quantile based optimization of noisy computer experiments with tunable precision this article addresses the issue of kriging based optimization of stochastic simulators many of these simulators depend on factors that tune the level of precision of the response the gain in accuracy being at a price of computational time the contribution of this work is two fold first we propose a quantile based criterion for the sequential design of experiments in the fashion of the classical expected improvement criterion which allows an elegant treatment of heterogeneous response precisions second we present a procedure for the allocation of the computational time given to each measurement allowing a better distribution of the computational effort and increased efficiency finally the optimization method is applied to an original application in nuclear criticality safety this article has supplementary material available online the proposed criterion is available in the r package diceoptim

coupling an eulerian fluid calculation to a lagrangian solid calculation with the ghost fluid method we propose a numerical method for modeling multimaterial flows where the domain is decomposed into separate eulerian and lagrangian subdomains that is the equations are written in eulerian form in one subdomain and in lagrangian form in the other subdomain this is of interest for example when considering the effect of underwater explosions on the hull of a ship or the impact of a low speed projectile on a soft explosive target on the one hand high speed fluid flows are traditionally modeled by applying shock capturing schemes to the compressible euler equations to avoid problems associated with tangling of a lagrangian mesh on the other hand solid dynamics calculations are traditionally carried out using lagrangian numerical methods to avoid problems associated with numerical smearing in eulerian calculations we use the ghost fluid method to create accurate discretizations across the eulerian lagrangian interface the numerical method is presented in both one and two spatial dimensions three dimensional extensions to the interface coupling method are straightforward

optimal orlicz domains in sobolev embeddings into marcinkiewicz spaces abstract in this paper we present a method for determining whether there exists a largest orlicz space l a satisfying the sobolev embedding w m l a y where y stands for an arbitrary so called marcinkiewicz endpoint space the tool developed in this work enables us to investigate the optimality of orlicz domain spaces in sobolev embeddings and also in sobolev trace embeddings on domains in r n with various regularity

contact with friction in multi material arbitrary lagrangian eulerian formulations using x fem a contact method with friction for the multi dimensional lagrangian step in multi material arbitrary lagrangian eulerian ale formulations is presented in our previous research the extended finite element method x fem was used to create independent fields i e velocity strain rate force mass etc for each material in the problem to model contact without friction the research presented here includes the extension to friction and improvements to the accuracy and robustness of our previous study the accelerations of the multi material nodes are obtained by coupling the material force and mass fields as a function of the prescribed contact similarly the velocities of the multi material nodes are recalculated using the conservation of momentum when the prescribed contact requires it the coupling procedures impose the same nodal velocity on the coupled materials in the direction normal to their interface during the time step update as a result the overlap of materials is prevented and unwanted separation does not occur three different types of contacts are treated perfectly bonded frictionless slip and slip with friction example impact problems are solved and the numerical solutions are presented copyright 2008 john wiley sons ltd

finite elements for materials with strain gradient effects a finite element implementation is reported of the fleck hutchinson phenomenological strain gradient theory this theory fits within the toupin mindlin framework and deals with first order strain gradients and the associated work conjugate higher order stresses in conventional displacement based approaches the interpolation of displacement requires c1 continuity in order to ensure convergence of the finite element procedure for higher order theories mixed type finite elements are developed herein for the fleck hutchinson theory these elements use standard c0 continuous shape functions and can achieve the same convergence as c1 elements these c0 elements use displacements and displacement gradients as nodal degrees of freedom kinematic constraints between displacement gradients are enforced via the lagrange multiplier method the elements developed all pass a patch test the resulting finite element scheme is used to solve some representative linear elastic boundary value problems and the comparative accuracy of various types of element is evaluated copyright 1999 john wiley sons ltd

a particle level set based sharp interface cartesian grid method for impact penetration and void collapse an eulerian sharp interface cartesian grid method is developed for the numerical simulation of the response of materials to impact shocks and detonations the mass momentum and energy equations are solved along with evolution equations for deviatoric stresses and equivalent plastic strain these equations are cast in eulerian conservation law form the mie gruneisen equation of state is used to obtain pressure and the material is modeled as a johnson cook solid the eno scheme is employed to capture shocks in combination with a hybrid particle level set technique to evolve sharp immersed boundaries the numerical technique is able to handle collisions between multiple materials and can accurately compute the dynamics of the immersed boundaries results of calculations for axisymmetric taylor bar impact and penetration of a tungsten rod into steel plate show good agreement with moving finite element solutions and experimental results qualitative agreement with theory is shown for the void collapse phenomenon in an impacted material containing a spherical void

