities with or without internal heat source annulus region with constant and quadratic conductivities as well as complex region with varying thermal conductivity and complex heat source are provided the accuracy and convergence of the proposed method are assessed by comparing the present solution with the existing results obtained by several other researchers or ansys results and excellent performance is achieved

voltage interval mappings for activity transitions in neuron models for elliptic bursters abstract we performed a thorough bifurcation analysis of a mathematical elliptic bursting model using a computer assisted reduction to equationless one dimensional poincare mappings for a voltage interval using the interval mappings we were able to examine in detail the bifurcations that underlie the complex activity transitions between tonic spiking and bursting bursting and mixed mode oscillations and finally mixed mode oscillations and quiescence in the fitzhugh nagumo rinzel model we illustrate the wealth of information qualitative and quantitative that was derived from the poincare mappings for the neuronal models and for similar electro chemical systems

fast subsystem bifurcations in a slowly varying lie nard system exhibiting bursting a perturbed lienard differential system is examined using local stability and hopf bifurcation analyses asymptotic techniques and melnikov s method the results of these analyses are applied to a simple cubic model that exhibits a variety of different oscillatory behaviors for different parameter values for a bounded region in fast parameter space the model exhibits square wave bursting patterns analogous to the bursting electrical activity observed in pancreatic beta cells under certain hypotheses solutions of the cubic model are known to have square wave patterns by using the theory developed for the more general lienard system each hypothesis is shown to correspond to a curve in parameter space together the curves bound a region in which the model exhibits square wave bursting patterns since the model is simple the curves that bound this region can all be determined analytically

isogeometric boundary element analysis with elasto plastic inclusions part 1 plane problems abstract in this work a novel approach is presented for the isogeometric boundary element analysis of domains that contain inclusions with different elastic properties than the ones used for computing the fundamental solutions in addition the inclusion may exhibit inelastic material behavior in this paper only plane stress strain problems are considered in our approach the geometry of the inclusion is described using nurbs basis functions the advantage over currently used methods is that no discretization into cells is required in order to evaluate the arising volume integrals the other difference to current approaches is that kernels of lower singularity are used in the domain term the implementation is verified on simple finite and infinite domain examples with various boundary conditions finally a practical application in geomechanics is presented

isogeometric analysis for surface pdes with extended loop subdivision abstract we investigate the isogeometric analysis for surface pdes based on the extended loop subdivision approach the basis functions consisting of quartic box splines corresponding to each subdivided control mesh are utilized to represent the geometry exactly and construct the solution space for dependent variables as well which is consistent with the concept of isogeometric analysis the subdivision process is equivalent to the h refinement of nurbs based isogeometric analysis the performance of the proposed method is evaluated by solving various surface pdes such as surface laplace beltrami harmonic biharmonic triharmonic equations which are defined on the limit surfaces of extended loop subdivision for different initial control meshes numerical experiments show that the proposed method has desirable performance in terms of the accuracy convergence and computational cost for solving the above surface pdes defined on both open and closed surfaces the proposed approach is proved to be second order accuracy in the sense of l 2 norm with theoretical and or numerical results which is also outperformed over the standard linear finite element by several numerical comparisons

bifurcations of mixed mode oscillations in a three variable autonomous van der pol duffing model with a cross shaped phase diagram abstract the bifurcation structure of a three variable van der pol duffing type model is studied in some detail with special attention to the mixed mode solutions a type of complex periodic behavior frequently encountered in oscillating chemical reactions the mixed mode oscillations in the model occur close to two hopf bifurcations which are arranged with the saddle node bifurcations in a so called cross shaped phase diagram a bifurcation diagram also typical for chemical reactions the mixed mode oscillations are shown to lie on isolated bifurcation curves which are all born in a single codimension two bifurcation known as the neutrally twisted homoclinic orbit or inclination switch with the introduction of an additional slow time scale the same model can exhibit more complex mixed mode oscillations and torus bifurcations

multimodal regimes in a compartmental model of the dopamine neuron we study chains of strongly electrically coupled relaxation oscillators modeling dopamine neurons when individual oscillators are in the regime close to an andronov hopf bifurcation ahb the coupled system exhibits a variety of oscillatory behavior we show that the proximity of individual oscillators to the ahb has a significant impact on the system dynamics in a wide range of parameters it manifests itself through a family of stable multimodal periodic solutions that are composed out of large amplitude relaxation oscillations and small amplitude oscillations this family of solutions has a rich bifurcation structure the waveform and the period vary greatly across the family the structure and bifurcations of the stable periodic solutions of the coupled system are investigated using numerical and analytic techniques

