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

rotation free isogeometric thin shell analysis using pht splines this paper presents a novel approach for isogeometric analysis of thin shells using polynomial splines over hierarchical t meshes pht splines the method exploits the flexibility of t meshes for local refinement the main advantage of the pht splines in the context of thin shell theory is that it achieves c1 continuity so the kirchhoff love theory can be used in pristine form no rotational degrees of freedom are needed numerical results show the excellent performance of the present method

the postulations a la d alembert and a la cauchy for higher gradient continuum theories are equivalent a review of existing results in order to found continuum mechanics two different postulations have been used the first introduced by lagrange and piola starts by postulating how the work expended by internal interactions in a body depends on the virtual velocity field and its gradients then by using the divergence theorem a representation theorem is found for the volume and contact interactions which can be exerted at the boundary of the considered body this method assumes an a priori notion of internal work regards stress tensors as dual of virtual displacements and their gradients deduces the concept of contact interactions and produces their representation in terms of stresses using integration by parts the second method conceived by cauchy and based on the celebrated tetrahedron argument starts by postulating the type of contact interactions which can be exerted on the boundary of every suitably regular part of a body then it proceeds by proving the existence of stress tensors from a balance type postulate in this paper we review some relevant literature on the subject discussing how the two postulations can be reconciled in the case of higher gradient theories finally we underline the importance of the concept of contact surface edge and wedge s order forces

nonlinear isogeometric spatial bernoulli beam abstract a new element formulation of a geometrically nonlinear spatial curved beam assuming bernoulli theory including torsion without warping is proposed the element formulation is derived directly from the 3d continuum by means of nonlinear kinematics thus accounting for large displacements the geometric description of the proposed element is adapted from the spatial rod of greco and cuomo 2013 and extended to a nonlinear element formulation the proposed formulation can handle arbitrary orientations of the cross section along the beam in this publication nurbs are used as basis functions for discretization since they can easily provide the required c 1 continuity between elements the presented element formulation has four degrees of freedom three for displacements and one for the rotation around the center line in order to prove the accuracy of the developed spatial bernoulli beam several numerical examples are presented and compared to analytic solutions and other element formulations

finite element simulation of non linear sound wave propagation abstract a numerical approach for non linear acoustic wave propagation is proposed in which a finite element method is used an approximate equation which can describe the effect of diffraction with reasonable accuracy is derived based on the equations of fluid dynamics only the generation of the second harmonic wave is considered with the higher order harmonics being neglected under the assumption of weak non linearity a set of uncoupled equations for the primary and secondary wave is discretized in space by a finite element method and then solved by using the newmark integration scheme for time only two dimensional cases are considered and some numerical examples are presented for sound propagation along a duct with stepped width for scattering from a circular rod and for a focusing source in which the diffraction effect is of interest some examples of the wave propagation fields are displayed

alternative ways for formulation of hybrid stress elements an element stiffness matrix can be derived by the conventional potential energy principle and indirectly also by generalized variational principles such as the hu washizu principle and the hellinger reissner principle the present investigation has the objective to show an approach which is concerned with the formulation of incompatible elements for solid continuum and for plate bending problems by the hellinger reissner principle it is found that the resulting scheme is equivalent to that considered by tong 1982 for the construction of hybrid stress elements in tong s scheme the inversion of a large flexibility matrix can be avoided it is concluded that the introduction of additional internal displacement modes in mixed finite element formulations by the hellinger reissner principle and the hu washizu principle can lead to element stiffness matrices which are equivalent to the assumed stress hybrid method

justification for power laws and fractional models in previous chapters it has been shown that the standard linear viscoelastic models can be generalized to their fractional counterparts it has also been demonstrated that the fractional zener kelvin voigt maxwell and newtonian models all satisfy physical criteria such as passivity and causality these models have temporal responses that are characterized by power laws either exactly or asymptotically and this corresponds to observed temporal characteristics in many media the character of the fractional derivative operator is also such that it describes temporal memory in the media

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

large deformations of planar extensible beams and pantographic lattices heuristic homogenization experimental and numerical examples of equilibrium the aim of this paper is to find a computationally efficient and predictive model for the class of systems that we call pantographic structures the interest in these materials was increased by the possibilities opened by the diffusion of technology of three dimensional printing they can be regarded once choosing a suitable length scale as families of beams also called fibres interconnected to each other by pivots and undergoing large displacements and large deformations there are however relatively few ready to use results in the literature of nonlinear beam theory in this paper we consider a discrete spring model for extensible beams and propose a heuristic homogenization technique of the kind first used by piola to formulate a continuum fully nonlinear beam model the homogenized energy which we obtain has some peculiar and interesting features which we start to describe by solving numerically some exemplary deformation problems furthermore we consider pantographic structures find the corresponding homogenized second gradient deformation energies and study some planar problems numerical solutions for these two dimensional problems are obtained via minimization of energy and are compared with some experimental measurements in which elongation phenomena cannot be neglected

