a unified approach for analyzing static and dynamic behaviors of functionally graded timoshenko and euler bernoulli beams abstract this paper presents a new unified approach for analyzing the static and dynamic behaviors of functionally graded beams fgb with the rotary inertia and shear deformation included as two special cases the euler bernoulli and rayleigh beam theories can be analytically reduced from the timoshenko beam theory all material properties are arbitrary functions along the beam thickness a single fourth order governing partial differential equation is derived and all physical quantities can be expressed in terms of the solution of the resulting equation the static result of deflection and stress distribution is presented for a cantilever fgb furthermore two branches of flexural waves propagating in fgb are obtained with different wave speeds the higher wave speed disappears when the effects of neither the rotary inertia nor shear deformation are considered free vibration of an fgb is analyzed and the frequency equation is given the natural frequencies and mode shapes of a simply supported beam are obtained for frequencies lower than equal to and higher than the cut off frequency numerical results are presented for an fgb with the power law gradient and a laminated beam the second frequency spectrum is found to exist when frequencies exceed the cut off frequency in addition double frequencies may occur for certain specified geometry of the beam previous results for a homogeneous timoshenko beam can be recovered from the present only letting the material properties be constant the suggested method is also applicable to layered timoshenko beams

locking free isogeometric collocation methods for spatial timoshenko rods in this work we present the application of isogeometric collocation techniques to the solution of spatial timoshenko rods the strong form equations of the problem are presented in both displacement based and mixed formulations and are discretized via nurbs based isogeometric collocation several numerical experiments are reported to test the accuracy and efficiency of the considered methods as well as their applicability to problems of practical interest in particular it is shown that mixed collocation schemes are locking free independently of the choice of the polynomial degrees for the unknown fields such an important property is also analytically proven

isogeometric collocation methods with generalized b splines we introduce isogeometric collocation methods based on generalized b splines and we analyze their performance through numerical examples for univariate and multivariate scalar and vector valued problems in particular advection diffusion and linear elasticity model problems are addressed the resulting method combines the favorable properties of isogeometric collocation and the geometrical and analytical problem oriented advantages of generalized b splines

an isogeometric collocation approach for bernoulli euler beams and kirchhoff plates abstract in this paper iga collocation methods are for the first time introduced for the solution of thin structural problems described by the bernoulli euler beam and kirchhoff plate models in particular a precise description of the proposed methods of the relevant implementation details and of the strategy to efficiently deal with different combinations of boundary conditions is given finally several numerical experiments confirm that the proposed formulations represent an efficient and geometrically flexible tool for the simulation of thin structures

mathematical properties of the jpeg2000 wavelet filters the legall 5 3 and daubechies 9 7 filters have risen to special prominence because they were selected for inclusion in the jpeg2000 standard we determine their key mathematical features riesz bounds order of approximation and regularity holder and sobolev we give approximation theoretic quantities such as the asymptotic constant for the l sup 2 error and the angle between the analysis and synthesis spaces which characterizes the loss of performance with respect to an orthogonal projection we also derive new asymptotic error formulae that exhibit bound constants that are proportional to the magnitude of the first nonvanishing moment of the wavelet the daubechies 9 7 stands out because it is very close to orthonormal but this turns out to be slightly detrimental to its asymptotic performance when compared to other wavelets with four vanishing moments

variational formulation and isogeometric analysis for fourth order boundary value problems of gradient elastic bar and plane strain stress problems abstract the fourth order boundary value problems of one parameter gradient elastic bar and plane strain stress models are formulated in a variational form within an h 2 sobolev space setting for both problems the existence and uniqueness of the solution is established by proving the continuity and coercivity of the associated symmetric bilinear form for completeness the full sets of boundary conditions of the problems are derived and in particular the new types of boundary conditions featured by the gradient elastic models are given the additional attributes singly and doubly by utilizing the continuity and coercivity of the continuous problems corresponding error estimates are formulated for conforming galerkin formulations finally numerical results with isogeometric c p 1 continuous discretizations for nurbs basis functions of order p 2 confirm the theoretical results and illustrate the essentials of both static and vibration problems

isogeometric collocation for elastostatics and explicit dynamics we extend the development of collocation methods within the framework of isogeometric analysis iga to multi patch nurbs configurations various boundary and patch interface conditions and explicit dynamic analysis the methods developed are higher order accurate stable with no hourglass modes and efficient in that they require a minimum number of quadrature evaluations the combination of these attributes has not been obtained previously within standard finite element analysis