shape optimization directly from cad an isogeometric boundary element approach using t splines abstract we develop a t spline isogeometric boundary element method igabem simpson et al 2012 scott et al 2013 simpson et al 2014 to shape sensitivity analysis and gradient based shape optimization in three dimensional linear elasticity contrary to finite element based isogeometric analysis iga approaches no parametrization of the volume is required hence the iterative optimization algorithm can be implemented directly from cad without any mesh generation or postprocessing step for returning the resulting structure to cad designers t splines also guarantee a water tight geometry without the manual geometrical repair work as with non uniform rational b splines nurbs we demonstrate the worth of the method by analysing problems with and without analytical solutions including engineering examples involving complex shapes additionally we provide all the derivations of the required sensitivities and the details pertaining to the geometries examined in the benchmarking to provide helpful reference problems for 3d shape optimization

on bifurcations of the lorenz attractor in the shimizu morioka model abstract bifurcations of lorenz like attractors are studied in the shimizu morioka model two scenarios for their destruction are suggested the first is connected with the appearance of a lacuna the second with a transition to a quasi attractor the effects of codimension two bifurcations are considered

synchronization of elliptic bursters periodic bursting behavior in neurons is a recurrent transition between a quiescent state and repetitive spiking when the transition to repetitive spiking occurs via a subcritical andronov hopf bifurcation and the transition to the quiescent state occurs via fold limit cycle bifurcation the burster is said to be of elliptic type also known as a subhopf fold cycle burster here we study the synchronization dynamics of weakly connected networks of such bursters we find that the behavior of such networks is quite different from the behavior of weakly connected phase oscillators and resembles that of strongly connected relaxation oscillators as a result such weakly connected bursters need few usually one bursts to synchronize and synchronization is possible for bursters having quite different quantitative features we also find that interactions between bursters depend crucially on the spiking frequencies namely the interactions are most effective when the presynaptic interspike frequency matches the frequency of postsynaptic oscillations finally we use the fitzhugh rinzel morris lecar and hodgkin huxley models to illustrate our major results

isogeometric analysis for explicit elastodynamics using a dual basis diagonal mass formulation abstract we propose a method to obtain diagonal mass matrices for nurbs based approximation spaces by a dual lumping method the use of lumped mass matrices is of great importance in elastodynamics problems as they can be employed in explicit time integration schemes which do not require the solution of a linear system in finite elements several well established methods such as row sum diagonal scaling or nodal quadrature methods have been used to obtain lumped mass matrices for different applications however for higher order and higher continuity approximation spaces such as those derived from nurbs these approaches have only limited second order accuracy in this work we derive a dual basis which has optimal approximation and dispersion properties while maintaining local support the dual space has discontinuities at the element boundaries knots and it is used to provide the test functions in the context of a petrov galerkin method this results in a general framework for the study of lumped mass matrices which can be employed in explicit time integration schemes with high order accuracy numerical experiments are presented to demonstrate the applicability of the method to problems with smooth solutions as well as to wave propagation problems with reduced regularity

d4 modular forms in this paper we study modular forms on two simply connected groups of type d 4 over q one group g s is a globally split group of type d 4 viewed as the group of isotopies of the split rational octonions the other g c is the isotopy group of the rational nonsplit octonions we study automorphic forms on g s in analogy to the work of gross gan and savin on g 2 namely we study automorphic forms whose component at infinity corresponds to a quaternionic discrete series representation we study automorphic forms on g c using gross s formalism of algebraic modular forms finally we follow work of gan savin gross rallis and others to study an exceptional theta correspondence connecting modular forms on g c and g s this can be thought of as an octonionic generalization of the jacquet langlands correspondence