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

continuum and discrete models for structures including quasi inextensible elasticae with a view to the design and modeling of composite reinforcements abstract inspired by some composite fiber reinforcements used in aeronautical engineering and by the need of conceiving new metamaterials some discrete models including quasi inextensible elasticae are considered a class of continuum models approximately describing the macroscopic mechanical behavior of introduced structures is then heuristically proposed some of these continuum models can be regarded as a special kind of second gradient elastic media in which the higher gradient elasticity is conferred by the flexural stiffnesses of elasticae constituting the microscopic lattice the discrete models are studied by means of suitably tailored numerical codes designed to avoid numerical instabilities and locking and a comparison of discrete versus continuum models is attempted the obtained results show that the theory of generalized continua may be useful in some engineering applications and it plays a relevant role in the mechanics of woven composites the introduced discrete and continuum models are used to describe the so called bias extension test on woven fabrics and it is shown that a good choice to correctly reproduce the targeted phenomenology is to use a second gradient continuum theory however as discussed throughout the paper in the context of rigorous micro macro identification procedures there still remain many open problems to be solved especially when dealing with systems subjected to particular constraints such as inextensibility

reissner mindlin shell implementation and energy conserving isogeometric multi patch coupling summary a shear flexible isogeometric reissner mindlin shell formulation using non uniform rational b splines basis functions is introduced which is used for the demonstration of a coupling approach for multiple non conforming patches the six degrees of freedom formulation uses the exact surface normal vectors and curvature the shell formulation is implemented in an isogeometric analysis framework for computation of structures composed of multiple geometric entities to enable local model refinement as well as non matching domains coupling a conservative multi patch approach using lagrange multipliers for structured non uniform rational b splines patches is presented here an additional border frame mesh is used to couple geometries during structural analyses this frame interface approach avoids the problem of excessive constraints when multiple patches are coupled at one point first the shell formulation is verified with several reference cases then the influence of the frame interface discretization and frame penalty stiffness on the smoothness of the results is investigated the effects of the perturbed lagrangian method in combination with the frame interface approach is shown in addition results of models with t joint interface connections and perpendicular stiffener patches are presented copyright 2016 john wiley sons ltd

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

implementation of sparse matrix algorithms in an advection diffusion chemistry module a two dimensional advection diffusion chemistry module of a large scale environmental model is taken the module is described mathematically by a system of partial differential equations sequential splitting is used in the numerical treatment the non linear chemistry is most time consuming part and it is handled by six implicit algorithms for solving ordinary differential equations this leads to the solution of very long sequences of systems of linear algebraic equations it is crucial to solve these systems efficiently this is achieved by applying four different algorithms the numerical results indicate that the algorithms based on a preconditioned sparse matrix technique and on a specially designed algorithm for the particular problem under consideration perform best

a causal and fractional all frequency wave equation for lossy media this work presents a lossy partial differential acoustic wave equation including fractional derivative terms it is derived from first principles of physics mass and momentum conservation and an equation of state given by the fractional zener stress strain constitutive relation for a derivative order in the fractional zener relation the resulting absorption k obeys frequency power laws as k 1 in a low frequency regime k 1 2 in an intermediate frequency regime and k 1 in a high frequency regime the value 1 corresponds to the case of a single relaxation process the wave equation is causal for all frequencies in addition the sound speed does not diverge as the frequency approaches infinity this is an improvement over a previously published wave equation building on the fractional kelvin voigt constitutive relation

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

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

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

three dimensional extension of non linear shell formulation based on the enhanced assumed strain concept conventional shell formulations such as 3 or 5 parameter theories or even 6 parameter theories including the thickness change as extra parameter require a condensation of the constitutive law in order to avoid a significant error due to the assumption of a linear displacement field across the thickness this means that the normal stress in thickness direction has to either vanish or be constant in general these extra constraints cannot be satisfied explicitly or they lead to elaborate strain expressions the main objective of the present study is to introduce directly a complete 3 d constitutive law without modification therefore a 7 parameter theory is utilized which includes a linear varying thickness stretch as extra variable allowing also large strain effects

accelerating with rank one updates abstract consider the iteration xk 1 xk h b axk for solving ax b a is n x n nonsingular we discuss rank one updates to improve h as an approximation to a 1 during the iteration the update kills and reduces singular values of i ah and thus speeds up the convergence the algorithm terminates after at most n sweeps and if all n sweeps are needed then a 1 has been computed

on the coefficient of nonlinearity β in nonlinear acoustics an analysis valid to second order in the acoustic variables is used to investigate the origin and form of the coefficient of nonlinearity two cases are considered first for progressive plane waves the well known formula is 1 b 2a where b 2a is the coefficient of the quadratic term in the isentropic pressure density relation it is shown that the component 1 which is associated with convection comes solely from nonlinear terms in the equation of continuity the second component comes from a quadratic term in the equation of state the momentum equation is linear for this type of wave motion and therefore contributes nothing to the expression for second for noncollinear interaction of two plane waves quasilinear analysis shows that the expression for appropriate for the sum and difference frequency waves has three components in this case the momentum equation does contribute to the formula for most practical problems the three component formula reduces to cos b 2a where is the angle of interaction the factor cos represents the combination of contributions cos2 2 and sin2 2 from the continuity and momentum equations respectively

