uncertainty propagation in reduced order models based on crystal plasticity abstract in this work an uncertainty propagation study is performed based on simulated ensembles of statistical volume elements sve used to inform a reduced order internal state variable model focusing on model form and uncertainties in numerical simulations a rate dependent polycrystal plasticity finite element model cp fem of cartridge brass is calibrated to room temperature uniaxial tension testing data of an annealed sample model forms are considered with and without the inclusion of back stress in the cp model three sizes of sve are explored for simulation uncertainty effects effects are explored with regard to the parameters of a bammann chiesa johnson bcj macroscopic viscoplasticity model calibrated element wise to sves from an ensemble modeled using cp fem simulations exploring quasi static uniaxial tension variability in stress strain response at multiple defined length scales is measured the selection of model form is discussed the numerical simulation uncertainty of the model reduction process is quantified

self avoiding polygons on the square l and manhattan lattices transfer matrix techniques are used to extend the self avoiding polygon generating function on the square lattice to terms in x46 corresponding to 46 step polygons these techniques are then extended to apply to directed square lattices such as the l and manhattan lattice and the self avoiding polygon generating function to x48 is found for these lattices series analysis confirms that the specific heat exponent alpha 1 2 for the self avoiding walk problem and gives the following estimates for the connective constants mu sq 2 638155 or 0 000025 mu l 1 5657 or 0 0019 and mu man 1 7328 or 0 0005 some evidence for a correction to scaling exponent delta approximately 0 84 is found from square lattice series

exact enumeration study of free energies of interacting polygons and walks in two dimensions we present analyses of substantially extended series for both interacting self avoiding walks isaw and polygons isap on the square lattice we argue that these provide good evidence that the free energies of both linear and ring polymers are equal above the temperature thus extending the application of a theorem of tesi et al to two dimensions below the temperature the conditions of this theorem break down in contradistinction to three dimensions but an analysis of the ratio of the partition functions for isap and isaw indicates that the free energies are in fact equal at all temperatures within at least any perceived difference can be interpreted as the difference in the size of corrections to scaling in both problems this may be used to explain the vastly different values of the crossover exponent previously estimated for isap to that predicted theoretically and numerically confirmed for isaw an analysis of newly extended neighbour avoiding self avoiding walk series is also given

two dimensional lattice vesicles and polygons we consider a lattice model of two dimensional vesicles in which the boundary of the vesicle is the perimeter of a self avoiding polygon embeddable in the square lattice with fixed boundary length m we incorporate an osmotic pressure difference by associating a fugacity with the area enclosed by the polygon we derive rigorous results concerning the behaviour of the associated free energy and the form of the phase diagram by deriving exact values ofthe numbers of polygons with m edges which enclose area n and analysing the resulting series we obtain the free energy the phase boundary and various scaling exponents and amplitudes numerically

crossover between random and self avoiding behaviour in a ring polymer the crossover between the random walk and the self avoiding walk limits of a large ring polymer is described using a direct renormalisation theory the methods of field theory renormalisation are applied directly to the ring polymer system to calculate the mean square radius of gyration rg2 ring through first order in epsilon 4 dimension of space the expansion factor alpha 2 for rg2 ring is not a universal function of the natural scaling variable x that characterises the strength of the self repelling excluded volume interaction the crossover exhibited by the radius of gyration ratio r x rg2 ring rg2 line is described by a universal scaling function of x and has a range space 1 2 or r x or 1 2 1 13 96 epsilon that is bounded below and above by finite universal numbers representing the random and self avoiding limits respectively

lee yang zeros in the scaling region of a two dimensional quasiperiodic ising model quasiperiodic planar ising models with ferromagnetic nearest neighbour interactions should show the same universal critical behaviour as the classical ising model on the square lattice we use the eightfold symmetric ammann beenker tiling to investigate this and employ the distribution of the lee yang and the temperature zeros of the partition function in the complex plane our results support as expected the existence of an onsager type phase transition i e a second order transition with critical exponents and