weakening the tight coupling between geometry and simulation in isogeometric analysis from sub and super geometric analysis to geometry independent field approximation gift this paper presents an approach to generalize the concept of isogeometric analysis by allowing different spaces for the parameterization of the computational domain and for the approximation of the solution field the method inherits the main advantage of isogeometric analysis ie preserves the original exact computer aided design geometry for example given by nonuniform rational b splines but allows pairing it with an approximation space which is more suitable flexible for analysis for example t splines lr splines truncated hierarchical b splines and pht splines this generalization offers the advantage of adaptive local refinement without the need to reparameterize the domain and therefore without weakening the link with the computer aided design model we demonstrate the use of the method with different choices of geometry and field spaces and show that despite the failure of the standard patch test the optimum convergence rate is achieved for nonnested spaces

homoclinic chaos in chemical systems abstract we first focus on complex dynamical phenomena generated by chemical kinetics in homogeneous media we comment on the alternating sequences of periodic and chaotic states observed in some experiments on the belousov zhabotinsky reaction when conducted in a continuously stirred tank reactor we present numerical results obtained with a seven variable oregonator model which reproduce most of the features of the experimental sequences we discuss the origin and the nature of the chemical chaos encountered along these sequences in terms of the chaotic orbits which exist on the way to homoclinicity we construct poincare map models which reduce to either multi humped or multi branched 1d maps in the strong area contraction limit we report on a recent experiment which provides evidence that the iteration scheme deduced from the time series of spiral type strange attractors satistifies the symbolic dynamics predicted by sil nikov theory of homoclinic chaos we then investigate spatio temporal pattern forming phenomena in a one dimensional reaction diffusion system with equal diffusion coefficients when imposing a concentration gradient through the system this model mimics the sustained stationary and periodically oscillating front structures observed in recent experiments conducted in an open couette flow reactor we emphasize the possibility that the oscillations of the spatial structure become chaotic we report on numerical simulations of a diffusion induced intermittent bursting phenomenon that is likely to be observed in bench experiments we elaborate on the interpretation of this intermittent occurrence of spatially localized structures in an extended system in terms of sil nikov s homoclinic chaos

a two dimensional isogeometric boundary element method for elastostatic analysis the concept of isogeometric analysis where functions that are used to describe geometry in cad software are used to approximate the unknown fields in numerical simulations has received great attention in recent years the method has the potential to have profound impact on engineering design since the task of meshing which in some cases can add significant overhead has been circumvented much of the research effort has been focused on finite element implementations of the isogeometric concept but at present little has been seen on the application to the boundary element method the current paper proposes an isogeometric boundary element method bem which we term igabem applied to two dimensional elastostatic problems using non uniform rational b splines nurbs we find it is a natural fit with the isogeometric concept since both the nurbs approximation and bem deal with quantities entirely on the boundary the method is verified against analytical solutions where it is seen that superior accuracies are achieved over a conventional quadratic isoparametric bem implementation

theta lifting from elliptic cusp forms to automorphic forms on sp 1 q tsuneo arakawa formulated a theta lifting from elliptic cusp forms to automorphic forms on sp 1 q in his unpublished note which was inspired by kudla lifting i e a theta lifting from elliptic modular forms to holomorphic automorphic forms on su 1 q we prove that the images of arakawa s theta lifting belong to the space of bounded automorphic forms generating quaternionic discrete series which are non holomorphic forms in the appendix we provide the construction of eisenstein series and poincare series generating such discrete series

complex dynamics of a single neuron model we study a mathematical model of a single neuron with self coupling the model is based on the fitzhugh nagumo oscillator and an equation describing synaptic properties of the neuron the analysis of the model is focused on its dynamics depending on parameters characterizing synaptic time constants and external signals that affect the neuron applying lyapunov exponents and bifurcation analysis we point out the occurrence of parameter regions with different behavior such as bursting chaotic or periodic spiking and multistable phenomena moreover we can describe the dynamics of the model using an analytical approximation of the one dimensional poincar e map extracted from the numerical simulations

local and global behavior near homoclinic orbits we study the local behavior of systems near homoclinic orbits to stationary points of saddle focus type we explicitly describe how a periodic orbit approaches homoclinicity and with the help of numerical examples discuss how these results relate to global patterns of bifurcations

