a critical evaluation of the resolution properties of b spline and compact finite difference methods resolution properties of b spline and compact finite difference schemes are compared using fourier analysis in periodic domains and tests based on solution of the wave and heat equations in finite domains with uniform and nonuniform grids results show that compact finite difference schemes have a higher convergence rate and in some cases better resolution however b spline schemes have a more straightforward and robust formulation particularly near boundaries on nonuniform meshes

orthogonalized generalized iso geometric analysis ogiga and its applications to problems of fracture mechanics abstract ill conditioning of system matrices is one of the drawbacks of generalized extended isogeometric analysis giga xiga which results in increase in computational cost in the case of iterative solvers or erroneous results in the case of direct solvers linear dependency between the basis functions which can be measured in terms of smallest angle among all the bases is one of the attributes of ill conditioning in xiga in the present study a simplified method to find the smallest angle among all the bases is introduced also to reduce the problem of ill conditioning orthogonalized generalized isogeometric analysis ogiga is proposed in which the standard and the corresponding enriched basis functions are orthogonalized in order to study the performance of present method various fracture mechanics problems like 2 d homogenous cracks bi material interface cracks inclined cracks and 3 d crack problems are chosen the effect of crack position and the radius of enrichment zone on accuracy scaled condition number and smallest angle among all the bases is studied the results obtained from the proposed ogiga method are compared with results obtained from standard xiga and xiga with enrichment function orthogonalization oe xiga agathos et al 2019 in all the cases considered the conditioning of system matrices is significantly improved by employing present methodology and it can be easily extended to wide variety of problems

the matching of a prescribed linear input output behavior in a nonlinear system in this paper we deal with the problem of finding a feedback under which the input output behavior of a nonlinear system becomes exactly the same as that of a specified linear model the solvability of this problem is shown to depend on a formal infinite zero structure associated with the system

discontinuous galerkin finite element methods for linear elasticity and quasistatic linear viscoelasticity we consider a finite element in space and quadrature in time discretization of a compressible linear quasistatic viscoelasticity problem the spatial discretization uses a discontinous galerkin finite element method based on polynomials of degree r termed dg r and the time discretization uses a trapezoidal rectangle rule approximation to the volterra history integral both semi and fully discrete a priori error estimates are derived without recourse to gronwall s inequality and therefore the error bounds do not show exponential growth in time moreover the convergence rates are optimal in both h and r providing that the finite element space contains a globally continuous interpolant to the exact solution e g when using the standard k polynomial basis on simplicies or tensor product polynomials k on quadrilaterals when this is not the case e g using k on quadri laterals the convergence rate is suboptimal in r but remains optimal in h we also consider a reduction of the problem to standard linear elasticity where similarly optimal a priori error estimates are derived for the dg r approximation

isogeometric analysis using t splines we explore t splines a generalization of nurbs enabling local refinement as a basis for isogeometric analysis we review t splines as a surface design methodology and then develop it for engineering analysis applications we test t splines on some elementary two dimensional and three dimensional fluid and structural analysis problems and attain good results in all cases we summarize the current status of t splines their limitations and future possibilities

stability analysis of constrained control systems an alternative approach abstract the purpose of the present paper is to study stability of constrained nonlinear control systems usually this is done by reducing the constrained control system to an unconstrained control system with respect to the constrained manifold different from these approaches an alternative is proposed here which allows to establish stability without an explicit knowledge of the constrained manifold and the reduced unconstrained control system the main result is a simple stability theorem despite its simplicity the theorem can be applied to a broad class of stability problems for example the stability analysis of unstructured nonlinear differential algebraic equations of higher index and to lasalle s invariance principle furthermore the theorem allows a computationally efficient stability analysis of the class of constrained polynomial control systems using semidefinite programming and the sum of squares decomposition

spectral approximation properties of isogeometric analysis with variable continuity abstract we study the spectral approximation properties of isogeometric analysis with local continuity reduction of the basis such continuity reduction results in a decrement in the interconnection between the degrees of freedom of the mesh which allows for large computational savings during the solution of the resulting linear system the continuity reduction results in extra degrees of freedom that modify the approximation properties of the method the convergence rate of such refined isogeometric analysis is equivalent to that of the maximum continuity basis we show how the breaks in continuity and inhomogeneity of the basis lead to artifacts in the frequency spectra such as stopping bands and outliers and present a unified description of these effects in finite element method isogeometric analysis and refined isogeometric analysis accuracy of the refined isogeometric analysis approximations can be improved by using non standard quadrature rules in particular optimal quadrature rules lead to large reductions in the eigenvalue errors and yield two extra orders of convergence as it occurs in standard isogeometric analysis

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

nonlinear decoupling via feedback a differential geometric approach the paper deals with the nonlinear decoupling and noninteracting control problems a complete solution to those problems is made possible via a suitable nonlinear generalization of several powerful geometric concepts already introduced in studying linear multivariable control systems the paper also includes algorithms concerned with the actual construction of the appropriate control laws

performance limitations of non minimum phase systems in the servomechanism problem abstract this paper studies the cheap regulator problem and the cheap servomechanism problem for systems which may be non minimum phase the study extends some well known properties of perfect regulation and the perfect tracking and disturbance rejection of minimum phase systems to non minimum phase systems it is shown that perfect rejection to disturbances applied to the plant input can be achieved no matter whether the system is minimum phase or non minimum phase whereas a fundamental limitation exists in the achievable transient performance of tracking and rejection to disturbances applied to the plant output for a non minimum phase system and that this limitation can be simply and completely characterized by the number and locations of those zeros of the system which lie in the right half of the complex plane furthermore this limitation provides a quantitative measure of the degree of difficulty which is inherent in the control of such non minimum phase systems

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 numerical dispersion analysis for two dimensional elastic wave propagation abstract in this paper we carry out a numerical dispersion analysis for the linear two dimensional elastodynamics equations approximated by means of nurbs based isogeometric analysis in the framework of the galerkin method specifically we consider the analysis of harmonic plane waves in an isotropic and homogeneous elastic medium we compare and discuss the errors associated with the compressional and shear wave velocities and we provide the anisotropic curves for numerical approximations obtained by considering b spline and nurbs basis functions of different regularity namely globally c 0 and c p 1 continuous p being the polynomial degree we conclude our analysis by numerically simulating the seismic wave propagation in a sinusoidal shaped valley with discontinuous elastic parameters across an internal interface

a mortar spectral finite element method for complex 2d and 3d elastodynamic problems abstract we present a hybrid spectral element finite element domain decomposition method for solving elastic wave propagation problems aim of the method is to exploit both the enhanced accuracy of spectral elements allowing significant reductions of the computational load and the flexibility of finite elements in treating irregular geometries and non linear media interfaces between finite and spectral elements are managed by the mortar method which enjoys optimal accuracy and allows to discretize the different regions independently several test cases illustrate the robustness of this method and its tolerance to a significant range of geometrical models characterized by different grid refinements