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

a more fundamental approach to the derivation of nonlinear acoustic wave equations with fractional loss operators l a corrected derivation of nonlinear wave propagation equations with fractional loss operators is presented the fundamental approach is based on fractional formulations of the stress strain and heat flux definitions but uses the energy equation and thermodynamic identities to link density and pressure instead of an erroneous fractional form of the entropy equation as done in prieur and holm nonlinear acoustic wave equations with fractional loss operators j acoust soc am 130 3 1125 1132 2011 the loss operator of the obtained nonlinear wave equations differs from the previous derivations as well as the dispersion equation but when approximating for low frequencies the expressions for the frequency dependent attenuation and velocity dispersion remain unchanged

subdivision surfaces a new paradigm for thin shell finite element analysis we develop a new paradigm for thin shell finite element analysis based on the use of subdivision surfaces for i describing the geometry of the shell in its undeformed configuration and ii generating smooth interpolated displacement fields possessing bounded energy within the strict framework of the kirchhoff love theory of thin shells the particular subdivision strategy adopted here is loop s scheme with extensions such as required to account for creases and displacement boundary conditions the displacement fields obtained by subdivision are h2 and consequently have a finite kirchhoff love energy the resulting finite elements contain three nodes and element integrals are computed by a one point quadrature the displacement field of the shell is interpolated from nodal displacements only in particular no nodal rotations are used in the interpolation the interpolation scheme induced by subdivision is non local i e the displacement field over one element depend on the nodal displacements of the element nodes and all nodes of immediately neighbouring elements however the use of subdivision surfaces ensures that all the local displacement fields thus constructed combine conformingly to define one single limit surface numerical tests including the belytschko et al 10 obstacle course of benchmark problems demonstrate the high accuracy and optimal convergence of the method

iterative solution of linear systems in the 20th century this paper sketches the main research developments in the area of iterative methods for solving linear systems during the 20th century although iterative methods for solving linear systems nd their origin in the early 19th century work by gauss the eld has seen an explosion of activity spurred by demand due to extraordinary technological advances in engineering and sciences the past ve decades have been particularly rich in new developments ending with the availability of large toolbox of specialized algorithms for solving the very large problems which arise in scientic and industrial computational models as in any other scientic area research in iterative methods has been a journey characterized by a chain of contributions building on each other it is the aim of this paper not only to sketch the most signicant of these contributions during the past century but also to relate them to one another c 2000 elsevier science b v all rights reserved

development of a mixed displacement stress formulation for the analysis of elastoplastic structures under small strains application to a locking free nurbs based solid shell element abstract in this paper we develop a general framework for the extension of mixed displacement stress types of formulations to elastoplastic problems under small strains the difficulty with such formulations is that contrary to the more usual mixed methods used for incompressible materials plasticity which involves the deviatoric part of the stress is affected by the static unknown thus the plastic flow is not governed by displacement induced strains alone the approach followed in this paper consists in choosing the total elastic stress as the static unknown of the mixed formulation then one determines a strain field which is used as input for the plastic projection we applied this formalism to the mixed nurbs based solid shell element of bouclier et al 2013 2015 which led to an efficient extension of the element into the elastoplastic domain we were able to verify the ability of this element to overcome shell locking problems through several test cases involving a plastic flow rule with linear isotropic strain hardening

fractal ladder models and power law wave equations extensive measurement of the ultrasonic attenuation coefficient in human and mammalian tissue has revealed a power law dependence on frequency to describe this power law behavior in tissue a hierarchical fractal network model is proposed the viscoelastic and self similar properties of tissue are captured by a constitutive equation based on a lumped parameter infinite ladder topology involving alternating springs and dashpots in the low frequency limit this ladder network yields a stress strain constitutive equation with a time fractional derivative by combining this constitutive equation with linearized conservation principles and an adiabatic equation of state the fractional partial differential equations previously proposed by m caputo linear models of dissipation whose q is almost frequency independent ii geophys j r astron soc 13 529 539 1967 and mg wismer finite element analysis of broadband acoustic pulses through inhomogeneous media with power law attenuation j acoust soc am 120 3493 3502 2006 are derived the resulting attenuation coefficient is a power law with exponents ranging between 1 and 2 while the phase velocity is in agreement with the kramers kronig relations the power law exponent is interpreted in terms of the mechanical structure and related to the spectral dimension of the underlying fractal geometry

anderson acceleration for fixed point iterations this paper concerns an acceleration method for fixed point iterations that originated in work of d g anderson j assoc comput mach 12 1965 pp 547 560 which we accordingly call anderson acceleration here this method has enjoyed considerable success and wide usage in electronic structure computations where it is known as anderson mixing however it seems to have been untried or underexploited in many other important applications moreover while other acceleration methods have been extensively studied by the mathematics and numerical analysis communities this method has received relatively little attention from these communities over the years a recent paper by h fang and y saad numer linear algebra appl 16 2009 pp 197 221 has clarified a remarkable relationship of anderson acceleration to quasi newton secant updating methods and extended it to define a broader anderson family of acceleration methods in this paper our goals are to shed additional light on anderson acceleration and to draw further attention to its usefulness as a general tool we first show that on linear problems anderson acceleration without truncation is essentially equivalent in a certain sense to the generalized minimal residual gmres method we also show that the type 1 variant in the fang saad anderson family is similarly essentially equivalent to the arnoldi full orthogonalization method we then discuss practical considerations for implementing anderson acceleration and illustrate its performance through numerical experiments involving a variety of applications

isogeometric kirchhoff love shell formulation for elasto plasticity abstract an isogeometric thin shell formulation allowing for large strain plastic deformation is presented a stress based approach is adopted which means that the constitutive equations are evaluated at different integration points through the thickness allowing the use of general 3d material models the plane stress constraint is satisfied by iteratively updating the thickness stretch at the integration points the deformation of the shell structure is completely described by the deformation of its midsurface and furthermore the formulation is rotation free which means that the discrete shell model involves only three degrees of freedom several numerical benchmark examples with comparison to fully 3d solid simulations confirm the accuracy and efficiency of the proposed formulation