multifidelity cokriging for high dimensional output functions with application to hypersonic airloads computation an accurate surrogate model is capable of reproducing high fidelity results at a fraction of the computational cost cokriging is an efficient method for constructing surrogate models when two leve

an extended finite element formulation for contact in multi material arbitrary lagrangian eulerian calculations a contact method for the multi dimensional lagrangian step in multi material arbitrary lagrangian eulerian ale formulations is presented the extended finite element method x fem is used to create independent fields i e velocity strain rate force mass etc for each material in the problem the accelerations of the multi material nodes are obtained by coupling the material force and mass fields as a function of the prescribed contact similarly the velocities of the multi material nodes are recalculated using the conservation of momentum when the prescribed contact requires it the coupling procedures impose the same nodal velocity on the coupled materials in the direction normal to their interface during the time step update as a result the overlap of materials is prevented and unwanted separation does not occur three different contacts are treated perfectly bonded frictionless slip and slip with friction example impact problems are solved and the numerical solutions are presented

contact in a multi material eulerian finite element formulation in a multi material eulerian formulation for problems in solid mechanics the default interaction between the surfaces of adjacent materials is perfect bonding in contrast for the lagrangian formulation the surfaces don t interact and may pass through each other freely unless a contact algorithm is introduced to enforce the contact inequality constraints sophisticated contact algorithms have been developed for lagrangian calculations and are routinely used in large deformation applications such as automobile crashworthiness in comparison the eulerian finite element method lags far behind the lagrangian finite element method in modeling contact a natural application for the eulerian formulation is machining which includes large deformations material failure and contact of a tool with a workpiece making its simulation a challenging problem the simulation of the isolated chips involving material failure due to shear bands is used as a test problem to investigate the strength and weakness of the different approaches to implementing contact in multi material eulerian calculations

a meshfree contact detection algorithm a new contact detection algorithm is proposed which is based on the moment matrix of the meshfree interpolation this new contact detection algorithm has several advantages over conventional contact detection algorithms they are ease of implementation accuracy in detection it is scalable in parallel computations and applicable to a wide variety of problems because the moment matrix is computed in constructing meshfree approximation the algorithm is very suitable for meshfree computations though it can be used in finite element computations as well moreover such contact detection algorithm can be conveniently implemented to detect self contact both algorithmic and mathematical aspects of the contact algorithm are discussed in the paper to validate the proposed contact algorithm the taylor bar problem is computed using both meshfree method and finite element method fem to simulate a viscoplastic projectile impacting with a rigid target at high speed numerical results obtained with the meshfree contact algorithm show that this new contact algorithm can accurately predict the contact as well as separation of projectile and target in addition the numerical results obtained show distinct and detailed shear band formations occurring in the impact event

treatment of contact separation in eulerian high speed multimaterial dynamic simulations summary a frictionless contact separation treatment in a sharp interface eulerian framework is presented to handle the general situation of high speed impact and separation of materials the algorithm has been developed for an established eulerian based cartesian grid multimaterial flow code in which the interfaces are tracked in a sharp manner using a standard narrow band level set approach boundary conditions have been applied using a modified ghost fluid method for elasto plastic materials the sharp interface treatment maintains the distinct interacting interfaces without smearing the contact zone while also removing the difficulties associated with lagrangian moving mesh entities in contact separation situations the algorithm has been tested and verified against experimental and numerical results for three different problems in the high strain rate regime which involve contact separation and sliding of materials copyright 2014 john wiley sons ltd

fractional maximal operator in orlicz spaces abstract this work studies the fractional maximal operator acting between orlicz spaces the necessary and sufficient conditions for the existence of optimal target and domain spaces are established and the explicit formulas for corresponding optimal young functions are presented also we characterize whether the operator is bounded between two given orlicz spaces

a mixture theory for contact in multi material eulerian formulations abstract multi material eulerian methods are attractive for solving a broad range of highly nonlinear problems in solid mechanics because they permit arbitrarily large deformations and automatically allow new free surfaces to evolve many of these problems involve the contact and separation of surfaces there is an extensive literature for including contact inequality constraints in lagrangian finite element methods but little consideration has been given to contact in eulerian methods the contact forces between adjacent bodies are governed by the mixture theory in the eulerian formulation making the mixture theory the natural starting point in the treatment of the contact inequalities a mixture theory that equilibrates the traction across the material interface and limits the shear stress according to a coulomb friction law is developed