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

new fractional step runge kutta nystrom methods up to order three abstract fractional step runge kutta nystro fsrkn methods have been revealed to be an excellent option to integrate numerically many multidimensional evolution models governed by second order in time partial differential equations these methods combined with suitable spatial discretizations lead to strong computational cost reductions respect to many classical implicit time integrators in this paper we present the construction process of several implicit fsrkn methods of two and three levels which attain orders up to three and satisfy adequate stability properties we have also performed some numerical experiments in order to show the unconditionally convergent behavior of these schemes as well as their computational advantages

isogeometric analysis of the advective cahn hilliard equation spinodal decomposition under shear flow we present a numerical study of the spinodal decomposition of a binary fluid undergoing shear flow using the advective cahn hilliard equation a stiff nonlinear parabolic equation characterized by the presence of fourth order spatial derivatives our numerical solution procedure is based on isogeometric analysis an approximation technique for which basis functions of high order continuity are employed these basis functions allow us to directly discretize the advective cahn hilliard equation without resorting to a mixed formulation we present steady state solutions for rectangular domains in two dimensions and for the first time in three dimensions we also present steady state solutions for the two dimensional taylor couette cell to enforce periodic boundary conditions in this curved domain we derive and utilize a new periodic bezier extraction operator we present an extensive numerical study showing the effects of shear rate surface tension and the geometry of the domain on the phase evolution of the binary fluid theoretical and experimental results are compared with our simulations

some estimates for h p k refinement in isogeometric analysis in this paper we propose a theoretical study of the approximation properties of nurbs spaces which are used in isogeometric analysis we obtain error estimates that are explicit in terms of the mesh size h the degree p and the global regularity measured by the parameter k our approach covers the approximation with global regularity from c 0 up to c k 1 with 2k 1 p notice that the interesting case of higher regularity up to k p is still open however our results give an indication of the role of the smoothness k in the approximation properties and offer a first mathematical justification of the potential of isogeometric analysis based on globally smooth nurbs

discontinuous galerkin finite element method for the wave equation the symmetric interior penalty discontinuous galerkin finite element method is presented for the numerical discretization of the second order wave equation the resulting stiffness matrix is symmetric positive definite and the mass matrix is essentially diagonal hence the method is inherently parallel and leads to fully explicit time integration when coupled with an explicit time stepping scheme optimal a priori error bounds are derived in the energy norm and the l 2 norm for the semidiscrete formulation in particular the error in the energy norm is shown to converge with the optimal order cal o h min s ell with respect to the mesh size h the polynomial degree ell and the regularity exponent s of the continuous solution under additional regularity assumptions the l 2 error is shown to converge with the optimal order cal o h min s ell numerical results confirm the expected convergence rates and illustrate the versatility of the method

a quadrature based superconvergent isogeometric frequency analysis with macro integration cells and quadratic splines abstract a quadrature based superconvergent isogeometric method is presented for efficient frequency analysis with macro integration cells and quadratic splines the present development particularly refers to the isogeometric analysis of scalar wave equations the single element two element and three element macro integration cells are proposed to develop a set of explicit superconvergent quadrature rules with a 6th order frequency accuracy in contrast to the 4th order frequency accuracy associated with the standard isogeometric formulation with consistent mass matrix the 1d quadrature rules with various macro integration cells are derived from the condition of exact integration of the higher order mass matrix as well as the stiffness matrix consequently 3 point 5 point and 7 point quadrature rules with identical precision are established for the single element two element and three element macro integration cells respectively by construction these quadrature rules exactly recover the isogeometric higher order mass matrix formulation with frequency superconvergence in 1d case the 2d and 3d superconvergent quadrature rules with versatile integration cells are directly constructed through the tensor product formulation of the 1d integration algorithms it is shown that the multidimensional isogeometric analysis employing the proposed quadrature rules for both mass and stiffness matrices does produce the frequency superconvergence simultaneously without the wave propagation direction dependence problem which needs special treatments for the multidimensional higher order mass matrix formulation the proposed approach is featured by its simplicity for numerical implementation and efficiency using macro integration cells numerical examples confirm the efficacy of the present methodology

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

inner outer factorization for nonlinear noninvertible systems this paper considers inner outer factorization of asymptotically stable nonlinear state space systems in continuous time that are noninvertible our approach will be via a nonlinear analogue of spectral factorization which concentrates on first finding the outer factor instead of the inner factor an application of the main result to control of nonminimum phase nonlinear systems is indicated

optimal and reduced quadrature rules for tensor product and hierarchically refined splines in isogeometric analysis abstract we continue the study initiated in hughes et al 2010 in search of optimal quadrature rules for tensor product and hierarchically refined splines in isogeometric analysis these rules are optimal in the sense that there exists no other quadrature rule that can exactly integrate the elements of the given spline space with fewer quadrature points we extend the algorithm presented in hughes et al 2010 with an improved starting guess which combined with arbitrary precision arithmetic results in the practical computation of quadrature rules for univariate non uniform splines up to any precision explicit constructions are provided in sixteen digits of accuracy for some of the most commonly used uniform spline spaces defined by open knot vectors we study the efficacy of the proposed rules in the context of full and reduced quadrature applied to two and three dimensional diffusion reaction problems using tensor product and hierarchically refined splines and prove a theorem rigorously establishing the stability and accuracy of the reduced rules

interior two dimensional acoustic modelling and modal analysis using isogeometric approach abstract in this paper the interior two dimensional acoustic modelling and modal analysis are presented in the framework of isogeometric analysis iga the acoustic domain governed by the helmholtz equation is parameterized and analysed by non uniform rational lagrange nurl basis which can construct exact geometry model inherently like non uniform rational b spines nurbs and can provide a more straightforward way than nurbs to impose dirichlet boundary conditions the verification and versatility of iga using nurl are assessed in the investigation of a unit square domain and a unit circular domain by comparison with analytical solutions furthermore a quadrilateral domain with concave or convex conics is studied and some innovative results are presented which can be the benchmark data for other algorithm researches the effects of geometric weight and aspect ratio on acoustic eigenvalue and eigenmode of the quadrilateral domain are investigated in detail