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

on the mathematical foundation of the 1 1 2 platemodel abstract the plate problem of three dimensional 3 d linearized elastostatics is considered inthe framework of the hierarchical modelling with the help of the energy projection method it isshown that the 1 1 2 bending model is the simplest asymptotically correct model in thehierarchical family i e the distance between the solution of the 3 d problem and themodel solution in the energy norm tends to zero for vanishing plate thickness h whileusing unmodified 3 d material laws the formulation of the 1 1 2 bending model and the mechanical significance of itsingredients are discussed we present error estimates for the deviation of the 1 1 2 solutionfrom the kirchhoff solution as well as from the 3 d solution by using the 1 1 2 energy normand the 3 d energy norm respectively the analysis leads to h as order of convergence the results are illustrated by a numerical example

two new triangular g1 conforming finite elements with cubic edge rotation for the analysis of kirchhoff plates abstract two triangular g 1 conforming elements based on the triangular gregory patch suitable for the analysis of the kirchhoff plate model are presented both have cubic normal derivative along the sides so that they can be effectively used in combination with generalized hermitian elements the gregory patch consists in a rational enhancement of the base polynomial spaces useful to design g 1 conforming elements on general c 0 conforming unstructured meshes because of the presence of the rational functions the second derivatives at the corners of the element present finite discontinuities that prevent the elements from passing the bending patch test the discontinuities are removed using a constrained version of the gregory patch with lagrange multipliers in this way the rational conforming space collapses into a conforming rearrangement of the original polynomial interpolant spaces the proposed formulation design elements that pass the bending patch test and present optimal rate of convergence on general unstructured meshes

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

bloch theorem for isogeometric analysis of periodic problems governed by high order partial differential equations abstract with the emergence of isogeometric analysis iga numerical rotation free solutions to high order partial differential equations pdes are facilitated indeed iga allows the use of high order continuous basis functions through the domain for periodic systems the bloch theorem is useful for analyzing wave propagation restricting the analysis to a single unit cell and applying periodic boundary conditions however in the literature boundary conditions are only accounting for c 0 continuity and we propose in the present paper to enforce the g 1 continuity as well or to maintain the angle in case of unit cells meeting at kinks while g 1 continuity is enforced by maintaining the collinearity between adjacent control points in the presence of kinks variations of the field derived with respect to opposite in plane edge normals are linked by the rodrigues rotation matrix this method applied at collocation points belonging to the unit cell edges can be viewed as a novel approach of coupling multi patch geometries with kinks avoiding the use of nonlinear multipliers or additional bending strip elements both transfer matrix and indirect methods of the bloch theorem are investigated for respectively quasi one and quasi two dimensional wave propagation in periodic structures discretized by nurbs based iga dispersion curves of a set of numerical problems treating isogeometric rotation free extensible elastica beams and kirchhoff love shells are then analyzed

reissner mindlin shell theory based on tangential differential calculus abstract the linear reissner mindlin shell theory is reformulated in the frame of the tangential differential calculus tdc using a global cartesian coordinate system the rotation of the normal vector is modelled with a difference vector approach the resulting equations are applicable to both explicitly and implicitly defined shells because the employed surface operators do not necessarily rely on a parametrization hence shell analysis on surfaces implied by level set functions is enabled but also the classical case of parametrized surfaces is captured as a consequence the proposed tdc based formulation is more general and may also be used in recent finite element approaches such as the tracefem and cutfem where a parametrization of the middle surface is not required herein the numerical results are obtained by isogeometric analysis using nurbs as trial and test functions for classical and new benchmark tests in the residual errors optimal higher order convergence rates are confirmed when the involved physical fields are sufficiently smooth

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

rotation free isogeometric analysis of an arbitrarily curved plane bernoulli euler beam abstract the present study elucidates linear static analysis for plane beam structures using the isogeometric approach a novel methodology for rotation free analysis of an arbitrarily curved bernoulli euler beam in the convective frame of reference is derived in detail the full degeneration of a 3d continuum beam to a 1d line has been presented and a fully applicable isogeometric finite element has been obtained the driving force behind developing the present research has been the derivation of the nurbs based isogeometric analysis which will enable an elegant formulation of the plane bernoulli euler beams being a function only of the global rectangular cartesian coordinates the verification and accuracy of the research are obtained via a thorough comparison between theory finite element analyses and relevant examples from literature an excellent agreement of results is achieved and usefulness for academic and practical purposes alike are proved the effects of the hpk refinements are illuminated and it is observed that the convergences for the most variables and refinement techniques are not monotonic a special attention is paid to the influence of the product of maximum curvature and thickness of beam on the accuracy of the solution the limits of applicability of the present approach are defined for a few specific types of analyses the derived formulation is geometrically exact and appropriate for the analysis of strongly curved bernoulli euler beams

isogeometric reissner mindlin shell analysis with exactly calculated director vectors abstract an isogeometric reissner mindlin shell derived from the continuum theory is presented the geometry is described by nurbs surfaces the kinematic description of the employed shell theory requires the interpolation of the director vector and of a local basis system hence the definition of nodal basis systems at the control points is necessary for the proposed formulation the control points are in general not located on the shell reference surface and thus several choices for the nodal values are possible the proposed new method uses the higher continuity of the geometrical description to calculate nodal basis system and director vectors which lead to geometrical exact interpolated values thereof thus the initial director vector coincides with the normal vector even for the coarsest mesh in addition to that a more accurate interpolation of the current director and its variation is proposed instead of the interpolation of nodal director vectors the new approach interpolates nodal rotations account is taken for the discrepancy between interpolated basis systems and the individual nodal basis systems with an additional transformation the exact evaluation of the initial director vector along with the interpolation of the nodal rotations lead to a shell formulation which yields precise results even for coarse meshes the convergence behavior is shown to be correct for k refinement allowing the use of coarse meshes with high orders of nurbs basis functions this is potentially advantageous for applications with high numerical effort per integration point the geometrically nonlinear formulation accounts for large rotations the consistent tangent matrix is derived various standard benchmark examples show the superior accuracy of the presented shell formulation a new benchmark designed to test the convergence behavior for free form surfaces is presented despite the higher numerical effort per integration point the improved accuracy yields considerable savings in computation cost for a predefined error bound