knotted periodic orbits in suspensions of smale s horseshoe torus knots and bifurcation sequences we construct a suspension of smale s horseshoe diffeomorphism of the two dimensional disc as a flow in an orientable three manifold such a suspension is natural in the sense that it occurs frequently in periodically forced nonlinear oscillators such as the duffing equation from this suspension we construct a knot holder or template a branched two manifold with a semiflow in such a way that the periodic orbits are isotopic to those in the full three dimensional flow we discuss some of the families of knotted periodic orbits carried by this template in particular we obtain theorems of existence uniqueness and non existence for families of torus knots we relate these families to resonant hamiltonian bifurcations which occur as horseshoes are created in a one parameter family of area preserving maps and we also relate them to bifurcations of families of one dimensional quadratic like maps which can be studied by kneading theory thus using knot theory kneading theory and hamiltonian bifurcation theory we are able to connect a countable subsequence of one dimensional bifurcations with a subsequence of area preserving bifurcations in a two parameter family of suspensions in which horseshoes are created as the parameters vary one implication is that infinitely many bifurcation sequences are reversed as one passes from the one dimensional to the area preserving family there are no universal routes to chaos

isogeometric analysis in electromagnetics b splines approximation abstract we introduce a new discretization scheme for maxwell equations in two space dimension inspired by the new paradigm of isogeometric analysis introduced in hughes et al 2005 16 we propose an algorithm based on the use of bivariate b splines spaces suitably adapted to electromagnetics we construct b splines spaces of variable interelement regularity on the parametric domain these spaces and their push forwards on the physical domain form a de rham diagram and we use them to solve the maxwell source and eigen problem our scheme has the following features i is adapted to treat complex geometries ii is spectral correct iii provides regular e g globally c 0 discrete solutions of maxwell equations

voltage interval mappings for an elliptic bursting model we employed poincare return mappings for a parameter interval to an exemplary elliptic bursting model the fitzhugh nagumo rinzel model using the interval mappings we were able to examine in detail the bifurcations that underlie the complex activity transitions between tonic spiking and bursting bursting and mixed mode oscillations and finally mixed mode oscillations and quiescence in the fitzhugh nagumo rinzel model we illustrate the wealth of information qualitative and quantitative that was derived from the poincare mappings for the neuronal models and for similar electro chemical systems

implementation of regularized isogeometric boundary element methods for gradient based shape optimization in two dimensional linear elasticity the present work addresses shape sensitivity analysis and optimization in two dimensional elasticity with a regularized isogeometric boundary element method igabem non uniform rational b splines are used both for the geometry and the basis functions to discretize the regularized boundary integral equations with the advantage of tight integration of design and analysis the application of igabem in shape optimization reduces the mesh generation regeneration burden greatly the work is distinct from the previous literatures in igabem shape optimization mainly in two aspects 1 the structural and sensitivity analysis takes advantage of the regularized form of the boundary integral equations eliminating completely the need of evaluating strongly singular integrals and jump terms and their shape derivatives which were the main implementation difficulty in igabem and 2 although based on the same computer aided design cad model the mesh for structural and shape sensitivity analysis is separated from the geometrical design mesh thus achieving a balance between less design variables for efficiency and refined mesh for accuracy this technique was initially used in isogeometric finite element method and was incorporated into the present igabem implementation

fast isogeometric boundary element method based on independent field approximation abstract an isogeometric boundary element method for problems in elasticity is presented which is based on an independent approximation for the geometry traction and displacement field this enables a flexible choice of refinement strategies permits an efficient evaluation of geometry related information a mixed collocation scheme which deals with discontinuous tractions along non smooth boundaries and a significant reduction of the right hand side of the system of equations for common boundary conditions all these benefits are achieved without any loss of accuracy compared to conventional isogeometric formulations the system matrices are approximated by means of hierarchical matrices to reduce the computational complexity for large scale analysis for the required geometrical bisection of the domain a strategy for the evaluation of bounding boxes containing the supports of nurbs basis functions is presented the versatility and accuracy of the proposed methodology are demonstrated by convergence studies showing optimal rates and real world examples in two and three dimensions

enriched isogeometric collocation for two dimensional time harmonic acoustics abstract in this work the isogeometric collocation iga c is paired with two types of enrichment plane wave pw and generalized harmonic polynomial ghp functions to solve 2d problems for the helmholtz equation a parametric study is conducted and a detailed assessment of the performance of the method in a number of benchmark problems is provided three different collocation methods are tested for the non enriched formulation namely greville abscissae ga approximated cauchy galerkin acg and superconvergent sc points showing that the acg scheme is the best choice in terms of overall error convergence rate and ease of solving the linear system then the influence of the number of shape functions in the original and enriched basis location and number of collocation points and wave number over both the convergence rate as well as the condition number of the stiffness matrix are studied the numerical results show that 1 there is an improvement over the non enriched formulation 2 the improvement depends on the choice of the number and type of enrichment and 3 the pollution error is not completely alleviated with the enriched formulations

