differentiable interpolants for high order runge kutta methods for a particular family of pairs of explicit runge kutta methods of orders p 1 and p sets of efficient continuously differentiable interpolants of several orders up to p are characters zed algorithmically in terms of several arbitrary parameters the approach can be applied in an obvious way to yield interpolants for other types of explicit runge kutta methods as well as families of other types of explicit and implicit methods derivative evaluations required for each pair of methods are reused and additional derivative evaluations are selected in an attempt to minimize the total number of stages required the analysis provides a lower bound on the number of stages required and indicates for example why twelve stages are required to provide interpolants for eight stage pairs of orders 5 and 6 in contrast to the eighteen stages used to obtain a known interpolant of order 7 for a pair of methods of orders 6 and 7 only sixteen stages are required by the proposed derivation details for interpola

on the physical interpretation of proper orthogonal modes in vibrations abstract the authors relate the proper orthogonal modes as applied in discrete vibration systems to normal modes of vibration in systems with a known mass matrix in the case of undamped free vibration the proper orthogonal modes converge to the linear normal modes as the amount of data increases this interpretation is also practical for lightly modally damped systems forced resonances lead to proper orthogonal modes which approximate vibration modes a particular case of non linear normal modes are looked at in which the motion of a single mode follows a curve in the co ordinate space in this case the proper orthogonal modes represent the principal axes of inertia formed by the distribution of data on the modal co ordinate curve more generally the proper orthogonal modes represent the principal axes of inertia of the data

approximating the exponential from a lie algebra to a lie group consider a differential equation y a t y y y 0 y0 with y 0 g and a r g g where g is a lie algebra of the matricial lie group g every b g can be mapped to g by the matrix exponential map exp tb with t r most numerical methods for solving ordinary differential equations odes on lie groups are based on the idea of representing the approximation y n of the exact solution y t n t n r by means of exact exponentials of suitable elements of the lie algebra applied to the initial value y 0 this ensures that y n g when the exponential is difficult to compute exactly as is the case when the dimension is large an approximation of exp tb plays an important role in the numerical solution of odes on lie groups in some cases rational or polynomial approximants are unsuitable and we consider alternative techniques whereby exp tb is approximated by a product of simpler exponentials in this paper we present some ideas based on the use of the strang splitting for the approximation of matrix exponentials several cases of g and g are considered in tandem with general theory order conditions are discussed and a number of numerical experiments conclude the paper

perturbed collocation and runge kutta methods it is well known thatsome implicit runge kutta methods are equivalent to collocation methods this fact permits very short and natural proofs of order anda b an bn stability properties for this subclass of methods see 9 and 10 the present paper answers the natural question ifall rk methods can be considered as a somewhat perturbed collocation after having introduced this notion we give a proof on the order of the method and discuss their stability properties much of known theory becomes simple and beautiful

galerkin lie group variational integrators based on unit quaternion interpolation abstract lie group variational integrators of arbitrary order are developed using the galerkin method based on unit quaternion interpolation to our knowledge quaternions have not been used before for this purpose though they allow a very simple and efficient way to perform the interpolation the resulting integrators are symplectic and structure preserving in the sense that certain symmetries in the lagrangian of the mechanical system are carried over to the discrete setting which leads to the preservation of the corresponding momentum maps the integrators furthermore exhibit a very good long time energy behavior i e energy is neither dissipated nor gained artificially at the same time the lie group structure is preserved by carefully defining the variations the interpolation method and by solving the non linear system of equations directly on the manifold rather than constraining it in a surrounding space using lagrange multipliers as a consequence we are able to show that lie group variational integrators based on the special orthogonal group are equivalent to the variational integrators for constrained systems using the discrete null space method employed e g in dmocc discrete mechanics and optimal control of constrained systems we show new numerical results on the convergence rates which are substantially higher than the known theoretical bounds and on the relation between accuracy and computational cost