on the variational formulation of stress constraints in isogeometric design the isogeometric paradigm is aimed at unifying the geometric and analysis descriptions of engineering problems this unification is brought about by employing the same basis functions to describe exact geometry and to approximate the physical response the isogeometric framework has recently been extended to include design optimisation a novel variational stress constraint formulation is proposed in this paper that further extends recent efforts in isogeometric design the weak form of the constraint is obtained by utilising the same basis functions not only to represent geometry response and design variables but also lagrange multipliers in this form the proposed variational formulation fits seamlessly with the underlying analysis and design environment the variational constraint formulation is compared to the widely utilised stress averaging and collocation approaches to this end volume minimisation of elastic arches subjected to stress constraints is considered results indicate that the proposed variational approach has the best convergence rates and utilises a much smaller number of discrete constraints than competing methods

a finiteness criterion for nonlinear input output differential systems we solve the following problem which has been dealt with in several publications when does a system have the same input output behaviour as a system depending linearly on the state variables precise definitions of this question are supplied in the text a particular case of the problem is the equivalence of a system with a bilinear one at the end we illustrate our methods and the scope of our results by a brief study of several examples

duality and unified analysis of discrete approximations in structural dynamics and wave propagation comparison of p method finite elements with k method nurbs we study the discretization behavior of classical finite element and nurbs approximations on problems of structural vibrations and wave propagation we find that on the basis of equal numbers of degrees of freedom and bandwidth nurbs have superior approximation properties in fact we observe that the high mode behavior of classical finite elements is divergent with the order of approximation a surprisingly negative result on the other hand nurbs offer almost spectral approximation properties and all modes converge with increasing order of approximation

reduced bezier element quadrature rules for quadratic and cubic splines in isogeometric analysis abstract we explore the use of various element based reduced quadrature strategies for bivariate and trivariate quadratic and cubic spline elements used in isogeometric analysis the rules studied encompass tensor product gauss and gauss lobatto rules and certain so called monomial rules that do no possess a tensor product structure the objective of the study is to determine quadrature strategies which enjoy the same accuracy and stability behavior as full gauss quadrature but with significantly fewer quadrature points several cases emerge that satisfy this objective and also demonstrate superior efficiency compared with standard c 0 continuous finite elements of the same order

three dimensional simulation of unstable gravity driven infiltration of water into a porous medium infiltration of water in dry porous media is subject to a powerful gravity driven instability although the phenomenon of unstable infiltration is well known its description using continuum mathematical models has posed a significant challenge for several decades the classical model of water flow in the unsaturated flow the richards equation is unable to reproduce the instability here we present a computational study of a model of unsaturated flow in porous media that extends the richards equation and is capable of predicting the instability and captures the key features of gravity fingering quantitatively the extended model is based on a phase field formulation and is fourth order in space the new model poses a set of challenges for numerical discretizations such as resolution of evolving interfaces stiffness in space and time treatment of singularly perturbed equations and discretization of higher order spatial partial differential operators we develop a numerical algorithm based on isogeometric analysis a generalization of the finite element method that permits the use of globally smooth basis functions leading to a simple and efficient discretization of higher order spatial operators in variational form we illustrate the accuracy efficiency and robustness of our method with several examples in two and three dimensions in both homogeneous and strongly heterogeneous media we simulate for the first time unstable gravity driven infiltration in three dimensions and confirm that the new theory reproduces the fundamental features of water infiltration into a porous medium our results are consistent with classical experimental observations that demonstrate a transition from stable to unstable fronts depending on the infiltration flux

fast formation and assembly of finite element matrices with application to isogeometric linear elasticity abstract recently a new formation and assembly strategy was proposed in 1 which resulted in significant speedups in the formation and assembly time of the galerkin mass matrix in isogeometric analysis the strategy relies on three key ingredients 1 assembly row by row instead of element by element and an efficient formation strategy based on 2 sum factorization and 3 weighted quadrature that is applied to each specific row of the matrix compared to traditional element procedures applied to three dimensional problems the computational complexity is lowered from o p 9 per degree of freedom to o p 4 where p is the order of polynomials this is close to the theoretical minimum of o p 3 attained by for example collocation consequently this type of formation and assembly scales favorably with polynomial degree which opens the way for high order isogeometric analysis employing k refinement that is use of maximally smooth higher order splines in this work we discuss various important details for the practical implementation of the weighted quadrature formation strategy proposed in 1 specifically we extend the weighted quadrature scheme to accurately integrate the elements of the stiffness matrix in linear elasticity and propose a means of distributing quadrature points for non uniform mixed continuity spline spaces furthermore we discuss efficient access and assignment into the prevalent sparse matrix data structures namely compressed sparse row csr and compressed sparse column csc in particular row by row or column by column assembly allows matrix rows or columns respectively to be formed contiguously in the storage order of the sparse matrix thereby minimizing the memory overhead and eliminating the addition assignment operation on sparse matrices several three dimensional benchmark problems illustrate the efficiency and efficacy of the proposed formation and assembly technique applied to isogeometric linear elasticity we show that the accuracy of full gauss quadrature is maintained while the computational burden of forming the matrix equations is significantly reduced

on the stability of the rayleigh ritz method for eigenvalues this paper studies global stability properties of the rayleigh ritz approximation of eigenvalues of the laplace operator the focus lies on the ratios hat lambda k lambda k of the kth numerical eigenvalue hat lambda k and the kth exact eigenvalue lambda k in the context of classical finite elements the maximal ratio blows up with the polynomial degree for b splines of maximum smoothness the ratios are uniformly bounded with respect to the degree except for a few instable numerical eigenvalues which are related to the presence of essential boundary conditions these phenomena are linked to the inverse inequalities in the respective approximation spaces

