a proper generalized decomposition for the solution of elliptic problems in abstract form by using a functional eckart young approach the proper generalized decomposition pgd is a methodology initially proposed for the solution of partial differential equations pde defined in tensor product spaces it consists in constructing a separated representation of the solution of a given pde in this paper we consider the mathematical analysis of this framework for a larger class of problems in an abstract setting in particular we introduce a generalization of eckart and young theorem which allows to prove the convergence of the so called progressive pgd for a large class of linear problems defined in tensor product hilbert spaces

gaussian wave packet transform based numerical scheme for the semi classical schrodinger equation with random inputs abstract in this work we study the semi classical limit of the schrodinger equation with random inputs and show that the semi classical schrodinger equation produces o e oscillations in the random variable space with the gaussian wave packet transform the original schrodinger equation is mapped to an ordinary differential equation ode system for the wave packet parameters coupled with a partial differential equation pde for the quantity w in rescaled variables further we show that the w equation does not produce e dependent oscillations and thus it is more amenable for numerical simulations we propose multi level sampling strategy in implementing the gaussian wave packet transform where in the most costly part i e simulating the w equation it is sufficient to use e independent samples we also provide extensive numerical tests as well as meaningful numerical experiments to justify the properties of the numerical algorithm and hopefully shed light on possible future directions

a new mesh generation scheme for arbitrary planar domains this paper describes a new algorithm to generate interior nodes within any arbitrary multi connected regions the boundary nodes and the interior nodes are then linked up to form the best possible triangular elements by a completely revised technique in an efficient and stable manner owing to the generality of the central generation program the global domain is allowed to be divided into as many irregular subdomains as desired in order to model closely the actual physical situation moreover the boundaries of the sub domains are updated from time to time when necessary to include the possibilities of progressive refinement around a sharp corner generating radiating mesh from a prescribed node generating mesh between two circular arcs etc despite its flexibility and capabilities data for triangulation have been kept to a minimum by a logical input module no connectivity information between subregions is needed and common boundaries are defined once only all these features have contributed to a powerful method to generate 3 node or 6 node triangular element meshes of great variety within the most irregular heterogeneous regions

a new indirect anisotropic quadrilateral mesh generation scheme with enhanced local mesh smoothing procedures an enhanced version of indirect advancing front technique is proposed for the construction of full quadrilateral anisotropic meshes on 3d surfaces the final quadrilateral mesh is constructed from a background triangular mesh and the merging procedure is carried out in the parametric space the proposed method employed the systemic merging technique for the formation of quadrilaterals and the metric specifications for controlling the anisotropic characteristics of the elements in order to generate well graded anisotropic quadrilaterals the selection criteria of base segment for merging are carefully defined furthermore in order to improve the shape quality of the final mesh a set of new local smoothing algorithms for anisotropic quadrilateral mesh quality enhancement has been proposed and tested in this study numerical examples obtained indicate that the proposed enhanced local smoothing schemes can effectively improve the quality of the quadrilateral meshes produced and increase the robustness of the mesh generation scheme copyright 2003 john wiley sons ltd

generalized 3 d paving an automated quadrilateral surface mesh generation algorithm this paper discusses the extension of the paving algorithm for all quadrilateral mesh generation to arbitrary three dimensional trimmed surfaces methods of calculating angles projecting elements and detecting collisions between paving boundaries for general surfaces are presented extensions of the smoothing algorithms for three dimensions are set forth advances in the use of scalar sizing functions are presented these functions can be used to better approximate internal mesh density from boundary densities and surface characteristics

h morph an indirect approach to advancing front hex meshing h morph is a new automatic algorithm for the generation of a hexahedral dominant finite element mesh for arbitrary volumes the h morph method starts with an initial tetrahedral mesh and systematically transforms and combines tetrahedral into hexahedra it uses an advancing front technique where the initial front consists of a set of prescribed quadrilateral surface facets fronts are individually processed by recovering each of the six quadrilateral faces of a hexahedron from the tetrahedral mesh recovery techniques similar to those used in boundary constrained delaunay mesh generation are used tetrahedral internal to the six hexahedral faces are then removed and a hexahedron is formed at any time during the h morph procedure a valid mixed hexahedral tetrahedral mesh is in existence within the volume the procedure continues until no tetrahedral remain within the volume or tetrahedral remain which cannot be transformed or combined into valid hexahedral elements any remaining tetrahedral are typically towards the interior of the volume generally a less critical region for analysis transition from tetrahedral to hexahedra in the final mesh is accomplished through pyramid shaped elements advantages of the proposed method include its ability to conform to an existing quadrilateral surface mesh its ability to mesh without the need to decompose or recognize special classes of geometry and its characteristic well aligned layers of elements parallel to the boundary example test cases are presented on a variety of models

blossom quad a non uniform quadrilateral mesh generator using a minimum cost perfect matching algorithm a new indirect way of producing all quad meshes is presented the method takes advantage of a wellknown algorithm of the graph theory namely the blossom algorithm that computes the minimum cost perfect matching in a graph in polynomial time the new blossom quad algorithm is compared with standard indirect procedures meshes produced by the new approach are better both in terms of element shape and in terms of size eld ef ciency

generation of tetrahedral mesh of variable element size by sphere packing over an unbounded 3d domain this paper describes an algorithm for the generation of tetrahedral mesh of specified element size over an unbounded three dimensional domain starting from an arbitrary point in space defined as the origin and guided by the concept of advancing front spheres of size compatible with the specified element size are packed tightly together one by one to form a cluster of spheres of different sizes the compactness of the cluster of spheres is achieved by packing spheres at a site closest to the origin in a densest manner with tangent but no overlapping with as many other spheres as possible in view of these criteria a rotational mechanism between spheres is innovated which allows the newly inserted sphere to follow the path by rotation between existing spheres until the lowest point is reached the centres of the packing spheres provide ideal locations for delaunay point insertion to form a triangulation of tetrahedral elements of size compatible with the specified value spheres of random size distribution or of size specified by a node spacing function are packed by the proposed algorithm in all test examples high quality tetrahedral elements of size consistent with the specified node spacing are generated the process is fast and robust and the time complexity for mesh generation is expected to be almost linear however in the present implementation a quasi linear time relationship is observed as a search for the nearest node on the front is proposed in the sphere packing process the memory requirement has been kept to a minimum as no additional data structure other than the adjacency relationship of the tetrahedral elements as required by the delaunay triangulation is stored the structure of the advancing frontal surface needs not be explicitly constructed nor updated as it is simply not required in the sphere packing process