isogeometric collocation neumann boundary conditions and contact abstract isogeometric collocation methods have been proposed recently and their accuracy and efficiency demonstrated for elastostatics and explicit dynamics this paper addresses two important aspects in the development of the isogeometric collocation technology namely the imposition of neumann boundary conditions and the enforcement of contact constraints which are both treated within the same framework it is shown that the strong imposition of neumann boundary conditions may lead to a significant loss of accuracy in some situations in particular when non uniform meshes are used two possible remedies are proposed to restore the desired level of accuracy while keeping the computational cost virtually unchanged i e a hybrid collocation galerkin approach and an enhanced collocation ec approach a frictionless contact formulation suitable for the collocation framework is further proposed and shown to pass the contact patch test to machine precision when combined with the ec approach the formulation is shown to deliver accurate results and to perform robustly also for highly non uniform meshes for all the collocation formulations contact pressures are greater than or equal to zero pointwise in contrast with standard lagrange finite elements

an isogeometric implicit g1 mixed finite element for kirchhoff space rods abstract in the paper we present a multi patch isogeometric mixed element with implicit geometric continuity at the ends for the analysis of non polar kirchhoff space rods rotations are introduced at the ends of the element analogously to the hermitian interpolation performing a change of basis for the configuration space a mixed method in which the internal forces are interpolated by means of an l 2 projection leading to a b formulation is used to cope with the membrane flexural locking that affects the solution the efficiency of the method in avoiding locking and instabilities is investigated the formulation adopted has the advantage with respect to other formulations that a non full global stiffness matrix is obtained this approach appears very appealing for efficient analyses of general assemblies of 3d non polar elements

explicit isogeometric collocation for the dynamics of three dimensional beams undergoing finite motions abstract we initiate the study of three dimensional shear deformable geometrically exact beam dynamics through explicit isogeometric collocation methods the formulation we propose is based on a natural combination of the chosen finite rotations representation with an explicit geometrically consistent lie group time integrator we focus on extending the integration scheme originally proposed for rigid body dynamics to our nonlinear initial boundary value problem where special attention is required by neumann boundary conditions the overall formulation is simple and only relies on a geometrically consistent procedure to compute the internal forces once control angular and linear accelerations of the beam cross sections are obtained from the previous time step the capabilities of the method are shown through numerical applications involving very large displacements and rotations and different boundary conditions

convergence of nonstationary cascade algorithms a nonstationary multiresolution of l 2 mathbb r s is generated by a sequence of scaling functions phi k in l 2 mathbb r s k in mathbb z we consider phi k that is the solution of the nonstationary refinement equations phi k m sum j h k 1 j phi k 1 m cdot j k in mathbb z where h k is finitely supported for each k and m is a dilation matrix we study various forms of convergence in l 2 mathbb r s of the corresponding nonstationary cascade algorithm phi k n m sum j h k 1 j phi k 1 n 1 m cdot j as k or n tends to infty it is assumed that there is a stationary refinement equation at infty with filter sequence h and that sum k h k j h j infty the results show that the convergence of the nonstationary cascade algorithm is determined by the spectral properties of the transition operator associated with h

cardinal exponential splines part i theory and filtering algorithms causal exponentials play a fundamental role in classical system theory starting from those elementary building blocks we propose a complete and self contained signal processing formulation of exponential splines defined on a uniform grid we specify the corresponding b spline basis functions and investigate their reproduction properties green function and exponential polynomials we also characterize their stability riesz bounds we show that the exponential b spline framework allows an exact implementation of continuous time signal processing operators including convolution differential operators and modulation by simple processing in the discrete b spline domain we derive efficient filtering algorithms for multiresolution signal extrapolation and approximation extending earlier results for polynomial splines finally we present a new asymptotic error formula that predicts the magnitude and the nth order decay of the l sub 2 approximation error as a function of the knot spacing t

isogeometric collocation methods for cosserat rods and rod structures abstract we present a novel method for the mechanical simulation of slender elastic spatial rods and rod structures subject to large deformation and rotation we develop an isogeometric collocation method for the geometrically exact nonlinear cosserat rod theory the rod centerlines are represented as spatial nurbs curves and cross section orientations are parameterized in terms of unit quaternions as 4 dimensional nurbs curves within the isogeometric framework the strong forms of the equilibrium equations of forces and moments of the discretized cosserat model are collocated leading to an efficient method for higher order discretizations for rod structures consisting of multiple connected rods we introduce a formulation with rigid quasi g 1 coupling it is based on the strong enforcement of continuity of displacement and change of cross section orientation at interfaces we also develop a mixed isogeometric formulation which is based on an independent discretization of internal forces and moments and alleviates shear locking for thin rods the novel rod simulation methods are verified by numerical convergence studies further computational examples include realistic applications with large deformations and rotations as well as a large scale rod structure with several hundreds of coupled rods and complex buckling behavior