polygons on the honeycomb lattice the authors enumerated the number of self avoiding polygons on the honeycomb lattice to 82 steps analysis of the resulting series gives for the connective constant mu 1 8477594 or 6 10 6 in agreement with nienhuis value 1984 of mu 2 2 12 1 84775906 the critical exponent alpha 0 4999 or 10 4 unbiased or alpha 0 49998 or 2 10 0 5 biased a non physical singularity on the negative real axis is accurately located and some curious results concerning the confluent singularity are obtained

directed self avoiding polygons the polygon exponent h associated with directed self avoiding walks in two dimensions has been obtained by exact enumeration and series extrapolation methods the value of h is found to be 1 40 or 0 05 and the connectivity constant associated with such polygons is seen to be less than that of the corresponding walks

aperiodic ising quantum chains some years ago luck proposed a relevance criterion for the effect of aperiodic disorder on the critical behaviour of ferromagnetic ising systems in this article we show how luck s criterion can be derived within an exact renormalization scheme for ising quantum chains with coupling constants modulated according to substitution rules luck s conjectures for this case are confirmed and refined among other outcomes we give an exact formula for the correlation length critical exponent for arbitrary two letter substitution sequences with marginal fluctuations of the coupling constants

a new stable basis for radial basis function interpolation it is well known that radial basis function interpolants suffer from bad conditioning if the basis of translates is used in the recent work by pazouki and schaback 2011 5 the authors gave a quite general way to build stable and orthonormal bases for the native space n f w associated to a kernel f on a domain w r s the method is simply based on the factorization of the corresponding kernel matrix starting from that setting we describe a particular basis which turns out to be orthonormal in n f w and in 2 w x where x is a set of data sites of the domain w the basis arises from a weighted singular value decomposition of the kernel matrix this basis is also related to a discretization of the compact operator t f n f w n f w t f f x w f x y f y dy x w and provides a connection with the continuous basis that arises from an eigendecomposition of t f finally using the eigenvalues of this operator we provide convergence estimates and stability bounds for interpolation and discrete least squares approximation

determination of the size of the representative volume element for random composites statistical and numerical approach the representative volume element rve plays a central role in the mechanics and physics of random heterogeneous materials with a view to predicting their effective properties a quantitative definition of its size is proposed in this work a rve size can be associated with a given precision of the estimation of the wanted overall property and the number of realizations of a given volume v of microstructure that one is able to consider it is shown to depend on the investigated morphological or physical property the contrast in the properties of the constituents and their volume fractions the methodology is applied to a specific random microstructure namely a two phase three dimensional voronoi mosaic finite element simulations of volumes of different sizes are performed in the case of linear elasticity and thermal conductivity the volumes are subjected to homogeneous strain stress or periodic boundary conditions the effective properties can be determined for large volumes and a small number of realizations conversely smaller volumes can be used providing that a sufficient number of realizations are considered a bias in the estimation of the effective properties is observed for too small volumes for all types of boundary conditions the variance of computed apparent properties for each volume size is used to define the precision of the estimation the key notion of integral range is introduced to relate this error estimation and the definition of the rve size for given wanted precision and number of realizations one is able to provide a minimal volume size for the computation of effective properties the results can also be used to predict the minimal number of realizations that must be considered for a given volume size in order to estimate the effective property for a given precision the rve sizes found for elastic and thermal properties but also for a geometrical property like volume fraction are compared

vesicles on a hierarchical lattice an exact renormalization group approach a model of two dimensional vesicles on a sierpinski gasket is introduced and solved by iterative methods the effect of osmotic pressure is obtained by associating a fugacity to each plaquette of the underlying triangular lattice enclosed by the vesicle several results concerning the critical behaviour and phase diagram are discussed in particular it is shown exactly that deflated critical vesicles fall into the universality class of branched polymers