a data driven stochastic collocation approach for uncertainty quantification in mems this work presents a data driven stochastic collocation approach to include the effect of uncertain design parameters during complex multi physics simulation of micro electromechanical systems mems the proposed framework comprises of two key steps first probabilistic characterization of the input uncertain parameters based on available experimental information and second propagation of these uncertainties through the predictive model to relevant quantities of interest the uncertain input parameters are modeled as independent random variables for which the distributions are estimated based on available experimental observations using a nonparametric diffusion mixing based estimator botev nonparametric density estimation via diffusion mixing technical report 2007 the diffusion based estimator derives from the analogy between the kernel density estimation kde procedure and the heat dissipation equation and constructs density estimates that are smooth and asymptotically consistent the diffusion model allows for the incorporation of the prior density and leads to an improved density estimate in comparison with the standard kde approach as demonstrated through several numerical examples following the characterization step the uncertainties are propagated to the output variables using the stochastic collocation approach based on sparse grid interpolation smolyak soviet math dokl 1963 4 240 243 the developed framework is used to study the effect of variations in young s modulus induced as a result of variations in manufacturing process parameters or heterogeneous measurements on the performance of a mems switch copyright 2010 john wiley sons ltd

automatic generation of anisotropic quadrilateral meshes on three dimensional surfaces using metric specifications a new algorithm for constructing full quadrilateral anisotropic meshes on 3d surfaces is proposed in this paper the proposed method is based on the advancing front and the systemic merging techniques full quadrilateral meshes are constructed by systemically converting triangular elements in the background meshes into quadrilateral elements by using the metric specifications to describe the element characteristics the proposed algorithm is applicable to convert both isotropic and anisotropic triangular meshes into full quadrilateral meshes special techniques for generating anisotropic quadrilaterals such as new selection criteria of base segment for merging new approaches for the modifications of the background mesh and construction of quadrilateral elements are investigated and proposed in this study since the final quadrilateral mesh is constructed from a background triangular mesh and the merging procedure is carried out in the parametric space the mesh generator is robust and no expensive geometrical computation that is commonly associated with direct quadrilateral mesh generation schemes is needed copyright 2002 john wiley sons ltd

model reduction based on proper generalized decomposition for the stochastic steady incompressible navier stokes equations in this paper we consider a proper generalized decomposition method to solve the steady incompressible navier stokes equations with random reynolds number and forcing term the aim of such a technique is to compute a low cost reduced basis approximation of the full stochastic galerkin solution of the problem at hand a particular algorithm inspired by the arnoldi method for solving eigenproblems is proposed for an efficient greedy construction of a deterministic reduced basis approximation this algorithm decouples the computation of the deterministic and stochastic components of the solution thus allowing reuse of preexisting deterministic navier stokes solvers it has the remarkable property of only requiring the solution of m uncoupled deterministic problems for the construction of an m dimensional reduced basis rather than m coupled problems of the full stochastic galerkin approximation space with m ll m up to one order of magnitude for the problem at hand in this work

a priori model reduction through proper generalized decomposition for solving time dependent partial differential equations over the past years model reduction techniques have become a necessary path for the reduction of computational requirements in the numerical simulation of complex models a family of a priori model reduction techniques called proper generalized decomposition pgd methods are receiving a growing interest these methods rely on the a priori construction of separated variables representations of the solution of models defined in tensor product spaces they can be interpreted as generalizations of proper orthogonal decomposition pod for the a priori construction of such separated representations in this paper we introduce and study different definitions of pgd for the solution of time dependent partial differential equations we review classical definitions of pgd based on galerkin or minimal residual formulations and we propose and discuss several improvements for these classical definitions we give an interpretation of optimal decompositions as the solution of pseudo eigenproblems we also introduce a new definition of pgd called minimax pgd which can be interpreted as a petrov galerkin model reduction technique where test and trial reduced basis functions are related by an adjoint problem this new definition improves convergence properties of separated representations with respect to a chosen metric it coincides with a classical pod for degenerate time dependent partial differential equations for the numerical construction of each pgd we propose algorithms inspired from the solution of eigenproblems several numerical examples illustrate and compare the different definitions of pgd on transient advection diffusion reaction equations

on a multiscale computational strategy with time and space homogenization for structural mechanics a new multiscale computational strategy was recently proposed for the analysis of structures described both on a fine space scale and a fine time scale this strategy which involves homogenization in space as well as in time could replace in several domains of application the standard homogenization techniques which are generally limited to the space domain it is an iterative strategy which calls for the resolution of problems on both a micro fine scale and a macro homogenized scale in this paper we review the bases of this approach and present improved approximation techniques to solve the micro and macro problems

minimal multi element stochastic collocation for uncertainty quantification of discontinuous functions we propose a multi element stochastic collocation method that can be applied in high dimensional parameter space for functions with discontinuities lying along manifolds of general geometries the key feature of the method is that the parameter space is decomposed into multiple elements defined by the discontinuities and thus only the minimal number of elements are utilized on each of the resulting elements the function is smooth and can be approximated using high order methods with fast convergence properties the decomposition strategy is in direct contrast to the traditional multi element approaches which define the sub domains by repeated splitting of the axes in the parameter space such methods are more prone to the curse of dimensionality because of the fast growth of the number of elements caused by the axis based splitting the present method is a two step approach firstly a discontinuity detector is used to partition parameter space into disjoint elements in each of which the function is smooth the detector uses an efficient combination of the high order polynomial annihilation technique along with adaptive sparse grids and this allows resolution of general discontinuities with a smaller number of points when the discontinuity manifold is low dimensional after partitioning an adaptive technique based on the least orthogonal interpolant is used to construct a generalized polynomial chaos surrogate on each element the adaptive technique reuses all information from the partitioning and is variance suppressing we present numerous numerical examples that illustrate the accuracy efficiency and generality of the method when compared against standard locally adaptive sparse grid methods the present method uses many fewer number of collocation samples and is more accurate

automatic metric 3d surface mesh generation using subdivision surface geometrical model part 1 construction of underlying geometrical model summary this paper proposes a new automatic mesh generation algorithm for 3d surface mesh generation the algorithm is base d on the metric specification approach and can generate anisotropic meshes on 3d surfaces it is based on a new geometrical model using the interpolating subdivision surface concept by using the subdivision surface concept the new mesh generator can generate finite element meshes to model a wide range of surfaces which may contain sharp features such as cusp and crease lines when comparing with other algorithms which use analytical surface patches as the underlying geometrical model the new mesh generation scheme can be used in applications such as large deformation or crack analyses in which the domains to be gridded are not well defined or involve changing boundary the presentation of the work is divided into two parts in part i i e the present paper a detailed description of the underlying geometrical model used will be given while in part ii attentions will be focused on the mesh generation algorithms and the performance of the mesh generator