a hierarchic family of isogeometric shell finite elements a hierarchic family of isogeometric shell finite elements based on nurbs shape functions is presented in contrast to classical shell finite element formulations inter element continuity of at least c1 enables a unique and continuous representation of the surface normal within one nurbs patch this does not only facilitate formulation of kirchhoff love type shell models for which the standard galerkin weak form has a variational index of 2 but it also offers significant advantages for shear deformable reissner mindlin type shells and higher order shell models for a 5 parameter shell formulation with reissner mindlin kinematics a hierarchic difference vector which accounts for shear deformations is superimposed onto the rotated kirchhoff love type director of the deformed configuration this split into bending and shear deformations in the shell kinematics results in an element formulation which is free from transverse shear locking without the need to apply further remedies like reduced integration assumed natural strains or mixed finite element formulations the third member of the hierarchy is a 7 parameter model including thickness change and allowing for application of unmodified three dimensional constitutive laws the phenomenon of curvature thickness locking coming along with this kinematic extension again is automatically avoided by the hierarchic difference vector concept without any further treatment membrane locking and in plane shear locking are removed by two different approaches firstly elimination via the discrete strain gap dsg method and secondly removal of parasitic membrane strains using a hybrid mixed method based on the hellinger reissner variational principle the hierarchic kinematic structure of the three different shell formulations allows a straightforward combination of these elements within one mesh and is thus the ideal basis for a model adaptive approach

biorthogonal bases of compactly supported wavelets orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide we show here that under fairly general conditions exact reconstruction schemes with synthesis filters different from the analysis filters give rise to two dual riesz bases of compactly supported wavelets we give necessary and sufficient conditions for biorthogonality of the corresponding scaling functions and we present a sufficient condition for the decay of their fourier transforms we study the regularity of these biorthogonal bases we provide several families of examples all symmetric corresponding to linear phase filters in particular we can construct symmetric biorthogonal wavelet bases with arbitrarily high preassigned regularity we also show how to construct symmetric biorthogonal wavelet bases close to a nonsymmetric orthonormal basis

isogeometric collocation methods for the reissner mindlin plate problem abstract within the general framework of isogeometric methods collocation schemes have been recently proposed as a viable and promising low cost alternative to standard isogeometric galerkin approaches in this paper isogeometric collocation methods for the numerical approximation of reissner mindlin plate problems are proposed for the first time locking free primal and mixed formulations are herein considered and the potential of isogeometric collocation as a geometrically flexible and computationally efficient simulation tool for shear deformable plates is shown through the solution of several numerical tests

single variable formulations and isogeometric discretizations for shear deformable beams abstract we present numerical formulations of timoshenko beams with only one unknown the bending displacement and it is shown that all variables of the beam problem can be expressed in terms of it and its derivatives we develop strong and weak forms of the problem the strong form of the problem involves the fourth derivative of the bending displacement whereas the symmetric weak form involves somewhat surprisingly third and second derivatives based on these we develop isogeometric collocation and galerkin formulations that are completely locking free and involve only half the degrees of freedom compared to standard timoshenko beam formulations several numerical tests are presented to demonstrate the performance of the proposed formulations

image coding using wavelet transform a scheme for image compression that takes into account psychovisual features both in the space and frequency domains is proposed this method involves two steps first a wavelet transform used in order to obtain a set of biorthogonal subclasses of images the original image is decomposed at different scales using a pyramidal algorithm architecture the decomposition is along the vertical and horizontal directions and maintains constant the number of pixels required to describe the image second according to shannon s rate distortion theory the wavelet coefficients are vector quantized using a multiresolution codebook to encode the wavelet coefficients a noise shaping bit allocation procedure which assumes that details at high resolution are less visible to the human eye is proposed in order to allow the receiver to recognize a picture as quickly as possible at minimum cost a progressive transmission scheme is presented it is shown that the wavelet transform is particularly well adapted to progressive transmission