evolution implementation and application of level set and fast marching methods for advancing fronts abstract a variety of numerical techniques are available for tracking moving interfaces in this review we concentrate on techniques that result from the link between the partial differential equations that describe moving interfaces and numerical schemes designed for approximating the solutions to hyperbolic conservation laws this link gives rise to computational techniques for tracking moving interfaces in two and three space dimensions under complex speed laws we discuss the evolution of these techniques the fundamental numerical approximations involved implementation details and applications in particular we review some work on three aspects of materials sciences semiconductor process simulations seismic processing and optimal structural topology design

design and analysis of noisy computer experiments of functions calculated by long running computer codes the literature in this area commonly assumes that the objective function is a smooth deterministic function of the inputs yet it is well known that many computer simulations especiallythoseofcomputational fluidandstructuraldynamicscodes oftendisplaywhatonemightcall numerical noise rather than lying on a smooth curve results appear to contain a random scatter about a smooth trend this paper extends previous optimization methods based on the interpolating method ofkriging to the case of such noisy computer experiments firstly we review how the kriging interpolation can be modified to filter out numerical noise we then show how to adjust the estimate of the error in a kriging prediction so that previous approaches to optimization such as the method of maximizing the expected improvement continue to work effectively we introduce the problems associated with noise and demonstrate our approach using computational fluid dynamics based problems

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

optimal sobolev imbeddings involving rearrangement invariant quasinorms let m and n be positive integers with n 2 and 1 m n 1 we study rearrangement invariant quasinorms r and d on functions f 0 1 view the mathml source such that to each bounded domain in view the mathml sourcen with lebesgue measure there corresponds c c 0 for which one has the sobolev imbedding inequality r u t c d mu t u cm0 involving the nonincreasing rearrangements of u and a certain mth order gradient of u when m 1 we deal in fact with a closely related imbedding inequality of talenti in which d need not be rearrangement invariant r u t c d d dt x view the mathml sourcen u x u t u x dx u c10 in both cases we are especially interested in when the quasinorms are optimal in the sense that r cannot be replaced by an essentially larger quasinorm and d cannot be replaced by an essentially smaller one our results yield best possible refinements of such limiting sobolev inequalities as those of trudinger strichartz hansson brezis and wainger

quasilinear elliptic equations on noncompact riemannian manifolds abstract the existence of solutions to a class of quasilinear elliptic problems on noncompact riemannian manifolds with finite volume is investigated boundary value problems with homogeneous neumann conditions in possibly irregular euclidean domains are included as a special instance a nontrivial solution is shown to exist under an unconventional growth condition on the right hand side which depends on the geometry of the underlying manifold the identification of the critical growth is a crucial step in our analysis and entails the use of the isocapacitary function of the manifold a condition involving its isoperimetric function is also provided

evaluation of multifidelity surrogate modeling techniques to construct closure laws for drag in shock particle interactions abstract meta or surrogate models constructed from meso scale simulations can be used in place of empirical correlations to close macro scale equations in shocked particulate flows surrogate models for drag are constructed as functions of shock mach number ma particle volume fraction reynolds number re etc the computational cost of the high fidelity meso scale simulations is a challenge in construction of surrogates in such hierarchical multi scale frameworks here multifidelity surrogate modeling techniques are evaluated as inexpensive alternatives to high fidelity surrogate models for obtaining closure laws for drag in shock particle interactions preliminary surrogates for drag as a function of ma and are constructed from ensembles of low fidelity coarse grid mesoscale computations the low fidelity surrogates are subsequently corrected using only a few n h f high fidelity computations to obtain multifidelity surrogate models the paper evaluates three different methods for correcting an initial low fidelity surrogate space mapping sm radial basis functions rbf and modified bayesian kriging mbkg of these methods mbkg is found to provide the best multi fidelity surrogate model simultaneously minimizing the computational cost and error in the constructed surrogate

using radial basis functions to approximate a function and its error bounds a novel network called the validity index network vi net is presented the vi net derived from radial basis function networks fits functions and calculates confidence intervals for its predictions indicating local regions of poor fit and extrapolation

optimal sobolev trace embeddings optimal target spaces are exhibited in arbitrary order sobolev type embeddings for traces of n dimensional functions on lower dimensional subspaces sobolev spaces built upon any rearrangement invariant norm are allowed a key step in our approach consists in showing that any trace embedding can be reduced to a one dimensional inequality for a hardy type operator depending only on n and on the dimension of the relevant subspace this can be regarded as an analogue for trace embeddings of a well known symmetrization principle for first order sobolev embeddings for compactly supported functions the stability of the optimal target space under iterations of sobolev trace embeddings is also established and is part of the proof of our reduction principle as a consequence we derive new trace embeddings with improved optimal target spaces for classical sobolev lorentz sobolev and orlicz sobolev spaces