a dynamical adaptive tensor method for the vlasov poisson system abstract a numerical method is proposed to solve the full eulerian time dependent vlasov poisson system the algorithm relies on the construction of a tensor decomposition of the solution whose rank is adapted at each time step this decomposition is obtained through the use of an efficient modified progressive generalized decomposition pgd method whose convergence is proved we suggest in addition a symplectic time discretization splitting scheme that preserves the hamiltonian properties of the system this scheme is naturally obtained by considering the tensor structure of the approximation the proposed approach is illustrated through time dependent 1d 1d 2d 2d and 3d 3d numerical examples

multi fidelity non intrusive polynomial chaos based on regression abstract in this paper we present a multi fidelity mf extension of non intrusive polynomial chaos based on regression point collocation for uncertainty quantification purposes the proposed method uses the principle of a global correction function from a previous similar method that uses spectral projection to estimate the coefficients due to its usage of regression to estimate the coefficients the present method offers high flexibility in the sampling and generation of the polynomial basis the method takes advantage of a nested sampling plan to create the samples for the low fidelity lf and correction expansions where the high fidelity hf samples are a subset of the lf ones to build the polynomial basis a total order or hyperbolic truncation strategy is used with a highly flexible combination of the lf and correction polynomial expansions the method is demonstrated on some artificial test problems and aerodynamic problems of the euler flow around an airfoil and common three dimensional research models in order to derive the strategies for successful mf approximation the effect of the correlation and the errors between the lf and hf functions is also studied the results show that high correlation and moderately low errors are important to improve the mf approximation s accuracy on a common research model problem the mf approach with partially converged simulations as the lf samples can successfully reduce the computational cost to about 40 for similar accuracy compared to an approach using a single hf expansion

a boundary focused quadrilateral mesh generation algorithm for multi material structures this paper describes a new boundary focused mesh generation algorithm the algorithm produces quadrilateral meshes from images of multi material structures the challenge of meshing such images is to create high quality elements which are aligned with complex material boundaries the approach proposed in this paper uses the following steps 1 extract boundaries according to pixel colours 2 smooth the jagged boundaries 3 generate geodesic isolines aligned with smoothed boundaries 4 generate a mesh from the boundaries and isolines 5 optimize the mesh using the pisces pattern which is introduced in this paper application examples are presented to contrast the reliability and effectiveness of the new algorithm with existing approaches

construction of three dimensional improved quality triangulations using local transformations three dimensional delaunay triangulation are the most common form of three dimensional triangulation known but they are not very suitable for tetrahedral finite element meshes because they tend to contain poorly shaped sliver tetrahedra in this paper we present an algorithm for constructing improved quality triangulation with respect to a tetrahedron shape measure this algorithm uses combination of two or more local transformations to improve a given triangulation toward an optimal triangulation experimental results on finite element meshes show that this algorithm is much more effective than previous methods at removing slivers from delaunay triangulation and producing nearly optimal triangulation a variation of this algorithm for improving a pseudo locally optimal non delaunay triangulationn toward a delaunay triangulation is also presented

the latin multiscale computational method and the proper generalized decomposition this paper deals with the synergy between the latin multiscale method and what is called the proper generalized decomposition pgd which is the key of its performances

unstructured grid generation using iterative point insertion and local reconnection a procedure is presented for efficient generation of high quality two or three dimensional unstructured grids of triangular or tetrahedral elements the present procedure uses an iterative point creation and insertion scheme wherein points are created using advancing front type point placement initially the connectivity for these generated points is obtained by directly subdividing the elements which contain them without regard to quality this connectivity is then improved by iteratively using local reconnection subject to a quality criterion for two dimensions a min max criterion is used and for three dimensions a delaunay in sphere criterion followed by a min max type criterion is used the overall procedure is applied repetitively until a complete field grid is generated with a desired point distribution grid quality and performance statistics are presented for a variety of two and three dimensional configurations the combined quality and efficiency attributes of this procedure appear to be a substantial improvement over existing methods

on the existence of a progressive variational vademecum based on the proper generalized decomposition for a class of elliptic parameterized problems abstract in this study we present the mathematical analysis needed to explain the convergence of a progressive variational vademecum based on the proper generalized decomposition pgd the pgd is a novel technique that was developed recently for solving problems with high dimensions and it also provides new approaches for obtaining the solutions of elliptic and parabolic problems via the abstract separation of variables method this new scenario requires a mathematical framework in order to justify its application to the solution of numerical problems and the pgd can help in the change to this paradigm the main aim of this study is to provide a mathematical environment for defining the notion of progressive variational vademecum we prove the convergence of this iterative procedure and we also provide the first order optimality conditions in order to construct the numerical approximations of the parameterized solutions in particular we illustrate this methodology based on a robot path planning problem this is one of the common tasks when designing the trajectory or path of a mobile robot the construction of a progressive variational vademecum provides a novel methodology for computing all the possible paths from any start and goal positions derived from a harmonic potential field in a predefined map

approximate counting uniform generation and rapidly mixing markov chains the paper studies effective approximate solutions to combinatorial counting and uniform generation problems using a technique based on the simulation of ergodic markov chains it is shown that for self reducible structures almost uniform generation is possible in polynomial time provided only that randomised approximate counting to within some arbitrary polynomial factor is possible in polynomial time it follows that for self reducible structures polynomial time randomised algorithms for counting to within factors of the form 1 n are available either for all fl e r or for no fi e r a substantial part of the paper is devoted to investigating the rate of convergence of finite ergodic markov chains and a simple but powerful characterisation of rapid convergence for a broad class of chains based on a structural property of the underlying graph is established finally the general techniques of the paper are used to derive an almost uniform generation procedure for labelled graphs with a given degree sequence which is valid over a much wider range of degrees than previous methods this in turn leads to randomised approximate counting algorithms for these graphs with very good

a posteriori error estimation and adaptivity in stochastic galerkin fem for parametric elliptic pdes beyond the affine case abstract we consider a linear elliptic partial differential equation pde with a generic uniformly bounded parametric coefficient the solution to this pde problem is approximated in the framework of stochastic galerkin finite element methods we perform a posteriori error analysis of galerkin approximations and derive a reliable and efficient estimate for the energy error in these approximations practical versions of this error estimate are discussed and tested numerically for a model problem with non affine parametric representation of the coefficient furthermore we use the error reduction indicators derived from spatial and parametric error estimators to guide an adaptive solution algorithm for the given parametric pde problem the performance of the adaptive algorithm is tested numerically for model problems with two different non affine parametric representations of the coefficient

probabilistic collocation for period 1 limit cycle oscillations in this paper probabilistic collocation for limit cycle oscillations pclco is proposed probabilistic collocation pc is a non intrusive approach to compute the polynomial chaos description of uncertainty numerically polynomial chaos can require impractical high orders to approximate long term time integration problems due to the fast increase of required polynomial chaos order with time pclco is a pc formulation for modeling the long term stochastic behavior of dynamical systems exhibiting a periodic response i e a limit cycle oscillation lco in the pc method deterministic time series are computed at collocation points in probability space in pclco pc is applied to a time independent parametrization of the periodic response of the deterministic solves instead of to the time dependent functions themselves due to the time independent parametrization the accuracy of pclco is independent of time the approach is applied to period 1 oscillations with one main frequency subject to a random parameter numerical results are presented for the harmonic oscillator a two dof airfoil flutter model and the fluid structure interaction of an elastically mounted cylinder