a computational formulation for constrained solid and liquid membranes considering isogeometric finite elements abstract a geometrically exact membrane formulation is presented that is based on curvilinear coordinates and isogeometric finite elements and is suitable for both solid and liquid membranes the curvilinear coordinate system is used to describe both the theory and the finite element equations of the membrane in the latter case this avoids the use of local cartesian coordinates at the element level consequently no transformation of derivatives is required the formulation considers a split of the in plane and out of plane membrane contributions which allows the construction of a stable formulation for liquid membranes with constant surface tension the proposed membrane formulation is general and accounts for dead and live loading as well as enclosed volume area and contact constraints the new formulation is illustrated by several challenging examples considering linear and quadratic lagrange elements as well as isogeometric elements based on quadratic nurbs and cubic t splines it is seen that the isogeometric elements are much more accurate than standard lagrange elements the gain is especially large for the liquid membrane formulation since it depends explicitly on the surface curvature

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

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

error estimate based adaptive integration for immersed isogeometric analysis abstract the finite cell method fcm together with isogeometric analysis iga has been applied successfully in various problems in solid mechanics in image based analysis fluid structure interaction and in many other applications a challenging aspect of the isogeometric finite cell method is the integration of cut cells in particular in three dimensional simulations the computational effort associated with integration can be the critical component of a simulation a myriad of integration strategies has been proposed over the past years to ameliorate the difficulties associated with integration but a general optimal integration framework that suits a broad class of engineering problems is not yet available in this contribution we provide a thorough investigation of the accuracy and computational effort of the octree integration scheme we quantify the contribution of the integration error using the theoretical basis provided by strang s first lemma based on this study we propose an error estimate based adaptive integration procedure for immersed isogeometric analysis additionally we present a detailed numerical investigation of the proposed optimal integration algorithm and its application to immersed isogeometric analysis using two and three dimensional linear elasticity problems

isogeometric analysis of nearly incompressible solids this paper describes the use of isogeometric methods to solve problems in finite deformation solid mechanics in which compressible and nearly incompressible behavior may be encountered the work is based on a three field variational structures in which displacements mean stress and volume effects are independently approximated using this approach and tensor product nurbs approximations it is shown how the solution can be conducted in a standard approach where main variables are organized at nodes control points this leads to an efficient procedure to solve large problems of a general form without the need to develop special solution strategies the approach is demonstrated on some standard test problems copyright 2010 john wiley sons ltd

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

lagrangian aspects of the kirchhoff elastic rod the relation between euler s planar elastic curves and vortex filaments evolving by the localized induction equation lie of hydrodynamics was discovered by hasimoto in 1971 basic facts about an integrable case of kirchhoff elastic rods are described here which amplify the connection between the variational problem for rods and the soliton equation lie in particular it is shown that the centerline of the kirchhoff rod is an equilibrium for a linear combination of the first three conserved hamiltonians in the lie hierarchy

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

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

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

the quasi newton least squares method a new and fast secant method analyzed for linear systems we present a new quasi newton method that can solve systems of equations of which no information is known explicitly and which requires no special structure of the system matrix like positive definiteness or sparseness the method builds an approximate jacobian based on input output combinations of a black box system uses a rank one update of this jacobian after each iteration and satisfies the secant equation while it has originally been developed for nonlinear equations we analyze its properties and performance when applied to linear systems analytically the method is shown to be convergent in n 1 iterations n being the number of unknowns irrespective of the nature of the system matrix the performance of this method is greatly superior to other quasi newton methods and comparable with gmres when tested on a number of standardized test cases

linking multiple relaxation power law attenuation and fractional wave equations the acoustic wave attenuation is described by an experimentally established frequency power law in a variety of complex media e g biological tissue polymers rocks and rubber recent papers present a variety of acoustical fractional derivative wave equations that have the ability to model power law attenuation on the other hand a multiple relaxation model is widely recognized as a physically based description of the acoustic loss mechanisms as developed by nachman et al j acoust soc am 88 1584 1595 1990 through assumption of a continuum of relaxation mechanisms each with an effective compressibility described by a distribution related to the mittag leffler function this paper shows that the wave equation corresponding to the multiple relaxation approach is identical to a given fractional derivative wave equation this work therefore provides a physically based motivation for use of fractional wave equations in acoustic modeling

isogeometric analysis on v reps first results abstract inspired by the introduction of volumetric modeling via volumetric representations v reps by massarwi and elber in 2016 in this paper we present a novel approach for the construction of isogeometric numerical methods for elliptic pdes on trimmed geometries seen as a special class of more general v reps we develop tools for approximation and local re parameterization of trimmed elements for three dimensional problems and we provide a theoretical framework that fully justifies our algorithmic choices we validate our approach both on two and three dimensional problems for diffusion and linear elasticity

