improving the k compressibility of hyper reduced order models with moving sources applications to welding and phase change problems abstract the simulation of engineering problems is quite often a complex task that can be time consuming in this context the use of hyper reduced order models hroms is a promising alternative for real time simulations in this work we study the design of hroms for non linear problems with a moving source applications to nonlinear phase change problems with temperature dependent thermophysical properties are particularly considered however the techniques developed can be applied in other fields as well a basic assumption in the design of hroms is that the quantities that will be hyper reduced are k compressible in a certain basis in the sense that these quantities have at most k non zero significant entries when expressed in terms of that basis to reach the computational speed required for a real time application k must be small this work examines different strategies for addressing hyper reduction of the nonlinear terms with the objective of obtaining k compressible signals with a notably small k to improve performance and robustness it is proposed that the different contributing terms to the residual are separately hyper reduced additionally the use of moving reference frames is proposed to simulate and hyper reduce cases that contain moving heat sources two application examples are presented the solidification of a cube in which no heat source is present and the welding of a tube in which the problem posed by a moving heat source is analysed

balanced model reduction via the proper orthogonal decomposition a new method for performing a balanced reduction of a high order linear system is presented the technique combines the proper orthogonal decomposition and concepts from balanced realization theory the method of snapshotsisused to obtainlow rank reduced rangeapproximationsto thesystemcontrollability and observability grammiansineitherthetimeorfrequencydomain theapproximationsarethenusedtoobtainabalancedreducedorder model the method is particularly effective when a small number of outputs is of interest it is demonstrated for a linearized high order system that models unsteady motion of a two dimensional airfoil computation of the exact grammians would be impractical for such a large system for this problem very accurate reducedorder models are obtained that capture the required dynamics with just three states the new models exhibit far superiorperformancethanthosederived using a conventionalproperorthogonal decomposition although further development is necessary the concept also extends to nonlinear systems

a spectral viscosity method for correcting the long term behavior of pod models low dimensional flow dynamical systems may converge to erroneous states after long time integration even if they are initialized with the correct state in this paper we investigate the accuracy of such two dimensional models constructed from karhunen loeve expansions for flows past a circular cylinder we first demonstrate that although the short term dynamics may be predicted accurately with only a handful of modes retained drifting of the solution may arise after a few hundred vortex shedding cycles we then propose a dissipative model based on a spectral viscosity sv diffusion convolution operator the parameters of the sv model are selected rigorously based on bifurcation analysis our results show that this is an effective way of improving the accuracy of long term predictions of low dimensional galerkin systems

batch size effects in the analysis of simulation output batching is a commonly used method for calculating confidence intervals on the mean of a sequence of correlated observations arising from a simulation experiment several recent papers have considered the effect of using batch sizes too small to satisfy assumptions of normality and or independence and the resulting incorrect probabilities of the confidence interval covering the mean this paper quantifies the effects of using batch sizes larger than necessary to satisfy normality and independence assumptions these effects include 1 correct probability of covering the mean 2 an increase in expected half length 3 an increase in the standard deviation and coefficient of variation of the half length and 4 an increase in the probability of covering points not equal to the mean for any sample size and independent and normal batch means the results are 1 the effects of less than 10 batches are large and the effects of more than 30 batches small and 2 additional batches have lesser effects on

a nonlinear pod galerkin reduced order model for compressible flows taking into account rigid body motions abstract the construction of a nonlinear reduced order model for fluid structure interaction problems is investigated in this paper for unsteady compressible flows excited by the rigid body motion of a structure the reduction is achieved by means of a galerkin projection of the navier stokes equations on the first pod modes resulting from the proper orthogonal decomposition in the first part of the paper the projection technique is carried out on a purely aerodynamic case in order i to validate an efficient iterative technique based on an updated qr decomposition to compute the pod modes and ii to discuss the merits of different correction methods introduced to improve the long term stability of the reduced order model the second and most original part of the paper deals with the construction of the reduced set of equations which arise from the projection of the compressible navier stokes equations formulated in a suitable moving frame representing the rigid body motion the expressions of the resulting non autonomous terms appearing in the reduced order model have also been optimized to reduce the computational costs

the variational multiscale method a paradigm for computational mechanics abstract we present a general treatment of the variational multiscale method in the context of an abstract dirichlet problem we show how the exact theory represents a paradigm for subgrid scale models and a posteriori error estimation we examine hierarchical p methods and bubbles in order to understand and ultimately approximate the fine scale green s function which appears in the theory we review relationships between residual free bubbles element green s functions and stabilized methods these suggest the applicability of the methodology to physically interesting problems in fluid mechanics acoustics and electromagnetics

galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics error estimates for galerkin proper orthogonal decomposition pod methods for nonlinear parabolic systems arising in fluid dynamics are proved for the time integration the backward euler scheme is considered the asymptotic estimates involve the singular values of the pod snapshot set and the grid structure of the time discretization as well as the snapshot locations

a semi continuous formulation for goal oriented reduced order models 1d problems a semi continuous formulation is introduced for finding bases that minimise the error in a specific output functional of a reduced order model the formulation is advantageous in that it can be used with arbitrary reference data and can be easily applied to nonlinear models and functionals a general description of the approach is given then explicit formulations are derived for the advection diffusion and burgers equations numerical results are given for both linear and nonlinear functionals these show substantial reductions in error when compared with pod modes depending on the functional considered the optimisation of bases for a reduced order model using an approximated governing equation is also described for which large increases in accuracy are obtained relative to pod modes copyright c 2015 john wiley sons ltd