normal modes for non linear vibratory systems a methodology is presented which extends to non linear systems the concept of normal modes of motion which is well developed for linear systems the method is constructive for weakly non linear systems and provides the physical nature of the normal modes along with the non linear differential equations which govern their dynamics it also provides the non linear co ordinate transformation which relates the original system co ordinates to the modal co ordinates using this transformation we demonstrate how an approximate non linear version of superposition can be employed to reconstruct the overall motion from the individual non linear modal dynamics the results presented herein for non linear systems reduce to modal analysis for the linearized system when non linearities are neglected even though the approach is entirely different from the traditional one the tools employed are from the theory of invariant manifolds for dynamical systems and were inspired by the center manifold reduction technique in this paper the basic ideas are outlined a few examples are presented and some natural extensions and applications of the method are briefly described in the conclusions

an optimal modal reduction of a system with frictional excitation abstract the proper orthogonal decomposition theorem is used in an optimal modal reduction of a frictionally excited system the chaotic dynamics are exploited in obtaining a basis that broadly represents the system s dynamics this basis is used in a galerkin s approximation to reduce the system order the reduced model is validated using qualitative and quantitative means and also from the occurrence of bifurcations the results confirm the validity of the method and expose the benefits of using proper orthogonal modes rather than the linear natural modes of the system

interpolation in lie groups we consider interpolation in lie groups based on points on the manifold together with tangent vectors at some of these points we construct hermite interpolation polynomials if the points and tangent vectors are produced in the process of integrating an ordinary differential equation in terms of lie algebra actions we use the truncated inverse of the differential of the exponential mapping and the truncated baker campbell hausdorff formula to relatively cheaply construct an interpolation polynomial much effort has lately been put into research on geometric integration i e the process of integrating differential equations in such a way that the configuration space of the true solution is respected by the numerical solution some of these methods may be viewed as generalizations of classical methods and we investigate the construction of intrinsic dense output devices as generalizations of the continuous runge kutta methods

order barriers for continuous explicit runge kutta methods in this paper we deal with continuous numerical methods for solving initial value problems for ordinary differential equations the need for which occurs frequently in applications whereas most of the commonly used multistep methods provide continuous extensions by means of an interpolant which is available without making extra function evaluations this is not always the case for one step methods we consider the class of explicit runge kutta methods and provide theorems used to obtain lower bounds for the number of stages required to construct methods of a given unifonn order p these bounds are similar to the butcher barriers known for the discrete case and are derived up to order p 5 as far as we know the examples we present of 8 stage continuous runge kutta methods of uniform order 5 are the first of their kind

scattered manifold valued data approximation we consider the problem of approximating a function f from an euclidean domain to a manifold m by scattered samples f xi i i in mathcal i f i i i where the data sites xi i i in mathcal i i i i are assumed to be locally close but can otherwise be far apart points scattered throughout the domain we introduce a natural approximant based on combining the moving least square method and the karcher mean we prove that the proposed approximant inherits the accuracy order and the smoothness from its linear counterpart the analysis also tells us that the use of karcher s mean dependent on a riemannian metric and the associated exponential map is inessential and one can replace it by a more general notion of center of mass based on a general retraction on the manifold consequently we can substitute the karcher mean by a more computationally efficient mean we illustrate our work with numerical results which confirm our theoretical findings

smoothness equivalence properties of manifold valued data subdivision schemes based on the projection approach interpolation of manifold valued data is a fundamental problem which has applications in many fields the linear subdivision method is an efficient and well studied method for interpolating or approximating real valued data in a multiresolution fashion a natural way to apply a linear subdivision scheme overline s to interpolate manifold valued data is to first embed the manifold at hand into an euclidean space and construct a projection operator p that maps points from the ambient space to a closest point on the embedded surface and then consider the nonlinear subdivision operator s p circ overline s when applied to symmetric spaces such as s n 1 so n sl n se n g n k the projection method can also be carried out in such a way that the resulting schemes enjoy natural coordinate invariance properties and robust numerical implementations despite such nice features the mathematical analysis of such nonlinear subdivision schemes is in its infancy in this article we attack the so called smoothness equivalence conjecture which asserts that the smoothness property of s is exactly the same as that of overline s we show that in the cases of s n 1 so n and related manifolds we have a proximity condition of the form s overline s y infty lesssim sum i 1 p 1 delta i y infty delta p i y infty where p is the accuracy order of overline s armed with this proximity condition and other known approximation theoretic results we can establish the result that the ho lder smoothness exponent of s is always as high as that of overline s no matter how high the latter is