invariant hermitian finite elements for thin kirchhoff rods ii the linear three dimensional case abstract this paper considers the formulation of finite elements for thin kirchhoff rods in the three dimensional case extending our work on plane rods presented in 5 the new issue to be addressed besides the basic consideration of the three dimensional geometry is the approximation of the rod s torsion still in the geometrically linear setting of infinitesimal strains in particular it is the goal of this paper to identify approximations that preserve the basic invariance properties of the underlying structural theory in the sense that the fundamental rigid body modes of translations and infinitesimal rotations are represented exactly without straining accounting fully for the rod s curvature and associated kirchhoff s kinematics in contrast to common straight framework element approximations based on the simplified euler bernoulli beam kinematics to this purpose different basic and new interpolations of the angle of twist rotation as well as interpolations of directors increments in the setting provided by a cosserat treatment of the rod s kinematics are explored and studied in detail including rigorous analytical characterizations of the aforementioned rigid body modes these different interpolations are combined with the new c 1 continuous hermitian interpolation developed in that reference for the axis displacements extended to general three dimensionally curved rods as well as with an assumed strain mixed treatment to improve the overall performance of the finite elements several representative numerical simulations are presented to illustrate the properties and the numerical performance of the different new finite elements

hierarchical analogues to fractional relaxation equations recently fractional calculus has become an important tool in the analysis of slow relaxation phenomena such as stress strain relationships in polymeric materials in the rheological constitutive equations this implies the replacement of the first order derivatives by fractional derivatives here we show that such procedures have hierarchical mechanical analogues we focus on the generalized dashpot and the generalized maxwell model and display the corresponding arrangements our models allow a transparent interpretation of the parameters which enter the fractional equations and reveal that the internal dynamics are hierarchically constrained

a reconstructed local b formulation for isogeometric kirchhoff love shells abstract an efficient assumed strain formulation for avoiding the membrane locking of non polar shells is developed in the context of b spline interpolation assumed membrane strains are introduced locally in each element with a local l 2 projection and then a spline reconstruction algorithm developed by thomas et al 2015 is employed for reconstructing the membrane strain at the global patch level in this way a banded stiffness matrix is obtained that strongly reduces the computational cost with respect to the standard b formulation for isogeometric analysis the main advantages of the proposed method are the reproduction of regular membrane strain fields an accurate membrane locking free solution and an high computational efficiency with respect to standard b formulation the method can be applied for any polynomial degree of the b spline interpolation the effectiveness of the proposed method is demonstrated analyzing some bending and membrane dominated problem commonly employed as benchmark in the shell literature

invariant isogeometric formulations for three dimensional kirchhoff rods abstract in the isogeometric formulation for three dimensional kirchhoff rods the invariance is required such that rigid body modes associated with translations and infinitesimal rotations can be exactly represented by enforcing the invariance in the isogeometric formulation of the rod the equilibrium of forces and moments at fixed ends is preserved and the performance is improved considerably for elements of finite sizes we recognize that not all existing isogeometric formulations are invariant for general geometry of rods though an alternative formulation based on the frenet serret frame and rotation angle representation was shown to be invariant it does not work for rods with vanishing curvature while its accuracy is affected by the low continuity of the angle of twist used motivated by this two new invariant isogeometric formulations are developed through different interpolations for the angle of twist one uses the derivatives of the nurbs functions as the basis for the angle of twist interpolation and the other uses a mixed interpolation incorporating both the lagrange and nurbs functions several representative numerical examples were prepared to verify the accuracy and robustness of the two new formulations particularly for rods of variable curvatures simulated by elements of finite or large sizes the proposed formulations are demonstrated to outperform the two existing ones mentioned in predicting the displacements end reactions and stress resultants

the shifted boundary method for hyperbolic systems embedded domain computations of linear waves and shallow water flows abstract we propose a new computational approach for embedded boundary simulations of hyperbolic systems and in particular the linear wave equations and the nonlinear shallow water equations the proposed approach belongs to the class of surrogate approximate boundary algorithms and is based on the idea of shifting the location where boundary conditions are applied from the true to a surrogate boundary accordingly boundary conditions enforced weakly are appropriately modified to preserve optimal error convergence rates this framework is applied here in the setting of a stabilized finite element method even though other spatial discretization techniques could have been employed accuracy stability and robustness of the proposed method are tested by means of an extensive set of computational experiments for the acoustic wave propagation equations and shallow water equations comparisons with standard weak boundary conditions imposed on body fitted grids which conform to the geometry of the computational domain boundaries are also presented

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

plane bias extension test for a continuum with two inextensible families of fibers a variational treatment with lagrange multipliers and a perturbation solution in the present paper a system constituted by two families of nearly inextensible fibers is studied the interest towards this kind of structures which can be suitably modeled by 2d continua with inextensible curves comes form both theoretical and applicative motivations indeed the advantageous weight strength ratio and the particularly safe behavior in fractures make them attractive in many engineering fields while the mechanical properties displayed by them are theoretically interesting and still not well understood in full generality we focused on dead loading traction boundary value problems assuming particular forms for the deformation energy associated to every node of the structure the main result of the paper consists in the determination of the nonlinear integral equations describing fibers directions for a specific class of deformation energies a perturbative analysis of a particular solution is also performed and some physical justification for one of the considered forms for the energy density are provided in the appendix