isogeometric approximation of cardiac electrophysiology models on surfaces an accuracy study with application to the human left atrium abstract we consider isogeometric analysis in the framework of the galerkin method for the spatial approximation of cardiac electrophysiology models defined on nurbs surfaces specifically we perform a numerical comparison between basis functions of degree p 1 and globally c k continuous with k 0 or p 1 to find the most accurate approximation of a propagating front with the minimal number of degrees of freedom we show that b spline basis functions of degree p 1 which are c p 1 continuous capture accurately the front velocity of the transmembrane potential even with moderately refined meshes similarly we show that for accurate tracking of curved fronts high order continuous b spline basis functions should be used finally we apply isogeometric analysis to an idealized human left atrial geometry described by nurbs with physiologically sound fiber directions and anisotropic conductivity tensor to demonstrate that the numerical scheme retains its favorable approximation properties also in a more realistic setting

optimal discontinuous galerkin methods for wave propagation we have developed and analyzed a new class of discontinuous galerkin methods dg which can be seen as a compromise between standard dg and the finite element fe method in the way that it is explicit like standard dg and energy conserving like fe in the literature there are many methods that achieve some of the goals of explicit time marching unstructured grid energy conservation and optimal higher order accuracy but as far as we know only our new algorithms satisfy all the conditions we propose a new stability requirement for our dg the stability analysis is based on the careful selection of the two fe spaces which verify the new stability condition the convergence rate is optimal with respect to the order of the polynomials in the fe spaces moreover the convergence is described by a series of numerical experiments

dissipativity of pseudorational behaviors this paper studies dissipativity for a class of infinite dimensional systems called pseudorational in the behavioral context extending the finite dimensional counterpart we show that a pseudorational behavior is dissipative if and only if it admits a storage function or a dissipation function for its proof we derive a new necessary and sufficient condition for entire functions of exponential type in the so called paley wiener class to allow a symmetric factorization characterization of dissipative behaviors and linear quadratic optimal behaviors are also given

robustness of isogeometric structural discretizations under severe mesh distortion abstract this paper investigates higher order and higher continuity functions in isogeometric structural analysis under distortion of the control and physical meshes first the concepts behind isogeometric analysis are briefly reviewed the effect of perturbing control points upon the geometrical mapping is observed in the one dimensional setting the concept is extended to multiple dimensions with a series of cases where the interior of a cube is distorted in different ways and patch tests are performed to assess accuracy a circular fillet patch is introduced through the appropriate degeneration of a control mesh in order to address the problem of modeling small features in finite element models the fillet patch is tested with two problems of linear elasticity and is shown to be accurate the investigation is further extended to the setting of large deformation nearly incompressible nonlinear elasticity with two examples which make use of the recently proposed f projection method

a new 3 d numerical approach to the solution of pdes with optimal accuracy on irregular domains and cartesian meshes abstract a new 3 d numerical approach for the time dependent wave and heat equations as well as for the time independent laplace equation on irregular domains with the dirichlet boundary conditions has been developed trivial cartesian meshes and simple 27 point uniform and nonuniform stencil equations are used for 3 d irregular domains the calculation of the coefficients of the stencil equations is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy of the new technique very small distances 0 1 h 1 0 9 h where h is the grid size between the grid points of a cartesian mesh and the boundary do not worsen the accuracy of the new technique at similar 27 point stencils the accuracy of the new approach is two orders higher than that for the linear finite elements the numerical results for irregular domains show that at the same number of degrees of freedom the new approach is even much more accurate than the high order up to the fifth order tetrahedral finite elements with much wider stencils the wave and heat equations can be uniformly treated with the new approach the order of the time derivative in these equations does not affect the coefficients of the stencil equations of the semi discrete systems the new approach can be directly applied to other partial differential equations

liquid vapor phase transition thermomechanical theory entropy stable numerical formulation and boiling simulations abstract we develop a new continuum mechanics modeling framework for liquid vapor flows with particular focus on the van der waals fluid by invoking microforce theory the coleman noll procedure is generalized to derive consistent constitutive relations in the presence of non local effects a new thermodynamically consistent algorithm for the van der waals model is designed using functional entropy variables and a new temporal scheme employing a family of new quadrature rules we show that the resulting fully discrete scheme is unconditionally stable in entropy and second order time accurate isogeometric analysis is utilized for spatial discretization the analytical properties of the formulation are corroborated by benchmark problems three sets of application problems are simulated to demonstrate the capability of the model and the algorithm our methodology provides a particularly useful predictive tool for boiling flows

isogeometric analysis of the electrophysiology in the human heart numerical simulation of the bidomain equations on the atria abstract we consider isogeometric analysis iga for the numerical solution of the electrophysiology of the atria which in this work is modeled by means of the bidomain equations on thin surfaces first we consider the bidomain equations coupled with the roger mcculloch ionic model on simple slabs here our goal is to evaluate the effects of the spatial discretization by iga and the use of different b spline basis functions on the accuracy of the approximation in particular regarding the accuracy of the front velocity and the dispersion error specifically we consider basis functions with high polynomial degree p and global high order continuity c p 1 in the computational domain our results show that the use of such basis functions is beneficial to the accurate approximation of the solution then we consider a realistic application of the bidomain equations coupled with the courtemanche ramirez nattel ionic model on the two human atria which are represented by means of two nurbs surfaces

isogeometric analysis of geometric partial differential equations abstract we consider the numerical approximation of geometric partial differential equations pdes defined on surfaces in the 3d space in particular we focus on the geometric pdes deriving from the minimization of an energy functional by l 2 gradient flow we analyze two energy functionals the area which leads to the mean curvature flow a nonlinear second order pde and the willmore energy leading to the willmore flow a nonlinear fourth order pde we consider surfaces represented by single patch tensor product nurbs and discretize the pdes by means of nurbs based isogeometric analysis in the framework of the galerkin method to approximate the high order geometric pdes we use high order continuous nurbs basis functions for the time discretization of the nonlinear geometric pdes we use backward differentiation formulas bdf with extrapolation of the geometric quantities involved in the weak formulation of the problem in this manner we solve a linear problem at each time step we report numerical results concerning the mean curvature and willmore flows on different geometries of interest and we show the accuracy and efficiency of the proposed approximation scheme

non conforming high order approximations of the elastodynamics equation in this paper we formulate and analyze two non conforming high order strategies for the approximation of elastic wave problems in heterogeneous media namely the mortar spectral element method and the discontinuous galerkin spectral element method starting from a common variational formulation we make a full comparison of the two techniques from the points of view of accuracy convergence grid dispersion and stability