geodesic finite elements for cosserat rods we introduce geodesic finite elements as a new way to discretize the non linear configuration space of a geometrically exact cosserat rod these geodesic finite elements naturally generalize standard one dimensional finite elements to spaces of functions with values in a riemannian manifold for the special orthogonal group our approach reproduces the interpolation formulas of crisfield and jelenic geodesic finite elements are conforming and lead to objective and path independent problem formulations we introduce geodesic finite elements for general riemannian manifolds discuss the relationship between geodesic finite elements and coefficient vectors and estimate the interpolation error then we use them to find static equilibria of hyperelastic cosserat rods using the riemannian trust region algorithm of absil et al we show numerically that the discretization error depends optimally on the mesh size copyright 2009 john wiley sons ltd

collocation and relaxed collocation for the fer and the magnus expansions we consider the fer and the magnus expansions for the numerical solution of the nonlinear matrix lie group ode y gamma t y y y 0 y 0 where y evolves in a matrix lie group g and gamma t y is in the lie algebra hbox fr g ghe departing from a geometrical approach that distinguishes between those operations performed in the group and those performed in the tangent space we construct lie group invariant methods based on collocation we prove that as long as the two expansions are correctly truncated the collocation nodes c 1 c 2 ldots c nu yield numerical methods whose order is the same as in the classical setting we also relax the collocation conditions thereby devising explicit methods of order 3 to conclude we discuss the proposed methods in a numerical experiment that arises in hamiltonian mechanics comparing the results with projection techniques

hybrid proper orthogonal decomposition formulation for linear structural dynamics abstract hybrid proper orthogonal decomposition podh formulation is a pod based reduced order modeling method where the continuous equation of the physical system is projected on the pod modes obtained from a discrete model of the system the aim of this paper is to evaluate the hybrid pod formulation and to compare it with other pod formulations on the simple case of a linear elastic rod subject to prescribed displacements in the perspective of building reduced order models for coupled fluid structure systems in the future in the first part of the paper the hybrid pod is compared to two other formulations for the response to an initial condition an approach based on the discrete finite elements equation of the rod called the discrete pod podd and an analytical approach using the exact solution of the problem and consequently called the analytical pod poda this first step is useful to ensure that the podh performs well with respect to the other formulations the podh is therefore used afterwards for the forced motion response where a displacement is imposed at the free end of the rod the main contribution of this paper lies in the comparison of three techniques used to take into account the non homogeneous dirichlet boundary condition with the hybrid pod the first method relies on control functions the second on the penalty method and the third on lagrange multipliers finally the robustness of the hybrid pod is investigated on two examples involving firstly the introduction of structural damping and secondly a nonlinear force applied at the free end of the rod

unitary integrators and applications to continuous orthonormalization techniques in this paper the issue of integrating matrix differential systems whose solutions are unitary matrices is addressed such systems have skew hermitian coefficient matrices in the linear case and a

sliding window proper orthogonal decomposition application to linear and nonlinear modal identification abstract the proper orthogonal decomposition pod method is a tool well adapted to analyze vector signals whereas continuous gabor transform cgt is suitable for scalar signal with multi frequency components in this paper a method named sliding window proper orthogonal decomposition swpod combining pod and cgt to analyze multi degrees of freedom mdof vibration system responses is presented swpod gives access to the evolution of the coherent spacial structures and their frequency components versus time the method is of principal interest in the case of swept sine excitation of linear or nonlinear systems to access the resonance frequencies mode shapes and modal damping ratios a procedure is proposed to extract the linear and nonlinear normal modes of weakly damped mdof mechanical systems and illustrated using numerical examples