on stability and monotonicity requirements of finite difference approximations of stochastic conservation laws with random viscosity on stability and monotonicity requirements of finite difference approximations of stochastic conservation laws with random viscosity

approximating probabilistic inference in bayesian belief networks is np hard abstract it is known that exact computation of conditional probabilities in belief networks is np hard many investigators in the ai community have tacitly assumed that algorithms for performing approximate inference with belief networks are of polynomial complexity indeed special cases of approximate inference can be performed in time polynomial in the input size however we have discovered that the general problem of approximating conditional probabilities with belief networks like exact inference resides in the np hard complexity class we develop a complexity analysis to elucidate the difficulty of approximate probabilistic inference more specifically we show that the existence of a polynomial time relative approximation algorithm for major classes of problem instances implies that np p we present our proof and explore the implications of the result

generalized spectral decomposition for stochastic nonlinear problems we present an extension of the generalized spectral decomposition method for the resolution of nonlinear stochastic problems the method consists in the construction of a reduced basis approximation of the galerkin solution and is independent of the stochastic discretization selected polynomial chaos stochastic multi element or multi wavelets two algorithms are proposed for the sequential construction of the successive generalized spectral modes they involve decoupled resolutions of a series of deterministic and low dimensional stochastic problems compared to the classical galerkin method the algorithms allow for significant computational savings and require minor adaptations of the deterministic codes the methodology is detailed and tested on two model problems the one dimensional steady viscous burgers equation and a two dimensional nonlinear diffusion problem these examples demonstrate the effectiveness of the proposed algorithms which exhibit convergence rates with the number of modes essentially dependent on the spectrum of the stochastic solution but independent of the dimension of the stochastic approximation space

the computational complexity of probabilistic inference using bayesian belief networks research note abstract bayesian belief networks provide a natural efficient method for representing probabilistic dependencies among a set of variables for these reasons numerous researchers are exploring the use of belief networks as a knowledge representation in artificial intelligence algorithms have been developed previously for efficient probabilistic inference using special classes of belief networks more general classes of belief networks however have eluded efforts to develop efficient inference algorithms we show that probabilistic inference using belief networks is np hard therefore it seems unlikely that an exact algorithm can be developed to perform probabilistic inference efficiently over all classes of belief networks this result suggests that research should be directed away from the search for a general efficient probabilistic inference algorithm and toward the design of efficient special case average case and approximation algorithms

an optimal approximation algorithm for bayesian inference abstract approximating the inference probability pr x x e e in any sense even for a single evidence node e is np hard this result holds for belief networks that are allowed to contain extreme conditional probabilities that is conditional probabilities arbitrarily close to 0 nevertheless all previous approximation algorithms have failed to approximate efficiently many inferences even for belief networks without extreme conditional probabilities we prove that we can approximate efficiently probabilistic inference in belief networks without extreme conditional probabilities we construct a randomized approximation algorithm the bounded variance algorithm that is a variant of the known likelihood weighting algorithm the bounded variance algorithm is the first algorithm with provably fast inference approximation on all belief networks without extreme conditional probabilities from the bounded variance algorithm we construct a deterministic approximation algorithm using current advances in the theory of pseudorandom generators in contrast to the exponential worst case behavior of all previous deterministic approximations the deterministic bounded variance algorithm approximates inference probabilities in worst case time that is subexponential 2 log n d for some integer d that is a linear function of the depth of the belief network

an indirect approach for automatic generation of quadrilateral meshes with arbitrary line constraints a new approach to the automatic generation of a quadrilateral mesh with arbitrary line constraints is proposed in this paper it is an indirect all quad mesh generation and presented in the following steps 1 discretizing the constrained lines within the domain 2 converting the above domain to a triangular mesh together with the line constraints 3 transforming the generated triangular mesh with line constraints to an all quad mesh through performing an advancing front algorithm from the line constraints which enables the construction of quadrilaterals layer by layer and roughly keeps the feature of the initial triangular mesh 4 optimizing the topology of the quadrilateral mesh to reduce the number of irregular nodes 5 smoothing the generated mesh toward high quality all quad mesh generation finally a few application examples are given to demonstrate the reliability and usefulness of the proposed algorithm copyright c 2011 john wiley sons ltd

low rank approximate inverse for preconditioning tensor structured linear systems in this paper we propose an algorithm for the construction of low rank approximations of the inverse of an operator given in low rank tensor format the construction relies on an updated greedy algorithm for the minimization of a suitable distance to the inverse operator it provides a sequence of approximations that are defined as the projections of the inverse operator in an increasing sequence of linear subspaces of operators these subspaces are obtained by the tensorization of bases of operators that are constructed from successive rank one corrections in order to handle high order tensors approximate projections are computed in low rank hierarchical tucker subsets of the successive subspaces of operators some desired properties such as symmetry or sparsity can be imposed on the approximate inverse operator during the correction step where an optimal rank one correction is searched as the tensor product of operators with the desired properties numerical examples illustrate the ability of this a

ingredients for a general purpose stochastic finite elements implementation abstract the stochastic finite element method has seen a number of incarnations over the past few years ranging from perturbation based methods and neumann expansion based methods to methods for evaluating bounds on the response variability and more recent functional expansions combined with error minimization techniques of these only the last class of methods provides an explicit expression to the solution of the problem as opposed to evaluating a global norm of this solution and can thus be viewed as a rational extension of concepts developed for the deterministic finite element method the main obstacle in developing a substantial user community of various stochastic finite element methods has been the lack of a general purpose formalism for addressing issues of general engineering interest this paper presents the main ingredients for developing a general purpose version of the spectral stochastic finite element method

modeling uncertainty in flow simulations via generalized polynomial chaos we present a new algorithm to model the input uncertainty and its propagation in incompressible flow simulations the stochastic input is represented spectrally by employing orthogonal polynomial functionals from the askey scheme as trial basis to represent the random space a standard galerkin projection is applied in the random dimension to obtain the equations in the weak form the resulting system of deterministic equations is then solved with standard methods to obtain the solution for each random mode this approach can be considered as a generalization of the original polynomial chaos expansion first introduced by wiener am j math 60 1938 897 the original method employs the hermite polynomials one of the 13 members of the askey scheme as the basis in random space the algorithm is applied to micro channel flows with random wall boundary conditions and to external flows with random freestream efficiency and convergence are studied by comparing with exact solutions as well as numerical solutions obtained by monte carlo simulations it is shown that the generalized polynomial chaos method promises a substantial speed up compared with the monte carlo method the utilization of different type orthogonal polynomials from the askey scheme also provides a more efficient way to represent general non gaussian processes compared with the original wiener hermite expansions