an unfitted hp adaptive finite element method based on hierarchical b splines for interface problems of complex geometry generating finite element discretizations with direct interface parameterizations constitutes a considerable computational expense in case of complex interface geometries the paper at hand introduces a b spline finite element method which circumvents parameterization of interfaces and offers fast and easy meshing irrespective of the geometric complexity involved its core idea is the adaptive approximation of discontinuities by hierarchical grid refinement which adds several levels of local basis functions in the close vicinity of interfaces but unfitted to their exact location so that a simple regular grid of knot span elements can be maintained numerical experiments show that an hp refinement strategy which simultaneously increases the polynomial degree of the b spline basis and the levels of refinement around interfaces achieves exponential rates of convergence despite the presence of discontinuities it is also demonstrated that the hierarchical b spline fem can be used to transfer the recently introduced finite cell concept to geometrically nonlinear problems its computational performance imposition of unfitted boundary conditions and fast hierarchical grid generation are illustrated for a set of benchmark problems in one two and three dimensions and the advantages of the regular grid approach for complex geometries are demonstrated by the geometrically nonlinear simulation of a voxel based foam composite

dispersion analysis and error estimation of galerkin finite element methods for the helmholtz equation when applying numerical methods for the computation of stationary waves from the helmholtz equation one obtains numerical waves that are dispersive also in non dispersive media the numerical wave displays a phase velocity that depends on the parameter k of the helmholtz equation in dispersion analysis the phase difference between the exact and the numerical solutions is investigated in this paper the authors recent result on the phase difference for one dimensional problems is numerically evaluated and discussed in the context of other work directed to this topic it is then shown that previous error estimates in h1 norm are of nondispersive character but hold for medium or high wavenumber on extremely refined mesh only on the other hand recently proven error estimates for constant resolution contain a pollution term with certain assumptions on the exact solution this term is of the order of the phase difference thus a link is established between the results of dispersion analysis and the results of numerical analysis throughout the paper the presentation and discussion of theoretical results is accompanied by numerical evaluation of several model problems special attention is given to the performance of the galerkin method with a higher order of polynomial approximation p h p version

isogeometric schwarz preconditioners for linear elasticity systems isogeometric schwarz preconditioners are constructed and analyzed for both compressible elasticity in primal formulation and almost incompressible elasticity in mixed formulation these preconditioners require the solution of local elasticity problems on overlapping subdomains forming a decomposition of the problem domain and the solution of a coarse elasticity problem associated with the subdomain coarse mesh an h analysis of the preconditioner for the primal formulation of compressible elasticity yields an optimal convergence rate bound that is scalable in the number of subdomains and is linear in the ratio between subdomain and overlap sizes extensive numerical experiments in 2d and 3d confirm this theoretical bound and show that an analogous bound holds for the mixed formulation of almost incompressible elasticity the numerical tests also show the good preconditioner performance with respect to the polynomial degree p and regularity k of the isogeometric basis functions as well as with respect to the presence of discontinuous elastic coefficients in composite materials and to domain deformation

dispersion minimizing quadrature rules for c1 quadratic isogeometric analysis abstract we develop quadrature rules for the isogeometric analysis of wave propagation and structural vibrations that minimize the discrete dispersion error of the approximation the rules are optimal in the sense that they only require two quadrature points per element to minimize the dispersion error barto et al 2017 1 and they are equivalent to the optimized blending rules we recently described our approach further simplifies the numerical integration instead of blending two three point standard quadrature rules we construct directly a single two point quadrature rule that reduces the dispersion error to the same order for uniform meshes with periodic boundary conditions also we present a 2 5 point rule for both uniform and non uniform meshes with arbitrary boundary conditions consequently we reduce the computational cost by using the proposed quadrature rules various numerical examples demonstrate the performance of these quadrature rules

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

discrete dispersion relation for hp version finite element approximation at high wave number the dispersive properties of high order finite element schemes are analyzed in the setting of the helmholtz equation and an explicit form of the discrete dispersion relation is obtained for elements of arbitrary order it is shown that the numerical dispersion displays three different types of behavior depending on the size of the order of the method relative to the mesh size and the wave number quantitative estimates are obtained for the behavior and rates of decay of the dispersion error in the differing regimes all estimates are fully explicit and are shown to be sharp limits are obtained where transitions between the different regimes occur and used to provide guidelines for the selection of the mesh size and the polynomial order in terms of the wave number so that the dispersion error is controlled

provably unconditionally stable second order time accurate mixed variational methods for phase field models we introduce provably unconditionally stable mixed variational methods for phase field models our formulation is based on a mixed finite element method for space discretization and a new second order accurate time integration algorithm the fully discrete formulation inherits the main characteristics of conserved phase dynamics namely mass conservation and nonlinear stability with respect to the free energy we illustrate the theory with the cahn hilliard equation but our method may be applied to other phase field models we also propose an adaptive time stepping version of the new time integration method we present some numerical examples that show the accuracy stability and robustness of the new method

studies of refinement and continuity in isogeometric structural analysis we investigate the effects of smoothness of basis functions on solution accuracy within the isogeometric analysis framework we consider two simple one dimensional structural eigenvalue problems and two static shell boundary value problems modeled with trivariate nurbs solids we also develop a local refinement strategy that we utilize in one of the shell analyses we find that increased smoothness that is the k method leads to a significant increase in accuracy for the problems of structural vibrations over the classical c0c0 continuous p method whereas a judicious insertion of c0c0 continuous surfaces about singularities in a mesh otherwise generated by the k method usually outperforms a mesh in which all basis functions attain their maximum level of smoothness we conclude that the potential for the k method is high but smoothness is an issue that is not well understood due to the historical dominance of c0c0 continuous finite elements and therefore further studies are warranted