a multi step transversal linearization mtl method in non linear structural dynamics an implicit family of multi step transversal linearization mtl methods is proposed for efficient and numerically stable integration of nonlinear oscillators of interest in structural dynamics the presently developed method is a multi step extension and further generalization of the locally transversal linearization ltl method proposed earlier by roy proceedings of the academy of the royal society london 457 2001 539 566 roy and ramachandra journal of sound and vibration 41 2001a 653 679 international journal for numerical methods in engineering 51 2001b 203 224 and roy international journal of numerical methods in engineering 61 2004 764 the mtl based linearization is achieved through a non unique replacement of the nonlinear part of the vector field by a conditionally linear interpolating expansion of known accuracy whose coefficients contain the discretized state variables defined at a set of grid points in the process the nonlinear part of the vector field becomes a conditionally determinable equivalent forcing function the mtl based linearized differential equations thus become explicitly integrable based on the linearized solution a set of algebraic constraint equations are so formed that transversal intersections of the linearized and nonlinearized solution manifolds occur at the multiple grid points the discretized state vectors are thus found as the zeros of the constraint equations simple error estimates for the displacement and velocity vectors are provided and in particular it is shown that the formal accuracy of the mtl methods as a function of the time step size depends only on the error of replacement of the nonlinear part of the vector field presently only two different polynomial based interpolation schemes are employed for transversal linearization viz the taylor like interpolation and the lagrangian interpolation while the taylor like interpolation leads to numerical ill conditioning as the order of interpolation increases the lagrangian interpolation is shown to overcome this numerical problem finally the family of mtl methods is illustrated through limited numerical results for a couple of harmonically driven workhorse oscillators viz the hardening duffing and the duffing holmes oscillators comparisons with results obtained via a sixth order runge kutta method with adaptive time step sizes are also provided

acceleration of the spectral stochastic fem using pod and element based discrete empirical approximation for a micromechanical model of heterogeneous materials with random geometry abstract the spectral stochastic fem with local basis functions in the stochastic domain sl fem is one of the most flexible and accurate stochastic methods however also the most computationally expensive these expenses are traditionally associated with the extra large tangent stiffness matrix and a huge number of elements which need to be re integrated in every iteration in this work we incorporate the proper orthogonal decomposition pod into the sl fem thus performing a drastic reduction of the stiffness matrix in order to reduce the integration costs by hyperreduction a novel element based modification of the discrete empirical interpolation the so called element based empirical approximation method edeam is developed and combined with the pod particular advantages of the sl fem for order reduction and hyperreduction compared to other stochastic techniques are discussed the new reduced order sl fem is applied to the computational homogenization of materials with random geometry of the microstructure i e to a general class of problems exhibiting strongly nonlinear non smooth and sometimes discontinuous dependency of the solution on some random parameters the reduced order sl fem demonstrates a high accuracy and a high solution speed whereby the solution time for the reduced order sl fem is comparable to the solution time of only one single monte carlo sample

a time varying identification method for mixed response measurements the proper orthogonal decomposition is a method that may be applied to linear and nonlinear structures for extracting important information from a measured structural response this method is often applied for model reduction of linear and nonlinear systems and has been applied recently for time varying system identification although methods have previously been developed to identify time varying models for simple linear and nonlinear structures using the proper orthogonal decomposition of a measured structural response the application of these methods has been limited to cases where the excitation is either an initial condition or an applied load but not a combination of the two this paper presents a method for combining previously published proper orthogonal decomposition based identification techniques for strictly free or strictly forced systems to identify predictive models for a system when only mixed response data are available i e response data resulting from initial conditions and loads that are applied together this method extends the applicability of the previous proper orthogonal decomposition based identification techniques to operational data acquired outside of a controlled laboratory setting the method is applied to response data generated by finite element models of simple linear time invariant time varying and nonlinear beams and the strengths and weaknesses of the method are discussed

physical interpretation of the proper orthogonal modes using the singular value decomposition abstract proper orthogonal decomposition is a statistical pattern analysis technique for finding the dominant structures called the proper orthogonal modes in an ensemble of spatially distributed data while the proper orthogonal modes are obtained through a statistical formulation they can be physically interpreted in the field of structural dynamics the purpose of this paper is thus to provide some insights into the physical interpretation of the proper orthogonal modes using the singular value decomposition

on the dominance of trivial knots among saps on a cubic lattice the knotting probability is defined by the probability with which an n step self avoiding polygon sap with a fixed type of knot appears in the configuration space we evaluate these probabilities for some knot types on a simple cubic lattice for the trivial knot we find that the knotting probability decays much slower for the sap on the cubic lattice than for continuum models of the sap as a function of n in particular the characteristic length of the trivial knot that corresponds to a half life of the knotting probability is estimated to be 2 5 105 on the cubic lattice