optimal gaussian sobolev embeddings abstract a reduction theorem is established showing that any sobolev inequality involving arbitrary rearrangement invariant norms with respect to the gauss measure in r n is equivalent to a one dimensional inequality for a suitable hardy type operator involving the same norms with respect to the standard lebesgue measure on the unit interval this result is exploited to provide a general characterization of optimal range and domain norms in gaussian sobolev inequalities applications to special instances yield optimal gaussian sobolev inequalities in orlicz and lorentz zygmund spaces point out new phenomena such as the existence of self optimal spaces and provide further insight into classical results

high order splitting methods for the incompressible navier stokes equations abstract a new pressure formulation for splitting methods is developed that results in high order time accurate schemes for the solution of the incompressible navier stokes equations in particular improved pressure boundary conditions of high order in time are introduced that minimize the effect of erroneous numerical boundary layers induced by splitting methods a new family of stiffly stable schemes is employed in mixed explicit implicit time intgration rules these schemes exhibit much broader stability regions as compared to adams family schemes typically used in splitting methods their stability properties remain almost constant as the accuracy of the integration increases so that robust third or higher order time accurate schemes can readily be constructed that remain stable at relatively large cfl number the new schemes are implemented within the framework of spectral element discretizations in space so that flexibility and accuracy is guaranteed in the numerical experimentation a model stokes problem is studied in detail and several examples of navier stokes solutions of flows in complex geometries are reported comparison is made with the previously used first order in time spectral element splitting and non splitting e g uzawa schemes high order splitting spectral element methods combine accuracy in space and time and flexibility in geometry and thus can be very efficient in direct simulations of turbulent flows in complex geometries

an adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations in recent years there has been a growing interest in analyzing and quantifying the effects of random inputs in the solution of ordinary partial differential equations to this end the spectral stochastic finite element method ssfem is the most popular method due to its fast convergence rate recently the stochastic sparse grid collocation method has emerged as an attractive alternative to ssfem it approximates the solution in the stochastic space using lagrange polynomial interpolation the collocation method requires only repetitive calls to an existing deterministic solver similar to the monte carlo method however both the ssfem and current sparse grid collocation methods utilize global polynomials in the stochastic space thus when there are steep gradients or finite discontinuities in the stochastic space these methods converge very slowly or even fail to converge in this paper we develop an adaptive sparse grid collocation strategy using piecewise multi linear hierarchical basis functions hierarchical surplus is used as an error indicator to automatically detect the discontinuity region in the stochastic space and adaptively refine the collocation points in this region numerical examples especially for problems related to long term integration and stochastic discontinuity are presented comparisons with monte carlo and multi element based random domain decomposition methods are also given to show the efficiency and accuracy of the proposed method

two level additive schwarz methods for a discontinuous galerkin approximation of second order elliptic problems we present some two level nonoverlapping and overlapping additive schwarz methods for a discontinuous galerkin method for solving second order elliptic problems it is shown that the condition numbers of the preconditioned systems are of the order o frac h h for the nonoverlapping schwarz methods and of the order o frac h delta for the overlapping schwarz methods where h and h stand for the fine mesh size and the coarse mesh size respectively and delta denotes the size of the overlaps between subdomains numerical experiments are provided to gauge the efficiency of the methods and to validate the theory

a comparison of three methods for selecting values of input variables in the analysis of output from a computer code two types of sampling plans are examined as alternatives to simple random sampling in monte carlo studies these plans are shown to be improvements over simple random sampling with respect to variance for a class of estimators which includes the sample mean and the empirical distribution function

factorial sampling plans for preliminary computational experiments a computational model is a representation of some physical or other system of interest first expressed mathematically and then implemented in the form of a computer program it may be viewed as a function of inputs that when evaluated produces outputs motivation for this article comes from computational models that are deterministic complicated enough to make classical mathematical analysis impractical and that have a moderate to large number of inputs the problem of designing computational experiments to determine which inputs have important effects on an output is considered the proposed experimental plans are composed of individually randomized one factor at a time designs and data analysis is based on the resulting random sample of observed elementary effects those changes in an output due solely to changes in a particular input advantages of this approach include a lack of reliance on assumptions of relative sparsity of important inputs monotonicity of outputs with respect to inputs or ad

gaussian functional regression for output prediction in this paper we introduce a gaussian functional regression gfr technique that integrates multi fidelity models with model reduction to efficiently predict the input output relationship of a high fidelity model the gfr method combines the high fidelity model with a low fidelity model to provide an estimate of the output of the high fidelity model in the form of a posterior distribution that can characterize uncertainty in the prediction a reduced basis approximation is constructed upon the low fidelity model and incorporated into the gfr method to yield an inexpensive posterior distribution of the output estimate as this posterior distribution depends crucially on a set of training inputs at which the high fidelity models are simulated we develop a greedy sampling algorithm to select the training inputs our approach results in an output prediction model that inherits the fidelity of the high fidelity model and has the computational complexity of the reduced basis approximation numerical results are presented to demonstrate the proposed approach