a robust bi orthogonal dynamically orthogonal method using the covariance pseudo inverse with application to stochastic flow problems abstract we develop a new robust methodology for the stochastic navier stokes equations based on the dynamically orthogonal do and bi orthogonal bo methods 1 2 3 both approaches are variants of a generalized karhunen loeve kl expansion in which both the stochastic coefficients and the spatial basis evolve according to system dynamics hence capturing the low dimensional structure of the solution the do and bo formulations are mathematically equivalent 3 but they exhibit computationally complimentary properties specifically the bo formulation may fail due to crossing of the eigenvalues of the covariance matrix while both bo and do become unstable when there is a high condition number of the covariance matrix or zero eigenvalues to this end we combine the two methods into a robust hybrid framework and in addition we employ a pseudo inverse technique to invert the covariance matrix the robustness of the proposed method stems from addressing the following issues in the do bo formulation i eigenvalue crossing we resolve the issue of eigenvalue crossing in the bo formulation by switching to the do near eigenvalue crossing using the equivalence theorem and switching back to bo when the distance between eigenvalues is larger than a threshold value ii ill conditioned covariance matrix we utilize a pseudo inverse strategy to invert the covariance matrix iii adaptivity we utilize an adaptive strategy to add remove modes to resolve the covariance matrix up to a threshold value in particular we introduce a soft threshold criterion to allow the system to adapt to the newly added removed mode and therefore avoid repetitive and unnecessary mode addition removal when the total variance approaches zero we show that the do bo formulation becomes equivalent to the evolution equation of the optimally time dependent modes 4 we demonstrate the capability of the proposed methodology with several numerical examples namely i stochastic burgers equation we analyze the performance of the method in the presence of eigenvalue crossing and zero eigenvalues ii stochastic kovasznay flow we examine the method in the presence of a singular covariance matrix and iii we examine the adaptivity of the method for an incompressible flow over a cylinder where for large stochastic forcing thirteen do bo modes are active

robust geometrically based automatic two dimensional mesh generation a technical description of the algorithms employed in the modified quadtree mesh generator is given although the basis of the mesh generator is the same as the original version developed by yerry and shephard 1 2 the actual algorithms on which it is built have been entirely changed for the purpose of ensuring the robustness of the technique as demonstrated in the paper the algorithmic changes made do ensure the robustness of the approach but introduce additional algorithmic difficulties the solutions of which are also presented in addition to examples showing the capability of the mesh generator the linear computational growth rate of the mesh generator is demonstrated

a least squares approximation of partial differential equations with high dimensional random inputs uncertainty quantification schemes based on stochastic galerkin projections with global or local basis functions and also stochastic collocation methods in their conventional form suffer from the so called curse of dimensionality the associated computational cost grows exponentially as a function of the number of random variables defining the underlying probability space of the problem in this paper to overcome the curse of dimensionality a low rank separated approximation of the solution of a stochastic partial differential spde with high dimensional random input data is obtained using an alternating least squares als scheme it will be shown that in theory the computational cost of the proposed algorithm grows linearly with respect to the dimension of the underlying probability space of the system for the case of an elliptic spde an a priori error analysis of the algorithm is derived finally different aspects of the proposed methodology are explored through its application to some numerical experiments

galerkin reduced order modeling scheme for time dependent randomly parametrized linear partial differential equations summary in this paper we consider the problem of constructing reduced order models of a class of time dependent randomly parametrized linear partial differential equations our objective is to efficiently construct a reduced basis approximation of the solution as a function of the spatial coordinates parameter space and time the proposed approach involves decomposing the solution in terms of undetermined spatial and parametrized temporal basis functions the unknown basis functions in the decomposition are estimated using an alternating iterative galerkin projection scheme numerical studies on the time dependent randomly parametrized diffusion equation are presented to demonstrate that the proposed approach provides good accuracy at significantly lower computational cost compared with polynomial chaos based galerkin projection schemes comparison studies are also made against nouy s generalized spectral decomposition scheme to demonstrate that the proposed approach provides a number of computational advantages copyright 2012 john wiley sons ltd

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

generation of finite element mesh with variable size over an unbounded 2d domain with the advance of the finite element general fluid dynamic and traffic flow problems with arbitrary boundary definition over an unbounded domain are tackled this paper describes an algorithm for the generation of finite element mesh of variable element size over an unbounded 2d domain by using the advancing front circle packing technique unlike the conventional frontal method the procedure does not start from the object boundary but starts from a convenient point within an open domain the sequence of construction of the packing circles is determined by the shortest distance from the fictitious centre in such a way that the generation front is more or less a circular loop with occasional minor concave parts due to element size variation as soon as a circle is added to the generation front finite elements are directly generated by properly connecting frontal segments with the centre of the new circle in contrast to other mesh generation schemes the domain boundary is not considered in the process of circle packing this reduces a lot of geometrical checks for intersection with frontal segments and a linear time complexity for mesh generation can be achieved in case the boundary of the domain is needed simply generate an unbounded mesh to cover the entire object as the element adjacency relationship of the mesh has already been established in the circle packing process insertion of boundary segments by neighbour tracing is fast and robust details of such a boundary recovery procedure are described and practical meshing problems are given to demonstrate how physical objects are meshed by the unbounded meshing scheme followed by the insertion of domain boundaries

proper generalized decomposition for nonlinear convex problems in tensor banach spaces tensor based methods are receiving a growing interest in scientific computing for the numerical solution of problems defined in high dimensional tensor product spaces a family of methods called proper generalized decompositions pgd methods have been recently introduced for the a priori construction of tensor approximations of the solution of such problems in this paper we give a mathematical analysis of a family of progressive and updated pgds for a particular class of problems associated with the minimization of a convex functional over a reflexive tensor banach space

approximating the permanent a randomised approximation scheme for the permanent of a 0 1s presented the task of estimating a permanent is reduced to that of almost uniformly generating perfect matchings in a graph the latter is accomplished by simulating a markov chain whose states are the matchings in the graph for a wide class of 0 1 matrices the approximation scheme is fully polynomial i e runs in time polynomial in the size of the matrix and a parameter that controls the accuracy of the output this class includes all dense matrices those that contain sufficiently many 1 s and almost all sparse matrices in some reasonable probabilistic model for 0 1 matrices of given density for the approach sketched above to be computationally efficient the markov chain must be rapidly mixing informally it must converge in a short time to its stationary distribution a major portion of the paper is devoted to demonstrating that the matchings chain is rapidly mixing apparently the first such result for a markov chain with genuinely c

a generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations we propose a new robust technique for solving stochastic partial differential equations the solution is approximated by a series of terms each of which being the product of a scalar stochastic function by a deterministic function none of these functions are fixed a priori but determined by solving a problem which can be interpreted as an extended eigenvalue problem this technique generalizes the classical spectral decomposition namely the karhunen loeve expansion ad hoc iterative techniques to build the approximation inspired by the power method for classical eigenproblems then transform the problem into the resolution of a few uncoupled deterministic problems and stochastic equations this method drastically reduces the calculation costs and memory requirements of classical resolution techniques used in the context of galerkin stochastic finite element methods finally this technique is particularly suitable to non linear and evolution problems since it enables the construction of a relevant reduced basis of deterministic functions which can be efficiently reused for subsequent resolutions

fictitious domain method and separated representations for the solution of boundary value problems on uncertain parameterized domains abstract a tensor based method is proposed for the solution of partial differential equations defined on uncertain parameterized domains it provides an accurate solution which is explicit with respect to parameters defining the shape of the domain thus allowing efficient a posteriori probabilistic or parametric analyses in the proposed method a fictitious domain approach is first adopted for the reformulation of the parametric problem on a fixed domain yielding a weak formulation in a tensor product space product of space functions and parametric functions the paper is limited to the case of neumann conditions on uncertain parts of the boundary the proper generalized decomposition method is then introduced for the construction of a tensor product approximation separated representation of the solution it can be seen as an a priori model reduction technique which automatically captures reduced bases of space functions and parametric functions which are optimal for the representation of the solution this tensor based method is made computationally tractable by introducing separated representations of variational forms resulting from separated representations of the parameterized indicator function of the uncertain domain for this purpose a method is proposed for the construction of a constrained tensor product approximation which preserves positivity and therefore ensures well posedness of problems associated with approximate indicator functions moreover a regularization of the geometry is introduced to speed up the convergence of these tensor product approximations

model order reduction based on proper generalized decomposition for the propagation of uncertainties in structural dynamics summary a priori model reduction methods based on separated representations are introduced for the prediction of the low frequency response of uncertain structures within a parametric stochastic framework the proper generalized decomposition method is used to construct a quasi optimal separated representation of the random solution at some frequency samples at each frequency an accurate representation of the solution is obtained on reduced bases of spatial functions and stochastic functions an extraction of the deterministic bases allows for the generation of a global reduced basis yielding a reduced order model of the uncertain structure which appears to be accurate on the whole frequency band under study and for all values of input random parameters this strategy can be seen as an alternative to traditional constructions of reduced order models in structural dynamics in the presence of parametric uncertainties this reduced order model can then be used for further analyses such as the computation of the response at unresolved frequencies or the computation of more accurate stochastic approximations at some frequencies of interest because the dynamic response is highly nonlinear with respect to the input random parameters a second level of separation of variables is introduced for the representation of functions of multiple random parameters thus allowing the introduction of very fine approximations in each parametric dimension even when dealing with high parametric dimension copyright 2011 john wiley sons ltd

ideal minimal residual based proper generalized decomposition for non symmetric multi field models application to transient elastodynamics in space time domain abstract it is now well established that separated representations built with the help of proper generalized decomposition pgd can drastically reduce computational costs associated with solution of a wide variety of problems however it is still an open question to know if separated representations can be efficiently used to approximate solutions of hyperbolic evolution problems in space time domain in this paper we numerically address this issue and concentrate on transient elastodynamic models for such models the operator associated with the space time problem is non symmetric and low rank approximations are classically computed by minimizing the space time residual in a natural l 2 sense yet leading to non optimal approximations in usual solution norms therefore a new algorithm has been recently introduced by one of the authors and allows to find a quasi optimal low rank approximation a priori with respect to a target norm we presently extend this new algorithm to multi field models the proposed algorithm is applied to elastodynamics formulated over space time domain with the time discontinuous galerkin method in displacement and velocity numerical examples demonstrate convergence of the proposed algorithm and comparisons are made with classical a posteriori and a priori approaches

towards essential improvement for the parareal tr and parareal gauss4 algorithms parareal is an iterative algorithm and is characterized by two propagators g and f which are respectively associated with large step size t and small step size t where t j t and j 2 is an integer for symmetric positive definite spd system u t a u t g t arising from semi discretizing time dependent pdes if we fix the g propagator to the backward euler method and choose for f some l stable time integrator it can be proven that the convergence factors of the corresponding parareal algorithms satisfy 1 3 j 2 and a 0 where a is the spectrum of the matrix a however this result does not hold when time integrators that lack l stability such as the trapezoidal rule and the 4th order gauss rk method are chosen as the f propagator the parareal algorithms using these two methods for the f propagator are denoted by parareal tr and parareal gauss4 in this paper we propose a strategy to let these two parareal algorithms possess such a uniform convergence property the idea is to choose an l stable propagator f and on each coarse time interval t n t n 1 we perform first two steps of f then followed by j 2 steps of f precisely for the trapezoidal rule we select the 2nd order sdirk method as the f propagator and for the 4th order gauss rk method we select the 4th order lobatto iii c method as the f propagator numerical results are given to support our theoretical conclusions

algebraic grid generation on trimmed parametric surface using non self overlapping planar coons patch using a coons patch mapping to generate a structured grid in the parametric region of a trimmed surface can avoid the singularity of elliptic pde methods when only c1 continuous boundary is given the error of converting generic parametric c1 boundary curves into a specified representation form is also avoided however overlap may happen on some portions of the algebraically generated grid when a linear or naive cubic blending function is used in the mapping this severely limits its usage in most of engineering and scientific applications where a grid system of non self overlapping is strictly required to solve the problem non trivial blending functions in a coons patch mapping should be determined adaptively by the given boundary so that self overlapping can be averted we address the adaptive determination problem by a functional optimization method the governing equation of the optimization is derived by adding a virtual dimension in the parametric space of the given trimmed surface both one and two parameter blending functions are studied to resolve the difficulty of guessing good initial blending functions for the conjugate gradient method used a progressive optimization algorithm is then proposed which has been shown to be very effective in a variety of practical examples also an extension is added to the objective function to control the element shape finally experiment results are shown to illustrate the usefulness and effectiveness of the presented method copyright 2004 john wiley sons ltd

random generation of combinatorial structures from a uniform distribution abstract the class of problems involving the random generation of combinatorial structures from a uniform distribution is considered uniform generation problems are in computational difficulty intermediate between classical existence and counting problems it is shown that exactly uniform generation of efficiently verifiable combinatorial structures is reducible to approximate counting and hence is within the third level of the polynomial hierarchy natural combinatorial problems are presented which exhibit complexity gaps between their existence and generation and between their generation and counting versions it is further shown that for self reducible problems almost uniform generation and randomized approximate counting are inter reducible and hence of similar complexity