quasi interpolation scheme for arbitrary dimensional scattered data approximation based on natural neighbors and rbf interpolation abstract a local quasi interpolation scheme is presented in the paper which can be applied to arbitrary dimensional scattered data approximation with high accuracy high efficiency and high stability under our new quasi interpolation scheme we treat the natural neighbor set of the point to be estimated in a given scattered node set as a local interpolation node set based on the local node set a local radial basis function interpolation with algebraic precision of any degree is constructed through a modified taylor expansion of sampled function at each natural neighbor to compute approximate values given the exclusion of the point to be estimated from its natural neighbor set i e interpolation node set the interpolation scheme we construct is a local quasi interpolation scheme and its approximation error is given the numerical experimental results show that when the derivative information is unavailable our method outperforms the famous natural neighbor interpolation method in both approximation accuracy and efficiency as algebraic precision increases the approximation accuracy can be further improved

self avoiding walks neighbour avoiding walks and trails on semiregular lattices we study self avoiding and neighbour avoiding walks and lattice trails on two semiregular lattices the lattice and the lattice for the lattice we find the exact connective constant for both self avoiding walks neighbour avoiding walks and trails for the lattice we generate long series which permit the accurate estimation of the connective constant for self avoiding walks and trails

high temperature expansion for ising models on quasiperiodic tilings we consider high temperature expansions for the free energy of zero field ising models on planar quasiperiodic graphs for the penrose and the octagonal ammann beenker tiling we compute the expansion coefficients up to 18th order as a by product we obtain exact vertex averaged numbers of self avoiding polygons on these quasiperiodic graphs in addition we analyse periodic approximants by computing the partition function via the kac ward determinant it turns out that the series expansions alone do not yield reliable estimates of the critical exponents this is due to the limitation on the order of the series caused by the number of graphs that have to be taken into account and more seriously to rather strong fluctuations in the behaviour of the coefficients nevertheless our results are compatible with the commonly accepted conjecture that the models under consideration belong to the same universality class as those on periodic two dimensional lattices

size of rings in two dimensions the authors report enumeration results for the radius of gyration and caliper size distribution of self avoiding unrooted polygons of up to 28 steps on the square lattice the second moment radius of gyration series is sufficiently smooth to allow verification of the theoretical prediction v rings v walks to 0 2 accuracy

self avoiding rings on the triangular lattice the finite lattice method of series expansion is used to extend the enumeration of self avoiding polygons on the triangular lattice through to rings of 35 steps the authors also give the enumeration of these triangular polygons grouped by perimeter and area of up to 21 unit triangles for perimeters up to 24 steps

universal amplitude in the sizes of rings in two dimensions for self avoiding rings of n steps in two dimensions the limiting value as n to infinity of the combination npn r2 nxcn where pn is the number of distinct rings r2 n is their mean square radius of gyration and xc is the critical fugacity is equal to a calculable lattice dependent number times a universal amplitude this latter quantity is calculated exactly using methods of conformal invariance the value is in good agreement with the results of enumeration studies

monte carlo results for projected self avoiding polygons a two dimensional model for knotted polymers we introduce a two dimensional lattice model for the description of knotted polymer rings a polymer configuration is modelled by a closed polygon drawn on the square diagonal lattice with possible crossings describing pairs of strands of polymer passing on top of each other each polygon configuration can be viewed as the two dimensional projection of a particular knot we study numerically the statistics of large polygons with a fixed knot type using a generalization of the bfacf algorithm for self avoiding walks this new algorithm incorporates both the displacement of crossings and the three types of reidemeister transformations preserving the knot topology its ergodicity within a fixed knot type is not proven here rigorously but strong arguments in favour of this ergodicity are given together with a tentative sketch of proof assuming this ergodicity we obtain numerically the following results for the statistics of knotted polygons in the limit of a low crossing fugacity we find a localization along the polygon of all the primary factors forming the knot increasing the crossing fugacity gives rise to a transition from a self avoiding walk to a branched polymer behaviour

diffraction pattern calculations for a certain class of n fold quasilattices using the cut and project method we obtain seven and nine fold quasiperiodic lattices for such quasilattices the structure factors are calculated analytically through integration in higher dimensional spaces the obtained formulae are compared with the direct numerical calculations of fourier transforms