a stable algorithm for flat radial basis functions on a sphere when radial basis functions rbfs are made increasingly flat the interpolation error typically decreases steadily until some point when runge type oscillations either halt or reverse this trend because the most obvious method to calculate an rbf interpolant becomes a numerically unstable algorithm for a stable problem in the case of near flat basis functions there will typically also be a separate point at which disastrous ill conditioning enters we introduce here a new method rbf qr which entirely eliminates such ill conditioning and we apply it in the special case when the data points are distributed over the surface of a sphere this algorithm works even for thousands of node points and it allows the rbf shape parameter to be optimized without the limitations imposed by stability concerns since interpolation in the flat rbf limit on a sphere is found to coincide with spherical harmonics interpolation new insights are gained as to why the rbf approach with nonflat basis functions often is the more accurate of the two methods

ghost fluid method for strong shock interactions part 2 immersed solid boundaries numerical simulation of shock waves interacting with multimaterial interface is immensely challenging particularly when the embedded interface is retained as a sharp entity the challenge lies in accurately capturing and representing the interface dynamics and the wave patterns at the interface in this regard the ghost fluid method has been successfully used to capture the interface conditions for both fluid fluid and solid fluid interfaces however the ghost fluid method results in over underheating errors when shocks impact interfaces and hence must be supplemented with numerical corrective measures to mitigate these errors such corrections typically fail for strong shock applications variants and extensions ofthe ghost fluid method have been proposed to remedy its shortcomings with mixed success in this paper the performance of approaches based on the ghost fluid method in the case of strong shocks impinging on immersed solid boundaries in compressible flows is evaluated it is found that from the viewpoint of simplicity robustness and accuracy a reflective boundary condition used in conjunction with a local riemann solver at the interface proves to be a good choice the method is found to be stable accurate and robust for wide range problems involving strong shocks interacting with embedded solid objects

higher order sobolev embeddings and isoperimetric inequalities abstract optimal higher order sobolev type embeddings are shown to follow via isoperimetric inequalities this establishes a higher order analogue of a well known link between first order sobolev embeddings and isoperimetric inequalities sobolev type inequalities of any order involving arbitrary rearrangement invariant norms on open sets in r n possibly endowed with a measure density are reduced to much simpler one dimensional inequalities for suitable integral operators depending on the isoperimetric function of the relevant sets as a consequence the optimal target space in the relevant sobolev embeddings can be determined both in standard and in non standard classes of function spaces and underlying measure spaces in particular our results are applied to any order sobolev embeddings in regular john domains of the euclidean space in maz ya classes of possibly irregular euclidean domains described in terms of their isoperimetric function and in families of product probability spaces of which the gauss space is a classical instance

two level additive schwarz preconditioners for c 0 interior penalty methods we study two level additive schwarz preconditioners that can be used in the iterative solution of the discrete problems resulting from c0 interior penalty methods for fourth order elliptic boundary value problems we show that the condition number of the preconditioned system is bounded by c 1 h3 3 where h is the typical diameter of a subdomain measures the overlap among the subdomains and the positive constant c is independent of the mesh sizes and the number of subdomains

two level additive schwarz preconditioners for nonconforming finite element methods two level additive schwarz preconditioners are developed for the nonconforming p1 finite element approximation of scalar second order symmetric positive definite elliptic boundary value problems the morley finite element approximation of the biharmonic equation and the divergence free nonconforming p1 finite element approximation of the stationary stokes equations the condition numbers of the preconditioned systems are shown to be bounded independent of mesh sizes and the number of subdomains in the case of generous overlap

robust bddc preconditioners for reissner mindlin plate bending problems and mitc elements a balancing domain decomposition method by constraints bddc is constructed and analyzed for the reissner mindlin plate bending problem discretized with mixed interpolation of tensorial components mitc finite elements this bddc algorithm is based on selecting the plate rotations and deflection degrees of freedom at the subdomain vertices as primal continuity constraints after the implicit elimination of the interior degrees of freedom in each subdomain the resulting plate schur complement is solved by the preconditioned conjugate gradient method the preconditioner is based on the solution of local reissner mindlin plate problems on each subdomain with clamping conditions at the primal degrees of freedom and on the solution of a coarse reissner mindlin plate problem for the primal degrees of freedom the main results of the paper are the proof and numerical verification that the proposed bddc plate algorithm is scalable quasi optimal and most important robust with respect to the plate thickness while this result is due to an underlying mixed formulation of the problem both the interface plate problem and the preconditioner are positive definite the numerical results also show that the proposed algorithm is robust with respect to discontinuities of the material properties