wavelet theory demystified we revisit wavelet theory starting from the representation of a scaling function as the convolution of a b spline the regular part of it and a distribution the irregular or residual part this formulation leads to some new insights on wavelets and makes it possible to rederive the main results of the classical theory including some new extensions for fractional orders n a self contained accessible fashion in particular we prove that the b spline component is entirely responsible for five key wavelet properties order of approximation reproduction of polynomials vanishing moments multiscale differentiation property and smoothness regularity of the basis functions we also investigate the interaction of wavelets with differential operators giving explicit time domain formulas for the fractional derivatives of the basis functions this allows us to specify a corresponding dual wavelet basis and helps us understand why the wavelet transform provides a stable characterization of the derivatives of a signal additional results include a new peeling theory of smoothness leading to the extended notion of wavelet differentiability in the l sub p sense and a sharper theorem stating that smoothness implies order

b spline interpolation of kirchhoff love space rods the paper deals with the isogeometric analysis via b splines of space rods under kirchhoff love hypotheses the approach was used by gontier and vollmer 12 for developing a plane curve element within the framework of the timoshenko rod model but they adopted only one patch to represent entirely the geometry of the rod furthermore the authors developed their theory only for plane elements in this work we develop an isogeometric approach for the numerical analysis of the 3d kirchhoff love rod theory we use b splines and bezier interpolations and we show that they are able to attain very good accuracy for rod structures particularly for developing a 3d exact curve element with geometric torsion the paper presents an original parametrization of the geometric torsion that proves to be very effective the use of b splines allows to avoid discontinuities on the geometrical quantities and particularly on the normal fields so that even relatively low order interpolation functions are able to yield accurate results

isogeometric analysis for nonlinear planar kirchhoff rods weighted residual formulation and collocation of the strong form abstract high order shape functions used in isogeometric analysis allow the direct solution not only to the weighted residual formulation of the strong form opening the door to new integration schemes e g reduced gauss lobatto quadrature integration at superconvergent sites but also to collocation approaches e g using greville or superconvergent collocation points the goal of the present work is to compare these different methods through the application to the planar kirchhoff rod a fourth order rotation free formulation used to model slender beams under large deformations robustness of the geometrically nonlinear solution is improved using the mixed integration point mip newton a method that combines the advantages of displacement and mixed formulations based on the observations of the convergence plots convergence order estimates are provided for each discretization method advantages and weaknesses in terms of robustness and computation cost are also discussed for a set of benchmarks

variational boundary conditions based on the nitsche method for fitted and unfitted isogeometric discretizations of the mechanically coupled cahn hilliard equation the primal variational formulation of the fourth order cahn hilliard equation requires c1 continuous finite element discretizations e g in the context of isogeometric analysis in this paper we explore the variational imposition of essential boundary conditions that arise from the thermodynamic derivation of the cahn hilliard equation in primal variables our formulation is based on the symmetric variant of nitsche s method does not introduce additional degrees of freedom and is shown to be variationally consistent in contrast to strong enforcement the new boundary condition formulation can be naturally applied to any mapped isogeometric parametrization of any polynomial degree in addition it preserves full accuracy including higher order rates of convergence which we illustrate for boundary fitted discretizations of several benchmark tests in one two and three dimensions unfitted cartesian b spline meshes constitute an effective alternative to boundary fitted isogeometric parametrizations for constructing c1 continuous discretizations in particular for complex geometries we combine our variational boundary condition formulation with unfitted cartesian b spline meshes and the finite cell method to simulate chemical phase segregation in a composite electrode this example involving coupling of chemical fields with mechanical stresses on complex domains and coupling of different materials across complex interfaces demonstrates the flexibility of variational boundary conditions in the context of higher order unfitted isogeometric discretizations

generalized daubechies wavelet families we present a generalization of the orthonormal daubechies wavelets and of their related biorthogonal flavors cohen daubechies feauveau 9 7 our fundamental constraint is that the scaling functions should reproduce a predefined set of exponential polynomials this allows one to tune the corresponding wavelet transform to a specific class of signals thereby ensuring good approximation and sparsity properties the main difference with the classical construction of daubechies is that the multiresolution spaces are derived from scale dependent generating functions however from an algorithmic standpoint mallat s fast wavelet transform algorithm can still be applied the only adaptation consists in using scale dependent filter banks finite support ensures the same computational efficiency as in the classical case we characterize the scaling and wavelet filters construct them and show several examples of the associated functions we prove that these functions are square integrable and that they converge to their classical counterparts of the corresponding order