approximants of quasiperiodic structures generated by the inflation mapping the problem of deriving explicit coordinates for quasicrystal approximants is solved in all cases where the quasicrystal has an inflation symmetry in the higher dimensional space rn from which the quasiperiodic pattern is obtained by the cut method the inflation symmetry is represented by a hyperbolic modular matrix with integer entries leaving the physical space invariant but this matrix also generates by iteration a sequence of rational approximant spaces which converges to the irrational space a simple algorithm is described providing the approximant periodic lattice and the set of vertices within a unit cell

better bases for radial basis function interpolation problems radial basis function interpolation involves two stages the first is fitting solving a linear system corresponding to the interpolation conditions the second is evaluation the systems occurring in fitting problems are often very ill conditioned changing the basis in which the radial basis function space is expressed can greatly improve the conditioning of these systems resulting in improved accuracy and in the case of iterative methods improved speed of solution the change of basis can also improve the accuracy of evaluation by reducing loss of significance errors in this paper new bases for the relevant space of approximants and associated preconditioning schemes are developed which are based on floater s mean value coordinates positivity results and scale independence results are shown for schemes of a general type numerical results show that the given preconditioning scheme usually improves conditioning of polyharmonic spline and multiquadric interpolation problems in r 2 and r 3 by several orders of magnitude the theory indicates that using the new basis elements evaluated indirectly for both fitting and evaluation will reduce loss of significance errors on evaluation numerical experiments confirm this showing that such an approach can improve overall accuracy by several significant figures

miscellaneous error bounds for multiquadric and related interpolators we establish several types of a a priori error bounds for multiquadric and related interpolators the results are stated and proven in the general multivariate case these estimates show for example that in many cases such interpolators converge very quickly and can be used in the recovery of band limited functions from discrete data we also include numerical experiments which illustrate the theoretical results

invaded cluster algorithm for critical properties of periodic and aperiodic planar ising models we demonstrate that the invaded cluster algorithm introduced by machta et al 1995 phys rev lett 75 2792 5 is a fast and reliable tool for determining the critical temperature and the magnetic critical exponent of periodic and aperiodic ferromagnetic ising models in two dimensions the algorithm is shown to reproduce the known values of the critical temperature on various periodic and quasiperiodic graphs with an accuracy of more than three significant digits but only modest computational effort on two quasiperiodic graphs which were not investigated in this respect before the 12 fold symmetric square triangle tiling and the 10 fold symmetric tubingen triangle tiling we determine the critical temperature furthermore a generalization of the algorithm to non identical coupling strengths is presented and applied to a class of ising models on the labyrinth tiling for generic cases in which the heuristic harris luck criterion predicts deviations from the onsager universality class we find a magnetic critical exponent different from the onsager value but notable exceptions to the criterion are found which consist not only of the exactly solvable cases in agreement with a recent exact result but also of the self dual ones and maybe more

the size and number of rings on the square lattice the authors have calculated the number of self avoiding polygons on the square lattice to 56 steps and the caliper size to 54 steps analysis of the generating function permits estimates of the connective constant mu 2 638 1585 or 10 6 and the critical exponents alpha 0 500 06 or 0 000 06 and v 0 753 or 0 007 the singularity structure of the polygon generating function is found to be consistent with a correction to the scaling exponent delta 1 5 as predicted by nienhuis 1982 1984 the confluent part however maps into the additive analytic term due to the value of the exponent alpha

a multiscale stochastic finite element method on elliptic problems involving uncertainties in this study a multiscale stochastic finite element method mssfem is developed to resolve scale coupling stochastic elliptic problems based on formulations of a stochastic variational approach and scale bridging multiscale shape functions by employing polynomial chaos expansions and lagrange polynomials stochastic galerkin method and stochastic collocation method are introduced and combined with deterministic multiscale methods i e variational multiscale method and multiscale finite element method the resulting spectral and pseudo spectral stochastic finite element methods are incorporated into the fine and coarse mesh computation of the mssfem a benchmark multiscale model and the numerical experiment based on the formulated mssfem are provided for illustration it is expected that the proposed mssfem can act as a paradigm for solving of general stochastic partial differential equations involving multiscale stochastic data