the use of the local truncation error to improve arbitrary order finite elements for the linear wave and heat equations abstract the local truncation error in space and time can be efficiently used for the analysis and the increase in accuracy of the linear and high order finite elements in the 1 d 2 d and 3 d cases on uniform and non uniform meshes several applications of the local truncation error are considered in the paper it is proven that for the 1 d wave equation with a piece wise constant wave velocity the local truncation error is zero if the linear finite elements with the element size proportional to the wave velocity the lumped mass matrix and the central difference method with the time increments equal to the stability limit are used it is shown in the 1 d and multidimensional cases that the optimal lumped mass matrix can be calculated by the minimization of the order of the local truncation error and yields the maximum possible order of accuracy the minimization of the order of the local truncation error allows us to develop the linear finite elements and the isogeometric high order elements with improved accuracy i e accuracy is improved from order 2 p the conventional elements to order 4 p the new elements where p is the order of the polynomial approximations new high order boundary conditions are developed in order to keep a high order accuracy of the developed technique the new elements can be equally applied to linear wave propagation and heat transfer problems it is also shown that non uniform meshes may lead to inaccurate results due to the increase in the local truncation error the difference in accuracy between the quadrilateral and triangular linear elements is analyzed with the suggested approach the presented numerical examples are in good agreement with the theoretical results the approach considered in the paper can be easily applied to the analysis of different aspects of finite elements techniques as well as other numerical approaches

overlapping schwarz methods for isogeometric analysis we construct and analyze an overlapping schwarz preconditioner for elliptic problems discretized with isogeometric analysis the preconditioner is based on partitioning the domain of the problem into overlapping subdomains solving local isogeometric problems on these subdomains and solving an additional coarse isogeometric problem associated with the subdomain mesh we develop an h analysis of the preconditioner showing in particular that the resulting algorithm is scalable and its convergence rate depends linearly on the ratio between subdomain and overlap sizes for fixed polynomial degree p and regularity k of the basis functions numerical results in two and three dimensional tests show the good convergence properties of the preconditioner with respect to the isogeometric discretization parameters h p k number of subdomains n overlap size and also jumps in the coefficients of the elliptic operator

spline based meshfree method summary in the present work the spline based meshfree method sbmfm is presented the sbmfm uses spline basis functions for field variables and boundary representation because of the inherent properties of splines that is nurbs nonuniform rational b splines the analysis domain used in the sbmfm has its own parametric domain unlike the classical concepts in the computer aided design cad community in which the nurbs patches are used to model the body the nurbs patch is treated as a background mesh in the present work the real analysis domain is created by trimming operation so that the domain boundaries are described by spline curves only on a single nurbs patch building the geometry in this way enables a convenient boundary representation of the domains because direct manipulation of spline curves becomes possible from a deformation point of view the mesh distortion problem is effectively avoided even if the analysis domain undergoes a large deformation in the sbmfm boundary spline curves freely move with complete independence from the quadrilateral background nurbs patch the so called element connectivity is unnecessary for geometry description in this paper geometrically nonlinear problems are considered implementation strategy is presented and the effectiveness of the present sbmfm is shown through various numerical examples copyright 2012 john wiley sons ltd

dispersion minimized mass for isogeometric analysis abstract we introduce the dispersion minimized mass for isogeometric analysis to approximate the structural vibration which we model as a second order differential eigenvalue problem the dispersion minimized mass reduces the eigenvalue error significantly from the optimum order of 2 p to the superconvergence order of 2 p 2 for the p th order isogeometric elements with maximum continuity which in return leads to a more accurate method we first establish the dispersion error where the leading error term is explicitly written in terms of the stiffness and mass entries for arbitrary polynomial order isogeometric elements we derive the dispersion minimized mass in one dimension by solving a p dimensional local matrix problem for the p th order approximation and then extend it to multiple dimensions on tensor product grids we show that the dispersion minimized mass can also be obtained by approximating the mass matrix using optimally blended quadratures we generalize the lower order quadrature blending results to arbitrary polynomial order isogeometric approximations as well as to arbitrary quadrature rules various numerical examples validate the eigenvalue and eigenfunction error estimates

optimal reduction of numerical dispersion for wave propagation problems part 1 application to 1 d isogeometric elements abstract a numerical technique with the optimal coefficients of the stencil equation has been suggested based on this approach new high order isogeometric elements with the reduced dispersion error have been developed for wave propagation problems in the 1 d case by the modification of the mass and stiffness matrices the order of the dispersion error is improved from order 2 p the conventional elements to order 4 p the new elements where p is the order of the polynomial approximations it was shown that the new approach yields the maximum order of the dispersion error for the stencil equations related to the high order isogeometric elements the analysis of the dispersion error of the high order isogeometric elements with the lumped mass matrix has also shown that independent of the procedures for the calculation of the lumped mass matrix the second order of the dispersion error cannot be improved with the conventional stiffness matrix however the dispersion error for the lumped mass matrix can be improved from the second order to order 2 p by the modification of the stiffness matrix the numerical examples confirm the computational efficiency of the new high order isogeometric elements with reduced dispersion we have also showed that numerical results obtained by the new and conventional high order isogeometric elements may include spurious oscillations due to the dispersion error these oscillations can be quantified and filtered by the two stage time integration technique recently developed in our papers the approach developed in the paper can be directly applied to other space discretization techniques with similar stencil equations

a phase field description of dynamic brittle fracture abstract in contrast to discrete descriptions of fracture phase field descriptions do not require numerical tracking of discontinuities in the displacement field this greatly reduces implementation complexity in this work we extend a phase field model for quasi static brittle fracture to the dynamic case we introduce a phase field approximation to the lagrangian for discrete fracture problems and derive the coupled system of equations that govern the motion of the body and evolution of the phase field we study the behavior of the model in one dimension and show how it influences material properties for the temporal discretization of the equations of motion we present both a monolithic and staggered time integration scheme we study the behavior of the dynamic model by performing a number of two and three dimensional numerical experiments we also introduce a local adaptive refinement strategy and study its performance in the context of locally refined t splines we show that the combination of the phase field model and local adaptive refinement provides an effective method for simulating fracture in three dimensions

numerical efficiency locking and unlocking of nurbs finite elements abstract the strategy of using finite elements with nurbs shape functions for approximation of both geometry and displacements isogeometric approach is investigated from the point of view of finite element technology convergence rates are compared to those of classical finite element approaches utilizing standard lagrange shape functions moreover typical locking phenomena are examined it is found that higher order inter element continuity within the nurbs approach results in identical convergence rates but smaller absolute errors compared to c 0 continuous approaches however nurbs finite elements suffer from the same locking problems as finite elements using lagrange shape functions the discrete shear gap dsg method a general framework for formulation of locking free elements is applied to develop a new class of nurbs finite elements the resulting nurbs dsg elements are absolutely free from locking and preserve the property of improved accuracy compared with standard locking free finite elements the method is exemplified for the timoshenko beam model but may be applied to more general cases