an isogeometric collocation method for frictionless contact of cosserat rods abstract a frictionless contact formulation for spatial rods is developed within the framework of isogeometric collocation the structural mechanics is described by the cosserat theory of geometrically nonlinear spatial rods the numerical discretization is based on an isogeometric collocation method where the geometry and solution fields are represented as nurbs curves and the strong forms of the equilibrium equations are collocated at greville points in this framework a frictionless rod to rod contact formulation is proposed contact points are detected by a coarse level and a refined search for close centerline points and reaction forces are computed by the actual penetration of rod surface points so that the enforcement of the contact constraints is performed with the penalty method an important aspect is the application of contact penalty forces as point loads within the collocation scheme and methods for this purpose are proposed and evaluated the overall contact algorithm is validated by and applied to several numerical examples

isogeometric collocation using analysis suitable t splines of arbitrary degree abstract this paper deals with the use of analysis suitable t splines of arbitrary degree in combination with isogeometric collocation methods for the solution of second and fourth order boundary value problems in fact analysis suitable t splines appear to be a particularly efficient locally refinable basis for isogeometric collocation able to conserve the cost of only one point evaluation per degree of freedom typical of standard nurbs based isogeometric collocation furthermore t splines allow to easily create highly non uniform meshes without introducing elements with high aspect ratios this makes it possible to avoid the numerical instabilities that may arise in the case of problems characterized by reduced regularity when neumann boundary conditions are imposed in strong form and elements with high aspect ratio are used the local refinement properties of t splines can be also successfully exploited to approximate problems where point loads are applied finally several numerical tests are herein presented in order to confirm all the above mentioned features as well as the good overall convergence properties of the combination of isogeometric collocation and analysis suitable t splines

locking free variational formulations and isogeometric analysis for the timoshenko beam models of strain gradient and classical elasticity abstract the timoshenko beam bending problem is formulated in the context of strain gradient elasticity for both static and dynamic analysis two non standard variational formulations in the sobolev space framework are presented in order to avoid the numerical shear locking effect pronounced in the strain gradient context both formulations are shown to be reducible to their locking free counterparts of classical elasticity conforming galerkin discretizations for numerical results are obtained by an isogeometric c p 1 continuous approach with b spline basis functions of order p 2 convergence analyses cover both statics and free vibrations as well as both strain gradient and classical elasticity parameter studies for the thickness and gradient parameters including micro inertia terms demonstrate the capability of the beam model in capturing size effects finally a model comparison between the gradient elastic timoshenko and euler bernoulli beam models justifies the relevance of the former confirmed by experimental results on nano beams from literature

collaborative design of fiber path and shape for complex composite shells based on isogeometric analysis abstract composite shells with complex geometry are widely used in aerospace structures due to the complexity of geometry and curvilinear fiber path the analysis and optimization based on finite element analysis fea for complex variable stiffness vs shells is extremely time consuming by comparing with fea isogeometric analysis iga exhibits higher prediction efficiency of buckling load in this work the formula of geometric stiffness matrix for complex vs shells is derived for the first time based on degenerated shell method using iga which is the basis of performing linear buckling analysis then a new variable curvature quasi linear function vcqlf to describe curvilinear fiber path is proposed which can further expand the design space of vs shells after that two frameworks for shape optimization of complex shells are put forward and then compared and it is found that the one based on loft function can provide representative control variables of shape and effectively reduces the number of design variables for complex shells finally a novel collaborative optimization framework of fiber path and shell shape using iga is established by comparison of traditional methods it is demonstrated that the proposed framework can greatly improve the efficiency of optimization and fully explore the buckling load of complex vs shells

a novel three variable shear deformation plate formulation theory and isogeometric implementation abstract this study brings to readers the generalized formulation of three variable plate theory and an efficient computational approach for analyzing plates the theory not only has three degree of freedoms dofs per node which complies with three dimensional space of full plate model as classical plate theory cpt but also accounts for the effect of shear deformation without any requirement of shear correction factors scf a complete set of strong forms weak form as well as classical and non classical boundary conditions bcs for linear and geometrically nonlinear analysis are consistently derived in this paper through a variational approach the strong forms are sixth order differential equations resulting in the symmetrical fourth order differential weak form it is known that isogeometric analysis iga arguably outweighs classical finite element method in terms of high continuity and high order differentiability thanks to its advantage an iga framework for the generalized three variable plate theory is formulated with completely locking free and only three dofs per node the classical bcs are strongly enforced to system equations as usual whilst the non classical bcs are weakly imposed by a penalty approach the new plate theory with only three variable is thereafter used for static linear and nonlinear analysis of isotropic and functionally graded material fgm plates to demonstrate its ability the reliability and accuracy of the present approach are ascertained by comparing the obtained results with other existing ones based on a robust formulation devoted in the paper the proposed approach can be further extended for numerous problems related to the shear deformable effect in the literature