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 new implementation of en method en method proposed by eirola and nevalinna can be more efficient than gmres in some cases but be not the case in general which has been studied by vuik and van der vorst comparison of the efficiency of the two methods in practice shows that gmres is preferable a new implementation of en method with significant saving both in the amount of work and memory requirements is introduced in the present paper numerical experiments indicate that this new version is competitive

an isogeometric method for the reissner mindlin plate bending problem we present a new isogeometric method for the discretization of the reissner mindlin plate bending problem the proposed scheme follows a recent theoretical framework that makes possible the construction of a space of smooth discrete deflections wh and a space of smooth discrete rotations h such that the kirchhoff constraint is exactly satisfied at the limit therefore we obtain a formulation which is natural from the theoretical mechanical viewpoint and locking free by construction we prove that the method is uniformly stable and satisfies optimal convergence estimates finally the theoretical results are fully supported by numerical tests

embedded structural entities in nurbs based isogeometric analysis the core idea of this article is nested parametrization in the context of isogeometric analysis the method has been inspired by trimming procedures and can be applied to different applications like local modifications and enhancements of thin walled structures or coupling of two overlying elements by embedding one in the other a remodeling for an explicit representation of the boundaries is avoided which would be contradictory to the aim of isogeometric analysis of using the original cad model the nested entity is directly linked to the super element however its element formulation is independent of the super element formulation within this article the derivation implementation and application is shown in particular for one dimensional entities embedded into 2d domains consequently the definition of an embedded curve in the surface is required and realized by using nurbs curves in the parameter space of the corresponding surface this curve with its respective predefined base vectors serves as the basis for new element formulations or adaptation of already developed element formulations which are based on the geometric description of a curve in detail an adapted formulation of the recently developed nonlinear isogeometric spatial bernoulli beam by the authors is presented in this paper furthermore those embedded curves are used for line supports and loads as well as a mass manipulation the accuracy of the proposed element formulation is verified by several benchmark examples and the potential for future applications is briefly revealed

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

a fully locking free isogeometric approach for plane linear elasticity problems a stream function formulation in this paper we study plane incompressible elastic problems by means of a stream function formulation such that a divergence free displacement field can be computed from a scalar potential the numerical scheme is constructed within the framework of nurbs based isogeometric analysis and we take advantage of the high continuity guaranteed by nurbs basis functions in order to obtain the displacement field from the potential differentiation as a consequence the obtained numerical scheme is automatically locking free in the presence of the incompressibility constraint a discontinuous galerkin approach is proposed to deal with multiple mapped possibly multiply connected domains extensive numerical results are provided to show the method capabilities 2007 elsevier b v all rights reserved

a consistent co rotational formulation for non linear three dimensional beam elements abstract this paper describes a co rotational formulation for three dimensional beams in which both the internal force vector and tangent stiffness matrix are consistently derived from the adopted strain measures the latter relate to standard beam theory but are embedded in a continuously rotating frame a set of numerical examples show that the element provides an excellent numerical performance

an implicit multi patch b spline interpolation for kirchhoff love space rod abstract the paper presents a novel formulation for the isogeometric analysis of assemblies of kirchhoff love space rod elements introducing a multi patch implicit g 1 formulation so that an automatic non singular stiffness operator is obtained without the need of adding continuity conditions the goal is achieved using a polar decomposition of the deformation of the first and last segments of the control polygon that allows to introduce directly the end rotations as degrees of freedom both parametric and geometric continuity can be obtained in this way we use bezier and b spline interpolations and we show that they are able to attain very good accuracy for developing a 3d exact curve element with geometric torsion pre twisted rod in the paper the performance of the multi patch elements is examined comparing the rates of convergence of the l 2 error norm for the multi patch and single patch formulations it is shown that the rate of convergence remains the same although in certain cases the accuracy is lower for the multi patch solutions

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

blended isogeometric shells abstract we propose a new isogeometric shell formulation that blends kirchhoff love theory with reissner mindlin theory this enables us to reduce the size of equation systems by eliminating rotational degrees of freedom while simultaneously providing a general and effective treatment of kinematic constraints engendered by shell intersections folds boundary conditions the merging of nurbs patches etc we illustrate the blended theory s performance on a series of test problems

weak boundary conditions for wave propagation problems in confined domains formulation and implementation using a variational multiscale method abstract we propose a new approach to the enforcement of dirichlet neumann or robin boundary conditions in finite element computations of wave propagation problems the key idea is to enforce the boundary conditions weakly as part of the variational formulation due to the hyperbolic structure of the problem considered the variational formulation does not require any penalty parameters in contrast with what typically happens in elliptic or advection diffusion parabolic problems this article presents the implementation of the proposed boundary condition framework using a variational multiscale method for the wave equation in mixed form we conclude with an extensive set of tests to validate the robustness and accuracy of the proposed approach

relaxation matched modeling of propagation through porous media including fractal pore structure the viscous and thermal dissipation of an acoustic wave propagating through a porous medium is shown to be characteristic of a relaxation process based on this interpretation a new model for the complex density dynamic permeability and bulk modulus which describe the viscous and thermal processes respectively is proposed the model is based on approximating the relaxational characteristic as opposed to previous models based on matching low and high frequency asymptotic behavior an advantage of the relaxation model is that one fewer parameter is required the relaxation model is also simpler than the commonly used zwikker kosten attenborough or biot allard models in the sense that it contains no bessel or kelvin functions and is not a modified form of the solution for uniform circular pores and unlike the delany bazley empirical equations the relaxation model is physically realistic for all frequencies extension of the relaxation model to fractal pore surfaces is also discussed