isogeometric analysis and proper orthogonal decomposition for parabolic problems we investigate the combination of isogeometric analysis iga and proper orthogonal decomposition pod based on the galerkin method for model order reduction of linear parabolic partial differential equations for the proposed fully discrete scheme the associated numerical error features three components due to spatial discretization by iga time discretization with the theta scheme and eigenvalue truncation by pod first we prove a priori error estimates of the spatial iga semi discrete scheme then we show stability and prove a priori error estimates of the space time discrete scheme and the fully discrete iga theta pod galerkin scheme numerical tests are provided to show efficiency and accuracy of nurbs based iga for model order reduction in comparison with standard finite element based pod techniques

explicit second order isogeometric discretizations for acoustic wave problems abstract this paper presents a stability analysis of isogeometric iga and explicit newmark discretizations for acoustic wave problems with absorbing boundary conditions in spite of the very ill conditioned iga mass and stiffness matrices especially with respect to the polynomial degree p of the b splines and non uniform rational b splines nurbs basis functions employed the stability bounds of the proposed newmark iga method depend linearly on the meshsize h of the iga mesh and inversely on the iga polynomial degree p several numerical tests in the plane confirm these stability bounds and additionally explore the newmark iga order of convergence with respect to h p t domain deformation and the absorbing boundary conditions performance in the presence of a ricker wavelet in a nurbs domain

right half plane poles and zeros and design tradeoffs in feedback systems this paper expresses limitations imposed by right half plane poles and zeros of the open loop system directly in terms of the sensitivity and complementary sensitivity functions of the closed loop system the limitations are determined by integral relationships which must be satisfied by these functions the integral relationships are interpreted in the context of feedback design

finite element and nurbs approximations of eigenvalue boundary value and initial value problems abstract we study the spectral approximation properties of finite element and nurbs spaces from a global perspective we focus on eigenfunction approximations and discover that the l 2 norm errors for finite element eigenfunctions exhibit pronounced spikes about the transition points between branches of the eigenvalue spectrum this pathology is absent in nurbs approximations by way of the pythagorean eigenvalue error theorem we determine that the squares of the energy norm errors of the eigenfunctions are the sums of the eigenvalue errors and the squares of the l 2 norm eigenfunction errors the spurious behavior of the higher eigenvalues for standard finite elements is well known and therefore inherited by the energy norm errors along with the spikes in the l 2 norm of the eigenfunction errors the eigenvalue pathology is absent for nurbs the implications of these results to the corresponding elliptic boundary value problem and parabolic and hyperbolic initial value problems are discussed

on the quadratic finite element approximation of one dimensional waves propagation observation and control we study the propagation observation and control properties of the quadratic p 2 classical finite element semidiscretization of the one dimensional wave equation on a bounded interval a careful fourier analysis of the discrete wave dynamics reveals two different branches in the spectrum the acoustic one of physical nature and the optic one related to the perturbations that this second order finite element approximation introduces with respect to the p 1 one on both modes there are high frequencies with vanishing group velocity as the mesh size tends to zero this shows that the classical property of continuous waves of being observable from the boundary fails to be uniform for this discretization scheme as a consequence of this the controls of the discrete waves may blow up as the mesh size tends to zero to remedy these high frequency pathologies we design filtering mechanisms based on the fourier truncation method or on a bi grid algorithm for which one can recover the uniformity of the

on the synthesis of linear input output responses for nonlinear systems necessary and sufficient conditions are given for the existence of a feedback control law under which the input dependent part of the response of a given nonlinear system becomes linear in the input and independent of the initial state

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

iterative solution of helmholtz problem with high order isogeometric analysis and finite element method at mid range frequencies abstract solving wave problems with isogeometric analysis has attracted a significant attention in the past few years it is well known that keeping a fixed number of degrees of freedom per wavelength leads to an increased error as higher wavenumbers are considered this behaviour often cited as the pollution error improves significantly with isogeometric analysis when compared to the conventional finite element method the improvement in handling pollution along with the ability to represent exact geometries has been the main reasons behind the attention that isogeometric analysis has received furthermore using high order elements also presents major advantages over low order elements for this range of frequencies however it remains to be studied how iterative linear solvers often necessary for solving high frequency wave problems perform when using isogeometric analysis compared to the finite element method especially at high polynomial orders this paper is one of the first studies in this direction in this work we investigate the generalised minimal residual method a standard krylov subspace iterative technique for solving the linear system resulting form isogeometric analysis furthermore we look into the use of some recently proposed preconditioners for helmholtz problem such as shifted laplace or ilu with a complex shift preconditioners and how they perform with high order isogeometric analysis and finite element method in general the results show improvement when using isogeometric analysis in terms of the number of iterations required for convergence compared to the finite element method for both preconditioned and non preconditioned linear systems we use eigenvalue spectra to understand this improvement

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

a hybrid variational collocation immersed method for fluid structure interaction using unstructured t splines summary we present a hybrid variational collocation immersed and fully implicit formulation for fluid structure interaction fsi using unstructured t splines in our immersed methodology we define an eulerian mesh on the whole computational domain and a lagrangian mesh on the solid domain which moves arbitrarily on top of the eulerian mesh mathematically the problem reduces to solving three equations namely the linear momentum balance mass conservation and a condition of kinematic compatibility between the lagrangian displacement and the eulerian velocity we use a weighted residual approach for the linear momentum and mass conservation equations but we discretize directly the strong form of the kinematic relation deriving a hybrid variational collocation method we use t splines for both the spatial discretization and the information transfer between the eulerian mesh and the lagrangian mesh t splines offer us two main advantages against non uniform rational b splines they can be locally refined and they are unstructured the generalized method is used for the time discretization we validate our formulation with a common fsi benchmark problem achieving excellent agreement with the theoretical solution an example involving a partially immersed solid is also solved the numerical examples show how the use of t junctions and extraordinary nodes results in an accurate efficient and flexible method copyright 2015 john wiley sons ltd

variationally consistent domain integration for isogeometric analysis abstract spline type approximations for solving partial differential equations are the basis of isogeometric analysis while the common approach of using integration cells defined by single knot spans using standard e g gaussian quadrature rules is sufficient for accuracy more efficient domain integration is still in high demand the recently introduced concept of variational consistency provides a guideline for constructing accurate and convergent methods requiring fewer quadrature points than standard integration techniques in this work variationally consistent domain integration is proposed for isogeometric analysis test function gradients are constructed to meet the consistency conditions which only requires solving small linear systems of equations the proposed approach allows for significant reduction in the number of quadrature points employed while maintaining the stability accuracy and optimal convergence properties of higher order quadrature rules several numerical examples are provided to illustrate the performance of the proposed domain integration technique