analysis of asymptotically equivalent binary subdivision schemes non stationary uniform subdivision schemes are presented including some important examples as the up function and exponential splines the multivariate analysis of subdivision schemes is extended to a class of non stationary schemes which are asymptotically equivalent to converging stationary or non stationary schemes

isogeometric analysis for sixth order boundary value problems of gradient elastic kirchhoff plates abstract sixth order boundary value problems of a one parameter gradient elastic kirchhoff plate model are formulated in a weak form within an h 3 sobolev space setting with the corresponding equilibrium equations and general boundary conditions the corresponding conforming galerkin method is proposed with error estimates for discretizations satisfying c 2 continuity requirements continuity coercivity and consistency of the corresponding bilinear form are utilized for proving the theoretical results numerical computations with conforming isogeometric discretizations of c p 1 continuous nurbs basis functions of order p 3 confirm the theoretical results and illustrate the features of the problem for both statics and free vibrations in particular the effects of the additional boundary conditions and parameter dependent boundary layers corresponding to the gradient elasticity theory are addressed by the numerical examples

order reduction method for locking free isogeometric analysis of timoshenko beams the order reduction or method is proposed to provide an effective and efficient locking free formulation in the isogeometric analysis of plane curved timoshenko beams the shear and membrane strains are represented by the reduced order b spline basis functions and the locking phenomenon can be prevented the modified strains are equivalent to the original ones in the sense of the least square approximation different order reduction strategies are discussed and it is suggested to use multiple sets of lower order b spline basis functions to improve the accuracy the theoretical analysis of the formulation and the computational cost is carried out to assess the efficiency of the or method the performance of the proposed method is tested by the straight beam curved beams with constant and varied curvatures problems and compared with other methods such as the selective and reduced integration method the local discrete shear gap method and the local b method without strain smoothing the numerical results show that the proposed method is competitive which leads to an efficient locking free formulation

accurate efficient and iso geometrically flexible collocation methods for phase field models we propose new collocation methods for phase field models our algorithms are based on isogeometric analysis a new technology that makes use of functions from computational geometry such as for example non uniform rational b splines nurbs nurbs exhibit excellent approximability and controllable global smoothness and can represent exactly most geometries encapsulated in computer aided design cad models these attributes permitted us to derive accurate efficient and geometrically flexible collocation methods for phase field models the performance of our method is demonstrated by several numerical examples of phase separation modeled by the cahn hilliard equation we feel that our method successfully combines the geometrical flexibility of finite elements with the accuracy and simplicity of pseudo spectral collocation methods and is a viable alternative to classical collocation methods

from differential equations to the construction of new wavelet like bases in this paper an approach is introduced based on differential operators to construct wavelet like basis functions given a differential operator l with rational transfer function elementary building blocks are obtained that are shifted replicates of the green s function of l it is shown that these can be used to specify a sequence of embedded spline spaces that admit a hierarchical exponential b spline representation the corresponding b splines are entirely specified by their poles and zeros they are compactly supported have an explicit analytical form and generate multiresolution riesz bases moreover they satisfy generalized refinement equations with a scale dependent filter and lead to a representation that is dense in l sub 2 this allows us to specify a corresponding family of semi orthogonal exponential spline wavelets which provides a major extension of earlier polynomial spline constructions these wavelets are completely characterized and it is proven that they satisfy the following remarkable properties 1 they are orthogonal across scales and generate riesz bases at each resolution level 2 they yield unconditional bases of l sub 2 either compactly supported b spline type or with exponential decay orthogonal or dual type 3 they have n vanishing exponential moments where n is the order of the differential operator 4 they behave like multiresolution versions of the operator l from which they are derived and 5 their order of approximation is n m where n and m give the number of poles and zeros respectively last but not least the new wavelet like decompositions are as computationally efficient as the classical ones they are computed using an adapted version of mallat s filter bank algorithm where the filters depend on the decomposition level

isogeometric collocation for three dimensional geometrically exact shear deformable beams abstract we extend the isogeometric collocation method to the geometrically nonlinear beams an exact kinematic formulation able to represent three dimensional displacements and rotations without any restriction in magnitude is presented without the introduction of the moving frame concept a displacement based formulation is adopted full linearization of the strong form of the governing equations is derived consistently with the underlying geometric structure of the configuration manifold incremental rotations are parametrized through eulerian rotation vectors and configuration updates are performed by means of the exponential map numerical tests demonstrate that the proposed combination of isogeometric collocation method with the chosen rotations parametrization results in an efficient computational scheme able to model complex problems with high accuracy