tensor based methods for numerical homogenization from high resolution images we present a complete numerical strategy based on tensor approximation techniques for the solution of numerical homogenization problems with geometrical data coming from high resolution images we first introduce specific numerical treatments for the translation of image based homogenization problems into a tensor framework it includes the tensor approximations in suitable tensor formats of fields of material properties or indicator functions of multiple material phases recovered from segmented images we then introduce some variants of proper generalized decomposition pgd methods for the construction of tensor decompositions in different tensor formats of the solution of boundary value problems a new definition of pgd is introduced which allows the progressive construction of a tucker decomposition of the solution this tensor format is well adapted to the present application and improves convergence properties of tensor decompositions finally we use a dual based error estimator on quantities of interest which was recently introduced in the context of pgd we exhibit its specificities when it is used for assessing the error on the homogenized properties of the heterogeneous material we also provide a complete goal oriented adaptive strategy for the progressive construction of tensor decompositions of primal and dual solutions yielding to predictions of homogenized quantities with a prescribed accuracy

development of an automatic two dimensional finite element mesh generator using quadrilateral elements and bezier curve boundary definition this paper describes the development of an automatic two dimensional quadrilateral element mesh generator the methodology is based on the looping algorithm developed in part by dr m l c sluiter bezier curves are used to define the boundaries of the parts to be meshed the required interaction by the user is reduced to specifying the boundary geometry the preferred element size and the refinement coefficient transition elements are automatically generated between regions of varying element sizes the ability to offset specified boundary curves insures well conditioned elements along the boundary meshes for several sample geometries are presented to illustrate the versatility of the mesh generator

efficient posterior exploration of a high dimensional groundwater model from two stage markov chain monte carlo simulation and polynomial chaos expansion 1 this study reports on two strategies for accelerating posterior inference of a highly parameterized and cpu demanding groundwater flow model our method builds on previous stochastic collocation approaches e g marzouk and xiu 2009 and marzouk and najm 2009 and uses generalized polynomial chaos gpc theory and dimensionality reduction to emulate the output of a large scale groundwater flow model the resulting surrogate model is cpu efficient and serves to explore the posterior distribution at a much lower computational cost using two stage mcmc simulation the case study reported in this paper demonstrates a two to five times speed up in sampling efficiency

a multiscale method with patch for the solution of stochastic partial differential equations with localized uncertainties abstract we here propose a multiscale numerical method for the solution of stochastic parametric partial differential equations with localized uncertainties described with a finite number of random variables it is based on a multiscale domain decomposition method that exploits the localized side of uncertainties and incidentally improves the conditioning of the problem by operating a separation of scales an efficient iterative algorithm is proposed that requires the solution of a sequence of simple global problems at a macro scale involving a deterministic operator and local problems at a micro scale for which we have the possibility to use fine approximation spaces global and local problems are solved using tensor approximation methods allowing the representation of high dimensional stochastic parametric solutions convergence properties of these tensor based methods which are closely related to spectral decompositions benefit from the separation of scales different types of uncertainties are considered at the micro level they may be associated with some variability in the operator or source terms or even with some geometrical variability in the latter case specific reformulations of local problems using fictitious domain methods are introduced

efficient three dimensional delaunay triangulation with automatic point creation and imposed boundary constraints a method is described which constructs three dimensional unstructured tetrahedral meshes using the delaunay triangulation criterion several automatic point creation techniques will be highlighted and an algorithm will be presented which can ensure that given an initial surface triangulation which bounds a domain a valid boundary conforming assembly of tetrahedra will be produced statistics of measures of grid quality are presented for several grids the efficiency of the proposed procedure reduces the computer time for the generation of realistic unstructured tetrahedral grids to the order of minutes on workstations of modest computational capabilities

approximation of constrained problems using the pgd method with application to pure neumann problems in this paper we introduce analyze and compare several approaches designed to incorporate a linear or affine constraint within the proper generalized decomposition framework we apply the considered methods and numerical strategies to two classes of problems the pure neumann case where the role of the constraint is to recover unicity of the solution and the robin case where the constraint forces the solution to move away from the already existing unique global minimizer of the energy functional

a new approach to the development of automatic quadrilateral mesh generation a methodology for the automatic mesh generation of quadrilateral elements is proposed the methodology is based on the facts that i a region can always be subdivided entirely into quadrilaterals if the polygon which forms the boundary of the region has an even number of sides ii a quadrilateral can be formed by two triangles which share a common side by taking advantage of the techniques developed in an existing automatic triangular mesh generator quadrilateral meshes can be generated according to the distribution of user specified element size over the domain of interest in particular an automatic quadrilateral mesh generator based on the advancing front technique for triangular mesh generation1 is described techniques to improve the quality of the generated quadrilateral meshes are introduced the robustness and versatility of the mesh generator are demonstrated by a variety of examples

finite element euler computations in three dimensions an explicit finite element solution procedure for the three dimensional euler equations is presented the solution domain is automatically meshed using a tetrahedral mesh generator which is an extension of our previous two dimensional work several examples are included to illustrate the performance of the generator and solver an adaptive mesh regeneration procedure is used for the first time in three dimensions

approximating probabilistic inference in bayesian belief networks a belief network comprises a graphical representation of dependencies between variables of a domain and a set of conditional probabilities associated with each dependency unless rho np an efficient exact algorithm does not exist to compute probabilistic inference in belief networks stochastic simulation methods which often improve run times provide an alternative to exact inference algorithms such a stochastic simulation algorithm d bnras which is a randomized approximation scheme is presented to analyze the run time belief networks are parameterized by the dependence value d sub xi which is a measure of the cumulative strengths of the belief network dependencies given background evidence xi this parameterization defines the class of f dependence networks the run time of d bnras is polynomial when f is a polynomial function thus the results prove the existence of a class of belief networks for which inference approximation is polynomial and hence provably faster than any exact algorithm

comparative study of projection schemes for stochastic finite element analysis we present a comparison of subspace projection schemes for stochastic finite element analysis in terms of accuracy and computational efficiency more specifically we compare the polynomial chaos projection scheme with reduced basis projection schemes based on the preconditioned stochastic krylov subspace numerical studies are presented for two problems 1 static analysis of a plate with random young s modulus and 2 settlement of a foundation supported on a randomly heterogeneous soil monte carlo simulation results based on exact structural analysis are used to generate benchmark results against which the projection schemes are compared we show that stochastic reduced basis methods require significantly less computer memory and execution time compared to the polynomial chaos approach particularly for large scale problems with many random variables for the class of problems considered we find that stochastic reduced basis methods can be up to orders of magnitude faster while providing results of comparable or better accuracy

effect of randomness on multi frequency aeroelastic responses resolved by unsteady adaptive stochastic finite elements the unsteady adaptive stochastic finite elements uasfe method resolves the effect of randomness in numerical simulations of single mode aeroelastic responses with a constant accuracy in time for a constant number of samples in this paper the uasfe framework is extended to multi frequency responses and continuous structures by employing a wavelet decomposition pre processing step to decompose the sampled multi frequency signals into single frequency components the effect of the randomness on the multi frequency response is then obtained by summing the results of the uasfe interpolation at constant phase for the different frequency components results for multi frequency responses and continuous structures show a three orders of magnitude reduction of computational costs compared to crude monte carlo simulations in a harmonically forced oscillator a flutter panel problem and the three dimensional transonic agard 445 6 wing aeroelastic benchmark subject to random fields and random parameters with various probability distributions

emulation of numerical models with over specified basis functions abstractmathematical models are frequently used to explore physical systems but can be computationally expensive to evaluate in such settings an emulator is used as a surrogate in this work we propose a basis function approach for computer model emulation to combine field observations with a collection of runs from the numerical model we use the proposed emulator within the kennedy o hagan framework of model calibration a novel feature of the approach is the use of an over specified set of basis functions where number of bases used and their inclusion probabilities are treated as unknown quantities the new approach is found to have smaller predictive uncertainty and computational efficiency than the standard gaussian process approach to emulation and calibration along with several simulation examples focusing on different model characteristics we also use the method to analyze a dataset on laboratory experiments related to astrophysics

automatic metric 3d surface mesh generation using subdivision surface geometrical model part 2 mesh generation algorithm and examples in this paper a new metric advancing front surface mesh generation scheme is suggested this new surface mesh generator is based on a new geometrical model employing the interpolating subdivision surface concept the target surfaces to be meshed are represented implicitly by interpolating subdivision surfaces which allow the presence of various sharp and discontinuous features in the underlying geometrical model while the main generation steps of the new generator are based on a robust metric surface triangulation kernel developed previously a number of specially designed algorithms are developed in order to combine the existing metric advancing front algorithm with the new geometrical model as a result the application areas of the new mesh generator are largely extended and can be used to handle problems involving extensive changes in domain geometry numerical experience indicates that by using the proposed mesh generation scheme high quality surface meshes with rapid varying element size and anisotropic characteristics can be generated in a short time by using a low end pc finally by using the pseudo curvature element size controlling metric to impose the curvature element size requirement in an implicit manner the new mesh generation procedure can also generate finite element meshes with high fidelity to approximate the target surfaces accurately copyright 2003 john wiley sons ltd

unconstrained plastering hexahedral mesh generation via advancing front geometry decomposition the generation of all hexahedral finite element meshes has been an area of ongoing research for the past two decades and remains an open problem unconstrained plastering is a new method for generating all hexahedral finite element meshes on arbitrary volumetric geometries starting from an unmeshed volume boundary unconstrained plastering generates the interior mesh topology without the constraints of a pre defined boundary mesh using advancing fronts unconstrained plastering forms partially defined hexahedral dual sheets by decomposing the geometry into simple shapes each of which can be meshed with simple meshing primitives by breaking from the tradition of previous advancing front algorithms which start from pre meshed boundary surfaces unconstrained plastering demonstrates that for the tested geometries high quality boundary aligned orientation insensitive all hexahedral meshes can be generated automatically without pre meshing the boundary examples are given for meshes from both solid mechanics and geotechnical applications copyright 2009 john wiley sons ltd

numerical studies of the stochastic korteweg de vries equation we present numerical solutions of the stochastic korteweg de vries equation for three cases corresponding to additive time dependent noise multiplicative space dependent noise and a combination of the two we employ polynomial chaos for discretization in random space and discontinuous galerkin and finite difference for discretization in physical space the accuracy of the stochastic solutions is investigated by comparing the first two moments against analytical and monte carlo simulation results of particular interest is the interplay of spatial discretization error with the stochastic approximation error which is examined for different orders of spatial and stochastic approximation

a frontal delaunay quad mesh generator using the l norm a new indirect way of producing all quad meshes is presented the method takes advantage of a well known algorithm of the graph theory namely the blossom algorithm that computes the minimum cost perfect matching in a graph in polynomial time then the triangulation itself is taylored with the aim of producing right triangles in the domain this is done using the infinity norm to compute distances in the meshing process the alignement of the triangles is controlled by a cross field that is defined on the domain meshes constructed this way have their points aligned with the cross field direction and their triangles are almost right everywhere then recombination with our blossom based approach yields quadrilateral meshes of excellent quality

a novel energy based approach for merging finite elements a novel approach for merging two intersecting finite elements is presented and demonstrated the solution mimics concepts from biology and uses principles rooted in continuum mechanics the problem of attaching or merging two coincident finite elements is common when using the plastering technique as part of the advancing front method this problem is particularly challenging for 3 d meshes of non convex shapes some automatic meshing methods require portions of the partially formed mesh to coincide and merge this problem is generally solved with heuristic rules which lack generality and may have difficulties with unforeseen situations the problem of merging two overlapping polyhedra may also appear in other applications such as computer graphics and cad software a new approach to address the problem of merging is presented here this solution does not utilize heuristic rules but rather uses an approach based on minimization of strain energy a fully automatic merging routine has been created that can address in an optimum way any situation of two nearby or overlapping elements that are to be merged this approach with minor adjustments is suitable for most types of 3 d elements

generalized spectral decomposition method for solving stochastic finite element equations invariant subspace problem and dedicated algorithms stochastic galerkin methods have become a significant tool for the resolution of stochastic partial differential equations spde however they suffer from prohibitive computational times and memory requirements when dealing with large scale applications and high stochastic dimensionality some alternative techniques based on the construction of suitable reduced deterministic or stochastic bases have been proposed in order to reduce these computational costs recently a new approach based on the concept of generalized spectral decomposition gsd has been introduced for the definition and the automatic construction of reduced bases in this paper the concept of gsd initially introduced for a class of linear elliptic spde is extended to a wider class of stochastic problems the proposed definition of the gsd leads to the resolution of an invariant subspace problem which is interpreted as an eigen like problem this interpretation allows the construction of efficient numerical algorithms for building optimal reduced bases which are associated with dominant generalized eigenspaces the proposed algorithms by separating the resolution of reduced stochastic and deterministic problems lead to drastic computational savings their efficiency is illustrated on several examples where they are compared to classical resolution techniques

formation of pyramid elements for hexahedra to tetrahedra transitions new algorithms are proposed for the modification of a mixed hexahedra tetrahedra element mesh to maintain compatibility by the insertion of pyramid elements several methods for generation of the pyramids are presented involving local tetrahedral transformations and or node insertion near the hex tet interface local smoothing and topological operations improve the quality of the transition region results show superior performance of the resulting elements in a commercial finite element code over non conforming interface conditions