the conley index for fast slow systems ii multidimensional slow variable we develop a qualitative theory for fast slow systems with a one dimensional slow variable using conley index theory for singularity perturbed systems conditions are given which imply that if one can construct heteroclinic connections and periodic orbits in systems with the derivative of the slow variable set to 0 these orbits persist when the derivative of the slow variable is small and nonzero

what can we learn from homoclinic orbits in chaotic dynamics state diagrams of two model systems involving three variables are constructed the parameter dependence of different forms of complex nonperiodic behavior and particularly of homoclinic orbits is analyzed it is shown that the onset of homoclinicity is reflected by deep changes in the qualitative behavior of the system

return map characterizations for a model of bursting with two slow variables various physiological systems display bursting electrical activity bea there exist numerous three variable models to describe this behavior however higher dimensional models with two slow processes have recently been used to explain qualitative features of the bea of some experimentally observed systems t chay and d cook math biosci 90 1988 pp 139 153 p smolen and j keizer j memb biol 127 1992 pp 9 19 r bertram et al biophys j 79 2000 pp 2880 2892 r bertram et al biophys j 68 1995 pp 2323 2332 j keizer and p smolen proc nat acad sci usa 88 1991 pp 3897 3901 in this paper we present a model with two slow and two fast variables for some parameter values the system has stable equilibria while for other values there exist bursting solutions singular perturbation methods are used to define a one dimensional return map wherein fixed points correspond to singular bursting solutions we analytically demonstrate that bursting solutions may exist

isogeometric analysis using t splines on two patch geometries we develop optimal approximation estimates for t splines in the case of geometries obtained by gluing two standard tensor product patches we derive results both for the t spline space in the parametric domain and the mapped t nurbs in the physical one a set of numerical tests in complete accordance with the theoretical developments is also presented

acoustic isogeometric boundary element analysis an isogeometric boundary element method based on t splines is used to simulate acoustic phenomena we restrict our developments to low frequency problems to establish the fundamental properties of the proposed approach using t splines the computer aided design cad and boundary element analysis are integrated without recourse to geometry clean up or mesh generation a regularized burton miller formulation is used resulting in integrals which are at most weakly singular we employ a collocation based approach to generate the linear system of equations the method is verified against closed form solutions and direct comparisons are made with conventional lagrangian discretizations it is demonstrated that the superior accuracy of the isogeometric approach emanates from the exact geometric description encapsulated in the t spline the method is then applied to a real world application to illustrate the potential for integrated engineering design and analysis

isogeometric boundary element methods for three dimensional static fracture and fatigue crack growth abstract we present a novel numerical method to simulate crack growth in 3d directly from the computer aided design cad geometry of the component without any mesh generation the method is an isogeometric boundary element method igabem based on non uniform rational b splines nurbs nurbs basis functions are used for the domain and crack representation as well as to approximate the physical quantities involved in the simulations a stable quadrature scheme for singular integration is proposed to enhance the robustness of the method in dealing with highly distorted elements convergence studies in the crack opening displacement is performed for a penny shaped crack and an elliptical crack two approaches to extract stress intensity factors sifs the contour m integral and the virtual crack closure integral are compared using dual integral equations the results show remarkable accuracy in the computed sifs leading to smooth crack paths and reliable fatigue lives without requiring the generation of any mesh from the cad model of the component under consideration

reduction of a model of an excitable cell to a one dimensional map we use qualitative methods for singularly perturbed systems of differential equations and the principle of averaging to compute the first return map for the dynamics of a slow variable calcium concentration in the model of an excitable cell the bifurcation structure of the system with continuous time endows the map with distinct features it is a unimodal map with a boundary layer corresponding to the homoclinic bifurcation in the original model this structure accounts for different periodic and aperiodic regimes and transitions between them all parameters in the discrete system have biophysical meaning which allows for precise interpretation of various dynamical patterns our results provide analytical explanation for the numerical studies reported previously