a general result on the stabilization of linear systems using bounded controls we present two constructions of controllers that globally stabilize linear systems subject to control saturation we allow essentially arbitrary saturation functions the only conditions imposed on the system are the obvious necessary ones namely that no eigenvalues of the uncontrolled system have positive real part and that the standard stabilizability rank condition hold one of the constructions is in terms of a neural network type one hidden layer architecture while the other one is in terms of cascades of linear maps and saturations

multidimensional a priori hyper reduction of mechanical models involving internal variables this paper concerns the adaptation of reduced order models during simulations of series of elastoviscoplastic problems in continuation with previous works this paper aimed at extending the a priori hyper reduction method aphr method for nonlinear thermal problems to nonlinear mechanical problems involving internal variables this method is an a priori approach because full incremental responses of detailed models are not forecasted in order to build reduced order models the recent extension of the hyper reduction method to reduction of mechanical models involving internal variables makes possible the reduction of degrees of freedom and the reduction of integration points a multi level formulation is introduced to focus on the capability of the method to perform efficient parallel computations to adapt reduced order models

optimal control of the cylinder wake in the laminar regime by trust region methods and pod reduced order models in this paper optimal control theory is used to minimize the total mean drag for a circular cylinder wake flow in the laminar regime re 200 the control parameters are the amplitude and the frequency of the time harmonic cylinder rotation in order to reduce the size of the discretized optimality system a proper orthogonal decomposition pod reduced order model rom is derived to be used as state equation we then propose to employ the trust region proper orthogonal decomposition trpod approach originally introduced by fahl m fahl trust region methods for flow control based on reduced order modeling ph d thesis trier university 2000 to update the reduced order models during the optimization process a lot of computational work is saved because the optimization process is now based only on low fidelity models a particular care was taken to derive a pod rom for the pressure and velocity fields with an appropriate balance between model accuracy and robustness the key enablers are the extension of the pod basis functions to the pressure data the use of calibration methods for the pod rom and the addition in the pod expansion of several non equilibrium modes to describe various operating conditions when the trpod algorithm is applied to the wake flow configuration this approach converges to the minimum predicted by an open loop control approach and leads to a relative mean drag reduction of 30 at reduced cost

reduced basis approximation for nonlinear parametrized evolution equations based on empirical operator interpolation we present a new approach to treating nonlinear operators in reduced basis approximations of parametrized evolution equations our approach is based on empirical interpolation of nonlinear differential operators and their frechet derivatives efficient offline online decomposition is obtained for discrete operators that allow an efficient evaluation for a certain set of interpolation functionals an a posteriori error estimate for the resulting reduced basis method is derived and analyzed numerically we introduce a new algorithm the podei greedy algorithm which constructs the reduced basis spaces for the empirical interpolation and for the numerical scheme in a synchronized way the approach is applied to nonlinear parabolic and hyperbolic equations based on explicit or implicit finite volume discretizations we show that the resulting reduced scheme is able to capture the evolution of both smooth and discontinuous solutions in case of symmetries of the problem the approach realizes an automatic and intuitive space compression or even space dimensionality reduction we perform empirical investigations of the error convergence and run times in all cases we obtain a good run time acceleration

α regularization of the pod galerkin dynamical systems of the kuramoto sivashinsky equation abstract in order to improve the dynamics and stability of the pod galerkin models of strongly stiff systems an like regularization is suggested and assessed in the present article in this method the pod eigenmodes of the non linear terms are replaced by their helmholtz filtered counterparts while the other terms are remained unchanged as an example the method is applied to the pod galerkin models of the one dimensional kuramoto sivashinsky ks equation in a full chaotic regime and the fidelity of the original and regularized models to the direct numerical simulations dns are investigated moreover the effects of regularization on the dynamics of various terms and whole of the systems are analyzed via eigenvalue analysis of each term separately and the total dynamical system as a whole the numerical experiments show definite effectiveness of the method and excellent improvements in the predicted dynamics and stability by minimum number of free parameters

on the use of pod based roms to analyze bifurcations in some dissipative systems abstract this paper deals with the use of pod based reduced order models to construct bifurcation diagrams which requires calculating steady and time dependent attractors in complex bifurcation problems involving dissipative systems the method proposed in the paper relies on the observation that pod manifolds resulting from snapshots calculated in time dependent runs for specific values of the parameters of the problem also contain the attractors for other values of the parameters the reason for this property is explained for a general class of dissipative systems which includes many problems of scientific industrial interest the consequence is that appropriate pod manifolds can be calculated in a quite computationally efficient way the method is illustrated considering both a simple bifurcation problem for a fisher like equation and a fairly complex bifurcation problem for the complex ginzburg landau equation

a domain decomposition method for the non intrusive reduced order modelling of fluid flow abstract in this paper we present a new domain decomposition non intrusive reduced order model ddnirom for the navier stokes equations the computational domain is partitioned into subdomains and a set of local basis functions is constructed in each subdomain using proper orthogonal decomposition pod a radial basis function rbf method is then used to generate a set of hypersurfaces for each subdomain each local hypersurface represents not only the fluid dynamics over the subdomain to which it belongs but also the interactions with the surrounding subdomains this implicit coupling between the subdomains provides the global coupling necessary to enforce incompressibility and is a means of providing boundary conditions for each subdomain the performance of this ddnirom is illustrated numerically by three examples flow past a cylinder and air flow over 2d and 3d street canyons the results show that the ddnirom exhibits good agreement with the high fidelity full model while the computational cost is reduced by several orders of magnitude the domain decomposition dd method provides the flexibility to choose different numbers of local basis functions for each subdomain depending on the complexity of the flow therein the fact that the rbf surface representation takes input only from its current subdomain and the surrounding subdomains means that crucially there is a reduction in the dimensionality of the hypersurface when compared with a more traditional global nirom this comes at the cost of having a larger number of hypersurfaces

reduced order subscales for pod models in this work the reduced order subscales for proper orthogonal decomposition models are presented the basic idea consists in splitting the full order solution into the part which can be captured by the reduced order model and the part which cannot the subscales for which a model is required the proposed model for the subscales is defined as a linear function of the solution of the reduced order model the coefficients of this linear function are obtained by comparing the solution of the full order model with the solution of the reduced order model for the same initial conditions which for convenience are evaluated in the snapshots used to train the original reduced order model the difference between both solutions are the subscales for which a model can be built using a least squares procedure the subscales are then introduced as a correction in the reduced order model resulting in an important improvement in accuracy the enhanced reduced order model is tested in several numerical examples these practical cases show that the use of the subscales leads to more accurate solutions successfully corrects errors introduced by hyper reduction and allows to solve complex flow problems using a reduced number of degrees of freedom

a reduced order discontinuous galerkin method based on pod for electromagnetic simulation this paper is concerned with the design of a reduced order discontinuous galerkin dg method based on the proper orthogonal decomposition pod method for electromagnetic simulation a centered flux approximation for surface integral and a second order leap frog scheme for advancing in time are applied in the classical dg method the pod basis is created by the eigensystem of the correlation matrix which is generated by the snapshot matrix whose columns are the snapshot vectors extracted from the high fidelity dg simulation the pod discontinuous galerkin time domain formulation with lower dimension and sufficiently high accuracy is established by applying a galerkin projection for the semidiscrete dg scheme the overall goal is to reduce the computational cost while maintaining an acceptable level of accuracy numerical experiments for electromagnetic problems illustrate the performance of the proposed reduction method

brief paper on the stabilizability condition in indirect adaptive control many recent studies on overall stability of adaptive control systems require among other conditions that the identified model be stabilizable at least asymptotically in fact the usual identification methods do not guarantee this property in the passive approach without exciting extra signals or if the model order is subject to over estimation in order to solve this problem it is shown in this paper that certain classical identification algorithms can be easily modified in order to ensure the above stabilizability condition without affecting the other properties

on coverage probability independence and normality in batch means algorithms abstract we consider the batching approach to interval estimation of the mean of a stochastic process based upon simulation output in particular using batch means processes generated from underlying ar 1 processes and m m 1 queueing systems we study the effects of undetected autocorrelation on both the probability of coverage and the performance of the shapiro wilk test of normality we also conduct power comparisons of two tests used to detect first order autocorrelation in batch means processes the von neumann ratio test and a rank version thereof proposed by bartels 1982 our results indicate that a large number of batches perhaps at least 100 be used when testing for the presence of autocorrelation

testing independence of simulation subruns a note on the power of the von neumann test abstract in simulation experiments confidence intervals may be based on subruns to test whether subruns are independent the von neumann statistic is often used this note shows that the power of this statistic may be very small the practical recommendation is to apply the statistic only if at least 100 subruns are available

adaptive pod based low dimensional modeling supported by residual estimates summary an adaptive low dimensional model is considered to simulate time dependent dynamics in nonlinear dissipative systems governed by pdes the method combines an inexpensive pod based galerkin system with short runs of a standard numerical solver that provides the snapshots necessary to first construct and then update the pod modes switching between the numerical solver and the galerkin system is decided on the fly by monitoring i a truncation error estimate and ii a residual estimate the latter estimate is used to control the mode truncation instability and highly improves former adaptive strategies that detected this instability by monitoring consistency with a second instrumental galerkin system based on a larger number of pod modes the most computationally expensive run of the numerical solver occurs at the outset when the whole set of pod modes is calculated this step is improved by using mode libraries which may either be generic or result from former applications of the method the outcome is a flexible robust computationally inexpensive procedure that adapts itself to the local dynamics by using the faster galerkin system for the majority of the time and few on demand short runs of a numerical solver the method is illustrated considering the complex ginzburg landau equation in one and two space dimensions copyright 2015 john wiley sons ltd

pod galerkin method for finite volume approximation of navier stokes and rans equations numerical simulation of fluid flows requires important computational efforts but it is essential in engineering applications reduced order model rom can be employed whenever fast simulations are required or in general whenever a trade off between computational cost and solution accuracy is a preeminent issue as in process optimization and control in this work the efforts have been put to develop a rom for computational fluid dynamics cfd application based on finite volume approximation starting from the results available in turbulent reynold averaged navier stokes simulations in order to enlarge the application field of proper orthogonal decomposition reduced order model pod rom technique to more industrial fields the approach is tested in the classic benchmark of the numerical simulation of the 2d lid driven cavity in particular two simulations at re 103 and re 105 have been considered in order to assess both a laminar and a turbulent case some quantities have been compared with the full order model in order to assess the performance of the proposed rom procedure i e the kinetic energy of the system and the reconstructed quantities of interest velocity pressure and turbulent viscosity in addition for the laminar case the comparison between the rom steady state solution and the data available in literature has been presented the results have turned out to be very satisfactory both for the accuracy and the computational times as a major outcome the approach turns out not to be affected by the energy blow up issue characterizing the results obtained by classic turbulent pod galerkin methods

pod deim reduced order strategies for efficient four dimensional variational data assimilation this work studies reduced order modeling rom approaches to speed up the solution of variational data assimilation problems with large scale nonlinear dynamical models it is shown that a key requirement for a successful reduced order solution is that reduced order karush kuhn tucker conditions accurately represent their full order counterparts in particular accurate reduced order approximations are needed for the forward and adjoint dynamical models as well as for the reduced gradient new strategies to construct reduced order based are developed for proper orthogonal decomposition pod rom data assimilation using both galerkin and petrov galerkin projections for the first time pod tensorial pod and discrete empirical interpolation method deim are employed to develop reduced data assimilation systems for a geophysical flow model namely the two dimensional shallow water equations numerical experiments confirm the theoretical framework for galerkin projection in the case of petrov galerkin projection stabilization strategies must be considered for the reduced order models the new reduced order shallow water data assimilation system provides analyses similar to those produced by the full resolution data assimilation system in one tenth of the computational time

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

a spectral vanishing viscosity method for large eddy simulations a new simulation approach for high reynolds number turbulent flows is developed combining concepts of monotonicity in nonlinear conservation laws with concepts of large eddy simulation the spectral vanishing viscosity svv first introduced by e tadmor siam j numer anal 26 30 1989 is incorporated into the navier stokes equations for controlling high wavenumber oscillations unlike hyperviscosity kernels the svv approach involves a second order operator which can be readily implemented in standard finite element codes in the work presented here discretization is performed using hierarchical spectral hp methods accommodating effectively an ab initio intrinsic scale separation the key result is that monotonicity is enforced via svv leading to stable discretizations without sacrificing the formal accuracy i e exponential convergence in the proposed discretization several examples are presented to demonstrate the effectiveness of the new approach including a comparison with eddy viscosity spectral les of turbulent channel flow in its current implementation the svv approach for controlling the small scales is decoupled from the large scales but a procedure is proposed that will provide coupling similar to the classical les formulation

accelerating pde constrained optimization by the reduced basis method application to batch chromatography summary in this work we show that the reduced basis method accelerates a partial differential equation constrained optimization problem where a nonlinear discretized system with a large number of degrees of freedom must be repeatedly solved during optimization such an optimization problem arises for example from batch chromatography to reduce the computational burden of repeatedly solving the large scale system under parameter variations a parametric reduced order model with a small number of equations is derived by using the reduced basis method as a result the small reduced order model rather than the full system is solved at each step of the optimization process an adaptive technique for selecting the snapshots is proposed so that the complexity and runtime for generating the reduced basis are largely reduced an output oriented error bound is derived in the vector space whereby the construction of the reduced model is managed automatically an early stop criterion is proposed to circumvent the stagnation of the error and to make the construction of the reduced model more efficient numerical examples show that the adaptive technique is very efficient in reducing the offline time the optimization based on the reduced model is successful in terms of the accuracy and the runtime for acquiring the optimal solution copyright 2015 john wiley sons ltd

fast divergence conforming reduced basis methods for steady navier stokes flow abstract reduced basis methods rb methods or rbms form one of the most promising techniques to deliver numerical solutions of parametrized pdes in real time with reasonable accuracy for incompressible flow problems rbms based on lbb stable velocity pressure spaces do not generally inherit the stability of the underlying high fidelity model and instead additional stabilization techniques must be introduced one way of bypassing the loss of lbb stability in the rbm is to inflate the velocity space with supremizer modes this however deteriorates the performance of the rbm in the performance critical online stage as additional dofs must be introduced to retain stability while these dofs do not effectively contribute to accuracy of the rb approximation in this work we consider a velocity only rb approximation exploiting a solenoidal velocity basis the solenoidal reduced basis emerges directly from the high fidelity velocity solutions in the offline stage by means of piola transforms the solenoidality of the velocity space is retained under geometric transformations making the proposed rb method suitable also for the investigation of geometric parameters to ensure exact solenoidality of the high fidelity velocity solutions that constitute the rb we consider approximations based on divergence conforming compatible b splines we show that the velocity only rb method leads to a significant improvement in computational efficiency in the online stage and that the pressure solution can be recovered a posteriori at negligible extra cost we illustrate the solenoidal rb approach by modeling steady two dimensional navier stokes flow around a naca0015 airfoil at various angles of attack

the rank version of von neumann s ratio test for randomness abstract although rank tests for randomness were proposed in the literature as early as 1943 no such test has gained wide acceptance comparable to say spearman s rho test this may be due to the lack of small sample theory and of tables of critical values to enable such a test to be carried out on small samples in this article we consider the rank version of the von neumann ratio statistic and we obtain the critical values of this statistic under the randomization hypothesis in a monte carlo experiment we then show that the resulting nonparametric test for randomness has far greater power than the test based on the number of runs up and down moreover under normality its power vis a vis the normal theory von neumann ratio test is also very good it is therefore suggested that with the tables presented in this article the rank von neumann ratio test for randomness provides an easy and powerful alternative to nonparametric tests now in common use

nonlinear model order reduction via lifting transformations and proper orthogonal decomposition this paper presents a structure exploiting nonlinear model reduction method for systems with general nonlinearities first the nonlinear model is lifted to a model with more structure via variable

stability and accuracy of periodic flow solutions obtained by a pod penalty method we develop a new penalty method to derive low dimensional galerkin models for fluid flows with time dependent boundary conditions we then outline a procedure based on bifurcation analysis in selecting the proper values of the penalty parameter s that yield asymptotically stable periodic solutions of the highest possible accuracy we illustrate this new approach by studying flow past a circular cylinder using direct numerical simulation dns data and a wave structure interaction problem using particle image velocimetry piv data

reduced order model for the bgk equation based on pod and optimal transport we present a reduced order approximation of the bgk equation leading to fast and accurate computations the bgk model describes the dynamics of a gas flow in both hydrodynamic and rarefied regimes the particles of the gas are represented by a density distribution function depending on physical space velocity space and time in this work the density distribution function is approximated in the velocity space by a small number of basis functions computed offline in the offline phase the bgk equation is sampled in order to collect information on the density distribution function to complete this sampling optimal transport is used to add new information by interpolating the samples of the density distribution function finally the basis functions are built by proper orthogonal decomposition during the online phase the offline knowledge is used to compute approximations of the density distribution function at low cost to do so the bgk equation is projected onto the basis functions leading to a system of partial differential equations the system obtained is hyperbolic by construction and is solved by an imex runge kutta scheme in time and a finite volume scheme in space to improve the accuracy the reduced oder model is modified to conserve mass momentum and energy of the gas numerical illustrations of 1d and 2d flows are given in particular we investigate the reconstruction and the prediction of shock waves boundary layers and vortices the results demonstrate the accuracy of the reduced order model and the significant reduction of the computational cost

non linearly stable reduced order models for incompressible flow with energy conserving finite volume methods abstract a novel reduced order model rom formulation for incompressible flows is presented with the key property that it exhibits non linearly stability independent of the mesh of the full order model the time step the viscosity and the number of modes the two essential elements to non linear stability are 1 first discretise the full order model and then project the discretised equations and 2 use spatial and temporal discretisation schemes for the full order model that are globally energy conserving in the limit of vanishing viscosity for this purpose as full order model a staggered grid finite volume method in conjunction with an implicit runge kutta method is employed in addition a constrained singular value decomposition is employed which enforces global momentum conservation the resulting velocity only rom is thus globally conserving mass momentum and kinetic energy for non homogeneous boundary conditions a one time poisson equation is solved that accounts for the boundary contribution the stability of the proposed rom is demonstrated in several test cases furthermore it is shown that explicit runge kutta methods can be used as a practical alternative to implicit time integration at a slight loss in energy conservation

two level discretizations of nonlinear closure models for proper orthogonal decomposition proper orthogonal decomposition has been successfully used in the reduced order modeling of complex systems its original promise of computationally efficient yet accurate approximation of coherent structures in high reynolds number turbulent flows however still remains to be fulfilled to balance the low computational cost required by reduced order modeling and the complexity of the targeted flows appropriate closure modeling strategies need to be employed since modern closure models for turbulent flows are generally nonlinear their efficient numerical discretization within a proper orthogonal decomposition framework is challenging this paper proposes a two level method for an efficient and accurate numerical discretization of general nonlinear closure models for proper orthogonal decomposition reduced order models the two level method computes the nonlinear terms of the reduced order model on a coarse mesh compared with a brute force computational approach in which the nonlinear terms are evaluated on the fine mesh at each time step the two level method attains the same level of accuracy while dramatically reducing the computational cost we numerically illustrate these improvements in the two level method by using it in three settings the one dimensional burgers equation with a small diffusion parameter n 10 3 the two dimensional flow past a cylinder at reynolds number re 200 and the three dimensional flow past a cylinder at reynolds number re 1000

adaptive tracking with saturating input and controller integral action the authors consider the problem of controlling stable linear plants subject to input saturation constraints and parametric uncertainties a globally stable adaptive controller which performs well in the presence of such constraints is designed using a specific pole placement that accounts for the saturation constraint an integral action to deal with the tracking problem and an adaptive control law with an ad hoc excitation capability together with a parameter adaptation to cope with the parametric uncertainty the proposed controller allows one to track slowly varying reference sequences particularly in the case of a constant reference the output tracking error vanishes asymptotically and the control input does stop saturating after a finite time

an optimal projection method for the reduced order modeling of incompressible flows a new method to build reduced order models roms for incompressible flows is addressed in this paper proper orthogonal decompositions pods are used to efficiently represent both the fluid velocity and pressure fields with a small number of spatial modes a minimum residual projection is then developed for building the reduced models in which the velocity and pressure temporal coefficients are fully coupled comparisons of numerical calculations based on the proposed approach with results arising from the classical galerkin projection are exposed on three fluid flows a transient ventilated cavity a periodic lid driven cavity and a transient mixed convection flow it is shown that both stability and accuracy of the roms are strongly enhance when the minimum residual projection is used

a galerkin free model reduction approach for the navier stokes equations galerkin projection of the navier stokes equations on proper orthogonal decomposition pod basis is predominantly used for model reduction in fluid dynamics the robustness for changing operating conditions numerical stability in long term transient behavior and the pressure term consideration are generally the main concerns of the galerkin reduced order models rom in this article we present a novel procedure to construct an off reference solution state by using an interpolated pod reduced basis a linear interpolation of the pod reduced basis is performed by using two reference solution states the pod basis functions are optimal in capturing the averaged flow energy the energy dominant pod modes and corresponding base flow are interpolated according to the change in operating parameter the solution state is readily built without performing the galerkin projection of the navier stokes equations on the reduced pod space modes as well as the following time integration of the resulted ordinary differential equations ode to obtain the pod time coefficients the proposed interpolation based approach is thus immune from the numerical issues associated with a standard pod galerkin rom in addition a posteriori error estimate and a stability analysis of the obtained rom solution are formulated a detailed case study of the flow past a cylinder at low reynolds numbers is considered for the demonstration of proposed method the rom results show good agreement with the high fidelity numerical flow simulation

discrete time saturation constrained adaptive pole assignment control the performance of a discrete time adaptive pole assignment controller subject to a saturation control constraint is analyzed closed loop system stability is established for a class of type i plants in the presence of plant modeling uncertainties and disturbances it is shown that the stability is maintained by implementing a simple fixed deadzone in the parameter estimator if the plant is free from the modeling uncertainties and disturbances the deadzone threshold can be set to zero then the closed loop system can be characterized by the desired poles when the control input is not saturated

data driven pod galerkin reduced order model for turbulent flows abstract in this work we present a reduced order model which is specifically designed to deal with turbulent flows in a finite volume setting the method used to build the reduced order model is based on the idea of merging combining projection based techniques with data driven reduction strategies in particular the work presents a mixed strategy that exploits a data driven reduction method to approximate the eddy viscosity solution manifold and a classical pod galerkin projection approach for the velocity and the pressure fields respectively the newly proposed reduced order model has been validated on benchmark test cases in both steady and unsteady settings with reynolds up to r e o 10 5

proper orthogonal decomposition closure models for turbulent flows a numerical comparison this paper puts forth two new closure models for the proper orthogonal decomposition reduced order modeling of structurally dominated turbulent flows the dynamic subgrid scale model and the variational multiscale model these models which are considered state of the art in large eddy simulation together with the mixing length and the smagorinsky closure models are tested in the numerical simulation of 3d turbulent flow past a circular cylinder at re 1000 five criteria are used to judge the performance of the proper orthogonal decomposition reduced order models the kinetic energy spectrum the mean velocity the reynolds stresses the root mean square values of the velocity fluctuations and the time evolution of the pod coefficients all the numerical results are benchmarked against a direct numerical simulation based on these numerical results we conclude that the dynamic subgrid scale and the variational multiscale models are the most accurate

centroidal voronoi tessellation based reduced order modeling of complex systems noindent a reduced order modeling methodology based on centroidal voronoi tessellations cvts is introduced cvts are special voronoi tessellations for which the generators of the voronoi diagram are also the centers of mass means of the corresponding voronoi cells for discrete data sets cvts are closely related to the h means and k means clustering techniques a discussion of reduced order modeling for complex systems such as fluid flows is given to provide a context for the application of reduced order bases then detailed descriptions of cvt based reduced order bases and how they can be constructed from snapshot sets and how they can be applied to the low cost simulation of complex systems are given subsequently some concrete incompressible flow examples are used to illustrate the construction and use of cvt based reduced order bases the cvt based reduced order modeling methodology is shown to be effective for these examples

a stabilized pod model for turbulent flows over a range of reynolds numbers optimal parameter sampling and constrained projection abstract we present a reduced basis technique for long time integration of parametrized incompressible turbulent flows the new contributions are threefold first we propose a constrained galerkin formulation that corrects the standard galerkin statement by incorporating prior information about the long time attractor for explicit and semi implicit time discretizations our statement reads as a constrained quadratic programming problem where the objective function is the euclidean norm of the error in the reduced galerkin algebraic formulation while the constraints correspond to bounds for the maximum and minimum value of the coefficients of the n term expansion second we propose an a posteriori error indicator which corresponds to the dual norm of the residual associated with the time averaged momentum equation we demonstrate that the error indicator is highly correlated with the error in mean flow prediction and can be efficiently computed through an offline online strategy third we propose a greedy algorithm for the construction of an approximation space procedure valid over a range of parameters the greedy is informed by the a posteriori error indicator developed in this paper we illustrate our approach and we demonstrate its effectiveness by studying the dependence of a two dimensional turbulent lid driven cavity flow on the reynolds number

computational reduction strategies for the detection of steady bifurcations in incompressible fluid dynamics applications to coanda effect in cardiology abstract we focus on reducing the computational costs associated with the hydrodynamic stability of solutions of the incompressible navier stokes equations for a newtonian and viscous fluid in contraction expansion channels in particular we are interested in studying steady bifurcations occurring when non unique stable solutions appear as physical and or geometric control parameters are varied the formulation of the stability problem requires solving an eigenvalue problem for a partial differential operator an alternative to this approach is the direct simulation of the flow to characterize the asymptotic behavior of the solution both approaches can be extremely expensive in terms of computational time we propose to apply reduced order modeling rom techniques to reduce the demanding computational costs associated with the detection of a type of steady bifurcations in fluid dynamics the application that motivated the present study is the onset of asymmetries i e symmetry breaking bifurcation in blood flow through a regurgitant mitral valve depending on the reynolds number and the regurgitant mitral valve orifice shape

a reduced order method for simulation and control of fluid flows this article presents a reduced order modeling approach for simulation and control of viscous incompressible flows the reduced order models suitable for control and which capture the essential physics are developed using the reduced basis method the so called lagrange approach is used to define reduced bases and the basis functions in this approach are obtained from the numerical solutions the feasibility of this method for flow control is demonstrated on boundary control problems in closed cavity and in wall bounded channel flows control action is effected through boundary surface movement on a part of the solid wall our formulation of the reduced order method applied to flow control problems leads to a constrained minimization problem and is solved by applying newton like methods to the necessary conditions of optimality through our computational experiments we demonstrate the feasibility and applicability of the reduced order method for simulation and control of fluid flows

a priori space time separated representation for the reduced order modeling of low reynolds number flows abstract reduced order models roms in fluid dynamics are nowadays mostly developed by performing a projection of the navier stokes equations onto a low dimensional space basis this basis is usually obtained through proper orthogonal decomposition pod which remains one the most efficient techniques to compress precomputed data the main drawback of a posteriori pod based roms is however their lacks of reliability as their parameters are varied preventing their direct uses within optimization algorithms the goal of the present article is to obtain an a priori low dimensional space time separated representation of the fluid fields without precomputed data the approach is based on the use of space time proper generalized decomposition pgd definitions which are successfully applied in several fields but whose uses in fluid dynamics remain scarce their applications to the navier stokes equations are indeed not straightforward due to the pressure velocity coupling the divergence free constraint and the non linear convective term the roms are built here from a space time weak formulation of the chorin temam prediction correction scheme more particularly a priori space time separated representations are obtained by applying the galerkin based progressive pgd definition the minimax pgd definition is moreover experimented on a linear stokes simplified case the related algorithms are explicitly given and illustrated on a transient lid driven cavity flow it is shown that the a priori space time separated representations converge to the full model solution as the decomposition order increases the ability of the resulting rom to learn iteratively from its own error is highlighted the progressive pgd algorithm can be used to effectively enrich incomplete pod and pgd precomputed space bases such as those obtained with different parameter values this may allow to ensure the accuracy of a rom as the parameter is varied which is of crucial interest for optimization problems

pod galerkin reduced order methods for combined navier stokes transport equations based on a hybrid fv fe solver abstract the purpose of this work is to introduce a novel pod galerkin strategy for the semi implicit hybrid high order finite volume finite element solver introduced in bermudez et al 2014 and busto et al 2018 the interest is into the incompressible navier stokes equations coupled with an additional transport equation the full order model employed in this article makes use of staggered meshes this feature will be conveyed to the reduced order model leading to the definition of reduced basis spaces in both meshes the reduced order model presented herein accounts for velocity pressure and a transport related variable the pressure term at both the full order and the reduced order level is reconstructed making use of a projection method more precisely a poisson equation for pressure is considered within the reduced order model results are verified against three dimensional manufactured test cases moreover a modified version of the classical cavity test benchmark including the transport of a species is analysed

stabilization of projection based reduced order models for linear time invariant systems via optimization based eigenvalue reassignment abstract a new approach for stabilizing unstable reduced order models roms for linear time invariant lti systems through an a posteriori post processing step applied to the algebraic rom system is developed the key idea is to modify the unstable eigenvalues of the rom system by moving these eigenvalues into the stable half of the complex plane it is demonstrated that this modification to the rom system eigenvalues can be accomplished using full state feedback a k a pole placement algorithms from control theory this approach ensures that the modified rom is stable provided the system s unstable poles are controllable and observable however the accuracy of the stabilized rom is not guaranteed to remedy this difficulty and guarantee an accurate stabilized rom a constrained nonlinear least squares optimization problem for the stabilized rom eigenvalues in which the error in the rom output is minimized is formulated this optimization problem is small and therefore computationally inexpensive to solve performance of the proposed algorithms is evaluated on two test cases for which roms constructed via the proper orthogonal decomposition pod galerkin method suffer from instabilities

parametric free form shape design with pde models and reduced basis method we present a coupling of the reduced basis methods and free form deformations for shape optimization and design of systems modelled by elliptic pdes the free form deformations give a parameterization of the shape that is independent of the mesh the initial geometry and the underlying pde model the resulting parametric pdes are solved by reduced basis methods an important role in our implementation is played by the recently proposed empirical interpolation method which allows approximating the non affinely parameterized deformations with affinely parameterized ones these ingredients together give rise to an efficient online computational procedure for a repeated evaluation design environment like the one for shape optimization the proposed approach is demonstrated on an airfoil inverse design problem

multilevel algorithm for obtaining the proper orthogonal decomposition a multilevel proper orthogonal decomposition mpod is introduced for obtaining the proper orthogonal decomposition pod of data sets with a large number of snapshots without requiring the solutio

goal oriented model constrained optimization for reduction of large scale systems optimization oriented reduced order models should target a particular output functional span an applicable range of dynamic and parametric inputs and respect the underlying governing equations of the system to achieve this goal we present an approach for determining a projection basis that uses a goal oriented model constrained optimization framework the mathematical framework permits consideration of general dynamical systems with general parametric variations and is applicable to both linear and nonlinear systems results for a simple linear model problem of the two dimensional heat equation demonstrate the ability of the goal oriented approach to target a particular output functional of interest application of the methodology to a more challenging example of a subsonic blade row governed by the unsteady euler flow equations shows a significant advantage of the new method over the proper orthogonal decomposition

error estimation in reduced basis method for systems with time varying and nonlinear boundary conditions many physical phenomena such as mass transport and heat transfer are modeled by systems of partial differential equations with time varying and nonlinear boundary conditions control inputs and disturbances typically affect the system dynamics at the boundaries and a correct numerical implementation of boundary conditions is therefore crucial however numerical simulations of high order discretized partial differential equations are often too computationally expensive for real time and many query analysis for this reason model complexity reduction is essential in this paper it is shown that the classical reduced basis method is unable to incorporate time varying and nonlinear boundary conditions to address this issue it is shown that by using a modified surrogate formulation of the reduced basis ansatz combined with a feedback interconnection and a input related term the effects of the boundary conditions are accurately described in the reduced order model the results are compared with the classical reduced basis method unlike the classical method the modified ansatz incorporates boundary conditions without generating unphysical results at the boundaries moreover a new approximation of the bound and a new estimate for the error induced by model reduction are introduced the effectiveness of the error measures is studied through simulation case studies and a comparison with existing error bounds and estimates is provided the proposed approximate error bound gives a finite bound of the actual error unlike existing error bounds that grow exponentially over time finally the proposed error estimate is more accurate than existing error estimates

enablers for robust pod models this paper focuses on improving the stability as well as the approximation properties of reduced order models rom based on proper orthogonal decomposition pod the rom is obtained by seeking a solution belonging to the pod subspace and that at the same time minimizes the navier stokes residuals we propose a modified rom that directly incorporates the pressure term in the model the rom is then stabilized making use of a method based on the fine scale equations an improvement of the pod solution subspace is performed thanks to an hybrid method that couples direct numerical simulations and reduced order model simulations the methods proposed are tested on the two dimensional confined square cylinder wake flow in laminar regime

calibrated reduced order pod galerkin system for fluid flow modelling to improve the behaviour of reduced order proper orthogonal decomposition pod galerkin systems two numerical methods are proposed these methods determine free parameters in the pod galerkin system from flow simulations via a minimization problem they give rise to linear systems and their computational costs are reasonable both methods are assessed for two flow configurations a two dimensional flow around a square cylinder for a reynolds number of 100 and a three dimensional flow past a backward facing step for a reynolds number of 7432 based on the step height and the streamwise velocity at the middle of the inlet for both configurations the methods are effective since accurate calibrated reduced order pod galerkin systems are obtained

an input output approach to achieve adaptive tracking for saturating stable systems the problem of controlling time invariant linear systems subject to parametric uncertainty and input saturation constraint is dealt with using a pole placement adaptive controller it is shown that the resulting adaptive control system can be described by a non linear feedback scheme then the l 2 stability condition for the control system turns out to be the positive realness of a transfer function involving the plant estimated model and the specified closed loop poles this makes it necessary to adapt on line the desired closed loop poles to the time varying estimated model the proposed adaptive controller keeps bounded all the closed loop sequences whatever the initial conditions furthermore it ensures a quite interesting output reference tracking behaviour more precisely the quality of the tracking depends on the reference sequence rate the slower the reference the better the tracking

achieving a confidence interval for parameters estimated by simulation this paper presents a procedure for determining the number of simulation observations required to achieve a preassigned confidence interval for means estimated by simulation this procedure which is simple to implement and efficient to use is compared with two other methods for determining the required sample size in a simulation run the empirical results show that this procedure gives good results in the precision of estimated means and in sample size requirement

variational multiscale residual based turbulence modeling for large eddy simulation of incompressible flows we present an les type variational multiscale theory of turbulence our approach derives completely from the incompressible navier stokes equations and does not employ any ad hoc devices such as eddy viscosities we tested the formulation on forced homogeneous isotropic turbulence and turbulent channel flows in the calculations we employed linear quadratic and cubic nurbs a dispersion analysis of simple model problems revealed nurbs elements to be superior to classical finite elements in approximating advective and diffusive processes which play a significant role in turbulence computations the numerical results are very good and confirm the viability of the theoretical framework

supremizer stabilization of pod galerkin approximation of parametrized steady incompressible navier stokes equations in this work we present a stable proper orthogonal decomposition pod galerkin approximation for parametrized steady incompressible navier stokes equations with low reynolds number supremizers solutions are added to the reduced velocity space in order to obtain a stable reduced order system considering in particular the fulfillment of an inf sup condition the stability analysis is first carried out from a theoretical standpoint then confirmed by numerical tests performed on a parametrized two dimensional backward facing step flow

equation free galerkin free pod assisted computation of incompressible flows we present a galerkin free proper orthogonal decomposition pod assisted computational methodology for numerical simulations of the long term dynamics of the incompressible navier stokes equations the approach is based on the equation free framework we use short appropriate initialized bursts of full direct numerical simulations dns of the navier stokes equations to observe estimate and accelerate through projective integration the evolution of the flow dynamics the main assumption is that the long term dynamics of the flow lie on a low dimensional attracting and invariant manifold which can be parametrized not necessarily spanned by a few pod basis functions we start with a discussion of the consistency and accuracy of the approach and then illustrate it through numerical examples two dimensional periodic and quasi periodic flows past a circular cylinder we demonstrate that the approach can successfully resolve complex flow dynamics at a reduced computational cost and that it can capture the long term asymptotic state of the flow in cases where traditional galerkin pod models fail the approach trades the overhead involved in developing pod galerkin and pod nonlinear galerkin codes for the repeated yet short and on demand use of an existing full dns simulator moreover since in this approach the pod modes are used to observe rather than span the true system dynamics the computation is much less sensitive than pod galerkin to values of the system parameters e g the reynolds number and the particular simulation data ensemble used to obtain the pod basis functions

projection based reduced order models for a cut finite element method in parametrized domains abstract this work presents a reduced order modeling technique built on a high fidelity embedded mesh finite element method such methods and in particular the cutfem method are attractive in the generation of projection based reduced order models thanks to their capabilities to seamlessly handle large deformations of parametrized domains and in general to handle topological changes the combination of embedded methods and reduced order models allows us to obtain fast evaluation of parametrized problems avoiding remeshing as well as the reference domain formulation often used in the reduced order modeling for boundary fitted finite element formulations the resulting novel methodology is presented on linear elliptic and stokes problems together with several test cases to assess its capability the role of a proper extension and transport of embedded solutions to a common background is analyzed in detail

proper orthogonal decomposition with supg stabilized isogeometric analysis for reduced order modelling of unsteady convection dominated convection diffusion reaction problems abstract we consider reduced order modelling of unsteady convection dominated convection diffusion reaction problems with proper orthogonal decomposition pod in combination with isogeometric analysis isogeometric analysis has potential advantages in exact geometry representations efficient mesh generation different h p and k refinements and smooth bspline nurbs basis functions in order to compensate the oscillations caused by the convection dominated effect the streamline upwind petrov galerkin supg stabilization method is used both in generation of snapshots and pod galerkin method based on the recent novel and promising discretization method isogeometric analysis we propose a new fully discrete supg stabilized scheme the associated numerical error features three components due to spatial discretization by isogeometric analysis with supg stabilization time discretization with the crank nicolson scheme and modes truncation by pod we show a priori error estimates of the fully discrete scheme and give suitable stabilization parameters numerically a variety of two and three dimensional benchmark tests and numerical examples are provided to show the effectiveness accuracy and efficiency of the reduced order modelling methods by virtue of potential advantages of isogeometric analysis