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

dispersion and pollution of the fem solution for the helmholtz equation in one two and three dimensions for high wave numbers the helmholtz equation suffers the so called pollution effect this effect is directly related to the dispersion a method to measure the dispersion on any numerical method related to the classical galerkin fem is presented this method does not require to compute the numerical solution of the problem and is extremely fast numerical results on the classical galerkin fem p method is compared to modified methods presented in the literature a study of the influence of the topology triangles is also carried out the efficiency of the different methods is compared the numerical results in two of the mesh and for square elements show that the high order elements control the dispersion well the most effective modified method is the qsfem 1 2 but it is also very complicated in the general setting the residual free bubble 3 4 is effective in one dimension but not in higher dimensions the least square method 1 5 approach lowers the dispersion but relatively little the results for triangular meshes show that the best topology is the criss cross pattern copyright 1999 john wiley sons ltd

second order convex maximum entropy approximants with applications to high order pde we present a new approach for second order maximum entropy max ent meshfree approximants that produces positive and smooth basis functions of uniform aspect ratio even for non uniform node sets and prescribes robustly feasible constraints for the entropy maximization program defining the approximants we examine the performance of the proposed approximation scheme in the numerical solution by a direct galerkin method of a number of partial differential equations pdes including structural vibrations elliptic second order pdes and fourth order pdes for kirchhoff love thin shells and for a phase field model describing the mechanics of biomembranes the examples highlight the ability of the method to deal with non uniform node distributions and the high accuracy of the solutions surprisingly the first order meshfree max ent approximants with large supports are competitive when compared to the proposed second order approach in all the tested examples even in the higher order pdes

a dissipation inequality for the minimum phase property the minimum phase property is an important notion in systems and control theory in this paper a characterization of the minimum phase property of nonlinear control systems in terms of a dissipation inequality is derived it is shown that this dissipation inequality is equivalent to the classical definition of the minimum phase property in the sense of byrnes and isidori if the control system is affine in the input and the so called input output normal form exists

b and f projection methods for nearly incompressible linear and non linear elasticity and plasticity using higher order nurbs elements abstract this paper presents projection methods to treat the incompressibility constraint in small and large deformation elasticity and plasticity within the framework of isogeometric analysis after reviewing some fundamentals of isogeometric analysis we investigate the use of higher order non uniform rational b splines nurbs within the b projection method the higher continuity property of such functions is explored in nearly incompressible applications and shown to produce accurate and robust results a new non linear f projection method based on a modified minimum potential energy principle and inspired by the b method is proposed for the large deformation case it leads to a symmetric formulation for which the consistent linearized operator for fully non linear elasticity is derived and used in a newton raphson iterative procedure the performance of the methods is assessed on several numerical examples and results obtained are shown to compare favorably with other published techniques

minimum phase properties for input nonaffine nonlinear systems for input nonaffine nonlinear control systems the minimum phase property of the system in general depends on the control law switching or discontinuous controllers may offer advantages in this context in particular there may not exist a continuous control law which would keep the output identically equal to zero and for which the zero output constrained dynamics are locally stable whereas a discontinuous controller which achieves this exists for single input single output input nonaffine nonlinear systems we give sufficient conditions for existence and present a method for the design of discontinuous switching controllers which yield locally stable zero dynamics

application of inverse system for linearization and decoupling abstract we show that for a special class of nonlinear systems the linearization and or decoupling of nonlinear dynamics by immersion under feedback is in fact an application of the right inverse system

a simple algorithm for obtaining nearly optimal quadrature rules for nurbs based isogeometric analysis we develop new quadrature rules for isogeometric analysis based on the solution of a local nonlinear problem a simple and robust algorithm is developed to determine the rules which are exact for important b spline spaces of uniform and geometrically stretched knot spacings we consider both periodic and open knot vector configurations and illustrate the efficiency of the rules on selected boundary value problems we find that the rules are almost optimally efficient but much easier to obtain than optimal rules which require the solution of global nonlinear problems that are often ill posed

output input stability and minimum phase nonlinear systems this paper introduces and studies the notion of output input stability which represents a variant of the minimum phase property for general smooth nonlinear control systems the definition of output input stability does not rely on a particular choice of coordinates in which the system takes a normal form or on the computation of zero dynamics in the spirit of the input to state stability philosophy it requires the state and input of the system to be bounded by a suitable function of the output and derivatives of the output modulo a decaying term depending on initial conditions the class of output input stable systems thus defined includes all affine systems in global normal form whose internal dynamics are input to state stable and also all left invertible linear systems whose transmission zeros have negative real parts as an application we explain how the new concept enables one to develop a natural extension to nonlinear systems of a basic result from linear adaptive control

a high order numerical approach with cartesian meshes for modeling of wave propagation and heat transfer on irregular domains with inhomogeneous materials abstract recently we have developed a new numerical approach for pdes with constant coefficients on irregular domains and cartesian meshes in this paper we extend it to a much more general case of pdes with variable coefficients that have a lot of applications e g the modeling of functionally graded materials the inhomogeneous materials obtained by 3 d printing and many others here we consider the 2 d wave and heat equations for isotropic and anisotropic inhomogeneous materials the idea of the extension to the case of pdes with variable coefficients is based on the representation of the stencil coefficients as functions of the mesh size this leads to the increase in the size of the local system of algebraic equations solved for each grid point of the new approach however this does not change the size of the global system of semidiscrete equations and practically does not increase the computational costs of the proposed technique similar to our previous technique the new 2 d approach with compact 9 point stencils uses trivial cartesian meshes for complex irregular domains and provides the fourth order of accuracy for the wave and heat equations with variable coefficients the calculation of the coefficients of the stencil equations is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy of the new technique at similar 9 point stencils the accuracy of the new approach is much higher than that for the linear finite elements the numerical results for irregular domains show that at the same number of degrees of freedom the new approach is even much more accurate than the high order up to the third order finite elements with much wider stencils the wave and heat equations are uniformly treated with the new approach