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

locking free isogeometric collocation formulation for three dimensional geometrically exact shear deformable beams with arbitrary initial curvature abstract this paper presents a displacement based and a mixed isogeometric collocation formulations for three dimensional geometrically exact shear deformable beams with arbitrarily curved initial geometry strong forms of both formulations are consistently linearized with respect to the underlying configuration manifold high efficiency is achieved by using a geometrically consistent kinematic model in which incremental rotations are parametrized through the spatial rotation vector several numerical tests demonstrate that the proposed mixed formulation is totally locking free for any choice of approximation degree mixed formulation also exhibits higher robustness in the incremental iterative algorithm elimination of shear locking and low computational cost assured by the combination of isogeometric collocation with a consistent kinematic model result in a very efficient accurate and geometrically powerful method able to handle geometrically complex beam problems

effects of parameterization and knot placement techniques on primal and mixed isogeometric collocation formulations of spatial shear deformable beams with varying curvature and torsion abstract we present a displacement based and a mixed isogeometric collocation iga c formulation for free form three dimensional shear deformable beams with high and rapidly varying curvature and torsion when such complex shapes are concerned the approach used to build the iga geometric model becomes relevant although iga c has been so far successfully applied to a wide range of problems the effects that different parameterization and knot placement techniques may have on the accuracy of collocation based formulations is still an unexplored field to fill this gap primal and mixed formulations are used combining two parameterization methods chord length and equally spaced with two knot placement techniques uniformly spaced and de boor with respect to the space varying frenet local frame we derive the strong form of the governing equations in a compact form through the definition of two matrix operators conveniently used to perform first and second order derivatives of the vector fields involved in the formulations this approach is very efficient and easy to implement within a collocation based scheme several challenging numerical experiments allow to test the different considered parameterizations and knot placement techniques revealing in particular that with the primal formulation an equally spaced parameterization is definitively the most recommended choice and it should always be used with an approximation degree of at least p 6 although some caution must be adopted when very high jacobians and small curvatures occur the same holds for the mixed formulation with the difference that p 4 is enough to yield accurate results

avoiding shear locking for the timoshenko beam problem via isogeometric collocation methods in this work we study isogeometric collocation methods for the timoshenko beam problem considering both mixed and displacement based formulations in particular we show that locking free solutions are obtained for mixed methods independently on the approximation degrees selected for the unknown fields moreover several numerical tests are provided in order to support our theoretical results and to show the good behavior and the flexibility of isogeometric collocation methods in this context

isogeometric collocation for the reissner mindlin shell problem abstract we present an isogeometric collocation formulation for the reissner mindlin shell problem after recalling the necessary basics on differential geometry and the shell governing equations we show that the standard approach of expressing the equilibrium equations in terms of the primal variables is not a suitable way for shells due to the complexity of the underlying equations we then propose an alternative approach based on a stepwise formulation and show its numerical implementation within an isogeometric collocation framework the formulation is tested successfully on a set of benchmark examples which comprise important aspects like locking and boundary layers these test show that locking effects can be conveniently avoided by using high polynomial degrees an accompanying study on the computational time also confirms that high polynomial degrees are preferable in terms of computational efficiency

non prismatic timoshenko like beam model the present paper combines an effective beam theory with a simple and accurate numerical technique opening the door to the prediction of the structural behavior of planar beams characterized by a continuous variation of the cross section geometry that in general deeply influences the stress distribution and therefore leads to non trivial constitutive relations accounting for these peculiar aspects the beam theory is described by a mixed formulation of the problem represented by six linear ordinary differential equations odes with non constant coefficients depending on both the cross section displacements and the internal forces due to the odes complexity the solution can be typically computed only numerically also for relatively simple geometries loads and boundary conditions however the use of classical numerical tools for this problem like a six field mixed finite element approach might entail several issues e g shear locking ill conditioned matrices etc conversely the recently proposed isogeometric collocation method consisting of the direct discretization of the odes in strong form and using the higher continuity properties typical of spline shape functions allows an equal order approximation of all unknown fields without affecting the stability of the solution this makes such an approach simple robust efficient and particularly suitable for solving the system of odes governing the non prismatic beam problem several numerical experiments confirm that the proposed mixed isogeometric collocation method is actually cost effective and able to attain high accuracy