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

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

a natural framework for isogeometric fluid structure interaction based on bem shell coupling abstract the interaction between thin structures and incompressible newtonian fluids is ubiquitous both in nature and in industrial applications in this paper we present an isogeometric formulation of such problems which exploits a boundary integral formulation of stokes equations to model the surrounding flow and a non linear kirchhoff love shell theory to model the elastic behavior of the structure we propose three different coupling strategies a monolithic fully implicit coupling a staggered elasticity driven coupling and a novel semi implicit coupling where the effect of the surrounding flow is incorporated in the non linear terms of the solid solver through its damping characteristics the novel semi implicit approach is then used to demonstrate the power and robustness of our method which fits ideally in the isogeometric paradigm by exploiting only the boundary representation b rep of the thin structure middle surface

a locking free finite element formulation and reduced models for geometrically exact kirchhoff rods in this work we suggest a locking free geometrically exact finite element formulation incorporating the modes of axial tension torsion and bending of thin kirchhoff beams with arbitrary initial curvatures and cross section shapes the proposed formulation has been designed in order to represent general load cases and three dimensional problem settings in the geometrically nonlinear regime of large deformations from this comprehensive theory we not only derive a general beam model but also several reduced formulations which deliver accurate solutions for special problem classes concerning the beam geometry and the external loads the advantages of these reduced models arise for example in terms of simplified finite element formulations less degrees of freedom per element and consequently a higher computational efficiency of the corresponding numerical models a second core topic of this publication is the treatment of membrane locking which is a locking phenomenon predominantly occurring in highly slender curved structures thus exactly in the prime field of application for kirchhoff theories in order to address the membrane locking effect we will propose a new interpolation strategy for the axial strain field and compare this method with common approaches such as assumed natural strains ans or reduced integration the effectiveness of this method as well as the consistency and accuracy of the general finite element formulation and the reduced beam models will be illustrated with selected numerical examples

an efficient blended mixed b spline formulation for removing membrane locking in plane curved kirchhoff rods abstract mixed formulations of the b type when applied to b spline representations bear significant computational cost and lead to a full stiffness matrix in this work we present an efficient mixed formulation for avoiding membrane locking of curved non polar plane rods in the context of isogeometric analysis an efficient spline reconstruction of the assumed axial strains obtained by means of a local projection at the element level is performed using the method proposed by thomas et al 2015 for the reconstruction of the geometry in this way a much smaller bandwidth of the stiffness matrix is obtained with respect to the non local b formulation with significant reduction of the computational cost the blended mixed formulation splits in two steps in the first step the local b operators are defined at the element level via projection of the strain measure both l 2 projection and discrete collocation approaches are considered successively by means of the spline reconstruction algorithm the global projection of the strain measures at the patch level is defined numerical experiments show that the proposed method in addition to completely remove membrane locking yields the same accuracy and rate of convergence as the non local b method

isogeometric shape optimization of shells using semi analytical sensitivity analysis and sensitivity weighting abstract we present isogeometric shape optimization for shell structures applying sensitivity weighting and semi analytical analysis we use a rotation free shell formulation and all involved geometry models i e initial design analysis model optimization model and final design use the same geometric basis in particular nurbs a sensitivity weighting scheme is presented which eliminates certain effects of the chosen discretization on the design update a multilevel design approach is applied such that the design space can be chosen independently from the analysis space the use of semi analytical sensitivities allows having different polynomial degrees for design and analysis model different numerical examples are performed which confirm the applicability of the proposed method furthermore a shape optimization example with an exact solution is presented which can serve as general benchmark for shape optimization methods

a comparison of some gmres like methods abstract gmres and cgs are well known iterative methods for the solution of certain sparse linear systems with a nonsymmetric matrix these methods have been compared experimentally in many studies and specific observations on their convergence behavior have been reported a new iterative method to solve a nonsymmetric system has been proposed by eirola and nevanlinna the purpose of this paper is to investigate this method and to compare it with gmres we have seen problems for which this method is more efficient than gmres the original method has as drawbacks that it is not scaling invariant and that it may suffer from numerical instability but it is shown that these deficiencies can be repaired a method proposed by broyden 1969 seems related to the new method and is therefore included in the comparison

rational approach for assumed stress finite elements a new method for the formulation of hybrid elements by the hellinger reissner principle is established by expanding the essential terms of the assumed stresses as complete polynomials in the natural coordinates of the element the equilibrium conditions are imposed in a variational sense through the internal displacements which are also expanded in the natural co ordinates the resulting element possesses all the ideal qualities i e it is invariant it is less sensitive to geometric distortion it contains a minimum number of stress parameters and it provides accurate stress calculations for the formulation of a 4 node plane stress element a small perturbation method is used to determine the equilibrium constraint equations the element has been proved to be always rank sufficient

a nitsche method for wave propagation problems in time domain we propose a new nitsche type approach for the weak enforcement of dirichlet and neumann boundary conditions in the context of time domain wave propagation problems in mixed form a peculiar feature of the proposed method is that due to the hyperbolic structure of the problem considered two penalty parameters are introduced corresponding to dirichlet and neumann conditions respectively a stability and convergence estimate is also provided in the case of a discontinuous in time galerkin space time integrator the spatial discretization used is based on a stabilized method with equal order interpolation for all solution components in principle however the proposed methodology is not confined to stabilized methods we conclude with an extensive set of tests to validate the robustness and accuracy of the proposed approach