a two level additive schwarz preconditioner for nonconforming plate elements a two level additive schwarz preconditioner is developed for the systems resulting from the discretizations of the plate bending problem by the morley finite element the fraeijs de veubeke finite element the zienkiewicz finite element and the adini finite element the condition numbers of the preconditioned systems are shown to be bounded independent of mesh sizes and the number of subdomains in the case of a generous overlap

gaussian process surrogate model with composite kernel learning for engineering design the composite kernel learning ckl method is introduced to efficiently construct composite kernels for gaussian process gp surrogate models with applications in engineering design the mixture o

evaluation of convergence behavior of metamodeling techniques for bridging scales in multi scale multimaterial simulation the effectiveness of several metamodeling techniques viz the polynomial stochastic collocation method adaptive stochastic collocation method a radial basis function neural network a kriging method and a dynamic kriging method is evaluated this is done with the express purpose of using metamodels to bridge scales between micro and macro scale models in a multi scale multimaterial simulation the rate of convergence of the error when used to reconstruct hypersurfaces of known functions is studied for sufficiently large number of training points stochastic collocation methods generally converge faster than the other metamodeling techniques while the dkg method converges faster when the number of input points is less than 100 in a two dimensional parameter space because the input points correspond to computationally expensive micro meso scale computations the dkg is favored for bridging scales in a multi scale solver

a feti dp preconditioner for a composite finite element and discontinuous galerkin method in this paper a nitsche type discretization based on a discontinuous galerkin dg method for an elliptic two dimensional problem with discontinuous coefficients is considered the problem is posed on a polygonal region omega which is a union of n disjoint polygonal subdomains omega i of diameter o h i the discontinuities of the coefficients possibly very large are assumed to occur only across the subdomain interfaces partial omega i inside each subdomain a conforming finite element space on a quasi uniform triangulation with mesh size o h i is introduced to handle the nonmatching meshes across the subdomain interfaces a dg discretization is applied only on the interfaces for solving the resulting discrete system a feti dp type method is proposed and analyzed it is established that the condition number of the preconditioned linear system is estimated by c 1 max i log h i h i 2 with a constant c independent of h i h i h j h i and the jumps of coefficients the

evaluation of kriging based surrogate models constructed from mesoscale computations of shock interaction with particles abstract macro scale computations of shocked particulate flows require closure laws that model the exchange of momentum energy between the fluid and particle phases closure laws are constructed in this work in the form of surrogate models derived from highly resolved mesoscale computations of shock particle interactions the mesoscale computations are performed to calculate the drag force on a cluster of particles for different values of mach number and particle volume fraction two kriging based methods viz the dynamic kriging method dkg and the modified bayesian kriging method mbkg are evaluated for their ability to construct surrogate models with sparse data i e using the least number of mesoscale simulations it is shown that if the input data is noise free the dkg method converges monotonically convergence is less robust in the presence of noise the mbkg method converges monotonically even with noisy input data and is therefore more suitable for surrogate model construction from numerical experiments this work is the first step towards a full multiscale modeling of interaction of shocked particle laden flows

optimization of multi fidelity computer experiments via the eqie criterion computer experiments based on mathematical models are powerful tools for understanding physical processes this article addresses the problem of kriging based optimization for deterministic computer experiments with tunable accuracy our approach is to use multi fidelity computer experiments with increasing accuracy levels and a nonstationary gaussian process model we propose an optimization scheme that sequentially adds new computer runs by following two criteria the first criterion called eqi scores candidate inputs with given level of accuracy and the second criterion called eqie scores candidate combinations of inputs and accuracy from simulation results and a real example using finite element analysis our method outperforms the expected improvement ei criterion that works for single accuracy experiments supplementary materials for this article are available online

an interior penalty method for distributed optimal control problems governed by the biharmonic operator in this paper a c 0 interior penalty method has been proposed and analyzed for distributed optimal control problems governed by the biharmonic operator the state and adjoint variables are discretized using continuous piecewise quadratic finite elements while the control variable is discretized using piecewise constant approximations a priori and a posteriori error estimates are derived for the state adjoint and control variables under minimal regularity assumptions numerical results justify the theoretical results obtained the a posteriori error estimators are useful in adaptive finite element approximation and the numerical results indicate that the sharp error estimators work efficiently in guiding the mesh refinement

a fully discrete c0 interior penalty galerkin approximation of the extended fisher kolmogorov equation a fully discrete c 0 interior penalty finite element method is proposed and analyzed for the extended fisher kolmogorov efk equation u email protected d email protected u 3 u 0 with appropriate initial and boundary conditions where c is a positive constant we derive a regularity estimate for the solution u of the efk equation that is explicit in c and as a consequence we derive a priori error estimates that are robust in c

multigrid algorithms for c 0 interior penalty methods multigrid algorithms for c 0 interior penalty methods for fourth order elliptic boundary value problems on polygonal domains are studied in this paper it is shown that v cycle f cycle and w cycle algorithms are contractions if the number of smoothing steps is sufficiently large the contraction numbers of these algorithms are bounded by cm alpha where m is the number of presmoothing and postsmoothing steps alpha is the index of elliptic regularity and the positive constant c is mesh independent these estimates are established for a smoothing scheme that uses a poisson solve as a preconditioner which can be easily implemented because the c 0 finite element spaces are standard spaces for second order problems furthermore the variable v cycle algorithm is also shown to be an optimal preconditioner

predicting the output from a complex computer code when fast approximations are available s we consider prediction and uncertainty analysis for complex computer codes which can be run at different levels of sophistication in particular we wish to improve efficiency by combining expensive runs of the most complex versions of the code with relatively cheap runs from one or more simpler approximations a bayesian approach is described in which prior beliefs about the codes are represented in terms of gaussian processes an example is presented using two versions of an oil reservoir simulator 1 c complex mathematical models implemented in large computer codes have been used to study real systems in many areas of scientific research sacks et al 1989 usually because physical experimentation is too costly and sometimes impossible as in the case of large environmental systems a computer experiment involves running the code with various input values for the purpose of learning something about the real system often a simulator can be run at different levels of complexity with versions ranging from the most sophisticated high level code to the most basic for example in 4 we consider two codes which simulate oil pressure at a well of a hydrocarbon reservoir both codes use finite element analysis in which the rocks comprising the reservoir are represented by small interacting grid blocks the flow of oil within the reservoir can be simulated by considering the interaction between the blocks the two codes differ in the resolution of the grid so that we have a very accurate slow version using many small blocks and a crude approximation using large blocks which runs much faster alternatively a mathematical model could be expanded to include more of the scientific laws underlying the physical processes simple fast versions of the code may well include the most important features and are useful for preliminary investigations in real time applications the number of runs from a high level simulator may be limited by expense then there is a need to trade off the complexity of the expensive code with the availability of the simpler approximations the purpose of the current paper is to explore ways in which runs from several levels of a code can be used to make inference about the output from the most complex code we may also have uncertainty about values for the input parameters which apply in any given application uncertainty analysis of computer codes describes how this uncertainty on the inputs affects our uncertainty about the output

continuous discontinuous finite element approximations of fourth order elliptic problems in structural and continuum mechanics with applications to thin beams and plates and strain gradient elasticity abstract a new finite element method for fourth order elliptic partial differential equations is presented and applied to thin bending theory problems in structural mechanics and to a strain gradient theory problem the method combines concepts from the continuous galerkin cg method the discontinuous galerkin dg method and stabilization techniques a brief review of the cg method the dg method and stabilization techniques highlights the advantages and disadvantages of these methods and suggests a new approach for the solution of fourth order elliptic problems a continuous discontinuous galerkin c dg method is proposed which uses c 0 continuous interpolation functions and is formulated in the primary variable only the advantage of this formulation over a more traditional mixed approach is that the introduction of additional unknowns and related difficulties can be avoided in the context of thin bending theory the c dg method leads to a formulation where displacements are the only degrees of freedom and no rotational degrees of freedom need to be considered the main feature of the c dg method is the weak enforcement of continuity of first and higher order derivatives through stabilizing terms on interior boundaries consistency stability and convergence of the method are shown analytically numerical experiments verify the theoretical results and applications are presented for bernoulli euler beam bending poisson kirchhoff plate bending and a shear layer problem using toupin mindlin strain gradient theory

on the abstract theory of additive and multiplicative schwarz algorithms in recent years it has been shown that many modern iterative algorithms multigrid schemes multilevel preconditioners domain decomposition methods etc for solving problems resulting from the discretization of pdes can be interpreted as additive jacobi like or multiplicative gauss seidel like subspace correction methods the key to their analysis is the study of certain metric properties of the underlying splitting of the discretization space v into a sum of subspaces v j and the splitting of the variational problem on v into auxiliary problems on these subspaces in this paper we propose a modification of the abstract convergence theory of the additive and multiplicative schwarz methods that makes the relation to traditional iteration methods more explicit the analysis of the additive and multiplicative schwarz iterations can be carried out in almost the same spirit as in the traditional block matrix situation making convergence proofs of multilevel and domain decomposition methods clearer or at least more classical in addition we present a new bound for the convergence rate of the appropriately scaled multiplicative schwarz method directly in terms of the condition number of the corresponding additive schwarz operator these results may be viewed as an appendix to the recent surveys x ys

orlicz sobolev boundary trace embeddings we prove a sharp trace embedding theorem for orlicz sobolev spaces into orlicz spaces on the boundary an optimal trace embedding into more general rearrangement invariant spaces on the boundary is also established

optimization with missing data engineering optimization relies routinely on deterministic computer based design evaluations typically comprising geometry creation mesh generation and numerical simulation simple optimization routines tend to stall and require user intervention if a failure occurs at any of these stages this motivated us to develop an optimization strategy based on surrogate modelling which penalizes the likely failure regions of the design space without prior knowledge of their locations a gaussian process based design improvement expectation measure guides the search towards the feasible global optimum

stable evaluation of gaussian radial basis function interpolants we provide a new way to compute and evaluate gaussian radial basis function interpolants in a stable way with a special focus on small values of the shape parameter i e for flat kernels this work is motivated by the fundamental ideas proposed earlier by bengt fornberg and his coworkers however following mercer s theorem an l 2 mathbb r d rho orthonormal expansion of the gaussian kernel allows us to come up with an algorithm that is simpler than the one proposed by fornberg larsson and flyer and that is applicable in arbitrary space dimensions d in addition to obtaining an accurate approximation of the radial basis function interpolant using many terms in the series expansion of the kernel we also propose and investigate a highly accurate least squares approximation based on early truncation of the kernel expansion

an extended eulerian method for contacts in godunov formulations summary this paper offers a new method to treat contacts in eulerian godunov formulations the inadequacy of classic eulerian formulations to describe contacts behavior is identified in the lack of degrees of freedom this limitation is overcome by employing a set of state variables per material for the calculation of multi material elements and faces the extra state variables allow for the independent analysis of the materials present in the problem materials interactions are characterized by introducing contact constraints in the formulation the locality of the method limits the memory burden copyright 2012 john wiley sons ltd

multifidelity quasi newton method for design optimization multifidelity approaches are frequently used in design when high fidelity models are too expensive to use directly and lower fidelity models of reasonable accuracy exist in optimization corrected

multi fidelity optimization via surrogate modelling this paper demonstrates the application of correlated gaussian process based approximations to optimization where multiple levels of analysis are available using an extension to the geostatistical method of co kriging an exchange algorithm is used to choose which points of the search space to sample within each level of analysis the derivation of the co kriging equations is presented in an intuitive manner along with a new variance estimator to account for varying degrees of computational noise in the multiple levels of analysis a multi fidelity wing optimization is used to demonstrate the methodology

the correlated knowledge gradient for simulation optimization of continuous parameters using gaussian process regression we extend the concept of the correlated knowledge gradient policy for the ranking and selection of a finite set of alternatives to the case of continuous decision variables we propose an approximate knowledge gradient for problems with continuous decision variables in the context of a gaussian process regression model in a bayesian setting along with an algorithm to maximize the approximate knowledge gradient in the problem class considered we use the knowledge gradient for continuous parameters to sequentially choose where to sample an expensive noisy function in order to find the maximum quickly we show that the knowledge gradient for continuous decisions is a generalization of the efficient global optimization algorithm proposed in d r jones m schonlau and w j welch j global optim 13 1998 pp 455 492

an overlapping domain decomposition preconditioner for a class of discontinuous galerkin approximations of advection diffusion problems we consider a scalar advection diffusion problem and a recently proposed discontinuous galerkin approximation which employs discontinuous finite element spaces and suitable bilinear forms containing interface terms that ensure consistency for the corresponding sparse non symmetric linear system we propose and study an additive two level overlapping schwarz preconditioner consisting of a coarse problem on a coarse triangulation and local solvers associated to suitable problems defined on a family of subdomains this is a generalization of the corresponding overlapping method for approximations on continuous finite element spaces related to the lack of continuity of our approximation spaces some interesting new features arise in our generalization which have no analog in the conforming case we prove an upper bound for the number of iterations obtained by using this preconditioner with gmres which is independent of the number of degrees of freedom of the original problem and the number of subdomains the performance of the method is illustrated by several numerical experiments for different test problems using linear finite elements in two dimensions