an improved quasilinearization method for second order nonlinear boundary value problems abstract solutions to the general second order nonlinear boundary value problems arising in many physical phenomena such as dynamic programming fluid flow over a solid object and heat transfer in a solid and or adjacent to a solid body in a liquid are obtained also existence uniqueness and convergence results are established by using quasilinearization and monotone iterative methods furthermore examples are presented to demonstrate the application of the solution techniques

isogeometric collocation cost comparison with galerkin methods and extension to adaptive hierarchical nurbs discretizations abstract we compare isogeometric collocation with isogeometric galerkin and standard c 0 finite element methods with respect to the cost of forming the matrix and residual vector the cost of direct and iterative solvers the accuracy versus degrees of freedom and the accuracy versus computing time on this basis we show that isogeometric collocation has the potential to increase the computational efficiency of isogeometric analysis and to outperform both isogeometric galerkin and standard c 0 finite element methods when a specified level of accuracy is to be achieved with minimum computational cost we then explore an adaptive isogeometric collocation method that is based on local hierarchical refinement of nurbs basis functions and collocation points derived from the corresponding multi level greville abscissae we introduce the concept of weighted collocation that can be consistently developed from the weighted residual form and the two scale relation of b splines using weighted collocation in the transition regions between hierarchical levels we are able to reliably handle coincident collocation points that naturally occur for multi level greville abscissae the resulting method combines the favorable properties of isogeometric collocation and hierarchical refinement in terms of computational efficiency local adaptivity robustness and straightforward implementation which we illustrate by numerical examples in one two and three dimensions

a shear deformable rotation free isogeometric shell formulation abstract a finite element formulation for a geometrically linear shear deformable reissner mindlin type shell theory is presented which exclusively uses displacement degrees of freedom the total displacement is subdivided into a part representing the membrane and bending deformation enriched by two extra shear displacements representing transverse shear deformation this rotation free approach is accomplished within the isogeometric concept using c 1 continuous quadratic nurbs as shape functions the particular displacement parametrization decouples transverse shear from bending and thus the formulation is free from transverse shear locking by construction i e locking is avoided on the theory level not by choice of a particular discretization compared to the hierarchic formulation proposed earlier within the group of the authors echter et al 2013 the method presented herein avoids artificial oscillations of the transverse shear forces up to now a similar displacement based method to avoid membrane locking has not been found thus in the present formulation the mixed method from echter et al 2013 is used to avoid membrane locking

a class of mixed assumed strain methods and the method of incompatible modes a three field mixed formulation in terms of displacements stresses and an enhanced strain field is presented which encompasses as a particular case the classical method of incompatible modes within this frame work incompatible elements arise as particular compatible mixed approximations of the enhanced strain field the conditions that the stress interpolation contain piece wise constant functions and be l2 ortho gonal to the enhanced strain interpolation ensure satisfaction of the patch test and allow the elimination of the stress field from the formulation the preceding conditions are formulated in a form particularly convenient for element design as an illustration of the methodology three new elements are developed and shown to exhibit good performance a plane 3d elastic plastic quad an axisymmetric element and a thick plate bending quad the formulation described herein is suitable for non linear analysis

isogeometric collocation for kirchhoff love plates and shells abstract with the emergence of isogeometric analysis iga the galerkin rotation free discretization of kirchhoff love shells is facilitated enabling more efficient thin shell structural analysis high order shape functions used in iga also allow the collocation of partial differential equations avoiding the time consuming numerical integration of the galerkin technique the goal of the present work is to apply this method to nurbs based isogeometric kirchhoff love plates and shells under the assumption of small deformations since kirchhoff love plate theory yields a fourth order formulation two boundary conditions are required at each location on the contour generating some conflicts at the corners where there are more equations than needed to remedy this overdetermination we provide priority and averaging rules that cover all the possible combinations of adjacent edge boundary conditions i e the clamped simply supported symmetric and free supports greville and alternative superconvergent points are used for nurbs basis of even and odd degrees respectively for square circular and annular flat plates convergence orders are found to be in agreement with a priori error estimates the proposed isogeometric collocation method is then validated and benchmarked against a galerkin implementation by studying a set of problems involving kirchhoff love shells

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed